leftri rightri


This is PART 1: Introduction and Centers X(1) - X(1000)

Introduction and Centers X(1) - X(1000) Centers X(1001) - X(3000) Centers X(3001) - X(5000)
Centers X(5001) - X(7000) Centers X(7001) - X(10000) Centers X(10001) - X(12000)
Centers X(12001) - X(14000) Centers X(14001) - X(16000) Centers X(16001) - X(18000)
Centers X(18001) - X(20000) Centers X(20001) - X(22000) Centers X(22001) - X(24000)
Centers X(24001) - X(26000) Centers X(26001) - X(28000) Centers X(28001) - X(30000)
Centers X(30001) - X(32000) Centers X(32001) - X(34000) Centers X(34001) - X(36000)
Centers X(36001) - X(38000) Centers X(38001) - X(40000) Centers X(40001) - X(42000)
Centers X(42001) - X(44000) Centers X(44001) - X(46000) Centers X(46001) - X(48000)
Centers X(48001) - X(50000) Centers X(50001) - X(52000) Centers X(52001) - X(54000)
Centers X(54001) - X(56000) Centers X(56001) - X(58000) Centers X(58001) - X(60000)
Centers X(60001) - X(62000) Centers X(62001) - X(64000) Centers X(64001) - X(66000)
Centers X(66001) - X(68000) Centers X(68001) - X(70000) Centers X(70001) - X(72000)


Long ago, someone drew a triangle and three segments across it. Each segment started at a vertex and stopped at the midpoint of the opposite side. The segments met in a point. The person was impressed and repeated the experiment on a different shape of triangle. Again the segments met in a point. The person drew yet a third triangle, very carefully, with the same result. He told his friends. To their surprise and delight, the coincidence worked for them, too. Word spread, and the magic of the three segments was regarded as the work of a higher power.

Centuries passed, and someone proved that the three medians do indeed concur in a point, now called the centroid. The ancients found other points, too, now called the incenter, circumcenter, and orthocenter. More centuries passed, more special points were discovered, and a definition of triangle center emerged. Like the definition of continuous function, this definition is satisfied by infinitely many objects, of which only finitely many will ever be published. The Encyclopedia of Triangle Centers (ETC) extends a list of 400 triangle centers published in the 1998 book Triangle Centers and Central Triangles. For subsequent developments, click Links (one of the buttons atop this page). In particular, Eric Weisstein's MathWorld, covers much of classical and modern triangle geometry, including sketches and references.

A site in which triangle centers play a central role is Bernard Gibert's Cubics in the Triangle Plane. Special points and properties of 4-sided plane figures are closely associated with triangle centers; see Chris van Tienhoven's Encyclopedia of Quadri-Figures (EQF).

HOW TO USE ETC

You won't have to scroll down very far to find well known centers. Other named centers can be found using your computer's searcher - for example, search for "Apollonius" to find "Apollonius point" as X(181).

To determine if a possibly new center is already listed, click Tables at the top of this page and scroll to "Search 6.9.13". If you're unsure of a term, click Glossary or Pierre Douillet's much expanded and very useful version: Translation of the Kimberling's Glossary into barycentrics.

For visual constructions of selected centers with text, click Sketches. To learn about the triangle geometry interest group, Hyacinthos and other resources, or to view acknowledgments or supplementary encyclopedic material, click Links, Thanks, or Tables.

Under Tables, you can find Search_13_6_9 (and two other Searches), which can be used to determine whether a newly discovered point is already in ETC. For such a search, be sure to visit Ron Knott's Triangle Convertor for Cartesian, Trilinear and Barycentric Coordinates.

If you have The Geometer's Sketchpad, you can view sketches of many of the triangle centers. These are dynamic sketches, meaning that you can vary the shape of the reference triangle A, B, C by dragging these vertices. (For information on Sketchpad, click Sketchpad.) The sketches are also useful for making your own Sketchpad tools, so that you can quickly construct X-of-T for many choices of X and T. For example, starting with ABC and point P, you could efficiently construct center X of the four triangles ABC, BCP, CAP, ABP.

The algebraic definition of triangle center (MathWorld) admits points whose geometric interpretation for fixed numerical sidelengths a,b,c is not "central." Roger Smyth offers this example: on the domain of scalene triangles, define f(a,b,c) = 1 for a>b and a>c and f(a,b,c) = 0 otherwise; then f(a,b,c) : f(b,c,a) : f(c,a,b) is a triangle center which picks out the vertex opposite the longest side. Such centers turn out to be useful, as, for example, when distinguishing between the Fermat point and the 1st isogonic center; see the note at X(13).

NOTATION AND COORDINATES

The reference triangle is ABC, with sidelengths a,b,c. Each triangle center P has homogeneous trilinear coordinates, or simply trilinears, of the form x : y : z. This means that there is a nonzero function h of (a,b,c) such that

x = hx', y = hy', z = hz',

where x', y', z' denote the directed distances from P to sidelines BC, CA, AB. Likewise, u : v : w are barycentrics if there is a function k of (a,b,c) such that

u = ku', v = kv', w = kw',

where u', v', w' denote the signed areas of triangles PBC, PCA, PAB. Both coordinate systems are widely used; if trilinears for a point are x : y : z, then barycentrics are ax : by : cz.

In order that every center should have its own name, in cases where no particular name arises from geometrical or historical considerations, the name of a star is used instead. For example, X(770) is POINT ACAMAR. For a list of star names, visit SkyEye - (Un)Common Star Names.


Introduced on November 22, 2024: Dynamic Perspective Fields: The Hidden Structure in Triangle Geometry

In two parts, by Chris van Tienhoven (van10hoven@gmail.com.nl)

Introduced on August 17, 2024: How to find Forum Geometricorum

The journal Forum Geometricorum, formerly published at Florida Atlantic University, can be accessed. For a file of all issues from 2001 to 2009, see single searchable file, 2001-2009. For individual articles, see individual articles.

Introduced on August 1, 2024: the new journal Geometry

The Encyclopedia of Triangle Centers is well represented as the first article in a new open-access journal named simply Geometry. See "Unary Operations on Homogeneous Coordinates in the Plane of a Triangle", by Peter J. C. Moses and Clark Kimberling: First Article.

Information for prospective authors in Geometry can be found in Information and Instructions.


Introduced on July 24, 2024: Loci of triangle centers in Poncelet porisms (circular caustic):

Dan Reznik presents observations and demonstrations of loci of certain triangle centers, over a Poncelet family of triangles inscribed in an ellipse E and circumscribed about a circular caustic K having center C. If K "closes Poncelet N=3", then References:

[1] D. Reznik, "Ellipse-inscribed Poncelet triangles: loci of triangle centers over all circular caustics", YouTube video, 2024.

[2] R. Schwartz and S. Tabachnikov, "Centers of Mass of Poncelet Polygons, 200 Years After", The Mathematical Intelligencer, Vol. 38, 2016, pp. 29-34.

[3] M. Helman, D. Laurain, R. Garcia, D. Reznik, "Poncelet triangles: a theory for locus ellipticity", Beiträge zur Algebra und Geometrie, vol.3, 2022, pp. 445-457.

[4] D. Reznik, "Ellipse-Inscribed Poncelet triangles with a circular caustic III: the loci of X2,X3,X4,X5", YouTube video, 2024.

[5] D. Reznik, "Ellipse-inscribed Poncelet triangles w/ circular caustic IV: foci of the circumcenter locus", YouTube video, 2024.

[6] D. Reznik, "Nifty loci of Poncelet triangles with a circular caustic", Google Docs, July 2024.


Introduced on August 30, 2023: Two introductions to barycentric coordinates in the plane of a triangle:

Notes on Barycentric Homogeneous Coordinates, by Wong Lan Loi.

Barycentric coordinates or Barycentrics, by Paris Pamfilos.


Published on December 31, 2021: "Location of Triangle Centers Relative to the Incircle and Circumcircle", by Stanley Rabinowitz

in International Journal of Computer Discovered Mathematics (IJCDM)


Introduced on December 8, 2020: Access to Hyacinthos and ADGEOM postings

ETC includes many postings to interest groups that are no longing sponsored by Yahoo. Fortunately, César Lozada has archived the thousands of postings. These can now be easily accessed as in the following example for Hyacinthos #28936: http://www.hyacinthos.epizy.com/message.php?msg=28936

The same procedure works for ADGEOM messages, following this example: http://www.adgeom.epizy.com/message.php?msg=900&i=1

For Quadrilateral Geometry and Polygon Geometry messages: http://www.qfg.epizy.com/message.php?msg=3200.

And for Anopolis messages: http://anopolis.epizy.com/message.php?msg=3001

Introduced on October 25, 2020: A New Electric Field Interpretation of Barycentric and Trilinear Coordinates

A New Electric Field Interpretation of Barycentric and Trilinear Coordinates, by Suren


Introduced on December 23, 2019: Alphabetical Index of Terms in ETC

Alphabetical Index of Terms in ETC, by César Lozada


Introduced on May 2, 2019: Writer

If you wish to submit one or more triangles centers for possible inclusion in ETC, please click Tables at the top of this page, then scroll to and click Search_13_6_9. There, find Writer, to be used for proper formatting.


Introduced on December 28, 2016: Index of Triangles Referenced in ETC

Many triangles are defined in the plane of a reference triangle ABC. Some of them have well-established names (e.g., medial, orthic, tangential), but many more have been discovered only recently.

The Index is authored and updated by César Lozada. You can access it here, and also from Glossary and Tables.


Introduced on March 21, 2015: Shinagawa coefficients for triangle centers on the Euler line

Suppose that X is a triangle center given by barycentric coordinates f(a,b,c) : f(b,c,a) : f(c,a,b). The Shinagawa coefficients of X are the functions G(a,b,c) and H(a,b,c) such that

f(a,b,c) = G(a,b,c)*S2 + H(a,b,c)*SBSC.

For many choices of X, G(a,b,c) and H(a,b,c) are conveniently expressed in terms of the following:

E = (SB + SC)(SC + SA)(SA + SB)/S2, so that E = (abc/S)2 = 4R2
F = SASBSC/S2, so that F = (a2 + b2 + c2)/2 - 4R2 = Sω - 4R2

Examples:
X(2) has Shinagawa coefficients (1, 0); i.e., X(2) = 1*S2 + 0*SBSC
X(3) has Shinagawa coefficients (1, -1)
X(4) has Shinagawa coefficients (0, 1)
X(5) has Shinagawa coefficients (1, 1)
X(23) has Shinagawa coefficients (E + 4F, -4E - 4F)
X(1113) has Shinagawa coefficients (R - |OH|, -3R + |OH|)

A cyclic sum notation, $...$, is introduced here especially for use with Shinagawa coefficients. For example, $aSBSC$ abbreviates aSBSC + bSCSA + cSASB.

Example: X(21) has Shinagawa coefficients ($aSA$, abc - $aSA$)

If a point X has Shinagawa coefficients (u,v) where u and v are real numbers (i.e, G(a,b,c) and H(a,b,c) are constants), then the segment joining X and X(2) is given by |GX| = 2v|GO|/(3u + v), where |GO| = (E - 8F)1/2/6. Then the equation |GX| = 2v|GO|/(3u + v) can be used to obtain these combos:

X = [(u + v)/2]*X(2) - (v/3)*X(3)
X = u*X(2) + (v/3)*X(4)
X = u*X(3) + [(u + v)/2]*X(4).

The function F is also given by these identities:
F = (4R2 - 36|GO|2)/8 and F = R2( 1 - J2)/2, where J = |OH|/R.


Introduced on November 1, 2011: Combos

Suppose that P and U are finite points having normalized barycentric coordinates (p,q,r) and (u,v,w). (Normalized means that p + q + r = 1 and u + v + w = 1.) Suppose that f = f(a,b,c) and g = g(a,b,c) are nonzero homogeneous functions having the same degree of homogeneity. Let x = fp + gu, y = fq + gv, z = fr + gw. The (f,g) combo of P and U, denoted by f*P + g*U, is introduced here as the point X = x : y : z (homogeneous barycentric coordinates); the normalized barycentric coordinates of X are (kx,ky,kz), where k=1/(x+y+z).

Note 1. If P and U are given by normalized trilinear coordinates (instead of barycentric), then f*P + g*U has homogeneous trilinears fp+gu : fq+gv : fr+gw, which is symbolically identical to the homogenous barycentrics for f*P + g*U. The normalized trilinear coordinates for X are (hx,hy,hz), where h=2*area(ABC)/(ax + by + cz).

Note 2. The definition of combo readily extends to finite sets of finite points. In particular, the (f,g,h) combo of P = (p,q,r), U = (u,v,w), J = (j,k,m) is given by fp + gu + hj : fq + gv + hk : fr + gw + hm and denoted by f*P + g*U + h*J.

Note 3. f*P + g*U is collinear with P and U, and its {P,Q}-harmonic conjugate is fp - gu : fq - gv : fr - gw.

Note 4. Suppose that f,g,h are homogeneous symmetric functions all of the same degree of homogeneity, and suppose that X, X', X" are triangle centers. Then f*X + g*X' + h*X'' is a triangle center.

Note 5. Suppose that X, X', X'', X''' are triangle centers and X', X'', X''' are not collinear. Then there exist f,g,h as in Note 4 such that X = f*X' + g*X'' + h*X'''. That is, loosely speaking, every triangle center is a linear combination of any other three noncollinear triangle centers.

Note 6. Continuing from Note 5, examples of f,g,h are conveniently given using Conway symbols for a triangle ABC with sidelengths a,b,c. Conway symbols and certain classical symbols are identified here:

S = 2*area(ABC)
SA = (b2 + c2 - a2)/2 = bc cos A
SB = (c2 + a2 - b2)/2 = ca cos B
SC = (a2 + b2 - c2)/2 = ab cos C
Sω = S cot ω
s = (a + b + c)/2
sa = (b + c - a)/2
sb = (c + a - b)/2
sc = (a + b - c)/2
r = inradius = S/(a + b + c)
R = circumradius = abc/(2S)
cot(ω) = (a2 + b2 + c2)/(2S), where ω is the Brocard angle

Note 7. The definition of combo along with many examples were developed by Peter Moses prior to November 1, 2011. After that combos have been further developed by Peter Moses, Randy Hutson, and Clark Kimberling.

Examples of two-point combos:
X(175) = 2s*X(1) - (r + 4R)*X(7)
X(176) = 2s*X(1) + (r + 4R)*X(7)
X(481) = s*X(1) - (r + 4R)*X(7)
X(482) = s*X(1) + (r + 4R)*X(7)

Examples of three-point combos: see below at X(1), X(2), etc.

Note 8. Suppose that T is a (central) triangle with vertices A',B',C' given by normalized barycentrics. Then T is represented by a 3x3 matrix with row sums equal to 1. Let NT denote the set of these matrices and let * denote matrix multiplication. Then NT is closed under *. Also, NT is closed under matrix inversion, so that (NT, *) is a group. Once normalized, any central T can be used to produce triangle centers as combos of the form Xcom(nT); see the preambles just before X(3663) and X(3739).



X(1) = INCENTER

Trilinears    1 : 1 : 1
Barycentrics   a : b : c
Barycentrics  sin A : sin B : sin C
Tripolars    Sqrt[b c (b + c - a)] : :
Tripolars    sec A' : :, where A'B'C' is the excentral triangle
X(1) = 3R*X(2) + r*X(3) + s*cot(ω)*X(6)
X(1) = [A]/Ra + [B]/Rb + [C]/Rc - X(176)/Rs, where Ra, Rb, Rc = radii of Soddy circles, Rs = radius of inner Soddy circle, [A], [B], [C] are the vertices of ABC
X(1) = [A]/Ra + [B]/Rb + [C]/Rc - X(175)/Rs', where Ra, Rb, Rc = radii of Soddy circles, Rs' = radius of outer Soddy circle, [A], [B], [C] are the vertices of ABC
X(1) = (sin A)*[A] + (sin B)*[B] + (sin C)*[C], where [A], [B], [C] are vertices of ABC
X(1) = a*[A] + b*[B] + c*[C], where [A], [B], [C] are vertices

X(1) is the point of concurrence of the interior angle bisectors of ABC; the point inside ABC whose distances from sidelines BC, CA, AB are equal. This equal distance, r, is the radius of the incircle, given by r = 2*area(ABC)/(a + b + c).

Three more points are also equidistant from the sidelines; they are given by these names and trilinears:

A-excenter = -1 : 1 : 1,     B-excenter = 1 : -1 : 1,     C-excenter = 1 : 1 : -1.

The radii of the excircles are 2*area(ABC)/(-a + b + c), 2*area(ABC)/(a - b + c), 2*area(ABC)/(a + b - c).

If you have The Geometer's Sketchpad, you can view Incenter.
If you have GeoGebra, you can view Incenter.

Writing the A-exradius as r(a,b,c), the others are r(b,c,a) and r(c,a,b). If these exradii are abbreviated as ra, rb, rc, then 1/r = 1/ra +1/rb + 1/rc. Moreover,

area(ABC) = sqrt(r*ra*rb*rc) and ra + rb + rc = r + 4R, where R denotes the radius of the circumcircle.

The incenter is the identity of the group of triangle centers under trilinear multiplication defined by (x : y : z)*(u : v : w) = xu : yv : zw.

A construction for * is easily obtained from the construction for "barycentric multiplication" mentioned in connection with X(2), just below.

The incenter and the other classical centers are discussed in these highly recommended books:

Paul Yiu, Introduction to the Geometry of the Triangle, 2002;
Nathan Altshiller Court, College Geometry, Barnes & Noble, 1952;
Roger A. Johnson, Advanced Euclidean Geometry, Dover, 1960.

Let OA be the circle tangent to side BC at its midpoint and to the circumcircle on the side of BC opposite A. Define OB and OC cyclically. Let LA be the external tangent to circles OB and OC that is nearest to OA. Define LB and LC cyclically. Let A' = LB ∩LC, and define B' and C' cyclically. Then triangle A'B'C' is homothetic to the medial triangle, and the center of homothety is X(1); see the reference at X(1001).

Let A'B'C' and A″B″C″ be the intouch and excentral triangles; X(1) is the radical center of the circumcircles of AA'″, BB'B″, CC'C″. (Randy Hutson, December 10, 2016)

Let A'B'C' be the Feuerbach triangle. Let A″ be the trilinear pole of line B'C', and define B″ and C″ cyclically. The lines A″, BB″, CC″ concur in X(1). (Randy Hutson, November 17, 2019)

Let A'B'C' be the mixtilinear excentral triangle. Let A″ be the cevapoint of B' and C', and define B″ and C″ cyclically. The lines A″, BB″, CC″ concur in X(1). (Randy Hutson, November 17, 2019)

Let A'B'C' be the mixtilinear excentral triangle. Let A″ be the trilinear pole of line B'C', and define B″ and C″ cyclically. The lines A″, BB″, CC″ concur in X(1). (Randy Hutson, November 17, 2019)

Let A'B'C' be the mixtilinear excentral triangle. Let LA be the trilinear polar of A', and define LB and LC cyclically. Let A″ = LB∩LC, and define B″ and C″ cyclically. The lines A″, BB″, CC″ concur in X(1). (Randy Hutson, November 17, 2019)

Let OA be the circle centered at the A-vertex of the excenters-midpoints triangle and passing through A; define OB and OC cyclically. X(1) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

Let OA be the circle centered at the A-vertex of the Gemini triangle 22 and passing through A; define OB and OC cyclically. X(1) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

"Conjecture for the locus of X(1) over Poncelet triangles: a conic iff the pair is confocal", by Dan Reznik, Video.

Let P be any point inside triangle ABC. Let Oa be the smaller of two circles through P tangent to lines AB and AC, and define Ob and Oc cyclically. Let O'a be the larger of two circles through P tangent to lines AB and AC, and define O'b and O'c cyclically. Let O be the center of the circle tangent to Oa, Ob, and Oc, and let O' be the center of the circle tangent to O'a, O'b, and O'c. Then, regardless of the choice of point P, line OO' passes through a fixed point, which is X(1). (Miłosz Płatek, July 22, 2024; see X(1) on line OO'.

In the plane of a triangle ABC, let
D = the point on line AC such that angle CAB = angle DAB and |AD| = |AB|
E = the point on line AB such that angle CAB = angle CAE and |AE| = |AC|
F = the point on line BC such that angle CBA = angle FBA and |BF| = |AB|
G = the point on line AB such that angle CBA = angle CBG and |BG| = |BC|
H = the point on line BC such that angle ACB = angle ACH and |CH| = |AC|
I = the point on line AC such that angle ACB = angle ICB and |CI| = |BC|
The circles through the quadruples {A,B,H,I}, {B,C,D,E}, and {A,C,F,G} concur in X(1).
(Benjamin Warren, October 9, 2024)

In the plane of a triangle ABC, let
D = the point on line AC such that angle CAB = angle DAB and |AD| = |AB|
E = the point on line AB such that angle CAB = angle CAE and |AE| = |AC|
F = the point on line BC such that angle CBA = angle FBA and |BF| = |AB|
G = the point on line AB such that angle CBA = angle CBG and |BG| = |BC|
H = the point on line BC such that angle ACB = angle ACH and |CH| = |AC|
I = the point on line AC such that angle ACB = angle ICB and |CI| = |BC|
M = circumcenter of AGI
N = circumcenter of CDF
O = circumcenter of BEH
The circles AFH, BDI, CEG, and MNO are concentric about X(1).
See figure. (Benjamin Warren, October 24, 2024)

In the plane of a triangle ABC, let
D = point on line AC with angle BAD = π - angle BAC, such that |AD| = |AB|
E = point on line AB with angle CAE = π - angle BAC, such that |AE|=|AC|
F = point on line BC with angle FCA = π - angle ACB, such that |CF| = |AC|
G = point on line AC with angle GCB = π - angle ACB, such that |CG| = |CB|
H = point on line AB with angle HBC = π - angle ABC, such that |HB| = |BC|
J = the point on line BC with angle JBA = π - angle ABC, such that |JB| = |AB|
K = circumcenter of ADE
L = circumcenter of CFG
M = circumcenter of BHJ
K' = reflection of K about A
L' = reflection of L about C
M' = reflection of M about B
Then K', L', M', and X(3) are concyclic about X(1). (Benjamin Warren, November 16, 2024)

X(1) lies on all Z-cubics (e.g., Thomson, Darboux, Napoleon, Neuberg) and these lines: 2,8   3,35   4,33   5,11   6,9   7,20   15,1251   16,1250   19,28   21,31   24,1061   25,1036   29,92   30,79   32,172   39,291   41,101   49,215   54,3460   60,110   61,203   62,202   64,1439   69,1245   71,579   74,3464   75,86   76,350   82,560   84,221   87,192   88,100   90,155   99,741   102,108   104,109   142,277   147,150   159,1486   163,293   164,258   166,1488   167,174   168,173   179,1142   181,970   182,983   184,1726   185,296   188,361   190,537   195,3467   196,207   201,212   204,1712   210,2334   224,377   227,1465   228,1730   229,267   256,511   257,385   280,1256   281,282   289,363   312,1089   318,1897   320,752   321,964   329,452   335,384   336,811   341,1050   344,1265   346,1219   357,1508   358,1507   364,365   371,1702   372,1703   376,553   378,1063   393,836   394,1711   399,3065   409,1247   410,1248   411,1254   442,1834   474,1339   475,1861   512,875   513,764   514,663   522,1459   528,1086   561,718   563,1820   564,1048   572,604   573,941   574,1571   594,1224   607,949   615,3300   631,1000   644,1280   647,1021   650,1643   651,1156   659,891   662,897   672,1002   689,719   704,1502   727,932   731,789   748,756   761,825   765,1052   810,1577   840,1308   905,1734   908,998   921,1800   939,1260   945,1875   947,1753   951,1435   969,1444   971,1419   989,1397   1013,1430   1037,1041   1053,1110   1057,1598   1059,1597   1073,3341   1075,1148   1106,1476   1157,3483   1168,1318   1170,1253   1185,1206   1197,1613   1292,1477   1333,1761   1342,1700   1343,1701   1361,1364   1389,1393   1399,1727   1406,1480   1409,1765   1437,1710   1472,1791   1719,1790   1855,1886   1859,1871   1872,1887   2120,3461   2130,3347   3183,3345   3342,3343   3344,3351   3346,3353   3348,3472   3350,3352   3354,3355   3462,3469

X(1) is the {X(2),X(8)}-harmonic conjugate of X(10). For a list of other harmonic conjugates of X(1), click Tables at the top of this page.

X(1) = midpoint of X(i) and X(j) for these (i,j):
(3, 1482), (7,390), (8,145), (55,2099), (56,2098)
X(1) = reflection of X(i) in X(j) for these (i,j):
(2,551), (3,1385), (4,946), (6,1386), (8,10), (9,1001), (10,1125), (11,1387), (36,1319), (40,3), (43,995), (46,56), (57,999), (63,993), (65,942), (72,960), (80,11), (100,214), (191,21), (200,997), (238,1279), (267,229), (291,1015), (355,5), (484,36), (984,37), (1046,58), (1054,106), (1478,226)
X(1) = isogonal conjugate of X(1)
X(1) = isotomic conjugate of X(75)
X(1) = cyclocevian conjugate of X(1029)
X(1) = circumcircle-inverse of X(36)
X(1) = Fuhrmann-circle-inverse of X(80)
X(1) = Bevan-circle-inverse of X(484)
X(1) = Spieker-radical-circle-inverse of X(38471)
X(1) = complement of X(8)
X(1) = anticomplement of X(10)
X(1) = anticomplementary conjugate of X(1330)
X(1) = complementary conjugate at X(1329)
X(1) = eigencenter of cevian triangle of X(i) for I = 1, 88, 100, 162, 190
X(1) = eigencenter of anticevian triangle of X(i) for I = 1, 44, 513
X(1) = exsimilicenter of inner and outer Soddy circles; insimilicenter is X(7)
X(1) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,9), (4,46), (6,43), (7,57), (8,40), (9,165), (10,191), (21,3), (29,4), (75,63), (77,223), (78,1490), (80,484), (81,6), (82,31), (85,169), (86,2), (88,44), (92,19), (100,513), (104,36), (105,238), (174,173), (188,164), (220,170), (259,503), (266,361), (280,84), (366,364), (508,362), (1492,1491)
X(1) = cevapoint of X(i) and X(j) for these (i,j):
(2,192), (6,55), (11,523), (15,202), (16,203), (19,204), (31,48), (34,207), (37,42), (50,215), (56,221), (65,73), (244,513)
X(1) = X(i)-cross conjugate of X(j) for these (i,j): (2,87), (3,90), (6,57), (31,19), (33,282), (37,2), (38,75), (42,6), (44,88), (48,63), (55,9), (56,84), (58,267), (65,4), (73,3), (192,43), (207,1490), (221,40), (244,513), (259,258), (266,505), (354,7), (367,366), (500,35), (513,100), (517,80), (518,291), (1491,1492)
X(1) = crosspoint of X(i) and X(j) for these (i,j): (2,7), (8,280), (21,29), (59,110), (75,92), (81,86)
X(1) = crosssum of X(i) and X(j) for these (i,j): (2,192), (4,1148), (6,55), (11,523), (15,202), (16,203), (19,204), (31,48), (34,207), (37,42), (44,678), (50,215), (56,221), (57,1419), (65,73), (214,758), (244,513), (500,942), (512,1015), (774,820), (999,1480)
X(1) = crossdifference of every pair of points on line X(44)X(513)
X(1) = X(i)-Hirst inverse of X(j) for these (i,j): (2,239), (4,243), (6,238), (9,518), (19,240), (57,241), (105,294), (291,292)
X(1) = X(6)-line conjugate of X(44)
X(1) = X(i)-aleph conjugate of X(j) for these (i,j):
(1,1), (2,63), (4,920), (21,411), (29,412), (88,88), (100,100), (162,162), (174,57), (188,40), (190,190), (266,978), (365,43), (366,9), (507,173), (508,169), (513,1052), (651, 651), (653,653), (655,655), (658,658), (660,660), (662,662), (673,673), (771,771), (799,799), (823,823), (897,897)
X(1) = X(i)-beth conjugate of X(j) for these (i,j): (1,56), (2,948), (8,8), (9,45), (21,1), (29,34), (55,869), (99,85), (100,1), (110,603), (162,208), (643,1), (644,1), (663,875), (664,1), (1043,78)
X(1) = insimilicenter of 1st & 2nd Johnson-Yff circles (the exsimilicenter is X(4))
X(1) = orthic-isogonal conjugate of X(46)
X(1) = excentral-isogonal conjugate of X(40)
X(1) = excentral-isotomic conjugate of X(2951)
X(1) = center of Conway circle
X(1) = center of Adams circle
X(1) = X(3) of polar triangle of Conway circle
X(1) = homothetic center of intangents triangle and reflection of extangents triangle in X(3)
X(1) = Hofstadter 1/2 point
X(1) = orthocenter of X(4)X(9)X(885)
X(1) = intersection of tangents at X(7) and X(8) to Lucas cubic K007
X(1) = trilinear product of vertices of 2nd mixtilinear triangle
X(1) = trilinear product of vertices of 2nd Sharygin triangle
X(1) = homothetic center of Mandart-incircle triangle and 2nd isogonal triangle of X(1); see X(36)
X(1) = trilinear pole of the antiorthic axis (which is also the Monge line of the mixtilinear excircles)
X(1) = pole wrt polar circle of trilinear polar of X(92) (line X(240)X(522))
X(1) = X(48)-isoconjugate (polar conjugate) of X(92)
X(1) = X(6)-isoconjugate of X(2)
X(1) = trilinear product of PU(i) for these i: 1, 17, 114, 115, 118, 119, 113
X(1) = barycentric product of PU(i) for these i: 6, 124
X(1) = vertex conjugate of PU(9)
X(1) = bicentric sum of PU(i) for these i: 28, 47, 51, 55, 64
X(1) = trilinear pole of PU(i) for these i: 33, 50, 57, 58, 74, 76, 78
X(1) = crossdifference of PU(i) for these i: 33, 50, 57, 58, 74, 76, 78
X(1) = midpoint of PU(i) for these i: 47, 51, 55
X(1) = PU(28)-harmonic conjugate of X(1023)
X(1) = PU(64)-harmonic conjugate of X(351)
X(1) = intersection of diagonals of trapezoid PU(6)PU(31)
X(1) = perspector circumconic centered at X(9)
X(1) = eigencenter of mixtilinear excentral triangle
X(1) = eigencenter of 2nd Sharygin triangle
X(1) = perspector of ABC and unary cofactor triangle of extangents triangle
X(1) = perspector of ABC and unary cofactor triangle of Feuerbach triangle
X(1) = perspector of ABC and unary cofactor triangle of Apollonius triangle
X(1) = perspector of ABC and unary cofactor triangle of 2nd mixtilinear triangle
X(1) = perspector of ABC and unary cofactor triangle of 4th mixtilinear triangle
X(1) = perspector of ABC and unary cofactor triangle of Apus triangle
X(1) = perspector of unary cofactor triangles of 6th and 7th mixtilinear triangles
X(1) = perspector of unary cofactor triangles of 2nd and 3rd extouch triangles
X(1) = perspector of 5th mixtilinear triangle and unary cofactor triangle of 2nd mixtilinear triangle
X(1) = perspector of 5th mixtilinear triangle and unary cofactor triangle of 4th mixtilinear triangle
X(1) = X(3)-of-reflection-triangle-of-X(1)
X(1) = X(1181)-of-2nd-extouch triangle
X(1) = perspector of ABC and orthic-triangle-of-2nd-circumperp-triangle
X(1) = X(4)-of-excentral triangle
X(1) = X(40)-of-Yff central triangle
X(1) = X(20)-of-1st circumperp triangle
X(1) = X(4)-of-2nd circumperp triangle
X(1) = X(4)-of-Fuhrmann triangle
X(1) = X(100)-of-X(1)-Brocard triangle
X(1) = antigonal image of X(80)
X(1) = trilinear pole wrt excentral triangle of antiorthic axis
X(1) = trilinear pole wrt incentral triangle of antiorthic axis
X(1) = Miquel associate of X(7)
X(1) = homothetic center of Johnson triangle and cross-triangle of ABC and 2nd isogonal triangle of X(1)
X(1) = homothetic center of 1st Johnson-Yff triangle and cross-triangle of ABC and 2nd isogonal triangle of X(1)
X(1) = homothetic center of 2nd Johnson-Yff triangle and cross-triangle of ABC and 2nd isogonal triangle of X(1)
X(1) = homothetic center of Mandart-incircle triangle and cross-triangle of ABC and 1st Johnson-Yff triangle
X(1) = homothetic center of medial triangle and cross-triangle of Aquila and anti-Aquila triangles
X(1) = homothetic center of outer Garcia triangle and cross-triangle of Aquila and anti-Aquila triangles
X(1) = X(8)-of-cross-triangle-of Aquila-and-anti-Aquila-triangles
X(1) = X(3)-of-Mandart-incircle-triangle
X(1) = X(100)-of-inner-Garcia-triangle
X(1) = Thomson-isogonal conjugate of X(165)
X(1) = X(8)-of-outer-Garcia-triangle
X(1) = X(486)-of-BCI-triangle
X(1) = X(164)-of-orthic-triangle if ABC is acute
X(1) = X(1593)-of-Ascella-triangle
X(1) = excentral-to-Ascella similarity image of X(1697)
X(1) = Dao image of X(1)
X(1) = X(40)-of-reflection of ABC in X(3)
X(1) = radical center of the tangent circles of ABC
X(1) = homothetic center of intangents triangle and anti-tangential midarc triangle
X(1) = K(X(15)) = K(X(16)), as defined at X(174)
X(1) = X(3)-of-hexyl-triangle
X(1) = eigencenter of trilinear obverse triangle of X(2)
X(1) = hexyl-isogonal conjugate of X(40)
X(1) = inverse-in-polar-circle of X(1785)
X(1) = inverse-in-orthoptic-circle-of-Steiner-inellipse of X(5121)
X(1) = inverse-in-OI-inverter of X(1155)
X(1) = inverse-in-Steiner-circumellipse of X(239)
X(1) = inverse-in-MacBeath-circumconic of X(2323)
X(1) = inverse-in-circumconic-centered-at-X(9) of X(44)
X(1) = excentral-to-ABC barycentric image of X(40)
X(1) = excentral-to-ABC functional image of X(164)
X(1) = excentral-to-ABC trilinear image of X(164)
X(1) = orthic-to-ABC functional image of X(4), if ABC is acute
X(1) = orthic-to-ABC trilinear image of X(4), if ABC is acute
X(1) = intouch-to-ABC barycentric image of X(1)
X(1) = excentral-to-intouch similarity image of X(40)
X(1) = ABC-to-excentral barycentric image of X(8)
X(1) = X(1)-vertex conjugate of X(56)
X(1) = perspector of ABC and reflection triangle of intangents triangle
X(1) = perspector of pedal and anticevian triangles of X(40)
X(1) = perspector of hexyl triangle and antipedal triangle of X(40)
X(1) = perspector of hexyl triangle and anticevian triangle of X(57)
X(1) = X(4)-of-Pelletier-triangle


X(2) = CENTROID

Trilinears    1/a : 1/b : 1/c
Trilinears    bc : ca : ab
Trilinears    csc A : csc B : csc C
Trilinears    cos A + cos B cos C : cos B + cos C cos A : cos C + cos A cos B
Trilinears    sec A + sec B sec C : sec B + sec C sec A : sec C + sec A sec B
Trilinears    cos A + cos(B - C) : cos B + cos(C - A) : cos C + cos(A - B)
Trilinears    cos B cos C - cos(B - C) : cos C cos A - cos(C - A) : cos A cos B - cos(A - B)
Trilinears    tan(A/2) + cot(A/2) : :
Trilinears    1 + csc A/2 sin B/2 sin C/2 : :
Barycentrics  1 : 1 : 1
Tripolars    Sqrt[2(b^2 + c^2) - a^2] : :
X(2) = (3 + J) X(1113) + (3 - J) X(1114)

As a point on the Euler line, X(2) has Shinagawa coefficients (1, 0).

X(2) is the point of concurrence of the medians of ABC, situated 1/3 of the distance from each vertex to the midpoint of the opposite side. More generally, if L is any line in the plane of ABC, then the distance from X(2) to L is the average of the distances from A, B, C to L. An idealized triangular sheet balances atop a pin head located at X(2) and also balances atop any knife-edge that passes through X(2). The triangles BXC, CXA, AXB have equal areas if and only if X = X(2).

If you have The Geometer's Sketchpad, you can view Centroid.
If you have GeoGebra, you can view Centroid.

X(2) is the centroid of the set of 3 vertices, the centroid of the triangle including its interior, but not the centroid of the triangle without its interior; that centroid is X(10).

X(2) is the identity of the group of triangle centers under "barycentric multiplication" defined by

(x : y : z)*(u : v : w) = xu : yv : zw.

X(2) is the unique point X (as a function of a,b,c) for which the vector-sum XA + XB + XC is the zero vector.

The Parry isodynamic circle is here introduced as the circle centered at X(2502) that passes through the isodynamic points, X(15) and X(16). This circle is orthogonal to both the circumcircle and Parry circle. (Randy Hutson, February 10, 2016)

Let A' be the trilinear pole of the perpendicular bisector of BC, and define B' and C' cyclically. A'B'C' is also the anticomplement of the anticomplement of the midheight triangle. A'B'C' is homothetic to the midheight triangle at X(2). (Randy Hutson, January 29, 2018)

Let A'B'C' be the excentral triangle. Let Oa be the A'-McCay circle of triangle A'BC, and define Ob, Oc cyclically. X(2) is the radical center of Oa, Ob, Oc. (Randy Hutson, June 27, 2018)

X(2) is the unique point that is the symmedian point of its antipedal triangle. (Randy Hutson, August 19, 2019)

Let A'B'C' be the midheight triangle. Let A″ be the cevapoint of B' and C', and define B″ and C″ cyclically. The lines A″, BB″, CC″ concur in X(2). (Randy Hutson, October 8, 2019)

In the plane of a triangle ABC, let
Oa = circle with diameter BC, and define Ob and Oc cyclically;
A′ = the reflection of A with in Oa, and define B′ and C′ cyclically.
The triangle A′B′C′ is perspective to ABC, and the perspector is X(2).
(Dasari Naga Vijay Krishna, April 19, 2021)

Let Na = reflection of ninepoint center(N) wrt to BC, define Nb, Nc cyclically; then NaNbNc is perspective to ABC, and the perspector is X(2). (Dasari Naga Vijay Krishna, June 8, 2021)

Let O be a point(not necessarily X(3)), and let AOB be a fixed-angle sector of a circle C=(O,R), rigidly rotating about center O. Let P be an arbitrary point. The locus of X(2)-of-PAB is a circle C'=(O',R') whose center O' lies on OP. The radius R', independent of P, is given by R'=sqrt(2+sqrt(2))(R/3). Figure. (Dan Reznik, December 11, 2021)

In the plane of a triangle ABC, let P be a point, and let
A'B'C' = pedal triangle of P;
O' = circumcenter of A'B'C';
A" = reflection of A' in O', and define B" and C" cyclically.
The triangles ABC and A"B"C" are perspective, and their pespector is named here the pedal antipodal perspector of P. (Randy Hutson, Hyacinthos 20403, Nov. 21, 2011).

Following are three videos by Dan Reznik (June 30, 2024) . These pertain to porisms involving X(2), X(3), and X(4):

1) Circular caustic (Chapple's porism): Video 1.

2) Generic inellipse: Video 2.

3) Concentric ellipse and MacBeath inellipse: Video 3.

X(2) lies on the Parry circle, Lucas cubic, Thomson cubic, and these lines: 1,8   3,4   6,69   7,9   11,55   12,56   13,16   14,15   17,62   18,61   19,534   31,171   32,83   33,1040   34,1038   35,1479   36,535  37,75   38,244   39,76   40,946   44,89   45,88   51,262   52,1216   54,68   58,540   65,959   66,206   71,1246   72,942   74,113   77,189   80,214   85,241   92,273   94,300   95,97   98,110   99,111   101,116   102,117   103,118   104,119   106,121   107,122   108,123   109,124   112,127   128,1141   129,1298   130,1303   131,1300   133,1294   136,925   137,930   154,1503   165,516   169,1763   174,236   176,1659   178,188   187,316   196,653   201,1393   210,354   216,232   220,1170   222,651   231,1273   242,1851   243,1857   252,1166   253,1073   254,847   257,1432   261,593   265,1511   271,1034   272,284   280,318   283,580   290,327   292,334   294,949   308,702   311,570   314,941   319,1100   322,1108   330,1107   341,1219   351,804   355,944   360,1115   366,367   371,486   372,485   392,517   476,842   480,1223   489,1132   490,1131   495,956   496,1058   514,1022   523,1649   525,1640   561,716   568,1154   572,1746   573,1730   578,1092   585,1336   586,1123   588,1504   589,1505   594,1255   647,850   648,1494   650,693   664,1121   668,1015   670,1084   689,733   743,789   799,873   812,1635   846,1054   914,1442   918,1638   927,1566   954,1260   1073,1249   968,1738   1000,1145   1043,1834   1060,1870   1074,1785   1076,1838   1089,1224   1093,1217   1124,1378   1143,1489   1155,1836   1171,1509   1186,1207   1257,1265   1284,1403   1335,1377   1340,1349   1341,1348   1500,1574   1501,1691   1672,1681   1673,1680   1674,1679   1675,1678   1697,1706   3343,3344   3349,3350   3351,3352

X(2) = midpoint of X(i) and X(j) for these (i,j): {1,3679}, {3,381}, {4,376}, {5,549}, {6,599}, {7,6172}, {8,3241}, {9,6173}, {10,551}, {11,6174}, {13,5463}, {14,5464}, {20,3543}, {21,6175}, {32,7818}, {37,4688}, {39,9466}, {51,3917}, {69,1992}, {75,4664}, {76,7757}, {98,6054}, {99,671}, {110,9140}, {114,6055}, {115,2482}, {125,5642}, {126,9172}, {140,547}, {141,597}, {148,8591}, {154,1853}, {165,1699}, {190,903}, {192,4740}, {210,354}, {329,2094}, {351,9148}, {355,3655}, {373,5650}, {384,7924}, {385,7840}, {392,3753}, {428,7667}, {591,1991}, {618,5459}, {619,5460}, {620,5461}, {631,5071}, {648,1494}, {664,1121}, {668,3227}, {670,3228}, {858,7426}, {1003,7841}, {1086,4370}, {1125,3828}, {1635,4728}, {1638,1639}, {1641,1648}, {1644,1647}, {1649,8371}, {1650,1651}, {2454,2455}, {2479,2480}, {2487,4677}, {2966,5641}, {2976,6161}, {2979,3060}, {3034,3875}, {3034,7292}, {3251,4162}, {3268,9979}, {3448,9143}, {3524,3545}, {3534,3830}, {3576,5587}, {3616,4521}, {3617,3676}, {3623,4468}, {3628,10124}, {3654,3656}, {3681,3873}, {3739,4755}, {3740,3742}, {3817,10164}, {3819,5943}, {3845,8703}, {3929,4654}, {4025,4808}, {4108,5996}, {4120,4750}, {4364,10022}, {4373,4776}, {4379,4893}, {4430,4661}, {4643,4795}, {4730,6332}, {4763,4928}, {5054,5055}, {5108,9169}, {5309,7801}, {5466,9168}, {5485,9741}, {5569,8176}, {5603,5657}, {5640,7998}, {5692,5902}, {5858,5859}, {5860,5861}, {5862,5863}, {5883,10176}, {5891,9730}, {5892,10170}, {5927,10167}, {6032,9829}, {6039,6040}, {6189,6190}, {6545,6546}, {6656,6661}, {6784,6786}, {7615,7618}, {7617,7622}, {7734,10128}, {7753,7810}, {7811,7812}, {7817,7880}, {8010,8011}, {8352,8598}, {8356,8370}, {8360,8368}, {8597,9855}, {8667,9766}, {9185,9191}, {9200,9204}, {9201,9205}, {9761,9763}, {9774,10033}, {9778,9812}, {10162,10163}, {10165,10175}

X(2) = reflection of X(i) in X(j) for these (i,j): (1,551), (3,549), (4,381), (5,547), (6,597), (7,6173), (8,3679), (10,3828), (13,5459), (14,5460), (20,376), (23,7426), (37,4755), (51,5943), (69,599), (75,4688), (76,9466), (98,6055), (99,2482), (100,6174), (110,5642), (111,9172), (115,5461), (140,10124), (144,6172), (145,3241), (147,6054), (148,671), (154,10192), (165,10164), (182,10168), (190,4370), (192,4664), (193,1992), (194,7757), (210,3740), (315,7818), (352,9127), (353,10166), (354,3742), (356,5455), (376,3), (381,5), (384,6661), (547,3628), (549,140), (551,1125), (568,5946), (597,3589), (599,141), (616,5463), (617,5464), (648,3163), (671,115), (903,1086), (944,3655), (1003,8369), (1121,1146), (1278,4740), (1635,4763), (1651,402), (1699,3817), (1962,10180), (1992,6), (2094,57), (2475,6175), (2479,2454), (2480,2455), (2482,620), (2979,3917), (3034,2321), (3060,51), (3091,5071), (3146,3543), (3227,1015), (3228,1084), (3241,1), (3448,9140), (3524,5054), (3534,8703), (3543,4), (3545,5055), (3576,10165), (3617,4521), (3623,3676), (3655,1385), (3676,3616), (3679,10), (3681,210), (3742,3848), (3817,10171), (3828,3634), (3830,3845), (3839,3545), (3845,5066), (3873,354), (3877,392), (3917,3819), (3929,5325), (4240,1651), (4363,10022), (4370,4422), (4430,3873), (4440,903), (4453,1638), (4468,3617), (4521,1698), (4644,4795), (4661,3681), (4664,37), (4669,4745), (4677,4669), (4688,3739), (4728,4928), (4740,75), (4755,4698), (4776,3161), (4795,4670), (4808,3239), (4808,8834), (5066,10109), (5071,1656), (5309,7817), (5459,6669), (5460,6670), (5461,6722), (5463,618), (5464,619), (5466,8371), (5468,1641), (5569,1153), (5587,10175), (5603,5886), (5640,373), (5642,5972), (5692,10176), (5731,3576), (5860,591), (5861,1991), (5862,5858), (5863,5859), (5883,3833), (5890,9730), (5891,10170), (5902,5883), (5918,10178), (5919,10179), (5927,10157), (5943,6688), (6031,9829), (6032,10162), (6054,114), (6055,6036), (6161,2505), (6172,9), (6173,142), (6174,3035), (6175,442), (6546,10196), (6655,7924), (6661,7819), (6688,10219), (6792,9169), (7426,468), (7615,7617), (7618,7622), (7620,7615), (7622,7619), (7671,10177), (7757,39), (7779,7840), (7801,7880), (7811,7810), (7812,7753), (7818,626), (7833,8356), (7840,325), (7924,6656), (7998,5650), (8182,5569), (8353,8354), (8354,8358), (8356,8359), (8368,8365), (8369,8368), (8591,99), (8596,148), (8597,8352), (8860,3054), (9123,9125), (9140,125), (9143,110), (9144,5465), (9147,351), (9168,1649), (9172,6719), (9185,9189), (9263,3227), (9466,3934), (9485,9123), (9730,5892), (9778,165), (9779,7988), (9812,1699), (9829,10163), (9855,8598), (9909,10154), (9939,7811), (9965,2094), (9979,1637), (10022,4472), (10056,10197), (10072,10199), (10162,10173), (10166,10160), (10175,10172)

X(2) = isogonal conjugate of X(6)
X(2) = isotomic conjugate of X(2)
X(2) = cyclocevian conjugate of X(4)
X(2) = circumcircle-inverse of X(23)
X(2) = Conway-circle-inverse of X(38473)
X(2) = nine-point-circle-inverse of X(858)
X(2) = Brocard-circle-inverse of X(110)
X(2) = complement of X(2)
X(2) = anticomplement of X(2)
X(2) = anticomplementary conjugate of X(69)
X(2) = complementary conjugate of X(141)
X(2) = insimilicenter of incircle and Spieker circle
X(2) = insimilicenter of incircle and AC-incircle
X(2) = exsimilicenter of Spieker circle and AC-incircle
X(2) = insimilicenter of Conway circle and Spieker radical circle
X(2) = insimilicenter of polar circle and de Longchamps circle
X(2) = harmonic center of pedal circles of X(13) and X(14) (which are also the pedal circles of X(15) and X(16))
X(2) = X(99)-of -1st-Parry-triangle
X(2) = X(98)-of-2nd-Parry-triangle
X(2) = X(2)-of-1st-Brocard-triangle
X(2) = X(111)-of-4th-Brocard-triangle
X(2) = X(110)-of-X(2)-Brocard-triangle
X(2) = X(110)-of-orthocentroidal-triangle
X(2) = X(353)-of-circumsymmedial-triangle
X(2) = X(165)-of-orthic-triangle if ABC is acute
X(2) = X(51)-of-excentral-triangle
X(2) = inverse-in-polar-circle of X(468)
X(2) = inverse-in-de-Longchamps-circle of X(858)
X(2) = inverse-in-MacBeath-circumconic of X(323)
X(2) = inverse-in-Feuerbach-hyperbola of X(390)
X(2) = inverse-in-circumconic-centered-at-X(1) of X(3935)
X(2) = inverse-in-circumconic-centered-at-X(9) of X(3218)
X(2) = inverse-in-excircles-radical-circle of X(5212)
X(2) = inverse-in-Parry-isodynamic-circle of X(353)
X(2) = barycentric product of (real or nonreal) circumcircle intercepts of the de Longchamps line
X(2) = barycentric product of circumcircle intercepts of line X(325)X(523)
X(2) = barycentric product of PU(3)
X(2) = barycentric product of PU(35)
X(2) = harmonic center of nine-point circle and Johnson circle
X(2) = pole wrt polar circle of trilinear polar of X(4) (orthic axis)
X(2) = polar conjugate of X(4)
X(2) = excentral-to-ABC functional image of X(165)
X(2) = excentral-to-ABC barycentric image of X(165)
X(2) = orthic-to-ABC functional image of X(51)
X(2) = orthic-to-ABC barycentric image of X(51)
X(2) = incentral-to-ABC functional image of X(1962)
X(2) = incentral-to-ABC barycentric image of X(1962)
X(2) = Feuerbach-to-ABC functional image of X(5947)
X(2) = Feuerbach-to-ABC barycentric image of X(5947)
X(2) = perspector of orthic triangle and polar triangle of the complement of the polar circle
X(2) = trilinear pole, wrt orthocentroidal triangle, of Fermat axis
X(2) = trilinear pole, wrt 1st Parry triangle, of line X(1499)X(8598)
X(2) = pole of Brocard axis wrt Stammler hyperbola
X(2) = pole of de Longchamps line wrt the nine-point circle
X(2) = pole of de Longchamps line wrt the de Longchamps circle
X(2) = pole of orthic axis wrt polar circle
X(2) = crosspoint of X(3) and X(6) wrt both the excentral and tangential triangles
X(2) = intersection of tangents at X(1) and X(9) to the hyperbola passing through X(1), X(9) and the excenters (the Jerabek hyperbola of the excentral triangle)
X(2) = crosspoint of X(1) and X(9) wrt excentral triangle
X(2) = crosspoint of X(3) and X(6) wrt excentral triangle
X(2) = crosspoint of X(7) and X(8) wrt 2nd Conway triangle
X(2) = antipode of X(3228) in hyperbola {{A,B,C,X(2),X(6)}}
X(2) = antipode of X(1494) in hyperbola {{A,B,C,X(2),X(69)}}
X(2) = perspector of pedal and anticevian triangles of X(20)
X(2) = homothetic center of the 2nd pedal triangle of X(4) and the 3rd pedal triangle of X(3)
X(2) = perspector of ABC and the reflection in X(6) of the pedal triangle of X(6)
X(2) = perspector of orthic triangle and polar triangle of the complement of the polar circle
X(2) = Moses-radical-circle-inverse of X(34235)
X(2) = X(6374)-cross conjugate of X(194)
X(2) = 1st-Brocard-isogonal conjugate of X(3734)
X(2) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,192), (4,193), (6,194), (7,145), (8,144), (30,1494), (69,20), (75,8), (76,69), (83,6), (85,7), (86,1), (87,330), (95,3), (98,385), (99,523), (190,514), (264,4), (274,75), (276, 264), (287,401), (290,511), (308,76), (312,329), (325,147), (333,63), (348,347), (491,487), (492,488), (523,148), (626,1502)
X(2) = cevapoint of X(i) and X(j) for these (i,j): (1,9), (3,6), (5,216), (10,37), (11,650), (32,206), (39,141), (44,214), (57,223), (114,230), (115,523), (125,647), (128,231), (132,232), (140,233), (188,236), (5408,5409)
X(2) = X(i)-cross conjugate of X(j) for these (i,j):
(1,7), (3,69), (4,253), (5,264), (6,4), (9,8), (10,75), (32,66), (37,1), (39,6), (44,80), (57,189), (75,330), (114,325), (140,95), (141,76), (142,85), (178,508), (187,67), (206,315), (214,320), (216,3), (223,329), (226,92), (230,98), (233,5), (281,280), (395,14), (396,13), (440,306), (511,290), (514,190), (523,99)
X(2) = crosspoint of X(i) and X(j) for these (i,j): (1,87), (75,85), (76,264), (83,308), (86,274), (95,276),(36308,36311)
X(2) = crosssum of X(i) and X(j) for these (i,j): (1,43), (2,194), (31,41), (32,184), (42,213), (51,217), (125,826), (649,1015), (688,1084), (902,1017), (1400,1409)
X(2) = crossdifference of every pair of points on line X(187)X(237)
X(2) = X(i)-Hirst inverse of X(j) for these (i,j):
(1,239), (3,401), (4,297), (6,385), (21,448), (27,447), (69,325), (75,350), (98,287), (115,148), (193,230), (291,335), (298,299), (449,452)
X(2) = X(3)-line conjugate of X(237) = X(316)-line conjugate of X(187)
X(2) = X(i)-aleph conjugate of X(j) for these (i,j): (1,1045), (2,191), (86,2), (174,1046), (333,20), (366,846)
X(2) = X(i)-beth conjugate of X(j) for these (i,j): (2,57), (21,995) (190,2), (312,312), (333,2), (643,55), (645,2), (646,2), (648,196), (662,222)
X(2) = one of two harmonic traces of the power circles; the other is X(858)
X(2) = one of two harmonic traces of the McCay circles; the other is X(111)
X(2) = orthocenter of X(i)X(j)X(k) for these (i,j,k): (4,6,1640), (4,10,4040)
X(2) = centroid of PU(1)X(76) (1st, 2nd, 3rd Brocard points)
X(2) = trilinear pole of PU(i) for these i: 24, 41
X(2) = crossdifference of PU(i) for these i: 2, 26
X(2) = trilinear product of PU(i) for these i: 6,124
X(2) = bicentric sum of PU(i) for these i: 116, 117, 118, 119, 138, 148
X(2) = midpoint of PU(i) for these i: 116, 117, 118, 119, 127
X(2) = intersection of diagonals of trapezoid PU(11)PU(45) (lines P(11)P(45) and U(11)U(45))
X(2) = X(5182) of 6th Brocard triangle (see X(384))
X(2) = PU(148)-harmonic conjugate of X(669)
X(2) = bicentric difference of PU(147)
X(2) = eigencenter of 2nd Brocard triangle
X(2) = perspector of ABC and unary cofactor triangle of Lucas central triangle
X(2) = perspector of ABC and unary cofactor triangle of Lucas(-1) central triangle
X(2) = perspector of ABC and unary cofactor triangle of Lucas tangents triangle
X(2) = perspector of ABC and unary cofactor triangle of Lucas(-1) tangents triangle
X(2) = perspector of ABC and unary cofactor triangle of Lucas inner triangle
X(2) = perspector of ABC and unary cofactor triangle of Lucas(-1) inner triangle
X(2) = perspector of ABC and unary cofactor triangle of 1st anti-Brocard triangle
X(2) = perspector of ABC and unary cofactor triangle of 1st Sharygin triangle
X(2) = perspector of ABC and unary cofactor triangle of 2nd Sharygin triangle
X(2) = perspector of ABC and unary cofactor triangle of 1st Pamfilos-Zhou triangle
X(2) = perspector of ABC and unary cofactor triangle of Artzt triangle
X(2) = perspector of 1st Parry triangle and unary cofactor of 3rd Parry triangle
X(2) = X(6032) of 4th anti-Brocard triangle
X(2) = orthocenter of X(3)X(9147)X(9149)
X(2) = exsimilicenter of Artzt and anti-Artzt circles; the insimilicenter is X(183)
X(2) = perspector of ABC and cross-triangle of inner- and outer-squares triangles
X(2) = perspector of ABC and 2nd Brocard triangle of 1st Brocard triangle
X(2) = perspector of half-altitude triangle and cross-triangle of ABC and half-altitude triangle
X(2) = 4th-anti-Brocard-to-anti-Artzt similarity image of X(111)
X(2) = homothetic center of Aquila triangle and cross-triangle of Aquila and anti-Aquila triangles
X(2) = X(551)-of-cross-triangle-of Aquila-and-anti-Aquila-triangles
X(2) = harmonic center of polar circle and circle O(PU(4))
X(2) = Thomson-isogonal conjugate of X(3)
X(2) = Lucas-isogonal conjugate of X(20)
X(2) = X(3679)-of-outer-Garcia-triangle
X(2) = Dao image of X(13)
X(2) = Dao image of X(14)
X(2) = center of equilateral triangle X(3)PU(5)
X(2) = center of equilateral triangle formed by the circumcenters of BCF, CAF, ABF, where F = X(13)
X(2) = center of equilateral triangle formed by the circumcenters of BCF', CAF', ABF', where F' = X(14)
X(2) = trisector nearest X(5) of segment X(3)X(5)
X(2) = trisector nearest X(4) of segment X(4)X(20)
X(2) = pedal antipodal perspector of X(15)
X(2) = pedal antipodal perspector of X(16)
X(2) = K(X(3)), as defined at X(174)
X(2) = Ehrmann-mid-to-Johnson similarity image of X(381)
X(2) = Kiepert hyperbola antipode of X(671)
X(2) = antigonal conjugate of X(671)
X(2) = trilinear square of X(366)
X(2) = intersection of diagonals of trapezoid X(1)X(7)X(8)X(9)
X(2) = Danneels point of X(99)
X(2) = Danneels point of X(648)
X(2) = perspector of Spieker circle
X(2) = orthic-isogonal conjugate of X(193)
X(2) = X(154)-of-intouch-triangle
X(2) = Vu circlecevian point V(X(13),X(14))


X(3) = CIRCUMCENTER

Trilinears    cos A : cos B : cos C
Trilinears    a(b2 + c2 - a2) : b(c2 + a2 - b2) : c(a2 + b2 - c2)
Barycentrics  sin 2A : sin 2B : sin 2C
Barycentrics  tan B + tan C : tan C + tan A: tan A + tan B
Barycentrics    S^2 - SB SC : :
Barycentrics    1 - cot B cot C : :
Tripolars    1 : 1 : 1

As a point on the Euler line, X(3) has Shinagawa coefficients (1, -1).

X(3) is the point of concurrence of the perpendicular bisectors of the sides of ABC. The lengths of segments AX, BX, CX are equal if and only if X = X(3). This common distance is the radius of the circumcircle, which passes through vertices A,B,C. Called the circumradius, it is given by R = a/(2 sin A) = abc/(4*area(ABC)).

The tangents at vertices of excentral triangle to the McCay cubic K003 concur in X(3). Also, the tangents at A,B,C to the orthocubic K006 concur in X(3). (Randy Hutson, November 18, 2015)

Let A'B'C' be the cevian triangle of X(4). Let A″ be X(4)-of-AB'C', and define B″, C″ cyclically. The lines A″, BB″, CC″ concur in X(3). (Randy Hutson, June 27, 2018)

Let P be a point in the plane of ABC. Let P' be the isogonal conjugate of P. Let P" be the pedal antipodal perspector of P. X(3) is the unique point P for which P' = P". (Randy Hutson, June 27, 2018)

Taking a reference triangle ABC and a variable point P on the plane, P=X(3) is the point of maximal area of its pedal triangle when considering all points P inside the circumcircle of ABC. There are points P far away from the circumcircle for which the area of their pedal triangles is much larger. However, if you consider the signed area of the pedal triangle of P (of which sign depends on whether the points are in clockwise or anti-clockwise order), you could just say that the area of the pedal triangle of P is always negative whenever P is outside of the circumcircle so that P=X(3) gives the global maximum. (Mark Helman, July 10, 2020)

A slightly similar thing happens regarding the area of the antipedal triangle of P. P=X(4) has the smallest area of its antipedal amongst all P in the interior of triangle ABC (when X(4) is in this interior). There are points P (on the circumcircle) for which this area goes to 0. However, if we consider the signed area of the antipedal, even though there are still regions of the plane outside of ABC where the signed area is positive, P=X(4) gives the smallest area of the antipedal among all P for which this area is positive (this works even when ABC is obtuse, and points close to the circumcircle (on both sides) have negative antipedal area). (Mark Helman, July 10, 2020)

View Extremal Area Pedal and Antipedal Triangles, by Mark Helman, Ronaldo Garcia, and Dan Reznik.

If you have The Geometer's Sketchpad, you can view Circumcenter.
If you have GeoGebra, you can view Circumcenter.

Let T be any one of these trianges: {Aries, X(3)-Ehrmann, X3-ABC reflections, 3rd pedal of X(3), 3rd antipedal of X(3), inner-Le Viet An, outer-Le Viet An}. Let OA be the circle centered at the A-vertex of T and passing through A; define OBand OC cyclically. X(3) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

In the plane of a triangle ABC, let
Oa = circle with diameter BC, and define Ob and Oc cyclically;
A′ = reflection of A in Oa, and define B′ and C′ cyclically;
Ab = BA'∩Oa, and define Bc and Ca cyclically;
Ac = CA'∩Oa, and define Ba and Cb cyclically;
A1 = BcBa∩CaCb, and define B1 and C1 cyclically.
The triangle A1B1C1 is perspective to ABC, and the perspector is X(3). (Dasari Naga Vijay Krishna, April 19, 2021)

Let O be a point (not necessarily X(3)), and let be a AOB be a fixed-angle sector of a circle C=(O,R), rigidly rotating about center O. Let P be an arbitrary point. The locus of X(3)-of-PAB is a conic E whose major axis is OP. This conic is an ellipse (resp. hyperbola) if P is interior (resp. exterior) to C. One of its foci is O. Figure (ellipse). Figure (hyperbola). (Dan Reznik, December 10, 2021)

Let A'B'C' be the anticevian triangle of X(3), and let Ea be the ellipse passing through A' and having foci B' and C'. Define Eb and Ec cyclically. The 6 major vertices of the three ellipses lie on a circle that is concentric with the circumcircle of A'B'C'. Figure. (Dan Reznik, December 19, 2021)

X(3) lies on the Thomson cubic, the Darboux cubic, the Napoleon cubic, the Neuberg cubic, the McCay cubic, then Darboux quintic, and these lines: {1,35}, {2,4}, {6,15}, {7,943}, {8,100}, {9,84}, {10,197}, {11,499}, {12,498}, {13,17}, {14,18}, {19,1871}, {31,601}, {33,1753}, {34,1465}, {37,975}, {38,976}, {41,218}, {42,967}, {43,5247}, {47,1399}, {48,71}, {49,155}, {51,3527}, {54,97}, {60,1175}, {63,72}, {64,154}, {66,141}, {67,542}, {68,343}, {69,332}, {73,212}, {74,110}, {76,98}, {77,1410}, {80,5445}, {81,5453}, {83,262}, {85,5088}, {86,1246}, {90,1898}, {95,264}, {96,5392}, {101,103}, {102,109}, {105,277}, {106,1293}, {107,1294}, {108,1295}, {111,1296}, {112,1297}, {113,122}, {114,127}, {115,2079}, {119,123}, {125,131}, {128,1601}, {142,516}, {143,1173}, {144,5843}, {145,1483}, {147,2896}, {149,1484}, {158,243}, {161,1209}, {164,3659}, {169,910}, {172,2276}, {191,1768}, {193,1353}, {194,385}, {200,963}, {201,1807}, {207,1767}, {214,2800}, {217,3289}, {223,1035}, {225,1074}, {226,4292}, {227,1455}, {230,2549}, {232,1968}, {238,978}, {248,3269}, {252,930}, {256,987}, {269,939}, {295,2196}, {296,820}, {298,617}, {299,616}, {302,621}, {303,622}, {305,1799}, {315,325}, {323,3431}, {329,2096}, {345,1791}, {347,1119}, {348,1565}, {351,2780}, {352,353}, {356,3278}, {358,6120}, {373,3066}, {380,2257}, {388,495}, {390,1058}, {392,3420}, {393,1217}, {395,398}, {396,397}, {476,477}, {480,5223}, {485,590}, {486,615}, {489,492}, {490,491}, {496,497}, {501,5127}, {513,3657}, {518,3433}, {519,3654}, {523,5664}, {524,5486}, {525,878}, {528,3813}, {532,5859}, {533,5858}, {539,3519}, {541,5642}, {543,5569}, {551,3653}, {595,995}, {604,2269}, {607,1951}, {608,1950}, {609,5280}, {611,1469}, {612,5322}, {613,1428}, {614,5310}, {618,635}, {619,636}, {623,629}, {624,630}, {639,641}, {640,642}, {653,1148}, {659,2826}, {662,1098}, {667,1083}, {669,1499}, {690,6334}, {691,842}, {692,2807}, {695,1613}, {732,6308}, {741,6010}, {758,5884}, {759,6011}, {805,2698}, {840,2742}, {843,2709}, {846,2944}, {847,925}, {895,4558}, {901,953}, {902,1201}, {905,1946}, {915,2969}, {917,1305}, {920,1858}, {927,2724}, {929,2723}, {934,972}, {935,2697}, {938,3488}, {945,1457}, {947,5399}, {950,1210}, {951,1407}, {955,1170}, {960,997}, {962,1621}, {968,6051}, {974,5504}, {984,3497}, {1000,1476}, {1014,3945}, {1015,2241}, {1018,4513}, {1033,1249}, {1037,1066}, {1046,4650}, {1047,2636}, {1054,1283}, {1055,1334}, {1056,3600}, {1057,1450}, {1069,6238}, {1072,3011}, {1075,1941}, {1093,1105}, {1104,3752}, {1107,4386}, {1124,2066}, {1131,3316}, {1132,3317}, {1135,6121}, {1137,6122}, {1138,3471}, {1139,3370}, {1140,3397}, {1167,1413}, {1177,1576}, {1180,1627}, {1184,1194}, {1196,1611}, {1199,1994}, {1203,5313}, {1211,5810}, {1213,5816}, {1247,2640}, {1263,3459}, {1270,5874}, {1271,5875}, {1276,5240}, {1277,5239}, {1290,2687}, {1298,1303}, {1301,5897}, {1304,2693}, {1308,2717}, {1309,2734}, {1330,4417}, {1331,1797}, {1335,2067}, {1337,3489}, {1338,3490}, {1348,2040}, {1349,2039}, {1364,1795}, {1386,3941}, {1389,2320}, {1397,1682}, {1398,1870}, {1400,2268}, {1406,1464}, {1411,1772}, {1412,2213}, {1425,3561}, {1427,1448}, {1433,2188}, {1445,5728}, {1446,3188}, {1447,3673}, {1452,1905}, {1453,2999}, {1471,2293}, {1475,2280}, {1495,3426}, {1500,2242}, {1506,5475}, {1568,3521}, {1575,4426}, {1587,3068}, {1588,3069}, {1602,2550}, {1603,2551}, {1612,4000}, {1625,1987}, {1630,3197}, {1632,2790}, {1633,5698}, {1661,2883}, {1672,3238}, {1673,3237}, {1676,5403}, {1677,5404}, {1696,3731}, {1698,4413}, {1699,3624}, {1709,3683}, {1714,5721}, {1723,2264}, {1724,3216}, {1728,1864}, {1737,1837}, {1745,1935}, {1762,2939}, {1770,1836}, {1779,1780}, {1788,3486}, {1794,3173}, {1796,3690}, {1808,4173}, {1810,4587}, {1811,4571}, {1813,3270}, {1834,5292}, {1901,5747}, {1914,2275}, {1916,3406}, {1918,2274}, {1939,6181}, {1960,2821}, {1986,2904}, {2007,3235}, {2008,3236}, {2053,2108}, {2120,3463}, {2121,3482}, {2130,3343}, {2131,3350}, {2133,5670}, {2163,2334}, {2174,2911}, {2183,2267}, {2197,2286}, {2222,2716}, {2292,3724}, {2329,3501}, {2346,3296}, {2407,2452}, {2548,3815}, {2688,2690}, {2689,2695}, {2691,2752}, {2692,2758}, {2694,2766}, {2696,2770}, {2699,2703}, {2700,2702}, {2701,2708}, {2704,2711}, {2705,2712}, {2706,2713}, {2707,2714}, {2710,2715}, {2718,2743}, {2719,2744}, {2720,2745}, {2721,2746}, {2722,2747}, {2725,2736}, {2726,2737}, {2727,2738}, {2728,2739}, {2729,2740}, {2730,2751}, {2731,2757}, {2732,2762}, {2733,2765}, {2735,2768}, {2783,4436}, {2792,4655}, {2797,6130}, {2801,3678}, {2810,3939}, {2814,3960}, {2827,4491}, {2854,5505}, {2886,4999}, {2888,3448}, {2916,3456}, {2951,3646}, {2971,3563}, {3006,5300}, {3058,4309}, {3061,3496}, {3065,3467}, {3092,5413}, {3093,5412}, {3100,6198}, {3101,6197}, {3165,5669}, {3166,5668}, {3177,3732}, {3200,3205}, {3201,3206}, {3218,3418}, {3219,3876}, {3224,6234}, {3229,3360}, {3272,3334}, {3276,3280}, {3277,3282}, {3305,5927}, {3306,5439}, {3332,4648}, {3341,3347}, {3351,3354}, {3366,3391}, {3367,3392}, {3373,3387}, {3374,3388}, {3381,5402}, {3382,5401}, {3399,3407}, {3413,6178}, {3414,6177}, {3417,3869}, {3436,5552}, {3437,5224}, {3440,5674}, {3441,5675}, {3447,6328}, {3452,6259}, {3460,3465}, {3461,3483}, {3462,5667}, {3464,3466}, {3474,3485}, {3555,3870}, {3582,4330}, {3583,4324}, {3584,4325}, {3585,4316}, {3589,5480}, {3614,5326}, {3620,5921}, {3632,5288}, {3647,3652}, {3667,4057}, {3679,5258}, {3681,4420}, {3687,5814}, {3694,5227}, {3705,5015}, {3710,3977}, {3711,5531}, {3733,6003}, {3734,3934}, {3740,5302}, {3824,5715}, {3849,6232}, {3874,4973}, {3877,4881}, {3889,3957}, {3901,4880}, {3925,6253}, {4001,4101}, {4317,4995}, {4338,4870}, {4340,5323}, {4549,4846}, {4653,6176}, {4720,5372}, {4850,5262}, {4993,4994}, {5226,5714}, {5260,5818}, {5268,5345}, {5275,5277}, {5284,5550}, {5286,5305}, {5306,5319}, {5346,5355}, {5436,5437}, {5441,5442}, {5443,5444}, {5530,5725}, {5541,6264}, {5590,5594}, {5591,5595}, {5606,5951}, {5638,6141}, {5639,6142}, {5640,5643}, {5656,6225}, {5658,5811}, {5672,6191}, {5673,6192}, {5735,6173}, {5962,5963}, {5971,6031}, {6082,6093}, {6118,6250}, {6119,6251}, {6228,6230}, {6229,6231}, {6233,6323}, {6236,6325}, {6294,6295}, {6296,6298}, {6297,6299}, {6300,6302}, {6301,6303}, {6304,6306}, {6305,6307}, {6311,6313}, {6312,6314}, {6315,6317}, {6316,6318}, {6391,6461}, {6413,6458}, {6414,6457}, {6581,6582}

X(3) is the {X(2),X(4)}-harmonic conjugate of X(5). For a list of other harmonic conjugates of X(3), click Tables at the top of this page. If triangle ABC is acute, then X(3) is the incenter of the tangential triangle and the Bevan point, X(40), of the orthic triangle.

X(3) = midpoint of X(i) and X(j) for these (i,j): (1,40), (2,376), (4,20), (22,378), (74,110), (98,99), (100,104), (101,103), (102,109), (476,477)
X(3) = reflection of X(i) in X(j) for these (i,j): (1,1385), (2,549), (4,5), (5,140), (6,182), (20,550), (52,389), (110,1511), (114,620), (145,1483), (149,1484), (155,1147), (193,1353), (195,54), (265,125), (355,10), (381,2), (382,4), (399,110), (550,548), (576,575), (946,1125), (1351,6), (1352,141), (1482,1)
X(3) = isogonal conjugate of X(4)
X(3) = isotomic conjugate of X(264)
X(3) = complement of X(4)
X(3) = anticomplement of X(5)
X(3) = complementary conjugate of X(5)
X(3) = anticomplementary conjugate of X(2888)
X(3) = nine-point-circle-inverse of X(2072)
X(3) = orthocentroidal-circle-inverse of X(5)
X(3) = 1st-Lemoine-circle-inverse of X(2456)
X(3) = 2nd-Lemoine-circle-inverse of X(1570)
X(3) = Conway-circle-inverse of X(38474)
X(3) = eigencenter of the medial triangle
X(3) = eigencenter of the tangential triangle
X(3) = exsimilicenter of 1st and 2nd Kenmotu circles
X(3) = exsimilicenter of nine-point circle and tangential circle
X(3) = X(1)-of-Trinh-triangle if ABC is acute
X(3) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,6), (4,155), (5,195), (20,1498), (21,1), (22,159), (30,399), (63,219), (69,394), (77,222), (95,2), (96,68), (99,525), (100,521), (110,520), (250, 110), (283,255)
X(3) = cevapoint of X(i) and X(j) for these (i,j): (6,154), (48,212), (55,198), (71,228), (185,417), (216,418)
X(3) = X(i)-cross conjugate of X(j) for these (i,j): (48,222), (55,268), (71,63), (73,1), (184,6), (185,4), (212,219), (216,2), (228,48), (520,110)
X(3) = crosspoint of X(i) and X(j) for these (i,j): (1,90), (2,69), (4,254), (21,283), (54,96), (59,100), (63,77), (78,271), (81,272), (95,97), (99,249), (110,250), (485,486)
X(3) = crosssum of X(i) and X(j) for these (i,j):
(1,46), (2,193), (3,155), (5,52), (6,25), (11,513), (19,33), (30,113), (37,209), (39, 211), (51,53), (65,225), (114,511), (115,512), (116,514), (117, 515), (118,516), (119,517), (120,518), (121,519), (122,520), (123,521), (124,522), (125,523), (126,524), (127,525), (128,1154), (136,924), (184,571), (185,235), (371,372), (487,488)
X(3) = crossdifference of every pair of points on the line X(230)X(231)
X(3) = X(i)-Hirst inverse of X(j) for these (i,j): (2, 401), (4,450), (6,511), (21,416), (194, 385)
X(3) = X(2)-line conjugate of X(468)
X(3) = X(i)-aleph conjugate of X(j) for these (i,j): (1,1046), (21,3), (188,191), (259,1045)
X(3) = X(i)-beth conjugate of X(j) for these (i,j): (3,603), (8,355), (21,56), (78,78), (100,3), (110,221), (271,84), (283,3), (333,379), (643,8)
X(3) = center of inverse-in-de-Longchamps-circle-of-anticomplementary-circle
X(3) = perspector of inner and outer Napoleon triangles
X(3) = Hofstadter 2 point
X(3) = trilinear product of vertices of 2nd Brocard triangle
X(3) = orthocenter of X(i)X(j)X(k) for these (i,j,k): (1,8,5556), (1,9,885), (2,6,1640), (2,10,4049), (3,6,879), (3,66,2435), (4,6,879), (7,8,885), (67,74,879), (6,64,2435), (4,66,2435)
X(3) = intersection of tangents at X(3) and X(4) to Orthocubic K006
X(3) = homothetic center of tangential triangle and 2nd isogonal triangle of X(4); see X(36)
X(3) = trilinear pole of line X(520)X(647)
X(3) = crossdifference of PU(4)
X(3) = trilinear product of PU(16)
X(3) = barycentric product of PU(22)
X(3) = midpoint of PU(i) for these i: 37, 44
X(3) = bicentric sum of PU(i) for these i: 37, 44, 63, 125
X(3) = vertex conjugate of PU(39)
X(3) = PU(63)-harmonic conjugate of X(351)
X(3) = PU(125)-harmonic conjugate of X(650)
X(3) = intersection of tangents to orthocentroidal circle at PU(5)
X(3) = X(3398) of 5th Brocard triangle (see X(32))
X(3) = X(182) of 6th Brocard triangle (see X(384))
X(3) = homothetic center of 1st anti-Brocard triangle and 6th Brocard triangle
X(3) = similitude center of antipedal triangles of the 1st and 2nd Brocard points (PU(1))
X(3) = inverse-in-polar-circle of X(403)
X(3) = inverse-in-{circumcircle, nine-point circle}-inverter of X(858)
X(3) = inverse-in-de-Longchamps-circle of X(3153)
X(3) = inverse-in-Steiner-circumellipse of X(401)
X(3) = inverse-in-Steiner-inellipse of X(441)
X(3) = inverse-in-MacBeath-circumconic of X(3284)
X(3) = radical trace of circumcircle and 8th Lozada circle
X(3) = perspector of medial triangle and polar triangle of the complement of the polar circle
X(3) = pole of line X(6)X(110) wrt Parry circle
X(3) = pole wrt polar circle of trilinear polar of X(2052) (line X(403)X(523))
X(3) = pole wrt {circumcircle, nine-point circle}-inverter of de Longchamps line
X(3) = polar conjugate of X(2052)
X(3) = X(i)-isoconjugate of X(j) for these (i,j): (6,92), (24,91), (25,75), (48,2052), (93,2964), (112,1577), (1101,2970), (2962,3518)
X(3) = X(30)-vertex conjugate of X(523)
X(3) = homothetic center of any 2 of {tangential, Kosnita, 2nd Euler} triangles
X(3) = X(5)-of-excentral-triangle
X(3) = X(26)-of-intouch-triangle
X(3) = antigonal image of X(265)
X(3) = X(2)-of-antipedal-triangle-of-X(6)
X(3) = perspector of the MacBeath Circumconic
X(3) = perspector of ABC and unary cofactor triangle of 5th Euler triangle
X(3) = intersection of trilinear polars of any 2 points on the MacBeath circumconic
X(3) = circumcevian isogonal conjugate of X(1)
X(3) = orthology center of ABC and orthic triangle
X(3) = orthology center of Fuhrmann triangle and ABC
X(3) = orthic isogonal conjugate of X(155)
X(3) = Miquel associate of X(2)
X(3) = X(40)-of-orthic-triangle if ABC is acute
X(3) = X(98)-of-1st-Brocard-triangle
X(3) = X(1380)-of-2nd-Brocard-triangle
X(3) = X(399)-of-orthocentroidal-triangle
X(3) = X(104)-of X(1)-Brocard-triangle
X(3) = X(74)-of X(2)-Brocard-triangle
X(3) = X(74)-of-X(4)-Brocard-triangle
X(3) = X(597)-of-antipedal-triangle-of-X(2)
X(3) = X(182)-of-1st-anti-Brocard-triangle
X(3) = X(381)-of-4th-anti-Brocard-triangle
X(3) = QA-P12 (Orthocenter of the QA-Diagonal Triangle)-of-quadrilateral X(98)X(99)X(110)X(111)
X(3) = orthocenter of X(2)X(9147)X(9149)
X(3) = perspector of ABC and 1st Brocard triangle of 6th Brocard triangle
X(3) = perspector of ABC and 1st Brocard triangle of circumorthic triangle
X(3) = perspector of ABC and 1st Brocard triangle of dual of orthic triangle
X(3) = perspector of ABC and cross-triangle of ABC and half-altitude triangle
X(3) = homothetic center of inner Yff triangle and cross-triangle of ABC and 1st Johnson-Yff triangle
X(3) = homothetic center of outer Yff triangle and cross-triangle of ABC and 2nd Johnson-Yff triangle
X(3) = anti-Artzt-to-4th-anti-Brocard similarity image of X(6)
X(3) = Thomson-isogonal conjugate of X(2)
X(3) = Lucas-isogonal conjugate of X(2979)
X(3) = X(4)-of-2nd-anti-extouch triangle
X(3) = X(185)-of-A'B'C', as described in ADGEOM #2697 (8/26/2015, Tran Quang Hung)
X(3) = X(5)-of-3rd-anti-Euler-triangle
X(3) = X(5)-of-4th-anti-Euler-triangle
X(3) = X(671)-of-McCay-triangle
X(3) = Dao image of X(4)
X(3) = centroid of ABCX(20)
X(3) = Kosnita(X(20),X(2)) point
X(3) = centroid of incenter and excenters
X(3) = X(265)-of-Fuhrmann-triangle
X(3) = intersection of tangents to 2nd Lemoine circle at intersections with Brocard circle
X(3) = perspector of ABC and antipedal triangle of X(64)
X(3) = trisector nearest X(5) of segment X(5)X(20)
X(3) = Ehrmann-vertex-to-Ehrmann-side similarity image of X(4)
X(3) = Ehrmann-mid-to-ABC similarity image of X(4)
X(3) = Ehrmann-mid-to-Johnson similarity image of X(5)
X(3) = Johnson-to-Ehrmann-mid similarity image of X(20)
X(3) = center of inverse similitude of AAOA triangle and Ehrmann side-triangle
X(3) = X(5)-of-hexyl-triangle
X(3) = X(175)-of-Lucas-central-triangle
X(3) = reflection of X(2080) in the Lemoine axis
X(3) = excentral-isogonal conjugate of X(191)
X(3) = excentral-isotomic conjugate of X(2938)
X(3) = crosssum of foci of orthic inconic
X(3) = crosspoint of foci of orthic inconic
X(3) = similicenter of antipedal triangles of PU(1)
X(3) = excentral-to-ABC functional image of X(40)
X(3) = orthic-to-ABC barycentric image of X(4)
X(3) = orthic-to-ABC functional image of X(5)
X(3) = Feuerbach-to-ABC functional image of X(5)
X(3) = intouch-to-ABC functional image of X(1)
X(3) = ABC-to-excentral barycentric image of X(10)
X(3) = concurrence of Euler lines of intouch triangle and A-, B-, and C-extouch triangles
X(3) = perspector of hexyl triangle and cevian triangle of X(21)
X(3) = perspector of pedal and anticevian triangles of X(1498)
X(3) = perspector of ABC and medial triangle of pedal triangle of X(20)
X(3) = perspector of ABC and the reflection in X(6) of the antipedal triangle of X(6)
X(3) = tangential-isotomic conjugate of tangential-isogonal conjugate of X(35225)
X(3) = Moses-radical-circle-inverse of X(35901)
X(3) = 1st-Brocard-isogonal conjugate of X(2782)
X(3) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,35,55), (1,36,56), (1,46,65), (1,55,3295), (1,56,999), (1,57,942), (1,165,40), (1,171,5711), (1,484,5903), (1,1038,1060), (1,1040,1062), (1,1754,5706), (1,2093,3340), (1,3333,5045), (1,3336,5902), (1,3338,354), (1,3361,3333), (1,3550,5255), (1,3576,1385), (1,3612,2646), (1,3746,3303), (1,5010,35), (1,5119,3057), (1,5131,3336), (1,5264,5710), (1,5563,3304), (1,5697,2098), (1,5903,2099), (2,4,5), (2,5,1656), (2,20,4), (2,21,405), (2,22,25), (2,23,1995), (2,24,6642), (2,25,5020), (2,140,3526), (2,186,6644), (2,377,442), (2,381,5055), (2,382,3851), (2,401,458), (2,404,474), (2,411,3149), (2,418,6638), (2,452,5084), (2,464,440), (2,546,5079), (2,548,1657), (2,549,5054), (2,550,382), (2,631,140), (2,858,5094), (2,859,4245), (2,1010,2049), (2,1113,1344), (2,1114,1345), (2,1370,427), (2,1599,1583), (2,1600,1584), (2,1656,5070), (2,1657,3843), (2,2071,378), (2,2475,2476), (2,2478,4187), (2,2554,2570), (2,2555,2571), (2,2675,2676), (2,3090,3628), (2,3091,3090), (2,3146,3091), (2,3151,469), (2,3152,5125), (2,3522,20), (2,3523,631), (2,3524,549), (2,3525,632), (2,3528,550), (2,3529,546), (2,3534,3830), (2,3543,3545), (2,3545,547), (2,3546,3548), (2,3547,3549), (2,3548,6640), (2,3549,6639), (2,3552,384), (2,3627,5072), (2,3832,5056), (2,3839,5071), (2,4184,1011), (2,4188,404), (2,4189,21), (2,4190,377), (2,4210,4191), (2,4216,859), (2,4226,1316), (2,5046,4193), (2,5056,5067), (2,5059,3832), (2,5189,5169), (2,6636,22), (4,5,381), (4,21,3560), (4,24,25), (4,25,1598), (4,140,1656), (4,186,24), (4,376,20), (4,378,1593), (4,381,3843), (4,382,3830), (4,548,3534), (4,549,3526), (4,550,1657), (4,631,2), (4,632,5079), (4,1006,405), (4,1593,1597), (4,1656,3851), (4,1657,5073), (4,1658,2070), (4,2937,5899), (4,3088,1595), (4,3089,1596), (4,3090,3091), (4,3091,546), (4,3146,3627), (4,3147,3542), (4,3515,3517), (4,3520,378), (4,3522,550), (4,3523,140), (4,3524,631), (4,3525,3090), (4,3526,5055), (4,3528,376), (4,3529,3146), (4,3530,5054), (4,3533,5056), (4,3541,427), (4,3542,235), (4,3543,3853), (4,3545,3832), (4,3548,2072), (4,3627,5076), (4,3628,5072), (4,3832,3845), (4,3839,3861), (4,3855,3839), (4,5054,5070), (4,5056,3850), (4,5067,3545), (4,5068,3858), (4,5071,3855), (4,6353,3089), (4,6621,6624), (4,6622,6623), (5,20,382), (5,26,25), (5,140,2), (5,376,1657), (5,381,3851), (5,382,3843), (5,427,5576), (5,546,3091), (5,547,5056), (5,548,20), (5,549,140), (5,631,3526), (5,632,3628), (5,1656,5055), (5,1657,3830), (5,1658,24), (5,3090,5079), (5,3091,5072), (5,3522,3534), (5,3523,5054), (5,3526,5070), (5,3529,5076), (5,3530,631), (5,3534,5073), (5,3627,546), (5,3628,3090), (5,3845,3850), (5,3850,3545), (5,3853,3832), (5,3858,5066), (5,3861,3855), (5,5066,5068), (5,5498,6143), (5,6642,5020), (5,6644,6642), (6,182,5050), (6,187,1384), (6,371,3311), (6,372,3312), (6,574,5024), (6,1151,371), (6,1152,372), (6,1351,5093), (6,1620,1192), (6,2076,5017), (6,3053,32), (6,3311,6417), (6,3312,6418), (6,3592,6419), (6,3594,6420), (6,4252,58), (6,4255,386), (6,4258,4251), (6,5013,39), (6,5022,4253), (6,5023,3053), (6,5085,182), (6,5102,5097), (6,5210,187), (6,5585,5210), (6,6200,6221), (6,6221,6199), (6,6396,6398), (6,6398,6395), (6,6409,1151), (6,6410,1152), (6,6411,6200), (6,6412,6396), (6,6417,6500), (6,6418,6501), (6,6419,6427), (6,6420,6428), (6,6425,3592), (6,6426,3594), (6,6433,6437), (6,6434,6438), (6,6451,6445), (6,6452,6446), (6,6455,6407), (6,6456,6408), (7,3487,6147), (7,5703,3487), (8,100,5687), (8,2975,956), (8,5657,5690), (8,5731,944), (9,936,5044), (9,1490,5777), (9,5438,936), (10,355,5790), (10,993,958), (10,5267,993), (10,5745,5791), (11,5433,499), (11,6284,1479), (12,5432,498), (15,16,6), (15,62,61), (15,3364,371), (15,3365,372), (15,5237,62), (15,5352,5238), (16,61,62), (16,3389,371), (16,3390,372), (16,5238,61), (16,5351,5237), (20,21,1012), (20,140,381), (20,186,26), (20,376,550), (20,381,5073), (20,404,3149), (20,417,6638), (20,549,1656), (20,550,3534), (20,631,5), (20,1006,3560), (20,1656,3830), (20,1658,2937), (20,2060,3079), (20,3090,3627), (20,3091,3146), (20,3146,3529), (20,3522,376), (20,3523,2), (20,3524,140), (20,3525,546), (20,3526,3843), (20,3528,548), (20,3530,3526), (20,3533,3845), (20,3543,5059), (20,3628,5076), (20,5054,3851), (20,5056,3543), (20,5067,3853), (21,404,2), (21,411,4), (21,416,1982), (21,1816,29), (21,1817,28), (21,3658,3109), (21,4188,474), (21,4203,4195), (21,4225,859), (22,24,26), (22,26,2937), (22,381,5899), (22,426,6638), (22,631,6642), (22,1599,3155), (22,1600,3156), (22,1995,23), (22,6644,2070), (23,1995,25), (24,25,3517), (24,26,2070), (24,186,3515), (24,378,4), (24,1593,1598), (24,1657,5899), (24,3516,1597), (24,3520,1593), (25,378,1597), (25,426,6617), (25,1593,4), (25,3515,24), (25,3516,1593), (26,140,6642), (26,378,382), (26,382,5899), (26,6642,3517), (26,6644,24), (28,4219,4), (29,412,4), (32,39,6), (32,182,3398), (32,187,3053), (32,574,39), (32,3053,1384), (32,5171,2080), (32,5206,187), 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X(4) = ORTHOCENTER

Trilinears    sec A : sec B : sec C
Trilinears    cos A - sin B sin C : cos B - sin C sin A : cos C - sin A sinB
Trilinears    cos A - cos(B - C) : cos B - cos(C - A) : cos C - cos(A - B)
Trilinears    sin B sin C - cos(B - C) : sin C sin A - cos(C - A) : sin A sin B - cos(A - B)
Trilinears    csc A tan 3A - 2 sec 3A : :
Trilinears    4 cos A - cos(B - C) - 3 sin B sin C : :
Trilinears    cos A + cos(B - C) + 5 cos B cos C - 2 sin B sin C : :
Barycentrics    1/SA : 1/SB : 1/SC
Barycentrics    tan A : tan B : tan C
Barycentrics    1/(b2 + c2 - a2) : 1/(c2 + a2 - b2) : 1/(a2 + b2 - c2)
Tripolars    |cos A| : :
Tripolars    |a(b^2 + c^2 - a^2)| : :
X(4) = (1 + J) X(1113) + (1 - J) X(1114)
X(4) = (tan A)*[A] + (tan B)*[B] + (tan C)*[C], where A, B, C are the angles and [A], [B], [C] are the vertices

As a point on the Euler line, X(4) has Shinagawa coefficients (0, 1).

X(4) is the point of concurrence of the altitudes of ABC.

The tangents at A,B,C to the McCay cubic K003 concur in X(4). Also, the tangents at A,B,C to the Lucas cubic K007 concur in X(4). (Randy Hutson, November 18, 2015)

Let P be a point in the plane of ABC. Let Oa be the circumcenter of BCP, and define Ob and Oc cyclically. Let Q be the circumcenter of OaObOc. P = Q only when P = X(4). (Randy Hutson, June 27, 2018)

Taking a reference triangle ABC and a variable point P on the plane, P=X(3) is the point of maximal area of its pedal triangle when considering all points P inside the circumcircle of ABC. There are points P far away from the circumcircle for which the area of their pedal triangles is much larger. However, if you consider the signed area of the pedal triangle of P (of which sign depends on whether the points are in clockwise or anti-clockwise order), you could just say that the area of the pedal triangle of P is always negative whenever P is outside of the circumcircle so that P=X(3) gives the global maximum. (Mark Helman, July 10, 2020)

A slightly similar thing happens regarding the area of the antipedal triangle of P. P=X(4) has the smallest area of its antipedal amongst all P in the interior of triangle ABC (when X(4) is in this interior). There are points P (on the circumcircle) for which this area goes to 0. However, if we consider the signed area of the antipedal, even though there are still regions of the plane outside of ABC where the signed area is positive, P=X(4) gives the smallest area of the antipedal among all P for which this area is positive (this works even when ABC is obtuse, and points close to the circumcircle (on both sides) have negative antipedal area). (Mark Helman, July 10, 2020)

View Extremal Area Pedal and Antipedal Triangles, by Mark Helman, Ronaldo Garcia, and Dan Reznik.

If you have The Geometer's Sketchpad, you can view Orthocenter.
If you have GeoGebra, you can view Orthocenter.

X(4) and the vertices A,B,C comprise an orthocentric system, defined as a set of four points, one of which is the orthocenter of the triangle of the other three (so that each is the orthocenter of the other three). The incenter and excenters also form an orthocentric system. If ABC is acute, then X(4) is the incenter of the orthic triangle.

Suppose P is not on a sideline of ABC. Let A'B'C' be the cevian triangle of P. Let D,E,F be the circles having diameters AA', BB',CC'. The radical center of D,E,F is X(4). (Floor van Lamoen, Hyacinthos #214, Jan. 24, 2000.)

Let A2B2C2 be the 2nd Conway triangle. Let A' be the crosspoint of B2 and C2, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(4). (Randy Hutson, December 10, 2016)

Let T be any one of these trianges: {anticevian of X(30), anti-Hutson-intouch, anti-incircle-circles, Ehrmann side, X(2)-Ehrmann, Gemini 15, Gemini 16, Kosnita, midheight, N-obverse of X(6), Schroeter, tangential, Trinh, 1st Zaniah, 2nd Zaniah}. Let OA be the circle centered at the A-vertex of T and passing through A; define OB and OC cyclically. X(4) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

See Ross Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, Mathematical Association of America, 1995. Chapter 2: The Orthocenter.

In the plane of a triangle ABC, let
Oa = circle with diameter BC, and define Ob and Oc cyclically;
A′ = reflection of A in Oa, and define B′ and C′ cyclically;
Ab = polar of A' with respect to Ob, and define Bc and Ca cyclically;
Ac = polar of A' with respect to OC, and define Ba and Cb cyclically;
″ = Ab∩Ac, and define B″ and C″ cyclically.
The triangle A″B″C″ is perspective to ABC, and the perspector is X(4). (Dasari Naga Vijay Krishna, April 19, 2021)

In the plane of a triangle ABC, let
Oa = circle with diameter BC, and define Ob and Oc cyclically;
Pa = polar of A with respect to Oa, and define Pb and Pc cyclically.
Then X(4)= Pa∩Pb∩Pc, (Dasari Naga Vijay Krishna, July 19, 2021)

For extensions of triangle geometry to 3-dimensional shapes called orthocentric tetrahedra, see

William Barker and Roger Howe, Continuous Symmetry From Euclid to Klein, American Mathematical Society, 2007, pages 306-309.

In the plane of a triangle ABC, let D, E, F be points on sides AB, AC, BC respectively. Let
G = circle with diameter AF;
H = circle with diameter BE;
I = circle with diameter CD.
Then X(4) = radical center of G, H, I. See X(4), radical center. (Benjamin Warren, July 24, 2024)

X(4) lies on the Thomson, Darboux, Napoleon, Lucas, McCay, and Neuberg cubics, and the Darboux septic, and on these lines: {1,33}, {2,3}, {6,53}, {7,273}, {8,72}, {9,10}, {11,56}, {12,55}, {13,61}, {14,62}, {15,17}, {16,18}, {31,3072}, {32,98}, {35,498}, {36,499}, {37,1841}, {39,232}, {41,2202}, {42,1860}, {46,90}, {48,1881}, {49,156}, {50,9220}, {51,185}, {52,68}, {54,184}, {57,84}, {58,5292}, {63,5709}, {64,459}, {65,158}, {66,9969}, {67,338}, {69,76}, {74,107}, {75,12689}, {78,908}, {79,1784}, {80,1825}, {81,5707}, {83,182}, {85,4872}, {93,562}, {94,143}, {95,8797}, {96,231}, {99,114}, {100,119}, {101,118}, {102,124}, {103,116}, {105,5511}, {106,5510}, {109,117}, {110,113}, {111,1560}, {120,1292}, {121,1293}, {122,1294}, {123,1295}, {126,1296}, {127,1289}, {128,930}, {129,1303}, {130,1298}, {131,135}, {137,933}, {141,1350}, {142,5732}, {144,2894}, {145,149}, {147,148}, {150,152}, {151,2818}, {154,8888}, {155,254}, {157,5593}, {160,3613}, {162,270}, {165,1698}, {171,601}, {175,10905}, {176,10904}, {177,8095}, {181,9553}, {183,3785}, {187,7607}, {189,5908}, {191,2949}, {193,1351}, {195,399}, {200,6769}, {201,7069}, {204,1453}, {210,7957}, {212,3074}, {214,12119}, {215,9652}, {216,8799}, {218,294}, {230,3053}, {233,10979}, {236,8128}, {238,602}, {240,256}, {250,1553}, {251,8879}, {252,1487}, {255,1935}, {276,327}, {279,1565}, {280,2968}, {282,3345}, {284,5747}, {290,6528}, {298,5864}, {299,5865}, {312,7270}, {325,1975}, {333,5788}, {339,10749}, {341,12397}, {345,7283}, {346,3695}, {347,6356}, {348,5088}, {354,3296}, {371,485}, {372,486}, {373,11465}, {385,7823}, {386,2051}, {390,495}, {391,2322}, {394,1217}, {477,1304}, {484,3460}, {487,489}, {488,490}, {493,8212}, {494,8213}, {496,999}, {512,879}, {518,6601}, {519,3680}, {523,1552}, {524,5485}, {525,2435}, {527,5735}, {528,3913}, {529,3813}, {532,5862}, {533,5863}, {535,8666}, {538,7758}, {539,9936}, {541,9140}, {542,576}, {543,5503}, {544,10710}, {551,9624}, {566,9221}, {567,7578}, {569,1179}, {572,1474}, {574,1506}, {575,598}, {579,1713}, {580,1714}, {584,8818}, {590,1151}, {595,8750}, {603,3075}, {604,7120}, {608,1518}, {615,1152}, {616,627}, {617,628}, {618,5473}, {619,5474}, {620,7862}, {625,3788}, {626,3734}, {635,3643}, {636,3642}, {639,5590}, {640,5591}, {641,12124}, {642,12123}, {651,3157}, {653,1156}, {674,12587}, {684,2797}, {685,2682}, {690,11005}, {693,8760}, {695,3981}, {754,7751}, {758,5693}, {774,1254}, {800,13380}, {801,1092}, {842,935}, {885,3309}, {912,3868}, {916,2997}, {936,3452}, {937,1534}, {940,1396}, {941,1880}, {953,1309}, {958,2886}, {960,5794}, {970,9534}, {972,5514}, {973,6145}, {974,7729}, {983,5255}, {990,4000}, {991,4648}, {993,11012}, {1000,3057}, {1015,9651}, {1029,2906}, {1032,5910}, {1034,5911}, {1036,1065}, {1037,1067}, {1038,1076}, {1039,1096}, {1040,1074}, {1041,2263}, {1043,4417}, {1046,2648}, {1060,4296}, {1062,3100}, {1073,2130}, {1078,5171}, {1089,3974}, {1104,3772}, {1111,4056}, {1123,7133}, {1125,3576}, {1131,3311}, {1132,3312}, {1138,2132}, {1139,3368}, {1140,3395}, {1157,2120}, {1160,1162}, {1161,1163}, {1164,3595}, {1165,3593}, {1175,5320}, {1177,5622}, {1192,3532}, {1209,4549}, {1216,2979}, {1248,2660}, {1251,1832}, {1260,5687}, {1317,12763}, {1319,7704}, {1327,6419}, {1328,6420}, {1329,1376}, {1336,2362}, {1340,1348}, {1341,1349}, {1342,1676}, {1343,1677}, {1353,5093}, {1379,2040}, {1380,2039}, {1383,8791}, {1384,8778}, {1385,1538}, {1389,2099}, {1392,3241}, {1393,7004}, {1399,5348}, {1420,4311}, {1430,1468}, {1435,3333}, {1440,7053}, {1441,4329}, {1445,3358}, {1448,7365}, {1469,12589}, {1483,3623}, {1484,12773}, {1495,3431}, {1499,1550}, {1500,9650}, {1510,10412}, {1511,12121}, {1521,7115}, {1562,6529}, {1566,2724}, {1609,9722}, {1621,10267}, {1670,5404}, {1671,5403}, {1682,9552}, {1689,2010}, {1690,2009}, {1691,3406}, {1697,7160}, {1715,1730}, {1716,1721}, {1717,1718}, {1726,1782}, {1729,8558}, {1764,10479}, {1768,3065}, {1773,2961}, {1781,2955}, {1798,13323}, {1840,4876}, {1903,2262}, {1942,2790}, {1957,5247}, {1970,1971}, {1973,2201}, {1987,3269}, {1994,2904}, {2077,3814}, {2080,7793}, {2092,3597}, {2093,4848}, {2095,9965}, {2098,10944}, {2121,3481}, {2131,3349}, {2133,8440}, {2181,4642}, {2217,3417}, {2275,9597}, {2276,9596}, {2278,5397}, {2287,5778}, {2331,3755}, {2332,4251}, {2353,3425}, {2355,3579}, {2361,7299}, {2393,5486}, {2456,10349}, {2457,3667}, {2477,9653}, {2482,12117}, {2536,2540}, {2537,2541}, {2574,2592}, {2575,2593}, {2646,4305}, {2651,2907}, {2679,2698}, {2687,2766}, {2697,10423}, {2734,10017}, {2752,10101}, {2770,10098}, {2771,9803}, {2778,10693}, {2783,10769}, {2784,11599}, {2787,10768}, {2791,4516}, {2793,9180}, {2801,3254}, {2802,12641}, {2814,3762}, {2817,13532}, {2822,4466}, {2823,4858}, {2826,10773}, {2827,10774}, {2828,10775}, {2830,10779}, {2831,10780}, {2840,4939}, {2889,6101}, {2896,6194}, {2900,3189}, {2905,6625}, {2908,7139}, {2917,8146}, {2929,2935}, {2972,10745}, {2975,5841}, {2995,8048}, {3023,12185}, {3024,12374}, {3027,12184}, {3028,12373}, {3054,5210}, {3056,12588}, {3058,3303}, {3062,3339}, {3094,3399}, {3096,3098}, {3101,8251}, {3120,3924}, {3162,5359}, {3164,9290}, {3172,3424}, {3180,5873}, {3181,5872}, {3184,6716}, {3190,3191}, {3212,7261}, {3216,5400}, {3218,5770}, {3255,5883}, {3270,11461}, {3304,5434}, {3305,3587}, {3306,7171}, {3314,7885}, {3320,12945}, {3329,7864}, {3338,7284}, {3340,3577}, {3342,3347}, {3344,3348}, {3352,3354}, {3356,3637}, {3364,3391}, {3365,3392}, {3366,3389}, {3367,3390}, {3369,3397}, {3370,3396}, {3371,3387}, {3372,3388}, {3373,3385}, {3374,3386}, {3379,5402}, {3380,5401}, {3381,3394}, {3382,3393}, {3398,3407}, {3413,3558}, {3414,3557}, {3416,3714}, {3426,13093}, {3430,3454}, {3438,3443}, {3439,3442}, {3440,5682}, {3441,5681}, {3461,7165}, {3463,5683}, {3466,3469}, {3479,3489}, {3480,3490}, {3495,8866}, {3497,7351}, {3499,8925}, {3500,7350}, {3502,8867}, {3521,5946}, {3527,8796}, {3580,11472}, {3582,4325}, {3584,4330}, {3589,5085}, {3590,6221}, {3591,6398}, {3601,4304}, {3611,11460}, {3614,5217}, {3617,5690}, {3620,7879}, {3621,5844}, {3622,5901}, {3624,7987}, {3629,5102}, {3632,4900}, {3633,11224}, {3634,10164}, {3648,3652}, {3668,8809}, {3671,5665}, {3679,4866}, {3701,5300}, {3704,5695}, {3706,10371}, {3738,10771}, {3741,10476}, {3746,4309}, {3753,9800}, {3812,5880}, {3815,5013}, {3819,13348}, {3820,6244}, {3822,5248}, {3825,10200}, {3826,11495}, {3829,11194}, {3841,7688}, {3847,6691}, {3849,7615}, {3870,5534}, {3871,10528}, {3877,7700}, {3885,12648}, {3887,10772}, {3911,6705}, {3916,5744}, {3917,7999}, {3925,5584}, {3933,7776}, {3934,5188}, {3940,5763}, {3947,4314}, {3972,7828}, {4008,12723}, {4045,7808}, {4048,5103}, {4277,4646}, {4308,7743}, {4313,5226}, {4316,7280}, {4317,5563}, {4324,5010}, {4339,5266}, {4355,10980}, {4357,10444}, {4423,7958}, {4425,8235}, {4444,6002}, {4512,10268}, {4645,7155}, {4654,11518}, {4658,5733}, {4692,4894}, {4721,4805}, {4723,12693}, {4768,9525}, {4846,5462}, {4847,12527}, {4863,12692}, {5007,5309}, {5008,5346}, {5032,11405}, {5038,11170}, {5044,10157}, {5045,5558}, {5050,5395}, {5092,7859}, {5097,7894}, {5119,7162}, {5121,11512}, {5123,13528}, {5173,12677}, {5204,5433}, {5206,6781}, {5221,10308}, {5223,12777}, {5249,10884}, {5253,10269}, {5265,10593}, {5273,5791}, {5278,9958}, {5281,10592}, {5377,6074}, {5418,6200}, {5420,6396}, {5424,5441}, {5435,5704}, {5437,9841}, {5438,6700}, {5439,9776}, {5440,5748}, {5447,7998}, {5449,7689}, {5461,10153}, {5505,10752}, {5513,9085}, {5533,10074}, {5535,6597}, {5536,6763}, {5542,6744}, {5550,11230}, {5553,7702}, {5556,10977}, {5557,12005}, {5559,5697}, {5561,11552}, {5597,8196}, {5598,8203}, {5599,11822}, {5600,11823}, {5601,8200}, {5602,8207}, {5606,5950}, {5609,5655}, {5623,8446}, {5624,8456}, {5627,6070}, {5670,8487}, {5671,8494}, {5672,8444}, {5673,8454}, {5674,8495}, {5675,8496}, {5676,8486}, {5677,7329}, {5678,8491}, {5679,8492}, {5680,7164}, {5685,8480}, {5688,12698}, {5689,12697}, {5705,5745}, {5708,12684}, {5795,9623}, {5848,10759}, {5853,6765}, {5854,13271}, {5860,6278}, {5861,6281}, {5874,11917}, {5875,11916}, {5885,10266}, {5892,11451}, {5933,10362}, {5934,8079}, {5935,7593}, {5943,9729}, {5951,5952}, {5965,7877}, {5984,7766}, {6020,12955}, {6032,12506}, {6036,7857}, {6055,9166}, {6073,11607}, {6075,10428}, {6082,6092}, {6114,9750}, {6115,9749}, {6128,8749}, {6130,9409}, {6147,11036}, {6196,8927}, {6204,8957}, {6217,6266}, {6218,6267}, {6219,6276}, {6220,6277}, {6224,6265}, {6233,13234}, {6235,8705}, {6238,10055}, {6285,7049}, {6292,7935}, {6323,12494}, {6326,6596}, {6339,10981}, {6407,9542}, {6409,8253}, {6410,8252}, {6453,9681}, {6462,8220}, {6463,8221}, {6467,12283}, {6519,9692}, {6680,7844}, {6704,9751}, {6735,12534}, {6752,8795}, {6777,11602}, {6778,11603}, {7017,7141}, {7028,8127}, {7059,7345}, {7060,7344}, {7149,8811}, {7161,11010}, {7264,7272}, {7320,9785}, {7325,8449}, {7326,8459}, {7327,8432}, {7352,10071}, {7587,8379}, {7588,8086}, {7589,8382}, {7595,12681}, {7603,11669}, {7605,13339}, {7617,8182}, {7618,8176}, {7666,10272}, {7676,7679}, {7677,7678}, {7693,13363}, {7703,11454}, {7712,10610}, {7720,7725}, {7721,7726}, {7723,12219}, {7730,7731}, {7739,7753}, {7757,7858}, {7769,7782}, {7777,7783}, {7778,7789}, {7779,7900}, {7786,7847}, {7792,7851}, {7794,7818}, {7796,7809}, {7798,7838}, {7799,7814}, {7801,7821}, {7804,7834}, {7813,7903}, {7815,7830}, {7820,7867}, {7822,7853}, {7829,7902}, {7831,7910}, {7832,7934}, {7835,7899}, {7836,7912}, {7839,7921}, {7845,7855}, {7846,7919}, {7854,7873}, {7856,12150}, {7863,7888}, {7875,7923}, {7883,10302}, {7889,7913}, {7891,7925}, {7905,7926}, {7906,7941}, {7932,10583}, {8068,10058}, {8069,10321}, {8075,8087}, {8076,8088}, {8077,8085}, {8080,8092}, {8099,9793}, {8100,9795}, {8105,8426}, {8106,8427}, {8107,8380}, {8108,8381}, {8109,8377}, {8110,8378}, {8117,8123}, {8118,8124}, {8125,8129}, {8126,8130}, {8141,9536}, {8144,9538}, {8172,8447}, {8173,8457}, {8193,9911}, {8197,12458}, {8204,12459}, {8222,11828}, {8223,11829}, {8224,8230}, {8225,8228}, {8372,12674}, {8431,8443}, {8433,8483}, {8434,8484}, {8435,8481}, {8436,8482}, {8437,8497}, {8438,8498}, {8445,8458}, {8448,8455}, {8450,8461}, {8451,8460}, {8452,8463}, {8453,8462}, {8488,8527}, {8489,8532}, {8490,8533}, {8501,8509}, {8502,8508}, {8515,8536}, {8516,8535}, {8517,8534}, {8538,11416}, {8582,10860}, {8583,10863}, {8588,10185}, {8591,8724}, {8596,12355}, {8674,10767}, {8679,12586}, {8719,10155}, {8864,8921}, {8868,8872}, {8878,10340}, {8983,9583}, {9147,11615}, {9300,9607}, {9530,10718}, {9627,9629}, {9628,9630}, {9638,10535}, {9646,9660}, {9647,9661}, {9648,9662}, {9649,9663}, {9658,9672}, {9659,9673}, {9705,13482}, {9783,12488}, {9787,12489}, {9789,12490}, {9791,9959}, {9845,12577}, {9857,12497}, {9874,12139}, {9897,11280}, {9919,13171}, {9934,13198}, {9942,10391}, {9967,12220}, {9973,13622}, {10042,10050}, {10043,10057}, {10052,10073}, {10088,12896}, {10187,10646}, {10188,10645}, {10202,11220}, {10264,10620}, {10293,12099}, {10305,11023}, {10309,12676}, {10313,10316}, {10363,10369}, {10415,10422}, {10434,10887}, {10435,12547}, {10455,10464}, {10529,10680}, {10546,10564}, {10547,10548}, {10627,13340}, {10634,11420}, {10635,11421}, {10707,11240}, {10791,12197}, {10797,10799}, {10798,12835}, {10831,10833}, {10873,10877}, {10882,10886}, {10897,11417}, {10898,11418}, {10912,13463}, {10915,12703}, {10916,12704}, {10923,10927}, {10924,10928}, {10956,10965}, {10957,10966}, {10958,11502}, {11082,11135}, {11087,11136}, {11171,11272}, {11177,11632}, {11270,11468}, {11402,11426}, {11408,11485}, {11409,11486}, {11423,13366}, {11449,12038}, {11557,11560}, {11587,13558}, {11646,13330}, {11649,11663}, {11698,12331}, {11703,12165}, {11755,11759}, {11764,11768}, {11773,11777}, {11782,11786}, {11792,13508}, {11800,12284}, {11869,11873}, {11870,11874}, {11891,12491}, {11900,12696}, {11905,11909}, {11930,11947}, {11931,11948}, {11990,11992}, {12006,13364}, {12061,12063}, {12120,12864}, {12146,12849}, {12166,12309}, {12168,12310}, {12169,12311}, {12170,12312}, {12171,12313}, {12172,12314}, {12175,12316}, {12223,12603}, {12224,12604}, {12226,12606}, {12271,12272}, {12273,12280}, {12350,12354}, {12369,13495}, {12387,12394}, {12388,12393}, {12507,13249}, {12515,12619}, {12516,12620}, {12517,12621}, {12518,12622}, {12519,12623}, {12520,12609}, {12521,12612}, {12522,12613}, {12523,12614}, {12524,12615}, {12556,13089}, {12624,13238}, {12739,12743}, {12837,13077}, {12859,12863}, {12941,13075}, {12942,13076}, {12944,13078}, {12946,13079}, {12947,13080}, {12948,13081}, {12949,13082}, {13007,13023}, {13008,13024}, {13009,13039}, {13010,13040}, {13321,13451}, {13353,13470}, {13418,13423}

X(4) is the {X(3),X(5)}-harmonic conjugate of X(2). For a list of other harmonic conjugates of X(4), click Tables at the top of this page.

X(4) = midpoint of X(i) and X(j) for these (i,j): (3,382), (146,3448), (147,148), (149,153), (150,152)
X(4) = reflection of X(i) in X(j) for these (i,j): (1,946), (2,381), (3,5), (5,546), (8,355), (20,3), (24,235), (25,1596), (40,10), (69,1352), (74,125), (98,115), (99,114), (100,119), (101,118), (102,124), (103,116), (104,11), (107,133), (109,117), (110,113), (112,132), (145,1482), (185,389), (186,403), (193,1351), (376,2), (378,427), (550,140), (917,5190), (925,131), (930,128), (944,1), (1113,1312), (1114,1313), (1141,137), (1292,120), (1293,121), (1294,122), (1295,123), (1296,126), (1297,127), (1298,130), (1299,135), (1300,136), (1303, 129), (1350,141), (1593,1595)
X(4) = isogonal conjugate of X(3)
X(4) = isotomic conjugate of X(69)
X(4) = cyclocevian conjugate of X(2)
X(4) = circumcircle-inverse of X(186)
X(4) = nine-point-circle-inverse of X(403)
X(4) = complement of X(20)
X(4) = anticomplement of X(3)
X(4) = complementary conjugate of X(2883)
X(4) = anticomplementary conjugate of X(20)
X(4) = 2nd-Brocard-circle-inverse of X(37991)
X(4) = Grebe-circle-inverse of X(37925)
X(4) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,1249), (7,196), (27,19), (29,1), (92,281), (107,523), (264,2), (273,278), (275,6), (286,92), (393, 459)
X(4) = cevapoint of X(i) and X(j) for these (i,j):
(1,46), (2,193), (3,155), (5,52), (6,25), (11,513), (19,33), (30,113), (34,208), (37,209), (39,211), (51,53), (65,225), (114,511), (115,512), (116,514), (117,515), (118,516), (119,517), (120,518), (121,519), (122,520), (123,521), (124,522), (125,523), (126,524), (127,525), (185,235), (371,372), (487,488)
X(4) = X(i)-cross conjugate of X(j) for these (i,j):
(3,254), (6,2), (19,278), (25,393), (33,281), (51,6), (52,24), (65,1), (113,403), (125,523), (185,3), (193,459), (225,158), (389,54), (397,17), (398,18), (407,225), (427,264), (512,112), (513,108), (523,107)
X(4) = crosspoint of X(i) and X(j) for these (i,j): (2,253), (7,189), (27,286), (92,273)
X(4) = crosssum of X(i) and X(j) for these (i,j): (4,1075), (6,154), (25,1033), (48,212), (55,198), (56,1035), (71,228), (184,577), (185,417), (216,418)
X(4) = crossdifference of every pair of points on line X(520)X(647)
X(4) = X(i)-Hirst inverse of X(j) for these (i,j):
(1,243), (2,297), (3,350), (19,242), (21,425), (24,421), (25,419), (27,423), (28,422), (29,415), (420,427), (424,451), (459,460), (470,471), (1249,1503)
X(4) = X(i)-aleph conjugate of X(j) for these (i,j): (1,1047), (29,4)
X(4) = X(i)-beth conjugate of X(j) for these (i,j):
(4,34), (8,40), (29,4), (162,56), (318,318), (811,331)
X(4) = intersection of tangents at X(3) and X(4) to McCay cubic K003
X(4) = intersection of tangents at X(4) and X(69) to Lucas cubic K007
X(4) = exsimilicenter of 1st & 2nd Johnson-Yff circles; the insimilicenter is X(1)
X(4) = trilinear pole of PU(4) (the orthic axis)
X(4) = trilinear pole wrt orthic triangle of orthic axis
X(4) = trilinear pole wrt intangents triangle of orthic axis
X(4) = trilinear pole wrt circumsymmedial triangle of orthic axis
X(4) = trilinear product of PU(15)
X(4) = barycentric product of PU(i) for these i: 21, 45
X(4) = bicentric sum of PU(i) for these i: 126, 131
X(4) = PU(126)-harmonic conjugate of X(652)
X(4) = midpoint of PU(131)
X(4) = crosspoint of polar conjugates of PU(4)
X(4) = cevapoint of foci of orthic inconic
X(4) = QA-P33 (Centroid of the Orthocenter Quadrangle) of quadrangle ABCX(2) (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/61-qa-p33.html)
X(4) = Hofstadter -1 point
X(4) = X(4)-of X(i)X(j)X(k) for these {i,j,k}: {1,8,5556}, {1,9,885}, {2,6,1640}, {2,10,4049}, {3,6,879}, {3,66,2435}, {7,8,885}
X(4) = homothetic center of these triangles: orthic, X(13)-Ehrmann, X(14)-Ehrmann (see X(25))
X(4) = perspector of anticomplementary circle
X(4) = pole wrt polar circle of trilinear polar of X(2) (line at infinity)
X(4) = pole wrt {circumcircle, nine-point circle}-inverter of Lemoine axis
X(4) = X(48)-isoconjugate (polar conjugate) of X(2)
X(4) = X(i)-isoconjugate of X(j) for these (i,j): (6,63), (75,184), (91,1147), (92,577), (1101,125), (2962,49), (2964,3519)
X(4) = X(1342)-vertex conjugate of X(1343)
X(4) = Zosma transform of X(1)
X(4) = X(1352) of 1st anti-Brocard triangle
X(4) = centroid of the union of X(8) and its 3 extraversions
X(4) = X(5) of extraversion triangle of X(8)
X(4) = homothetic center of orthic triangle and reflection of tangential triangle in X(5)
X(4) = homothetic center of 2nd circumperp and 3rd Euler triangles
X(4) = trilinear product of vertices of half-altitude triangle
X(4) = trilinear product of vertices of orthocentroidal triangle
X(4) = trilinear product of vertices of reflection triangle
X(4) = trilinear product of vertices of 4th Brocard triangle
X(4) = center of conic that is the locus of orthopoles of lines passing through X(4)
X(4) = perspector of circumanticevian triangle of X(4) and unary cofactor triangle of circumanticevian triangle of X(3)
X(4) = X(3)-of-2nd-extouch-triangle
X(4) = perspector of ABC and 2nd and 3rd extouch triangles
X(4) = perspector of ABC and 1st Brocard triangle of anticomplementary triangle
X(4) = perspector of ABC and 1st Brocard triangle of Johnson triangle
X(4) = perspector of ABC and mid-triangle of 2nd and 3rd extouch triangles
X(4) = perspector of extouch triangle and cross-triangle of ABC and 2nd extouch triangle
X(4) = perspector of 2nd Hyacinth triangle and cross-triangle of ABC and 2nd Hyacinth triangle
X(4) = homothetic center of 2nd Johnson-Yff triangle and cross-triangle of ABC and 1st Johnson-Yff triangle
X(4) = homothetic center of 1st Johnson-Yff triangle and cross-triangle of ABC and 2nd Johnson-Yff triangle
X(4) = X(1)-of-orthic-triangle if ABC is acute, and an excenter of orthic triangle otherwise
X(4) = X(52)-of-excentral triangle
X(4) = X(65)-of-tangential-triangle if ABC is acute
X(4) = X(155)-of-intouch-triangle
X(4) = X(110)-of-Fuhrmann-triangle
X(4) = X(147)-of-1st-Brocard-triangle
X(4) = X(1296)-of-4th-Brocard-triangle
X(4) = X(74)-of-orthocentroidal-triangle
X(4) = X(110)-of-X(4)-Brocard-triangle
X(4) = harmonic center of circle O(PU(4)) and orthoptic circle of Steiner inellipse
X(4) = Thomson-isogonal conjugate of X(154)
X(4) = Lucas-isogonal conjugate of X(11206)
X(4) = perspector of ABC and cross-triangle of 1st and 2nd Neuberg triangles
X(4) = perspector of circumconic centered at X(1249)
X(4) = center of circumconic that is locus of trilinear poles of lines passing through X(1249)
X(4) = circumcevian isogonal conjugate of X(4)
X(4) = orthic-isogonal conjugate of X(4)
X(4) = X(1)-of-circumorthic-triangle if ABC is acute
X(4) = isogonal conjugate wrt half-altitude triangle of X(185)
X(4) = Miquel associate of X(4)
X(4) = crosspoint of X(3) and X(155) wrt both the excentral and tangential triangles
X(4) = crosspoint of X(487) and X(488) wrt both the excentral and anticomplementary triangles
X(4) = X(3)-of-Ehrmann-mid-triangle
X(4) = X(110)-of-X(3)-Fuhrmann-triangle
X(4) = barycentric product X(112)*X(850)
X(4) = Kosnita(X(20),X(20)) point
X(4) = perspector of ABC and the reflection of the excentral triangle in X(10)
X(4) = pedal antipodal perspector of X(3)
X(4) = Ehrmann-side-to-Ehrmann-vertex similarity image of X(3)
X(4) = Ehrmann-vertex-to-orthic similarity image of X(4)
X(4) = Ehrmann-side-to-orthic similarity image of X(3)
X(4) = Ehrmann-mid-to-ABC similarity image of X(5)
X(4) = perspector of hexyl triangle and cevian triangle of X(27)
X(4) = perspector of hexyl triangle and anticevian triangle of X(19)
X(4) = perspector of ABC and medial triangle of pedal triangle of X(64)
X(4) = perspector of ABC and the reflection in X(2) of the antipedal triangle of X(2)
X(4) = perspector of hexyl triangle and tangential triangle wrt excentral triangle of the excentral-hexyl ellipse
X(4) = inverse-in-Steiner-circumellipse of X(297)
X(4) = {X(2479),X(2480)}-harmonic conjugate of X(297)
X(4) = symgonal of every point on the nine-point circle
X(4) = center of bianticevian conic of PU(4) (this conic being the polar circle)
X(4) = orthoptic-circle-of-Steiner-inellipse inverse of X(468)
X(4) = de-Longchamps-circle inverse of X(2071)
X(4) = center of inverse-in-de-Longchamps-circle of circumcircle
X(4) = inner-Napoleon circle-inverse of X(32460)
X(4) = outer-Napoleon circle-inverse of X(32461)
X(4) = excentral-to-ABC functional image of X(1)
X(4) = orthic-to-ABC functional image of X(52)
X(4) = incentral-to-ABC functional image of X(500)
X(4) = Feuerbach-to-ABC functional image of X(5948)
X(4) = excentral-to-intouch similarity image of X(84)
X(4) = barycentric product of circumcircle intercepts of line X(297)(525)
X(4) = trilinear product of vertices of infinite altitude triangle


X(5) = NINE-POINT CENTER

Trilinears    cos(B - C) : cos(C - A) : cos(A - B)
Trilinears    cos A + 2 cos B cos C : :
Trilinears    cos A - 2 sin B sin C
Trilinears    bc[a2(b2 + c2) - (b2 - c2)2] : :
Trilinears    2 sec A + sec B sec C : :
Trilinears    cos B cos C + sin B sin C : :
Trilinears    cos^2(B/2 - C/2) - sin^2(B/2 - C/2) : :
Trilinears    cos A' : :, where A' is the angle formed by internal tangents to incircle and A-excircle
Trilinears    cos A'' : : , where A'' is the angle formed by external tangents to B- and C-excircles
Barycentrics    a cos(B - C) : b cos(C - A) : c cos(A - B)
Barycentrics    a2(b2 + c2) - (b2 - c2)2
Barycentrics    S^2 + SB SC : :
Barycentrics    1 + cot B cot C : :
Tripolars    Sqrt[a^6-3 a^4 b^2+3 a^2 b^4-b^6-3 a^4 c^2+3 a^2 b^2 c^2+b^4 c^2+3 a^2 c^4+b^2 c^4-c^6] : :
X(5) = 3*X(2) + X(4) = 3*X(2) - X(3) = X(3) + X(4)
X(5) = (2 + J) X(1113) + (2 - J) X(1114)

As a point on the Euler line, X(5) has Shinagawa coefficients (1, 1).

X(5) is the center of the nine-point circle. Euler showed in that this circle passes through the midpoints of the sides of ABC and the feet of the altitudes of ABC, hence six of the nine points. The other three are the midpoints of segments A-to-X(4), B-to-X(4), C-to-X(4). The radius of the nine-point circle is one-half the circumradius.

Dan Pedoe, Circles: A Mathematical View, Mathematical Association of America, 1995.

Let Pa be the parabola with focus A and directrix BC. Let La be the polar of X(3) wrt Pa. Define Lb and Lc cyclically. Let A' = Lb∩Lc, and define B' and C; cyclically. Then X(5) = X(597)-of-A'B'C'. (Randy Hutson, December 10, 2016)

Let A'B'C' be the half-altitude triangle. Let A″ be the trilinear pole, wrt A'B'C', of line BC, and define B″ and C″ cyclically. The lines A'″, B'B″, C'C″ concur in X(5). (Randy Hutson, December 10, 2016)

Let A' be the nine-point center of BCX(13), and define B' and C' cyclically. Let A″ be the nine-point center of BCX(14), and define B″ and C″ cyclically. The lines A'A″, B'B″, C'C″ concur in X(5). (Randy Hutson, December 10, 2016)

Let A'B'C' be the Euler triangle. Let A″ be the centroid of AB'C', and define B″ and C″ cyclically. The lines A'″, B'B″, C'C″ concur in X(5). (Randy Hutson, December 10, 2016)

Let A'B'C' be any equilateral triangle inscribed in the circumcircle of ABC. The Simson lines of A', B', C' form an equilateral triangle with center X(5). If A'B'C' is the circumtangential triangle, the Simson lines of A', B', C' concur in X(5). (Randy Hutson, December 10, 2016)

Let OA be the circle centered at the A-vertex of the Steiner triangle and passing through A; define OB and OC cyclically. X(5) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

Let OA be the circle centered at the A-vertex of the 1st excosine triangle and passing through A; define OB and OC cyclically. X(5) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

If you have The Geometer's Sketchpad, you can view these sketches: Nine-point center, Euler Line, Roll Circle, MacBeath Inconic


If you have GeoGebra, you can view Nine-point center and Euler line.

Let A'B'C' be the Feuerbach triangle. Let A″ be the cevapoint of B' and C', and define B″ and C″ cyclically. The lines A″, BB″, CC″ concur in X(5). (Randy Hutson, July 20, 2016)

Let A'B'C' be the reflection triangle. Let A″ be the trilinear pole of line B'C', and define B″and C″ cyclically. The lines A″, BB″, CC″ concur in X(5). (Randy Hutson, July 20, 2016)

Let A'B'C' be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung). X(5) = X(6146)-of-A'B'C'.

Let A'B'C' be the cevian triangle of X(5). Let A″ be X(5)-of-AB'C', and define B″, C″ cyclically. The lines A″, BB″, CC″ concur in X(5). (Randy Hutson, June 27, 2018)

Let Na, Nb, Nc be the nine-point centers of BCF, CAF, ABF, resp., where F = X(13). Let Na', Nb', Nc' be the nine-point centers of BCF', CAF', ABF', resp., where F' = X(14). The lines NaNa', NbNb', NcNc' concur in X(5). (Randy Hutson, June 27, 2018)

Let Na, Nb, Nc be the nine-point centers of BCX, CAX, ABX, resp., where X = X(17). Let Na', Nb', Nc' be the nine-point centers of BCX', CAX', ABX', resp., where X' = X(18). The lines NaNa', NbNb', NcNc' concur in X(5). (Randy Hutson, June 27, 2018)

Let T be a family of Poncelet triangles inscribed in an ellipse E and circumscribing a fixed incircle. The line X(1)X(5) remains stationary iff the family contains an equilateral triangle. For this to happen, X(1) must lie on the locus E' of all equilateral centroids inscribed in E, which is an ellipse concentric, aligned with, and interior to E, derived in M. Stanev, "Locus of the centroid of the equilateral triangle inscribed in an ellipse", Intl. J. Comp. Disc. Math., vol. 4, 2019, pp.54-65. (Dan Reznik, August 16, 1024)

X(5) lies on the Napoleon cubic (also known as the Feuerbach cubic) and these lines: {1,11}, {2,3}, {6,68}, {7,5704}, {8,1389}, {9,1729}, {10,517}, {13,18}, {14,17}, {15,2913}, {16,2912}, {19,8141}, {32,230}, {33,1062}, {34,1060}, {35,3583}, {36,3585}, {39,114}, {40,1698}, {46,1836}, {47,5348}, {49,54}, {51,52}, {53,216}, {55,498}, {56,499}, {57,1728}, {60,5397}, {64,4846}, {65,1737}, {67,9970}, {69,1351}, {72,908}, {74,3521}, {76,262}, {78,3419}, {79,1749}, {83,98}, {84,5437}, {85,1565}, {93,6344}, {94,9221}, {96,1166}, {97,4994}, {99,5966}, {100,10738}, {101,10739}, {102,10740}, {103,10741}, {104,5253}, {105,10743}, {106,10744}, {107,10745}, {108,10746}, {109,10747}, {111,10748}, {112,10749}, {113,125}, {116,118}, {117,124}, {120,5511}, {121,2885}, {122,133}, {126,5512}, {127,132}, {128,137}, {129,130}, {131,136}, {141,211}, {142,971}, {145,10247}, {146,10620}, {147,3329}, {148,7783}, {149,3871}, {153,12773}, {154,9833}, {156,184}, {165,7965}, {169,6506}, {171,3073}, {181,10407}, {182,206}, {183,315}, {187,3054}, {191,5535}, {193,5093}, {194,7777}, {195,1994}, {214,6246}, {217,1625}, {222,8757}, {225,1465}, {226,912}, {236,8130}, {238,3072}, {252,1157}, {264,1093}, {273,6356}, {275,2055}, {276,6528}, {298,634}, {299,633}, {302,622}, {303,621}, {311,1225}, {312,3695}, {316,1078}, {318,2968}, {324,6663}, {329,2095}, {339,1235}, {354,13407}, {356,3608}, {371,590}, {372,615}, {385,7762}, {386,1834}, {388,999}, {389,5448}, {390,7678}, {392,1512}, {394,10982}, {399,3448}, {484,5445}, {487,12313}, {488,12314}, {491,637}, {492,638}, {493,8220}, {494,8221}, {497,3085}, {515,1125}, {516,3579}, {518,10916}, {519,3813}, {523,6757}, {524,576}, {528,8715}, {529,8666}, {538,7764}, {539,1493}, {542,575}, {543,9771}, {551,5882}, {566,9220}, {568,3567}, {570,1879}, {572,2126}, {573,1213}, {574,3055}, {577,6748}, {578,1147}, {579,1901}, {580,5127}, {581,5453}, {582,1754}, {583,8818}, {598,7607}, {599,11477}, {601,750}, {602,748}, {611,12589}, {613,12588}, {616,13103}, {617,13102}, {618,629}, {619,630}, {620,6721}, {641,6250}, {642,6251}, {671,7608}, {698,8149}, {754,7780}, {758,5694}, {804,11615}, {842,1287}, {920,1454}, {925,2383}, {930,6592}, {938,3487}, {944,3616}, {950,13411}, {956,3436}, {958,10526}, {962,5657}, {986,3944}, {993,4999}, {997,5794}, {1001,10198}, {1007,3926}, {1056,5261}, {1058,5274}, {1069,10055}, {1071,5249}, {1073,1217}, {1087,2599}, {1089,3703}, {1090,1091}, {1092,5651}, {1111,3665}, {1112,12358}, {1117,3470}, {1131,3317}, {1132,3316}, {1139,3393}, {1140,3370}, {1145,7704}, {1151,5418}, {1152,5420}, {1155,1770}, {1158,5880}, {1160,5590}, {1161,5591}, {1173,1487}, {1181,1899}, {1199,3410}, {1211,5752}, {1212,5179}, {1214,1838}, {1249,8888}, {1270,11917}, {1271,11916}, {1297,12918}, {1327,6426}, {1328,6425}, {1350,3763}, {1376,10525}, {1393,7069}, {1420,9613}, {1441,3007}, {1447,4911}, {1490,5787}, {1495,11572}, {1498,1853}, {1499,11182}, {1511,5972}, {1519,3753}, {1537,7705}, {1538,8582}, {1539,2777}, {1587,3069}, {1588,3068}, {1601,3432}, {1614,5012}, {1621,11491}, {1697,9614}, {1706,12700}, {1709,12679}, {1714,4383}, {1724,5398}, {1750,8726}, {1768,7701}, {1788,4295}, {1843,9967}, {1848,1871}, {1861,1872}, {1916,3399}, {1935,3075}, {1936,3074}, {1975,6390}, {1986,7723}, {1990,5158}, {1992,11482}, {2052,13599}, {2066,9646}, {2067,9661}, {2077,11826}, {2086,9490}, {2098,12647}, {2099,10573}, {2120,2121}, {2241,9665}, {2242,9650}, {2482,9880}, {2486,2783}, {2549,5013}, {2550,9709}, {2551,9708}, {2595,3460}, {2601,2602}, {2607,2957}, {2635,4303}, {2646,10572}, {2682,10568}, {2771,5883}, {2781,6698}, {2792,4672}, {2794,6036}, {2797,8552}, {2800,3754}, {2801,12005}, {2802,10284}, {2826,3837}, {2829,5450}, {2883,5892}, {2887,3831}, {2896,7616}, {2963,2965}, {2971,2974}, {2975,5080}, {2979,7999}, {3006,3701}, {3035,5840}, {3053,7737}, {3057,10039}, {3058,3584}, {3060,6243}, {3096,7934}, {3098,7914}, {3120,5492}, {3157,10071}, {3167,6193}, {3258,11749}, {3272,3609}, {3284,6749}, {3303,10056}, {3304,10072}, {3306,13226}, {3314,7912}, {3333,5290}, {3338,10404}, {3357,5893}, {3359,12705}, {3368,5401}, {3381,3382}, {3406,3407}, {3434,5552}, {3462,3463}, {3468,3469}, {3488,5703}, {3576,3624}, {3581,10545}, {3582,5270}, {3586,3601}, {3617,8148}, {3618,5050}, {3619,10519}, {3622,7967}, {3626,11278}, {3629,5097}, {3630,7882}, {3631,7896}, {3636,13607}, {3649,5693}, {3654,7991}, {3656,3679}, {3670,3782}, {3673,7179}, {3687,5295}, {3705,4385}, {3734,3788}, {3737,8819}, {3739,12490}, {3742,12675}, {3785,9752}, {3812,3838}, {3819,5447}, {3823,12393}, {3833,6701}, {3867,11574}, {3874,6583}, {3880,10915}, {3911,4292}, {3913,11235}, {3917,10625}, {3931,5530}, {3947,5045}, {3972,7857}, {4004,10273}, {4030,4894}, {4045,6683}, {4293,5229}, {4294,5218}, {4297,10165}, {4299,5204}, {4302,5217}, {4311,5126}, {4323,11041}, {4354,9629}, {4413,10310}, {4417,10449}, {4420,5178}, {4425,9959}, {4511,5086}, {4550,7689}, {4662,12612}, {4668,11224}, {4855,9945}, {4861,5176}, {4885,8760}, {5007,5306}, {5010,5326}, {5015,7081}, {5024,7738}, {5038,11646}, {5041,5355}, {5082,7080}, {5092,6704}, {5099,13162}, {5119,12701}, {5131,5442}, {5171,7761}, {5181,8263}, {5188,6249}, {5221,11544}, {5223,6067}, {5224,10446}, {5233,9534}, {5237,5350}, {5238,5349}, {5248,5842}, {5251,11012}, {5257,10445}, {5259,6253}, {5286,7736}, {5309,7772}, {5334,11485}, {5335,11486}, {5339,10654}, {5340,10653}, {5354,10339}, {5395,7612}, {5412,10897}, {5413,10898}, {5422,7592}, {5471,6783}, {5472,6782}, {5544,5656}, {5550,5731}, {5597,8200}, {5598,8207}, {5599,8196}, {5600,8203}, {5601,11875}, {5602,11876}, {5643,5655}, {5658,9799}, {5745,12572}, {5814,11679}, {5890,11451}, {5895,10606}, {5908,5909}, {5913,6032}, {5925,8567}, {5944,6689}, {5947,5948}, {5950,5952}, {5961,8146}, {5962,13557}, {6043,11992}, {6054,7827}, {6055,10991}, {6118,8180}, {6119,8184}, {6130,9517}, {6150,10615}, {6152,12606}, {6153,11692}, {6174,10993}, {6179,7812}, {6191,7345}, {6192,7344}, {6194,7938}, {6221,6459}, {6223,12684}, {6224,12747}, {6225,13093}, {6237,11435}, {6238,11436}, {6241,10574}, {6256,10200}, {6291,12603}, {6329,12007}, {6361,9812}, {6398,6460}, {6406,12604}, {6407,9692}, {6417,7582}, {6418,7581}, {6419,8960}, {6433,12819}, {6434,12818}, {6449,9541}, {6462,11949}, {6463,11950}, {6515,9777}, {6523,10002}, {6599,12660}, {6662,13409}, {6669,6694}, {6670,6695}, {6671,6673}, {6672,6674}, {6692,6705}, {6703,13323}, {6735,10914}, {6736,13600}, {6769,8580}, {7013,10400}, {7028,8129}, {7160,12856}, {7198,7272}, {7280,7294}, {7596,8228}, {7615,11184}, {7620,11165}, {7691,11016}, {7693,12307}, {7694,9756}, {7703,11439}, {7709,7864}, {7743,9957}, {7754,7774}, {7758,9766}, {7760,7858}, {7766,7921}, {7768,7809}, {7771,7802}, {7778,7795}, {7779,7941}, {7784,7800}, {7786,7790}, {7787,7806}, {7793,7823}, {7794,7821}, {7801,7888}, {7803,7851}, {7810,7873}, {7811,7860}, {7818,7854}, {7820,7874}, {7822,7867}, {7826,7845}, {7830,7842}, {7831,7911}, {7832,7899}, {7835,7940}, {7836,7925}, {7846,7942}, {7852,7889}, {7855,7903}, {7856,7878}, {7859,7919}, {7875,7932}, {7877,7926}, {7893,7900}, {7898,7904}, {7935,8722}, {7998,13340}, {8014,11555}, {8015,11556}, {8069,10320}, {8085,8087}, {8086,8088}, {8121,8123}, {8122,8124}, {8158,8165}, {8212,8222}, {8213,8223}, {8280,8855}, {8281,8854}, {8351,8379}, {8377,8380}, {8378,8381}, {8538,8541}, {8591,12355}, {8725,9751}, {8798,13157}, {8800,8905}, {8909,8966}, {8918,10218}, {8919,10217}, {8929,11581}, {8930,11582}, {8961,8963}, {8985,8990}, {9159,11639}, {9172,10162}, {9512,11061}, {9535,9566}, {9538,9642}, {9539,9641}, {9542,9691}, {9543,9690}, {9544,9704}, {9545,9703}, {9782,9809}, {9786,9815}, {9862,10583}, {9874,12872}, {9919,13203}, {9964,12528}, {10037,10832}, {10038,10874}, {10040,10925}, {10041,10926}, {10046,10831}, {10047,10873}, {10048,10923}, {10049,10924}, {10053,12185}, {10054,12351}, {10058,12764}, {10059,12860}, {10060,12950}, {10061,12951}, {10062,12952}, {10063,12836}, {10064,12954}, {10065,12374}, {10066,12956}, {10067,12958}, {10068,12959}, {10069,12184}, {10070,12350}, {10074,12763}, {10075,12859}, {10076,12940}, {10077,12941}, {10078,12942}, {10079,12837}, {10080,12944}, {10081,12373}, {10082,12946}, {10083,12948}, {10084,12949}, {10085,12678}, {10086,13183}, {10087,13274}, {10088,12904}, {10089,13182}, {10090,13273}, {10091,12903}, {10168,11645}, {10187,12816}, {10188,12817}, {10266,12919}, {10278,10279}, {10311,10316}, {10312,10317}, {10524,10530}, {10528,10596}, {10529,10597}, {10546,11464}, {10575,11381}, {10584,10785}, {10585,10786}, {10586,10805}, {10587,10806}, {10628,11557}, {10634,10641}, {10635,10642}, {10733,12121}, {10797,10802}, {10798,10801}, {10984,11550}, {11236,12513}, {11264,13366}, {11392,11399}, {11393,11398}, {11411,11431}, {11425,12118}, {11429,12428}, {11449,12278}, {11455,12279}, {11456,11457}, {11501,11508}, {11502,11507}, {11536,12234}, {11576,12363}, {11649,12061}, {11671,13512}, {11746,12236}, {11754,11755}, {11763,11764}, {11772,11773}, {11781,11782}, {11869,11879}, {11870,11880}, {11871,11877}, {11872,11878}, {11905,11913}, {11906,11912}, {11930,11953}, {11931,11954}, {11932,11951}, {11933,11952}, {12099,12827}, {12308,12317}, {12309,12318}, {12310,12319}, {12311,12320}, {12312,12321}, {12316,12325}, {12383,12902}, {12384,13115}, {12494,13234}, {12599,12864}, {12600,13089}, {12613,12621}, {12614,12622}, {12615,12623}, {12624,13249}, {12849,13126}, {12945,13117}, {12947,13129}, {12955,13116}, {12957,13128}, {13023,13025}, {13024,13026}, {13039,13051}, {13040,13052}, {13219,13310}, {13296,13312}, {13297,13311}, {13348,13570}, {13507,13508}

X(5) is the {X(2),X(4)}-harmonic conjugate of X(3). For a list of other harmonic conjugates of X(5), click Tables at the top of this page.

X(5) = midpoint of X(i) and X(j) for these (i,j):
(1,355), (2,381), (3,4), (11,119), (20,382), (68,155), (110,265), (113,125), (114,115), (116,118), (117,124), (122,133), (127,132), (128,137), (129,130), (131,136), (399,3448)

X(5) = reflection of X(i) in X(j) for these (i,j): (2,547), (3,140), (4,546), (20,548), (52,143), (549,2), (550,3), (1263,137), (1353,6), (1385,1125), (1483,1), (1484,11)

X(5) = isogonal conjugate of X(54)
X(5) = isotomic conjugate of X(95)
X(5) = circumcircle-inverse of X(2070)
X(5) = orthocentroidal-circle-inverse of X(3)
X(5) = de-Longchamps-circle-inverse of anticomplement of X(37943)
X(5) = circle-O(PU(4))-inverse of X(37969)
X(5) = complement of X(3)
X(5) = anticomplement of X(140)
X(5) = complementary conjugate of X(3)
X(5) = eigencenter of anticevian triangle of X(523)
X(5) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,216), (4,52), (110,523), (264, 324), (265,30), (311,343), (324,53)
X(5) = cevapoint of X(i) and X(j) for these (i,j): (3,195), (51,216)
X(5) = X(i)-cross conjugate of X(j) for these (i,j): (51,53), (54, 2121), (216,343), (233,2)
X(5) = crosspoint of X(i) and X(j) for these (i,j): (2,264), (311,324)
X(5) = crosssum of X(i) and X(j) for these (i,j): (3,1147), (6,184)
X(5) = crossdifference of every pair of points on line X(50)X(647)
X(5) = X(1)-aleph conjugate of X(1048) X(5) = radical center of Stammler circles
X(5) = center of inverse-in-circumcircle-of-tangential-circle
X(5) = harmonic center of 1st & 2nd Hutson circles
X(5) = homothetic center of circumorthic triangle and 2nd isogonal triangle of X(4); see X(36)
X(5) = X(3)-of-X(4)-Brocard-triangle
X(5) = X(4)-of-Schroeter-triangle
X(5) = X(5)-of-Fuhrmann-triangle
X(5) = X(5)-of-complement-of-excentral-triangle (or extraversion triangle of X(10))
X(5) = X(114)-of-1st-Brocard-triangle
X(5) = X(143)-of-excentral-triangle
X(5) = X(156)-of-intouch-triangle
X(5) = X(1511)-of-orthocentroidal-triangle
X(5) = bicentric sum of PU(i) for these i: 5, 7, 38, 65, 173
X(5) = midpoint of PU(i) for these i: 5, 7, 38
X(5) = trilinear product of PU(69)
X(5) = PU(65)-harmonic conjugate of X(351)
X(5) = perspector of circumconic centered at X(216)
X(5) = center of circumconic that is locus of trilinear poles of lines passing through X(216)
X(5) = trilinear pole of line X(2081)X(2600)
X(5) = pole wrt polar circle of trilinear polar of X(275) (line X(186)X(523))
X(5) = X(48)-isoconjugate (polar conjugate) of X(275)
X(5) = X(252)-isoconjugate of X(2964)
X(5) = homothetic center of medial triangle and Euler triangle
X(5) = homothetic center of ABC and the triangle obtained by reflecting X(3) in the points A, B, C
X(5) = radical center of the Stammler circles
X(5) = centroid of {A, B, C, X(4)}
X(5) = antigonal image of X(1263)
X(5) = crosspoint of X(627) and X(628) wrt both the excentral and anticomplementary triangles
X(5) = intersection of tangents to Evans conic at X(15) and X(16)
X(5) = polar-circle-inverse of X(186)
X(5) = inverse-in-{circumcircle, nine-point circle}-inverter of X(23)
X(5) = inverse-in-Kiepert-hyperbola of X(39)
X(5) = inverse-in-Steiner-inellipse of X(297)
X(5) = {X(2009),X(2010)}-harmonic conjugate of X(39)
X(5) = {X(2454),X(2455)}-harmonic conjugate of X(297)
X(5) = perspector of medial triangles of ABC, orthic and half-altitude triangles
X(5) = X(6)-isoconjugate of X(2167)
X(5) = orthic-isogonal conjugate of X(52)
X(5) = Thomson-isogonal conjugate of X(6030)
X(5) = X(1)-of-submedial triangle if ABC is acute
X(5) = harmonic center of circumcircles of Euler and anti-Euler triangles
X(5) = perspector of Feuerbach triangle and cross-triangle of ABC and Feuerbach triangle
X(5) = Kosnita(X(4),X(2)) point
X(5) = Kosnita(X(4),X(3)) point
X(5) = Kosnita(X(4),X(20)) point
X(5) = X(4)-of-Ehrmann-mid-triangle
X(5) = homothetic center of Ehrmann vertex-triangle and Kosnita triangle
X(5) = homothetic center of Ehrmann side-triangle and circumorthic triangle
X(5) = perspector of Ehrmann mid-triangle and submedial triangle
X(5) = Ehrmann-side-to-orthic similarity image of X(4)
X(5) = Johnson-to-Ehrmann-mid similarity image of X(3)
X(5) = excentral-to-ABC functional image of X(3)
X(5) = QA-P32 center (Centroid of the Circumcenter Quadrangle) of quadrangle ABCX(4) (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/60-qa-p32.html)


X(6) = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT)

Trilinears    a : b : c
Trilinears    sin A : sin B : sin C
Trilinears    (1 + cos A)(1 - cos A + cos B + cos C) : :
Trilinears    cot B/2 + cot C/2 : :
Barycentrics    a2 : b2 : c2
Barycentrics    SB + SC : :
Barycentrics    SA - SW : :
Barycentrics    cot A - cot ω : :
Barycentrics    cot B + cot C - cot A + cot ω : :
Tripolars    b c Sqrt[2(b^2 + c^2) - a^2] : :
X(6) = (r2 + s2 + 2rR)*X(1) - 6rR*X(2) -2r2*X(3)
X(6) = (1 + sqrt(3)*tan(ω))*X(13) + (1 - sqrt(3)*tan(ω))*X(14)
X(6) = (1 + sqrt(3)*tan(ω))*X(15) + (1 - sqrt(3)*tan(ω))*X(16)
X(6) = (3 + 5*sqrt(3)*tan(ω))*X(17) + (3 - 5*sqrt(3)*tan(ω))*X(18)
(The above four combos for X(6) found by Peter Moses, November, 2011)

X(6) is the point of concurrence of the symmedians (i.e., reflections of medians in corresponding angle bisectors). X(6) is the point which, when given by actual trilinear distances x,y,z, minimizes x2 + y2 + z2.

X(6) in Navigation: A talk about the symmedian point, by William Lionheart.

X(6) and other triangle centers play a fundamental part in Yuri I. Loginov's "Energy methods for single-position passive radar based on special points of a triangle", downloadable in Russian or as an English translation.

If you have The Geometer's Sketchpad, you can view Symmedian point.
If you have GeoGebra, you can view Symmedian point.

Let A'B'C' be the pedal triangle of an arbitrary point X, and let S(X) be the vector sum XA' + XB' + XC'. Then

S(X) = (0 vector) if and only if X = X(6).

The "if" implication is equivalent to the well known fact that X(6) is the centroid of its pedal triangle, and the converse was proved by Barry Wolk (Hyacinthos #19, Dec. 23, 1999).

X(6) is the radical trace of the 1st and 2nd Lemoine circles. (Peter J. C. Moses, 8/24/03)

X(6) is the perspector of ABC and the medial triangle of the orthic triangle of ABC. (Randy Hutson, 8/23/2011)

Ross Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, Mathematical Association of America, 995. Chapter 7: The Symmedian Point.

Let A' be the center of the conic through the contact points of the incircle and the A-excircle with the sidelines of ABC. Define B' and C' cyclically. Let A″ be the center of the conic through the contact points of the B- and C- excircles with the sidelines of ABC. Define B″ and C″ cyclically. The triangles A'B'C' and A″B″C″ are perspective at X(6). See also X(25), X(218), X(222), X(940), X(1743). (Randy Hutson, July 23, 2015)

The tangents at A,B,C to the Thomson cubic K002 concur in X(6). Let Ha be the foot of the A-altitude. Let Ba, Ca be the feet of perpendiculars from Ha to CA, AB, resp. Let A' be the orthocenter of HaBaCa, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(6). (Randy Hutson, November 18, 2015)

Let Ja, Jb, Jc be the excenters and I the incenter of ABC. Let Ka be the symmedian point of JbJcI, and define Kb and Kc cyclically. Then KaKbKc is perspective to JaJbJc at X(6). (Randy Hutson, February 10, 2016)

Let A'B'C' be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung); then X(6) = X(6467)-of-A'B'C'. (Randy Hutson, June 27, 2018)

X(6) is the perspector of every pair of these triangles: anticevian triangle of X(3), submedial triangle, unary cofactor triangle of submedial triangle, unary cofactor triangle of the intangents triangle, unary cofactor triangle of the extangents triangle. (Randy Hutson, June 27, 2018)

Let A'B'C' be the tangential triangle of the Jerabek hyperbola. Let A″ be the pole wrt circumcircle of line B'C', and define B″, C″ cyclically. The lines A'″, B'B″, C'C″ concur in X(6). (Randy Hutson, November 30, 2018)

Let A'B'C' be the half-altitude (midheight) triangle. Let LA be the line through A parallel to B'C', and define LB and LC cyclically. Let A″ = LB∩LC, and define B″, C″ cyclically. The lines A'″, B'B″, C'C″ concur in X(6). (Randy Hutson, November 30, 2018)

X(6) is the unique point that is the centroid of its pedal triangle. (Randy Hutson, June 7, 2019)

Let A'B'C' be any one of {Lucas(t) central triangle, Lucas(t) tangents triangle, Lucas(t) inner triangle} (for arbitrary t). Let A″ be the trilinear pole of line B'C', and define B″ and C″ cyclically. The lines A″, BB″, CC″ concur in X(6). (Randy Hutson, July 11, 2019)

Let A'B'C' be the medial triangle, and A″B″C″ the orthic triangle. Let A* be the centroid of AA'A″, and define B* and C* cyclically. A*B*C* is inversely similar to ABC, and the lines A'A*, B'B*, C'C* concur in X(6). (Randy Hutson, July 11, 2019)

X(6) is the intersection of the isotomic conjugate of the polar conjugate of the Euler line (i.e., line X(2)X(6)), and the polar conjugate of the isotomic conjugate of the Euler line (i.e., line X(4)X(6)). (Randy Hutson, July 11, 2019)

X(6) is the pole of the Euler line wrt each conic passing through each of the following sets of four points: {X(13), X(14), X(15), X(16)}, {X(13), X(14), X(17), X(18)}, {X(13), X(14), X(61), X(62)}, {X(15), X(16), X(17), X(18)}, {X(17), X(18), X(61), X(62)}. (Randy Hutson, January 17, 2020)

Let OA be the circle centered at the A-vertex of the 2nd Brocard triangle and passing through A; define OB and OC cyclically. X(6) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

In the plane of a triangle ABC, let
Oa = circle with diameter BC, and define Ob and Oc cyclically;
A' = apex of equilateral triangle on side BC, and define B′ and C′ cyclically;
Ab = BA′∩Oa, and define Bc and Ca cyclically;
Ac = CA′∩Oa, and define Ba and Cb cyclically;
A″= BcBa∩CaCb, and define B″ and C″ cyclically.
The triangle A″B″C″ is perspective to ABC, and the perspector is X(6). (Dasari Naga Vijay Krishna, April 19, 2021)

Let P and Q be distinct points in a plane, and let T be a triangle rigidly rotating about P. Let X be a triangle center, and let
XT = X-of-T;
O = a circle with center Q';
{A,B,C} = vertices of T;
A' = inverse of A in O, and define B' and C' cyclically;
T' = A'B'C';
X'T = X-of-T'.
(1) If X = X(k) for k = 3, 6, 15, 16, 61, 62, then the locus of X'T is a conic.
(2) If X = X(15) or X = X(16), the conic is a circle.
(3) If P is one of the points X(k), for k = 3, 6, 15, 16, 61, 62, then the locus of X'T is a single point, and the points O, XT, X'T are collinear. For example, if P = X(6), then the locus of X(6)-of-T' is collinear with O and X(6)-of-T.
Videos:
Surprisingly Stationary Symmedian Point of the Inversive Image of an X(6)-Pivoting Triangle
Inversive Image of Pivoting Triangle, Part I: Stationary Symmedian Point X(6) of Inversive
Inversive Image of Pivoting Triangle, Part II: Conic Loci of X(3) and X(6) of Inversive (Dan Reznik, August 15, 2021)

Let Γ denote the circumcircle. The trilinear polar of every point on Γ passes through X(6); conversely, the trilinear pole of every line through X(6) lies on Γ. The trilinear polar of every point on the Lemoine axis (the polar of X(6) with respect to Γ) is a line tangent to the Brocard inellipse; conversely, the trilinear pole of every line tangent to the Brocard inellipse lies on the Lemoine axis. (Dan Reznik, February 3, 2023)

Let P be a point in the plane of T = ABC, and let T'=A'B'C' be the cevian triangle of P. Let A'' = BC∩B'C', and define B'' and C'' cycllically. By Desargues' theorem, the points A",B",C" lie on the perspectrix of the ktriangles T and T'. Let Ka be the circle that passes through the points A,A',A'', and define Kb and Kc cyclically. As is well known, if P=X(1), then the circles Ka,Kb,Kc meet in X(15) and X(16), and their radical axis is the Brocard axis. We offer the following new observation. The locus of P such that Ka,Kb,Kc intersect in exactly two points, Z1 and Z2, is the curve Q066, which is the Stammler quartic (the isogonal image of the Stammler hyperbola). The curve Q066 passes through the excenters and the triangle centers X(k) for k=1, 2, 4, 254, and others, listed in Q066, Stammler quartic) . Moreover, the line Z1Z2 passes through the point X(6)-of-T. See the figures in Q066 and line Z1Z2 . (Bernard Gibert and Dan Reznik, February 12, 2023)

X(6) lies on the Walsmith rectangular hyperbola, the Thomson cubic, and these lines: 1,9   2,69   3,15   4,53   5,68   7,294   8,594   10,1377   13,14   17,18   19,34   21,941   22,251   23,353   24,54   25,51   26,143   27,1246   31,42   33,204   36,609  40,380   41,48   43,87   57,222   60,1169   64,185   66,427   67,125   70,1594   74,112   75,239   76,83   77,241   88,89   98,262   99,729   100,739   101,106   105,1002   110,111   145,346   157,248   160,237   162,1013   169,942   181,197   190,192   194,384   210,612   226,1751   256,1580   264,287   274,1218   279,1170   281,1146   282,1256   291,985   292,869   297,317   305,1241   314,981   330,1258   344,1332   354,374   442,1714   493,1583   494,1584   513,1024   517,998   519,996   523,879   560,1631   561,720   588,1599   589,1600   593,1171   595,1126   598,671   603,1035   644,1120   657,1459   662,757   688,882   689,703   691,843   692,1438   694,1084   706,1502   717,789   750,899   753,825   755,827   840,919   846,1051   893,1403   909,1415   911,1461   939,1802   943,1612   947,1622   959,961   963,1208   967,1790   971,990   986,1046   1073,3343   1096,1859   1112,1177   1131,1132   1139,1140   1166,1601   1173,1614   1174,1617   1195,1399   1201,1696   1214,1708   1327,1328   1362,1416   1398,1425   1423,1429   1718,1781   1826,1837   1836,1839   1854,1858   3342,3351   3344,3350

X(6) is the {X(15),X(16)}-harmonic conjugate of X(3). For a list of other harmonic conjugates of X(6), click Tables at the top of this page.

X(6) = midpoint of X(i) and X(j) for these (i,j): (32,5028), (39,5052), (69,193), (125,5095), (187,5107), (1689, 1690)
X(6) = reflection of X(i) in X(j) for these (i,j): (1,1386), (2,597), (3,182), (67,125), (69,141), (159,206), (182,575), (592,2), (694,1084), (1350,3), (1351,576), (1352,5), (32113,468)

X(6) = isogonal conjugate of X(2)
X(6) = isotomic conjugate of X(76)
X(6) = cyclocevian conjugate of X(1031)
X(6) = circumcircle-inverse of X(187)
X(6) = orthocentroidal-circle-inverse of X(115)
X(6) = 1st-Lemoine-circle-inverse of X(1691)
X(6) = complement of X(69)
X(6) = anticomplement of X(141)
X(6) = anticomplementary conjugate of X(1369)
X(6) = complementary conjugate of X(1368)
X(6) = crossdifference of every pair of points on line X(30)X(511)
X(6) = insimilicenter of 1st and 2nd Kenmotu circles
X(6) = exsimilicenter of circumcircle and (1/2)-Moses circle
X(6) = harmonic center of circumcircle and Gallatly circle
X(6) = perspector of polar circle wrt Schroeter triangle
X(6) = X(i)-Ceva conjugate of X(j) for these (i,j):
(1,55), (2,3), (3,154), (4,25), (7,1486), (8,197), (9,198), (10,199), (54,184), (57, 56), (58,31), (68,161), (69,159), (74,1495), (76,22), (81,1), (83,2), (88,36), (95,160), (98,237), (110,512), (111,187), (222,221), (249,110), (251,32), (264,157), (275,4), (284,48), (287,1503), (288,54), (323,399), (394,1498), (1613,3360)

X(6) = cevapoint of X(i) and X(j) for these (i,j): (1,43), (2,194), (31,41), (32,184), (42,213), (51,217)
X(6) = X(i)-cross conjugate of X(j) for these (i,j): (25,64), (31,56), (32,25), (39,2), (41,55), (42,1), (51,4), (184,3), (187,111), (213,31), (217,184), (237,98), (512,110)
X(6) = crosspoint of X(i) and X(j) for these (i,j):
(1,57), (2,4), (9,282), (54,275), (58,81), (83, 251), (110,249), (266,289)
X(6) = crosssum of X(i) and X(j) for these (i,j): (1,9), (3,6), (4,1249), (5,216), (10,37), (11,650), (32,206), (39,141), (44,214), (56,478), (57,223), (114,230), (115,523), (125,647), (128,231), (132,232), (140,233), (142,1212), (188,236), (226,1214), (244,661), (395,619), (396, 618), (512,1084), (513,1015), (514,1086), (522,1146), (570,1209), (577,1147), (590,641), (615,642), (1125,1213), (1196,1368), (5408,5409)

X(6) = X(i)-Hirst inverse of X(j) for these (i,j): (1,238), (2,385), (3,511), (15,16), (25,232), (56,1458), (58,1326), (523,1316), (1423,1429)
X(6) = X(i)-line conjugate of X(j) for these (i,j): (1,518), (2,524), (3,511)
X(6) = X(i)-aleph conjugate of X(j) for these (i,j): (1,846), (81,6), (365,1045), (366,191), (509,1046)
X(6) = X(i)-beth conjugate of X(j) for these (i,j): (6,604), (9,9), (21,1001), (101,6), (284,6), (294,6), (644,6), (645,6), (651,6), (652,7), (666,6)
X(6) = exsimilicenter of circumcircle and (1/2)-Moses circle; the insimilicenter is X(5013)
X(6) = homothetic center of outer Napoleon triangle and pedal triangle of X(15)
X(6) = homothetic center of inner Napoleon triangle and pedal triangle of X(16)
X(6) = trilinear product of vertices of Thomson triangle
X(6) = orthocenter of X(i)X(j)X(k) for these (i,j,k): (2,4,1640), (3,4,879), (3,64,2435)
X(6) = intersection of tangents at X(3) and X(4) to Darboux cubic K004
X(6) = radical trace of circumcircle and Ehrmann circle
X(6) = one of two harmonic traces of Ehrmann circles; the other is X(23)
X(6) = X(3734)-of-1st anti-Brocard-triangle
X(6) = X(182)-of-anti-McCay triangle
X(6) = intersection of tangents to 2nd Brocard circle at PU(1) (i.e., pole of line X(39)X(512) wrt 2nd Brocard circle)
X(6) = intersection of diagonals of trapezoid PU(1)PU(39)
X(6) = intersection of diagonals of trapezoid PU(6)PU(33)
X(6) = intersection of diagonals of trapezoid PU(31)PU(33)
X(6) = the point in which the extended legs P(6)U(31) and U(6)P(31) of the trapezoid PU(6)PU(31) meet
X(6) = trilinear pole of PU(i) for these i: 2, 26
X(6) = crosssum of PU(4)
X(6) = trilinear product of PU(8)
X(6) = barycentric product of PU(i) for these i: 1, 17, 113, 114, 115, 118, 119
X(6) = crossdifference of PU(i) for these i: 24, 41
X(6) = midpoint of PU(i) for these i: 45, 46, 54
X(6) = bicentric sum of PU(i) for these i: 45, 46, 54, 62
X(6) = Zosma transform of X(19)
X(6) = trilinear square of X(365)
X(6) = radical center of {circumcircle, Parry circle, Parry isodynamic circle}; see X(2)
X(6) = PU(62)-harmonic conjugate of X(351)
X(6) = vertex conjugate of PU(118)
X(6) = eigencenter of orthocentroidal triangle
X(6) = eigencenter of Stammler triangle
X(6) = eigencenter of outer Grebe triangle
X(6) = eigencenter of inner Grebe triangle
X(6) = eigencenter of submedial triangle
X(6) = perspector of unary cofactor triangles of every pair of homothetic triangles
X(6) = perspector of ABC and unary cofactor triangle of any triangle homothetic to ABC
X(6) = perspector of Stammler triangle and unary cofactor triangle of circumtangential triangle
X(6) = perspector of Stammler triangle and unary cofactor triangle of circumnormal triangle
X(6) = perspector of submedial triangle and unary cofactor triangle of orthic triangle
X(6) = perspector of unary cofactor triangles of extraversion triangles of X(7) and X(9)
X(6) = X(3)-of-reflection-triangle-of-X(2)
X(6) = center of the orthic inconic
X(6) = orthic isogonal conjugate of X(25)
X(6) = center of bicevian conic of X(371) and X(372)
X(6) = center of bicevian conic of X(6) and X(25)
X(6) = perspector of ABC and mid-triangle of Mandart-incircle and Mandart-excircles triangles
X(6) = X(381)-of-anti-Artzt-triangle
X(6) = homothetic center of medial triangle and cross-triangle of ABC and inner Grebe triangle
X(6) = homothetic center of medial triangle and cross-triangle of ABC and outer Grebe triangle
X(6) = 4th-anti-Brocard-to-anti-Artzt similarity image of X(3)
X(6) = perspector of pedal and anticevian triangles of X(3)
X(6) = X(9)-of-orthic-triangle if ABC is acute
X(6) = X(7)-of-tangential-triangle if ABC is acute
X(6) = X(53)-of-excentral-triangle
X(6) = Thomson-isogonal conjugate of X(376)
X(6) = perspector of ABC and mid-triangle of 1st and 2nd anti-Conway triangles
X(6) = X(193)-of-3rd-tri-squares-central-triangle
X(6) = X(193)-of-4th-tri-squares-central-triangle
X(6) = X(6)-of-circumsymmedial-triangle
X(6) = X(6)-of-inner-Grebe-triangle
X(6) = X(6)-of-outer-Grebe-triangle
X(6) = X(157)-of-intouch-triangle
X(6) = perspector, wrt Schroeter triangle, of polar circle
X(6) = center of the perspeconic of these triangles: ABC and Ehrmann vertex
X(6) = barycentric square of X(1)
X(6) = pole, wrt circumcircle, of Lemoine axis
X(6) = pole wrt polar circle of trilinear polar of X(264) (line X(297)X(525))
X(6) = polar conjugate of X(264)
X(6) = X(i)-isoconjugate of X(j) for these {i,j}: {1,2}, {6,75}, {31,76}, {91,1993}, {110, 1577}, {338,1101}, {1994,2962}
X(6) = inverse-in-2nd-Brocard-circle of X(39)
X(6) = inverse-in-Steiner-inellipse of X(230)
X(6) = inverse-in-Steiner-circumellipse of X(385)
X(6) = inverse-in-Kiepert-hyperbola of X(381)
X(6) = inverse-in-circumconic-centered-at-X(9) of X(238)
X(6) = perspector of medial triangle and half-altitude triangle
X(6) = intersection of tangents to Kiepert hyperbola at X(2) and X(4)
X(6) = antigonal conjugate of X(67)
X(6) = vertex conjugate of foci of Steiner inellipse
X(6) = X(99)-of-1st-Brocard-triangle
X(6) = X(1379)-of-2nd-Brocard-triangle
X(6) = X(6)-of-4th-Brocard-triangle
X(6) = X(6)-of-orthocentroidal-triangle
X(6) = reflection of X(2453) in the Euler line
X(6) = similitude center of ABC and orthocentroidal triangle
X(6) = similitude center of 4th Brocard and circumsymmedial triangles
X(6) = tangential isogonal conjugate of X(22)
X(6) = tangential isotomic conjugate of X(1498)
X(6) = barycentric product of (nonreal) circumcircle intercepts of the line at infinity
X(6) = eigencenter of anti-orthocentroidal triangle
X(6) = perspector of Aquarius conic
X(6) = trilinear pole wrt tangential triangle of Lemoine axis
X(6) = trilinear pole wrt symmedial triangle of Lemoine axis
X(6) = trilinear pole wrt circumsymmedial triangle of Lemoine axis
X(6) = crosspoint of X(2) and X(194) wrt both the excentral and anticomplementary triangles
X(6) = pedal antipodal perspector of X(5004) and of X(5005)
X(6) = vertex conjugate of Jerabek hyperbola intercepts of Lemoine axis
X(6) = hyperbola {{A,B,C,X(2),X(6)}} antipode of X(694)
X(6) = perspector of orthic triangle and tangential triangle, wrt orthic triangle, of the circumconic of the orthic triangle centered at X(4) (the bicevian conic of X(4) and X(459))
X(6) = perspector of excentral triangle and extraversion triangle of X(9)
X(6) = excentral-to-ABC functional image of X(9)
X(6) = orthic-to-ABC functional image of X(53)
X(6) = intouch-to-ABC functional image of X(7)
X(6) = 1st-Brocard-isogonal conjugate of X(804)
X(6) = center of the MacBeath circumconic
X(6) = center of the cosine circle (the 2nd Lemoine circle)
X(6) = one of the foci of the Lemoine inellipse (the other being X(2))
X(6) = antipode of X(32113) in Walsmith rectangular hyperbola
X(6) = orthocenter of X(74)X(110)X(3569)
X(6) = orthocenter of X(113)X(125)X(3569)
X(6) = QA-P23 (Inscribed Square Axes Crosspoint) of quadrangle ABCX(2); see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/51-qa-p23.html


X(7) = GERGONNE POINT

Trilinears    bc/(b + c - a) : ca/(c + a - b) : ab/(a + b - c)
Trilinears    sec2(A/2) : sec2(B/2) : sec2(C/2)
Trilinears    1/(tan(B/2) + tan(C/2)) : 1/(tan(C/2) + tan(A/2)) : 1/(tan(A/2) + tan(B/2))
Trilinears    (bc - SA)/a : (ca - SB)/b : (ab - SC)/c
Trilinears    (1 - cos A)a-2 : :
Barycentrics   1/(b + c - a) : 1/(c + a - b) : 1/(a + b - c)
Barycentrics    tan A/2 : tan B/2 : tan C/2
Barycentrics    bc - SA : ca - SB : ab - SC
Barycentrics    ra : rb : rc, where ra, rb, rc are the exradii
Tripolars    (a - b - c)*Sqrt[-a*(a - b - c)*(a^2 + a*b - 2*b^2 + a*c + 4*b*c - 2*c^2)] : :
X(7) = (2r + 4R)*X(1) + 3r*X(2) - 4r*X(3)
X(7) = [A]/Ra + [B]/Rb + [C]/Rc, where Ra, Rb, Rc = radii of Soddy circles

Let A', B', C' denote the points in which the incircle meets the sides BC, CA, AB, respectively. The lines AA', BB', CC' concur in X(7).

An exarc circle is a circle tangent to two sides of a triangle ABC and externally tangent to the circumcircle of ABC. The point whose distance to the sides BC, CA, AB are proportional to the respective radii of the exarc circles is X(279). The point having distances to the sides proportional to the radii of the inarc circles is X(7). See Martin Lukarevski, "Exarc radii and the Finsler-Hadwiger inequality", The Mathematical Gazette 106, issue 565, March 2022, pp. 138-143.

If you have The Geometer's Sketchpad, you can view Gergonne point.
If you have GeoGebra, you can view Gergonne point.

X(7) lies on the Lucas cubic and these lines: 1,20   2,9   3,943   4,273   6,294   8,65   11,658   12,1268   21,56   27,81   37,241   33,1041   34,1039   55,2346   58,272   59,1275   72,443   73,1246   76,1479   80,150   92,189   100,1004   104,934   108,1013   109,675   145,1266   171,983   174,234   177,555   190,344   192,335   193,239   218,277   220,1223   225,969   238,1471   253,280   256,982   274,959   281,653   286,331   310,314   330,1432   349,1269   354,479   404,1259   452,1467   464,1214   480,1376   492,1267   513,885   517,1000   528,664   554,1082   594,599   604,1429   757,1414   840,927   857,1901   870,1431   940,1407   941,1427   944,1389   952,1159   986,1254   987,1106   1002,1362   1020,1765   1061,1870   1210,3091   1354,1367   1365,1366   1386,1456   1419,1449   1435,1848   1486,1602   1617,1621   2475,2893  

X(7) is the {X(69),X(75)}-harmonic conjugate of X(8). For a list of other harmonic conjugates of X(7), click Tables at the top of this page.

X(7) = reflection of X(i) in X(j) for these (i,j): (9,142), (144,9), (390,1), (673,1086), (1156,11)
X(7) = isogonal conjugate of X(55)
X(7) = isotomic conjugate of X(8)
X(7) = cyclocevian conjugate of X(7)
X(7) = circumcircle-inverse of (32624)
X(7) = incircle-inverse of (1323)
X(7) = complement of X(144)
X(7) = anticomplement of X(9)
X(7) = complementary conjugate of X(2884)
X(7) = anticomplementary conjugate of X(329)
X(7) = X(i)-Ceva conjugate of X(j) for these (i,j): (75,347), (85,2), (86,77), (286,273), (331,278)
X(7) = cevapoint of X(i) and X(j) for these (i,j):
(1,57), (2,145), (4,196), (11,514), (55,218), (56,222), (63,224), (65,226), (81,229), (177,234)
X(7) = X(i)-cross conjugate of X(j) for these (i,j):
(1,2), (4,189), (11,514), (55,277), (56,278), (57,279), (65,57), (177,174), (222,348), (226,85), (354,1), (497,8), (517,88)
X(7) = crosspoint of X(i) and X(j) for these (i,j): (75,309), (86,286)
X(7) = crosssum of X(i) and X(j) for these (i,j): (41,1253), (42,228)
X(7) = crossdifference of every pair of points on line X(657)X(663)
X(7) = X(57)-Hirst inverse of X(1447)
X(7) = insimilicenter of inner and outer Soddy circles; the exsimilicenter is X(1)
X(7) = X(i)-beth conjugate of X(j) for these (i,j):
(7,269), (21,991), (75,75), (86,7), (99,7), (190,7), (290,7), (314,69), (645,344), (648,7), (662,6), (664,7), (666,7), (668,7), (670,7), (671,7), (811,264), (886,7), (889,7), (892,7), (903,7)
X(7) = vertex conjugate of foci of inellipse that is isotomic conjugate of isogonal conjugate of incircle (centered at X(2886))
X(7) = trilinear product of vertices of Hutson-extouch triangle
X(7) = orthocenter of X(4)X(8)X(885)
X(7) = trilinear cube of X(506)
X(7) = barycentric product of PU(47)
X(7) = trilinear product of PU(94)
X(7) = vertex conjugate of PU(95)
X(7) = bicentric sum of PU(120)
X(7) = perspector of ABC and its intouch triangle. X(7) = perspector of ABC and the reflection in X(57) of the pedal triangle of X(57)
X(7) = perspector of AC-incircle
X(7) = X(6)-of-extraversion triangle-of-X(8)
X(7) = X(6)-of-intouch-triangle; X(7) is the only point X inside ABC Such that X(ABC) = X(A'B'C'), where A'B'C' = cevian triangle of X
X(7) = {X(2),X(63)}-harmonic conjugate of X(5273)
X(7) = {X(9),X(57)}-harmonic conjugate of X(1445)
X(7) = {X(1371),X(1372)}-harmonic conjugate of X(1)
X(7) = {X(1373),X(1374)}-harmonic conjugate of X(1)
X(7) = trilinear pole of Gergonne line
X(7) = trilinear pole, wrt intouch triangle, of Gergonne line
X(7) = pole of Gergonne line wrt incircle
X(7) = pole wrt polar circle of trilinear polar of X(281) (line X(3064)X(3700))
X(7) = X(48)-isoconjugate (polar conjugate) of X(281)
X(7) = X(6)-isoconjugate of X(9)
X(7) = X(75)-isoconjugate of X(2175)
X(7) = X(1101)-isoconjugate of X(4092)
X(7) = perspector of circumconic centered at X(3160)
X(7) = center of circumconic that is locus of trilinear poles of lines passing through X(3160)
X(7) = X(2)-Ceva conjugate of X(3160)
X(7) = antigonal image of X(1156)
X(7) = homothetic center of intouch triangle and anticomplement of the excentral triangle
X(7) = X(6)-of-intouch-triangle; X(7) is the only point X inside ABC such that X(ABC) = X(A'B'C'), where A'B'C' = cevian triangle of X
X(7) = perspector of ABC and cross-triangle of inner and outer Soddy triangles
X(7) = perspector of excentral triangle and cross-triangle of ABC and Honsberger triangle
X(7) = perspector of inverse-in-excircles triangle and cross-triangle of ABC and inverse-in-excircles triangle
X(7) = perspector of inverse-in-incircle triangle and cross-triangle of ABC and inverse-in-incircle triangle
X(7) = X(1843)-of-excentral-triangle
X(7) = Cundy-Parry Phi transform of X(943)
X(7) = Cundy-Parry Psi transform of X(942)
X(7) = {X(1),X(1742)}-harmonic conjugate of X(2293)
X(7) = barycentric square of X(508)
X(7) = perspector of ABC and cross-triangle of ABC and Gemini triangle 40
X(7) = barycentric product of vertices of Gemini triangle 40
X(7) = excentral-to-intouch similarity image of X(9)
X(7) = circumconic-centered-at-X(9)-inverse of X(37787)
X(7) = endo-homothetic center of Ehrmann vertex-triangle and Ehrmann mid-triangle; the homothetic center is X(3818)


X(8) = NAGEL POINT

Trilinears    (b + c - a)/a : (c + a - b)/b : (a + b - c)/c
Trilinears    csc2(A/2) : csc2(B/2) : csc2(C/2)
Trilinears    (bc + SA)/a : (ca + SB)/b : (ab + SC)/c
Trilinears    (1 + cos A)a-2 : :
Trilinears    (r/R) - sin B sin C : :
Barycentrics    b + c - a : c + a - b : a + b - c
Barycentrics   cot A/2 : cot B/2 : cot C/2
Barycentrics    square of semi-minor axis of A-Soddy ellipse : :
Barycentrics    bc + SA : ca + SB : ab + SC
Tripolars    Sqrt[-a*(a^2 - a*b - 2*b^2 - a*c + 4*b*c - 2*c^2)] : :
X(8) = 2*X(1) - 3*X(2) = 3*X(2) - 4*X(10) = 4X(10) - X(145)
X(8) = [A]*Ra + [B]*Rb + [C]*Rc, where Ra, Rb, Rc = radii of Soddy circles

Let A'B'C' be the points in which the A-excircle meets BC, the B-excircle meets CA, and the C-excircle meets AB, respectively. The lines AA', BB', CC' concur in X(8). Another construction of A' is to start at A and trace around ABC half its perimeter, and similarly for B' and C'. Also, X(8) is the incenter of the anticomplementary triangle.

X(8) = perspector of ABC and the intouch triangle of the medial triangle of ABC. (Randy Hutson, 9/23/2011)

Let Ab, Ac, Bc, Ba, Ca, Cb be as in the construction of the Conway circle (http://mathworld.wolfram.com/ConwayCircle.html). Let Oa be the circumcircle of ABcCb, and define Ob, Oc cyclically. X(8) is the radical center of Oa, Ob, Oc. see also X(21) and X(274). (Randy Hutson, April 9, 2016)

Let A'B'C' be Triangle T(-2,1). Let A″ be the trilinear product B'*C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(519). The lines A'A″, B'B″, C'C″ concur in X(8). (Randy Hutson, November 18, 2015)

Let Ia be the reflection of X(1) in the perpendicular bisector of BC, and define Ib and Ic cyclically. X(8) = X(1)-of-IaIbIc. (Randy Hutson, September 14, 2016)

Let A28B28C28 be the Gemini triangle 28. Let LA be the line through A28 parallel to BC, and define LB and LC cyclically. Let A'28 = LB∩LC, and define B'28, C'28 cyclically. Triangle A'28B'28C'28 is homothetic to ABC at X(8). (Randy Hutson, November 30, 2018)

For yet another construction of X(8), see Dasari Naga Vijay Krishna, "On A Simple Construction of Triangle Centers X(8), X(197), X(K) & X(594)", Scientific Inquiry and Review, Vol. 2, Issue 3, July 2018.

Another construction is given by Xavier Dussau: Elementary construction of the Nagel point. (April 29, 2020)

If you have The Geometer's Sketchpad, you can view Nagel point.
If you have GeoGebra, you can view Nagel point.

Let T be any one of these trianges: {Ascella, 1st circumperp, 2nd circumperp}. Let OA be the circle centered at the A-vertex of T and passing through A; define OB and OC cyclically. X(8) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

For extensions of X(8) to non-Euclidean geometries, see Thomas E. Cooper, "Finagling a Nagel Point in Taxicab Geometry and Beyond", Mathematics Magazine 97, no. 4, October 2024.

X(8) lies on these curves:
Feuerbach hyperbola, anticomplementary Feuerbach hyperbola, Mandart hyperbola, Fuhrmann circle, K007, K013, K028, K033, K034, K154, K157, K199, K200, K201, K259, K308, K311, K338, K366, K386, K387, K455, K461, K506, K521, K623, K651, K654, K680, K692, K696, K697, K702, K744, K767, Q045

X(8) lies on the Lucas cubic and these lines: {1,2}, {3,100}, {4,72}, {5,1389}, {6,594}, {7,65}, {9,346}, {11,1320}, {12,2099}, {19,1891}, {20,40}, {21,55}, {22,8193}, {23,8185}, {25,7718}, {29,219}, {30,3578}, {31,987}, {32,5291}, {33,1039}, {34,1041}, {35,993}, {36,4188}, {37,941}, {38,986}, {39,7976}, {41,2329}, {44,4217}, {45,3943}, {46,3218}, {56,404}, {57,1219}, {58,996}, {60,7058}, {76,668}, {79,758}, {80,149}, {81,1010}, {86,2334}, {90,2994}, {101,1311}, {109,2370}, {113,7978}, {114,7970}, {115,7983}, {119,6941}, {125,7984}, {140,1483}, {141,3242}, {142,3243}, {144,516}, {147,9864}, {150,1930}, {151,2817}, {153,2800}, {165,3522}, {171,1468}, {172,4386}, {175,1270}, {176,1271}, {177,556}, {178,236}, {181,959}, {188,2090}, {190,528}, {191,3065}, {192,256}, {193,894}, {194,730}, {197,1603}, {201,1937}, {210,312}, {213,981}, {214,5445}, {215,9701}, {220,294}, {221,651}, {224,914}, {226,3340}, {238,983}, {244,3976}, {253,307}, {274,1002}, {277,1280}, {278,1257}, {279,7273}, {291,330}, {304,3263}, {313,2997}, {314,1264}, {315,760}, {326,1442}, {344,480}, {348,664}, {350,3789}, {354,3698}, {376,3579}, {381,8148}, {392,1000}, {394,3562}, {405,943}, {406,1061}, {411,3428}, {442,495}, {443,942}, {474,999}, {475,1063}, {479,7182}, {484,4299}, {491,1267}, {492,5391}, {496,3820}, {514,4546}, {521,4397}, {522,4474}, {523,4774}, {524,4363}, {527,4454}, {529,3474}, {535,3245}, {536,4419}, {537,4440}, {573,3588}, {595,1724}, {596,4674}, {599,1086}, {618,7975}, {619,7974}, {631,1385}, {637,7595}, {641,7981}, {642,7980}, {643,1098}, {645,4092}, {646,3271}, {663,4147}, {672,3501}, {673,4437}, {704,8264}, {712,4805}, {726,1278}, {860,1068}, {885,3900}, {901,2757}, {908,946}, {912,5553}, {961,1460}, {965,2256}, {971,9961}, {982,4457}, {1015,1574}, {1016,1083}, {1018,3730}, {1019,4807}, {1034,1895}, {1036,1183}, {1046,4418}, {1054,9457}, {1071,6916}, {1104,3744}, {1106,9363}, {1107,2276}, {1120,3445}, {1124,1377}, {1126,6539}, {1147,9933}, {1159,6147}, {1191,4383}, {1209,7979}, {1211,1834}, {1212,3693}, {1215,4865}, {1229,6601}, {1237,4485}, {1251,5239}, {1266,4346}, {1279,6687}, {1281,1282}, {1309,2745}, {1312,2103}, {1313,2102}, {1317,1388}, {1319,6049}, {1331,2988}, {1332,8759}, {1335,1378}, {1361,3042}, {1364,3040}, {1386,3618}, {1392,5048}, {1397,2985}, {1420,3911}, {1449,4982}, {1453,5294}, {1467,8732}, {1500,1573}, {1512,5720}, {1575,2275}, {1656,5901}, {1672,1680}, {1673,1681}, {1674,1678}, {1675,1679}, {1682,9564}, {1699,3832}, {1738,3620}, {1739,3953}, {1743,4058}, {1748,6197}, {1757,3923}, {1759,5011}, {1783,8743}, {1812,3193}, {1836,3962}, {1857,1896}, {1869,5307}, {1897,7358}, {1914,4426}, {1943,4296}, {1953,3949}, {1959,7379}, {1992,3758}, {1997,3816}, {2007,2013}, {2008,2014}, {2053,8851}, {2077,5450}, {2093,4001}, {2122,2123}, {2170,3061}, {2175,4157}, {2176,2238}, {2242,5277}, {2310,9365}, {2318,2654}, {2320,2646}, {2335,3694}, {2363,6043}, {2399,8058}, {2463,2467}, {2464,2468}, {2477,9702}, {2482,9884}, {2564,2568}, {2565,2569}, {2647,4332}, {2650,4938}, {2785,4088}, {2787,4730}, {2796,8596}, {2801,5696}, {2810,3888}, {2883,7973}, {2891,3754}, {2893,2897}, {2894,6839}, {2896,9857}, {2901,3995}, {2943,9355}, {3021,3039}, {3022,3041}, {3038,6018}, {3056,4110}, {3058,3715}, {3068,7969}, {3069,7968}, {3090,5886}, {3096,9997}, {3152,6360}, {3158,3601}, {3174,7675}, {3177,4712}, {3247,5257}, {3254,4858}, {3304,4413}, {3305,5129}, {3306,3333}, {3309,4462}, {3336,4317}, {3339,4298}, {3361,4315}, {3427,6836}, {3452,3680}, {3467,4309}, {3475,3925}, {3496,5282}, {3523,3576}, {3524,3655}, {3545,3656}, {3583,3899}, {3585,4067}, {3619,3844}, {3629,7227}, {3631,7232}, {3647,5441}, {3649,6175}, {3663,4452}, {3664,4924}, {3666,4646}, {3670,3987}, {3671,5290}, {3672,3755}, {3683,5302}, {3716,4895}, {3721,3959}, {3731,3950}, {3735,3954}, {3738,4768}, {3739,4648}, {3740,3983}, {3742,4731}, {3746,5248}, {3760,6381}, {3762,3887}, {3772,4952}, {3775,4085}, {3814,5154}, {3817,5068}, {3823,4864}, {3826,4966}, {3829,7173}, {3841,5425}, {3879,3945}, {3881,3918}, {3884,3992}, {3891,4972}, {3892,3968}, {3894,3919}, {3896,3931}, {3898,3956}, {3901,4084}, {3904,4528}, {3907,4041}, {3928,5128}, {3929,7285}, {3947,5726}, {3963,9052}, {3967,4005}, {3977,4304}, {3978,6382}, {3986,4898}, {3993,4704}, {4002,5045}, {4003,4706}, {4004,5551}, {4018,4980}, {4026,4360}, {4036,8702}, {4054,9612}, {4080,4792}, {4082,4866}, {4086,7253}, {4087,4531}, {4125,4857}, {4160,4761}, {4163,6332}, {4181,4182}, {4208,5249}, {4234,4921}, {4312,5850}, {4314,4512}, {4342,4900}, {4364,4748}, {4373,4862}, {4404,6003}, {4407,4743}, {4421,5217}, {4424,7226}, {4432,4473}, {4439,4527}, {4470,4670}, {4534,6558}, {4542,4582}, {4595,8299}, {4657,4852}, {4658,8025}, {4667,4747}, {4672,4753}, {4675,4688}, {4694,9335}, {4699,4732}, {4729,6002}, {4736,6758}, {4756,9668}, {4767,9669}, {4867,5141}, {4922,9508}, {4999,5432}, {5010,5267}, {5056,8227}, {5059,5493}, {5187,10043}, {5221,5434}, {5284,6767}, {5285,7520}, {5286,9620}, {5429,8258}, {5534,6908}, {5584,7411}, {5590,5604}, {5591,5605}, {5592,6546}, {5597,5600}, {5598,5599}, {5714,9654}, {5791,6857}, {5985,10053}, {6001,6223}, {6062,7068}, {6144,7277}, {6154,9963}, {6174,10031}, {6193,9928}, {6245,6282}, {6260,7971}, {6261,6838}, {6265,6949}, {6292,7977}, {6326,6960}, {6462,8214}, {6463,8215}, {6553,8056}, {6653,6655}, {6691,7231}, {6739,6742}, {6828,7680}, {6835,7686}, {6856,8164}, {6945,7681}, {6995,7713}, {7003,7020}, {7018,7033}, {7028,8242}, {7043,7126}, {7048,8422}, {7090,7133}, {7161,7206}, {7279,9723}, {7373,9342}, {7486,9624}, {7987,9588}, {8092,8125}, {8094,9795}, {8126,8351}, {8162,8167}, {8163,8169}, {8210,8222}, {8211,8223}, {8372,9787}, {8591,9881}, {8972,8983}, {9317,9451}, {9783,9805}, {9859,9943}, {10087,10093}, {10090,10094}

X(8) is the {X(69),X(75)}-harmonic conjugate of X(7). For a list of other harmonic conjugates of X(8), click Tables at the top of this page.

X(8) = midpoint of X(i) and X(j) for these {i,j}: {1,3632}, {10,3625}, {40,5881}, {145,3621}, {3057,3893}, {3626,4701}, {3679,4677}, {4474,4814}, {4668,4816}, {4900,8275}, {5541,9897}, {5691,7991}, {5903,5904}
X(8) = reflection of X(i) in X(j) for these (i,j):
(1,10), (2,3679), (3,5690), (4,355), (7,2550), (8,8), (10,3626), (11,3036), (20,40), (56,8256), (65,5836), (69,3416), (75,3696), (78,6736), (86,4733), (100,1145), (144,5223), (145,1), (147,9864), (149,80), (192,984), (193,3751), (210,4711), (315,4769), (329,3421), (346,4901), (376,3654), (388,5794), (390,9), (551,4745), (663,4147), (944,3), (950,5795), (960,4662), (962,4), (1019,4807), (1043,3704), (1120,3756), (1125,4691), (1280,4904), (1317,3035), (1320,11), (1361,3042), (1364,3040), (1392,7705), (1482,5), (1483,140), (1697,5837), (2098,1329), (2099,2886), (2102,1313), (2103,1312), (3021,3039), (3022,3041), (3057,960), (3146,5691), (3161,10005), (3189,3913), (3241,2), (3242,141), (3243,142), (3244,1125), (3434,3419), (3476,1376), (3486,958), (3488,9708), (3555,942), (3600,1706), (3616,3617), (3617,4668), (3621,3632), (3623,1698), (3625,4701), (3626,4746), (3632,3625), (3633,3244), (3635,3634), (3679,4669), (3685,3717), (3786,4111), (3868,65), (3869,72), (3872,4847), (3873,3753), (3874,3754), (3875,3755), (3877,210), (3878,3678), (3881,3918), (3883,3686), (3884,4015), (3885,3057), (3886,2321), (3889,3698), (3890,3697), (3892,3968), (3894,3919), (3898,3956), (3899,4134), (3901,4084), (3902,3706), (3952,4738), (4318,1861), (4344,2345), (4360,4026), (4363,4665), (4419,4643), (4430,5902), (4454,4659), (4511,6735), (4560,4041), (4643,4690), (4644,4363), (4673,3714), (4693,4439), (4720,4046), (4861,6734), (4864,3823), (4895,3716), (4922,9508), (5048,5123), (5080,5176), (5180,5080), (5263,594), (5441,3647), (5603,5790), (5697,3878), (5698,5220), (5710,5835), (5716,5793), (5731,5657), (5734,5818), (5882,6684), (5905,1478), (5919,3740), (5984,9860), (6018,3038), (6193,9928), (6224,100), (6327,4680), (6332,4163), (6737,6743), (6740,6741), (6742,6739), (6758,4736), (7192,4761), (7253,4086), (7962,3452), (7970,114), (7971,6260), (7972,214), (7973,2883), (7974,619), (7975,618), (7976,39), (7977,6292), (7978,113), (7979,1209), (7980,642), (7981,641), (7982,946), (7983,115), (7984,125), (8241,2090), (8591,9881), (8596,9875), (8834,6552), (9263,291), (9780,4678), (9785,2551), (9791,1654), (9797,938), (9802,149), (9809,153), (9856,9947), (9884,2482), (9933,1147), (9957,5044), (9963,6154), (9965,2093), (10031,6174)
X(8) = isogonal conjugate of X(56)
X(8) = isotomic conjugate of X(7)
X(8) = cyclocevian conjugate of X(189)
X(8) = complement of X(145)
X(8) = anticomplement of X(1)
X(8) = complementary conjugate of X(2885)
X(8) = anticomplementary conjugate of X(8)
X(8) = circumcircle-inverse of X(17100)
X(8) = Conway-circle-inverse of X(38475)
X(8) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,3161), (4,2899), (7,8055), (69,329), (75,2), (190,3239), (290,3948), (312,346), (314,312), (318,5552), (319,2895), (333,9), (341,7080), (643,7253), (645,3700), (646,650), (664,6332), (668,4391), (765,3952), (1016,644), (1043,78), (1219,3616), (1222,1), (1494,3936), (1909,1655), (2319,7155), (2985,6), (3596,345), (3699,522), (4076,3699), (4102,2321), (4373,6557), (4554,4130), (4555,3904), (4582,1639), (4997,2325), (4998,190), (6063,344), (6064,645), (6079,900), (7017,281), (7033,192), (7155,4903), (7257,4560), (8817,7)
X(8) = X(i)-cross conjugate of X(j) for these (i,j):
(1,280), (4,1034), (9,2), (10,318), (11,522), (40,7080), (55,281), (56,2123), (72,78), (200,346), (210,9), (219,345), (312,7155), (346,6557), (497,7), (521,100), (522,3699), (650,646), (950,29), (960,21), (1145,6735), (1146,4391), (1639,4582), (1837,4), (1857,8805), (1864,282), (2170,4560), (2321,312), (2325,4997), (2968,4397), (3057,1), (3059,200), (3239,190), (3271,650), (3680,6553), (3683,7110), (3686,333), (3687,4451), (3688,55), (3700,645), (3703,3596), (3704,3701), (3706,314), (3717,4518), (3877,2320), (3880,1320), (3885,1392), (3893,3680), (3900,644), (3907,7257), (4012,5423), (4046,2321), (4051,330), (4060,4102), (4081,3239), (4086,3952), (4092,3700), (4111,210), (4124,885), (4130,4554), (4152,2325), (4180,4182), (4531,41), (4534,514), (4542,1639), (4546,6558), (4847,75), (4853,1219), (4863,6601), (4875,274), (4965,7192), (5245,7026), (5246,7043), (5795,1220), (6062,7359), (6068,6745), (6555,3161), (6736,341), (6737,1043), (6741,4086), (7063,3709), (7067,3712), (8058,1897), (8611,4552), (9785,5558)
X(8) = cevapoint of X(i) and X(j) for these (i,j):
{1,40}, {2,144}, {4,3176}, {6,197}, {9,200}, {10,72}, {11,522}, {34,8899}, {42,3588}, {55,219}, {56,2122}, {65,5930}, {175,176}, {210,2321}, {312,4110}, {346,6555}, {497,4012}, {513,3756}, {514,4904}, {519,1145}, {521,2968}, {523,8286}, {650,3271}, {758,6739}, {960,3704}, {966,4859}, {1125,3650}, {1146,3900}, {1639,4542}, {2170,4041}, {2175,4548}, {2325,4152}, {3057,6736}, {3059,4847}, {3161,4859}, {3239,4081}, {3686,4046}, {3688,3703}, {3700,4092}, {3706,4111}, {3709,7063}, {3712,7067}, {3893,4895}, {4136,4531}, {4180,4181}, {4530,4543}, {4534,4546}, {6062,7359}, {6068,6745}, {7358,8058}
X(8) = crosspoint of X(i) and X(j) for these (i,j):
{1,979}, {2,4373}, {7,8051}, {9,2319}, {75,312}, {190,4998}, {314,333}, {643,765}, {645,6064}, {668,1016}, {3596,7017}, {3699,4076}
X(8) = crosssum of X(i) and X(j) for these (i,j): {1,978}, {6,3052}, {25,3209}, {31,604}, {57,1423}, {244,4017}, {649,3271}, {663,7117}, {667,1015}, {1042,1410}, {1284,8850}, {1400,1402}
X(8) = crossdifference of every pair of points on line X(649)X(854)
X(8) = X(1)-aleph conjugate of X(1050)
X(8) = X(i)-beth conjugate of X(j) for these (i,j): (8,1), (341,341), (643,3), (668,8), (1043,8)
X(8) = exsimilicenter of incircle and Spieker circle
X(8) = exsimilicenter of Conway circle and Spieker radical circle
X(8) = trilinear product of vertices of Hutson-intouch triangle
X(8) = trilinear product of vertices of Caelum triangle
X(8) = orthocenter of X(i)X(j)X(k) for these (i,j,k): (1,4,5556), (4,7,885)
X(8) = perspector of ABC and pedal triangle of X(40)
X(8) = perspector of ABC and reflection of medial triangle in X(10)
X(8) = perspector of ABC and reflection of intouch triangle in X(1)
X(8) = pedal antipodal perspector of X(1)
X(8) = pedal antipodal perspector of X(36)
X(8) = X(1498)-of-intouch-triangle
X(8) = X(185)-of-excentral-triangle
X(8) = X(74)-of-Fuhrmann-triangle
X(8) = X(5992)-of-Brocard-triangle
X(8) = perspector of circumconic with center X(3161)
X(8) = center of circumconic that is locus of trilinear poles of lines passing through X(3161)
X(8) = X(2)-Ceva conjugate of X(3161)
X(8) = trilinear pole of line X(522)X(650) (the radical axis of circumcircle and excircles radical circle)
X(8) = pole wrt polar circle of trilinear polar of X(278) (line X(513)X(1835))
X(8) = X(48)-isoconjugate (polar conjugate) of X(278)
X(8) = X(6)-isoconjugate of X(57)
X(8) = X(75)-isoconjugate of X(1397)
X(8) = X(1101)-isoconjugate of X(1365)
X(8) = antigonal image of X(1320)
X(8) = {X(1),X(2)}-harmonic conjugate of X(3616)
X(8) = {X(1),X(10)}-harmonic conjugate of X(2)
X(8) = {X(2),X(10)}-harmonic conjugate of X(667)
X(8) = inverse-in-polar-circle of X(1878)
X(8) = inverse-in-Steiner-circumellipse of X(3912)
X(8) = inverse-in-Mandart-inellipse of X(2325)
X(8) = inverse-in-circumconic-centered-at-X(1) of X(4511)
X(8) = X(4) of 2nd Conway triangle (the extraversion triangle of X(8))
X(8) = trilinear square root of X(341)
X(8) = perspector of 5th extouch triangle and anticevian triangle of X(7)
X(8) = centroid of cross-triangle of Gemini triangles 20 and 28
X(8) = X(i)-aleph conjugate of X(j) for these (i,j): (1,1050), (188,2943), (1222,8)
X(8) = X(i)-beth conjugate of X(j) for these (i,j): (8,1), (99,3160), (200,4517), (333,5222), (341,341), (346,4873), (643,3), (644,3730), (668,8), (1043,8), (2287,4266), (3699,8), (6558,8), (7256,8), (7257,76), (7259,220), (8706,8)
X(8) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (1,8), (2,69), (3,20), (5,2888), (6,2), (7,3434), (8,3436), (9,829), (10,1330), (13,621), (14, 622), (15,616), (16,617), (18,634), (19,59015), (20,6225), (21,3869), (22,5596), (24,6193), (25,192), (28,3868), (30,146), (31,192), (32,194), (54,3), (55,144), (56,145), (57,7), (58,1), (59,100), (74,30), (81,75), (98,511), (99,512), and many others
X(8) = X(i)-complementary conjugate of X(j) for these (i,j): (1,2885), (31,3161), (513,5510), (1293,513), (3445,10), (3680,1329), (4373,2887), (8056,141)
X(8) = perspector of ABC and mid-triangle of excentral and 2nd extouch triangles
X(8) = perspector of 5th extouch triangle and cross-triangle of ABC and 5th extouch triangle
X(8) = X(1593)-of-2nd-extouch-triangle
X(8) = excentral-to-2nd-extouch similarity image of X(1697)
X(8) = Cundy-Parry Phi transform of X(104)
X(8) = Cundy-Parry Psi transform of X(517)
X(8) = perspector of ABC and cross-triangle of ABC and Gemini triangle 39
X(8) = barycentric product of vertices of Gemini triangle 39
X(8) = perspector of Gemini triangle 13 and cross-triangle of Gemini triangles 1 and 13
X(8) = excentral-to-ABC barycentric image of X(1)
X(8) = X(4)-of-Yff-contact-triangle
X(8) = Yff-contact-isogonal conjugate of X(5592)
X(8) = perspector (Brianchon point) of the Mandart inellipse
X(8) = isoconjugate of X(i) and X(j) for these {i,j}: {1,56}, {2,604}, {3,34}, {4,603}, {6,57}, {7,31}, {8,1106}, {9,1407}, {10,1408}, {12,849}, {19,222}, {21,1042}, {25,77}, {27,1409}, {28,73}, {29,1410}, {32,85}, {33,7053}, {36,1411}, {37,1412}, {40,1413}, {41,279}, {42,1014}, {48,278}, {54,1393}, {55,269}, {58,65}, {59,244}, {60,1254}, {63,608}, {64,1394}, {69,1395}, {71,1396}, {75,1397}, {78,1398}, {79,1399}, {81,1400}, {82,1401}, {84,221}, {86,1402}, {87,1403}, {88,1404}, {89,1405}, {90,1406}, {101,3669}, {102,1455}, {103,1456}, {104,1457}, {105,1458}, {106,1319}, {108,1459}, {109,513}, {110,4017}, {154,8809}, {158,7335}, {163,7178}, {171,1431}, {172,1432}, {181,757}, {184,273}, {189,2199}, {198,1422}, {200,7023}, {208,1433}, {212,1119}, {213,1434}, {219,1435}, {220,738}, {223,1436}, {225,1437}, {226,1333}, {241,1438}, {255,1118}, {259,7370}, {267,8614}, {270,1425}, {281,7099}, {282,6611}, {283,1426}, {284,1427}, {291,1428}, {292,1429}, {296,1430}, {307,2203}, {326,7337}, {331,9247}, {346,7366}, {347,2208}, {348,1973}, {388,1472}, {393,7125}, {479,1253}, {512,1414}, {514,1415}, {518,1416}, {519,1417}, {552,872}, {560,6063}, {593,2171}, {607,7177}, {614,1037}, {643,7250}, {649,651}, {650,1461}, {657,4617}, {658,3063}, {661,4565}, {662,7180}, {663,934}, {664,667}, {669,4625}, {672,1462}, {692,3676}, {727,1463}, {741,1284}, {756,7341}, {759,1464}, {765,1357}, {798,4573}, {893,7175}, {896,7316}, {904,7176}, {909,1465}, {923,7181}, {937,1466}, {939,1467}, {951,1104}, {959,1468}, {961,1193}, {983,7248}, {985,1469}, {998,1470}, {1002,1471}, {1015,4564}, {1019,4559}, {1020,7252}, {1027,2283}, {1035,3345}, {1036,4320}, {1041,1473}, {1073,3213}, {1086,2149}, {1088,2175}, {1089,7342}, {1096,1804}, {1098,7143}, {1101,1365}, {1110,1358}, {1149,8686}, {1170,1475}, {1174,1418}, {1191,7091}, {1201,1476}, {1214,1474}, {1245,5323}, {1262,2170}, {1279,1477}, {1420,3445}, {1421,3446}, {1423,2162}, {1424,3224}, {1439,2299}, {1440,2187}, {1441,2206}, {1442,6186}, {1443,6187}, {1447,1911}, {1453,2213}, {1576,4077}, {1616,2137}, {1617,2191}, {1631,7213}, {1769,2720}, {1790,1880}, {1795,1875}, {1813,6591}, {1919,4554}, {1922,10030}, {1974,7182}, {1980,4572}, {2003,2160}, {2006,7113}, {2099,2163}, {2114,9500}, {2150,6354}, {2159,6357}, {2176,7153}, {2194,3668}, {2207,7183}, {2210,7233}, {2212,7056}, {2218,4306}, {2221,2285}, {2260,2982}, {2263,3423}, {2275,7132}, {2279,5228}, {2291,6610}, {2306,2307}, {2310,7339}, {2324,6612}, {2334,3361}, {2353,7210}, {2362,6502}, {3121,4620}, {3212,7121}, {3248,4998}, {3271,7045}, {3451,3752}, {3709,4637}, {3733,4551}, {3777,8685}, {3900,6614}, {3911,9456}, {3937,7012}, {3942,7115}, {4252,5665}, {4296,8615}, {4557,7203}, {4626,8641}, {4822,5545}, {5018,8852}, {5546,7216}, {6129,8059}, {6180,9315}, {7011,7129}, {7013,7151}, {7051,7052}, {7054,7147}, {7084,7195}, {7104,7196}, {7117,7128}, {7122,7249}, {9309,9316}, {9363,9435}, {9364,9432}


X(9) = MITTENPUNKT

Trilinears    b + c - a : c + a - b : a + b - c
Trilinears    cot(A/2) : cot(B/2) : cot(C/2)
Trilinears     a - s : b - s : c - s
Trilinears     csc A + cot A : :
Trilinears     csc A (1 + cos A) : :
Trilinears     tan A' : : , where A'B'C' = excentral triangle
Trilinears     d(a,b,c) : : , where d(a,b,c) = distance from A to the Gergonne line
Trilinears    square of semi-minor axis of A-Soddy ellipse : :
Barycentrics   a(b + c - a) : b(c + a - b) : c(a + b - c)
Barycentrics   1 + cos A : 1 + cos B : 1 + cos C
Tripolars    Sqrt[-(b*c*(a^4 - 2*a^2*b^2 + b^4 - 4*b^3*c - 2*a^2*c^2 + 6*b^2*c^2 - 4*b*c^3 + c^4))] : :
X(9) = (r + 2R)*X(1) - 6R*X(2) -2r*X(3) = 3*X(2) - X(7)

X(9) is the symmedian point of the excentral triangle.

X(9) = perspector of ABC and the medial triangle of the extouch triangle of ABC. (Randy Hutson, 9/23/2011)

Let A' be the orthocorrespondent of the A-excenter, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(9). (Randy Hutson, November 18, 2015)

Let A'B'C' be the orthic triangle. Let La be the antiorthic axis of AB'C', and define Lb and Lc cyclically. Let A″ = Lb∩Lc. B″ = Lc∩La, C″ = La∩Lb. Triangle A″B″C″ is inversely similar to ABC, with similitude center X(9). (Randy Hutson, November 18, 2015)

Let E be the locus of the trilinear pole of a line that passes through X(1). The center of E is X(9). Moreover, E passes through the points X(100), X(658), X(662), X(799), X(1821), X(2580), X(2581) and the bicentric pairs PU(34), PU(75), PU(77), PU(79). Also, E is a circumellipse of ABC and an inellipse of the excentral triangle. (Randy Hutson, February 10, 2016)

The locus E also passes through the vertices of Gemini triangle 2. (Randy Hutson, November 30, 2018)

Let A' be the intersection of the tangents to the A-excircle at the intercepts with the circumcircle, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(9). (Randy Hutson, December 2, 2017)

Let A' be the perspector of the A-mixtilinear excircle, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(9). (Randy Hutson, December 2, 2017)

Let E be the circumellipse of T = ABC with center X(9). Then ABC is a billiard orbit of E (3-periodic). If we fix E in the plane, all its triangular orbits (a set of "rotating" triangles T) have the same X(9). Note that X(9) is the point of concurrence of lines drawn from each excenter to the midpoint of the corresponding side of T. (Dan Reznik, June 30, 2019) See [1] Triangular Orbits in Elliptic Billiards: the Mittenpunkt X(9) is stationary at the origin and [2] Triangular Orbits in Elliptic Billiards: the Mittenpunkt X(9) is stationary at the origin.

NEW in 2020: A particularly fine article is recommended: 'Can the Ellliptic Billiard Still Surprise Us?', by Dan Reznik, Ronaldo Garcia, and Jair Koiller, in Mathematical Intelligencer 42 (2020) 6-17. An online pdf is available: Click here.

Notes from Peter Moses (July 1, 2019) about the circumellipse E with center X(9):
E passes through the vertices of these triangles: ABC; Honsberger (see X(7670)); Inner Conway (see X(11677)); Gemini 2, Gemini 30.
E passes through the point X(i) for these i:
     88, 100, 162, 190, 651, 653, 655, 658, 660, 662, 673, 771, 799,
     823, 897, 1156, 1492, 1821, 2349, 2580, 2581, 3257, 4598, 4599
     4604, 4606, 4607, 8052, 20332, 23707, 24624, 27834, 32680.
The perspector of E is X(1); the major axis of E passes through X(i) for these i: 9, 2590, 3307, 24646, and the minor axis, for these i: 9, 2591, 3309, 24647. The ellipse E is the isogonal conjugate of the antiorthic axis.
Let g = length of semi-major axis of E; then g = Sqrt[(2 R (r + R + Sqrt[R(R - 2 r)]) s^2)/(r + 4 R)^2].
Let h = length of semi-minor axis of E: then h = Sqrt[(2 R (r + R - Sqrt[R(R - 2 r)]) s^2)/(r + 4 R)^2].
h/g = Sqrt[(OI - R) (OI + 3 R) / ((OI - 3 R) (OI + R))] = Sqrt[(r + R - Sqrt[R (R - 2 r)]) / (r + R + Sqrt[R (R - 2 r)])].
eccentricity of E: 2 Sqrt[OI R / ((3 R - OI) (OI + R))].
If F is a focus of E, then |FX(9)|^2 = 4 s^2 R Sqrt[R (R - 2 r)] / (r + 4 R)^2/.
The axes of E are the asymptotes of the Feuerbach hyperbola.
The area of E is π R S /(r (r + 4 R))^(3/2) = 4 π a b c / ( 2 b c + 2 c a + 2 a b - a^2 - b^2 - c^2 )^(3/2) * area(ABC).
The triangle tangent to the vertices of E is the excentral triangle.

Let A'B'C' be the intouch triangle of the anticomplementary triangle of ABC. The ellipse E passes through A', B', C'. See Three orbits in elliptic billiard. (Dan Reznik, July 1, 2019)

The circumellipse with center X(9) meets the circumhyperbola with center X(11) (i.e., the Feuerbach hyperbola) in X(1156). (Dan Reznik, January 20, 2020.)

Each of the following cubics passes through the four foci (two real and two imaginary) of the ellipse E: K040, K351, K352, K710, K716, K1060. The two real foci are a pair of isogonal conjugates, and likewise for the two imaginary foci. Moreover, if p : q : r is on the circumcircle, then p/a : q/b : r/c is on E. (Peter Moses, July 2 and 3, 2019)

The ellipse E is the isogonal conjugate of the antiorthic axis, X(44)X(513), with barycentric equation x + y + z = 0, and E is the isotomic conjugate of the X(514)X(661), with barycentric equation a x + b y + c z = 0. This line, denoted by L(31), is the perspectrix of the anticomplementary triangle and the inner Conway triangle (which is the intouch triangle of the anticomplementary triangle). Points lying on X(514)X(661) include X(i) for i = 514, 661, 693, 857, 908, 914, 1577, 1959, 2084, 2582, 2583, 3239, 3250, 3762, 3766, 3835, 3904, 3912, 3936, 3948, 4129, 4358, 4391, 4462, 4468, 4486, 4728, 4766, 4776, 4789, 4791, 4801, 4823, 4978, 5074, 5179, 6332, 6381, 6590, 8045, 14206, 14207, 14208, 14209, 14210, 14281, 14349, 14350, 14963, 18669, 18715, 24018, 30565, 30566, 30804, 30806, 32679. (Peter Moses, July 2 and 3, 2019)

For a dynamic graphic with many options, based on a triangle inscribed in the circumellipse with center X(9), see Elliptic Billiard: Loci of Triangular Centers. (Dan Reznik, February 4, 2020)

If you have The Geometer's Sketchpad, you can view Mittenpunkt.
If you have GeoGebra, you can view Mittenpunkt.

X(9) lies on the these conics: Feuerbach circumhyperbola, Feuerbach circumhyperbola of the medial triangle, Jerabek circumhyperbola of the excentral triangle, Mandart hyperbola

X(9) lies on these curves: K002, K132, K202, K207, K220, K251, K294, K332, K343, K345, K351, K352, K363, K384, K387, K453, K637, K696, K697, K710, K716, K717, K760, K761, K817, K880, K950, K970, K977, K980, K982, K984, K1025, K1038, K1044, K1055, K1059, K1060, K1077, K1079, K1080, K1081, K1082, K1083, K1084, K1090, Q151

X(9) lies on these lines: {1, 6}, {2, 7}, {3, 84}, {4, 10}, {5, 1729}, {8, 346}, {11, 3254}, {12, 5857}, {20, 10429}, {21, 41}, {22, 5314}, {25, 5285}, {30, 3587}, {31, 612}, {32, 987}, {33, 212}, {34, 201}, {35, 90}, {36, 2178}, {38, 614}, {39, 978}, {42, 941}, {43, 256}, {46, 79}, {48, 101}, {51, 3690}, {55, 200}, {56, 1696}, {58, 975}, {65, 4047}, {69, 344}, {75, 190}, {77, 651}, {80, 528}, {81, 5287}, {85, 10509}, {86, 2279}, {87, 292}, {92, 6358}, {99, 35106}, {100, 1005}, {105, 4712}, {108, 3213}, {109, 2199}, {114, 24469}, {119, 2950}, {123, 20623}, {124, 5513}, {140, 5843}, {141, 4422}, {145, 4029}, {164, 168}, {165, 910}, {171, 1707}, {172, 5019}, {173, 177}, {182, 7193}, {184, 26885}, {192, 239}, {193, 3879}, {194, 16827}, {205, 2359}, {216, 13006}, {222, 17811}, {223, 1073}, {228, 1011}, {241, 269}, {244, 17125}, {255, 25063}, {257, 17743}, {258, 7028}, {259, 15997}, {261, 645}, {264, 1948}, {273, 26003}, {275, 26941}, {291, 17065}, {294, 1253}, {306, 5739}, {312, 314}, {318, 1896}, {319, 17233}, {320, 17234}, {321, 1751}, {326, 6518}, {335, 17000}, {341, 4095}, {342, 653}, {345, 2339}, {348, 738}, {350, 17026}, {354, 4423}, {355, 31789}, {362, 2090}, {363, 5934}, {364, 366}, {374, 517}, {375, 22276}, {377, 9579}, {379, 18655}, {381, 18482}, {386, 10467}, {388, 12527}, {393, 1785}, {394, 2003}, {404, 4652}, {418, 26901}, {427, 21015}, {440, 1211}, {443, 4292}, {474, 3916}, {478, 1038}, {484, 5036}, {497, 4847}, {498, 920}, {499, 25522}, {511, 3781}, {513, 3126}, {514, 23760}, {515, 3427}, {519, 1000}, {521, 4130}, {522, 657}, {524, 4851}, {536, 4361}, {545, 7263}, {551, 18490}, {581, 3682}, {583, 3338}, {584, 15175}, {595, 28594}, {597, 17045}, {599, 17231}, {604, 1420}, {607, 1039}, {608, 1041}, {609, 1333}, {631, 6700}, {644, 1320}, {646, 4110}, {649, 4521}, {650, 15737}, {652, 3239}, {654, 1639}, {660, 3252}, {661, 35354}, {664, 31169}, {668, 17786}, {726, 16825}, {750, 896}, {758, 2294}, {759, 29127}, {765, 5377}, {813, 2726}, {857, 17052}, {869, 2309}, {899, 4414}, {900, 22108}, {905, 20318}, {912, 3211}, {919, 2751}, {934, 2371}, {937, 2336}, {938, 5129}, {940, 4641}, {942, 3927}, {943, 1802}, {946, 5758}, {948, 3668}, {952, 7966}, {964, 2198}, {970, 15877}, {976, 5037}, {980, 27623}, {981, 1918}, {982, 3290}, {983, 1914}, {986, 1722}, {989, 5255}, {990, 13329}, {991, 1818}, {995, 19246}, {1012, 6282}, {1014, 24557}, {1015, 21826}, {1016, 35957}, {1020, 6356}, {1021, 3700}, {1026, 9319}, {1030, 2932}, {1040, 5452}, {1046, 10381}, {1050, 2275}, {1054, 3038}, {1055, 18450}, {1062, 35194}, {1071, 8726}, {1084, 35095}, {1086, 4859}, {1088, 1223}, {1096, 7076}, {1098, 2150}, {1125, 1732}, {1150, 4358}, {1155, 4413}, {1158, 5811}, {1167, 7129}, {1174, 1621}, {1181, 2910}, {1193, 5105}, {1195, 5552}, {1200, 5281}, {1202, 10578}, {1210, 5084}, {1215, 24511}, {1220, 31359}, {1247, 9560}, {1249, 1712}, {1250, 7126}, {1251, 7127}, {1259, 11344}, {1266, 20073}, {1278, 16816}, {1282, 3041}, {1331, 2000}, {1332, 28978}, {1377, 1703}, {1378, 1702}, {1384, 37589}, {1385, 20818}, {1389, 1953}, {1392, 4861}, {1404, 30318}, {1405, 2171}, {1409, 37558}, {1418, 7271}, {1424, 25918}, {1435, 17917}, {1441, 25001}, {1443, 33633}, {1465, 25939}, {1471, 4327}, {1473, 7484}, {1474, 30733}, {1475, 3616}, {1479, 1752}, {1486, 12329}, {1500, 4263}, {1571, 1574}, {1572, 1573}, {1575, 3551}, {1592, 16028}, {1593, 26935}, {1598, 26938}, {1615, 10178}, {1630, 6261}, {1633, 24309}, {1635, 13266}, {1654, 2893}, {1655, 17033}, {1656, 37532}, {1678, 1705}, {1679, 1704}, {1680, 1701}, {1681, 1700}, {1695, 9565}, {1699, 2886}, {1720, 20224}, {1721, 9441}, {1730, 6708}, {1737, 24005}, {1738, 24248}, {1740, 2235}, {1742, 6184}, {1745, 3330}, {1755, 7413}, {1760, 5224}, {1764, 5737}, {1768, 3035}, {1776, 5218}, {1783, 2331}, {1788, 8582}, {1793, 2341}, {1824, 11323}, {1827, 28044}, {1829, 37318}, {1836, 3925}, {1837, 6598}, {1848, 21062}, {1863, 28120}, {1868, 4185}, {1898, 37601}, {1909, 34283}, {1921, 3403}, {1936, 9817}, {1937, 1945}, {1947, 15466}, {1958, 21511}, {1959, 15988}, {1966, 6376}, {1974, 26924}, {1992, 29574}, {2006, 26611}, {2013, 2018}, {2014, 2017}, {2049, 19859}, {2066, 7133}, {2078, 17615}, {2093, 3753}, {2099, 31165}, {2108, 20603}, {2111, 33701}, {2112, 4579}, {2124, 2125}, {2137, 24150}, {2173, 3647}, {2174, 2278}, {2175, 2330}, {2177, 21805}, {2209, 3728}, {2214, 28615}, {2220, 5301}, {2223, 16688}, {2225, 29828}, {2242, 5042}, {2252, 5553}, {2271, 37573}, {2272, 5658}, {2293, 2340}, {2295, 4274}, {2305, 5277}, {2308, 5311}, {2312, 4220}, {2317, 22356}, {2318, 2335}, {2319, 7081}, {2320, 2364}, {2326, 11107}, {2355, 37385}, {2432, 14298}, {2478, 6734}, {2503, 21381}, {2509, 21189}, {2568, 2573}, {2569, 2572}, {2590, 3307}, {2591, 3308}, {2629, 2634}, {2640, 2645}, {2648, 17963}, {2802, 4752}, {2887, 4703}, {2895, 32858}, {2947, 2954}, {2957, 24250}, {2959, 20666}, {2999, 3666}, {3008, 3663}, {3009, 7032}, {3013, 35069}, {3037, 5539}, {3056, 3271}, {3057, 3680}, {3058, 4863}, {3060, 26911}, {3063, 24307}, {3068, 5393}, {3069, 5405}, {3083, 15890}, {3084, 15889}, {3085, 21075}, {3175, 19723}, {3177, 9312}, {3182, 18641}, {3185, 10434}, {3187, 3995}, {3189, 4314}, {3197, 6001}, {3207, 7987}, {3212, 27288}, {3216, 4261}, {3244, 4098}, {3255, 5432}, {3257, 37131}, {3262, 28974}, {3287, 3709}, {3293, 4277}, {3295, 3991}, {3336, 5356}, {3337, 5043}, {3339, 3812}, {3341, 3344}, {3343, 3352}, {3349, 3351}, {3361, 5022}, {3416, 3932}, {3421, 31397}, {3434, 9580}, {3436, 9578}, {3467, 7301}, {3474, 26040}, {3476, 34716}, {3486, 6737}, {3503, 25994}, {3525, 26877}, {3526, 37612}, {3550, 4386}, {3560, 31837}, {3567, 26915}, {3579, 9709}, {3584, 17699}, {3588, 31330}, {3589, 4364}, {3596, 3975}, {3617, 5175}, {3618, 17023}, {3619, 29596}, {3620, 29579}, {3621, 12630}, {3622, 17474}, {3623, 4982}, {3625, 4072}, {3626, 4058}, {3629, 17390}, {3632, 3943}, {3633, 4898}, {3634, 5714}, {3644, 17160}, {3648, 10123}, {3652, 16005}, {3664, 4644}, {3671, 28629}, {3672, 3946}, {3673, 17681}, {3675, 20275}, {3676, 25924}, {3695, 5814}, {3696, 5695}, {3697, 5687}, {3698, 37567}, {3699, 36798}, {3703, 3966}, {3706, 4042}, {3708, 25095}, {3712, 4023}, {3720, 32912}, {3726, 29820}, {3732, 21232}, {3735, 9620}, {3738, 14427}, {3739, 4363}, {3741, 4011}, {3742, 8167}, {3746, 4006}, {3752, 23511}, {3757, 21101}, {3759, 4360}, {3760, 29433}, {3761, 3770}, {3763, 17237}, {3764, 21035}, {3765, 3963}, {3772, 4415}, {3779, 20683}, {3782, 23681}, {3784, 3819}, {3814, 5535}, {3817, 8166}, {3834, 7232}, {3836, 4655}, {3840, 20785}, {3842, 4672}, {3846, 4438}, {3868, 3951}, {3871, 32635}, {3873, 4666}, {3874, 20116}, {3880, 4900}, {3888, 25279}, {3890, 36846}, {3893, 31509}, {3897, 17440}, {3899, 17443}, {3900, 23351}, {3913, 4515}, {3917, 26892}, {3920, 17127}, {3937, 5650}, {3940, 16418}, {3941, 20990}, {3942, 25097}, {3944, 17064}, {3945, 4667}, {3952, 26227}, {3955, 9306}, {3957, 4661}, {3971, 4362}, {3974, 4082}, {3977, 17740}, {3983, 33576}, {3984, 16865}, {3989, 17017}, {3997, 30116}, {3998, 16368}, {4005, 37080}, {4015, 8715}, {4020, 25079}, {4030, 4126}, {4033, 29712}, {4063, 6008}, {4067, 12559}, {4070, 5233}, {4071, 4388}, {4077, 26017}, {4078, 5847}, {4086, 4529}, {4090, 29670}, {4092, 23902}, {4111, 4433}, {4119, 4514}, {4123, 9447}, {4124, 30224}, {4148, 4768}, {4153, 30172}, {4154, 15628}, {4171, 35057}, {4180, 4182}, {4189, 4855}, {4191, 22060}, {4255, 8951}, {4260, 16850}, {4268, 7113}, {4269, 25516}, {4272, 5312}, {4287, 37616}, {4295, 19855}, {4297, 10864}, {4301, 6766}, {4304, 11111}, {4310, 16020}, {4313, 11106}, {4315, 34610}, {4328, 5228}, {4329, 5813}, {4333, 7700}, {4336, 28125}, {4346, 17067}, {4359, 25734}, {4389, 16706}, {4392, 7292}, {4393, 4704}, {4402, 4452}, {4417, 33116}, {4418, 26037}, {4421, 31508}, {4425, 25453}, {4429, 24723}, {4430, 29817}, {4432, 32941}, {4435, 4526}, {4440, 29628}, {4441, 24592}, {4445, 4690}, {4454, 4488}, {4461, 32087}, {4466, 31261}, {4480, 24199}, {4534, 12641}, {4552, 25243}, {4554, 30988}, {4557, 8053}, {4559, 24806}, {4568, 30108}, {4650, 16570}, {4651, 32929}, {4665, 28634}, {4668, 7300}, {4670, 4698}, {4675, 4888}, {4676, 5263}, {4677, 4908}, {4681, 4852}, {4683, 25957}, {4686, 17119}, {4688, 17118}, {4699, 16815}, {4708, 17327}, {4711, 8168}, {4713, 21264}, {4715, 17313}, {4731, 5183}, {4741, 17232}, {4748, 29604}, {4755, 28639}, {4759, 36480}, {4798, 6707}, {4869, 21296}, {4872, 24694}, {4911, 33838}, {4929, 9053}, {4967, 26998}, {4974, 32921}, {4981, 24552}, {4997, 30608}, {5020, 37581}, {5021, 37607}, {5024, 37599}, {5046, 21029}, {5057, 30311}, {5082, 10624}, {5087, 5536}, {5088, 27472}, {5110, 5529}, {5122, 16417}, {5124, 7280}, {5126, 35272}, {5128, 5177}, {5153, 5313}, {5217, 7285}, {5232, 7291}, {5252, 34606}, {5256, 16579}, {5266, 30435}, {5267, 22054}, {5274, 24386}, {5286, 13161}, {5290, 25466}, {5297, 9330}, {5307, 22001}, {5320, 37316}, {5423, 7172}, {5424, 5426}, {5433, 24954}, {5439, 16842}, {5440, 16370}, {5530, 31402}, {5534, 10267}, {5550, 30340}, {5551, 19862}, {5555, 24982}, {5560, 7297}, {5575, 25891}, {5584, 12565}, {5586, 28646}, {5651, 26884}, {5691, 5794}, {5693, 19350}, {5703, 17558}, {5708, 16853}, {5726, 11236}, {5736, 28627}, {5741, 33113}, {5743, 19542}, {5790, 18499}, {5836, 7991}, {5854, 8275}, {5881, 6936}, {5886, 20330}, {5887, 7971}, {5903, 17098}, {6009, 21385}, {6048, 21857}, {6056, 11429}, {6181, 17601}, {6223, 37108}, {6245, 6865}, {6259, 37424}, {6350, 30675}, {6377, 16576}, {6505, 16585}, {6506, 8068}, {6536, 29647}, {6542, 17242}, {6586, 21173}, {6626, 18784}, {6675, 11374}, {6687, 17235}, {6690, 8255}, {6701, 13159}, {6705, 6926}, {6706, 30494}, {6726, 7014}, {6735, 30513}, {6769, 11496}, {6832, 8227}, {6843, 10175}, {6857, 13411}, {6905, 21165}, {6910, 27385}, {6918, 37623}, {6939, 7682}, {6976, 12703}, {6986, 10884}, {6990, 24045}, {6996, 10444}, {7003, 7008}, {7004, 25096}, {7066, 19366}, {7098, 10588}, {7146, 18726}, {7176, 27340}, {7177, 10004}, {7183, 17095}, {7190, 24554}, {7191, 7226}, {7222, 31211}, {7229, 24590}, {7244, 18068}, {7273, 8898}, {7277, 17392}, {7282, 37448}, {7283, 9534}, {7293, 7485}, {7377, 24702}, {7489, 37533}, {7595, 8231}, {7670, 11691}, {7673, 14923}, {7678, 11680}, {7679, 11681}, {7736, 24239}, {7741, 37359}, {7957, 12651}, {7963, 8572}, {7992, 9943}, {8056, 16602}, {8069, 8573}, {8125, 8388}, {8126, 8389}, {8169, 9814}, {8237, 11687}, {8238, 11688}, {8245, 8424}, {8273, 12680}, {8385, 11685}, {8386, 11686}, {8387, 11690}, {8632, 14408}, {8666, 15179}, {8680, 18698}, {8730, 15348}, {8750, 23050}, {8771, 21508}, {8822, 16054}, {8835, 15856}, {8915, 35666}, {8941, 31459}, {9028, 26130}, {9342, 9352}, {9351, 34543}, {9365, 14936}, {9367, 11512}, {9470, 9499}, {9576, 9640}, {9577, 9639}, {9582, 9679}, {9583, 9678}, {9584, 9689}, {9585, 9688}, {9586, 9702}, {9587, 9701}, {9588, 9711}, {9589, 9710}, {9590, 9713}, {9591, 9712}, {9592, 31449}, {9599, 29676}, {9605, 37592}, {9614, 24390}, {9616, 30354}, {9619, 31456}, {9624, 31458}, {9785, 21627}, {9799, 37423}, {9843, 17559}, {9846, 12125}, {9955, 31493}, {9957, 12629}, {10012, 31269}, {10039, 21074}, {10050, 10058}, {10157, 19541}, {10167, 10857}, {10198, 21077}, {10246, 22147}, {10266, 12639}, {10268, 11500}, {10383, 10391}, {10387, 19589}, {10388, 17658}, {10446, 24705}, {10449, 21071}, {10455, 27164}, {10461, 11110}, {10476, 15825}, {10479, 34831}, {10638, 19551}, {10645, 11791}, {10646, 11790}, {10708, 34925}, {10855, 16411}, {10856, 15509}, {10865, 11678}, {10882, 23361}, {10902, 17857}, {11008, 29601}, {11019, 24477}, {11036, 17554}, {11194, 13462}, {11248, 11434}, {11343, 25083}, {11375, 24953}, {11433, 26872}, {11519, 30337}, {11520, 16859}, {11526, 11682}, {11604, 21014}, {11683, 16609}, {11684, 16133}, {12047, 19854}, {12389, 12399}, {12396, 12397}, {12435, 22299}, {12436, 17582}, {12511, 31871}, {12519, 13089}, {12520, 31803}, {12529, 12706}, {12530, 12718}, {12531, 12730}, {12532, 12755}, {12533, 12846}, {12534, 12847}, {12535, 12850}, {12575, 15998}, {12650, 31786}, {12659, 12693}, {12675, 22153}, {12699, 31419}, {12782, 24478}, {13143, 13144}, {13205, 36868}, {13388, 15891}, {13389, 15892}, {13405, 21060}, {13426, 13427}, {13442, 29181}, {13454, 13456}, {13567, 26942}, {14021, 18650}, {14151, 17439}, {14224, 14400}, {14319, 14321}, {14497, 16200}, {14543, 27039}, {14552, 34255}, {14621, 31323}, {14740, 34894}, {14829, 18743}, {14963, 22073}, {14996, 17021}, {14997, 17012}, {15066, 22128}, {15487, 26034}, {15496, 22080}, {15507, 31394}, {15669, 24315}, {15934, 16857}, {15935, 36867}, {16058, 20760}, {16286, 22458}, {16345, 35612}, {16367, 20769}, {16374, 23206}, {16408, 37582}, {16555, 21366}, {16565, 27688}, {16568, 31144}, {16575, 16592}, {16605, 24440}, {16608, 25964}, {16610, 17595}, {16704, 31035}, {16713, 17183}, {16726, 18186}, {16732, 17885}, {16738, 27261}, {16822, 17760}, {16823, 20459}, {16826, 17120}, {16863, 37545}, {16887, 30110}, {17002, 26247}, {17046, 17671}, {17050, 17753}, {17063, 18193}, {17080, 36636}, {17107, 24797}, {17113, 36888}, {17134, 36023}, {17137, 29966}, {17152, 30036}, {17156, 32864}, {17175, 25508}, {17227, 17273}, {17228, 17271}, {17230, 17268}, {17238, 17252}, {17239, 17251}, {17240, 17295}, {17241, 17297}, {17244, 17300}, {17246, 17301}, {17247, 17302}, {17249, 17305}, {17250, 17307}, {17309, 17372}, {17310, 17373}, {17311, 17374}, {17312, 17375}, {17315, 17377}, {17317, 17378}, {17320, 17380}, {17322, 17381}, {17323, 17382}, {17324, 17383}, {17325, 17384}, {17391, 20090}, {17394, 29597}, {17438, 37518}, {17444, 21398}, {17499, 26110}, {17542, 24473}, {17550, 24995}, {17605, 31245}, {17614, 37519}, {17616, 37309}, {17625, 25893}, {17687, 25500}, {17691, 25242}, {17718, 34917}, {17720, 35466}, {17728, 31249}, {17737, 24892}, {17757, 31434}, {17792, 18788}, {17793, 30546}, {17889, 33099}, {18040, 29396}, {18044, 18133}, {18046, 29561}, {18134, 33066}, {18139, 32859}, {18398, 25542}, {18483, 31418}, {18589, 24316}, {18596, 34823}, {18725, 34371}, {18755, 37574}, {18758, 23863}, {19261, 23169}, {19372, 37591}, {19523, 37522}, {19555, 31090}, {19582, 23640}, {19604, 25731}, {19785, 26723}, {19804, 32939}, {19872, 32632}, {20080, 29583}, {20092, 24184}, {20106, 26934}, {20174, 29773}, {20205, 21370}, {20247, 25261}, {20248, 26653}, {20257, 27304}, {20305, 34176}, {20370, 34832}, {20430, 37510}, {20444, 35550}, {20456, 22172}, {20486, 25613}, {20544, 24329}, {20608, 21250}, {20662, 21320}, {20667, 26069}, {20678, 23868}, {20719, 31785}, {20930, 28980}, {20973, 21858}, {20979, 21389}, {20980, 21348}, {20984, 22174}, {21010, 36635}, {21026, 31134}, {21066, 26793}, {21090, 24298}, {21281, 30030}, {21286, 26581}, {21319, 37319}, {21367, 32782}, {21368, 24611}, {21379, 33681}, {21405, 27390}, {21514, 37597}, {21759, 23566}, {21827, 23543}, {21832, 24121}, {21956, 32865}, {22003, 24435}, {22076, 37324}, {22758, 37611}, {23062, 23618}, {23073, 30389}, {23354, 25292}, {23529, 28118}, {23649, 28352}, {24067, 27368}, {24154, 24156}, {24155, 24157}, {24158, 24242}, {24172, 28090}, {24210, 33137}, {24268, 25252}, {24325, 32935}, {24343, 25107}, {24346, 36008}, {24398, 27918}, {24411, 34361}, {24512, 26102}, {24542, 33122}, {24547, 24612}, {24549, 33821}, {24586, 30758}, {24603, 28827}, {24633, 24993}, {24690, 30822}, {24935, 25669}, {25010, 30312}, {25057, 31171}, {25343, 30779}, {25457, 25682}, {25570, 25571}, {25660, 29456}, {25690, 25693}, {25716, 36628}, {25842, 25856}, {25850, 25858}, {25875, 34489}, {25885, 34036}, {25958, 29873}, {25960, 33119}, {25961, 33067}, {26006, 26668}, {26035, 31339}, {26068, 27020}, {26107, 26959}, {26592, 34388}, {26658, 34497}, {26724, 33146}, {26730, 26731}, {26756, 27073}, {26919, 26940}, {26933, 30739}, {27036, 27102}, {27044, 27136}, {27108, 27514}, {27253, 36854}, {27398, 37265}, {27507, 28789}, {28365, 37596}, {28604, 29576}, {29085, 36661}, {29381, 29536}, {29388, 29504}, {29395, 29423}, {29431, 29514}, {29570, 31313}, {29616, 32099}, {29632, 33065}, {29642, 33064}, {29643, 32843}, {29653, 32946}, {29664, 33107}, {29667, 33166}, {29674, 33082}, {29679, 33083}, {29681, 33153}, {29687, 33080}, {29711, 29716}, {29767, 30939}, {29826, 32944}, {29850, 32776}, {29851, 33069}, {29854, 32949}, {29855, 32775}, {29856, 34997}, {30295, 35990}, {30416, 30429}, {30417, 30430}, {30676, 30701}, {30695, 31994}, {30942, 34589}, {31316, 31343}, {31408, 31533}, {31534, 34495}, {31535, 34494}, {31547, 31565}, {31548, 31566}, {31937, 35239}, {32577, 33589}, {32771, 32938}, {32773, 33118}, {32778, 33164}, {32849, 33077}, {32860, 32936}, {32861, 33092}, {32862, 33075}, {32914, 32925}, {32947, 33117}, {33084, 33158}, {33096, 33111}, {33100, 33131}, {33101, 33130}, {33129, 33151}, {33132, 33154}, {33134, 33139}, {33700, 36906}, {33863, 37608}, {33934, 33951}, {34256, 34293}, {34926, 34932}, {35091, 35113}, {35128, 35129}, {35614, 35617}, {35659, 35668}, {36531, 36554}, {37234, 37585}, {37248, 37583}, {37507, 37609}

X(9) lies on the following circumconics: Feuerbach circumhyperbola, Feuerbach circumhyperbola of the medial triangle, Jerabek circumhyperbola of the excentral triangle, Mandart hyperbola, and these:
{{A,B,C,2,9,200,281,282,346,2184,2287,2297,4183,6605,7097,7110,14943,15889,15890,15891,15892,16016,19605,21446,23617,28071,28132,30705,31618,34525,36101,36627,36629,36796,36910,36916}}
{{A,B,C,3,9,40,271,1167,1819,3342,3347,7013,7078,34902}}
{{A,B,C,9,10,72,78,307,318,1793,3694,3710,3718,8806}}
{{A,B,C,9,33,37,210,226,312,1826,1903,2250,2321,2341,8818,13455,25430,27475,35144,35354,36800}}
{{A,B,C,9,41,213,1334,1400,2212,2279,2311,2333,18784,18785,35106}}
{{A,B,C,9,63,219,268,1073,1260,1815,2322,2327,2328,2983,3692,7123,15629,36631}}
{{A,B,C,9,75,518,522,765,1861,2751,3693,3717,9436,33676,36819}}
{{A,B,C,9,87,238,242,261,1447,2726,3684,3685,3737,4076,7220,9282,9499,18786,36815}}

X(9) lies on the inellipse through X(i) for i = 9,57,604,2171,2289,3083,3084,6602,34544; the perspector of this ellipse is X(4564).

X(9) lies on the following cubics: K002, K132, K202, K207, K220, K251, K294, K332, K343, K345, K351, K352, K363, K384, K387, K453, K637, K696, K697, K710, K716, K717, K760, K761, K817, K880, K950, K970, K977, K980, K982, K984, K1025, K1038, K1044, K1055, K1059, K1060, K1077, K1079, K1080, K1081, K1082, K1083, K1084, K1090

X(9) = midpoint of X(i) and X(j) for these (i,j): midpoint of X(i) and X(j) for these {i,j}: {1, 5223}, {2, 6172}, {3, 5779}, {4, 5759}, {7, 144}, {8, 390}, {11, 6068}, {20, 36991}, {40, 11372}, {57, 36973}, {63, 8545}, {72, 5728}, {100, 1156}, {190, 673}, {329, 12848}, {346, 5838}, {651, 36101}, {1001, 5220}, {2262, 21871}, {2294, 3958}, {2550, 5698}, {2951, 3062}, {3059, 14100}, {3257, 37131}, {3434, 36976}, {3587, 18540}, {3621, 12630}, {3681, 7671}, {3869, 7672}, {4361, 17262}, {4915, 9819}, {5817, 21168}, {5839, 17314}, {7670, 11691}, {7673, 14923}, {9846, 12125}, {11495, 16112}, {11684, 16133}, {12389, 12399}, {12526, 12560}, {12527, 12573}, {12528, 12669}, {12529, 12706}, {12530, 12718}, {12531, 12730}, {12532, 12755}, {12533, 12846}, {12534, 12847}, {12535, 12850}, {13359, 13360}, {15254, 15481}, {30628, 34784}, {31547, 31565}, {31548, 31566}, {34926, 34932}, {35659, 35668}

X(9) = reflection of X(i) in X(j) for these {i,j}: {1, 1001}, {3, 31658}, {7, 142}, {8, 24393}, {57, 8257}, {84, 3358}, {100, 6594}, {101, 28345}, {142, 6666}, {1001, 15254}, {2294, 25081}, {2550, 10}, {2951, 11495}, {3062, 16112}, {3174, 6600}, {3243, 1}, {3254, 11}, {3874, 20116}, {4312, 5880}, {4361, 17348}, {4851, 17243}, {5220, 15481}, {5223, 5220}, {5528, 100}, {5542, 1125}, {5732, 3}, {5735, 5805}, {5805, 5}, {5833, 5791}, {5880, 3826}, {6173, 2}, {6601, 24389}, {8255, 6690}, {9623, 9708}, {10427, 3035}, {13159, 6701}, {14943, 35508}, {15185, 5572}, {15298, 15296}, {15299, 15297}, {16593, 4422}, {17151, 4361}, {17314, 3950}, {17668, 15587}, {18443, 6883}, {20195, 18230}, {31657, 140}, {31671, 18482}, {36867, 15935}

X(9) = isogonal conjugate of X(57)
X(9) = isotomic conjugate of X(85)
X(9) = complement of X(7)
X(9) = anticomplement of X(142)
X(9) = polar conjugate of X(273)
X(9) = antigonal image of X(3254)
X(9) = antitomic image of X(14943)
X(9) = symgonal image of X(6594)
X(9) = circumcircle-inverse of X(32625)
X(9) = Spieker-circle-inverse of X(5199)
X(9) = Bevan-circle-inverse of X(5011)
X(9) = Stevanovic-circle inverse of X(15737)
X(9) = Thomson-isogonal conjugate of X(3576)
X(9) = medial-isogonal conjugate of X(2886)
X(9) = anticomplementary-isogonal conjugate of X(2890)
X(9) = excentral-isogonal conjugate of X(165)
X(9) = tangential-isogonal conjugate of X(2921)
X(9) = orthic-isogonal conjugate of X(2900)

X(9) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 3158}, {2, 1}, {4, 2900}, {6, 3169}, {7, 3174}, {8, 200}, {21, 55}, {27, 3189}, {29, 3190}, {31, 32468}, {41, 7075}, {55, 3208}, {57, 2136}, {63, 40}, {78, 7070}, {81, 3913}, {85, 3870}, {88, 3880}, {92, 3811}, {100, 3900}, {105, 19589}, {144, 2951}, {189, 6765}, {190, 522}, {241, 9451}, {257, 3961}, {294, 3684}, {312, 78}, {318, 33}, {329, 1490}, {333, 8}, {346, 2324}, {348, 8270}, {527, 5528}, {643, 4041}, {644, 650}, {645, 3737}, {650, 4919}, {651, 521}, {653, 8058}, {655, 2804}, {660, 926}, {664, 4105}, {672, 24578}, {673, 5853}, {765, 3939}, {799, 3907}, {894, 1045}, {897, 24394}, {908, 6326}, {1121, 3935}, {1156, 15733}, {1220, 42}, {1223, 142}, {1252, 1018}, {1261, 4513}, {1320, 3689}, {1751, 12625}, {1791, 5285}, {1821, 740}, {2053, 2319}, {2167, 8715}, {2184, 11523}, {2185, 3871}, {2287, 219}, {2297, 1449}, {2319, 4050}, {2322, 281}, {2339, 1697}, {2346, 6600}, {2349, 758}, {2975, 15621}, {3218, 5541}, {3219, 191}, {3257, 3738}, {3305, 3646}, {3596, 4149}, {3699, 663}, {3903, 4477}, {4076, 4069}, {4102, 4420}, {4564, 100}, {4997, 4511}, {5279, 18598}, {5546, 1021}, {6558, 4521}, {6605, 220}, {7123, 3501}, {7131, 57}, {8056, 3680}, {9776, 12658}, {10509, 7674}, {14942, 2340}, {15889, 30557}, {15890, 30556}, {17484, 13146}, {17743, 43}, {21446, 3243}, {23617, 6}, {23618, 7}, {27065, 5506}, {27834, 513}, {28659, 4123}, {30608, 3872}, {30705, 8271}, {30711, 4882}, {31343, 4162}, {31359, 612}, {32008, 2}, {32015, 3957}, {32635, 210}

X(9) = cevapoint of X(i) and X(j) for these (i,j): (1,165), (6,198), (31,205), (37,71), (41,212), (55,220). (2066.5414)
X(9) = X(i)-cross conjugate of X(j) for these (i,j): {1, 19605}, {6, 282}, {8, 2319}, {37, 281}, {41, 33}, {42, 10570}, {55, 1}, {57, 2125}, {71, 219}, {198, 2324}, {200, 3680}, {210, 8}, {212, 78}, {220, 200}, {518, 14943}, {650, 644}, {652, 101}, {654, 5548}, {657, 3939}, {663, 3699}, {672, 2338}, {1146, 1021}, {1212, 2}, {1334, 55}, {1400, 30457}, {1864, 4}, {1903, 8805}, {2082, 57}, {2170, 650}, {2183, 15629}, {2238, 15628}, {2245, 15627}, {2264, 1172}, {2269, 284}, {2310, 522}, {2340, 14942}, {2347, 6}, {2348, 294}, {3059, 6601}, {3119, 3900}, {3271, 3737}, {3287, 645}, {3683, 21}, {3689, 1320}, {3691, 333}, {3693, 4876}, {3700, 1018}, {3709, 4069}, {3711, 4900}, {3715, 4866}, {3900, 100}, {4041, 643}, {4105, 664}, {4162, 31343}, {4266, 2364}, {4326, 10390}, {4477, 3903}, {4517, 7220}, {4936, 3158}, {7008, 3347}, {7069, 318}, {7082, 90}, {7156, 7070}, {8012, 220}, {9404, 5546}, {10382, 5665}, {11429, 3469}, {11436, 3362}, {14100, 7}, {14298, 1783}, {14547, 29}, {15733, 3254}, {15837, 2346}, {18235, 2329}, {20665, 9439}, {21033, 2321}, {21811, 37}, {23544, 893}, {27538, 3208}, {28070, 728}, {30223, 84}, {30456, 27382}, {33299, 312}

X(9) = crosspoint of X(i) and X(j) for these (i,j): {1, 8056}, {2, 8}, {21, 333}, {55, 2053}, {57, 2137}, {63, 271}, {100, 4564}, {188, 7028}, {190, 765}, {236, 24158}, {258, 24242}, {312, 318}, {645, 4076}, {651, 7012}, {1016, 8706}, {1252, 5546}, {1275, 6606}, {2287, 2322}, {3161, 24150}, {6605, 32008}, {24154, 24155}

X(9) = crosssum of X(i) and X(j) for these (i,j): {1, 1743}, {2, 3210}, {3, 3211}, {6, 56}, {7, 3212}, {9, 2136}, {19, 208}, {25, 21058}, {34, 3213}, {37, 3214}, {41, 21059}, {44, 17460}, {48, 3215}, {63, 8897}, {65, 1400}, {173, 8078}, {244, 649}, {269, 17106}, {294, 9453}, {513, 2170}, {514, 24237}, {518, 19593}, {603, 604}, {650, 7004}, {661, 2611}, {798, 4128}, {1015, 6363}, {1086, 7178}, {1357, 7180}, {1407, 6611}, {1418, 1475}, {1575, 20366}, {2082, 28017}, {2446, 2590}, {2447, 2591}, {2488, 14936}, {4000, 28110}, {4466, 21124}, {10490, 18888}

X(9) = crossdifference of every pair of points on line X(513)X(663)
X(9) = X(i)-Hirst inverse of X(j) for these (i,j): {1, 518}, {2, 10025}, {8, 3685}, {43, 8844}, {55, 3684}, {57, 6168}, {192, 239}, {200, 3693}, {294, 28071}, {1282, 8299}, {1575, 16557}, {2170, 4919}, {2348, 3158}, {3307, 24646}, {3308, 24647}, {4876, 7077}, {5239, 5240}, {17792, 18788}
X(9) = X(i)-aleph conjugate of X(j) for these (i,j): {1, 43}, {2, 9}, {4, 1711}, {7, 16572}, {8, 10860}, {9, 170}, {75, 1759}, {76, 21366}, {99, 21383}, {174, 1743}, {188, 165}, {190, 1018}, {259, 32462}, {291, 18787}, {365, 1740}, {366, 1}, {507, 361}, {508, 57}, {509, 978}, {522, 2958}, {556, 1766}, {4146, 169}, {4182, 2951}, {5374, 1745}, {6728, 1053}, {7025, 503}, {14087, 1018}, {14089, 21383}, {18297, 63}, {20034, 1716}
X(9) = X(i)-beth conjugate of X(j) for these (i,j):
(9,6), (190,6), (346,346), (644,9), (645,75)
X(9) = X(1)-line conjugate of X(1279)
X(9) = X(i)-vertex conjugate of X(j) for these (i,j): {1, 3420}, {9, 1436}, {269, 1037}, {3900, 32625}
X(9) = perspector of ABC and extraversion triangle of X(57)
X(9) = X(6)-of-excentral-triangle
X(9) = X(159)-of-intouch-triangle
X(9) = X(6)-of-2nd-extouch-triangle
X(9) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 2886}, {2, 17046}, {3, 34822}, {6, 142}, {8, 2887}, {9, 141}, {19, 16608}, {21, 3741}, {25, 1210}, {31, 1}, {32, 3752}, {33, 5}, {37, 17052}, {40, 20307}, {41, 2}, {42, 442}, {43, 20338}, {48, 17073}, {55, 10}, {56, 11019}, {57, 21258}, {58, 3742}, {63, 18639}, {71, 18642}, {75, 17047}, {78, 1368}, {81, 17050}, {82, 17049}, {100, 17072}, {101, 4885}, {109, 3900}, {163, 17069}, {184, 17102}, {192, 20547}, {198, 20206}, {200, 1329}, {210, 3454}, {212, 3}, {213, 17056}, {219, 18589}, {220, 3452}, {228, 18641}, {251, 17048}, {281, 20305}, {282, 21239}, {284, 3739}, {294, 20335}, {312, 626}, {318, 21243}, {333, 21240}, {346, 21244}, {512, 8286}, {513, 17059}, {522, 21252}, {560, 17053}, {604, 4000}, {607, 226}, {643, 512}, {644, 3835}, {646, 21262}, {649, 4904}, {650, 116}, {657, 26932}, {662, 17066}, {663, 11}, {667, 3756}, {669, 16613}, {672, 17060}, {692, 522}, {756, 34829}, {798, 17058}, {893, 17062}, {902, 1145}, {904, 24239}, {923, 17070}, {983, 17792}, {1106, 5573}, {1110, 3035}, {1172, 34830}, {1174, 6706}, {1252, 21232}, {1253, 9}, {1260, 34823}, {1320, 21241}, {1333, 3946}, {1334, 1211}, {1395, 17054}, {1400, 18635}, {1402, 1834}, {1409, 18643}, {1415, 7658}, {1820, 18638}, {1911, 1738}, {1918, 2092}, {1946, 2968}, {1964, 17055}, {1973, 3772}, {1974, 20227}, {1980, 16614}, {2053, 3840}, {2148, 17043}, {2149, 17044}, {2150, 17045}, {2155, 18634}, {2156, 18636}, {2157, 18637}, {2158, 18640}, {2159, 18644}, {2162, 20257}, {2163, 17051}, {2172, 17068}, {2175, 37}, {2176, 20528}, {2177, 17057}, {2187, 7952}, {2192, 946}, {2194, 1125}, {2195, 518}, {2200, 18592}, {2208, 3086}, {2212, 6}, {2258, 25466}, {2268, 10472}, {2287, 21246}, {2289, 6389}, {2299, 942}, {2316, 3834}, {2318, 21530}, {2319, 20255}, {2320, 21242}, {2321, 21245}, {2328, 960}, {2332, 6708}, {2340, 120}, {2342, 517}, {2344, 21264}, {2361, 214}, {2364, 34824}, {3052, 12640}, {3063, 1086}, {3158, 2885}, {3195, 20264}, {3208, 21250}, {3433, 24388}, {3445, 24386}, {3596, 21235}, {3684, 20333}, {3685, 20542}, {3688, 21249}, {3689, 121}, {3693, 20540}, {3699, 21260}, {3700, 21253}, {3709, 8287}, {3711, 21251}, {3724, 6739}, {3900, 124}, {3939, 513}, {4041, 125}, {4069, 31946}, {4105, 5514}, {4162, 5510}, {4166, 20334}, {4182, 20543}, {4183, 34831}, {4548, 16582}, {4814, 15614}, {4845, 5087}, {4876, 20541}, {4895, 3259}, {5546, 4369}, {5547, 4892}, {5548, 4928}, {6059, 24005}, {6065, 24003}, {6066, 24036}, {6187, 1737}, {6602, 6554}, {6603, 31844}, {6614, 17427}, {7037, 6245}, {7054, 21233}, {7069, 1209}, {7070, 2883}, {7071, 20262}, {7072, 21616}, {7073, 25639}, {7074, 6260}, {7077, 3836}, {7084, 1376}, {7104, 28358}, {7110, 21236}, {7118, 57}, {7121, 17063}, {7139, 20268}, {7156, 20207}, {7252, 17761}, {7257, 23301}, {7339, 24009}, {7367, 20205}, {8611, 127}, {8641, 1146}, {8750, 521}, {8851, 20340}, {9439, 3816}, {9447, 39}, {9448, 16584}, {9456, 17067}, {10482, 3740}, {13455, 639}, {14827, 1212}, {14942, 20544}, {15374, 21629}, {18265, 1575}, {18757, 33135}, {18889, 527}, {19624, 6594}, {21059, 6600}, {21789, 34589}, {23990, 16578}, {32652, 8058}, {32666, 676}, {32739, 905}, {33299, 21248}, {33635, 17239}, {34067, 25380}, {34248, 17065}, {34446, 31397}, {36086, 926}, {36797, 21259}, {36799, 20549}, {36910, 21237}

X(9) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 2890}, {1170, 3434}, {1174, 8}, {2346, 69}, {6605, 3436}, {10482, 329}, {21453, 21285}, {31618, 21280}, {32008, 6327}
X(9) = Thomson-isogonal conjugate of X(3576)
X(9) = medial-isogonal conjugate of X(2886)
X(9) = anticomplementary-isogonal conjugate of X(2890)
X(9) = excentral-isogonal conjugate of X(165)
X(9) = tangential-isogonal conjugate of X(2921)
X(9) = orthic-isogonal conjugate of X(2900)
X(9) = trilinear square root of X(200)
X(9) = trilinear product of extraversions of X(57)
X(9) = trilinear product of PU(112)
X(9) = inverse-in-circumconic-centered-at-X(1) of X(6603)
X(9) = orthocenter of X(1)X(4)X(885)
X(9) = bicentric sum of PU(56)
X(9) = midpoint of PU(56)
X(9) = crossdifference of PU(96)
X(9) = perspector of circumconic centered at X(1)
X(9) = the point in which the extended legs P(6)P(33) and U(6)U(33) of the trapezoid PU(6)PU(33) meet
X(9) = trilinear pole of line X(650)X(663)
X(9) = pole wrt polar circle of trilinear polar of X(273) (line X(514)X(3064))
X(9) = X(48)-isoconjugate (polar conjugate) of X(273)
X(9) = perspector of ABC and unary cofactor triangle of 1st mixtilinear triangle
X(9) = perspector of ABC and unary cofactor triangle of 3rd mixtilinear triangle
X(9) = homothetic center of excentral triangle and 2nd extouch triangle
X(9) = perspector of ABC and medial triangle of extouch triangle
X(9) = perspector of pedal and anticevian triangles of X(1490)
X(9) = SS(A→A') of X(19), where A'B'C' is the excentral triangle
X(9) = homothetic center of medial triangle and tangential triangle of excentral triangle
X(9) = homothetic center of excentral triangle and complement of the intouch triangle
X(9) = Cundy-Parry Phi transform of X(84)
X(9) = Cundy-Parry Psi transform of X(40)
X(9) = trilinear product of circumcircle intercepts of excircles radical circle
X(9) = perspector of Gemini triangle 4 and unary cofactor triangle of Gemini triangle 3
X(9) = eigencenter of Gemini triangle 5
X(9) = perspector of Gemini triangle 35 and cross-triangle of ABC and Gemini triangle 35
X(9) = perspector of ABC and unary cofactor triangle of Gemini triangle 35
X(9) = perspector of ABC and unary cofactor triangle of Gemini triangle 40
X(9) = internal center of similitude of the Bevan circle and Spieker circle; the external center is X(1706)
X(9) = excentral-to-ABC barycentric image of X(9)
X(9) = intouch-to-ABC barycentric image of X(7)
X(9) = intouch-to-excentral similarity image of X(7)
X(9) = ABC-to-excentral barycentric image of X(9)

X(9) = barycentric product X(i)*X(j) for these {i,j}: {1, 8}, {3, 318}, {4, 78}, {6, 312}, {7, 200}, {10, 21}, {11, 765}, {12, 1098}, {19, 345}, {25, 3718}, {27, 3694}, {28, 3710}, {29, 72}, {31, 3596}, {32, 28659}, {33, 69}, {34, 1265}, {37, 333}, {40, 280}, {41, 76}, {42, 314}, {43, 7155}, {44, 4997}, {45, 30608}, {48, 7017}, {55, 75}, {56, 341}, {57, 346}, {58, 3701}, {59, 24026}, {60, 1089}, {63, 281}, {65, 1043}, {66, 4123}, {71, 31623}, {77, 7046}, {79, 4420}, {80, 4511}, {81, 2321}, {82, 3703}, {83, 33299}, {84, 7080}, {85, 220}, {86, 210}, {87, 27538}, {88, 2325}, {89, 4873}, {90, 5552}, {92, 219}, {95, 7069}, {99, 4041}, {100, 522}, {101, 4391}, {104, 6735}, {105, 3717}, {106, 4723}, {109, 4397}, {110, 4086}, {142, 6605}, {144, 19605}, {145, 3680}, {158, 1259}, {171, 4451}, {174, 6731}, {188, 188}, {189, 2324}, {190, 650}, {192, 2319}, {212, 264}, {213, 28660}, {222, 7101}, {225, 1792}, {226, 2287}, {236, 7028}, {238, 4518}, {239, 4876}, {241, 6559}, {244, 4076}, {253, 7070}, {256, 7081}, {257, 2329}, {259, 556}, {261, 756}, {266, 7027}, {269, 5423}, {270, 3695}, {271, 7952}, {273, 1260}, {274, 1334}, {278, 3692}, {279, 728}, {282, 329}, {284, 321}, {285, 21075}, {286, 2318}, {291, 3685}, {292, 3975}, {294, 3912}, {304, 607}, {305, 2212}, {306, 1172}, {307, 4183}, {309, 7074}, {313, 2194}, {319, 7073}, {322, 2192}, {326, 1857}, {330, 3208}, {331, 1802}, {332, 1824}, {335, 3684}, {348, 7079}, {350, 7077}, {366, 4182}, {391, 25430}, {393, 3719}, {480, 1088}, {483, 3082}, {492, 13455}, {512, 7257}, {513, 3699}, {514, 644}, {518, 14942}, {519, 1320}, {521, 1897}, {523, 643}, {561, 2175}, {594, 2185}, {596, 3871}, {612, 30479}, {645, 661}, {646, 649}, {648, 8611}, {651, 3239}, {652, 6335}, {657, 4554}, {658, 4130}, {660, 3716}, {662, 3700}, {663, 668}, {664, 3900}, {673, 3693}, {679, 4152}, {693, 3939}, {726, 8851}, {749, 4387}, {757, 6057}, {758, 6740}, {799, 3709}, {860, 1793}, {870, 4517}, {872, 18021}, {873, 7064}, {885, 1026}, {893, 17787}, {897, 3712}, {898, 14430}, {901, 4768}, {903, 3689}, {932, 4147}, {934, 4163}, {941, 11679}, {943, 6734}, {947, 23528}, {950, 1257}, {958, 31359}, {960, 1220}, {979, 19582}, {983, 3705}, {985, 3790}, {996, 3877}, {997, 30513}, {1000, 3872}, {1002, 3886}, {1014, 4082}, {1016, 2170}, {1018, 4560}, {1019, 30730}, {1021, 4552}, {1022, 30731}, {1025, 28132}, {1034, 1490}, {1036, 4385}, {1096, 1264}, {1100, 4102}, {1111, 6065}, {1120, 3880}, {1121, 6603}, {1125, 32635}, {1126, 3702}, {1146, 4564}, {1156, 6745}, {1174, 1229}, {1212, 32008}, {1214, 2322}, {1219, 1697}, {1222, 3057}, {1231, 2332}, {1240, 20967}, {1252, 4858}, {1253, 6063}, {1255, 3686}, {1261, 3663}, {1267, 13456}, {1268, 3683}, {1275, 3119}, {1280, 5853}, {1318, 4738}, {1332, 3064}, {1333, 30713}, {1390, 3883}, {1392, 3632}, {1407, 30693}, {1420, 6556}, {1434, 4515}, {1435, 30681}, {1441, 2328}, {1476, 6736}, {1492, 4522}, {1502, 9447}, {1577, 5546}, {1635, 4582}, {1639, 3257}, {1743, 6557}, {1751, 27396}, {1783, 6332}, {1785, 1809}, {1807, 5081}, {1812, 1826}, {1896, 3682}, {1903, 27398}, {1911, 4087}, {1928, 9448}, {1937, 7360}, {1959, 15628}, {1978, 3063}, {2052, 2289}, {2053, 6376}, {2057, 10309}, {2082, 30701}, {2125, 30695}, {2136, 6553}, {2137, 6552}, {2149, 23978}, {2150, 28654}, {2161, 32851}, {2162, 4110}, {2171, 7058}, {2176, 27424}, {2184, 27382}, {2195, 3263}, {2269, 30710}, {2279, 28809}, {2297, 18228}, {2298, 3687}, {2299, 20336}, {2310, 4998}, {2311, 3948}, {2316, 4358}, {2320, 3679}, {2323, 18359}, {2326, 26942}, {2330, 7018}, {2334, 4673}, {2335, 5271}, {2339, 2345}, {2340, 2481}, {2341, 3936}, {2342, 3262}, {2344, 3661}, {2346, 4847}, {2347, 32017}, {2349, 7359}, {2361, 20566}, {2363, 3704}, {2364, 4671}, {2643, 6064}, {2968, 7012}, {2985, 17452}, {2997, 3190}, {2998, 7075}, {3056, 7033}, {3059, 21453}, {3061, 17743}, {3083, 13454}, {3084, 13426}, {3112, 3688}, {3158, 4373}, {3161, 8056}, {3198, 5931}, {3219, 7110}, {3241, 4900}, {3247, 30711}, {3254, 3935}, {3271, 7035}, {3287, 27805}, {3452, 23617}, {3467, 27529}, {3478, 4737}, {3495, 26752}, {3615, 3678}, {3616, 4866}, {3623, 31509}, {3667, 31343}, {3669, 6558}, {3676, 4578}, {3691, 32009}, {3714, 5331}, {3715, 30598}, {3737, 3952}, {3762, 5548}, {3869, 10570}, {3870, 6601}, {3903, 3907}, {3996, 13476}, {3998, 8748}, {4007, 25417}, {4017, 7256}, {4024, 4612}, {4033, 7252}, {4034, 27789}, {4036, 4636}, {4069, 7192}, {4079, 4631}, {4081, 7045}, {4092, 24041}, {4105, 4569}, {4124, 5378}, {4140, 4603}, {4146, 6726}, {4149, 7357}, {4166, 18297}, {4171, 4573}, {4319, 8817}, {4433, 18827}, {4435, 4562}, {4494, 30650}, {4512, 5936}, {4513, 9311}, {4516, 4600}, {4521, 27834}, {4524, 4625}, {4526, 4607}, {4530, 5376}, {4534, 5382}, {4543, 4618}, {4551, 7253}, {4555, 4895}, {4557, 18155}, {4561, 18344}, {4567, 21044}, {4571, 7649}, {4572, 8641}, {4587, 17924}, {4597, 4814}, {4604, 4944}, {4606, 4765}, {4614, 4843}, {4624, 4827}, {4645, 7281}, {4791, 5549}, {4811, 8694}, {4845, 30806}, {4853, 7320}, {4861, 5559}, {4882, 5558}, {4919, 6630}, {4936, 27818}, {4985, 8701}, {5205, 9365}, {5239, 7043}, {5240, 7026}, {5380, 14432}, {5391, 13427}, {5430, 24242}, {5547, 14210}, {5665, 20007}, {6358, 7054}, {6554, 7131}, {6555, 19604}, {6606, 6608}, {6615, 8706}, {6737, 17097}, {7004, 15742}, {7020, 7078}, {7068, 24000}, {7071, 7182}, {7072, 20930}, {7090, 30556}, {7097, 27540}, {7105, 7283}, {7162, 10527}, {7178, 7259}, {7180, 7258}, {7220, 24349}, {7285, 27525}, {8012, 31618}, {8058, 13138}, {8707, 17420}, {8806, 13614}, {9368, 9369}, {9404, 15455}, {9436, 28071}, {9442, 28058}, {10025, 14943}, {10482, 20880}, {11609, 17763}, {12644, 12646}, {14121, 30557}, {14206, 15627}, {14534, 21033}, {14624, 17185}, {14827, 20567}, {15416, 32674}, {15891, 30413}, {15892, 30412}, {17780, 23838}, {18265, 18891}, {18750, 30457}, {19607, 21078}, {23705, 23836}, {24150, 24151}, {24152, 24154}, {24153, 24155}, {28070, 30705}

X(9) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 7}, {2, 85}, {3, 77}, {4, 273}, {6, 57}, {7, 1088}, {8, 75}, {10, 1441}, {11, 1111}, {19, 278}, {21, 86}, {22, 7210}, {25, 34}, {29, 286}, {31, 56}, {32, 604}, {33, 4}, {34, 1119}, {35, 1442}, {36, 1443}, {37, 226}, {38, 3665}, {40, 347}, {41, 6}, {42, 65}, {43, 3212}, {44, 3911}, {45, 5219}, {48, 222}, {51, 1393}, {55, 1}, {56, 269}, {57, 279}, {58, 1014}, {59, 7045}, {60, 757}, {63, 348}, {64, 8809}, {65, 3668}, {69, 7182}, {71, 1214}, {72, 307}, {73, 1439}, {75, 6063}, {76, 20567}, {77, 7056}, {78, 69}, {80, 18815}, {81, 1434}, {84, 1440}, {90, 7318}, {92, 331}, {99, 4625}, {100, 664}, {101, 651}, {109, 934}, {110, 1414}, {144, 31627}, {154, 1394}, {163, 4565}, {165, 3160}, {171, 7176}, {172, 7175}, {173, 18886}, {174, 555}, {181, 1254}, {184, 603}, {188, 4146}, {190, 4554}, {192, 30545}, {197, 21147}, {198, 223}, {200, 8}, {201, 6356}, {205, 478}, {210, 10}, {212, 3}, {213, 1400}, {218, 1445}, {219, 63}, {222, 7177}, {223, 14256}, {226, 1446}, {228, 73}, {238, 1447}, {239, 10030}, {244, 1358}, {255, 1804}, {256, 7249}, {258, 21456}, {259, 174}, {261, 873}, {266, 7371}, {269, 479}, {278, 1847}, {279, 23062}, {280, 309}, {281, 92}, {282, 189}, {283, 1444}, {284, 81}, {291, 7233}, {294, 673}, {306, 1231}, {312, 76}, {314, 310}, {318, 264}, {321, 349}, {326, 7055}, {330, 7209}, {333, 274}, {341, 3596}, {344, 21609}, {345, 304}, {346, 312}, {350, 18033}, {354, 10481}, {391, 19804}, {394, 7183}, {461, 5342}, {480, 200}, {497, 3673}, {512, 4017}, {513, 3676}, {514, 24002}, {517, 22464}, {518, 9436}, {521, 4025}, {522, 693}, {523, 4077}, {560, 1397}, {572, 17074}, {573, 17080}, {577, 7125}, {594, 6358}, {603, 7053}, {604, 1407}, {607, 19}, {608, 1435}, {610, 18623}, {612, 388}, {614, 7195}, {643, 99}, {644, 190}, {645, 799}, {646, 1978}, {649, 3669}, {650, 514}, {651, 658}, {652, 905}, {653, 13149}, {654, 3960}, {656, 17094}, {657, 650}, {661, 7178}, {662, 4573}, {663, 513}, {664, 4569}, {668, 4572}, {672, 241}, {678, 1317}, {692, 109}, {728, 346}, {750, 7223}, {756, 12}, {757, 552}, {765, 4998}, {798, 7180}, {846, 17084}, {849, 7341}, {862, 1874}, {869, 1469}, {872, 181}, {884, 1027}, {893, 1432}, {894, 7196}, {896, 7181}, {902, 1319}, {904, 1431}, {906, 1813}, {923, 7316}, {926, 2254}, {934, 4626}, {950, 17863}, {958, 10436}, {960, 4357}, {968, 3485}, {982, 7185}, {984, 7179}, {1018, 4552}, {1019, 17096}, {1021, 4560}, {1026, 883}, {1040, 17170}, {1043, 314}, {1054, 17089}, {1055, 6610}, {1096, 1118}, {1098, 261}, {1100, 553}, {1106, 7023}, {1107, 30097}, {1110, 59}, {1146, 4858}, {1155, 1323}, {1158, 31600}, {1170, 10509}, {1172, 27}, {1174, 1170}, {1193, 24471}, {1201, 1122}, {1212, 142}, {1220, 31643}, {1229, 1233}, {1250, 1082}, {1251, 1081}, {1252, 4564}, {1253, 55}, {1254, 6046}, {1259, 326}, {1260, 78}, {1261, 1222}, {1265, 3718}, {1318, 679}, {1320, 903}, {1331, 6516}, {1333, 1412}, {1334, 37}, {1376, 9312}, {1395, 1398}, {1397, 1106}, {1400, 1427}, {1402, 1042}, {1407, 738}, {1414, 4616}, {1415, 1461}, {1419, 9533}, {1436, 1422}, {1438, 1462}, {1445, 17093}, {1449, 21454}, {1460, 4320}, {1461, 4617}, {1469, 7204}, {1474, 1396}, {1475, 1418}, {1490, 5932}, {1500, 2171}, {1613, 1424}, {1615, 2124}, {1617, 4350}, {1635, 30725}, {1639, 3762}, {1697, 3672}, {1707, 17081}, {1721, 2898}, {1722, 31598}, {1731, 33129}, {1740, 17082}, {1742, 31526}, {1743, 5435}, {1754, 3188}, {1759, 17075}, {1760, 17076}, {1781, 18625}, {1783, 653}, {1792, 332}, {1802, 219}, {1812, 17206}, {1824, 225}, {1837, 17861}, {1857, 158}, {1859, 1838}, {1864, 1210}, {1897, 18026}, {1903, 8808}, {1909, 7205}, {1914, 1429}, {1918, 1402}, {1935, 6359}, {1936, 5088}, {1946, 1459}, {1962, 3649}, {1964, 1401}, {1973, 608}, {1974, 1395}, {2053, 87}, {2066, 13389}, {2082, 4000}, {2098, 4862}, {2115, 9499}, {2124, 17113}, {2136, 4452}, {2149, 1262}, {2150, 593}, {2161, 2006}, {2162, 7153}, {2170, 1086}, {2171, 6354}, {2173, 6357}, {2174, 2003}, {2175, 31}, {2176, 1423}, {2177, 2099}, {2183, 1465}, {2185, 1509}, {2187, 221}, {2188, 1433}, {2192, 84}, {2193, 1790}, {2194, 58}, {2195, 105}, {2199, 6611}, {2200, 1409}, {2204, 1474}, {2206, 1408}, {2208, 1413}, {2209, 1403}, {2210, 1428}, {2212, 25}, {2223, 1458}, {2238, 16609}, {2241, 7225}, {2244, 7214}, {2245, 18593}, {2246, 5723}, {2251, 1404}, {2258, 959}, {2259, 2982}, {2268, 940}, {2269, 3666}, {2276, 7146}, {2280, 5228}, {2284, 1025}, {2285, 7365}, {2287, 333}, {2289, 394}, {2293, 354}, {2295, 4032}, {2299, 28}, {2308, 32636}, {2310, 11}, {2316, 88}, {2318, 72}, {2319, 330}, {2321, 321}, {2322, 31623}, {2323, 3218}, {2324, 329}, {2325, 4358}, {2327, 1812}, {2328, 21}, {2329, 894}, {2330, 171}, {2331, 196}, {2332, 1172}, {2333, 1880}, {2340, 518}, {2341, 24624}, {2342, 104}, {2344, 14621}, {2346, 21453}, {2347, 3752}, {2348, 3008}, {2352, 4306}, {2356, 1876}, {2361, 36}, {2364, 89}, {2427, 24029}, {2632, 1367}, {2638, 1364}, {2640, 17085}, {2643, 1365}, {2646, 3664}, {2900, 12649}, {2911, 1708}, {2939, 18631}, {2951, 31527}, {2968, 17880}, {2997, 15467}, {3009, 1463}, {3022, 2310}, {3024, 7266}, {3052, 1420}, {3056, 982}, {3057, 3663}, {3058, 7264}, {3059, 4847}, {3061, 3662}, {3063, 649}, {3064, 17924}, {3083, 13453}, {3084, 13436}, {3100, 4872}, {3119, 1146}, {3158, 145}, {3161, 18743}, {3169, 3210}, {3172, 3213}, {3185, 10571}, {3190, 3868}, {3195, 208}, {3198, 5930}, {3207, 1419}, {3208, 192}, {3217, 4383}, {3218, 17078}, {3219, 17095}, {3239, 4391}, {3248, 1357}, {3270, 7004}, {3271, 244}, {3287, 4369}, {3295, 7190}, {3303, 4328}, {3304, 7271}, {3306, 17079}, {3309, 31605}, {3445, 19604}, {3452, 26563}, {3496, 17086}, {3596, 561}, {3601, 3945}, {3666, 3674}, {3680, 4373}, {3681, 33298}, {3683, 1125}, {3684, 239}, {3685, 350}, {3686, 4359}, {3687, 20911}, {3688, 38}, {3689, 519}, {3690, 201}, {3691, 3739}, {3692, 345}, {3693, 3912}, {3694, 306}, {3699, 668}, {3700, 1577}, {3701, 313}, {3702, 1269}, {3703, 1930}, {3704, 18697}, {3706, 20888}, {3707, 24589}, {3709, 661}, {3710, 20336}, {3711, 3679}, {3712, 14210}, {3713, 11679}, {3715, 1698}, {3716, 3766}, {3717, 3263}, {3718, 305}, {3719, 3926}, {3720, 4059}, {3721, 16888}, {3723, 3982}, {3724, 1464}, {3731, 5226}, {3733, 7203}, {3737, 7192}, {3738, 4453}, {3745, 4298}, {3746, 7269}, {3747, 1284}, {3786, 30966}, {3870, 6604}, {3871, 4360}, {3876, 5224}, {3877, 4389}, {3880, 1266}, {3883, 26234}, {3885, 4398}, {3886, 4441}, {3900, 522}, {3907, 4374}, {3910, 4509}, {3913, 3875}, {3920, 7247}, {3938, 30617}, {3939, 100}, {3949, 26942}, {3957, 32007}, {3965, 3687}, {3974, 4385}, {3975, 1921}, {3985, 3948}, {3996, 17143}, {4007, 28605}, {4009, 6381}, {4041, 523}, {4042, 32092}, {4046, 4647}, {4050, 1278}, {4055, 22341}, {4060, 4980}, {4069, 3952}, {4073, 3705}, {4076, 7035}, {4081, 24026}, {4082, 3701}, {4086, 850}, {4087, 18891}, {4092, 1109}, {4094, 3027}, {4095, 3963}, {4097, 3896}, {4102, 32018}, {4105, 3900}, {4110, 6382}, {4111, 21020}, {4117, 1356}, {4118, 7217}, {4119, 20432}, {4123, 315}, {4130, 3239}, {4136, 20234}, {4147, 20906}, {4149, 6327}, {4152, 4738}, {4162, 3667}, {4163, 4397}, {4166, 366}, {4167, 21442}, {4171, 3700}, {4178, 20627}, {4182, 18297}, {4183, 29}, {4253, 17092}, {4254, 5256}, {4258, 1449}, {4266, 4850}, {4319, 497}, {4320, 7197}, {4326, 10580}, {4336, 1836}, {4361, 7243}, {4387, 3760}, {4390, 4363}, {4391, 3261}, {4394, 30719}, {4420, 319}, {4421, 25716}, {4433, 740}, {4435, 812}, {4451, 7018}, {4474, 4411}, {4477, 3907}, {4501, 4382}, {4511, 320}, {4512, 3616}, {4513, 3729}, {4515, 2321}, {4516, 3120}, {4517, 984}, {4518, 334}, {4521, 4462}, {4524, 4041}, {4526, 4728}, {4528, 4768}, {4531, 3778}, {4548, 2172}, {4551, 4566}, {4557, 4551}, {4559, 1020}, {4560, 7199}, {4564, 1275}, {4565, 4637}, {4567, 4620}, {4571, 4561}, {4573, 4635}, {4578, 3699}, {4579, 6649}, {4587, 1332}, {4606, 4624}, {4612, 4610}, {4662, 4967}, {4723, 3264}, {4730, 30572}, {4765, 4801}, {4790, 30723}, {4814, 4777}, {4820, 4823}, {4827, 4765}, {4843, 4815}, {4845, 1156}, {4847, 20880}, {4849, 4848}, {4853, 31995}, {4858, 23989}, {4860, 21314}, {4861, 7321}, {4866, 5936}, {4873, 4671}, {4875, 24199}, {4876, 335}, {4877, 5333}, {4882, 32087}, {4895, 900}, {4901, 31130}, {4903, 20943}, {4907, 5274}, {4919, 4440}, {4936, 3161}, {4944, 4791}, {4953, 4939}, {4959, 4926}, {4976, 4978}, {4979, 30724}, {4990, 4985}, {4995, 7278}, {4997, 20568}, {5048, 4887}, {5089, 5236}, {5250, 17321}, {5269, 3600}, {5285, 4296}, {5289, 17274}, {5311, 10404}, {5320, 1451}, {5414, 13388}, {5423, 341}, {5452, 169}, {5532, 1090}, {5546, 662}, {5547, 897}, {5548, 3257}, {5549, 4604}, {5552, 20930}, {5795, 24993}, {5802, 19788}, {5837, 24547}, {6003, 31603}, {6056, 255}, {6057, 1089}, {6058, 1091}, {6059, 1096}, {6060, 1097}, {6061, 1098}, {6062, 1099}, {6064, 24037}, {6065, 765}, {6066, 1110}, {6139, 14413}, {6187, 1411}, {6198, 7282}, {6332, 15413}, {6558, 646}, {6600, 3870}, {6602, 220}, {6603, 527}, {6605, 32008}, {6607, 6608}, {6608, 6362}, {6726, 188}, {6731, 556}, {6735, 3262}, {6736, 20895}, {6740, 14616}, {6741, 17886}, {6745, 30806}, {7004, 1565}, {7007, 7149}, {7014, 558}, {7017, 1969}, {7032, 7248}, {7037, 3345}, {7046, 318}, {7050, 7091}, {7054, 2185}, {7062, 23996}, {7064, 756}, {7067, 24038}, {7068, 17879}, {7069, 5}, {7070, 20}, {7071, 33}, {7072, 90}, {7073, 79}, {7074, 40}, {7075, 194}, {7076, 1940}, {7077, 291}, {7078, 7013}, {7079, 281}, {7080, 322}, {7081, 1909}, {7082, 499}, {7083, 614}, {7084, 1037}, {7085, 1038}, {7087, 7213}, {7101, 7017}, {7110, 30690}, {7115, 7128}, {7117, 3942}, {7118, 1436}, {7123, 7131}, {7124, 7289}, {7131, 30705}, {7133, 1659}, {7154, 7129}, {7155, 6384}, {7156, 1249}, {7177, 30682}, {7180, 7216}, {7252, 1019}, {7253, 18155}, {7256, 7257}, {7257, 670}, {7259, 645}, {7281, 7261}, {7290, 3598}, {7322, 5261}, {7339, 24013}, {7359, 14206}, {7367, 282}, {7368, 2324}, {7675, 14548}, {7707, 234}, {7952, 342}, {7962, 4346}, {8012, 1212}, {8056, 27818}, {8058, 17896}, {8540, 18201}, {8545, 1996}, {8551, 8012}, {8580, 31994}, {8606, 7100}, {8611, 525}, {8641, 663}, {8647, 1279}, {8653, 4822}, {8676, 23800}, {8750, 108}, {8835, 15913}, {8851, 3226}, {9310, 6180}, {9404, 14838}, {9439, 9309}, {9440, 9446}, {9441, 14189}, {9447, 32}, {9448, 560}, {9629, 3583}, {10382, 938}, {10387, 3677}, {10393, 5738}, {10482, 2346}, {10501, 10491}, {10502, 10489}, {10570, 2995}, {10581, 21127}, {10582, 32086}, {10638, 559}, {11075, 26743}, {11124, 21105}, {11193, 21201}, {11429, 3075}, {11934, 21185}, {11997, 24210}, {11998, 24237}, {12329, 8270}, {13427, 1336}, {13455, 485}, {13456, 1123}, {14077, 30181}, {14100, 11019}, {14298, 14837}, {14308, 17898}, {14392, 6366}, {14427, 1639}, {14547, 942}, {14827, 41}, {14936, 2170}, {14942, 2481}, {15627, 2349}, {15628, 1821}, {15733, 26015}, {15837, 13405}, {16011, 2091}, {16012, 177}, {16283, 9310}, {16502, 28017}, {16545, 18626}, {16546, 18627}, {16552, 17077}, {16556, 17083}, {16566, 17087}, {16568, 17088}, {16569, 17090}, {16571, 17091}, {16572, 8732}, {16588, 17451}, {16601, 21617}, {16666, 4031}, {16686, 1421}, {16713, 16708}, {16721, 18176}, {16777, 4654}, {16780, 28079}, {16885, 31231}, {17185, 16705}, {17194, 17169}, {17197, 16727}, {17412, 13401}, {17420, 3004}, {17452, 3782}, {17453, 7251}, {17469, 7198}, {17742, 28739}, {17744, 28780}, {17787, 1920}, {17798, 5018}, {18098, 18097}, {18163, 18600}, {18191, 17205}, {18265, 1911}, {18344, 7649}, {18594, 18624}, {18595, 18628}, {18596, 18629}, {18597, 18630}, {18598, 18632}, {18599, 18633}, {18887, 21624}, {18888, 14596}, {18889, 2291}, {19605, 10405}, {19624, 2078}, {20229, 1475}, {20359, 24215}, {20663, 8850}, {20665, 2275}, {20672, 2114}, {20684, 3721}, {20753, 3784}, {20967, 1193}, {21010, 4334}, {21033, 1211}, {21039, 3925}, {21044, 16732}, {21059, 1617}, {21104, 23599}, {21127, 21104}, {21333, 4920}, {21334, 24214}, {21677, 18698}, {21789, 3737}, {21795, 21808}, {21803, 7211}, {21809, 4415}, {21811, 17056}, {21832, 7212}, {21840, 5244}, {21859, 4605}, {21879, 27691}, {22074, 22097}, {22079, 22053}, {23207, 4303}, {23344, 23703}, {23544, 28358}, {23638, 24443}, {23838, 6548}, {23990, 2149}, {24010, 4081}, {24012, 3022}, {24027, 7339}, {24041, 7340}, {24151, 27828}, {24394, 4442}, {24430, 17181}, {25082, 17234}, {25128, 23807}, {25268, 21580}, {26885, 1935}, {27382, 18750}, {27396, 18134}, {27424, 6383}, {27508, 20921}, {27523, 20923}, {27538, 6376}, {27540, 20914}, {27549, 30758}, {27958, 8033}, {28043, 2550}, {28070, 6554}, {28071, 14942}, {28125, 5880}, {28659, 1502}, {28660, 6385}, {28808, 20925}, {28809, 21615}, {30223, 3086}, {30457, 2184}, {30568, 18135}, {30608, 20569}, {30618, 17353}, {30706, 2082}, {30713, 27801}, {30730, 4033}, {30731, 24004}, {32008, 31618}, {32462, 31604}, {32635, 1268}, {32652, 8059}, {32666, 32735}, {32674, 32714}, {32739, 1415}, {32851, 20924}, {33299, 141}

X(9) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 6, 1449}, {1, 37, 3247}, {1, 44, 16670}, {1, 45, 16676}, {1, 72, 11523}, {1, 238, 7290}, {1, 405, 5436}, {1, 960, 15829}, {1, 984, 7174}, {1, 1723, 2257}, {1, 1724, 1453}, {1, 1728, 10396}, {1, 1743, 6}, {1, 1757, 3751}, {1, 3731, 37}, {1, 3973, 1743}, {1, 5234, 958}, {1, 10398, 5728}, {1, 16468, 16475}, {1, 16469, 1386}, {1, 16552, 21384}, {1, 16667, 1100}, {1, 16673, 16777}, {1, 17744, 17742}, {1, 30330, 5572}, {2, 7, 142}, {2, 57, 5437}, {2, 63, 57}, {2, 142, 20195}, {2, 144, 7}, {2, 226, 25525}, {2, 307, 18634}, {2, 329, 226}, {2, 672, 17754}, {2, 894, 10436}, {2, 908, 5219}, {2, 3218, 3306}, {2, 3219, 63}, {2, 3305, 7308}, {2, 3452, 30827}, {2, 3662, 17282}, {2, 3911, 31190}, {2, 3929, 3928}, {2, 4357, 17306}, {2, 5273, 5745}, {2, 5282, 3509}, {2, 5296, 5257}, {2, 5435, 6692}, {2, 5744, 3911}, {2, 5749, 5750}, {2, 5905, 5249}, {2, 6646, 3662}, {2, 9965, 9776}, {2, 17236, 17291}, {2, 17257, 4357}, {2, 17333, 17274}, {2, 17350, 894}, {2, 17483, 27186}, {2, 17484, 31019}, {2, 17781, 4654}, {2, 18228, 3452}, {2, 18230, 6666}, {2, 20347, 30949}, {2, 20348, 30097}, {2, 26125, 25521}, {2, 26685, 17353}, {2, 26792, 31053}, {2, 26806, 27147}, {2, 26836, 26997}, {2, 27065, 3305}, {2, 27131, 30852}, {2, 27184, 25527}, {2, 27420, 27384}, {2, 28287, 27626}, {2, 29696, 29740}, {2, 30414, 5242}, {2, 30415, 5243}, {2, 30946, 20335}, {2, 31018, 908}, {2, 31053, 31266}, {2, 31300, 26806}, {3, 84, 9841}, {3, 936, 5438}, {3, 5044, 936}, {3, 5777, 1490}, {3, 7330, 84}, {3, 12684, 31805}, {3, 24320, 3220}, {3, 31445, 31424}, {3, 31658, 21153}, {4, 21168, 5759}, {5, 5791, 5705}, {5, 5812, 5715}, {5, 26921, 5709}, {6, 37, 1}, {6, 44, 1743}, {6, 45, 37}, {6, 219, 2323}, {6, 220, 219}, {6, 1001, 16503}, {6, 1100, 16667}, {6, 1743, 16670}, {6, 2176, 2300}, {6, 3731, 3247}, {6, 8557, 2257}, {6, 8609, 3554}, {6, 15492, 3973}, {6, 16672, 16884}, {6, 16675, 16777}, {6, 16677, 3723}, {6, 16777, 1100}, {6, 16814, 3731}, {6, 16884, 16666}, {6, 16885, 44}, {6, 16969, 21785}, {6, 16970, 7290}, {6, 16972, 16475}, {6, 21769, 20228}, {7, 142, 6173}, {7, 1445, 57}, {7, 6172, 144}, {7, 6666, 20195}, {7, 8232, 226}, {7, 18230, 2}, {7, 29007, 8545}, {8, 346, 2321}, {8, 391, 3686}, {8, 452, 950}, {8, 950, 12625}, {8, 1334, 3208}, {8, 1697, 2136}, {8, 2269, 3169}, {8, 2321, 4007}, {8, 2325, 4873}, {8, 3161, 346}, {8, 3208, 4050}, {8, 3685, 3886}, {8, 3686, 4034}, {8, 3717, 4901}, {8, 5250, 1697}, {8, 5686, 24393}, {8, 27549, 3717}, {10, 40, 1706}, {10, 3730, 3501}, {10, 6554, 23058}, {10, 12514, 40}, {10, 12572, 4}, {10, 12618, 1861}, {10, 17355, 2345}, {10, 18250, 2551}, {10, 31594, 14121}, {10, 31595, 7090}, {19, 71, 40}, {19, 169, 16547}, {19, 1766, 16548}, {19, 2183, 2270}, {19, 7079, 281}, {21, 78, 3601}, {21, 2287, 284}, {21, 3876, 78}, {25, 7085, 5285}, {25, 26867, 7085}, {31, 612, 5269}, {31, 756, 612}, {33, 212, 7070}, {37, 44, 6}, {37, 45, 3731}, {37, 72, 22021}, {37, 220, 2324}, {37, 1100, 16777}, {37, 1743, 1449}, {37, 2911, 3553}, {37, 3723, 16672}, {37, 3731, 16676}, {37, 3973, 16670}, {37, 4426, 5336}, {37, 15479, 15829}, {37, 15492, 44}, {37, 16517, 7174}, {37, 16666, 3723}, {37, 16669, 1100}, {37, 16671, 16884}, {37, 16777, 16673}, {37, 16814, 45}, {37, 16885, 1743}, {37, 21873, 4053}, {37, 21879, 21810}, {38, 614, 3677}, {38, 748, 614}, {39, 21796, 2277}, {39, 31442, 31429}, {41, 2268, 284}, {41, 33299, 78}, {43, 846, 17594}, {43, 21369, 21387}, {44, 45, 1}, {44, 1100, 16669}, {44, 3723, 16671}, {44, 3731, 1449}, {44, 8557, 16572}, {44, 15492, 16885}, {44, 16675, 16667}, {44, 16814, 37}, {44, 16885, 3973}, {45, 1743, 3247}, {45, 3973, 1449}, {45, 15492, 1743}, {45, 16669, 16673}, {45, 16777, 16675}, {45, 16884, 16677}, {45, 16885, 6}, {48, 2265, 2261}, {48, 2267, 572}, {51, 3690, 26893}, {55, 200, 3158}, {55, 210, 200}, {55, 480, 6600}, {55, 1864, 10382}, {55, 2264, 380}, {55, 3059, 3174}, {55, 3683, 4512}, {55, 3711, 3689}, {55, 3715, 210}, {55, 7082, 30223}, {55, 14100, 4326}, {56, 8581, 4321}, {56, 25917, 8583}, {57, 63, 3928}, {57, 3929, 63}, {57, 7308, 2}, {63, 3219, 3929}, {63, 3305, 2}, {63, 3306, 3218}, {63, 7308, 5437}, {63, 21371, 16574}, {63, 25880, 28017}, {63, 25894, 28039}, {63, 27065, 7308}, {69, 344, 3912}, {69, 3912, 17296}, {71, 2183, 573}, {71, 2345, 3501}, {72, 405, 1}, {72, 10396, 6762}, {72, 14054, 5904}, {75, 190, 3729}, {75, 3729, 4659}, {75, 17277, 4384}, {75, 17335, 17277}, {75, 17336, 190}, {75, 20927, 20236}, {77, 651, 1419}, {85, 32024, 30625}, {85, 32088, 32008}, {85, 32100, 32024}, {86, 4687, 16831}, {86, 18206, 18164}, {101, 572, 48}, {101, 2265, 16554}, {101, 12034, 2265}, {105, 4712, 9451}, {141, 4422, 17279}, {141, 4643, 17272}, {141, 17279, 17284}, {141, 17332, 4643}, {142, 6666, 2}, {144, 6666, 6173}, {144, 18230, 142}, {165, 1709, 10860}, {165, 1750, 7580}, {165, 2951, 11495}, {165, 3062, 2951}, {165, 8580, 1376}, {165, 30326, 1750}, {165, 30393, 8580}, {169, 573, 2270}, {169, 1766, 19}, {169, 3730, 40}, {169, 4456, 7713}, {169, 7079, 23058}, {169, 12514, 3496}, {171, 7262, 1707}, {173, 8078, 18888}, {188, 236, 16016}, {190, 4384, 4659}, {190, 17277, 75}, {190, 17335, 4384}, {190, 17336, 25728}, {191, 1698, 46}, {191, 1781, 1761}, {192, 239, 3875}, {192, 17349, 239}, {193, 17316, 3879}, {198, 1436, 1604}, {198, 1903, 1490}, {198, 2182, 610}, {200, 4326, 3174}, {200, 4512, 55}, {200, 10382, 2900}, {210, 3683, 55}, {210, 3689, 3711}, {210, 4512, 3158}, {210, 13615, 2900}, {210, 14100, 3059}, {210, 15837, 480}, {212, 7069, 33}, {213, 5283, 1}, {218, 16601, 1}, {220, 958, 2329}, {220, 1212, 1}, {220, 8557, 15286}, {226, 329, 28609}, {226, 1708, 57}, {238, 984, 1}, {238, 16517, 1449}, {239, 17261, 192}, {241, 6180, 269}, {261, 645, 27958}, {281, 20262, 23058}, {284, 4877, 21}, {312, 333, 11679}, {319, 17233, 17294}, {319, 17264, 17233}, {320, 17234, 17298}, {320, 17263, 17234}, {321, 5278, 5271}, {329, 5749, 5746}, {333, 17185, 18163}, {344, 4416, 17296}, {345, 14555, 3687}, {346, 391, 8}, {346, 2269, 3208}, {346, 2321, 4873}, {346, 2347, 3169}, {346, 3161, 2325}, {346, 3686, 4007}, {346, 3692, 728}, {346, 3707, 4034}, {354, 4423, 10582}, {374, 21871, 2262}, {380, 3694, 3158}, {381, 31671, 18482}, {390, 5686, 8}, {390, 5809, 950}, {390, 5825, 10392}, {391, 452, 5802}, {391, 1334, 3169}, {391, 2321, 4034}, {391, 2325, 4007}, {391, 3161, 2321}, {392, 956, 1}, {405, 954, 1001}, {405, 15650, 72}, {474, 3916, 15803}, {480, 3059, 200}, {480, 4326, 3158}, {480, 14100, 3174}, {497, 4847, 24392}, {573, 1766, 40}, {573, 3730, 71}, {573, 17355, 3501}, {579, 5750, 17754}, {579, 8558, 1741}, {594, 4370, 17340}, {594, 17275, 3679}, {594, 17330, 17275}, {594, 17340, 17281}, {599, 17267, 17231}, {612, 756, 7322}, {644, 2170, 4919}, {672, 1400, 579}, {672, 5282, 63}, {728, 1697, 3208}, {728, 2082, 2136}, {894, 17260, 2}, {894, 21371, 57}, {894, 27420, 1944}, {908, 31018, 31142}, {936, 31424, 3}, {950, 10392, 5809}, {954, 5223, 11523}, {954, 5728, 1}, {954, 5729, 5728}, {958, 960, 1}, {958, 5302, 5234}, {958, 15479, 1449}, {960, 1212, 3061}, {960, 5302, 958}, {960, 30618, 220}, {965, 5782, 5783}, {966, 2345, 10}, {966, 6554, 20262}, {982, 5272, 5573}, {982, 17123, 5272}, {984, 16970, 3247}, {992, 2277, 978}, {993, 997, 3576}, {993, 10176, 997}, {1001, 5223, 3243}, {1001, 15481, 5223}, {1006, 18446, 3576}, {1038, 1935, 1394}, {1086, 17276, 4862}, {1086, 17278, 4859}, {1086, 17334, 17276}, {1086, 17337, 17278}, {1100, 16667, 1449}, {1100, 16669, 6}, {1100, 16675, 16673}, {1100, 16777, 1}, {1107, 2176, 1}, {1108, 2256, 1}, {1124, 8965, 1}, {1156, 6594, 5528}, {1212, 30618, 2329}, {1213, 1901, 442}, {1213, 7359, 7110}, {1213, 17303, 1698}, {1213, 17369, 17303}, {1253, 2310, 4319}, {1253, 21039, 28043}, {1260, 1864, 2900}, {1260, 10382, 3158}, {1260, 13615, 55}, {1276, 1277, 40}, {1276, 6192, 1277}, {1277, 6191, 1276}, {1278, 16816, 17117}, {1279, 3242, 1}, {1329, 18253, 26066}, {1329, 26066, 1698}, {1334, 2082, 1697}, {1334, 2347, 2269}, {1334, 3691, 8}, {1376, 3740, 8580}, {1376, 4640, 165}, {1376, 5574, 19605}, {1376, 16112, 17668}, {1376, 30624, 5574}, {1400, 2285, 57}, {1400, 5279, 3509}, {1400, 5749, 17754}, {1436, 1604, 32625}, {1445, 3305, 6666}, {1445, 8545, 7}, {1449, 3247, 1}, {1449, 16670, 6}, {1449, 16676, 3247}, {1621, 3681, 3870}, {1621, 3870, 10389}, {1621, 6605, 6602}, {1652, 1653, 57}, {1654, 3661, 17270}, {1654, 4473, 17280}, {1654, 17280, 3661}, {1654, 17339, 17286}, {1697, 10384, 390}, {1698, 9612, 442}, {1707, 5268, 171}, {1723, 1728, 1713}, {1723, 1743, 16572}, {1726, 21361, 1763}, {1728, 15298, 5728}, {1731, 4266, 2082}, {1743, 2324, 2323}, {1743, 3731, 1}, {1743, 3973, 44}, {1743, 16673, 16667}, {1743, 16814, 16676}, {1743, 17744, 5227}, {1743, 21061, 21384}, {1750, 30326, 5927}, {1757, 20372, 24727}, {1759, 16549, 46}, {1778, 2303, 58}, {1826, 26063, 5587}, {1864, 3683, 13615}, {1903, 2182, 5776}, {1903, 7367, 282}, {2082, 3692, 3169}, {2161, 4370, 16561}, {2170, 4390, 3872}, {2170, 21809, 17452}, {2220, 5301, 7031}, {2223, 20992, 16688}, {2238, 2276, 43}, {2245, 17369, 16549}, {2246, 14439, 100}, {2257, 5227, 6762}, {2264, 3965, 3684}, {2268, 3217, 41}, {2268, 21033, 78}, {2269, 2347, 4266}, {2269, 3691, 3686}, {2270, 2345, 1706}, {2278, 3204, 2174}, {2280, 3930, 3870}, {2284, 5701, 1}, {2285, 28070, 27382}, {2287, 27396, 78}, {2310, 4319, 4907}, {2318, 14547, 3190}, {2321, 2325, 346}, {2321, 3169, 4050}, {2321, 3686, 8}, {2321, 3707, 3686}, {2321, 4266, 3169}, {2321, 5802, 12625}, {2324, 9119, 11523}, {2325, 3686, 2321}, {2325, 3707, 8}, {2325, 4266, 3208}, {2329, 3061, 1}, {2345, 5819, 2550}, {2345, 6554, 281}, {2347, 3161, 3208}, {2347, 3691, 391}, {2348, 3693, 3684}, {2478, 6734, 9581}, {2886, 24703, 1699}, {2949, 5715, 5709}, {2975, 19861, 1420}, {3008, 3663, 4000}, {3009, 22343, 7032}, {3056, 4517, 3688}, {3057, 4853, 3680}, {3057, 4875, 4051}, {3059, 15837, 6600}, {3062, 8580, 15587}, {3068, 6351, 5393}, {3069, 6352, 5405}, {3161, 3686, 4873}, {3161, 3707, 4007}, {3174, 6600, 3158}, {3218, 3306, 57}, {3219, 3305, 57}, {3219, 7308, 3928}, {3219, 17260, 16574}, {3219, 27065, 2}, {3230, 16975, 1}, {3230, 20228, 21769}, {3247, 16670, 1449}, {3247, 16676, 37}, {3271, 3688, 3056}, {3271, 7064, 3688}, {3287, 3709, 3737}, {3294, 16552, 1}, {3294, 21061, 37}, {3299, 3301, 16473}, {3305, 3929, 5437}, {3333, 3646, 1125}, {3419, 11113, 3586}, {3436, 24987, 9578}, {3452, 5325, 5745}, {3452, 5745, 2}, {3452, 20258, 21246}, {3487, 16845, 1125}, {3496, 3501, 40}, {3509, 17754, 57}, {3586, 3679, 3419}, {3589, 4364, 4657}, {3589, 4657, 29598}, {3596, 17787, 4494}, {3618, 17321, 17023}, {3624, 6763, 3338}, {3633, 4898, 17388}, {3661, 17280, 17286}, {3661, 17331, 1654}, {3661, 17339, 17280}, {3662, 6646, 17274}, {3662, 17333, 6646}, {3662, 17338, 2}, {3664, 25072, 29571}, {3664, 29571, 4648}, {3666, 4383, 2999}, {3672, 5222, 3946}, {3678, 5248, 3811}, {3683, 3715, 200}, {3686, 3707, 391}, {3688, 7064, 4517}, {3689, 3711, 200}, {3693, 3965, 3694}, {3700, 9404, 1021}, {3713, 3965, 200}, {3717, 3883, 8}, {3723, 16666, 16884}, {3723, 16671, 16666}, {3723, 16884, 1}, {3729, 4384, 75}, {3729, 25728, 190}, {3731, 3973, 6}, {3731, 16667, 16673}, {3731, 16885, 16670}, {3739, 4363, 25590}, {3739, 17259, 16832}, {3739, 17351, 4363}, {3740, 4640, 1376}, {3740, 18227, 18236}, {3758, 4687, 86}, {3759, 4360, 16834}, {3759, 4664, 4360}, {3763, 17253, 17237}, {3782, 24789, 23681}, {3834, 17345, 7232}, {3869, 5260, 19860}, {3869, 19860, 3340}, {3872, 3877, 7962}, {3873, 5284, 4666}, {3876, 25082, 33299}, {3877, 4390, 4919}, {3883, 27549, 4901}, {3911, 5316, 2}, {3912, 4416, 69}, {3912, 25101, 344}, {3913, 4662, 4882}, {3927, 11108, 942}, {3928, 5437, 57}, {3929, 7308, 57}, {3940, 16418, 24929}, {3943, 17362, 17299}, {3944, 33138, 17064}, {3951, 5047, 11518}, {3961, 8616, 3749}, {3970, 16783, 1}, {3970, 17746, 5904}, {3973, 16673, 16669}, {3973, 16814, 3247}, {3975, 17787, 3596}, {3995, 19742, 3187}, {4000, 4419, 3663}, {4007, 4034, 8}, {4007, 4873, 2321}, {4029, 4700, 145}, {4030, 4126, 30615}, {4034, 4873, 4007}, {4042, 4387, 3706}, {4053, 21873, 24048}, {4067, 30143, 12559}, {4271, 17340, 1018}, {4313, 20007, 12437}, {4314, 6743, 3189}, {4357, 17353, 2}, {4361, 17348, 16833}, {4363, 17259, 3739}, {4370, 17330, 17281}, {4384, 25728, 3729}, {4386, 17735, 3550}, {4389, 16706, 17304}, {4389, 17352, 16706}, {4393, 4704, 17319}, {4416, 25101, 3912}, {4422, 4643, 17284}, {4422, 17332, 141}, {4445, 17269, 17229}, {4452, 24599, 4402}, {4473, 17280, 17339}, {4473, 17331, 17286}, {4488, 31995, 4454}, {4513, 4875, 4853}, {4520, 4875, 3057}, {4557, 8053, 15624}, {4640, 15587, 11495}, {4643, 17279, 141}, {4644, 4648, 3664}, {4670, 4698, 15668}, {4675, 17365, 4888}, {4681, 4852, 17318}, {4690, 17229, 4445}, {4708, 17385, 17327}, {4741, 17232, 17288}, {4851, 17243, 29573}, {4853, 4936, 4513}, {4859, 4862, 1086}, {4859, 31183, 17278}, {4862, 31183, 4859}, {4866, 4882, 4662}, {4999, 25681, 3624}, {5044, 5779, 5785}, {5044, 31424, 5438}, {5044, 31445, 3}, {5219, 31142, 908}, {5220, 15254, 1}, {5224, 17289, 17308}, {5224, 17354, 17289}, {5239, 5240, 1}, {5242, 5243, 2}, {5245, 5246, 8}, {5249, 5905, 4654}, {5249, 17781, 5905}, {5251, 5526, 16788}, {5251, 5692, 1}, {5257, 5746, 25525}, {5257, 5750, 2}, {5259, 5904, 1}, {5259, 17745, 16783}, {5269, 7322, 612}, {5273, 18228, 2}, {5279, 5749, 2285}, {5296, 5749, 2}, {5316, 5744, 31190}, {5325, 5745, 5273}, {5436, 11523, 1}, {5440, 16370, 30282}, {5506, 6763, 3624}, {5584, 12688, 12565}, {5686, 5838, 3686}, {5692, 18397, 72}, {5705, 31446, 5791}, {5728, 5729, 10398}, {5729, 15650, 5220}, {5732, 5785, 5784}, {5732, 21153, 3}, {5735, 5833, 5832}, {5739, 17776, 306}, {5742, 5830, 5831}, {5745, 18228, 30827}, {5758, 6846, 946}, {5759, 5817, 4}, {5766, 5809, 390}, {5766, 5825, 5809}, {5779, 31658, 5732}, {5785, 31424, 5732}, {5785, 31658, 5438}, {5795, 5837, 8}, {5809, 5838, 5802}, {5811, 6908, 6260}, {5904, 10399, 14054}, {5927, 7580, 1750}, {6172, 18230, 7}, {6173, 20195, 142}, {6191, 6192, 40}, {6203, 6204, 57}, {6210, 6211, 40}, {6211, 7609, 6210}, {6212, 6213, 40}, {6260, 6684, 6908}, {6601, 24389, 24392}, {6646, 17338, 17282}, {6687, 17235, 17356}, {6986, 12528, 10884}, {7001, 7010, 188}, {7026, 7043, 3679}, {7081, 20665, 2319}, {7090, 14121, 10}, {7174, 7290, 1}, {7174, 15601, 7290}, {7232, 17265, 3834}, {7290, 15601, 238}, {8165, 18231, 9780}, {8232, 12848, 7}, {8257, 8545, 6173}, {8580, 30393, 3740}, {9330, 17126, 5297}, {9776, 9965, 553}, {10177, 15185, 5572}, {10396, 31435, 5436}, {10398, 15299, 10396}, {10436, 16574, 57}, {10456, 18229, 10472}, {10857, 30304, 10167}, {11106, 20007, 4313}, {11679, 30568, 312}, {11683, 26671, 16609}, {11752, 11789, 3576}, {12572, 17355, 8804}, {13405, 21060, 25568}, {14100, 15837, 55}, {14829, 18743, 30567}, {15296, 15297, 1001}, {15298, 15299, 1}, {15492, 16814, 6}, {16482, 23343, 1}, {16517, 16970, 1}, {16519, 16974, 1}, {16547, 16548, 19}, {16566, 20602, 1760}, {16572, 17742, 6762}, {16666, 16671, 6}, {16666, 16672, 1}, {16667, 16673, 1}, {16667, 16675, 3247}, {16668, 16674, 1}, {16669, 16673, 1449}, {16669, 16675, 1}, {16669, 16777, 16667}, {16669, 16814, 16675}, {16670, 16676, 1}, {16671, 16677, 1}, {16672, 16677, 37}, {16672, 16884, 3723}, {16673, 16777, 3247}, {16675, 16777, 37}, {16675, 16885, 16669}, {16677, 16884, 16672}, {16706, 17258, 4389}, {16713, 17183, 17197}, {16779, 16973, 1449}, {16788, 21078, 3553}, {16814, 16885, 1}, {16815, 17116, 4699}, {16816, 25269, 1278}, {16826, 17120, 17379}, {16832, 25590, 3739}, {16833, 17151, 4361}, {16969, 17448, 1}, {17121, 17319, 4393}, {17227, 17329, 17273}, {17227, 17341, 17283}, {17228, 17328, 17271}, {17228, 17342, 17285}, {17230, 17343, 17287}, {17231, 17344, 599}, {17233, 17346, 319}, {17234, 17347, 320}, {17235, 17356, 17290}, {17237, 17357, 3763}, {17238, 17358, 17292}, {17239, 17359, 17293}, {17240, 17360, 17295}, {17241, 17361, 17297}, {17242, 17363, 6542}, {17244, 17364, 17300}, {17245, 17365, 4675}, {17246, 17366, 17301}, {17247, 17367, 17302}, {17248, 17368, 2}, {17249, 17370, 17305}, {17250, 17371, 17307}, {17251, 17293, 17239}, {17252, 17292, 17238}, {17254, 17291, 17236}, {17255, 17290, 17235}, {17256, 17289, 5224}, {17256, 17354, 17308}, {17257, 17353, 17306}, {17257, 26685, 2}, {17257, 28287, 1423}, {17257, 29497, 29740}, {17258, 17352, 17304}, {17259, 17351, 25590}, {17260, 17350, 10436}, {17261, 17349, 3875}, {17262, 17348, 17151}, {17263, 17347, 17298}, {17264, 17346, 17294}, {17266, 17288, 17232}, {17268, 17287, 17230}, {17270, 17286, 3661}, {17271, 17285, 17228}, {17272, 17284, 141}, {17273, 17283, 17227}, {17274, 17282, 3662}, {17275, 17281, 594}, {17276, 17278, 1086}, {17276, 17337, 4859}, {17277, 17336, 3729}, {17277, 25728, 4659}, {17278, 17334, 4862}, {17278, 17337, 31183}, {17279, 17332, 17272}, {17280, 17331, 17270}, {17281, 17330, 3679}, {17299, 17362, 3632}, {17300, 20072, 17364}, {17315, 17377, 29605}, {17322, 17381, 29603}, {17324, 29630, 17383}, {17328, 17342, 17228}, {17329, 17341, 17227}, {17330, 17340, 594}, {17331, 17339, 3661}, {17333, 17338, 3662}, {17334, 17337, 1086}, {17335, 17336, 75}, {17353, 29497, 1423}, {17375, 29572, 17312}, {17379, 27268, 16826}, {17484, 31019, 31164}, {18228, 26059, 21246}, {18249, 18250, 10}, {20090, 29569, 17391}, {20196, 31231, 2}, {20683, 21746, 3779}, {21296, 29627, 4869}, {21616, 26363, 8227}, {21811, 33299, 27396}, {24152, 24153, 200}, {24313, 24314, 8}, {24477, 26105, 11019}, {24697, 33159, 32784}, {24909, 24952, 2}, {25019, 28739, 18634}, {25447, 25651, 2}, {25525, 28609, 226}, {25760, 33115, 29857}, {25842, 25856, 25860}, {26059, 27420, 10436}, {26885, 26890, 184}, {27254, 29967, 5219}, {27509, 28731, 63}, {27819, 27834, 19604}, {28606, 32911, 5256}, {29382, 29492, 3662}, {29395, 29423, 29511}, {29492, 29698, 17274}, {29509, 29541, 29396}, {30324, 30325, 226}, {30327, 30328, 226}, {30412, 30413, 2}, {30414, 30415, 2}, {30556, 30557, 1}, {30557, 31438, 1449}, {30557, 31453, 18991}, {31561, 31562, 4}, {31594, 31595, 10}, {32008, 32024, 85}, {32008, 32100, 30625}, {32088, 32100, 85}, {32555, 32556, 3}, {32784, 33159, 1698}, {32864, 32915, 17156}, {32917, 32931, 29828}, {33076, 33165, 3679}


X(10) = SPIEKER CENTER

Trilinears    bc(b + c) : ca(c + a) : ab(a + b)
Trilinears    1/(r cos A - s sin A) : :
Trilinears    csc(A - U) : :, U as at X(572) and X(573)
Trilinears    (cos B + cos C)/(1 - cos A) : :
Trilinears    1 + 2 csc A/2 sin B/2 sin C/2 : :
Trilinears    |AP(1)| + |AU(1)| : :
Trilinears    (r/R) - 2 sin B sin C : :
Barycentrics  b + c : c + a : a + b
Barycentrics    semi-major axis of A-Soddy ellipse : :
Tripolars    X(10) = Sqrt[-a^3 + 2*a*b^2 + b^3 - a*b*c + 2*a*c^2 + c^3] : :
X(10) = X(1) - 3X(2) = 3X(2) + X(8) = X(7) + X(8) + 2 X(9)

Let A' be the intersection of these three lines: the perpendicular from midpoint of CA to line BX(1), the perpendicular from midpoint of AB to line CX(1), the perpendicular from midpoint of AX(1) to line BC, and define B' and C' cyclically. The orthocenter of A'B'C' is X(10), and X(10) is also the perspector of A'B'C' and the medial triangle. Note that A'B'C' is the complement of the excentral triangle, and the extraversion triangle of X(10). (Randy Hutson, December 2, 2017)

The Spieker circle is the incircle of the medial triangle; its center, X(10), is the centroid of the perimeter of ABC. If you have The Geometer's Sketchpad, you can view Spieker center.
If you have GeoGebra, you can view Spieker center.

A construction of X(10) is given at 24163. (Antreas Hatzipolakis, August 29, 2016)

Let A'B'C' be the excentral triangle. X(10) is the radical center of the polar circles of triangles A'BC, B'CA, C'AB. (Randy Hutson, July 31 2018)

Let A20B20C20 be the Gemini triangle 20. Let LA be the line through A20 parallel to BC, and define LB and LC cyclically. Let A'20 = LB∩LC, and define B'20 and C'20 cyclically. Triangle A'20B'20C'20 is homothetic to ABC at X(10). (Randy Hutson, November 30, 2018)

Let OA be the circle centered at the A-excenter and passing through A; define OB and OC cyclically. X(10) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

Let OA be the circle centered at the A-vertex of the hexyl triangle and passing through A; define OB and OC cyclically. X(10) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

In the plane of a triangle ABC, let
Ba = reflection of A with in the external angular bisector of B, and define Cb and Ac cyclically;
Ca = reflection of A with in the external angular bisector of C, and define Ab and Bc cyclically;
Va = CBc∩BCb, and define Vb and Vc cyclically.
The triangle VaVbVc is perspective to ABC, and the perspector is X(10).
(Dasari Naga Vijay Krishna, April 19, 2021)

X(10) lies on the Kiepert hyperbola and these lines: 1,2   3,197   4,9   5,517   6,1377   11,121   12,65   20,165   21,35   28,1891   29,1794   31,964   33,406   34,475   36,404   37,594   38,596   39,730   44,752   46,63   55,405   56,474   57,388   58,171   69,969   75,76   81,1224   82,83   86,319   87,979   92,1838   98,101   106,1222   116,120   117,123   119,124   140,214   141,142   150,1282   153,1768   158,318   182,1678   190,671   191,267   201,225   219,965   227,1214   235,1902   255,1771   257,1581   261,1326   274,291   307,1254   321,756   348,1323   391,1743   407,1867   427,1829   429,1824   480,954   485,1686   486,1685   497,1697   514,764   535,1155   537,1086   626,760   631,944   632,1483   750,1150   774,1736   775,801   846,1247   894,1046   908,994   962,1695   1018,1334   1074,1735   1146,1212   1482,1656   1587,1703   1588,1702   1762,1782   1828,1883   1900,1904

X(10) is the {X(1),X(2)}-harmonic conjugate of X(1125). For a list of other harmonic conjugates of X(10), click Tables at the top of this page. X(10) is the internal center of similitude of the Apollonius and nine-points circles.

Let A'B'C' be the 2nd extouch triangle. Let A″ be the trilinear product B'*C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(10). Also, let A''B''C'' be the 1st circumperp triangle. The Simson lines of A'', B'', C'' concur in X(10). (Randy Hutson, November 18, 2015)

X(10) = midpoint of X(i) and X(j) for these (i,j): (1,8), (3,355), (4,40), (6,3416), (10,3421), (55,3419), (65,72), (80,100), (2948,3448)
X(10) = reflection of X(i) in X(j) for these (i,j): (1,1125), (551,2), (946,5), (1385,140)
X(10) = isogonal conjugate of X(58)
X(10) = isotomic conjugate of X(86)
X(10) = circumcircle-inverse of X(1324)
X(10) = nine-point-circle-inverse of X(3814)
X(10) = Conway-circle-inverse of X(38476)
X(10) = complement of X(1)
X(10) = anticomplement of X(1125)
X(10) = complementary conjugate of X(10)
X(10) = anticomplementary conjugate of X(2891)
X(10) = X(15319)-complementary conjugate of X(32767)
X(10) = radical center of the excircles.
X(10) = radical center of extraversions of Conway circle
X(10) = radical center of the polar circles of triangles BCI, CAI, ABI
X(10) = X(20)-of-3rd-Euler-triangle
X(10) = X(4)-of-4th-Euler-triangle
X(10) = perspector of ABC and the tangential triangle of the Feuerbach triangle
X(10) = X(2)-Hirst inverse of X(6542)
X(10) = inverse-in-Steiner-circumellipse of X(6542)
X(10) = SS(a→a') of X(5), where A'B'C' is the excentral triangle (barycentric substitution)
X(10) = orthocenter of X(2)X(4)X(4049)
X(10) = midpoint of PU(10)
X(10) = bicentric sum of PU(i) for these i: 10, 66
X(10) = PU(66)-harmonic conjugate of X(351)
X(10) = crosssum of X(i) and X(j) for these (i,j): (6,31), (56,603)
X(10) = crossdifference of every pair of points on line X(649)X(834)
X(10) = X(i)-beth conjugate of X(j) for these (i,j): (8,10), (10,65), (100,73), (318,225), (643,35), (668,349)
X(10) = radical trace of Bevan circle and anticomplementary circle
X(10) = insimilicenter of Bevan circle and anticomplementary circle
X(10) = insimilicenter of nine-point circle and Apollonius circle
X(10) = X(i)-Ceva conjugate of X(j) for these (i,j):
(2,37), (8,72), (75,321), (80,519), (100,522), (313,306)
X(10) = cevapoint of X(i) and X(j) for these (i,j): (1,191), (6,199), (12,201), (37,210), (42,71), (65,227)
X(10) = X(i)-cross conjugate of X(j) for these (i,j): (37,226), (71,306), (191,502), (201,72)
X(10) = crosspoint of X(i) and X(j) for these (i,j): (2,75), (8,318)
X(10) = centroid of ABCX(8)
X(10) = Kosnita(X(8),X(2)) point
X(10) = X(578)-of-2nd-extouch-triangle
X(10) = X(389)-of-excentral triangle
X(10) = X(125)-of-Fuhrmann triangle
X(10) = perspector of ABC and triangle formed from orthocenters of JaBC, JbCA, JcAB, where Ja, Jb, Jc are excenters
X(10) = perspector of circumconic centered at X(37)
X(10) = center of circumconic that is locus of trilinear poles of lines passing through X(37)
X(10) = trilinear pole of line X(523)X(661) (the polar of X(27) wrt polar circle)
X(10) = pole wrt polar circle of trilinear polar of X(27) (line X(242)X(514))
X(10) = X(48)-isoconjugate (polar conjugate)-of-X(27)
X(10) = X(6)-isoconjugate of X(81)
X(10) = X(75)-isoconjugate of X(2206)
X(10) = X(1101)-isoconjugate of X(3120)
X(10) = X(1)-of-X(1)-Brocard triangle
X(10) = perspector of medial triangle and Ayme triangle
X(10) = homothetic center of Ayme triangle and anticevian triangle of X(37)
X(10) = perspector of Ayme triangle and Danneels-Bevan triangle
X(10) = X(1)-of-Danneels-Bevan-triangle
X(10) = homothetic center of medial triangle and Danneels-Bevan triangle
X(10) = homothetic center of ABC and anticomplementary triangle of Danneels-Bevan triangle
X(10) = {X(2),X(8)}-harmonic conjugate of X(1)
X(10) = inverse-in-polar-circle of X(242)
X(10) = inverse-in-{circumcircle, nine-point circle}-inverter of X(5205)
X(10) = inverse-in-Steiner-inellipse of X(3912)
X(10) = inverse-in-Feuerbach-hyperbola of X(3057)
X(10) = perspector of Feuerbach and Apollonius triangles
X(10) = perspector of symmedial triangles of Feuerbach and Apollonius triangles
X(10) = perspector of circumsymmedial triangles of Feuerbach and Apollonius triangles
X(10) = perspector of tangential triangles of Feuerbach and Apollonius triangles
X(10) = X(214)-of-inner-Garcia-triangle
X(10) = Cundy-Parry Phi transform of X(13478)
X(10) = Cundy-Parry Psi transform of X(573)
X(10) = perspector of Ayme and 4th Euler triangles
X(10) = barycentric product X(101)*X(850)
X(10) = perspector of Gemini triangle 12 and cross-triangle of ABC and Gemini triangle 12
X(10) = perspector of ABC and cross-triangle of ABC and Gemini triangle 15
X(10) = trilinear product of vertices of Gemini triangle 15
X(10) = homothetic center of Ayme triangle and Gemini triangle 16
X(10) = center of the {ABC, Gemini 18}-circumconic
X(10) = Gemini-triangle-19-to-ABC parallelogic center
X(10) = centroid of Gemini triangle 20
X(10) = perspector of ABC and cross-triangle of ABC and Gemini triangle 25
X(10) = perspector of ABC and Gemini triangle 26
X(10) = perspector of Gemini triangle 39 and cross-triangle of ABC and Gemini triangle 39
X(10) = excentral-to-ABC barycentric image of X(3)
X(10) = incentral-to-ABC barycentric image of X(1)


X(11) = FEUERBACH POINT

Trilinears    1 - cos(B - C) : 1 - cos(C - A) : 1 - cos(A - B)
Trilinears    sin2(B/2 - C/2) : :
Trilinears    bc(b + c - a)(b - c)2 : :
Trilinears    1 - cos A - 2 cos B cos C : :
Trilinears    1 + cos A - 2 sin B sin C : :
Barycentrics    a(1 - cos(B - C)) : b(1 - cos(C - A)) : c(1 - cos(A -B))
Barycentrics    (b + c - a)(b - c)2 : :
Tripolars    Sqrt[-((a - b - c)*(a^4 - 2*a^2*b^2 + b^4 + a^2*b*c + a*b^2*c - 2*b^3*c - 2*a^2*c^2 + a*b*c^2 + 2*b^2*c^2 - 2*b*c^3 + c^4))] : :
X(11) = R*X(1) - 3rX(2) + r*X(3)
X(11) = 3 X[1] - X[7972], 3 X[1] + X[9897], X[1] + 2 X[12019], 3 X[1] - 2 X[12735], X[1] - 3 X[16173], X[1] + 3 X[37718], and many others

X(11) is the point of tangency of the nine-point circle and the incircle. The nine-point circle is the circumcenter of the medial triangle, as well as the orthic triangle. Feuerbach's famous theorem states that the nine-point circle is tangent to the incircle and the three excircles.

Let LA be the line through A parallel to X(1)X(3), and define LB and LC cyclically. Let MA be the reflection of BC in LA, and define MB and MC cyclically. Let A' = MB∩MC, and define cyclically B' and C' cyclically. The triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. Let A″B″C″ be the reflection of A'B'C' in line X(1)X(3). The triangle A″B″C″ is homothetic to ABC, with center of homothety X(11); see Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, March 25, 2016)

The circumcircle of the incentral triangle intersects the incircle at 2 points, X(11) and X(3024), and the nine-point circle at 2 points, X(11) and X(115). (Randy Hutson, April 9, 2016)

X(11) lies on the bicevian conic of X(1) and X(2), which is also QA-Co1 (Nine-point Conic) of quadrangle ABCX(1) (see http://www.chrisvantienhoven.nl/other-quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/76-qa-co1.html) (Randy Hutson, April 9, 2016)

Let Na = X(5) of BCX(1), Nb = X(5) of CAX(1), Nc = X(5) of ABX(1). Then X(11) = X(186) of NaNbNc. (Randy Hutson, April 9, 2016)

Let JaJbJc be the excentral triangle and FaFbFc be the Feuerbach triangle. Let Fa' = {X(5),Ja}-harmonic conjugate of Fa, and define Fb', Fc' cyclically. The lines AFa', BFb', CFc' concur in X(11).

Let A'B'C' be the orthic triangle. Let La be the antiorthic axis of AB'C', and define Lb and Lc cyclically. Let A″ = Lb∩Lc, and define B'' and C'' cyclically. The triangle A″B″C″ is inversely similar to ABC, with similitude center X(9), and X(11) = X(55)-of-A″B″C″. (Randy Hutson, December 10, 2016)

Let A'B'C' be the orthic triangle. Let Na be the Nagel line of AB'C', and define Nb and Nc cyclically. Let A″ = Nb∩Nc, and define B'' and C'' cyclically. The triangle A″B″C″ is inversely similar to ABC, and X(11) = X(36)-of-A″B″C″. (Randy Hutson, December 10, 2016)

Let A'B'C' be the orthic triangle. The lines IO of AB'C', BC'A', CA'B' concur in X(11). (Randy Hutson, December 10, 2016)

Let A'B'C' be the orthic triangle. The lines IO of AB'C', BC'A', CA'B' concur in X(11). (Randy Hutson, June 27, 2018)

Let A'B'C' be a homothetic triangle with ABC at X(2). Let Ab=AB?B'C', and define Bc and Ca cyclically. Let Ac=CA?B'C', and define Ba and Cb cyclically. Let (I), (Ia), (Ib), (Ic) be the incircles of ABC, A'BcCb, B'CaAc, C'AbBa, respectively. Then, there exists a circle (K) simultaneously tangent to (I), (Ia), (Ib), (Ic). The touchpoint of (I) and (K) is X(11). See Construction of X(11). (Keita Miyamoto, January 14, 2024)

If you have The Geometer's Sketchpad, you can view Feuerbach point.
If you have GeoGebra, you can view Feuerbach point.

In the plane of a triangle ABC, let
D = the point on line AC such that angle CAB = angle DAB and |AD| = |AB|
E = the point on line AB such that angle CAB = angle CAE and |AE| = |AC|
F = the point on line BC such that angle CBA = angle FBA and |BF| = |AB|
G = the point on line AB such that angle CBA = angle CBG and |BG| = |BC|
H = the point on line BC such that angle ACB = angle ACH and |CH| = |AC|
I = the point on line AC such that angle ACB = angle ICB and |CI| = |BC|
M = circumcenter of AGI
N = circumcenter of CDF
O = circumcenter of BEH
The nine point circles of triangles AFH, BDI, and CEG concur in X(11).
See figure. (Benjamin Warren, October 24, 2024)

X(11) lies on the following curves:

incircle, nine point circle, 2nd Lester circle (Yiu), Mandart circle
cevian circle of every point on the Feuerbach hyperbola
Mandart inconic
circumellipse of the medial and incentral triangles (see X(34585))
K672, K877, K925, K1051, Q015

X(11) lies on the following lines: {1, 5}, {2, 55}, {3, 499}, {4, 56}, {6, 1985}, {7, 658}, {8, 1320}, {9, 3254}, {10, 121}, {13, 202}, {14, 203}, {15, 11755}, {16, 11773}, {17, 7005}, {18, 7006}, {19, 37432}, {20, 5204}, {21, 4996}, {22, 9673}, {24, 9672}, {25, 10829}, {28, 1852}, {29, 40980}, {30, 36}, {31, 5348}, {32, 9665}, {33, 427}, {34, 235}, {35, 140}, {37, 5513}, {38, 4415}, {39, 13077}, {40, 6922}, {42, 37662}, {43, 33141}, {46, 12515}, {54, 2477}, {57, 1360}, {58, 37357}, {59, 33562}, {60, 3615}, {63, 7082}, {64, 12920}, {65, 117}, {67, 32287}, {68, 1069}, {69, 10755}, {72, 10395}, {74, 10767}, {75, 16067}, {76, 12836}, {78, 25681}, {79, 3065}, {81, 14008}, {83, 10799}, {84, 12676}, {86, 14009}, {88, 19636}, {90, 17437}, {98, 10768}, {99, 10769}, {101, 10770}, {102, 10771}, {103, 10772}, {106, 10774}, {107, 10775}, {109, 10777}, {110, 215}, {111, 10779}, {112, 10780}, {113, 942}, {114, 2783}, {115, 1015}, {116, 3022}, {118, 226}, {122, 2803}, {123, 2804}, {124, 1364}, {125, 3024}, {126, 2805}, {127, 2806}, {128, 7159}, {131, 37361}, {132, 1848}, {133, 1838}, {137, 3327}, {141, 3056}, {142, 5580}, {145, 5154}, {150, 24203}, {153, 388}, {155, 10071}, {165, 9580}, {171, 33106}, {172, 7745}, {174, 8086}, {177, 12614}, {181, 2051}, {182, 38119}, {184, 9667}, {185, 26955}, {190, 17777}, {192, 7777}, {197, 37366}, {198, 37367}, {200, 4863}, {210, 3452}, {212, 748}, {214, 442}, {222, 34029}, {225, 37368}, {229, 37369}, {230, 1914}, {233, 21860}, {238, 1936}, {239, 26019}, {241, 33331}, {243, 17923}, {244, 676}, {255, 7299}, {262, 12837}, {265, 10091}, {273, 6046}, {278, 1857}, {279, 15511}, {312, 3703}, {314, 11609}, {321, 21333}, {325, 350}, {329, 5825}, {333, 37373}, {334, 18149}, {344, 30741}, {345, 4387}, {371, 9661}, {372, 13977}, {377, 22768}, {381, 999}, {382, 4299}, {384, 26686}, {386, 9555}, {395, 7127}, {398, 2307}, {402, 11903}, {403, 1870}, {404, 6691}, {405, 19755}, {428, 5322}, {429, 1104}, {430, 40956}, {431, 40985}, {440, 23207}, {443, 31418}, {452, 30478}, {474, 2932}, {479, 36620}, {480, 6601}, {481, 31555}, {482, 31556}, {484, 28174}, {485, 1124}, {486, 1335}, {493, 10945}, {494, 10946}, {498, 1656}, {500, 10035}, {511, 15974}, {512, 46671}, {513, 3025}, {514, 3328}, {515, 1319}, {516, 1155}, {517, 1737}, {518, 908}, {519, 3814}, {521, 34940}, {522, 3326}, {523, 1090}, {524, 4396}, {527, 41555}, {529, 5080}, {537, 21093}, {546, 3585}, {547, 3584}, {548, 4324}, {549, 5010}, {550, 7280}, {551, 3822}, {553, 13159}, {559, 51749}, {573, 9554}, {574, 9664}, {578, 9653}, {590, 2066}, {594, 17452}, {595, 45939}, {596, 44040}, {597, 38090}, {609, 18907}, {611, 14561}, {612, 37439}, {613, 1352}, {615, 5414}, {618, 13076}, {619, 13075}, {620, 15452}, {625, 5148}, {626, 6029}, {629, 22910}, {630, 22865}, {631, 4294}, {632, 10386}, {641, 13082}, {642, 13081}, {644, 26074}, {650, 1566}, {656, 38981}, {659, 33311}, {661, 20974}, {662, 19642}, {671, 12348}, {672, 17747}, {693, 23989}, {740, 51411}, {750, 33104}, {756, 29690}, {758, 11813}, {759, 3109}, {774, 1393}, {799, 19643}, {851, 20470}, {858, 3100}, {899, 33136}, {901, 31512}, {905, 15612}, {912, 5570}, {920, 37532}, {936, 24954}, {938, 3485}, {940, 26098}, {943, 24298}, {944, 1388}, {953, 40437}, {956, 17556}, {958, 2478}, {960, 6734}, {962, 1788}, {971, 1538}, {982, 3782}, {984, 29676}, {988, 50065}, {993, 11113}, {997, 3419}, {1000, 24297}, {1011, 19720}, {1012, 1470}, {1038, 6823}, {1040, 1368}, {1043, 14011}, {1054, 24715}, {1056, 3545}, {1058, 3085}, {1060, 15760}, {1061, 24300}, {1062, 11585}, {1071, 1898}, {1082, 51750}, {1100, 50036}, {1108, 1826}, {1111, 1358}, {1113, 10781}, {1114, 10782}, {1118, 40836}, {1122, 24213}, {1146, 1639}, {1147, 9666}, {1151, 9662}, {1168, 14629}, {1172, 43735}, {1191, 5230}, {1193, 1834}, {1201, 21935}, {1209, 13079}, {1211, 3741}, {1212, 28052}, {1213, 2269}, {1215, 29655}, {1250, 23303}, {1253, 17125}, {1259, 43740}, {1279, 3011}, {1284, 8229}, {1290, 36175}, {1297, 12925}, {1312, 2589}, {1313, 2588}, {1318, 36590}, {1327, 13693}, {1328, 13813}, {1346, 2463}, {1347, 2464}, {1348, 1674}, {1349, 1675}, {1354, 4292}, {1356, 44950}, {1357, 2827}, {1365, 4934}, {1366, 3664}, {1385, 6842}, {1386, 38050}, {1397, 13478}, {1398, 37197}, {1399, 3073}, {1402, 45189}, {1420, 5691}, {1425, 43820}, {1427, 1856}, {1428, 1503}, {1435, 7008}, {1440, 7023}, {1447, 4872}, {1455, 1877}, {1456, 34049}, {1457, 51421}, {1458, 2635}, {1460, 2050}, {1465, 8758}, {1466, 6847}, {1469, 5480}, {1482, 6971}, {1491, 42771}, {1500, 1506}, {1511, 12896}, {1513, 38646}, {1519, 6001}, {1560, 40941}, {1575, 21956}, {1587, 18995}, {1588, 18996}, {1591, 3083}, {1592, 3084}, {1594, 6198}, {1598, 9913}, {1617, 19541}, {1650, 11909}, {1657, 38754}, {1672, 1676}, {1673, 1677}, {1697, 1698}, {1728, 5812}, {1731, 7359}, {1738, 5121}, {1739, 45269}, {1746, 44085}, {1751, 6056}, {1758, 43056}, {1770, 22793}, {1776, 3218}, {1785, 39535}, {1795, 10747}, {1804, 7318}, {1827, 16580}, {1844, 18402}, {1853, 2192}, {1854, 17054}, {1855, 46830}, 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30957}, {25958, 33172}, {26001, 26002}, {26005, 26010}, {26102, 33111}, {26128, 29668}, {26326, 26380}, {26327, 26404}, {26328, 26433}, {26329, 26434}, {26330, 26435}, {26331, 26436}, {26332, 26437}, {26351, 26359}, {26352, 26360}, {26355, 26361}, {26356, 26362}, {26365, 48501}, {26366, 48502}, {26367, 49410}, {26368, 49409}, {26369, 49068}, {26370, 49069}, {26386, 45373}, {26389, 26401}, {26410, 45374}, {26413, 26425}, {26466, 45614}, {26467, 45613}, {26468, 49032}, {26469, 49033}, {26483, 26501}, {26484, 26510}, {26485, 26519}, {26486, 26524}, {26580, 46909}, {26725, 33857}, {26904, 26906}, {26942, 44707}, {26949, 26951}, {27020, 32992}, {27064, 33121}, {27739, 50316}, {27747, 50305}, {28186, 36975}, {28224, 37006}, {28234, 36920}, {28530, 32845}, {28606, 29680}, {28795, 30826}, {28797, 30847}, {29349, 34583}, {29677, 31237}, {29685, 31264}, {29820, 33130}, {29827, 32784}, {29830, 30834}, {29835, 46897}, {29840, 32926}, {29844, 32920}, {29846, 32943}, {29849, 32915}, {29861, 33159}, {29872, 33157}, {30117, 45272}, {30304, 41706}, {30312, 30332}, {30313, 30333}, {30314, 30334}, {30315, 30337}, {30316, 30338}, {30317, 30339}, {30375, 30380}, {30376, 30381}, {30377, 30382}, {30378, 30383}, {30447, 50757}, {30823, 49768}, {30824, 36479}, {30831, 33173}, {30963, 37664}, {30971, 36635}, {31003, 39691}, {31137, 33087}, {31145, 50894}, {31221, 37586}, {31263, 48696}, {31393, 31434}, {31401, 31448}, {31402, 31404}, {31408, 31412}, {31409, 31415}, {31426, 31428}, {31433, 31441}, {31451, 31455}, {31459, 31463}, {31461, 31467}, {31471, 31481}, {31473, 31484}, {31477, 31489}, {31500, 35255}, {31557, 31567}, {31558, 31568}, {31586, 31588}, {31587, 31589}, {31654, 44048}, {31657, 38124}, {31658, 38131}, {31659, 37621}, {31775, 37561}, {31946, 38979}, {31996, 33033}, {32243, 32274}, {32288, 32310}, {32336, 32369}, {32351, 32378}, {32379, 32404}, {32381, 32406}, {32390, 32391}, {32631, 43757}, {32911, 33142}, {32913, 33096}, {32918, 32947}, {32930, 33119}, {32931, 33120}, {33112, 37633}, {33139, 37680}, {33170, 41242}, {33178, 38336}, {33504, 44049}, {33650, 34234}, {33898, 41426}, {33925, 42884}, {34030, 34040}, {34467, 34948}, {34595, 41859}, {34699, 45701}, {34747, 50893}, {34855, 51364}, {34903, 34904}, {35065, 42337}, {35094, 45320}, {35104, 51407}, {35250, 35252}, {35508, 38973}, {35580, 44047}, {35581, 44050}, {35582, 44051}, {35587, 44052}, {35591, 44053}, {35663, 35671}, {35664, 35669}, {35678, 35679}, {35762, 35852}, {35763, 35853}, {35798, 35818}, {35799, 35819}, {36436, 36451}, {36439, 36443}, {36457, 36461}, {36473, 36493}, {36487, 36526}, {36495, 36513}, {36508, 36557}, {36530, 36542}, {36545, 36547}, {36561, 36574}, {36577, 36579}, {36999, 50701}, {37520, 50307}, {37525, 38028}, {37592, 44954}, {37596, 44951}, {37607, 49745}, {37660, 50295}, {37681, 38293}, {37686, 41324}, {37756, 50533}, {37764, 49704}, {37787, 38454}, {38018, 38238}, {38074, 50907}, {38076, 50906}, {38163, 45977}, {38188, 51150}, {38472, 51377}, {38941, 43038}, {38985, 47601}, {38992, 41224}, {39004, 41220}, {39816, 39822}, {39845, 39851}, {39890, 39903}, {40964, 46878}, {41697, 49177}, {41877, 45223}, {42312, 44316}, {42770, 47123}, {43728, 46041}, {43817, 43819}, {43821, 43858}, {43860, 43862}, {43986, 44001}, {44229, 45630}, {44620, 44645}, {44621, 44646}, {44661, 51410}, {44663, 51423}, {45305, 49537}, {45398, 49337}, {45399, 49338}, {45404, 45440}, {45405, 45441}, {45456, 45496}, {45457, 45497}, {45470, 45472}, {45471, 45473}, {45506, 45544}, {45507, 45545}, {45554, 45582}, {45555, 45583}, {45558, 45586}, {45559, 45587}, {46436, 47019}, {46659, 47020}, {48641, 50117}, {49658, 49659}, {50194, 51709}, {50605, 50621}

X(11) = midpoint of X(i) and X(j) for these {i,j}: {1, 80}, {2, 10707}, {3, 10738}, {4, 104}, {5, 1484}, {7, 1156}, {8, 1320}, {9, 3254}, {10, 21630}, {20, 10724}, {21, 11604}, {36, 3583}, {40, 14217}, {56, 12764}, {65, 17638}, {69, 10755}, {74, 10767}, {76, 32454}, {79, 3065}, {84, 46435}, {98, 10768}, {99, 10769}, {100, 149}, {101, 10770}, {102, 10771}, {103, 10772}, {105, 10773}, {106, 10774}, {107, 10775}, {108, 10776}, {109, 10777}, {110, 10778}, {111, 10779}, {112, 10780}, {119, 37726}, {145, 12531}, {153, 38669}, {314, 11609}, {355, 12737}, {382, 38753}, {390, 20119}, {551, 50889}, {901, 31512}, {908, 26015}, {943, 24298}, {946, 10265}, {1000, 24297}, {1061, 24300}, {1109, 2611}, {1113, 10781}, {1114, 10782}, {1172, 43735}, {1290, 36175}, {1387, 12019}, {1479, 10090}, {1482, 19914}, {1650, 13268}, {1699, 11219}, {1737, 30384}, {1768, 34789}, {1837, 12740}, {2481, 14947}, {3057, 17636}, {3218, 5057}, {3241, 50890}, {3259, 6075}, {3632, 26726}, {3679, 50891}, {3680, 12641}, {3828, 50892}, {3868, 12532}, {3874, 47320}, {3937, 38389}, {4010, 13277}, {4106, 42322}, {4440, 36237}, {5176, 38460}, {5533, 39692}, {5559, 13143}, {5620, 13604}, {6246, 11715}, {6264, 12751}, {6326, 49176}, {6595, 10266}, {6596, 6598}, {6597, 6599}, {6601, 34894}, {7972, 9897}, {9318, 24712}, {9809, 13243}, {10309, 34256}, {10609, 12690}, {10698, 12247}, {10728, 12248}, {10742, 12773}, {10914, 17652}, {11256, 32049}, {12114, 12761}, {12515, 12699}, {12672, 17654}, {12868, 15998}, {13205, 13271}, {13266, 46403}, {13272, 22560}, {13463, 32198}, {15635, 44013}, {16173, 37718}, {17668, 36868}, {17763, 32844}, {22799, 51529}, {22938, 38602}, {24302, 43731}, {24851, 24852}, {31145, 50894}, {32843, 32919}, {34501, 34503}, {34747, 50893}, {39144, 39145}, {40565, 40566}, {43728, 46041}, {43740, 45393}, {45981, 45982}, {51402, 51442}
reflection of X(i) in X(j) for these {i,j}: {1, 1387}, {2, 45310}, {3, 6713}, {8, 3036}, {10, 6702}, {12, 8068}, {20, 38759}, {36, 15325}, {59, 33562}, {65, 12736}, {72, 18254}, {80, 12019}, {100, 3035}, {104, 20418}, {105, 33970}, {119, 5}, {153, 38757}, {214, 1125}, {500, 10035}, {650, 10006}, {908, 5087}, {946, 16174}, {1071, 15528}, {1125, 33709}, {1145, 10}, {1155, 3911}, {1317, 1}, {1319, 44675}, {1519, 22835}, {1537, 946}, {1768, 13226}, {2720, 28347}, {3025, 14115}, {3035, 6667}, {3057, 15558}, {3649, 33593}, {3689, 6745}, {4996, 4999}, {5083, 18240}, {5298, 3582}, {5520, 47399}, {5948, 10277}, {6068, 9}, {6154, 100}, {6174, 2}, {6265, 11729}, {6594, 6666}, {6735, 5123}, {7972, 12735}, {10036, 8143}, {10427, 142}, {10609, 214}, {10993, 33814}, {11274, 51103}, {11570, 942}, {11700, 29008}, {12611, 9955}, {12665, 5777}, {12743, 950}, {12831, 226}, {12832, 1210}, {13257, 21635}, {13996, 1145}, {14115, 33646}, {14740, 46694}, {15326, 36}, {17660, 5083}, {17757, 3814}, {18239, 34293}, {18801, 8255}, {18802, 8256}, {19907, 5901}, {21578, 5126}, {22799, 546}, {24466, 3}, {24685, 25342}, {25485, 13464}, {25558, 25557}, {25606, 8257}, {27778, 17660}, {30384, 7743}, {31235, 31272}, {31525, 11735}, {33667, 10122}, {33812, 3636}, {33814, 140}, {34123, 32557}, {35204, 6675}, {35604, 34949}, {37725, 119}, {37726, 1484}, {38389, 38390}, {38752, 38319}, {38760, 34126}, {38761, 38602}, {39144, 45981}, {39145, 45982}, {39776, 5836}, {39778, 11281}, {40663, 1737}, {41541, 13411}, {41553, 13405}, {41556, 11019}, {41558, 6738}, {41684, 11545}, {44238, 17009}, {46409, 6714}, {50841, 3828}, {50842, 3679}, {50843, 551}, {50846, 3241}, {51007, 141}, {51008, 597}, {51062, 37}, {51157, 3589}, {51158, 20582}, {51198, 6}, {51377, 38472}, {51390, 24250}, {51409, 11813}, {51463, 26015}, {51569, 6701}
X(11) = isogonal conjugate of X(59)
X(11) = isotomic conjugate of X(4998)
X(11) = complement of X(100)
X(11) = anticomplement of X(3035)
X(11) = complementary conjugate of X(513)
X(11) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,523), (4,513), (7,514), (8,522), (262,1491)
X(11) = crosspoint of X(i) and X(j) for these (i,j): (7,514), (8,522)
X(11) = crosssum of X(i) and X(j) for these (i,j): (6,692), (55,101), (56,109), (1381,1382), (1397,1415)
X(11) = crossdifference of every pair of points on line X(101)X(109)
X(11) = circumcircle-inverse of X(14667)
X(11) = Fuhrmann-circle-inverse of X(1837)
X(11) = Stevanovic-circle-inverse of X(1566)
X(11) = Conway-circle-inverse of X(13244)
X(11) = Spieker-radical-circle-inverse of X(3030)
X(11) = polar-circle-inverse of X(108)
X(11) = orthoptic-circle-of-Steiner-inellipse-inverse of X(105)
X(11) = orthoptic-circle-of-Steiner-circumellipse-inverse of X(20097)
X(11) = inverse-in-{circumcircle, nine-point circle}-inverter of X(105)
X(11) = inverse-in-excircles-radical-circle of X(3030)
X(11) = polar conjugate of X(46102)
X(11) = anticomplement of the isogonal conjugate of X(18771)
X(11) = complement of the isogonal conjugate of X(513)
X(11) = complement of the isotomic conjugate of X(693)
X(11) = isogonal conjugate of the anticomplement of X(46100)
X(11) = isotomic conjugate of the anticomplement of X(46101)
X(11) = isotomic conjugate of the complement of X(17036)
X(11) = isotomic conjugate of the isogonal conjugate of X(3271)
X(11) = isogonal conjugate of the isotomic conjugate of X(34387)
X(11) = isotomic conjugate of the polar conjugate of X(8735)
X(11) = polar conjugate of the isotomic conjugate of X(26932)
X(11) = polar conjugate of the isogonal conjugate of X(7117)
X(11) = medial-isogonal conjugate of X(513)
X(11) = orthic-isogonal conjugate of X(513)
X(11) = psi-transform of X(18343)
X(11) = X(i)-beth conjugate of X(j) for these (i,j): (11,244), (522,11), (693,11)
X(11) = orthopole of line X(1)X(3)
X(11) = anticenter of cyclic quadrilateral ABCX(104)
X(11) = perspector of ABC and extraversion triangle of X(12)
X(11) = homothetic center of intouch and 3rd Euler triangles
X(11) = trilinear square root of X(6728)
X(11) = perspector of Feuerbach triangle and Schroeter triangle
X(11) = X(110)-of-intouch-triangle
X(11) = X(403) of Fuhrmann triangle
X(11) = perspector of circumconic centered at X(650)
X(11) = center of circumconic that is locus of trilinear poles of lines passing through X(650)
X(11) = X(2)-Ceva conjugate of X(650)
X(11) = trilinear pole wrt intouch triangle of Soddy line
X(11) = trilinear pole wrt extouch triangle of line X(8)X(9)
X(11) = midpoint of PU(i) for these i: 121, 123
X(11) = bicentric sum of PU(i) for these i: 121, 123
X(11) = homothetic center of medial triangle and Mandart-incircle triangle
X(11) = X(100) of Mandart-incircle triangle
X(11) = X(3659) of orthic triangle if ABC is acute
X(11) = homothetic center of intangents triangle and reflection of extangents triangle in X(100)
X(11) = homothetic center of 3rd Euler triangle and intouch triangle
X(11) = QA-P2 (Euler-Poncelet Point) of quadrangle ABCX(1) (see http://www.chrisvantienhoven.nl/quadrangle-objects/10-mathematics/quadrangle-objects/12-qa-p2.html)
X(11) = intersection of tangents to Steiner inellipse at X(1086) and X(1146)
X(11) = crosspoint wrt medial triangle of X(1086) and X(1146)
X(11) = perspector of orthic triangle and tangential triangle of hyperbola {{A,B,C,X(1),X(2)}}
X(11) = homothetic center of cyclic quadrilateral ABCX(104) and congruent quadrilateral formed by orthocenters of vertices taken 3 at a time
X(11) = perspector of ABC and cross-triangle of ABC and Feuerbach triangle
X(11) = homothetic center of medial triangle and cross-triangle of ABC and inner Johnson triangle
X(11) = homothetic center of Euler triangle and cross-triangle of ABC and 1st Johnson-Yff triangle
X(11) = homothetic center of medial triangle and cross-triangle of ABC and 2nd Johnson-Yff triangle
X(11) = orthic-isogonal conjugate of X(513)
X(11) = trilinear pole of line X(4530)X(14393)
X(11) = point of concurrence of cevian circles of vertices of anticevian triangle of X(7)
X(11) = homothetic center of cyclic quadrilateral ABCX(104) and congruent quadrilateral formed by orthocenters of vertices taken 3 at a time
X(11) = homothetic center of Ursa-minor and Ursa-major triangles
X(11) = homothetic center of ABC and inner Johnson triangle
X(11) = trilinear product X(57)*X(1146)
X(11) = barycentric product X(7)*X(1146)
X(11) = homothetic center of Garcia reflection triangle (aka Gemini triangle 8) and 2nd Schiffler triangle
X(11) = excentral-to-ABC functional image of X(3659)
X(11) = excentral-to-ABC barycentric image of X(100)
X(11) = Pelletier-isogonal conjugate of X(35604)
X(11) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {18771, 8}, {31628, 20295}, {38809, 21272}
X(11) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 513}, {2, 3835}, {3, 20315}, {4, 20316}, {6, 514}, {7, 17072}, {9, 20317}, {10, 31946}, {11, 124}, {25, 3239}, {27, 30476}, {28, 8062}, {31, 650}, {32, 6586}, {34, 521}, {37, 4129}, {42, 661}, {55, 4521}, {56, 522}, {57, 4885}, {58, 523}, {75, 21260}, {76, 21262}, {81, 4369}, {86, 512}, {87, 4083}, {88, 4928}, {89, 47779}, {100, 24003}, {101, 4422}, {105, 3716}, {106, 900}, {109, 3035}, {111, 45661}, {190, 27076}, {223, 20314}, {239, 27854}, {244, 11}, {256, 21051}, {269, 3900}, {274, 42327}, {278, 46396}, {279, 46399}, {282, 20318}, {286, 21259}, {291, 3837}, {292, 812}, {310, 23301}, {330, 21191}, {335, 21261}, {512, 1213}, {513, 10}, {514, 141}, {521, 34823}, {522, 1329}, {523, 3454}, {593, 21196}, {596, 44316}, {604, 905}, {608, 14837}, {614, 17115}, {647, 440}, {649, 2}, {650, 3452}, {651, 21232}, {656, 21530}, {657, 6554}, {659, 17793}, {661, 1211}, {663, 9}, {665, 16593}, {667, 37}, {669, 21838}, {676, 118}, {692, 24036}, {693, 2887}, {739, 4763}, {741, 9508}, {749, 23815}, {798, 16589}, {810, 18591}, {812, 20333}, {849, 31947}, {870, 788}, {875, 1575}, {876, 3836}, {884, 40869}, {893, 25666}, {899, 14434}, {900, 121}, {902, 6544}, {904, 3709}, {905, 18589}, {918, 20540}, {977, 832}, {985, 4874}, {987, 6133}, {996, 9002}, {998, 9001}, {1002, 24720}, {1015, 1086}, {1019, 3739}, {1022, 3834}, {1024, 34852}, {1027, 518}, {1042, 656}, {1073, 20319}, {1086, 116}, {1096, 14298}, {1106, 6129}, {1111, 21252}, {1120, 6085}, {1126, 4977}, {1169, 8045}, {1193, 50330}, {1201, 6615}, {1220, 6371}, {1222, 6363}, {1245, 47842}, {1252, 10196}, {1255, 48049}, {1333, 14838}, {1357, 3756}, {1358, 17059}, {1390, 48050}, {1395, 6588}, {1397, 6589}, {1398, 21172}, {1400, 1577}, {1407, 7658}, {1411, 3738}, {1412, 17069}, {1413, 8058}, {1415, 16578}, {1416, 676}, {1431, 3907}, {1434, 17066}, {1438, 918}, {1458, 3126}, {1459, 3}, {1461, 17044}, {1474, 525}, {1477, 4925}, {1577, 21245}, {1635, 16594}, {1638, 31844}, {1647, 3259}, {1769, 119}, {1911, 665}, {1914, 27929}, {1919, 39}, {1960, 4370}, {1973, 2509}, {1977, 6377}, {1980, 16584}, {2006, 46397}, {2107, 46390}, {2162, 31286}, {2163, 4777}, {2170, 26932}, {2191, 3309}, {2203, 16612}, {2206, 647}, {2215, 23882}, {2254, 120}, {2279, 4762}, {2291, 45326}, {2297, 8712}, {2308, 4988}, {2309, 40627}, {2310, 5514}, {2334, 4778}, {2350, 693}, {2382, 45666}, {2384, 45684}, {2423, 3911}, {2424, 516}, {2530, 21249}, {2605, 3647}, {2983, 29162}, {3063, 1212}, {3064, 41883}, {3120, 125}, {3121, 16592}, {3122, 115}, {3123, 5518}, {3124, 6627}, {3125, 8287}, {3223, 25142}, {3226, 6373}, {3248, 1015}, {3261, 626}, {3270, 40616}, {3271, 1146}, {3415, 4522}, {3445, 3667}, {3572, 3912}, {3667, 2885}, {3669, 142}, {3676, 2886}, {3699, 3038}, {3709, 38930}, {3720, 50497}, {3733, 1125}, {3737, 960}, {3756, 5510}, {3766, 20542}, {3768, 13466}, {3835, 21250}, {3837, 20551}, {3937, 2968}, {3939, 3039}, {4010, 45162}, {4017, 442}, {4025, 1368}, {4040, 40607}, {4057, 4075}, {4079, 6537}, {4083, 34832}, {4091, 6389}, {4105, 5574}, {4107, 39080}, {4164, 19563}, {4367, 51575}, {4391, 21244}, {4444, 20541}, {4455, 35068}, {4466, 127}, {4556, 620}, {4560, 21246}, {4581, 3831}, {4584, 40548}, {4598, 40562}, {4603, 40546}, {4724, 3789}, {4750, 126}, {4775, 16590}, {4777, 21251}, {4817, 21264}, {5029, 6651}, {5331, 8672}, {6129, 6260}, {6164, 11814}, {6186, 3700}, {6187, 1639}, {6373, 20532}, {6385, 21263}, {6548, 21241}, {6591, 226}, {6729, 2090}, {7004, 123}, {7050, 43061}, {7117, 16596}, {7121, 21348}, {7123, 11068}, {7151, 14331}, {7178, 17052}, {7180, 17056}, {7192, 3741}, {7199, 21240}, {7203, 3742}, {7216, 18635}, {7250, 1834}, {7252, 5745}, {7649, 5}, {7658, 2884}, {8050, 36951}, {8578, 39026}, {8632, 17755}, {8643, 3161}, {8656, 36911}, {8700, 45679}, {8747, 520}, {9262, 190}, {9268, 6550}, {9309, 4147}, {9315, 31287}, {9456, 3960}, {10013, 6005}, {10566, 3934}, {10579, 8713}, {11125, 113}, {13476, 50337}, {14413, 10427}, {14419, 16597}, {14936, 13609}, {16079, 4943}, {16726, 17761}, {16732, 21253}, {16757, 21247}, {16892, 21248}, {17096, 17050}, {17187, 3005}, {17924, 20305}, {17925, 34830}, {17954, 2787}, {17962, 2786}, {18001, 10026}, {18014, 20546}, {18098, 29512}, {18108, 1215}, {18191, 34589}, {18210, 34846}, {18344, 20262}, {20974, 40618}, {20979, 6376}, {21102, 1209}, {21109, 15116}, {21122, 40938}, {21123, 6292}, {21132, 46100}, {21143, 6547}, {21172, 2883}, {21758, 16586}, {21832, 46842}, {22350, 42769}, {22383, 1214}, {23189, 34851}, {23345, 519}, {23351, 5199}, {23355, 726}, {23472, 41771}, {23493, 798}, {23572, 6374}, {23838, 5123}, {23892, 536}, {24002, 17046}, {24012, 17426}, {25417, 4932}, {25426, 28840}, {25430, 4940}, {25576, 25107}, {26721, 23305}, {27789, 48041}, {27846, 38989}, {28615, 48003}, {30571, 4806}, {30650, 47778}, {30651, 48008}, {32674, 36949}, {32735, 24980}, {32739, 23988}, {34080, 25097}, {34444, 8714}, {34893, 2832}, {34916, 4160}, {35058, 23803}, {35348, 5087}, {35355, 3823}, {35365, 50752}, {36123, 8677}, {36127, 3042}, {36598, 29226}, {37129, 891}, {37627, 3880}, {38266, 3669}, {38346, 40619}, {38357, 46663}, {38986, 40610}, {39949, 4132}, {39961, 47996}, {39965, 48399}, {39972, 29198}, {40148, 649}, {40433, 6372}, {40495, 21235}, {40746, 824}, {40763, 3805}, {40958, 40628}, {41434, 28209}, {41436, 6006}, {43051, 20528}, {43531, 834}, {43922, 1647}, {43923, 1210}, {43924, 1}, {43925, 40940}, {43926, 50755}, {43927, 50605}, {43928, 4871}, {43929, 3008}, {43931, 3840}, {43932, 11019}, {46018, 4791}, {46107, 21243}, {47947, 17239}, {48032, 40609}, {48131, 51571}, {48144, 10472}, {48306, 51573}, {48340, 51572}, {50344, 3634}, {50521, 16587}, {50525, 28651}, {51223, 47843}, {51443, 4913}, {51476, 2490}, {51640, 18642}, {51642, 2092}, {51648, 35204}, {51650, 41540}, {51654, 6600}, {51658, 12640}, {51686, 23874}
X(11) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 523}, {2, 650}, {4, 513}, {5, 8819}, {7, 514}, {8, 522}, {9, 6362}, {10, 17420}, {75, 21120}, {79, 4977}, {80, 900}, {83, 3287}, {98, 659}, {100, 15914}, {261, 4560}, {262, 1491}, {273, 7178}, {278, 3064}, {312, 3700}, {314, 3910}, {333, 4976}, {497, 11934}, {514, 42462}, {522, 21132}, {596, 21119}, {650, 42454}, {693, 40166}, {941, 784}, {1000, 4777}, {1086, 4534}, {1088, 21104}, {1111, 1086}, {1156, 2826}, {1222, 40500}, {1320, 2804}, {1440, 3669}, {1751, 652}, {2051, 661}, {2298, 29142}, {2321, 48264}, {2481, 918}, {2997, 525}, {3254, 6366}, {3296, 4802}, {3434, 11927}, {3596, 4391}, {3615, 3737}, {3680, 42337}, {3701, 4985}, {4518, 50333}, {4858, 1146}, {4997, 1639}, {5397, 50349}, {5551, 28199}, {5556, 4778}, {5557, 28175}, {5558, 28147}, {5559, 28183}, {5560, 28217}, {5561, 28209}, {6384, 20508}, {6557, 3239}, {6601, 3900}, {7155, 3810}, {7249, 3004}, {7261, 812}, {7317, 28205}, {7318, 905}, {7319, 3667}, {7320, 28161}, {7607, 48226}, {7608, 47827}, {9221, 18116}, {11604, 3738}, {13478, 649}, {13606, 28187}, {14458, 50358}, {14484, 2526}, {14492, 50328}, {14554, 46393}, {15314, 29162}, {17197, 2170}, {17501, 39386}, {17758, 21127}, {18101, 3271}, {18490, 28151}, {18815, 10015}, {23836, 6550}, {24026, 38357}, {24298, 8674}, {24624, 654}, {26721, 21133}, {26856, 11998}, {30101, 23755}, {30479, 23880}, {30710, 47135}, {32016, 24113}, {32023, 693}, {34387, 26932}, {35097, 8677}, {36590, 23838}, {36620, 3676}, {38254, 7658}, {38255, 4521}, {39768, 6370}, {40419, 17494}, {40450, 1}, {40451, 244}, {40836, 7649}, {41527, 824}, {42318, 14330}, {43531, 17418}, {43672, 2254}, {43731, 28221}, {43732, 28213}, {43733, 28195}, {43734, 4926}, {43740, 521}, {43741, 35057}, {43749, 3907}, {43759, 4773}, {43972, 21106}, {44426, 42455}, {44733, 4841}, {45100, 48026}, {46103, 7252}, {51685, 4990}
X(11) = X(i)-cross conjugate of X(j) for these (i,j): {2170, 1086}, {2310, 1146}, {3259, 35015}, {3271, 8735}, {4516, 2170}, {4542, 4530}, {4953, 4534}, {5532, 42462}, {7117, 26932}, {7336, 21132}, {18210, 7004}, {21044, 4858}, {21132, 522}, {23615, 3064}, {42462, 514}, {42547, 43974}, {46101, 2}
X(11) = cevapoint of X(i) and X(j) for these (i,j): {1, 2957}, {2, 17036}, {650, 11193}, {1146, 4953}, {1647, 3259}, {2170, 2310}, {3120, 18210}, {3271, 7117}, {4516, 21044}, {4530, 4542}, {5532, 42462}, {7336, 21132}
X(11) = crosspoint of X(i) and X(j) for these (i,j): {1, 3737}, {2, 693}, {4, 44426}, {7, 514}, {8, 522}, {261, 4560}, {278, 3676}, {312, 18155}, {513, 34434}, {1111, 4858}, {3307, 3308}, {3596, 4391}, {4397, 6556}, {40437, 43728}
X(11) = crosssum of X(i) and X(j) for these (i,j): {1, 4551}, {3, 36059}, {6, 692}, {55, 101}, {56, 109}, {100, 2975}, {108, 41227}, {181, 4559}, {215, 1983}, {219, 3939}, {906, 6056}, {1110, 2149}, {1362, 2283}, {1381, 1382}, {1397, 1415}, {7066, 23067}, {17455, 23344}, {23981, 34586}
X(11) = trilinear pole of line {4530, 14393}
X(11) = crossdifference of every pair of points on line {101, 109}
X(11) = X(i)-line conjugate of X(j) for these (i,j): {10770, 101}, {10777, 109}
X(11) = X(i)-isoconjugate of X(j) for these (i,j): {1, 59}, {2, 2149}, {3, 7012}, {6, 4564}, {7, 1110}, {8, 24027}, {9, 1262}, {12, 1101}, {19, 44717}, {31, 4998}, {41, 1275}, {48, 46102}, {55, 7045}, {56, 765}, {57, 1252}, {63, 7115}, {65, 4570}, {73, 5379}, {85, 23990}, {100, 109}, {101, 651}, {108, 1331}, {110, 4551}, {162, 23067}, {163, 4552}, {181, 24041}, {190, 1415}, {200, 7339}, {201, 250}, {213, 4620}, {219, 7128}, {249, 2171}, {269, 6065}, {312, 23979}, {480, 24013}, {603, 15742}, {604, 1016}, {644, 1461}, {649, 31615}, {650, 4619}, {653, 906}, {655, 1983}, {662, 4559}, {664, 692}, {728, 23971}, {872, 7340}, {883, 32666}, {901, 23703}, {919, 1025}, {934, 3939}, {1018, 4565}, {1020, 5546}, {1026, 32735}, {1088, 6066}, {1106, 4076}, {1259, 24033}, {1319, 9268}, {1332, 32674}, {1397, 7035}, {1400, 4567}, {1402, 4600}, {1404, 5376}, {1405, 5385}, {1414, 4557}, {1428, 5378}, {1458, 5377}, {1469, 5384}, {1783, 1813}, {1897, 36059}, {2190, 44710}, {2223, 39293}, {2283, 36086}, {2284, 36146}, {2289, 23984}, {2397, 32669}, {2427, 37136}, {3719, 23985}, {3869, 15386}, {3882, 8687}, {3888, 8685}, {4554, 32739}, {4556, 21859}, {4578, 6614}, {4579, 29055}, {4585, 32675}, {4587, 32714}, {5383, 41526}, {6056, 24032}, {6335, 32660}, {6358, 23357}, {6516, 8750}, {6602, 23586}, {7066, 24000}, {16945, 44724}, {17439, 38809}, {18026, 32656}, {18315, 35307}, {19614, 44699}, {23592, 34544}, {23981, 36037}, {23995, 34388}, {24029, 32641}, {32643, 42718}, {36074, 37211}, {36075, 37212}
X(11) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 765}, {2, 4998}, {3, 59}, {4, 44699}, {5, 44710}, {6, 44717}, {7, 514}, {8, 522}, {8, 44724}, {9, 4564}, {11, 100}, {12, 523}, {55, 17115}, {56, 513}, {57, 661}, {63, 40628}, {65, 50330}, {69, 905}, {75, 1577}, {78, 656}, {99, 40625}, {101, 38991}, {108, 5521}, {109, 8054}, {115, 4552}, {125, 23067}, {181, 3005}, {190, 1146}, {200, 6608}, {223, 7045}, {226, 4988}, {244, 4551}, {345, 3239}, {478, 1262}, {480, 3900}, {518, 3126}, {521, 1259}, {644, 35508}, {647, 26942}, {651, 1015}, {653, 5190}, {658, 40615}, {664, 1086}, {665, 34253}, {668, 40624}, {676, 50441}, {692, 39025}, {798, 41526}, {883, 35094}, {900, 1317}, {918, 35509}, {934, 40617}, {1016, 3161}, {1025, 38980}, {1084, 4559}, {1249, 46102}, {1252, 5452}, {1275, 3160}, {1331, 38983}, {1332, 35072}, {1376, 4885}, {1400, 40627}, {1402, 50497}, {1403, 3835}, {1639, 51583}, {1788, 20315}, {1813, 39006}, {1897, 20620}, {2149, 32664}, {2170, 21362}, {2283, 38989}, {2284, 39014}, {2295, 3709}, {2968, 3699}, {3119, 35341}, {3162, 7115}, {3259, 23981}, {3667, 6049}, {3676, 17093}, {3700, 3969}, {3716, 8299}, {3738, 4996}, {3756, 43290}, {3837, 8850}, {3882, 17419}, {3911, 6544}, {3939, 14714}, {3952, 6741}, {3960, 41801}, {4076, 6552}, {4242, 13999}, {4417, 6589}, {4516, 22280}, {4521, 5435}, {4534, 25737}, {4554, 40619}, {4557, 40608}, {4561, 40626}, {4566, 40622}, {4567, 40582}, {4570, 40602}, {4571, 7358}, {4573, 40620}, {4585, 35128}, {4600, 40605}, {4620, 6626}, {5125, 7649}, {5375, 31615}, {5511, 40576}, {5515, 14594}, {6065, 6600}, {6067, 6362}, {6068, 6366}, {6129, 7080}, {6332, 51612}, {6516, 26932}, {6586, 33298}, {6609, 7339}, {6649, 16592}, {6735, 23757}, {7012, 36103}, {7952, 15742}, {14838, 40999}, {15632, 35014}, {17072, 34247}, {17780, 51402}, {18314, 34388}, {18838, 42769}, {21044, 22003}, {21232, 21320}, {23703, 38979}, {34467, 36059}
X(11) = barycentric product X(i)*X(j) for these {i,j}: {1, 4858}, {4, 26932}, {6, 34387}, {7, 1146}, {8, 1086}, {9, 1111}, {10, 17197}, {12, 26856}, {19, 17880}, {21, 16732}, {29, 4466}, {55, 23989}, {56, 23978}, {57, 24026}, {60, 338}, {69, 8735}, {75, 2170}, {76, 3271}, {85, 2310}, {86, 21044}, {92, 7004}, {100, 40166}, {115, 261}, {124, 13478}, {125, 46103}, {141, 18101}, {189, 38357}, {190, 21132}, {210, 16727}, {219, 2973}, {222, 21666}, {244, 312}, {257, 4459}, {264, 7117}, {270, 20902}, {273, 34591}, {274, 4516}, {278, 2968}, {279, 4081}, {281, 1565}, {284, 21207}, {314, 3125}, {318, 3942}, {321, 18191}, {331, 3270}, {333, 3120}, {335, 4124}, {339, 2189}, {345, 2969}, {346, 1358}, {348, 42069}, {479, 23970}, {513, 4391}, {514, 522}, {521, 17924}, {523, 4560}, {646, 764}, {649, 35519}, {650, 693}, {651, 42455}, {652, 46107}, {658, 23615}, {661, 18155}, {663, 3261}, {664, 42462}, {850, 7252}, {885, 918}, {903, 4530}, {905, 44426}, {1015, 3596}, {1016, 7336}, {1019, 4086}, {1021, 4077}, {1022, 4768}, {1088, 3119}, {1090, 4564}, {1109, 2185}, {1118, 23983}, {1275, 5532}, {1364, 2052}, {1365, 7058}, {1367, 36421}, {1440, 5514}, {1459, 46110}, {1461, 23104}, {1509, 4092}, {1577, 3737}, {1639, 6548}, {1647, 4997}, {1826, 17219}, {1977, 40363}, {2051, 34589}, {2150, 23994}, {2218, 17878}, {2320, 4957}, {2321, 17205}, {2325, 6549}, {2401, 2804}, {2481, 17435}, {2618, 39177}, {2995, 38345}, {3023, 40099}, {3035, 31611}, {3063, 40495}, {3064, 4025}, {3121, 40072}, {3122, 28660}, {3123, 27424}, {3124, 18021}, {3239, 3676}, {3248, 28659}, {3452, 40451}, {3613, 27010}, {3615, 8287}, {3669, 4397}, {3675, 36796}, {3699, 6545}, {3700, 7192}, {3701, 16726}, {3716, 4444}, {3756, 6557}, {3762, 23838}, {3790, 43266}, {3900, 24002}, {3910, 4581}, {3937, 7017}, {3939, 23100}, {4041, 7199}, {4089, 36910}, {4373, 4534}, {4451, 7200}, {4518, 27918}, {4522, 4817}, {4551, 40213}, {4582, 6550}, {4608, 4976}, {4811, 47915}, {4904, 6601}, {4939, 8056}, {4953, 27818}, {4985, 47947}, {6063, 14936}, {6332, 7649}, {6506, 7318}, {6556, 40617}, {6591, 35518}, {7068, 36419}, {7155, 21138}, {7178, 7253}, {8748, 17216}, {10015, 43728}, {10566, 48278}, {13426, 22106}, {13454, 22107}, {13609, 36620}, {14618, 23189}, {15413, 18344}, {15416, 43923}, {15633, 34050}, {15634, 40869}, {16082, 35014}, {16596, 40836}, {17094, 17926}, {17877, 39943}, {18210, 31623}, {21140, 36799}, {23062, 24010}, {26563, 40528}, {28132, 43042}, {31628, 42547}, {34234, 35015}, {34434, 40624}, {34896, 37771}, {35174, 46384}, {36795, 42753}, {37222, 51442}, {38347, 40216}, {38362, 44189}, {40086, 47793}, {40437, 46398}, {42754, 51565}, {43735, 46533}
X(11) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 4564}, {2, 4998}, {3, 44717}, {4, 46102}, {6, 59}, {7, 1275}, {8, 1016}, {9, 765}, {19, 7012}, {21, 4567}, {25, 7115}, {31, 2149}, {34, 7128}, {41, 1110}, {55, 1252}, {56, 1262}, {57, 7045}, {60, 249}, {86, 4620}, {100, 31615}, {109, 4619}, {115, 12}, {116, 33298}, {124, 4417}, {125, 26942}, {149, 31633}, {216, 44710}, {220, 6065}, {244, 57}, {261, 4590}, {281, 15742}, {284, 4570}, {294, 5377}, {312, 7035}, {314, 4601}, {333, 4600}, {338, 34388}, {346, 4076}, {479, 23586}, {512, 4559}, {513, 651}, {514, 664}, {521, 1332}, {522, 190}, {523, 4552}, {604, 24027}, {647, 23067}, {649, 109}, {650, 100}, {652, 1331}, {657, 3939}, {661, 4551}, {663, 101}, {665, 2283}, {667, 1415}, {673, 39293}, {693, 4554}, {738, 24013}, {764, 3669}, {884, 919}, {885, 666}, {905, 6516}, {918, 883}, {926, 2284}, {1015, 56}, {1019, 1414}, {1021, 643}, {1024, 36086}, {1027, 36146}, {1086, 7}, {1090, 4858}, {1109, 6358}, {1111, 85}, {1118, 23984}, {1146, 8}, {1172, 5379}, {1249, 44699}, {1320, 5376}, {1357, 1407}, {1358, 279}, {1364, 394}, {1365, 6354}, {1397, 23979}, {1407, 7339}, {1459, 1813}, {1509, 7340}, {1565, 348}, {1566, 50441}, {1635, 23703}, {1639, 17780}, {1647, 3911}, {1769, 24029}, {1946, 906}, {1977, 1397}, {2087, 1319}, {2150, 1101}, {2170, 1}, {2175, 23990}, {2185, 24041}, {2189, 250}, {2254, 1025}, {2310, 9}, {2316, 9268}, {2320, 5385}, {2344, 5384}, {2423, 2720}, {2488, 35326}, {2611, 16577}, {2638, 2289}, {2643, 2171}, {2804, 2397}, {2968, 345}, {2969, 278}, {2973, 331}, {3022, 220}, {3023, 6645}, {3063, 692}, {3064, 1897}, {3119, 200}, {3120, 226}, {3121, 1402}, {3122, 1400}, {3123, 1423}, {3124, 181}, {3125, 65}, {3161, 44724}, {3239, 3699}, {3248, 604}, {3261, 4572}, {3269, 7066}, {3270, 219}, {3271, 6}, {3287, 4579}, {3310, 23981}, {3326, 26611}, {3328, 35110}, {3596, 31625}, {3669, 934}, {3675, 241}, {3676, 658}, {3680, 5382}, {3699, 6632}, {3700, 3952}, {3708, 201}, {3709, 4557}, {3716, 3570}, {3733, 4565}, {3737, 662}, {3738, 4585}, {3756, 5435}, {3810, 33946}, {3900, 644}, {3907, 18047}, {3937, 222}, {3942, 77}, {4014, 6180}, {4017, 1020}, {4041, 1018}, {4081, 346}, {4086, 4033}, {4089, 17078}, {4091, 6517}, {4092, 594}, {4124, 239}, {4130, 4578}, {4147, 4595}, {4163, 6558}, {4171, 4069}, {4369, 6649}, {4391, 668}, {4397, 646}, {4403, 7223}, {4435, 3573}, {4459, 894}, {4466, 307}, {4474, 4482}, {4475, 7146}, {4516, 37}, {4521, 43290}, {4522, 3807}, {4526, 23343}, {4528, 30731}, {4530, 519}, {4534, 145}, {4542, 4370}, {4546, 30720}, {4560, 99}, {4581, 6648}, {4582, 6635}, {4705, 21859}, {4768, 24004}, {4814, 4752}, {4820, 4756}, {4834, 36074}, {4858, 75}, {4876, 5378}, {4895, 1023}, {4904, 6604}, {4939, 18743}, {4944, 4767}, {4953, 3161}, {4965, 3759}, {4976, 4427}, {4990, 30729}, {5190, 5125}, {5432, 43986}, {5514, 7080}, {5532, 1146}, {5548, 6551}, {6075, 43043}, {6332, 4561}, {6377, 1403}, {6506, 5552}, {6545, 3676}, {6550, 30725}, {6590, 14594}, {6591, 108}, {6608, 35341}, {6615, 21362}, {6729, 6733}, {6741, 3969}, {7004, 63}, {7023, 23971}, {7058, 6064}, {7063, 7109}, {7117, 3}, {7155, 5383}, {7178, 4566}, {7192, 4573}, {7199, 4625}, {7200, 7176}, {7202, 1442}, {7203, 4637}, {7252, 110}, {7253, 645}, {7336, 1086}, {7337, 23985}, {7649, 653}, {8034, 7180}, {8042, 7203}, {8287, 40999}, {8648, 1983}, {8735, 4}, {8754, 8736}, {11124, 14589}, {11193, 5375}, {11927, 30626}, {11998, 2975}, {14304, 42718}, {14413, 23890}, {14430, 23891}, {14442, 39771}, {14827, 6066}, {14935, 7123}, {14936, 55}, {15280, 28743}, {15615, 39686}, {15635, 34051}, {15914, 43991}, {16726, 1014}, {16732, 1441}, {17059, 17234}, {17096, 4616}, {17197, 86}, {17205, 1434}, {17219, 17206}, {17420, 3882}, {17435, 518}, {17880, 304}, {17924, 18026}, {17926, 36797}, {18021, 34537}, {18101, 83}, {18155, 799}, {18191, 81}, {18210, 1214}, {18344, 1783}, {18771, 38809}, {20975, 2197}, {20982, 2594}, {21044, 10}, {21104, 35312}, {21120, 21272}, {21123, 46153}, {21127, 35338}, {21132, 514}, {21138, 3212}, {21139, 9312}, {21140, 43040}, {21143, 43924}, {21207, 349}, {21666, 7017}, {21789, 5546}, {21950, 4848}, {21963, 28387}, {22106, 13436}, {22107, 13453}, {22383, 36059}, {23062, 24011}, {23189, 4558}, {23615, 3239}, {23760, 31605}, {23761, 47676}, {23764, 30719}, {23838, 3257}, {23970, 5423}, {23978, 3596}, {23983, 1264}, {23989, 6063}, {24002, 4569}, {24010, 728}, {24012, 6602}, {24026, 312}, {24031, 3719}, {24840, 17350}, {26856, 261}, {26932, 69}, {26956, 26872}, {27010, 1078}, {27846, 1429}, {27918, 1447}, {28132, 36802}, {33573, 6745}, {34387, 76}, {34589, 14829}, {34591, 78}, {35015, 908}, {35072, 1259}, {35091, 6068}, {35092, 1317}, {35128, 4996}, {35348, 37139}, {35505, 1362}, {35506, 1682}, {35508, 480}, {35519, 1978}, {36123, 39294}, {36197, 210}, {36798, 5381}, {38345, 3869}, {38347, 1621}, {38357, 329}, {38358, 3681}, {38362, 196}, {38365, 4251}, {38374, 14256}, {38375, 3870}, {38389, 34048}, {38986, 41526}, {38989, 34253}, {39687, 6056}, {39786, 1284}, {40166, 693}, {40213, 18155}, {40451, 40420}, {40528, 23617}, {40608, 2295}, {40615, 17093}, {40621, 6049}, {40626, 51612}, {42067, 608}, {42069, 281}, {42337, 25268}, {42454, 17494}, {42455, 4391}, {42462, 522}, {42753, 1465}, {42754, 22464}, {43728, 13136}, {43921, 1462}, {43923, 32714}, {43924, 1461}, {43929, 32735}, {43932, 4617}, {44311, 32939}, {44426, 6335}, {46101, 3035}, {46103, 18020}, {46107, 46404}, {46384, 3738}, {47694, 14612}, {48278, 4568}, {50333, 42720}, {50512, 36075}, {51402, 51583}, {51442, 30566}
X(11) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 5, 12}, {1, 12, 15888}, {1, 119, 10956}, {1, 355, 10944}, {1, 496, 37722}, {1, 1837, 10950}, {1, 2006, 45946}, {1, 5219, 17718}, {1, 5400, 4551}, {1, 5443, 37737}, {1, 5531, 37736}, {1, 5587, 5252}, {1, 5727, 37740}, {1, 5881, 37738}, {1, 5886, 15950}, {1, 6264, 20586}, {1, 6326, 12739}, {1, 7173, 3614}, {1, 7741, 5}, {1, 7951, 495}, {1, 7972, 12735}, {1, 7988, 5219}, {1, 7989, 9578}, {1, 8227, 11375}, {1, 9581, 1837}, {1, 9897, 7972}, {1, 10593, 7173}, {1, 10826, 355}, {1, 10943, 10949}, {1, 10950, 37734}, {1, 10958, 10955}, {1, 16173, 1387}, {1, 17717, 5718}, {1, 17718, 37703}, {1, 17719, 17724}, {1, 17720, 17602}, {1, 17722, 17726}, {1, 23708, 5886}, {1, 26470, 10957}, {1, 26475, 10959}, {1, 26476, 10958}, {1, 37692, 11374}, {1, 37701, 5719}, {1, 37702, 37730}, {1, 37706, 1483}, {1, 37711, 37727}, {1, 37714, 37709}, {1, 37717, 5724}, {1, 37718, 80}, {1, 37720, 496}, {1, 37721, 37739}, {1, 37732, 2594}, {1, 37735, 5901}, {1, 39692, 119}, {1, 50443, 11376}, {1, 50444, 50443}, {2, 55, 5432}, {2, 100, 3035}, {2, 149, 100}, {2, 390, 5218}, {2, 497, 55}, {2, 673, 26007}, {2, 1621, 6690}, {2, 2550, 4413}, {2, 2886, 3925}, {2, 3035, 31235}, {2, 3058, 4995}, {2, 3434, 1376}, {2, 4366, 26629}, {2, 5274, 497}, {2, 5432, 5326}, {2, 11235, 34612}, {2, 11238, 3058}, {2, 11680, 2886}, {2, 23541, 25882}, {2, 26007, 31192}, {2, 26105, 4423}, {2, 26139, 25531}, {2, 26795, 27072}, {2, 26846, 27009}, {2, 31272, 6667}, {2, 33108, 3826}, {3, 499, 5433}, {3, 1479, 6284}, {3, 6284, 15338}, {3, 6713, 21154}, {3, 9668, 4302}, {3, 9669, 1479}, {3, 10525, 11826}, {3, 11928, 10525}, {3, 51517, 10738}, {4, 56, 7354}, {4, 3086, 56}, {4, 4293, 12943}, {4, 10591, 10896}, {4, 10598, 10893}, {4, 10785, 12114}, {4, 12248, 10728}, {4, 47743, 3086}, {5, 12, 3614}, {5, 495, 7951}, {5, 496, 1}, {5, 1837, 10958}, {5, 5533, 1317}, {5, 7741, 7173}, {5, 8227, 7958}, {5, 10593, 7741}, {5, 10943, 355}, {5, 10948, 10944}, {5, 10959, 10955}, {5, 26475, 10950}, {5, 32214, 10942}, {5, 37720, 37722}, {5, 37722, 15888}, {5, 37726, 37725}, {6, 12589, 39873}, {6, 44618, 19024}, {6, 44619, 19023}, {6, 44623, 19030}, {6, 44624, 19029}, {7, 7678, 42356}, {8, 1329, 21031}, {8, 4193, 1329}, {8, 5233, 4023}, {10, 3825, 4187}, {10, 6702, 34122}, {10, 11814, 24003}, {10, 12053, 3057}, {10, 24387, 24390}, {10, 49600, 10914}, {12, 1317, 10956}, {12, 7173, 5}, {12, 10949, 10944}, {12, 10950, 10955}, {12, 10959, 37734}, {12, 37722, 1}, {20, 5225, 12953}, {20, 7288, 5204}, {20, 38693, 38759}, {31, 29662, 37646}, {35, 4857, 15171}, {36, 3582, 15325}, {36, 15325, 5298}, {38, 7069, 24431}, {40, 6922, 50031}, {40, 9614, 12701}, {55, 497, 3058}, {55, 5432, 4995}, {55, 11238, 497}, {56, 10896, 4}, {56, 12943, 4293}, {57, 1699, 1836}, {57, 1836, 11246}, {65, 20118, 12832}, {75, 21580, 21404}, {79, 3337, 24470}, {80, 1387, 1317}, {80, 2006, 14204}, {80, 5443, 45764}, {80, 5533, 37726}, {80, 7741, 39692}, {80, 7972, 9897}, {80, 8068, 119}, {80, 10057, 355}, {80, 16173, 1}, {80, 37718, 12019}, {80, 37720, 5533}, {90, 17437, 24467}, {100, 3035, 6174}, {100, 6667, 31235}, {100, 10707, 149}, {100, 31272, 2}, {104, 10728, 12248}, {104, 13273, 7354}, {116, 17761, 4904}, {119, 23513, 5}, {124, 34589, 26932}, {125, 3270, 26956}, {137, 3327, 14101}, {140, 15171, 35}, {140, 33814, 38760}, {145, 5154, 11681}, {145, 11681, 12607}, {149, 497, 13274}, {149, 3035, 6154}, {149, 3434, 13271}, {149, 6667, 6174}, {149, 31272, 3035}, {149, 45310, 31235}, {153, 388, 12763}, {181, 2051, 10406}, {200, 24392, 4863}, {214, 1125, 34123}, {214, 32557, 1125}, {226, 3817, 17605}, {226, 11019, 354}, {238, 1936, 2361}, {238, 33140, 35466}, {244, 1647, 3756}, {244, 2310, 7004}, {244, 3120, 1086}, {312, 3703, 6057}, {312, 3705, 3703}, {354, 17604, 1864}, {354, 17605, 226}, {354, 17660, 5083}, {355, 10523, 12}, {355, 10948, 10949}, {355, 11373, 1}, {371, 9661, 18965}, {381, 999, 1478}, {381, 10072, 5434}, {381, 12773, 10742}, {381, 18519, 18516}, {388, 3091, 10895}, {388, 14986, 3304}, {390, 5218, 55}, {390, 45043, 20119}, {405, 26363, 24953}, {485, 1124, 19028}, {486, 1335, 19027}, {495, 7951, 12}, {496, 1484, 5533}, {496, 1837, 10959}, {496, 7173, 15888}, {496, 7741, 12}, {496, 10593, 5}, {496, 10826, 10949}, {496, 10943, 10948}, {496, 12019, 37726}, {496, 26476, 10950}, {496, 39692, 1317}, {497, 3434, 10947}, {497, 5218, 390}, {497, 5274, 11238}, {497, 10589, 2}, {499, 1479, 3}, {499, 9669, 6284}, {499, 10058, 6713}, {546, 18990, 3585}, {547, 15170, 3584}, {590, 2066, 13901}, {590, 48714, 13922}, {613, 1352, 39897}, {615, 5414, 13958}, {615, 48715, 13991}, {631, 4294, 5217}, {631, 13199, 34474}, {650, 38347, 14936}, {693, 40619, 23989}, {936, 25522, 24954}, {938, 6828, 15844}, {942, 9955, 12047}, {942, 12047, 3649}, {944, 6941, 18242}, {946, 1210, 65}, {946, 12616, 12672}, {946, 16174, 38038}, {950, 1125, 2646}, {950, 2646, 10543}, {960, 6734, 21677}, {962, 1788, 37567}, {962, 5704, 1788}, {982, 3944, 3782}, {999, 1478, 5434}, {999, 12773, 10074}, {1056, 3545, 10590}, {1056, 10590, 11237}, {1058, 3085, 3303}, {1058, 3090, 3085}, {1086, 3756, 244}, {1111, 1565, 1358}, {1125, 17647, 17614}, {1125, 25639, 442}, {1125, 33709, 32557}, {1145, 34122, 10}, {1146, 2170, 4534}, {1146, 6506, 5514}, {1146, 13609, 3119}, {1210, 10265, 20118}, {1317, 23513, 3614}, {1329, 3813, 8}, {1329, 3847, 4193}, {1376, 3434, 34612}, {1376, 11235, 3434}, {1376, 13205, 100}, {1387, 5533, 37722}, {1387, 10593, 23513}, {1387, 39692, 12}, {1421, 2006, 15253}, {1478, 10072, 999}, {1479, 4302, 9668}, {1479, 5433, 15338}, {1484, 8068, 1317}, {1484, 10057, 10949}, {1484, 10593, 39692}, {1484, 16173, 37722}, {1484, 23513, 37725}, {1500, 1506, 31460}, {1537, 38038, 946}, {1647, 3120, 244}, {1656, 3295, 498}, {1656, 12331, 38752}, {1699, 1768, 34789}, {1699, 8727, 7965}, {1699, 17728, 11246}, {1738, 5121, 16610}, {1768, 11219, 13226}, {1836, 17728, 57}, {1837, 11376, 1}, {1837, 37740, 5727}, {2170, 3119, 38375}, {2170, 21044, 1146}, {2170, 33573, 43960}, {2310, 3120, 38357}, {2310, 4459, 24840}, {2476, 3616, 25466}, {2478, 10527, 958}, {2607, 21381, 14985}, {2886, 3816, 2}, {2886, 3826, 33108}, {2886, 3829, 11680}, {2886, 15842, 1376}, {2886, 25652, 27692}, {2968, 24026, 4081}, {3006, 4358, 3932}, {3035, 6667, 2}, {3035, 45310, 6667}, {3057, 17606, 10}, {3058, 5432, 55}, {3073, 3075, 1399}, {3086, 10591, 4}, {3086, 10598, 18961}, {3086, 10896, 7354}, {3090, 38665, 20400}, {3091, 14986, 388}, {3091, 38669, 38757}, {3100, 9629, 5160}, {3119, 33573, 13609}, {3149, 48482, 6253}, {3295, 12331, 10087}, {3297, 42265, 31472}, {3298, 42262, 44622}, {3304, 10895, 388}, {3333, 9612, 10404}, {3434, 10584, 2}, {3436, 10529, 12513}, {3452, 4847, 210}, {3452, 24386, 4847}, {3486, 3616, 34471}, {3582, 3583, 36}, {3583, 15325, 15326}, {3585, 5563, 18990}, {3600, 3832, 5229}, {3600, 5229, 9657}, {3614, 15888, 12}, {3628, 51525, 38763}, {3666, 24210, 4854}, {3673, 17181, 3665}, {3685, 32851, 3712}, {3687, 3706, 4046}, {3705, 20545, 20487}, {3720, 33105, 17056}, {3741, 3846, 1211}, {3742, 3838, 5249}, {3813, 3847, 1329}, {3813, 4193, 21031}, {3816, 3829, 2886}, {3816, 11680, 3925}, {3816, 15845, 55}, {3817, 11019, 226}, {3825, 24387, 10}, {3826, 33108, 3925}, {3841, 19862, 17529}, {3851, 7373, 9654}, {3912, 20544, 20486}, {3936, 29824, 4966}, {4187, 24390, 10}, {4202, 26094, 25914}, {4293, 12943, 7354}, {4302, 9668, 6284}, {4304, 10165, 37600}, {4413, 31140, 2550}, {4423, 31245, 2}, {4514, 7081, 4030}, {4551, 5400, 45885}, {4858, 4939, 24026}, {4871, 21241, 3836}, {4995, 5326, 5432}, {5055, 6767, 31479}, {5083, 18240, 354}, {5083, 21635, 12831}, {5187, 10529, 3436}, {5204, 9671, 12953}, {5204, 12953, 20}, {5211, 37759, 32922}, {5217, 9670, 4294}, {5225, 7288, 20}, {5226, 10580, 3475}, {5272, 17064, 24789}, {5274, 10589, 55}, {5274, 11680, 15845}, {5274, 45043, 10707}, {5281, 10385, 55}, {5298, 15326, 36}, {5433, 6284, 3}, {5435, 9812, 3474}, {5531, 7988, 15017}, {5531, 15017, 5660}, {5531, 37736, 41701}, {5533, 7741, 119}, {5533, 8068, 1}, {5533, 23513, 10956}, {5587, 6264, 12751}, {5587, 37704, 1}, {5603, 6830, 7680}, {5603, 12247, 10698}, {5603, 18391, 2099}, {5697, 15079, 18395}, {5697, 18395, 5690}, {5722, 5886, 1}, {5722, 23708, 15950}, {5727, 37740, 10950}, {5745, 40998, 3683}, {5748, 36845, 25568}, {5886, 6265, 11729}, {5901, 8070, 12}, {5901, 37730, 1}, {5902, 18393, 39542}, {5927, 17626, 17625}, {6154, 6174, 100}, {6154, 31235, 6174}, {6174, 31235, 3035}, {6224, 32558, 3616}, {6667, 10707, 6154}, {6667, 45310, 31272}, {6684, 10624, 37568}, {6690, 49736, 1621}, {6702, 21630, 1145}, {6713, 10738, 24466}, {6734, 41012, 960}, {6736, 21627, 3893}, {6767, 31479, 10056}, {6833, 10531, 11496}, {6834, 12116, 11500}, {6928, 11249, 11827}, {7004, 35015, 38357}, {7173, 37722, 12}, {7191, 33133, 17061}, {7491, 26286, 30264}, {7741, 8068, 23513}, {7741, 9581, 26476}, {7741, 16173, 8068}, {7741, 26475, 10958}, {7741, 37702, 8070}, {7741, 37720, 1}, {7741, 37722, 3614}, {7951, 9897, 11698}, {7956, 8727, 1699}, {7958, 26481, 3614}, {7972, 12735, 1317}, {8068, 37726, 10956}, {8068, 39692, 5}, {8086, 8379, 174}, {8104, 13267, 174}, {8256, 13463, 14923}, {8286, 8287, 125}, {8976, 31474, 13905}, {9580, 31231, 165}, {9581, 11376, 10950}, {9581, 37720, 26475}, {9581, 50443, 1}, {9581, 50444, 11376}, {9671, 12953, 5225}, {9779, 10883, 42356}, {9956, 9957, 10039}, {9957, 10039, 45081}, {10057, 12737, 10944}, {10058, 10090, 3}, {10058, 10738, 6284}, {10073, 11729, 1317}, {10073, 23708, 11729}, {10265, 12736, 12832}, {10265, 16174, 1537}, {10283, 37728, 1}, {10427, 38205, 142}, {10483, 18514, 3627}, {10523, 10943, 10944}, {10523, 10948, 1}, {10523, 10949, 15888}, {10525, 26492, 3}, {10576, 35808, 9646}, {10589, 11238, 5432}, {10589, 15845, 3925}, {10591, 47743, 56}, {10592, 37719, 12}, {10593, 37720, 12}, {10598, 10785, 4}, {10609, 34123, 214}, {10707, 31272, 100}, {10707, 45310, 6174}, {10724, 38693, 20}, {10826, 11373, 10944}, {10826, 37720, 10948}, {10863, 11019, 8581}, {10893, 12114, 4}, {10914, 17619, 10}, {10914, 17622, 3057}, {10916, 21616, 72}, {10919, 10920, 12586}, {10925, 10926, 12589}, {10925, 19029, 39873}, {10926, 19030, 39873}, {10942, 32214, 37727}, {10948, 11373, 37722}, {10958, 10959, 10950}, {10958, 37722, 37734}, {10993, 38760, 33814}, {11019, 21635, 5083}, {11108, 31493, 19854}, {11219, 34789, 1768}, {11230, 18527, 24929}, {11235, 13205, 13271}, {11236, 11240, 34749}, {11375, 26481, 12}, {11376, 26475, 37722}, {11376, 26476, 12}, {11496, 12332, 12775}, {11755, 11764, 15}, {11773, 11782, 16}, {11814, 24003, 16594}, {11928, 26492, 11826}, {12019, 16173, 1317}, {12114, 18961, 7354}, {12433, 37737, 1}, {12433, 45764, 1317}, {12586, 45454, 10920}, {12586, 45455, 10919}, {12589, 45460, 10926}, {12589, 45461, 10925}, {12690, 34123, 10609}, {12701, 24914, 40}, {12764, 13273, 4}, {13257, 41556, 17660}, {13274, 31272, 5432}, {13898, 19038, 3068}, {13955, 19037, 3069}, {14740, 46694, 210}, {14923, 25005, 8256}, {15251, 15252, 15253}, {15252, 15253, 45946}, {15280, 15283, 650}, {15842, 15845, 10947}, {16732, 17463, 23772}, {17063, 17889, 40688}, {17111, 23304, 427}, {17435, 36197, 38358}, {17527, 31419, 1698}, {17533, 17757, 3814}, {17556, 45700, 34606}, {17618, 17626, 226}, {17625, 17626, 354}, {17626, 17661, 5083}, {17658, 18236, 210}, {17669, 26801, 26558}, {17717, 24217, 1}, {17720, 17721, 1}, {17724, 37691, 17719}, {18480, 24928, 45287}, {18516, 18519, 34697}, {19023, 19024, 6}, {19029, 19030, 6}, {19642, 25533, 662}, {19907, 38044, 5901}, {21077, 49627, 3555}, {21154, 24466, 3}, {21630, 34122, 13996}, {22793, 37582, 1770}, {22799, 38141, 546}, {23477, 23517, 1}, {23513, 37726, 119}, {23821, 24237, 4014}, {24210, 24239, 3666}, {24216, 24231, 3999}, {24392, 30827, 200}, {24646, 24647, 55}, {24916, 24955, 2}, {25448, 25652, 2}, {25568, 36845, 41711}, {25760, 30942, 141}, {25960, 31330, 5743}, {26010, 26013, 26005}, {26475, 26476, 1837}, {26476, 37720, 10959}, {26966, 27027, 2}, {27134, 27190, 2}, {27256, 29969, 16593}, {29631, 32944, 3589}, {29845, 32772, 6703}, {30306, 30307, 42356}, {30309, 30310, 42356}, {30993, 31126, 20531}, {31141, 34625, 34689}, {31477, 31489, 31497}, {31517, 37720, 37728}, {31870, 40259, 946}, {32850, 37758, 5205}, {32918, 32947, 44419}, {32930, 33119, 44416}, {32931, 33120, 49524}, {33814, 34126, 140}, {37691, 45920, 12}, {37702, 37735, 1}, {37718, 37720, 1484}, {37718, 50443, 12740}, {38026, 50843, 551}, {38034, 39542, 18393}, {38055, 41556, 354}, {38090, 51008, 597}, {38099, 50842, 3679}, {38104, 50841, 3828}, {38141, 51529, 22799}, {40942, 40963, 2264}, {44618, 44619, 6}, {44623, 44624, 6}, {44623, 45460, 39873}, {44624, 45461, 39873}, {45454, 45455, 12586}, {45460, 45461, 12589}, {47743, 47744, 20418}


X(12) = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11)

Trilinears   1 + cos(B - C) : 1 + cos(C - A) : 1 + cos(A -B) : :
Trilinears    cos2(B/2 - C/2) : :
Trilinears    1 + cos A + 2 cos B cos C : :
Trilinears    1 - cos A + 2 sin B sin C : :
Trilinears    bc(b + c)2/(b + c - a) : :
Barycentrics   a(1 + cos(B - C)) : :
Barycentrics   (b + c)2/(b + c - a) : :
Tripolars    (a - b - c)*Sqrt[-(a^3 + a^2*b - a*b^2 - b^3 + a^2*c + 3*a*b*c + b^2*c - a*c^2 + b*c^2 - c^3)*(a^4 - 2*a^2*b^2 + b^4 + a^2*b*c - a*b^2*c - 2*b^3*c - 2*a^2*c^2 - a*b*c^2 + 2*b^2*c^2 - 2*b*c^3 + c^4)] : :
X(12) = R*X(1) + 3r*X(2) - r*X(3)
X(12) = X[1] - 3 X[37701], 2 X[5] - 3 X[38109], 2 X[355] - 3 X[38157], 4 X[496] - 3 X[10959], 2 X[1387] - 3 X[38063], 2 X[5901] - 3 X[38045], X[26470] - 3 X[38109], and many others

Let A'B'C' be the touch points of the nine-point circle with the A-, B-, C- excircles, respectively. The lines AA', BB', CC' concur in X(12).

X(12) lies on the cubics on K589, K672, K720, K877, K1058, K1187 and these lines: {1, 5}, {2, 56}, {3, 498}, {4, 55}, {6, 5230}, {7, 1268}, {8, 2099}, {9, 5857}, {10, 65}, {13, 7006}, {14, 7005}, {17, 203}, {18, 202}, {19, 37376}, {20, 5217}, {21, 5080}, {22, 9658}, {24, 9659}, {25, 10830}, {28, 20989}, {30, 35}, {31, 7299}, {32, 9650}, {33, 235}, {34, 427}, {36, 140}, {37, 225}, {38, 1393}, {39, 18982}, {40, 1836}, {42, 1834}, {43, 47514}, {45, 4331}, {46, 11246}, {54, 215}, {57, 1224}, {58, 17734}, {59, 3615}, {63, 1454}, {64, 12930}, {67, 32288}, {68, 3157}, {71, 1901}, {73, 3142}, {74, 12372}, {75, 21405}, {76, 12837}, {78, 5794}, {79, 484}, {83, 12835}, {84, 12677}, {85, 120}, {86, 14011}, {90, 17699}, {98, 10799}, {99, 13181}, {100, 2475}, {104, 6952}, {108, 451}, {109, 2372}, {110, 2477}, {112, 13295}, {113, 3024}, {114, 3023}, {115, 1500}, {116, 1362}, {117, 1364}, {118, 3022}, {121, 1357}, {122, 3324}, {123, 1359}, {124, 1361}, {125, 1425}, {126, 3325}, {127, 3320}, {128, 3327}, {132, 6020}, {133, 7158}, {137, 7159}, {141, 1469}, {142, 8581}, {144, 18231}, {145, 3813}, {153, 6888}, {155, 10055}, {165, 9579}, {171, 1399}, {172, 230}, {174, 8088}, {177, 12622}, {182, 38120}, {184, 9652}, {185, 26956}, {190, 24835}, {197, 4185}, {198, 37384}, {201, 756}, {208, 1360}, {214, 18976}, {219, 5747}, {221, 1853}, {222, 34030}, {223, 10366}, {228, 407}, {229, 37158}, {255, 5348}, {261, 31620}, {262, 12836}, {265, 10088}, {278, 5142}, {281, 1118}, {306, 3714}, {313, 349}, {318, 4081}, {321, 3704}, {325, 1909}, {330, 7777}, {341, 29641}, {344, 2899}, {348, 7223}, {354, 1210}, {371, 9646}, {372, 13958}, {377, 1259}, {381, 1479}, {382, 4302}, {384, 26629}, {386, 9552}, {390, 3832}, {392, 21616}, {396, 2307}, {397, 7127}, {402, 11904}, {403, 6198}, {404, 3035}, {405, 10198}, {406, 7337}, {428, 5310}, {431, 1824}, {443, 1466}, {474, 1470}, {478, 20029}, {480, 2550}, {481, 31557}, {482, 31558}, {485, 1335}, {486, 1124}, {493, 10951}, {494, 10952}, {497, 3091}, {499, 999}, {502, 14873}, {515, 2646}, {516, 15837}, {517, 6842}, {518, 6067}, {519, 4870}, {523, 2599}, {524, 4400}, {528, 3871}, {535, 5267}, {546, 3583}, {547, 3582}, {548, 4316}, {549, 7280}, {550, 5010}, {551, 3825}, {553, 3828}, {573, 9553}, {574, 9651}, {578, 9666}, {590, 2067}, {594, 2171}, {597, 38091}, {603, 750}, {604, 17398}, {611, 1352}, {613, 14561}, {614, 37439}, {615, 6502}, {618, 18974}, {619, 18975}, {625, 5194}, {629, 18973}, {630, 18972}, {631, 4293}, {641, 18988}, {642, 18989}, {651, 8614}, {664, 17084}, {671, 12349}, {740, 21927}, {774, 7069}, {858, 4296}, {894, 25978}, {908, 960}, {912, 13750}, {938, 3475}, {940, 5788}, {942, 1209}, {944, 6830}, {946, 1532}, {950, 8226}, {956, 26363}, {962, 6932}, {964, 28776}, {968, 1904}, {970, 10406}, {975, 21147}, {978, 37663}, {984, 37591}, {986, 3782}, {993, 7483}, {1000, 16615}, {1001, 2478}, {1010, 19839}, {1012, 6256}, {1014, 17551}, {1015, 1506}, {1018, 36637}, {1038, 1368}, {1040, 6823}, {1043, 14009}, {1056, 3086}, {1058, 3545}, {1060, 11585}, {1062, 15760}, {1069, 5654}, {1071, 12616}, {1086, 24443}, {1089, 3695}, {1091, 1109}, {1104, 3011}, {1106, 17124}, {1107, 37661}, {1125, 1319}, {1126, 3017}, {1146, 17451}, {1147, 9653}, {1151, 9649}, {1155, 4292}, {1193, 37662}, {1213, 1400}, {1214, 21530}, {1250, 5318}, {1279, 28027}, {1284, 4026}, {1297, 12935}, {1324, 37227}, {1327, 13694}, {1328, 13814}, {1334, 17747}, {1346, 2464}, {1347, 2463}, {1348, 1675}, {1349, 1674}, {1356, 45162}, {1365, 4013}, {1367, 20618}, {1385, 6882}, {1386, 38051}, {1388, 3476}, {1397, 43531}, {1398, 5094}, {1402, 4205}, {1403, 16062}, {1405, 17330}, {1406, 6180}, {1408, 5061}, {1415, 5277}, {1420, 3624}, {1423, 32784}, {1428, 3589}, {1429, 29633}, {1447, 7198}, {1450, 27627}, {1452, 1892}, {1457, 10459}, {1458, 17245}, {1460, 2049}, {1463, 3836}, {1464, 37558}, {1467, 41867}, {1468, 37646}, {1471, 17337}, {1482, 6980}, {1486, 4186}, {1503, 2330}, {1511, 18968}, {1512, 7686}, {1519, 45776}, {1537, 17662}, {1587, 19037}, {1588, 19038}, {1591, 3084}, {1592, 3083}, {1594, 1870}, {1598, 10833}, {1617, 11108}, {1621, 5046}, {1647, 46190}, {1650, 18958}, {1672, 1677}, {1673, 1676}, {1682, 2051}, {1696, 15849}, {1697, 1699}, {1709, 6259}, {1714, 37543}, {1722, 24789}, {1725, 35194}, {1727, 3652}, {1770, 3579}, {1785, 6750}, {1825, 37982}, {1829, 40635}, {1838, 37528}, {1846, 11105}, {1857, 7952}, {1858, 5777}, {1861, 1887}, {1869, 3198}, {1875, 25985}, {1877, 1883}, {1891, 37362}, {1894, 3185}, {1898, 5927}, {1899, 19349}, {1900, 44670}, {1906, 4319}, {1914, 7745}, {1940, 25987}, {1961, 34119}, {2003, 37559}, {2007, 2010}, {2008, 2009}, {2066, 3071}, {2072, 18447}, {2077, 31775}, {2078, 5259}, {2089, 8087}, {2098, 5603}, {2140, 4904}, {2222, 3109}, {2241, 5475}, {2242, 7746}, {2263, 23305}, {2275, 3815}, {2276, 5254}, {2283, 37165}, {2285, 17303}, {2292, 4415}, {2294, 21011}, {2295, 4559}, {2361, 3072}, {2362, 13973}, {2482, 18969}, {2534, 2541}, {2535, 2540}, {2548, 16502}, {2549, 31448}, {2564, 2567}, {2565, 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6922}, {3586, 18492}, {3596, 40828}, {3601, 5691}, {3612, 18481}, {3617, 9710}, {3622, 3847}, {3625, 39777}, {3626, 36920}, {3627, 18513}, {3628, 5563}, {3634, 3911}, {3638, 49565}, {3639, 49567}, {3647, 18977}, {3656, 30323}, {3666, 5530}, {3669, 19947}, {3674, 43037}, {3683, 12572}, {3690, 7144}, {3696, 21926}, {3703, 4385}, {3710, 3967}, {3712, 7283}, {3720, 47513}, {3721, 21965}, {3743, 30446}, {3745, 5717}, {3752, 23536}, {3755, 21955}, {3756, 28096}, {3761, 3933}, {3767, 31409}, {3812, 5123}, {3817, 5919}, {3837, 24099}, {3838, 5836}, {3839, 10385}, {3842, 4032}, {3843, 4309}, {3844, 24471}, {3845, 10386}, {3846, 28389}, {3850, 4857}, {3851, 6767}, {3853, 4330}, {3855, 9671}, {3869, 31053}, {3872, 32049}, {3878, 51409}, {3884, 11813}, {3885, 13463}, {3914, 4646}, {3920, 5133}, {3924, 33127}, {3930, 21029}, {3931, 4854}, {3935, 5178}, {3936, 17751}, {3944, 37598}, {3952, 27690}, {3969, 27558}, {3991, 21073}, {4017, 31946}, {4023, 9534}, {4030, 5015}, {4046, 5295}, {4059, 9436}, {4071, 4095}, {4092, 4605}, {4136, 21101}, {4202, 26030}, {4208, 26040}, {4222, 20988}, {4295, 5657}, {4297, 37374}, {4304, 31673}, {4305, 6845}, {4308, 5550}, {4311, 10165}, {4312, 5128}, {4313, 10883}, {4314, 12558}, {4315, 17575}, {4320, 30739}, {4321, 20195}, {4322, 30950}, {4327, 17278}, {4351, 37452}, {4354, 5160}, {4366, 16044}, {4423, 5084}, {4424, 30449}, {4429, 26125}, {4648, 21239}, {4653, 37357}, {4662, 25006}, {4679, 31435}, {4692, 30171}, {4861, 38455}, {4863, 6765}, {4868, 36250}, {5013, 9597}, {5016, 26227}, {5018, 39977}, {5025, 26590}, {5048, 13464}, {5050, 39901}, {5055, 7373}, {5056, 10589}, {5066, 15170}, {5068, 5274}, {5071, 47743}, {5082, 31140}, {5083, 6702}, {5086, 34772}, {5087, 41012}, {5099, 6023}, {5119, 12699}, {5122, 31776}, {5125, 34247}, {5143, 30362}, {5173, 34790}, {5174, 37371}, {5179, 16601}, {5183, 43174}, {5187, 10587}, {5192, 28741}, {5193, 31263}, {5231, 6762}, {5247, 35466}, {5248, 10197}, {5250, 24703}, {5251, 5427}, {5262, 17061}, {5280, 5305}, {5284, 37162}, {5286, 31402}, {5288, 31262}, {5294, 25982}, {5302, 44256}, {5303, 20067}, {5306, 7296}, {5312, 48847}, {5321, 10638}, {5322, 7499}, {5323, 14005}, {5353, 11542}, {5357, 11543}, {5393, 31532}, {5405, 31533}, {5425, 11545}, {5435, 19877}, {5436, 14022}, {5439, 17619}, {5440, 17647}, {5441, 16160}, {5478, 13076}, {5479, 13075}, {5490, 26433}, {5491, 26434}, {5510, 6018}, {5512, 6019}, {5513, 33966}, {5537, 31777}, {5542, 38056}, {5584, 6908}, {5590, 18960}, {5591, 18959}, {5597, 11867}, {5598, 11868}, {5599, 18955}, {5600, 18956}, {5613, 10061}, {5617, 10062}, {5620, 14219}, {5687, 17532}, {5690, 5903}, {5697, 18393}, {5704, 11037}, {5710, 26098}, {5711, 5810}, {5713, 7078}, {5728, 10395}, {5731, 6943}, {5736, 21270}, {5743, 31339}, {5745, 12527}, {5790, 10573}, {5805, 15298}, {5820, 45729}, {5837, 31165}, {5840, 11849}, {5844, 11009}, {5878, 10060}, {5880, 6068}, {5883, 12832}, {5884, 12831}, {5885, 11570}, {5902, 6147}, {5904, 17057}, {5920, 12612}, {5972, 46683}, {5993, 36098}, {6022, 44949}, {6025, 25642}, {6026, 44947}, {6027, 16188}, {6033, 10053}, {6042, 11992}, {6046, 6356}, {6058, 6538}, {6092, 44048}, {6145, 32381}, {6154, 8715}, {6173, 38096}, {6174, 11112}, {6175, 49732}, {6200, 9647}, {6201, 10928}, {6202, 10927}, {6214, 10041}, {6215, 10040}, {6221, 9648}, {6238, 22660}, {6245, 12680}, {6246, 12743}, {6247, 7355}, {6248, 13077}, {6249, 13078}, {6250, 13082}, {6251, 13081}, {6260, 12688}, {6283, 45861}, {6286, 20424}, {6287, 10064}, {6288, 10066}, {6289, 10068}, {6290, 10067}, {6292, 18983}, {6321, 10086}, {6405, 45860}, {6506, 9310}, {6526, 41088}, {6535, 21712}, {6541, 36632}, {6564, 35809}, {6565, 35808}, {6593, 32243}, {6656, 27020}, {6666, 12573}, {6706, 51400}, {6713, 37535}, {6736, 51416}, {6738, 44840}, {6745, 37363}, {6757, 15065}, {6776, 39890}, {6796, 15865}, {6797, 33593}, {6824, 10321}, {6826, 10629}, {6833, 12114}, {6834, 10532}, {6841, 10543}, {6847, 12667}, {6850, 10310}, {6862, 10320}, {6863, 11249}, {6865, 8273}, {6879, 10785}, {6883, 7742}, {6905, 37564}, {6911, 8071}, {6912, 38757}, {6917, 11499}, {6919, 26105}, {6923, 11248}, {6924, 14793}, {6928, 10267}, {6929, 11508}, {6933, 10527}, {6945, 15845}, {6946, 20400}, {6958, 10269}, {6959, 22767}, {6968, 10531}, {6971, 10246}, {6977, 37002}, {6985, 40292}, {6998, 17798}, {7031, 18907}, {7051, 23302}, {7071, 37197}, {7081, 7270}, {7090, 30325}, {7146, 29674}, {7160, 12858}, {7176, 7181}, {7178, 21051}, {7191, 37990}, {7248, 25961}, {7249, 40033}, {7279, 37294}, {7302, 37899}, {7334, 35971}, {7352, 12359}, {7353, 23312}, {7356, 21230}, {7362, 23311}, {7363, 21028}, {7380, 21010}, {7395, 10832}, {7491, 32613}, {7503, 9672}, {7507, 11393}, {7529, 10046}, {7557, 41230}, {7580, 37601}, {7677, 17536}, {7678, 8236}, {7682, 17642}, {7683, 10544}, {7687, 46687}, {7697, 10079}, {7705, 25557}, {7728, 10065}, {7743, 31792}, {7748, 31451}, {7765, 31462}, {7819, 30104}, {7956, 7962}, {7987, 37364}, {7992, 41706}, {8085, 8241}, {8086, 8242}, {8113, 8380}, {8114, 8381}, {8144, 15761}, {8163, 8166}, {8196, 11873}, {8200, 11877}, {8203, 11874}, {8207, 11878}, {8212, 11947}, {8213, 11948}, {8220, 11951}, {8221, 11952}, {8222, 18963}, {8223, 18964}, {8228, 8239}, {8229, 8240}, {8230, 8243}, {8255, 10394}, {8287, 30436}, {8361, 30103}, {8377, 8390}, {8378, 8392}, {8379, 11924}, {8422, 12614}, {8583, 24954}, {8724, 10054}, {8976, 13904}, {9540, 9663}, {9548, 24310}, {9565, 51407}, {9598, 31477}, {9599, 16781}, {9614, 31393}, {9630, 13160}, {9660, 35821}, {9661, 10576}, {9667, 10539}, {9673, 10594}, {9708, 19854}, {9709, 17528}, {9779, 9785}, {9842, 9848}, {9843, 9850}, {9880, 12354}, {9927, 12428}, {9947, 11018}, {9948, 41561}, {9955, 9957}, {9970, 32307}, {9993, 10877}, {9996, 10038}, {10024, 18455}, {10058, 10742}, {10059, 12856}, {10069, 38224}, {10071, 14852}, {10081, 15061}, {10087, 10738}, {10089, 15561}, {10090, 38752}, {10091, 14643}, {10104, 10802}, {10113, 12896}, {10129, 13996}, {10248, 30311}, {10264, 19470}, {10265, 12005}, {10266, 12937}, {10356, 10874}, {10358, 10798}, {10368, 11347}, {10371, 11679}, {10392, 38158}, {10401, 10436}, {10448, 29678}, {10473, 10479}, {10474, 41014}, {10475, 19863}, {10478, 10480}, {10481, 24798}, {10506, 21622}, {10514, 10925}, {10515, 10926}, {10516, 12589}, {10525, 10679}, {10535, 16252}, {10571, 15666}, {10577, 35769}, {10584, 10586}, {10596, 10598}, {10624, 18483}, {10749, 13311}, {10794, 10803}, {10796, 10801}, {10829, 10834}, {10863, 10866}, {10871, 10878}, {10902, 31789}, {10910, 13389}, {10911, 13388}, {10912, 12648}, {10914, 10915}, {10919, 10929}, {10920, 10930}, {10945, 11955}, {10946, 11956}, {10980, 30315}, {11010, 28174}, {11019, 17609}, {11109, 25968}, {11230, 24928}, {11231, 37582}, {11235, 11239}, {11390, 11400}, {11429, 12241}, {11436, 12233}, {11479, 16541}, {11544, 11552}, {11551, 31794}, {11571, 33668}, {11662, 30424}, {11682, 34647}, {11684, 17484}, {11753, 11755}, {11762, 11764}, {11771, 11773}, {11780, 11782}, {11865, 11881}, {11866, 11882}, {11897, 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36574}, {17618, 17622}, {17638, 21635}, {17670, 27091}, {17682, 26007}, {17700, 24467}, {17705, 45122}, {17768, 29007}, {17783, 36561}, {17874, 21807}, {17889, 24440}, {18004, 18006}, {18440, 39900}, {18491, 44229}, {18516, 18542}, {18519, 18545}, {18524, 37230}, {18583, 38169}, {18588, 30445}, {18635, 20305}, {18641, 22341}, {18761, 34697}, {18915, 23291}, {18978, 22966}, {19023, 19047}, {19024, 19048}, {19175, 23295}, {19365, 23292}, {19367, 23293}, {19368, 23294}, {19370, 23298}, {19371, 23299}, {19373, 23303}, {19469, 23306}, {19471, 23307}, {19472, 23308}, {19473, 23309}, {19474, 23310}, {19475, 23313}, {19476, 23314}, {19505, 23315}, {19542, 50037}, {19795, 37091}, {19860, 28628}, {19861, 25681}, {20323, 44675}, {20420, 44425}, {20470, 28238}, {20541, 25102}, {20612, 41550}, {20691, 21956}, {21025, 45208}, {21044, 21049}, {21052, 51640}, {21081, 46676}, {21154, 32554}, {21258, 30949}, {21320, 23542}, {21321, 51558}, {21454, 46932}, {21531, 40790}, {21714, 51645}, {21746, 48888}, {21842, 38028}, {21891, 34973}, {21896, 21949}, {21941, 43534}, {22000, 40966}, {22076, 22299}, {22300, 51377}, {22342, 36195}, {22466, 22957}, {22682, 22711}, {22703, 22731}, {22831, 22865}, {22832, 22910}, {22833, 22965}, {22857, 22886}, {22902, 22931}, {22955, 22980}, {22956, 22982}, {23332, 32065}, {23361, 27622}, {23868, 37390}, {24174, 40688}, {24210, 37548}, {24393, 38212}, {24431, 44706}, {24466, 26285}, {24797, 32086}, {24806, 33111}, {24828, 24840}, {24833, 24845}, {24834, 24847}, {24880, 25444}, {24984, 25882}, {25005, 26579}, {25010, 28402}, {25264, 47286}, {25328, 32259}, {25385, 51411}, {25516, 40980}, {25641, 33965}, {26015, 34791}, {26036, 37658}, {26102, 37355}, {26326, 26351}, {26327, 26352}, {26328, 26353}, {26329, 26354}, {26330, 26355}, {26331, 26356}, {26333, 26358}, {26359, 26380}, {26360, 26404}, {26361, 26435}, {26362, 26436}, {26386, 45371}, {26390, 26402}, {26410, 45372}, {26414, 26426}, {26466, 45612}, {26467, 45611}, {26468, 49030}, {26469, 49031}, {26488, 45615}, {26489, 26511}, {26490, 26520}, {26491, 26525}, {26543, 29967}, {26582, 26752}, {26903, 26906}, {26932, 30493}, {26948, 26951}, {26959, 32992}, {27525, 37161}, {27555, 27577}, {27568, 27583}, {28109, 36568}, {28740, 30825}, {28757, 30839}, {29640, 37370}, {29815, 37353}, {30306, 30333}, {30307, 30334}, {30308, 30337}, {30309, 30338}, {30310, 30339}, {30617, 40719}, {31254, 37797}, {31493, 34689}, {31522, 42422}, {31555, 31567}, {31556, 31568}, {31657, 38125}, {31658, 38132}, {31844, 47006}, {31870, 40260}, {31893, 50930}, {31996, 33034}, {32274, 32297}, {32287, 32309}, {32336, 32391}, {32350, 32351}, {32369, 32390}, {32379, 32403}, {32380, 32405}, {32423, 47378}, {32557, 41554}, {32760, 37290}, {32778, 50325}, {32917, 49728}, {33073, 41261}, {33106, 37588}, {33144, 37549}, {33330, 44042}, {33331, 44043}, {33333, 44053}, {34029, 34040}, {34046, 37674}, {34194, 42425}, {34749, 45700}, {34773, 37525}, {34879, 37428}, {35000, 47032}, {35018, 37602}, {35197, 50708}, {35249, 35251}, {35663, 35669}, {35664, 35671}, {35796, 35816}, {35797, 35817}, {35970, 47023}, {35972, 47024}, {36436, 36441}, {36439, 36442}, {36454, 36459}, {36457, 36460}, {36473, 36481}, {36477, 37576}, {36488, 36526}, {36495, 36501}, {36509, 36557}, {36530, 36541}, {36544, 36546}, {36576, 36578}, {36668, 49589}, {36669, 49588}, {36926, 41878}, {36928, 46176}, {36975, 37616}, {37160, 41339}, {37308, 51506}, {37321, 39585}, {37360, 37539}, {37368, 40950}, {37375, 49736}, {37563, 40273}, {37586, 49132}, {37607, 37634}, {37868, 49658}, {38106, 45310}, {38178, 41684}, {38189, 51150}, {38208, 44848}, {39535, 44044}, {39815, 39816}, {39844, 39845}, {39889, 39902}, {41690, 41694}, {41695, 49178}, {41858, 41864}, {42051, 49636}, {42455, 42758}, {42885, 43740}, {43040, 49769}, {43817, 43820}, {43821, 43857}, {43859, 43861}, {43924, 44316}, {44618, 44643}, {44619, 44644}, {44956, 47020}, {45404, 45472}, {45405, 45473}, {45440, 45470}, {45441, 45471}, {45454, 45494}, {45455, 45495}, {45544, 45570}, {45545, 45571}, {45554, 45580}, {45555, 45581}, {45556, 45584}, {45557, 45585}, {49600, 49626}, {50605, 50626}

If you have The Geometer's Sketchpad, you can view X(12).
If you have GeoGebra, you can view X(12).

X(12) = circumcircle-inverse of X(32626)
X(12) = polar conjugate of X(46103)
X(12) = anticomplement of the isogonal conjugate of X(18772)
X(12) = complement of the isogonal conjugate of X(34434)
X(12) = isogonal conjugate of the anticomplement of X(34829)
X(12) = isotomic conjugate of the isogonal conjugate of X(181)
X(12) = isogonal conjugate of the isotomic conjugate of X(34388)
X(12) = isotomic conjugate of the polar conjugate of X(8736)
X(12) = polar conjugate of the isotomic conjugate of X(26942)
X(12) = polar conjugate of the isogonal conjugate of X(2197)
X(12) = orthic-isogonal conjugate of X(15443)
X(12) = X(i)-beth conjugate of X(j) for these (i,j): (10,12), (1089,1089)
X(12) = insimilicenter of incircle and nine-point circle
X(12) = X(1594)-of-Fuhrmann triangle
X(12) = homothetic center of Euler and Mandart-incircle triangles
X(12) = homothetic center of intouch and 4th Euler triangles
X(12) = X(6)-isoconjugate of X(2185)
X(12) = trilinear pole of line X(2610)X(4024)
X(12) = trilinear square of X(6724)
X(12) = homothetic center of medial triangle and cross-triangle of ABC and outer Johnson triangle
X(12) = homothetic center of medial triangle and cross-triangle of ABC and 1st Johnson-Yff triangle
X(12) = homothetic center of Euler triangle and cross-triangle of ABC and 2nd Johnson-Yff triangle
X(12) = homothetic center of ABC and triangular hull of circumcircles of BCX(4), CAX(4), and ABX(4); i.e., the outer Johnson triangle
X(12) = centroid of curvatures of nine-point circle and excircles
X(12) = X(18772)-anticomplementary conjugate of X(8)
X(12) = X(i)-complementary conjugate of X(j) for these (i,j): {{649, 40624}, {2051, 141}, {20028, 3741}, {34434, 10}, {40453, 49598}, {46880, 21246}}
X(12) = X(i)-Ceva conjugate of X(j) for these (i,j): {{4, 15443}, {10, 201}, {226, 2171}, {1441, 6358}, {4551, 523}, {4605, 4024}, {4998, 4552}, {6358, 594}, {34388, 26942}, {46102, 4559}}
X(12) = X(i)-cross conjugate of X(j) for these (i,j): {{125, 4036}, {181, 8736}, {756, 594}, {1109, 523}, {2171, 6354}, {2197, 26942}, {3690, 41508}, {4024, 4605}, {4092, 4024}, {4705, 21859}, {20708, 335}, {21671, 41506}, {21674, 10}, {21810, 321}}
X(12) = X(i)-isoconjugate of X(j) for these (i,j): {{1, 60}, {2, 2150}, {3, 270}, {6, 2185}, {8, 849}, {9, 593}, {11, 1101}, {21, 58}, {27, 2193}, {28, 283}, {29, 1437}, {31, 261}, {41, 1509}, {48, 46103}, {55, 757}, {56, 1098}, {57, 7054}, {63, 2189}, {81, 284}, {86, 2194}, {110, 3737}, {162, 23189}, {163, 4560}, {200, 7341}, {222, 2326}, {249, 2170}, {250, 7004}, {269, 6061}, {285, 2360}, {314, 2206}, {332, 2203}, {333, 1333}, {341, 7342}, {513, 4636}, {552, 1253}, {560, 18021}, {604, 7058}, {643, 3733}, {649, 4612}, {650, 4556}, {654, 37140}, {662, 7252}, {763, 1334}, {873, 2175}, {1014, 2328}, {1019, 5546}, {1021, 4565}, {1043, 1408}, {1169, 17185}, {1172, 1790}, {1364, 24000}, {1396, 2327}, {1412, 2287}, {1414, 21789}, {1444, 2299}, {1474, 1812}, {1576, 18155}, {1625, 39177}, {1919, 4631}, {2149, 26856}, {2204, 17206}, {2289, 36419}, {2316, 30576}, {2330, 7303}, {2363, 4267}, {3056, 7305}, {3063, 4610}, {3248, 6064}, {3271, 24041}, {3615, 17104}, {3668, 23609}, {3719, 36420}, {3738, 36069}, {3794, 38813}, {3904, 32671}, {4282, 24624}, {4570, 18191}, {4858, 23357}, {5317, 6514}, {7125, 36421}, {8748, 18604}, {9456, 30606}, {18180, 35196}, {23995, 34387}, {30581, 33635}}
X(12) = X(i)-Dao conjugate of X(j) for these (i,j): {{1, 1098}, {2, 261}, {3, 60}, {8, 4075}, {9, 2185}, {10, 21}, {11, 523}, {12, 2975}, {27, 47345}, {37, 333}, {55, 40607}, {56, 15267}, {58, 40611}, {65, 4225}, {81, 40590}, {86, 1214}, {115, 4560}, {125, 23189}, {181, 16872}, {223, 757}, {226, 1444}, {244, 3737}, {270, 36103}, {283, 40591}, {284, 40586}, {314, 40603}, {478, 593}, {552, 17113}, {647, 26932}, {650, 26856}, {758, 4996}, {873, 40593}, {960, 4267}, {1014, 36908}, {1084, 7252}, {1249, 46103}, {1509, 3160}, {1812, 51574}, {2150, 32664}, {2171, 21363}, {2189, 3162}, {2194, 40600}, {2287, 40599}, {3005, 3271}, {3161, 7058}, {3738, 38982}, {4370, 30606}, {4610, 10001}, {4612, 5375}, {4631, 9296}, {4636, 39026}, {4858, 18155}, {4988, 17197}, {5324, 18589}, {5452, 7054}, {6061, 6600}, {6374, 18021}, {6609, 7341}, {6741, 7253}, {7192, 40622}, {16587, 27958}, {16591, 33295}, {17045, 41002}, {18191, 50330}, {18314, 34387}, {21233, 21321}, {21789, 40608}}
X(12) = cevapoint of X(i) and X(j) for these (i,j): {{10, 3178}, {115, 4705}, {181, 2197}, {226, 27691}, {756, 2171}, {3120, 50330}, {4024, 4092}}
X(12) = crosspoint of X(i) and X(j) for these (i,j): {{10, 41013}, {226, 1441}, {4552, 4998}}
X(12) = crosssum of X(i) and X(j) for these (i,j): {{1, 37732}, {6, 20986}, {58, 1437}, {215, 4282}, {284, 2194}, {1364, 23189}, {3271, 7252}, {4560, 27010}}
X(12) = trilinear pole of line {2610, 4024}
X(12) = crossdifference of every pair of points on line {654, 4282}
X(12) = barycentric product X(i)*X(j) for these {i,j}: {{1, 6358}, {4, 26942}, {6, 34388}, {7, 594}, {8, 6354}, {10, 226}, {37, 1441}, {42, 349}, {56, 28654}, {57, 1089}, {59, 338}, {65, 321}, {69, 8736}, {72, 40149}, {75, 2171}, {76, 181}, {85, 756}, {92, 201}, {115, 4998}, {125, 46102}, {210, 1446}, {222, 7141}, {225, 306}, {257, 7211}, {264, 2197}, {273, 3949}, {278, 3695}, {279, 6057}, {281, 6356}, {307, 1826}, {312, 1254}, {313, 1400}, {318, 37755}, {331, 3690}, {335, 7235}, {339, 7115}, {341, 7147}, {346, 6046}, {348, 7140}, {522, 4605}, {523, 4552}, {553, 6538}, {651, 4036}, {653, 4064}, {655, 6370}, {664, 4024}, {693, 21859}, {850, 4559}, {872, 20567}, {1016, 1365}, {1018, 4077}, {1020, 4086}, {1042, 30713}, {1091, 2185}, {1109, 4564}, {1214, 41013}, {1231, 1824}, {1275, 4092}, {1402, 27801}, {1425, 7017}, {1427, 3701}, {1434, 6535}, {1500, 6063}, {1509, 6058}, {1577, 4551}, {1880, 20336}, {2052, 7066}, {2149, 23994}, {2321, 3668}, {2610, 35174}, {2970, 44717}, {3027, 40098}, {3649, 6539}, {3676, 4103}, {3678, 43682}, {3700, 4566}, {3911, 4013}, {3952, 7178}, {4017, 4033}, {4037, 7233}, {4052, 4848}, {4053, 18815}, {4079, 4572}, {4080, 40663}, {4554, 4705}, {4620, 21043}, {4736, 34535}, {4999, 31612}, {5552, 7363}, {6757, 16577}, {7012, 20902}, {7046, 20618}, {7058, 7314}, {7068, 23984}, {7080, 13853}, {7109, 41283}, {7148, 30545}, {7180, 27808}, {7185, 43265}, {7249, 21021}, {8808, 21075}, {8818, 40999}, {14618, 23067}, {14624, 41003}, {15065, 18593}, {15523, 18097}, {15556, 43683}, {16603, 40718}, {16609, 43534}, {17097, 42708}, {20565, 21794}, {20616, 40216}, {21810, 31643}, {24002, 40521}, {36804, 51645}, {40447, 41393}, {41538, 43675}, {42666, 46405}}
X(12) = barycentric quotient X(i)/X(j) for these {i,j}: {{1, 2185}, {2, 261}, {4, 46103}, {6, 60}, {7, 1509}, {8, 7058}, {9, 1098}, {10, 333}, {11, 26856}, {19, 270}, {25, 2189}, {31, 2150}, {33, 2326}, {37, 21}, {42, 284}, {55, 7054}, {56, 593}, {57, 757}, {59, 249}, {65, 81}, {71, 283}, {72, 1812}, {73, 1790}, {76, 18021}, {85, 873}, {100, 4612}, {101, 4636}, {109, 4556}, {115, 11}, {125, 26932}, {181, 6}, {201, 63}, {210, 2287}, {213, 2194}, {220, 6061}, {225, 27}, {226, 86}, {227, 1817}, {228, 2193}, {279, 552}, {306, 332}, {307, 17206}, {313, 28660}, {321, 314}, {338, 34387}, {349, 310}, {512, 7252}, {519, 30606}, {523, 4560}, {553, 30593}, {594, 8}, {604, 849}, {647, 23189}, {661, 3737}, {664, 4610}, {668, 4631}, {756, 9}, {762, 210}, {872, 41}, {1014, 763}, {1016, 6064}, {1018, 643}, {1020, 1414}, {1042, 1412}, {1089, 312}, {1091, 6358}, {1109, 4858}, {1118, 36419}, {1214, 1444}, {1215, 27958}, {1254, 57}, {1275, 7340}, {1319, 30576}, {1334, 2328}, {1356, 1977}, {1365, 1086}, {1400, 58}, {1402, 1333}, {1407, 7341}, {1409, 1437}, {1425, 222}, {1426, 1396}, {1427, 1014}, {1432, 7303}, {1434, 6628}, {1441, 274}, {1500, 55}, {1577, 18155}, {1824, 1172}, {1826, 29}, {1840, 14006}, {1857, 36421}, {1867, 44734}, {1874, 31905}, {1880, 28}, {1893, 31926}, {1903, 285}, {2092, 4267}, {2149, 1101}, {2171, 1}, {2197, 3}, {2222, 37140}, {2292, 17185}, {2318, 2327}, {2321, 1043}, {2333, 2299}, {2594, 40214}, {2610, 3738}, {2616, 39177}, {2643, 2170}, {3027, 4366}, {3120, 17197}, {3124, 3271}, {3125, 18191}, {3178, 40605}, {3212, 7304}, {3269, 1364}, {3649, 8025}, {3668, 1434}, {3671, 42028}, {3682, 6514}, {3690, 219}, {3694, 1792}, {3695, 345}, {3700, 7253}, {3708, 7004}, {3709, 21789}, {3721, 3794}, {3724, 4282}, {3925, 16713}, {3947, 25507}, {3949, 78}, {3952, 645}, {4013, 4997}, {4017, 1019}, {4024, 522}, {4032, 17103}, {4033, 7257}, {4036, 4391}, {4037, 3685}, {4041, 1021}, {4053, 4511}, {4064, 6332}, {4069, 7259}, {4077, 7199}, {4079, 663}, {4092, 1146}, {4103, 3699}, {4155, 4435}, {4365, 4483}, {4415, 17183}, {4466, 17219}, {4551, 662}, {4552, 99}, {4554, 4623}, {4557, 5546}, {4559, 110}, {4564, 24041}, {4566, 4573}, {4583, 36806}, {4605, 664}, {4642, 18163}, {4705, 650}, {4848, 41629}, {4998, 4590}, {6046, 279}, {6057, 346}, {6058, 594}, {6354, 7}, {6355, 34400}, {6356, 348}, {6358, 75}, {6367, 4976}, {6370, 3904}, {6378, 2053}, {6535, 2321}, {6537, 41002}, {6538, 4102}, {7064, 220}, {7066, 394}, {7068, 23983}, {7109, 2175}, {7115, 250}, {7132, 7305}, {7138, 7125}, {7140, 281}, {7141, 7017}, {7143, 1407}, {7147, 269}, {7148, 2319}, {7178, 7192}, {7180, 3733}, {7206, 42033}, {7211, 894}, {7216, 7203}, {7230, 4387}, {7235, 239}, {7237, 3061}, {7276, 3758}, {7314, 6354}, {7337, 36420}, {7363, 7318}, {7668, 27010}, {8013, 3686}, {8736, 4}, {8754, 8735}, {8818, 3615}, {8898, 5323}, {13853, 1440}, {15232, 19607}, {16583, 5324}, {16603, 30966}, {16609, 33295}, {16886, 3705}, {16888, 33947}, {17094, 15419}, {20616, 1621}, {20617, 17074}, {20618, 7056}, {20653, 3687}, {20902, 17880}, {20975, 7117}, {21015, 27509}, {21021, 7081}, {21033, 46877}, {21043, 21044}, {21051, 27527}, {21056, 25128}, {21075, 27398}, {21077, 31631}, {21131, 21132}, {21674, 5745}, {21675, 6734}, {21699, 3691}, {21721, 20293}, {21741, 17104}, {21794, 35}, {21803, 2329}, {21804, 16699}, {21808, 17194}, {21810, 960}, {21813, 7083}, {21816, 3683}, {21833, 4516}, {21853, 3193}, {21854, 1816}, {21859, 100}, {21958, 21300}, {22229, 23864}, {22341, 18604}, {23067, 4558}, {26942, 69}, {26955, 26871}, {27691, 6626}, {27801, 40072}, {28654, 3596}, {30730, 7256}, {32636, 30581}, {32675, 36069}, {34294, 18101}, {34388, 76}, {34857, 2341}, {35069, 4996}, {35307, 2617}, {37755, 77}, {39793, 18166}, {40149, 286}, {40521, 644}, {40590, 4225}, {40663, 16704}, {40952, 46882}, {40966, 46889}, {40999, 34016}, {41003, 16705}, {41013, 31623}, {41393, 18607}, {41508, 43740}, {41538, 40571}, {41539, 41610}, {42666, 654}, {43534, 36800}, {45196, 16739}, {45208, 18169}, {46102, 18020}, {50487, 3063}, {51645, 3960}}
X(12) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {{1, 5, 11}, {1, 11, 37722}, {1, 80, 37730}, {1, 119, 10958}, {1, 355, 10950}, {1, 495, 15888}, {1, 3614, 7173}, {1, 4551, 2594}, {1, 5219, 11375}, {1, 5252, 10944}, {1, 5443, 5901}, {1, 5587, 1837}, {1, 5726, 9578}, {1, 5727, 37724}, {1, 5881, 37740}, {1, 7741, 496}, {1, 7951, 5}, {1, 7988, 50443}, {1, 7989, 9581}, {1, 8227, 11376}, {1, 9578, 5252}, {1, 10592, 3614}, {1, 10826, 5722}, {1, 10827, 355}, {1, 10942, 10955}, {1, 10944, 1317}, {1, 10957, 10949}, {1, 11375, 15950}, {1, 23708, 11373}, {1, 26470, 10959}, {1, 26481, 10957}, {1, 26482, 10956}, {1, 37692, 5886}, {1, 37701, 37737}, {1, 37702, 12433}, {1, 37706, 37728}, {1, 37707, 1483}, {1, 37708, 37727}, {1, 37709, 37738}, {1, 37711, 37739}, {1, 37714, 5727}, {1, 37719, 495}, {1, 37731, 5719}, {1, 37735, 1387}, {2, 56, 5433}, {2, 388, 56}, {2, 958, 24953}, {2, 2975, 4999}, {2, 3436, 958}, {2, 3600, 7288}, {2, 4999, 31260}, {2, 5253, 6691}, {2, 5261, 388}, {2, 5433, 7294}, {2, 5434, 5298}, {2, 6645, 26686}, {2, 11236, 34606}, {2, 11237, 5434}, {2, 11681, 1329}, {2, 20060, 2975}, {2, 41245, 43053}, {2, 43053, 31221}, {3, 498, 5432}, {3, 1478, 7354}, {3, 7354, 15326}, {3, 9654, 1478}, {3, 9655, 4299}, {3, 10526, 11827}, {3, 11929, 10526}, {3, 31479, 498}, {3, 31659, 21155}, {4, 55, 6284}, {4, 3085, 55}, {4, 4294, 12953}, {4, 8164, 3085}, {4, 10590, 10895}, {4, 10599, 10894}, {4, 10786, 11500}, {4, 11500, 6253}, {4, 41227, 1852}, {5, 11, 7173}, {5, 495, 1}, {5, 496, 7741}, {5, 5252, 10957}, {5, 7951, 3614}, {5, 10592, 7951}, {5, 10942, 355}, {5, 10954, 10950}, {5, 10956, 10949}, {5, 11698, 18357}, {5, 15888, 37722}, {5, 26482, 10944}, {5, 32213, 10943}, {5, 37719, 15888}, {6, 12588, 39897}, {6, 31472, 19028}, {6, 44620, 19026}, {6, 44621, 19025}, {6, 44622, 19027}, {7, 1788, 5221}, {7, 7679, 3826}, {7, 9780, 1788}, {7, 38057, 41712}, {8, 2476, 2886}, {8, 3485, 2099}, {8, 5226, 3485}, {10, 65, 40663}, {10, 72, 21677}, {10, 226, 65}, {10, 442, 3925}, {10, 3671, 4848}, {10, 3822, 442}, {10, 3947, 226}, {10, 10408, 181}, {10, 11263, 3754}, {10, 12609, 3753}, {10, 17757, 21031}, {10, 21075, 210}, {10, 21077, 72}, {11, 3614, 5}, {11, 10944, 10949}, {11, 10955, 10950}, {11, 10956, 1317}, {11, 15888, 1}, {11, 37734, 10959}, {20, 5218, 5217}, {20, 5229, 12943}, {36, 5270, 18990}, {39, 31476, 31460}, {40, 9612, 1836}, {42, 21935, 1834}, {55, 10895, 4}, {55, 12953, 4294}, {56, 388, 5434}, {56, 5433, 5298}, {56, 11237, 388}, {57, 1698, 24914}, {57, 5290, 10404}, {65, 210, 41538}, {65, 226, 3649}, {75, 21581, 21405}, {80, 37731, 1}, {85, 3665, 1358}, {85, 7179, 3665}, {125, 1425, 26955}, {140, 18990, 36}, {145, 5141, 11680}, {145, 11680, 3813}, {171, 1935, 1399}, {225, 1826, 1882}, {226, 4848, 3671}, {226, 16609, 5244}, {354, 17606, 1210}, {355, 10523, 11}, {355, 10942, 37725}, {355, 10954, 10955}, {355, 11374, 1}, {355, 37739, 37711}, {371, 9646, 13901}, {377, 5552, 1376}, {381, 3295, 1479}, {381, 10056, 3058}, {381, 18518, 18517}, {388, 3436, 18962}, {388, 5261, 11237}, {388, 7288, 3600}, {388, 10588, 2}, {390, 3832, 5225}, {390, 5225, 9670}, {404, 27529, 3035}, {442, 17757, 10}, {485, 1335, 19030}, {486, 1124, 19029}, {495, 3614, 37722}, {495, 5252, 10956}, {495, 7951, 11}, {495, 10592, 5}, {495, 10827, 10955}, {495, 10942, 10954}, {495, 26481, 10944}, {496, 7741, 11}, {497, 3091, 10896}, {498, 1478, 3}, {498, 9654, 7354}, {546, 15171, 3583}, {590, 2067, 18965}, {611, 1352, 39873}, {615, 6502, 18966}, {631, 4293, 5204}, {631, 31410, 9657}, {756, 1254, 201}, {908, 24987, 960}, {942, 9956, 1737}, {946, 31397, 3057}, {950, 13405, 37080}, {958, 3436, 34606}, {958, 11236, 3436}, {999, 1656, 499}, {1056, 3086, 3304}, {1056, 3090, 3086}, {1058, 3545, 10591}, {1058, 10591, 11238}, {1089, 3695, 6057}, {1125, 3814, 4187}, {1125, 10106, 1319}, {1210, 10175, 17606}, {1210, 21620, 354}, {1329, 15843, 958}, {1329, 25466, 2}, {1387, 39692, 11}, {1447, 7247, 7198}, {1478, 4299, 9655}, {1478, 5432, 15326}, {1478, 31479, 5432}, {1479, 3295, 3058}, {1479, 10056, 3295}, {1682, 10407, 2051}, {1697, 1699, 12701}, {1698, 3820, 50038}, {1698, 5290, 57}, {1698, 41229, 5791}, {1737, 13407, 942}, {1785, 39574, 42385}, {1837, 17718, 1}, {2294, 21011, 21933}, {2886, 12607, 8}, {2975, 4999, 31157}, {2975, 6668, 31260}, {3057, 17605, 946}, {3057, 31397, 45081}, {3069, 31408, 18995}, {3072, 3074, 2361}, {3085, 10590, 4}, {3085, 10599, 10953}, {3085, 10895, 6284}, {3297, 42262, 44624}, {3298, 42265, 44623}, {3303, 10896, 497}, {3340, 3679, 41687}, {3421, 6856, 19843}, {3434, 10528, 3913}, {3436, 10585, 2}, {3476, 3616, 1388}, {3487, 5818, 18391}, {3555, 10916, 51463}, {3583, 3746, 15171}, {3584, 3585, 35}, {3600, 7288, 56}, {3614, 15888, 11}, {3616, 4193, 3816}, {3617, 33108, 9710}, {3634, 4298, 3911}, {3649, 40663, 65}, {3671, 4848, 65}, {3753, 12709, 65}, {3812, 5123, 24982}, {3820, 8728, 1698}, {3822, 17757, 3925}, {3826, 9711, 9780}, {3826, 9780, 34501}, {3843, 31480, 4309}, {3851, 6767, 9669}, {3911, 4298, 32636}, {3925, 21031, 10}, {3949, 21675, 594}, {4197, 9711, 34501}, {4197, 9780, 3826}, {4292, 6684, 1155}, {4294, 12953, 6284}, {4295, 5657, 37567}, {4299, 9655, 7354}, {4311, 10165, 37605}, {4312, 9588, 5128}, {4995, 15338, 35}, {4999, 6668, 2}, {5010, 10483, 550}, {5051, 26115, 4026}, {5056, 14986, 10589}, {5082, 31418, 31140}, {5083, 6702, 20118}, {5177, 7080, 2550}, {5204, 9657, 4293}, {5217, 9656, 12943}, {5217, 12943, 20}, {5218, 5229, 20}, {5219, 5252, 15950}, {5219, 5726, 5252}, {5219, 9578, 1}, {5249, 24982, 3812}, {5252, 11375, 1}, {5252, 15950, 1317}, {5252, 37738, 37709}, {5261, 10588, 56}, {5261, 11681, 15844}, {5298, 7294, 5433}, {5432, 7354, 3}, {5433, 5434, 56}, {5530, 13161, 3666}, {5552, 10522, 1259}, {5587, 17857, 355}, {5603, 6941, 7681}, {5657, 5714, 4295}, {5690, 39542, 5903}, {5697, 18393, 22791}, {5719, 18357, 37730}, {5719, 37730, 1}, {5777, 50195, 1858}, {5901, 8070, 11}, {5927, 12711, 1898}, {6198, 9627, 10149}, {6668, 20060, 31157}, {6833, 12115, 12114}, {6834, 10532, 22753}, {6856, 19843, 31245}, {6871, 10528, 3434}, {6879, 10805, 10785}, {6923, 11248, 11826}, {6968, 10531, 10893}, {7173, 37722, 11}, {7176, 17095, 7181}, {7680, 18242, 4}, {7951, 8068, 38109}, {7951, 9578, 26481}, {7951, 15888, 7173}, {7951, 26482, 10957}, {7951, 37701, 8068}, {7951, 37719, 1}, {8068, 37710, 26470}, {8088, 8382, 174}, {8164, 10590, 55}, {8583, 30827, 24954}, {9553, 31496, 573}, {9578, 11375, 10944}, {9578, 37719, 26482}, {9597, 31497, 5013}, {9612, 31434, 40}, {9647, 31499, 6200}, {9649, 31500, 1151}, {9651, 31501, 574}, {9654, 31479, 3}, {9656, 12943, 5229}, {9955, 9957, 30384}, {10175, 21620, 1210}, {10404, 24914, 57}, {10523, 10942, 10950}, {10523, 10954, 1}, {10523, 10955, 37722}, {10526, 26487, 3}, {10572, 24929, 10543}, {10576, 35768, 9661}, {10585, 11236, 24953}, {10588, 11237, 5433}, {10592, 37719, 11}, {10592, 37737, 38109}, {10593, 37720, 11}, {10599, 10786, 4}, {10786, 10894, 6253}, {10827, 10954, 37725}, {10827, 11374, 10950}, {10827, 37719, 10954}, {10894, 11500, 4}, {10921, 10922, 12587}, {10923, 10924, 12588}, {10923, 19027, 39897}, {10924, 19028, 39897}, {10943, 32213, 37727}, {10944, 15950, 1}, {10953, 11500, 6284}, {10954, 11374, 15888}, {10956, 10957, 10944}, {10957, 15888, 1317}, {10957, 15950, 37722}, {11235, 11239, 34699}, {11375, 26481, 11}, {11375, 26482, 15888}, {11376, 26476, 11}, {11698, 12738, 37725}, {11929, 26487, 11827}, {12019, 12433, 37702}, {12587, 45456, 10922}, {12587, 45457, 10921}, {12588, 45458, 10924}, {12588, 45459, 10923}, {13405, 19925, 950}, {13897, 18996, 3068}, {13954, 18995, 3069}, {15843, 15844, 18962}, {15844, 25466, 56}, {16886, 21021, 594}, {17530, 24390, 25639}, {17532, 45701, 34612}, {18357, 37730, 80}, {18480, 24929, 10572}, {18517, 18518, 34746}, {18542, 37234, 18516}, {18641, 51368, 22341}, {19025, 19026, 6}, {19027, 19028, 6}, {19860, 31266, 28628}, {19861, 30852, 25681}, {20067, 37291, 5303}, {20616, 21859, 1500}, {21021, 21057, 16886}, {21044, 21808, 21049}, {21155, 30264, 3}, {23477, 23517, 80}, {24953, 34606, 958}, {24982, 25962, 25973}, {26470, 38109, 5}, {26481, 26482, 5252}, {26481, 37719, 10956}, {26752, 33841, 26582}, {27283, 30001, 30847}, {28742, 33839, 16593}, {30313, 30314, 3826}, {30316, 30317, 3826}, {31140, 34619, 34720}, {31157, 31260, 4999}, {31418, 34619, 5082}, {31472, 44622, 6}, {31472, 45458, 39897}, {31477, 44518, 9598}, {37691, 45920, 11}, {37701, 37710, 1}, {37701, 41689, 5719}, {37705, 37728, 37706}, {37709, 37738, 10944}, {37710, 37737, 37734}, {37711, 37739, 10950}, {37734, 38109, 7173}, {38027, 51112, 551}, {38062, 51111, 1125}, {38105, 51113, 3828}, {44620, 44621, 6}, {44622, 45459, 39897}, {45456, 45457, 12587}, {45458, 45459, 12588}}


X(13) = 1st ISOGONIC CENTER (FERMAT POINT, TORRICELLI POINT)

Trilinears    csc(A + π/3) : csc(B + π/3) : csc(C + π/3)
Trilinears    sec(A - π/6) : sec(B - π/6) : sec(C - π/6)
Barycentrics   a4 - 2(b2 - c2)2 + a2(b2 + c2 + 4*sqrt(3)*Area(ABC)) : :
Barycentrics    (SA + Sqrt[3] S) (SB + SC) + 4 SB SC : :
Tripolars    1/(a^4+a^2 b^2-2 b^4+a^2 c^2+4 b^2 c^2-2 c^4+2 Sqrt[3] a^2 S) : :
X(13) = 31/2(r2 + 2rR + s2)*X(1) - 6r(31/2R - 2s)*X(2) - 2r(31/2r + 3s)*X(3)
   (Peter Moses, April 2, 2013)

Construct the equilateral triangle BA'C having base BC and vertex A' on the negative side of BC; similarly construct equilateral triangles CB'A and AC'B based on the other two sides. The lines AA', BB', CC' concur in X(13). If each of the angles A, B, C is < 2π/3, then X(13) is the only center X for which the angles BXC, CXA, AXB are equal, and X(13) minimizes the sum |AX| + |BX| + |CX|. The antipedal triangle of X(13) is equilateral.

If, however, A> 2π/3, then the Fermat point, defined geometrically as the minimizer of |AX| + |BX| + |CX|, is not the 1st isogonic center (which is defined by the above trilinears). Trilinears for the Fermat point when A> 2π/3 are simply 1:0:0. To represent the Fermat point in the form f(a,b,c) : f(b,c,a) : f(c,a,b), one must use Boolean variables, as shown at Fermat point.

If you have The Geometer's Sketchpad, you can view these sketches:
Fermat Dynamic
1st isogonic center
Kiepert Hyperbola, showing X(13) and X(14) on the hyperbola, with midpoint X(115)
Evans Conic, passing through X(13), X(14), X(15), X(16), X(17), X(18), X(3070), X(3071).
X(3054), center of the Evans Conic and 19 other triangle centers.
If you have GeoGebra, you can view 1st isogonic center.

The Evans conic is introduced in Evans, Lawrence S., "A Conic Through Six Triangle Centers," Forum Geometricorum 2 (2002) 89-92.

Let NaNbNc, Na'Nb'Nc' be the outer and inner Napoleon triangles, respectively. Let A' be the isogonal conjugate of Na', wrt NaNbNc, and define B' and C' cyclically. The lines NaA', NbB', NcC' concur in X(13). (Randy Hutson, January 29, 2015)

Let P be a point inside triangle ABC such that the line AP bisects angle BPC, and NBP bisets CPA, and CP bisects APB. Then P = X(13). The locus of P such that AP bisects BPC is the circumcubic given by the barycentric equation c2xy2 - b2xz2 + (a2 - b2 + c2)y2z - (a2 + b2 - c2)yz2 = 0, and the other two cubics are given cyclically. Bernard Gibert discusses these cubics as K053A, K053B, K053C; see Apollonian strophoids. (Paul Hanna and Peter Moses, August 6, 2017)

The line X(13)X(15) is parallel to the Euler line, and the distance between the two lines is (SB - SC)(SC - SA)(SA - SB)/|(cot(ω) + 31/2)|*((E - 8F)S2)1/2. (Kiminari Shinagawa, February 20, 2018)

Let A'B'C' be the outer Napoleon triangle and A″B″C″ the inner Napoleon triangle. Let A* be the isogonal conjugate, wrt A'B'C', of A″, and define B* and C* cyclically. The lines A'A*, B'B*, C'C* concur in X(13). (Randy Hutson, December 2, 2017)

Let F be X(13) or X(14). Let L and L' be lines through F such that the angle between them is π/3; if you have GeoGebra, see Figure 13A. Let LBC = L∩BC, and define LCA and LAB cyclically. Let L'BC = L'∩BC, and define L'CA and L'AB cyclically. The lines LBCL'CA, LCAL'AB, LABL'BC concur. (Dao Thanh Oai, 2014)

Let F be X(13) or X(14). Let A0, B0, C0 be points on BC, CA, AB, respectively, such that the directed angles FA0-to-FC0 = π/3 and FC0-to-FB0 = π/3; if you have GeoGebra, see Figure 13B. The points A0, B0, C0 are collinear. (Dao Thanh Oai, 2014)

Video showing circular porisim-orbits of X(13), X(14), X(15), and X(16): 3-Periodics in a Concentric Homothetic Poncelet Pair: Circular Loci of Four Triangle Centers. (Dan Reznik, August 9, 2020) See also Loci of Centers of Ellipse-Mounted Triangles. (Dan Reznik, August 26, 2020)

If P is a point not on the line X(13)X(14), then the circle {P, X(13), X(14)}} is orthogonal to the orthocentroidal circle. (Peter Moses, April 22, 2021)

Several notable circles pass through X(13) and X(14). For each circle listed here, the appearance of (i; [name], [list]) means that the center is X(i), and the points with listed indices lie on the circle:

(1116; Lester circle, 3, 5, 13, 14, 1117, 5671, 14854, 15475, 15535, 15536, 15537, 15538, 15539, 15540, 15541, 15542, 15543, 15544, 15545, 15546, 15547, 15548, 15549, 15550, 15551, 15552, 15553, 15554, 15555, 34365)

(1637; Dao-Moses-Telv circle; 13, 14, 5000, 5001, 6104, 6105, 6106, 6107, 6108, 6109, 6110, 6111, 24007, 24008)

(1640; 13, 14, 10653, 10654, 32618, 32619)

(8371; Hutson-Parry circle, 2, 13, 14, 111, 476, 5466, 5640, 6032, 6792, 7698, 9140, 9159, 11628, 11639, 11640, 13636, 13722, 14846, 14932, 34320)

(9200; 13, 14, 16, 5623, 11586, 30439)

(9201; 13, 14, 15, 5624, 15743, 30440)

(9202; 13, 14, 16, 5623, 11586, 30439)

(14446; 13, 14, 5616, 5668, 6779, 8172, 11600, 38943)

(14447; 13, 14, 5612, 5669, 6780, 8173, 11601, 38944)

(30574; 13, 14, 80,484, 3464, 5540, 5902, 34301, 37718)

(42731; 13, 14, 112, 1141, 1157, 5667, 5890, 6761, 14644, 14651)

(42732; 13, 14, 112, 1141, 1157, 5667, 5890, 6761, 14644, 14651)

(42733; 4, 13, 14, 2132, 2394, 6794, 22265, 34298)

(42734; 13, 14, 616, 621, 5675, 16260, 39133)

(42735; 13, 14, 617, 621, 5674, 16259, 39132)

(42736; 13, 14, 125, 1637, 11657, 14847, 34310)

(42737; 13, 14, 110, 3448, 34306, 34308)

(42738; 13, 14, 98, 11005, 14223, 23969)

(42730; 13, 14, 74, 5627, 5670, 18331)

(42740; Dao-Parry circle of X(1), 1, 13, 14, 79, 5677, 42748, 42749)

Related circles are discussed in trhe preamble just before X(42740).

X(13) lies on the Neuberg cubic and these lines: 2,16   3,17   4,61   5,18   6,14   11,202   15,30   76,299   80,1251   98,1080   99,303   148,617   203,1478   226,1081   262,383   275,472   298,532   484,1277   531,671   533,621   634,635

X(13) is the {X(6),X(381)}-harmonic conjugate of X(14). For a list of other harmonic conjugates of X(13), click Tables at the top of this page.

X(13) = reflection of X(i) in X(j) for these (i,j): (14,115), (15,396), (99,619), (298,623), (616,618)
X(13) = isogonal conjugate of X(15)
X(13) = isotomic conjugate of X(298)
X(13) = circumcircle-inverse of X(6104)
X(13) = orthocentroidal-circle inverse of X(14)
X(13) = complement of X(616)
X(13) = anticomplement of X(618)
X(13) = cevapoint of X(15) and X(62)
X(13) = X(i)-cross conjugate of X(j) for these (i,j): (15,18), (30,14), (396,2)
X(13) = trilinear pole of line X(395)X(523) (polar of X(470) wrt polar circle)
X(13) = pole wrt polar circle of trilinear polar of X(470)
X(13) = X(48)-isoconjugate (polar conjugate) of X(470)
X(13) = antigonal image of X(14)
X(13) = reflection of X(14) in line X(115)X(125)
X(13) = X(15)-of-4th-Brocard-triangle
X(13) = X(15)-of-orthocentroidal-triangle
X(13) = orthocorrespondent of X(13)
X(13) = homothetic center of outer Napoleon triangle and antipedal triangle of X(13)
X(13) = inner-Napoleon-to-outer-Napoleon similarity image of X(15)
X(13) = outer-Napoleon-isogonal conjugate of X(3)
X(13) = outer-Napoleon-to-inner-Napoleon similarity image of X(14)
X(13) = orthocenter of X(14)X(98)X(2394)
X(13) = X(15)-of-pedal-triangle of X(13)
X(13) = {X(265),X(1989)}-harmonic Conjugate of X(14)
X(13) = homothetic center of (equilateral) antipedal triangle of X(13) and triangle formed by circumcenters of BCX(13), CAX(13), ABX(13)
X(13) = homothetic center of triangle formed by circumcenters of BCX(14), CAX(14), ABX(14) and triangle formed by nine-point centers of BCX(13), CAX(13), ABX(13)
X(13) = Cundy-Parry Phi transform of X(17)
X(13) = Cundy-Parry Psi transform of X(61)
X(13) = Kosnita(X(13),X(1)) point
X(13) = Kosnita(X(13),X(13)) point
X(13) = Thomson-isogonal conjugate of X(34317)


X(14) = 2nd ISOGONIC CENTER

Trilinears       csc(A - π/3) : csc(B - π/3) : csc(C - π/3)
                        = sec(A + π/6) : sec(B + π/6) : sec(C + π/6)

Barycentrics  f(a,b,c) : f(b,c,a) : f(c,a,b), where
                        f(a,b,c) = a4 - 2(b2 - c2)2 + a2(b2 + c2 - 4*sqrt(3)*Area(ABC))
Barycentrics    (SA - Sqrt[3] S) (SB + SC) + 4 SB SC : :
Tripolars    1/(a^4+a^2 b^2-2 b^4+a^2 c^2+4 b^2 c^2-2 c^4-2 Sqrt[3] a^2 S) : :
X(14) = 31/2(r2 + 2rR + s2)*X(1) - 6r(31/2R + 2s)*X(2) - 2r(31/2r - 3s)*X(3)   (Peter Moses, April 2, 2013)

Construct the equilateral triangle BA'C having base BC and vertex A' on the positive side of BC; similarly construct equilateral triangles CB'A and AC'B based on the other two sides. The lines AA',BB',CC' concur in X(14). The antipedal triangle of X(14) is equilateral.

Let NaNbNc, Na'Nb'Nc' be the outer and inner Napoleon triangles, resp. Let A' be the isogonal conjugate of Na, wrt Na'Nb'Nc', and define B' and C' cyclically. The lines Na'A', Nb'B', Nc'C' concur in X(14). (Randy Hutson, January 29, 2015)

Let A'B'C' be the outer Napoleon triangle and A″B″C″ the inner Napoleon triangle. Let A* be the isogonal conjugate, wrt A″B″C″, of A', and define B* and C* cyclically. The lines A″A*, B″B*, C″C* concur in X(14). (Randy Hutson, December 2, 2017)

The line X(14)X(16) is parallel to the Euler line, and the distance between the two lines is (SB - SC)(SC - SA)(SA - SB)/|(cot(ω) - 31/2)|((E - 8F)S2)1/2. (Kiminari Shinagawa, February 20, 2018)

If you have The Geometer's Sketchpad, you can view 2nd isogonic center
If you have GeoGebra, you can view 2nd isogonic center.

Let O be a point (not necessarily X(3)), let H=ABCDEF be a regular hexagon with center O, and let P be a point. Define 6 triangles T1=ABP, T2=BCP, ... , T6=FAP.
Claim 1: The points X(14)-of-Ti lie on OP, i=1...6.
Next, define 6 reflection triangles T1' = reflection of T1 in AB, T2' = reflection of T2 in BC, ..., T6' = reflection of T6 in FA.
Claim 2: If P is interior to H, the points X(14)-of-Ti' lie on a rectangular hyperbola centered at O: Figure. (Dan Reznik, December 10, 2021)

X(14) lies on the Neuberg cubic and these lines: 2,15   3,18   4,62   5,17   6,13   11,203   16,30   76,298   98,383   99,302   148,616   202,1478   226,554   262,1080   275,473   299,533   397,546   484,1276   530,671   532,622   633,636

X(14) is the {X(6),X(381)}-harmonic conjugate of X(13). For a list of other harmonic conjugates of X(14), click Tables at the top of this page.

X(14) = reflection of X(i) in X(j) for these (i,j): (13,115), (16,395), (99,618), (299,624), (617,619)
X(14) = isogonal conjugate of X(16)
X(14) = isotomic conjugate of X(299)
X(14) = complement of X(617)
X(14) = anticomplement of X(619)
X(14) = circumcircle-inverse of X(6105)
X(14) = orthocentroidal-circle-inverse of X(13)
X(14) = cevapoint of X(16) and X(61)
X(14) = X(i)-cross conjugate of X(j) for these (i,j): (16,17), (30,13), (395,2)
X(14) = trilinear pole of line X(396)X(523) (polar of X(471) wrt polar circle)
X(14) = pole wrt polar circle of trilinear polar of X(471)
X(14) = X(48)-isoconjugate (polar conjugate) of X(471)
X(14) = antigonal image of X(13)
X(14) = reflection of X(13) in line X(115)X(125)
X(14) = X(16)-of-4th-Brocard triangle
X(14) = X(16)-of-orthocentroidal-triangle
X(14) = orthocorrespondent of X(14)
X(14) = homothetic center of inner Napoleon triangle and antipedal triangle of X(14)
X(14) = inner-Napoleon-isogonal conjugate of X(3)
X(14) = outer-Napoleon-to-inner-Napoleon similarity image of X(16)
X(14) = inner-Napoleon-to-outer-Napoleon similarity image of X(13)
X(14) = orthocenter of X(13)X(98)X(2394)
X(14) = X(16)-of-pedal-triangle of X(14)
X(14) = {X(265),X(1989)}-harmonic Conjugate of X(13)
X(14) = homothetic center of (equilateral) antipedal triangle of X(14) and triangle formed by circumcenters of BCX(14), CAX(14), ABX(14)
X(14) = homothetic center of triangle formed by circumcenters of BCX(13), CAX(13), ABX(13) and triangle formed by nine-point centers of BCX(14), CAX(14), ABX(14)
X(14) = Cundy-Parry Phi transform of X(18)
X(14) = Cundy-Parry Psi transform of X(62)
X(14) = Kosnita(X(14),X(1)) point
X(14) = Kosnita(X(14),X(14)) point
X(14) = Thomson-isogonal conjugate of X(34318)


X(15) = 1st ISODYNAMIC POINT

Trilinears    sin(A + π/3) : sin(B + π/3) : sin(C + π/3)
Trilinears    cos(A - π/6) : cos(B - π/6) : cos(C - π/6)
Trilinears    3 cos A + sqrt(3) sin A : :
Barycentrics  a sin(A + π/3) : b sin(B + π/3) : c sin(C + π/3)
Barycentrics    (SB + SC)/((SA + Sqrt[3] S) (SB + SC) + 4 SB SC) : :
Barycentrics    (S + Sqrt[3] SA) (SB + SC) : :
Barycentrics    a^2*(Sqrt[3]*(a^2 - b^2 - c^2) - 2*S) : :
Tripolars    b c : :

X(15) = (r2 + 2rR + s2)*X(1) - 6rR*X(2) - 2r(r - 31/2s)*X(3) = sqrt(3)*X(3) + (cot ω)*X(6)    (Peter Moses, April 2, 2013)
X(15) = 3 X[2] - 4 X[6671], 3 X[13] - 2 X[5318], 3 X[13] - 4 X[11542], 2 X[13] - 3 X[16267], 3 X[13] - 5 X[16960], X[13] - 3 X[16962], 3 X[13] - X[19106], 3 X[14] - 2 X[33518], 3 X[17] - 2 X[31705], 3 X[18] - X[22849], 2 X[115] - 3 X[22510], 4 X[230] - 3 X[22511], 3 X[396] - X[5318], 3 X[396] - 2 X[11542], 4 X[396] - 3 X[16267], 6 X[396] - 5 X[16960], 2 X[396] - 3 X[16962], 6 X[396] - X[19106], X[621] - 4 X[6671], 3 X[3129] - X[3440], 4 X[5318] - 9 X[16267], 2 X[5318] - 5 X[16960], 2 X[5318] - 9 X[16962], 3 X[5469] - X[25166], 3 X[5470] - 2 X[31709], 2 X[6109] + X[6780], 3 X[6109] - X[33518], X[6778] - 3 X[16529], 3 X[6780] + 2 X[33518], 2 X[6783] - 3 X[16529], X[10675] - 3 X[11243], 8 X[11542] - 9 X[16267], 4 X[11542] - 5 X[16960], 4 X[11542] - 9 X[16962], 4 X[11542] - X[19106], 3 X[14138] - X[31705], 9 X[16267] - 10 X[16960], 9 X[16267] - 2 X[19106], 5 X[16960] - 9 X[16962], 5 X[16960] - X[19106], 9 X[16962] - X[19106], 3 X[22489] - 2 X[31693], X[22495] + 2 X[35931], 3 X[22510] - X[23004], 3 X[22571] - 2 X[31695], 3 X[22602] - 2 X[31697], 3 X[22631] - 2 X[31699], 3 X[22688] - 2 X[31701], 3 X[22846] - 2 X[31703], 3 X[25151] - 2 X[31707], 3 X[25157] - 2 X[31711], 3 X[25158] - 2 X[31713], 3 X[25159] - 2 X[31715], 3 X[25160] - 2 X[31717], 3 X[25217] - 2 X[31719]

Let U and V be the points on sideline BC met by the interior and exterior bisectors of angle A. The circle having diameter UV is the A-Apollonian circle. The B- and C- Apollonian circles are similarly constructed. Each circle passes through one vertex and both isodynamic points. (This configuration, based on the cevians of X(1), generalizes to arbitrary cevians; see TCCT, p. 98, problem 8.)

Let A'B'C' be the 4th Brocard triangle and A″B″C″ be the 4th anti-Brocard triangle. The circumcircles of AA'A″, BB'B″, CC'C″ concur in two points, X(15) and X(16). (Randy Hutson, July 20, 2016)

The line X(13)X(15) is parallel to the Euler line, and the distance between the two lines is (SB - SC)(SC - SA)(SA - SB)/(cot(ω) + 31/2)|*((E - 8F)S2)1/2. (Kiminari Shinagawa, February 20, 2018)

The pedal triangle of X(15) is equilateral. If you have The Geometer's Sketchpad, you can view 1st isodynamic point and X(15)&X(16), with Brocard axis and Lemoine axis.

Video showing circular porisim-orbits of X(13), X(14), X(15), and X(16): 3-Periodics in a Concentric Homothetic Poncelet Pair: Circular Loci of Four Triangle Centers. (Dan Reznik, August 9, 2020) See also Loci of Centers of Ellipse-Mounted Triangles. (Dan Reznik, August 26, 2020)

If you have GeoGebra, you can view 1st isodynamic point.

Several notable circles pass through X(15) and X(16). For each circle listed here, the appearance of (i; [name], [list]) means that the center is X(i), and the points with listed indices lie on the circle:

(187; Schoute circle, 15,16)

(351; Parry circle, 2, 15, 16, 23, 110, 111, 352, 353, 5638, 5639, 6141, 6142, 7598, 7599, 7601, 7602, 7711, 9138, 9147, 9153, 9156, 9157, 9158, 9162, 9163, 9212, 9213, 9978, 9980, 9998, 9999, 11199, 11673, 13114, 13242, 14660, 14704, 14705, 32072, 32073, 32074, 32526, 33502, 33503)

(647; Moses radical circle, 15, 16, 5000, 5001, 6112, 6113, 6114, 6115, 6116, 6117)

(649; Gheorghe circle, 15, 16, 1276, 1277, 32622, 32623)

(663; 1, 15, 16, 36, 3465, 4040, 5526, 5529)

(665; 15, 16, 32753, 32754)

(669; 15, 16, 5004, 5005, 5980, 5981)

(887; 15, 16, 99, 729, 13210, 14691)

(890; 15, 16, 100, 739))

(1960; Moses isodynamic circle, 15, 16, 101, 106, 214, 9321, 11716, 38013, 38014, 41183, 41184, 41185, 41186, 41187, 41188, 41189, 41190, 41191, 41192, 41193)

(2488; 15, 16, 3513, 3514)

(2502; Parry isodynamic circle, 15, 16)

(3005; 15, 16, 5002, 5003)

(3250; 15, 16, 32763, 32764)

(3569; 15, 16, 32618, 32619, 39665, 39666)

(5027; 15, 16, 18773, 18774, 22687, 22689)

(5075; 15, 16, 846, 1054, 1283, 5197)

(5638; 15, 16, 39164, 39165)

(5639; 15, 16, 39162, 39163)

(6137; 13, 15, 16, 3165, 5616, 5669, 6104, 10658)

(6138; 14, 15, 16, 3166, 5612, 5668, 6105, 10657)

(6139; Terzic circle, 15, 16, 55, 109, 654, 1155, 2291, 41155, 41156, 41157, 41158, 41159, 41160, 41161, 41162, 41163, 41164, 41165, 41166)

(6140; 15, 16, 115, 128, 399, 1263, 1511, 2079, 10277, 14367)

(8644; 15, 16, 38001, 38002)

(9409; 15, 16, 74, 112, 5667, 9862, 11587, 40894, 40895)

(9420; 15, 16, 98, 13236, 15920, 26714)

(15451; 4, 15, 16, 186, 3484, 11674, 13509, 15412)

(17414; 15, 16, 11629, 11630)

(9527; 15, 16, 351, 2502, 9129, 10166)

(42649; 15, 16, 35, 484, 3483, 14102)

(42650; 15, 16, 54, 1157, 3482, 18335)

(42651; 15, 16, 125, 184, 2081, 13558)

(42652; 15, 16, 385, 805, 5970, 32531)

(42653; 15, 16, 501, 3743, 5127, 14838, 14873, 39149)

(42654; 15, 16, 647, 1495, 14685, 16319, 35901)

(42655; 15, 16, 667, 1083, 3230, 11650, 11651, 11652)

(42656; 15, 16, 1138, 2132, 6794, 12112, 14254)

(42657; 15, 16, 3065, 3464, 5540, 6126)

This list of circles was contributed by Peter Moses, April 17, 2021, with the following notes. Starting with a circle X(15), X(16), and a point P = p : q : r, the center of the circle is given by

a^2*(c^2*(a^2 + b^2 - c^2)*p*q + a^2*c^2*q^2 - b^2*(a^2 - b^2 + c^2)*p*r - a^2*b^2*r^2) : : ,

and the power of vertex A with repect to the circle is

(b^2*c^2*(-(c^2*(a^2 + b^2 - c^2)*p*q) - a^2*c^2*q^2 + b^2*(a^2 - b^2 + c^2)*p*r + a^2*b^2*r^2))/((p + q + r)*(-(b^2*c^2*(b^2 - c^2)*p) + a^2*c^2*(a^2 - c^2)*q - a^2*b^2*(a^2 - b^2)*r)).

If P lies on the Lemoine axis, then the power of A with respect to to the circle is

-b^2*c^2*(c^2*q + b^2*r)/(c^2*(a^2 - b^2)*q + b^2*(a^2 - c^2)*r)).

Consider this experiment, in which 3 regular hexagons, HA, Hb, HC are erected on the sides of T = ABC. Let the hexagon vertices be labeled as
Ha = {B,A1,A2,A3,A4,C), Hb = {C,B1,B2,B3,B4,A}, Hc={A,C1,C2,C3,C4,B},
and the vertices of the 3 "flank-triangles", by Fa = {A,C1,B4}, Fb = "{B,A1,C4}, Fc = {C,B1,A4}.
Between each pair of consecutive hexagons, define 3 "flank" triangles Fa = {A,C1,B4}, Fb = {B,A1,C4}, Fc={C,Ba,A4}. Let T' be the triangle with vertices X(15)-of-Fa, X(15)-of-Fb, X(15)-of-Fc. Claim (1): T' is perspective to T, and the perspector is X(6). Claim (2): X(16)-of-Fa = X(16)-of-Fb = X(15)-of-Fc . (Dan Reznick, November 11, 2021)

X(15) lies on the Evans conic, Parry circle, Moses radical circle, Schoutte circle, Parry isodynamic circle, the cubics K001, K018, K048, K050, K073, K114, K129b, K148, K193, K206, K261a, K261b, K262a, K262b, K263, K290, K291, K292, K303a, K304, K341a, K390, K435, K438a, K438b, K439, K440, K441, K458, K463, K468, K469, K471, K505, K508, K513, K514, K523, K524, K639, K640, K641, K730, K802, K803, K881, K882, K883, K884, K885, K894, K900, K909, K912, K940, K942, K944, K946, K1052, K1064, K1099, K1105, K1132b, K1133a and the curves Q002, Q016, Q037, Q039, Q043, Q049, Q054, Q067, Q075, Q076, Q090, Q092, Q097, Q123, Q136, Q137, Q138, Q139, Q140, Q142, Q143, as well as these lines: {1, 1251}, {2, 14}, {3, 6}, {4, 17}, {5, 2913}, {11, 11755}, {13, 30}, {18, 140}, {20, 3412}, {21, 5362}, {23, 11629}, {24, 10642}, {35, 1250}, {36, 202}, {40, 10636}, {44, 11790}, {45, 11791}, {51, 3132}, {54, 10678}, {55, 203}, {56, 7005}, {57, 11760}, {74, 5668}, {86, 21898}, {98, 33388}, {99, 22687}, {110, 2378}, {111, 9202}, {115, 6771}, {128, 11600}, {183, 25167}, {184, 2903}, {185, 21647}, {186, 3165}, {214, 5240}, {230, 21156}, {237, 14186}, {298, 533}, {299, 3643}, {302, 34508}, {303, 316}, {323, 3170}, {351, 9162}, {376, 10653}, {378, 8740}, {381, 16644}, {383, 9993}, {385, 5980}, {395, 549}, {397, 550}, {399, 5612}, {404, 5367}, {465, 13567}, {466, 23292}, {470, 6110}, {484, 8444}, {485, 2041}, {486, 2042}, {512, 9163}, {523, 16181}, {524, 5463}, {530, 22495}, {532, 616}, {542, 9117}, {590, 18585}, {597, 35303}, {615, 15765}, {622, 9989}, {625, 11306}, {627, 22901}, {628, 636}, {630, 31706}, {631, 11489}, {633, 7836}, {635, 7832}, {691, 2379}, {740, 5699}, {842, 5994}, {843, 9203}, {846, 2946}, {940, 21476}, {1080, 6115}, {1082, 16577}, {1138, 5624}, {1147, 3205}, {1154, 2902}, {1157, 8447}, {1181, 19363}, {1263, 8173}, {1277, 8482}, {1337, 2981}, {1338, 2381}, {1495, 3129}, {1498, 17826}, {1511, 6105}, {1513, 9749}, {1593, 11408}, {1656, 5339}, {1657, 5340}, {1658, 11268}, {1682, 11758}, {1724, 11098}, {2043, 6560}, {2044, 6561}, {2045, 5420}, {2046, 5418}, {2058, 13391}, {2070, 2923}, {2132, 8445}, {2133, 8448}, {2380, 10409}, {2549, 5474}, {2777, 10681}, {2854, 13859}, {2926, 10329}, {2927, 2937}, {2952, 2959}, {3065, 5673}, {3070, 14814}, {3071, 14813}, {3096, 11290}, {3124, 14705}, {3130, 34417}, {3200, 11137}, {3231, 14178}, {3334, 14146}, {3411, 3530}, {3441, 8478}, {3464, 7326}, {3465, 7059}, {3479, 8451}, {3480, 8175}, {3483, 16883}, {3484, 8479}, {3515, 11409}, {3524, 16963}, {3631, 22845}, {3734, 25157}, {3849, 9763}, {3850, 5349}, {3923, 5700}, {3972, 35918}, {4383, 21475}, {5010, 7127}, {5054, 16268}, {5056, 5343}, {5059, 5344}, {5066, 12817}, {5068, 5365}, {5366, 22235}, {5459, 31710}, {5469, 25166}, {5470, 31709}, {5471, 6774}, {5472, 6781}, {5473, 9112}, {5529, 11789}, {5613, 9981}, {5614, 17403}, {5617, 6777}, {5623, 8491}, {5663, 10657}, {5667, 6111}, {5672, 8501}, {5675, 8456}, {5679, 8455}, {5681, 8462}, {5873, 22746}, {5916, 23895}, {5999, 22691}, {6000, 10675}, {6137, 9213}, {6138, 9138}, {6151, 21462}, {6241, 11466}, {6294, 23009}, {6296, 23019}, {6297, 23010}, {6300, 22611}, {6301, 22610}, {6304, 22640}, {6305, 22639}, {6564, 18587}, {6565, 18586}, {6581, 8177}, {6642, 10644}, {6694, 7859}, {6695, 10583}, {6759, 10676}, {6770, 6778}, {7060, 7089}, {7164, 8449}, {7325, 8508}, {7327, 8476}, {7329, 8472}, {7426, 34315}, {7488, 11421}, {7502, 11135}, {7622, 9761}, {7709, 32466}, {7751, 33466}, {7790, 11303}, {7844, 11305}, {7846, 11308}, {7865, 11297}, {7877, 35689}, {7880, 11301}, {7914, 11312}, {8172, 8495}, {8291, 9865}, {8431, 8453}, {8434, 8454}, {8438, 8457}, {8441, 8471}, {8458, 8535}, {8463, 8490}, {8473, 8486}, {8474, 8487}, {8475, 8494}, {8477, 8496}, {8483, 16882}, {8598, 12155}, {8839, 13367}, {8884, 19190}, {8919, 23721}, {9113, 21157}, {9147, 14447}, {9744, 9750}, {9754, 16652}, {9886, 22579}, {9932, 10660}, {10187, 22237}, {10188, 35018}, {10282, 30403}, {10546, 16259}, {10637, 10902}, {10661, 13754}, {10663, 17702}, {10664, 12893}, {10682, 13289}, {10788, 22696}, {11003, 14169}, {11004, 11126}, {11008, 22844}, {11145, 15018}, {11202, 11244}, {11449, 11453}, {11452, 12111}, {11464, 11467}, {11540, 33606}, {11676, 22701}, {11761, 11770}, {12367, 14173}, {12584, 32302}, {12816, 15682}, {12972, 12981}, {12973, 12983}, {12980, 13058}, {12982, 13057}, {13049, 13059}, {13050, 13060}, {13102, 22891}, {13704, 23011}, {13706, 23020}, {13824, 23012}, {13826, 23021}, {13860, 22693}, {13881, 16630}, {14137, 16940}, {14182, 23017}, {14188, 23022}, {14369, 14972}, {14704, 20998}, {15080, 34009}, {15412, 23872}, {15640, 33607}, {15743, 18776}, {15764, 32788}, {16319, 32460}, {16460, 16639}, {16807, 32628}, {17277, 21869}, {17821, 17827}, {18400, 32397}, {18762, 35738}, {18909, 18929}, {18925, 18930}, {18980, 19450}, {18981, 19451}, {19185, 19191}, {19357, 19364}, {19440, 19452}, {19441, 19453}, {22113, 22895}, {22489, 31693}, {22571, 31695}, {22602, 31697}, {22631, 31699}, {22688, 31701}, {22702, 22714}, {22707, 22715}, {22796, 22892}, {22843, 22862}, {22962, 22975}, {22999, 25220}, {23007, 25178}, {23259, 35732}, {23358, 32398}, {25151, 31707}, {25158, 31713}, {25159, 31715}, {25160, 31717}, {25217, 31719}, {30461, 30467}, {30468, 36185}, {31378, 36210}, {31694, 33475}, {31696, 33477}, {31698, 33447}, {31700, 33446}, {31702, 33479}, {31708, 33480}, {31712, 33483}, {31714, 33485}, {31716, 33489}, {31718, 33487}, {31720, 33490}, {32171, 32208}, {32465, 32515}, {35725, 35727}, {35730, 35740}

X(15) = midpoint of X(i) and X(j) for these {i,j}: {3, 5611}, {14, 6780}, {616, 3180}, {622, 14712}, {2378, 5610}, {6777, 25236}
X(15) = reflection of X(i) in X(j) for these {i,j}: {1, 11707}, {4, 7684}, {13, 396}, {14, 6109}, {16, 187}, {17, 14138}, {298, 618}, {316, 624}, {621, 623}, {623, 6671}, {2902, 11136}, {5318, 11542}, {5463, 35304}, {5668, 5995}, {5978, 619}, {6778, 6783}, {9162, 351}, {10409, 33526}, {11600, 15609}, {16267, 16962}, {19106, 5318}, {20428, 5}, {22997, 9117}, {22999, 25220}, {23004, 115}, {23007, 25178}, {34315, 7426}, {34317, 14170}
X(15) = reflection of X(i) in X(j) for these (i,j): (13,396), (16,187), (298,618), (316,624), (621,623)
X(15) = isogonal conjugate of X(13)
X(15) = isotomic conjugate of X(300)
X(15) = complement of X(621)
X(15) = anticomplement of X(623)
X(15) = circumcircle-inverse of X(16)
X(15) = nine-point-circle-inverse of X(6112)
X(15) = Brocard-circle-inverse of X(16)
X(15) = polar-circle-inverse of X(6116)
X(15) = orthoptic-circle-of-Steiner-inellipse-inverse of X(6114)
X(15) = 2nd-Brocard-circle-inverse of X(3105)
X(15) = circumcircle-of-inner-Napoleon-triangle-inverse of X(14)
X(15) = Lucas-inner-circle-inverse of X(16)
X(15) = Lucas-circles-radical-circle inverse of X(16)
X(15) = outer-Montesdeoca-Lemoine circle-inverse of X(16)
X(15) = inner-Montesdeoca-Lemoine-circle-inverse of X(16)
X(15) = antigonal image of X(11600)
X(15) = symgonal image of X(33526)
X(15) = complement of the isogonal conjugate of X(3438)
X(15) = complement of the isotomic conjugate of X(2992)
X(15) = isogonal conjugate of the anticomplement of X(618)
X(15) = isogonal conjugate of the complement of X(616)
X(15) = isotomic conjugate of the isogonal conjugate of X(34394)
X(15) = isogonal conjugate of the isotomic conjugate of X(298)
X(15) = isotomic conjugate of the polar conjugate of X(8739)
X(15) = isogonal conjugate of the polar conjugate of X(470)
X(15) = Thomson-isogonal conjugate of X(5463)
X(15) = excentral-isogonal conjugate of X(2945)
X(15) = tangential-isogonal conjugate of X(2925)
X(15) = orthic-isogonal conjugate of X(2902)
X(15) = psi-transform of X(16)
X(15) = X(i)-complementary conjugate of X(j) for these (i,j): {2992, 2887}, {3438, 10}
X(15) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 202}, {3, 3165}, {4, 2902}, {6, 3170}, {13, 62}, {14, 5616}, {30, 5668}, {54, 3200}, {74, 16}, {470, 8739}, {2981, 6}, {11117, 11126}, {17402, 6137}, {32036, 35443}
X(15) = X(i)-cross conjugate of X(j) for these (i,j): {74, 8445}, {1094, 7006}, {1154, 11600}, {1511, 16}, {3200, 62}, {6137, 17402}, {14816, 13}, {19295, 323}, {34327, 11146}, {34394, 8739}
X(15) = X(i)-isoconjugate of X(j) for these (i,j): {1, 13}, {2, 2153}, {16, 2166}, {18, 3383}, {31, 300}, {63, 8737}, {75, 3457}, {94, 2152}, {661, 23895}, {662, 20578}, {1081, 19551}, {1577, 5995}, {2154, 11078}, {2306, 7026}, {3179, 14358}, {6138, 32680}, {23871, 32678}, {24041, 30452}
X(15) = crosspoint of X(i) and X(j) for these (i,j): {2, 2992}, {13, 18}, {249, 10409}, {298, 470}, {2380, 16460}, {11600, 36210}
X(15) = crosssum of X(i) and X(j) for these (i,j): {1, 3179}, {2, 3180}, {3, 10661}, {6, 3129}, {15, 62}, {16, 5612}, {395, 30459}, {396, 8014}, {523, 30465}, {532, 619}, {6104, 36208}, {6111, 6116}, {9200, 30467}, {11542, 11555}, {18777, 30466}, {20578, 30452}, {23283, 30460}
X(15) = X(i)-line conjugate of X(j) for these (i,j): {13, 11537}, {549, 395}, {9138, 6138}, {16181, 523}
X(15) = X(i)-vertex conjugate of X(j) for these (i,j): {4, 16257}, {13, 3457}, {16, 512}, {3458, 32906}
X(15) = trilinear pole of line {526, 6137}
X(15) = crossdifference of every pair of points on line {395, 523}
X(15) = X(6)-Hirst inverse of X(16)
X(15) = X(15)-of-2nd-Brocard-triangle
X(15) = X(15)-of-circumsymmedial-triangle
X(15) = {X(371),X(372)}-harmonic conjugate of X(61)
X(15) = X(75)-isoconjugate of X(3457)
X(15) = X(1577)-isoconjugate of X(5995)
X(15) = outer-Napoleon-to-inner-Napoleon similarity image of X(13)
X(15) = orthocentroidal-to-ABC similarity image of X(13)
X(15) = 4th-Brocard-to-circumsymmedial similarity image of X(13)
X(15) = X(2378)-of-2nd-Parry triangle
X(15) = radical center of Lucas(2/sqrt(3)) circles
X(15) = homothetic center of (equilateral) 1st isogonal triangle of X(13) and pedal triangle of X(15)
X(15) = homothetic center of (equilateral) 1st isogonal triangle of X(14) and triangle formed by circumcenters of BCX(14), CAX(14), ABX(14)
X(15) = eigencenter of inner Napoleon triangle
X(15) = X(13)-of-4th-anti-Brocard-triangle
X(15) = X(15)-of-X(3)PU(1)
X(15) = Thomson-isogonal conjugate of X(5463)
X(15) = X(10657)-of-orthocentroidal-triangle
X(15) = {X(16),X(61)}-harmonic conjugate of X(6)
X(15) = Cundy-Parry Phi transform of X(61)
X(15) = Cundy-Parry Psi transform of X(17)
X(15) = X(1277)-of-orthic-triangle if ABC is acute
X(15) = barycentric product X(i)*X(j) for these {i,j}: {3, 470}, {6, 298}, {13, 11131}, {14, 323}, {16, 11092}, {17, 11146}, {18, 11127}, {50, 301}, {54, 33529}, {62, 19778}, {69, 8739}, {75, 2151}, {76, 34394}, {97, 6117}, {99, 6137}, {110, 23870}, {249, 30465}, {299, 11086}, {302, 8603}, {523, 17402}, {526, 23896}, {533, 6151}, {618, 2981}, {691, 9204}, {1082, 5240}, {2380, 14922}, {2987, 6782}, {3165, 19774}, {3170, 11121}, {3268, 5994}, {3457, 11129}, {3458, 7799}, {5616, 13582}, {6110, 14919}, {10409, 35443}, {10410, 14447}, {10411, 20579}, {10677, 11143}, {11078, 36209}, {11117, 19294}, {11120, 19295}, {11126, 11600}, {11130, 36210}, {11133, 21462}, {11136, 34390}, {11137, 34389}, {17403, 23284}
X(15) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 300}, {6, 13}, {14, 94}, {16, 11078}, {25, 8737}, {31, 2153}, {32, 3457}, {50, 16}, {61, 8838}, {62, 16770}, {110, 23895}, {186, 471}, {298, 76}, {301, 20573}, {323, 299}, {470, 264}, {512, 20578}, {526, 23871}, {1154, 33530}, {1250, 7026}, {1576, 5995}, {2088, 30468}, {2151, 1}, {2154, 2166}, {2981, 11119}, {3124, 30452}, {3165, 19772}, {3170, 3180}, {3200, 11127}, {3457, 11080}, {3458, 1989}, {5994, 476}, {6105, 8836}, {6117, 324}, {6137, 523}, {6138, 23283}, {6151, 11118}, {8603, 17}, {8604, 11601}, {8738, 6344}, {8739, 4}, {9204, 35522}, {10633, 472}, {10677, 11144}, {11062, 6116}, {11081, 36211}, {11083, 11581}, {11086, 14}, {11092, 301}, {11127, 303}, {11131, 298}, {11135, 6104}, {11136, 62}, {11137, 61}, {11146, 302}, {11243, 8919}, {14270, 6138}, {17402, 99}, {19294, 532}, {19295, 619}, {19373, 1081}, {19627, 34395}, {19778, 34390}, {20579, 10412}, {21461, 11139}, {21462, 11082}, {23870, 850}, {23896, 35139}, {30465, 338}, {32729, 9206}, {33529, 311}, {34327, 629}, {34394, 6}, {34395, 11081}, {34397, 8740}, {36209, 11092}
X(15) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 11754, 11753}, {1, 11763, 11762}, {2, 617, 3642}, {2, 621, 623}, {2, 5334, 18581}, {2, 10654, 14}, {2, 18581, 16967}, {3, 6, 16}, {3, 16, 10646}, {3, 61, 62}, {3, 62, 5237}, {3, 371, 3364}, {3, 372, 3365}, {3, 3390, 35739}, {3, 5238, 5352}, {3, 5615, 9736}, {3, 5865, 14540}, {3, 10634, 11515}, {3, 11480, 10645}, {3, 11485, 6}, {3, 11486, 11481}, {3, 13350, 21158}, {3, 15793, 15784}, {3, 18468, 10634}, {3, 22236, 61}, {3, 22238, 5351}, {4, 10632, 10641}, {4, 11488, 18582}, {4, 18582, 16808}, {5, 5321, 16809}, {5, 23302, 16966}, {6, 16, 62}, {6, 10645, 10646}, {6, 10646, 34755}, {6, 11480, 3}, {6, 11481, 11486}, {6, 11485, 61}, {6, 19781, 32}, {6, 22236, 11485}, {13, 396, 16267}, {13, 15441, 11581}, {13, 16960, 11542}, {13, 16962, 396}, {13, 19106, 5318}, {14, 16241, 2}, {14, 16967, 18581}, {14, 33417, 16967}, {16, 61, 6}, {16, 62, 34755}, {16, 5238, 10645}, {16, 6396, 35739}, {16, 10645, 3}, {16, 10646, 5237}, {16, 34754, 61}, {17, 16808, 18582}, {17, 19107, 16808}, {17, 22906, 31704}, {18, 33416, 23303}, {32, 3098, 16}, {32, 3105, 62}, {35, 2307, 7006}, {35, 5353, 1250}, {36, 5357, 19373}, {39, 5092, 16}, {50, 3581, 16}, {61, 3389, 3365}, {61, 3390, 3364}, {61, 5238, 3}, {61, 5352, 5237}, {61, 10645, 16}, {61, 11480, 10646}, {61, 14539, 3107}, {61, 34754, 11485}, {62, 5352, 3}, {62, 10646, 16}, {140, 398, 18}, {140, 11543, 23303}, {140, 23303, 33416}, {182, 574, 16}, {182, 3106, 62}, {182, 9735, 3}, {187, 11480, 21158}, {216, 11430, 16}, {298, 30471, 7799}, {299, 11299, 3643}, {303, 11304, 624}, {323, 11146, 11131}, {323, 34394, 3170}, {371, 372, 61}, {371, 3389, 62}, {371, 5238, 35739}, {371, 6200, 16}, {372, 3390, 62}, {372, 6396, 16}, {389, 22052, 16}, {395, 549, 16242}, {396, 5318, 11542}, {396, 11542, 16960}, {398, 23303, 11543}, {485, 2041, 3391}, {486, 2042, 3392}, {500, 17454, 16}, {566, 14805, 16}, {569, 14806, 16}, {572, 4256, 16}, {573, 4257, 16}, {575, 8589, 16}, {576, 8588, 16}, {576, 9736, 5615}, {577, 11438, 16}, {578, 10979, 16}, {621, 5334, 33518}, {623, 6671, 2}, {628, 22861, 22850}, {991, 4262, 16}, {1151, 6221, 16}, {1152, 6398, 16}, {1250, 2307, 5353}, {1250, 5353, 7006}, {1340, 1341, 16}, {1350, 1384, 16}, {1351, 5210, 16}, {1379, 1380, 16}, {1620, 33636, 16}, {1670, 1671, 3105}, {1689, 1690, 3106}, {1691, 35002, 16}, {2030, 18860, 16}, {2076, 5611, 3105}, {2076, 9301, 16}, {2076, 19781, 187}, {2080, 5104, 16}, {2903, 3166, 3201}, {3003, 10564, 16}, {3053, 33878, 16}, {3094, 26316, 16}, {3105, 3106, 3094}, {3311, 6411, 16}, {3312, 6412, 16}, {3364, 3365, 62}, {3365, 35739, 5237}, {3366, 3367, 5}, {3371, 3372, 3390}, {3385, 3386, 3389}, {3389, 3390, 3}, {3430, 33628, 16}, {3592, 6451, 16}, {3594, 6452, 16}, {5008, 14810, 16}, {5013, 12017, 16}, {5024, 5085, 16}, {5030, 13329, 16}, {5033, 9737, 16}, {5093, 5585, 16}, {5237, 34755, 16}, {5238, 10645, 11480}, {5238, 11485, 10646}, {5238, 22236, 62}, {5238, 34754, 16}, {5318, 11542, 13}, {5321, 16772, 23302}, {5321, 23302, 5}, {5334, 18581, 14}, {5352, 11485, 34755}, {5352, 30560, 21158}, {5357, 19373, 202}, {5473, 9112, 23006}, {5611, 13350, 14538}, {5611, 21401, 21158}, {6105, 36209, 11086}, {6199, 6409, 16}, {6200, 6396, 10645}, {6200, 11485, 3365}, {6295, 22689, 5981}, {6303, 6307, 14905}, {6395, 6410, 16}, {6396, 11485, 3364}, {6407, 6468, 16}, {6408, 6469, 16}, {6425, 6445, 16}, {6426, 6446, 16}, {6429, 9690, 16}, {6437, 6449, 16}, {6438, 6450, 16}, {6439, 9691, 16}, {6453, 6480, 16}, {6454, 6481, 16}, {6671, 33518, 16967}, {6778, 16529, 6783}, {7051, 10638, 1}, {9675, 9738, 16}, {10632, 32585, 8837}, {10641, 11475, 4}, {10645, 11480, 5352}, {10645, 11485, 62}, {10645, 34754, 6}, {10654, 18581, 5334}, {10667, 10671, 6}, {10676, 30402, 6759}, {11127, 11131, 323}, {11137, 22115, 3200}, {11477, 15655, 16}, {11480, 11485, 16}, {11480, 22236, 6}, {11480, 34754, 62}, {11481, 11486, 16}, {11485, 22236, 34754}, {11488, 18582, 17}, {11542, 16960, 16267}, {11543, 23303, 18}, {11581, 11586, 15441}, {11753, 11762, 1}, {11754, 11763, 1}, {11755, 11764, 11}, {11756, 11765, 55}, {11757, 11766, 1}, {11758, 11767, 1682}, {11759, 11768, 56}, {11760, 11769, 57}, {11761, 11770, 11993}, {12054, 12055, 16}, {14538, 21158, 3}, {15037, 15109, 16}, {16241, 16967, 33417}, {16808, 19107, 4}, {16809, 16964, 5321}, {16809, 16966, 5}, {16960, 19106, 13}, {16962, 19106, 16960}, {16964, 16966, 16809}, {16967, 33417, 2}, {17851, 17852, 16}, {21309, 31884, 16}, {22510, 23004, 115}, {33442, 33443, 6299}, {35207, 35208, 36}


X(16) = 2nd ISODYNAMIC POINT

Trilinears    sin(A- π/3) : sin(B - π/3) : sin(C - π/3)
Trilinears    cos(A + π/6) : cos(B + π/6) : cos(C + π/6)
Trilinears    3 cos A - sqrt(3) sin A : :
Barycentrics  a sin(A - π/3) : b sin(B - π/3) : c sin(C- π/3)
Barycentrics    (SB + SC)/((SA - Sqrt[3] S) (SB + SC) + 4 SB SC) : :
Barycentrics    (S - Sqrt[3] SA) (SB + SC) : :
Barycentrics    a^2*(Sqrt[3]*(a^2 - b^2 - c^2) + 2*S) : :
Tripolars    b c : :
X(16) = -(r2 + 2rR + s2)*X(1) + 6rR*X(2) + 2r(r + 31/2s)*X(3) = sqrt(3)*X(3) - (cot ω)*X(6)    (Peter Moses, April 2, 2013)
X(16) = 3 X[2] - 4 X[6672], 3 X[13] - 2 X[33517], 3 X[14] - 2 X[5321], 3 X[14] - 4 X[11543], 2 X[14] - 3 X[16268], 3 X[14] - 5 X[16961], X[14] - 3 X[16963], 3 X[14] - X[19107], 3 X[17] - X[22895], 3 X[18] - 2 X[31706], 2 X[115] - 3 X[22511], 4 X[230] - 3 X[22510], 3 X[395] - X[5321], 3 X[395] - 2 X[11543], 4 X[395] - 3 X[16268], 6 X[395] - 5 X[16961], 2 X[395] - 3 X[16963], 6 X[395] - X[19107], X[622] - 4 X[6672], 3 X[3130] - X[3441], 4 X[5321] - 9 X[16268], 2 X[5321] - 5 X[16961], 2 X[5321] - 9 X[16963], 3 X[5469] - 2 X[31710], 3 X[5470] - X[25156], 2 X[6108] + X[6779], 3 X[6108] - X[33517], X[6777] - 3 X[16530], 3 X[6779] + 2 X[33517], 2 X[6782] - 3 X[16530], X[10676] - 3 X[11244], 8 X[11543] - 9 X[16268], 4 X[11543] - 5 X[16961], 4 X[11543] - 9 X[16963], 4 X[11543] - X[19107], 3 X[14139] - X[31706], 9 X[16268] - 10 X[16961], 9 X[16268] - 2 X[19107], 5 X[16961] - 9 X[16963], 5 X[16961] - X[19107], 9 X[16963] - X[19107], 3 X[22490] - 2 X[31694], X[22496] + 2 X[35932], 3 X[22511] - X[23005], 3 X[22572] - 2 X[31696], 3 X[22604] - 2 X[31698], 3 X[22633] - 2 X[31700], 3 X[22690] - 2 X[31702], 3 X[22891] - 2 X[31704], 3 X[25161] - 2 X[31708], 3 X[25167] - 2 X[31712], 3 X[25168] - 2 X[31714], 3 X[25169] - 2 X[31718], 3 X[25170] - 2 X[31716], 3 X[25214] - 2 X[31720]

Let U and V be the points on sideline BC met by the interior and exterior bisectors of angle A. The circle having diameter UV is the A-Apollonian circle. The B- and C- Apollonian circles are similarly constructed. Each circle passes through a vertex and both isodynamic points. The pedal triangle of X(16) is equilateral. If you have The Geometer's Sketchpad, you can view 2nd isodynamic point.
If you have GeoGebra, you can view 2nd isodynamic point.

Let A'B'C' be the 4th Brocard triangle and A″B″C″ be the 4th anti-Brocard triangle. The circumcircles of AA'A″, BB'B″, CC'C″ concur in two points, X(15) and X(16). (Randy Hutson, July 20, 2016)

The line X(14)X(16) is parallel to the Euler line, and the distance between the two lines is (SB - SC)(SC - SA)(SA - SB)/|(cot(ω) - 31/2)|((E - 8F)S2)1/2. (Kiminari Shinagawa, February 20, 2018)

Consider this picture, in which (purple) hexagons are erected on the sides of ABC, with (green) flank-triangles:
(1) X(16) of the 3 flank-triangles coincide with X(16) of ABC.
(2) the (red) triangle of the centroids of the hexagons is perspective to ABC, and the perspector is X(13).
(3) the circumcircle of the centroids (apices of equilaterals erect4d on the sides of ABC) has center X(627).
(4) the (green) triangle with vertices on X(15)-of-flank-triangles is perspective to ABC, and the perspector is X(6).
(Dan Reznick, October 18, 2021)

X(16) lies on the Evans conic, Parry circle, Moses Radical circle, Schoutte circle, Parry isodynamic circle, the cubics K001, K018, K048, K050, K073, K114, K129a, K148, K193, K206, K261a, K261b, K262a, K262b, K263, K290, K291, K292, K303b, K304, K341b, K390, K435, K438a, K438b, K439, K440, K441, K458, K463, K468, K469, K471, K505, K508, K513, K514, K523, K524, K639, K640, K641, K730, K802, K803, K881, K882, K883, K884, K885, K894, K900, K909, K912, K940, K942, K944, K946, K1052, K1064, K1099, K1105, K1132a, K1133b, and the curves on Q002, Q016, Q037, Q039, Q043, Q049, Q054, Q067, Q075, Q076, Q090, Q092, Q097, Q123, Q136, Q137, Q138, Q139, Q140, Q142, Q143, as well as these lines: {1, 1250}, {2, 13}, {3, 6}, {4, 18}, {5, 2912}, {11, 11773}, {14, 30}, {17, 140}, {20, 3411}, {21, 5367}, {23, 11630}, {24, 10641}, {35, 5357}, {36, 203}, {40, 10637}, {44, 11791}, {45, 11790}, {51, 3131}, {54, 10677}, {55, 202}, {56, 7006}, {57, 11778}, {74, 5669}, {86, 21869}, {98, 33389}, {99, 22689}, {110, 2379}, {111, 9203}, {115, 6774}, {128, 11601}, {183, 25157}, {184, 2902}, {185, 21648}, {186, 3166}, {214, 5239}, {230, 21157}, {237, 14188}, {298, 3642}, {299, 532}, {302, 316}, {303, 34509}, {323, 3171}, {351, 9163}, {358, 1135}, {376, 10654}, {378, 8739}, {381, 16645}, {383, 6114}, {385, 5981}, {396, 549}, {398, 550}, {399, 5616}, {404, 5362}, {465, 23292}, {466, 13567}, {471, 6111}, {484, 7052}, {485, 2042}, {486, 2041}, {512, 9162}, {523, 16182}, {524, 5464}, {531, 22496}, {533, 617}, {542, 9115}, {559, 16577}, {590, 15765}, {597, 35304}, {615, 18585}, {621, 9988}, {625, 11305}, {627, 635}, {628, 22855}, {629, 31705}, {631, 11488}, {634, 7836}, {636, 7832}, {691, 2378}, {740, 5700}, {842, 5995}, {843, 9202}, {846, 2945}, {940, 21475}, {1080, 9993}, {1138, 5623}, {1147, 3206}, {1154, 2903}, {1157, 8457}, {1181, 19364}, {1263, 8172}, {1276, 8481}, {1337, 2380}, {1338, 3458}, {1495, 3130}, {1498, 17827}, {1511, 6104}, {1513, 9750}, {1593, 11409}, {1656, 5340}, {1657, 5339}, {1658, 11267}, {1682, 11776}, {1724, 11097}, {2043, 6561}, {2044, 6560}, {2045, 5418}, {2046, 5420}, {2059, 13391}, {2070, 2924}, {2132, 8455}, {2133, 8458}, {2307, 7280}, {2381, 10410}, {2549, 5473}, {2777, 10682}, {2854, 13858}, {2925, 10329}, {2928, 2937}, {2953, 2959}, {2981, 21461}, {3065, 5672}, {3070, 14813}, {3071, 14814}, {3096, 11289}, {3124, 14704}, {3129, 34417}, {3201, 11134}, {3231, 14182}, {3412, 3530}, {3440, 8470}, {3464, 7325}, {3465, 7060}, {3479, 8174}, {3480, 8461}, {3483, 16882}, {3484, 8471}, {3515, 11408}, {3524, 16962}, {3631, 22844}, {3734, 25167}, {3849, 9761}, {3850, 5350}, {3923, 5699}, {3972, 35917}, {4383, 21476}, {5054, 16267}, {5056, 5344}, {5059, 5343}, {5066, 12816}, {5068, 5366}, {5365, 22237}, {5460, 31709}, {5469, 31710}, {5470, 25156}, {5471, 6781}, {5472, 6771}, {5474, 9113}, {5529, 11752}, {5610, 17402}, {5613, 6778}, {5617, 9982}, {5624, 8492}, {5663, 10658}, {5667, 6110}, {5673, 8502}, {5674, 8446}, {5678, 8445}, {5682, 8452}, {5872, 22745}, {5917, 23896}, {5999, 22692}, {6000, 10676}, {6137, 9138}, {6138, 9213}, {6241, 11467}, {6294, 8177}, {6296, 23001}, {6297, 23025}, {6300, 22609}, {6301, 22612}, {6304, 22638}, {6305, 22641}, {6564, 18586}, {6565, 18587}, {6581, 23000}, {6642, 10643}, {6694, 10583}, {6695, 7859}, {6759, 10675}, {6773, 6777}, {7059, 7088}, {7164, 8459}, {7326, 8509}, {7327, 8468}, {7329, 8464}, {7426, 34316}, {7488, 11420}, {7502, 11136}, {7622, 9763}, {7709, 32465}, {7751, 33467}, {7790, 11304}, {7844, 11306}, {7846, 11307}, {7865, 11298}, {7877, 35688}, {7880, 11302}, {7914, 11311}, {8173, 8496}, {8292, 9865}, {8431, 8463}, {8433, 8444}, {8437, 8447}, {8442, 8479}, {8448, 8536}, {8453, 8489}, {8465, 8486}, {8466, 8487}, {8467, 8494}, {8469, 8495}, {8484, 16883}, {8598, 12154}, {8837, 13367}, {8884, 19191}, {8918, 23722}, {9112, 21156}, {9147, 14446}, {9744, 9749}, {9754, 16653}, {9885, 22580}, {9932, 10659}, {10187, 35018}, {10188, 22235}, {10282, 30402}, {10546, 16260}, {10636, 10902}, {10662, 13754}, {10663, 12893}, {10664, 17702}, {10681, 13289}, {10788, 22695}, {11003, 14170}, {11004, 11127}, {11008, 22845}, {11146, 15018}, {11202, 11243}, {11449, 11452}, {11453, 12111}, {11464, 11466}, {11540, 33607}, {11586, 18777}, {11676, 22702}, {11779, 11788}, {12367, 14179}, {12584, 32301}, {12817, 15682}, {12972, 12980}, {12973, 12982}, {12981, 13060}, {12983, 13059}, {13049, 13057}, {13050, 13058}, {13103, 22846}, {13704, 23026}, {13706, 23002}, {13824, 23027}, {13826, 23003}, {13860, 22694}, {13881, 16631}, {14136, 16941}, {14178, 23023}, {14186, 23028}, {14368, 14972}, {14705, 20998}, {15080, 34008}, {15412, 23873}, {15640, 33606}, {15764, 32787}, {16319, 32461}, {16459, 16638}, {16806, 32627}, {17277, 21898}, {17821, 17826}, {18400, 32398}, {18538, 35738}, {18909, 18930}, {18925, 18929}, {18980, 19452}, {18981, 19453}, {19185, 19190}, {19357, 19363}, {19440, 19450}, {19441, 19451}, {22114, 22849}, {22490, 31694}, {22572, 31696}, {22604, 31698}, {22633, 31700}, {22690, 31702}, {22701, 22715}, {22708, 22714}, {22797, 22848}, {22890, 22906}, {22962, 22974}, {23008, 25219}, {23014, 25173}, {23249, 35732}, {23267, 35733}, {23358, 32397}, {25161, 31708}, {25168, 31714}, {25169, 31718}, {25170, 31716}, {25214, 31720}, {30464, 30470}, {30465, 36186}, {31378, 36211}, {31693, 33474}, {31695, 33476}, {31697, 33445}, {31699, 33444}, {31701, 33478}, {31707, 33481}, {31711, 33482}, {31713, 33484}, {31715, 33486}, {31717, 33488}, {31719, 33491}, {32171, 32207}, {32466, 32515}, {32785, 35730}, {35726, 35727}

X(16) = midpoint of X(i) and X(j) for these {i,j}: {3, 5615}, {13, 6779}, {617, 3181}, {621, 14712}, {2379, 5614}, {6778, 25235}
X(16) = reflection of X(i) in X(j) for these (i,j): (14,395), (15,187), (299,619), (316,623), (622,624)
X(16) = isogonal conjugate of X(14)
X(16) = isotomic conjugate of X(301)
X(16) = complement of X(622)
X(16) = anticomplement of X(624)
X(16) = circumcircle-inverse of X(15)
X(16) = nine-point-circle-inverse of X(6113)
X(16) = Brocard-circle-inverse of X(15)
X(16) = polar-circle-inverse of X(6117)
X(16) = orthoptic-circle-of-Steiner-inellipse-inverse of X(6115)
X(16) = 2nd-Brocard-circle-inverse of X(3104)
X(16) = circumcircle-of-outer-Napoleon-triangle-inverse of X(13)
X(16) = Lucas-inner-circle-inverse of X(15)
X(16) = Lucas-circles-radical-circle-inverse of X(15)
X(16) = outer-Montesdeoca-Lemoine-circle-inverse of X(15)
X(16) = inner-Montesdeoca-Lemoine-circle-inverse of X(15)
X(16) = antigonal image of X(11601)
X(16) = symgonal image of X(33527)
X(16) = complement of the isogonal conjugate of X(3439)
X(16) = complement of the isotomic conjugate of X(2993)
X(16) = isogonal conjugate of the anticomplement of X(619)
X(16) = isogonal conjugate of the complement of X(617)
X(16) = isotomic conjugate of the isogonal conjugate of X(34395)
X(16) = isogonal conjugate of the isotomic conjugate of X(299)
X(16) = isotomic conjugate of the polar conjugate of X(8740)
X(16) = isogonal conjugate of the polar conjugate of X(471)
X(16) = Thomson-isogonal conjugate of X(5464)
X(16) = excentral-isogonal conjugate of X(2946)
X(16) = tangential-isogonal conjugate of X(2926)
X(16) = orthic-isogonal conjugate of X(2903)
X(16) = psi-transform of X(15)
X(16) = X(i)-complementary conjugate of X(j) for these (i,j): {2993, 2887}, {3439, 10}
X(16) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 203}, {3, 3166}, {4, 2903}, {6, 3171}, {13, 5612}, {14, 61}, {30, 5669}, {54, 3201}, {74, 15}, {471, 8740}, {6151, 6}, {7150, 7005}, {11118, 11127}, {17403, 6138}, {32037, 35444}
X(16) = X(i)-cross conjugate of X(j) for these (i,j): {74, 8455}, {1095, 7005}, {1154, 11601}, {1511, 15}, {3201, 61}, {6138, 17403}, {14817, 14}, {19294, 323}, {34328, 11145}, {34395, 8740}
X(16) = X(i)-isoconjugate of X(j) for these (i,j): {1, 14}, {2, 2154}, {15, 2166}, {17, 3376}, {31, 301}, {63, 8738}, {75, 3458}, {94, 2151}, {554, 7126}, {661, 23896}, {662, 20579}, {1577, 5994}, {2153, 11092}, {6137, 32680}, {7043, 33654}, {23870, 32678}, {24041, 30453}
X(16) = crosspoint of X(i) and X(j) for these (i,j): {1, 7150}, {2, 2993}, {14, 17}, {249, 10410}, {299, 471}, {2381, 16459}, {11601, 36211}
X(16) = crosssum of X(i) and X(j) for these (i,j): {2, 3181}, {3, 10662}, {6, 3130}, {15, 5616}, {16, 61}, {395, 8015}, {396, 30462}, {523, 30468}, {533, 618}, {6105, 36209}, {6110, 6117}, {9201, 30470}, {11543, 11556}, {18776, 30469}, {20579, 30453}, {23284, 30463}
X(16) = X(i)-line conjugate of X(j) for these (i,j): {14, 11549}, {549, 396}, {9138, 6137}, {16182, 523}
X(16) = X(i)-vertex conjugate of X(j) for these (i,j): {4, 16258}, {14, 3458}, {15, 512}, {3457, 32908}
X(16) = trilinear pole of line {526, 6138}
X(16) = crossdifference of every pair of points on line {396, 523}
X(16) = X(6)-Hirst inverse of X(15)
X(16) = X(16)-of-2nd-Brocard triangle
X(16) = X(16)-of-circumsymmedial-triangle
X(16) = {X(371),X(372)}-harmonic conjugate of X(62)
X(16) = X(75)-isoconjugate of X(3458)
X(16) = X(1577)-isoconjugate of X(5994)
X(16) = inner-Napoleon-to-outer-Napoleon similarity image of X(14)
X(16) = orthocentroidal-to-ABC similarity image of X(14)
X(16) = 4th-Brocard-to-circumsymmedial similarity image of X(14)
X(16) = X(2379)-of-2nd-Parry-triangle
X(16) = homothetic center of (equilateral) 1st isogonal triangle of X(14) and pedal triangle of X(16)
X(16) = homothetic center of (equilateral) 1st isogonal triangle of X(13) and triangle formed by circumcenters of BCX(13), CAX(13), ABX(13)
X(16) = radical center of Lucas(-2/sqrt(3)) circles
X(16) = eigencenter of outer Napoleon triangle
X(16) = X(14) of 4th anti-Brocard triangle
X(16) = X(16)-of-X(3)PU(1)
X(16) = Thomson-isogonal conjugate of X(5464)
X(16) = X(10658)-of-orthocentroidal-triangle
X(16) = {X(15),X(62)}-harmonic conjugate of X(6)
X(16) = Cundy-Parry Phi transform of X(62)
X(16) = Cundy-Parry Psi transform of X(18)
X(16) = X(1276)-of-orthic-triangle if ABC is acute
X(16) = barycentric product X(i)*X(j) for these {i,j}: {3, 471}, {6, 299}, {13, 323}, {14, 11130}, {15, 11078}, {17, 11126}, {18, 11145}, {50, 300}, {54, 33530}, {61, 19779}, {69, 8740}, {75, 2152}, {76, 34395}, {97, 6116}, {99, 6138}, {110, 23871}, {249, 30468}, {298, 11081}, {303, 8604}, {523, 17403}, {526, 23895}, {532, 2981}, {559, 5239}, {619, 6151}, {691, 9205}, {2987, 6783}, {3166, 19775}, {3171, 11122}, {3268, 5995}, {3457, 7799}, {3458, 11128}, {5612, 13582}, {6111, 14919}, {10409, 14446}, {10410, 35444}, {10411, 20578}, {10678, 11144}, {11092, 36208}, {11118, 19295}, {11119, 19294}, {11127, 11601}, {11131, 36211}, {11132, 21461}, {11134, 34390}, {11135, 34389}, {14922, 16459}, {17402, 23283}
X(16) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 301}, {6, 14}, {13, 94}, {15, 11092}, {25, 8738}, {31, 2154}, {32, 3458}, {50, 15}, {61, 16771}, {62, 8836}, {110, 23896}, {186, 470}, {299, 76}, {300, 20573}, {323, 298}, {471, 264}, {512, 20579}, {526, 23870}, {1154, 33529}, {1576, 5994}, {2088, 30465}, {2152, 1}, {2153, 2166}, {2981, 11117}, {3124, 30453}, {3166, 19773}, {3171, 3181}, {3201, 11126}, {3457, 1989}, {3458, 11085}, {5995, 476}, {6104, 8838}, {6116, 324}, {6137, 23284}, {6138, 523}, {6151, 11120}, {7051, 554}, {8603, 11600}, {8604, 18}, {8737, 6344}, {8740, 4}, {9205, 35522}, {10632, 473}, {10638, 7043}, {10678, 11143}, {11062, 6117}, {11078, 300}, {11081, 13}, {11086, 36210}, {11088, 11582}, {11126, 302}, {11130, 299}, {11134, 62}, {11135, 61}, {11136, 6105}, {11145, 303}, {11244, 8918}, {14270, 6137}, {17403, 99}, {19294, 618}, {19295, 533}, {19627, 34394}, {19779, 34389}, {20578, 10412}, {21461, 11087}, {21462, 11138}, {23871, 850}, {23895, 35139}, {30468, 338}, {32729, 9207}, {33530, 311}, {34328, 630}, {34394, 11086}, {34395, 6}, {34397, 8739}, {36208, 11078}
X(16) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 11772, 11771}, {1, 11781, 11780}, {2, 616, 3643}, {2, 622, 624}, {2, 5335, 18582}, {2, 10653, 13}, {2, 18582, 16966}, {3, 6, 15}, {3, 15, 10645}, {3, 61, 5238}, {3, 62, 61}, {3, 371, 3389}, {3, 372, 3390}, {3, 1152, 35739}, {3, 5237, 5351}, {3, 5611, 9735}, {3, 5864, 14541}, {3, 10635, 11516}, {3, 11481, 10646}, {3, 11485, 11480}, {3, 11486, 6}, {3, 13349, 21159}, {3, 15794, 15785}, {3, 18470, 10635}, {3, 22236, 5352}, {3, 22238, 62}, {4, 10633, 10642}, {4, 11489, 18581}, {4, 18581, 16809}, {5, 5318, 16808}, {5, 23303, 16967}, {6, 15, 61}, {6, 10645, 34754}, {6, 10646, 10645}, {6, 11480, 11485}, {6, 11481, 3}, {6, 11486, 62}, {6, 19780, 32}, {6, 22238, 11486}, {13, 16242, 2}, {13, 16966, 18582}, {13, 33416, 16966}, {14, 395, 16268}, {14, 15442, 11582}, {14, 16961, 11543}, {14, 16963, 395}, {14, 19107, 5321}, {15, 61, 34754}, {15, 62, 6}, {15, 5237, 10646}, {15, 10645, 5238}, {15, 10646, 3}, {15, 34755, 62}, {15, 35739, 6396}, {17, 33417, 23302}, {18, 16809, 18581}, {18, 19106, 16809}, {18, 22862, 31703}, {32, 3098, 15}, {32, 3104, 61}, {35, 5357, 10638}, {36, 5353, 7051}, {36, 7127, 203}, {39, 5092, 15}, {50, 3581, 15}, {61, 5351, 3}, {61, 10645, 15}, {62, 3364, 3390}, {62, 3365, 3389}, {62, 5237, 3}, {62, 5351, 5238}, {62, 10646, 15}, {62, 11481, 10645}, {62, 14538, 3106}, {62, 34755, 11486}, {62, 35739, 3365}, {140, 397, 17}, {140, 11542, 23302}, {140, 23302, 33417}, {182, 574, 15}, {182, 3107, 61}, {182, 9736, 3}, {187, 11481, 21159}, {216, 11430, 15}, {298, 11300, 3642}, {299, 30472, 7799}, {302, 11303, 623}, {323, 11145, 11130}, {323, 34395, 3171}, {371, 372, 62}, {371, 3364, 61}, {371, 6200, 15}, {372, 3365, 61}, {372, 6396, 15}, {389, 22052, 15}, {395, 5321, 11543}, {395, 11543, 16961}, {396, 549, 16241}, {397, 23302, 11542}, {485, 2042, 3366}, {486, 2041, 3367}, {500, 17454, 15}, {566, 14805, 15}, {569, 14806, 15}, {572, 4256, 15}, {573, 4257, 15}, {575, 8589, 15}, {576, 8588, 15}, {576, 9735, 5611}, {577, 11438, 15}, {578, 10979, 15}, {622, 5335, 33517}, {624, 6672, 2}, {627, 22907, 22894}, {991, 4262, 15}, {1151, 6221, 15}, {1152, 6398, 15}, {1250, 19373, 1}, {1340, 1341, 15}, {1350, 1384, 15}, {1351, 5210, 15}, {1379, 1380, 15}, {1620, 33636, 15}, {1670, 1671, 3104}, {1689, 1690, 3107}, {1691, 35002, 15}, {2030, 18860, 15}, {2076, 5615, 3104}, {2076, 9301, 15}, {2076, 19780, 187}, {2080, 5104, 15}, {2902, 3165, 3200}, {3003, 10564, 15}, {3053, 33878, 15}, {3094, 26316, 15}, {3104, 3107, 3094}, {3311, 6411, 15}, {3312, 6412, 15}, {3364, 3365, 3}, {3371, 3372, 3365}, {3385, 3386, 3364}, {3389, 3390, 61}, {3391, 3392, 5}, {3430, 33628, 15}, {3592, 6451, 15}, {3594, 6452, 15}, {5008, 14810, 15}, {5013, 12017, 15}, {5024, 5085, 15}, {5030, 13329, 15}, {5033, 9737, 15}, {5093, 5585, 15}, {5237, 10646, 11481}, {5237, 11486, 10645}, {5237, 22238, 61}, {5237, 34755, 15}, {5238, 34754, 15}, {5318, 16773, 23303}, {5318, 23303, 5}, {5321, 11543, 14}, {5335, 18582, 13}, {5351, 11486, 34754}, {5351, 30559, 21159}, {5353, 7051, 203}, {5357, 10638, 7005}, {5474, 9113, 23013}, {5615, 13349, 14539}, {5615, 21402, 21159}, {6104, 36208, 11081}, {6199, 6409, 15}, {6200, 6396, 10646}, {6200, 11486, 3390}, {6302, 6306, 14904}, {6395, 6410, 15}, {6396, 11486, 3389}, {6407, 6468, 15}, {6408, 6469, 15}, {6425, 6445, 15}, {6426, 6446, 15}, {6429, 9690, 15}, {6437, 6449, 15}, {6438, 6450, 15}, {6439, 9691, 15}, {6453, 6480, 15}, {6454, 6481, 15}, {6582, 22687, 5980}, {6672, 33517, 16966}, {6777, 16530, 6782}, {7051, 7127, 5353}, {9675, 9738, 15}, {10633, 32586, 8839}, {10642, 11476, 4}, {10646, 11481, 5351}, {10646, 11486, 61}, {10646, 34755, 6}, {10653, 18582, 5335}, {10668, 10672, 6}, {10675, 30403, 6759}, {11126, 11130, 323}, {11134, 22115, 3201}, {11477, 15655, 15}, {11480, 11485, 15}, {11481, 11486, 15}, {11481, 22238, 6}, {11481, 34755, 61}, {11486, 22238, 34755}, {11489, 18581, 18}, {11542, 23302, 17}, {11543, 16961, 16268}, {11582, 15743, 15442}, {11771, 11780, 1}, {11772, 11781, 1}, {11773, 11782, 11}, {11774, 11783, 55}, {11775, 11784, 1}, {11776, 11785, 1682}, {11777, 11786, 56}, {11778, 11787, 57}, {11779, 11788, 11993}, {12054, 12055, 15}, {14539, 21159, 3}, {15037, 15109, 15}, {16242, 16966, 33416}, {16808, 16965, 5318}, {16808, 16967, 5}, {16809, 19106, 4}, {16961, 19107, 14}, {16963, 19107, 16961}, {16965, 16967, 16808}, {16966, 33416, 2}, {17851, 17852, 15}, {21309, 31884, 15}, {22511, 23005, 115}, {33440, 33441, 6298}, {35209, 35210, 36}


X(17) = 1st NAPOLEON POINT

Trilinears    csc(A + π/6) : csc(B + π/6) : csc(C + π/6)
Trilinears    sec(A - π/3) : sec(B - π/3) : sec(C - π/3)
Barycentrics    a csc(A + π/6) : b csc(B + π/6) : c csc(C + π/6)
Tripolars    (a^2-b^2-c^2-2 Sqrt[3] S) Sqrt[(a^2-3 b^2-3 c^2-2 Sqrt[3] S)] : :

Let X,Y,Z be the centers of the equilateral triangles in the construction of X(13). The lines AX, BY, CZ concur in X(17).

Dao Thanh Oai, Equilateral Triangles and Kiepert Perspectors in Complex Numbers, Forum Geometricorum 15 (2015) 105-114.

Dao Thanh Oai, A family of Napoleon triangles associated with the Kiepert configuration, The Mathematical Gazette 99 (March 2015) 151-153.

John Rigby, "Napoleon revisited," Journal of Geometry,33 (1988) 126-146.

If you have The Geometer's Sketchpad, you can view 1st Napoleon point.
If you have GeoGebra, you can view 1st Napoleon point.

X(17) lies on the Napoleon cubic and these lines: 2,62   3,13   4,15   5,14   6,18   12,203   16,140   76,303   83,624   202,499   275,471   299,635   623,633

X(17) is the {X(231),X(1209)}-harmonic conjugate of X(18). For a list of other harmonic conjugates of X(17), click Tables at the top of this page.

X(17) = reflection of X(627) in X(629)
X(17) = isogonal conjugate of X(61)
X(17) = isotomic conjugate of X(302)
X(17) = complement of X(627)
X(17) = anticomplement of X(629)
X(17) = circumcircle-inverse of X(32627)
X(17) = X(i)-cross conjugate of X(j) for these (i,j): (16,14), (140,18), (397,4)
X(17) = polar conjugate of X(473)
X(17) = trilinear product of vertices of outer Napoleon triangle
X(17) = Kosnita(X(13),X(3)) point
X(17) = Kosnita(X(17),X(17)) point
X(17) = Cundy-Parry Phi transform of X(13)
X(17) = Cundy-Parry Psi transform of X(15)
X(17) = trilinear pole of line X(523)X(14446)
X(17) = X(63)-isoconjugate of X(10642)


X(18) = 2nd NAPOLEON POINT

Trilinears    csc(A - π/6) : csc(B - π/6) : csc(C - π/6)
Trilinears    sec(A + π/3) : sec(B + π/3) : sec(C + π/3)
Barycentrics    a csc(A - π/6) : b csc(B - π/6) : c csc(C - π/6)
Tripolars    (a^2-b^2-c^2+2 Sqrt[3] S) Sqrt[(a^2-3 b^2-3 c^2+2 Sqrt[3] S)] : :

Let X,Y,Z be the centers of the equilateral triangles in the construction of X(14). The lines AX, BY, CZ concur in X(18).

If you have The Geometer's Sketchpad, you can view 2nd Napoleon point.
If you have GeoGebra, you can view 2nd Napoleon point.

X(18) lies on the Napoleon cubic and these lines: 2,61   3,14   4,16   5,13   6,17   12,202   15,140   76,302   83,623   203,499   275,470   298,636   624,634

X(18) is the {X(231),X(1209)}-harmonic conjugate of X(17). For a list of other harmonic conjugates of X(18), click Tables at the top of this page.

X(18) = reflection of X(628) in X(630)
X(18) = isogonal conjugate of X(62)
X(18) = isotomic conjugate of X(303)
X(18) = complement of X(628)
X(18) = anticomplement of X(630)
X(18) = circumcircle-inverse of X(32628)
X(18) = X(i)-cross conjugate of X(j) for these (i,j): (15,13), (140,17), (398,4)
X(18) = polar conjugate of X(472)
X(18) = trilinear product of vertices of inner Napoleon triangle
X(18) = Kosnita(X(14),X(3)) point
X(18) = Cundy-Parry Phi transform of X(14)
X(18) = Cundy-Parry Psi transform of X(16)
X(18) = trilinear pole of line X(523)X(14447)
X(18) = X(63)-isoconjugate of X(10641)


X(19) = CLAWSON POINT

Trilinears    tan A : tan B : tan C
Trilinears    sin 2B + sin 2C - sin 2A : :
Trilinears    1/(b2 + c2 - a2) : :
Trilinears    a2 - S2 + SA : :
Trilinears    SB*SC : :
Barycentrics    a*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) : :
Barycentrics    a tan A : b tan B : c tan C
Tripolars    (-a^2+b^2+c^2) Sqrt[b c (a^6-a^4 b^2-a^2 b^4+b^6-2 a^4 b c-2 b^5 c-a^4 c^2+2 a^2 b^2 c^2-b^4 c^2+4 b^3 c^3-a^2 c^4-b^2 c^4-2 b c^5+c^6)] : :
X(19) = (r + 2R - s)(r + 2R + s)*X(1) - 6R(r + 2R)*X(2) - 2(r2 + 2rR - s2)*X(3)    (Peter Moses, April 2, 2013)
X(19) = 3 X[2] + X[20061], 6 X[2] - 5 X[31261], 2 X[3] - 3 X[21160], 2 X[4329] - 3 X[31158], 2 X[4329] - 5 X[31261], 2 X[18589] + X[20061], 4 X[18589] - 3 X[31158], 4 X[18589] - 5 X[31261], 2 X[20061] + 3 X[31158], 2 X[20061] + 5 X[31261], 3 X[21160] - X[30265], 3 X[31158] - 5 X[31261]

X(19) is the homothetic center of the orthic and extangents triangles. The Ayme triangle, constructed at X(3610), is perspective to ABC with perspector X(19).

If you have The Geometer's Sketchpad, you can view Clawson point.
If you have GeoGebra, you can view Clawson point.

Although John Clawson studied this point in 1925, it was studied earlier by Lemoine:

Emile Lemoine, "Quelques questions se rapportant à l'étude des antiparallèles des côtes d'un triangle", Bulletin de la S. M. F., tome 14 (1886), p. 107-128, specifically, on page 114. This article is available online at Numdam.

Let A'B'C' be the 4th Brocard triangle. Let A″ be the trilinear product B'*C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(19). (Randy Hutson, November 18, 2015)

Let A'B'C' be the orthic triangle. Let A″ be the trilinear product of the (real or imaginary) circumcircle intercepts of line B'C'. Define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(19). (Randy Hutson, December 26, 2015)

Let A'B'C' be the excentral triangle. Let A″ be the pole, wrt the polar circle, of line B'C', and define B″ and C″ cyclically. The lines A'A″, B'B″, C'C″ concur in X(19). (Randy Hutson, December 2, 2017)

Let A'B'C' be the hexyl triangle. Let La be the trilinear polar of A', and define Lb and Lc cyclically. Let A″ = Lb∩Lc, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(19). (Randy Hutson, December 2, 2017)

Let A'B'C' be the hexyl triangle. Let Ab = BC∩C'A', Ac = BC∩A'B', and define Bc, Ba, Ca, Cb cyclically. Then Ab, Ac, Bc, Ba, Ca, Cb lie on an ellipse. Let A″ be the intersection of the tangents to the ellipse at Ba and Ca, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(19). (Randy Hutson, December 2, 2017)

Let A'B'C' be the intouch triangle of the extangents triangle, if ABC is acute. Then A'B'C' is homothetic to the anti-Ara triangle at X(19). (Randy Hutson, December 2, 2017)

Let La be the A-extraversion of line X(650)X(663) (the trilinear polar of X(9)), and define Lb and Lc cyclically. Let A' = Lb∩Lc, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(19). (Randy Hutson, December 2, 2017)

Let La be the A-extraversion of line X(661)X(663) (the trilinear polar of X(19)), and define Lb and Lc cyclically. Let A' = Lb∩Lc, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(19). (Randy Hutson, December 2, 2017)

X(19) lies on the 2nd Lester circle (Yiu), the cubics K109, K145, K175, K343, K391, K431, K445, K457, K605, K678, K696, K697, K750, K864, K968, K999, K1039, K1042, K1090, the curve Q121, and these lines: {1, 28}, {2, 534}, {3, 1871}, {4, 9}, {5, 8141}, {6, 34}, {7, 5236}, {8, 1891}, {21, 4288}, {24, 2337}, {25, 33}, {27, 63}, {29, 5250}, {30, 15940}, {31, 204}, {32, 5336}, {35, 14017}, {38, 4211}, {41, 1825}, {44, 1828}, {45, 1900}, {46, 579}, {47, 921}, {51, 3611}, {52, 6237}, {53, 1846}, {54, 16031}, {56, 207}, {57, 196}, {64, 1903}, {77, 2002}, {81, 969}, {84, 3183}, {86, 14013}, {91, 920}, {100, 7466}, {101, 913}, {102, 282}, {107, 2249}, {108, 2291}, {109, 8775}, {112, 759}, {113, 12661}, {117, 5190}, {125, 10119}, {143, 32158}, {155, 12417}, {158, 1712}, {162, 897}, {163, 563}, {165, 2954}, {184, 10536}, {185, 6254}, {186, 16553}, {191, 31902}, {200, 3949}, {208, 225}, {216, 27622}, {219, 517}, {220, 1902}, {226, 1763}, {232, 444}, {264, 21371}, {269, 3942}, {270, 2363}, {273, 653}, {275, 19181}, {294, 1041}, {297, 25978}, {307, 24316}, {318, 1840}, {326, 1958}, {331, 10030}, {346, 3610}, {347, 24604}, {355, 7511}, {378, 7688}, {379, 1441}, {381, 18453}, {403, 7110}, {406, 5257}, {407, 1865}, {418, 26908}, {423, 2905}, {427, 3925}, {428, 5101}, {429, 1213}, {469, 3305}, {475, 5750}, {484, 5146}, {487, 12662}, {488, 12663}, {511, 15975}, {518, 5781}, {523, 14119}, {560, 1910}, {577, 1950}, {583, 2969}, {587, 3536}, {594, 5090}, {596, 14964}, {604, 909}, {614, 3162}, {648, 18827}, {649, 3064}, {650, 2432}, {656, 8768}, {662, 8771}, {672, 1851}, {774, 2155}, {775, 4100}, {798, 24006}, {800, 1945}, {823, 1821}, {846, 17038}, {851, 18591}, {876, 2489}, {908, 28807}, {911, 17463}, {960, 965}, {962, 27382}, {990, 3220}, {993, 4227}, {1013, 35258}, {1024, 7649}, {1030, 20832}, {1039, 2298}, {1040, 4224}, {1075, 7554}, {1086, 28017}, {1100, 11396}, {1104, 3172}, {1109, 2157}, {1122, 2097}, {1125, 7521}, {1158, 1715}, {1174, 2266}, {1212, 1593}, {1214, 11347}, {1247, 2959}, {1319, 37519}, {1333, 2217}, {1334, 28076}, {1350, 5784}, {1375, 17073}, {1405, 1866}, {1422, 7099}, {1423, 3512}, {1449, 1870}, {1457, 22063}, {1460, 7337}, {1461, 34492}, {1482, 20818}, {1572, 2300}, {1580, 17891}, {1581, 1740}, {1598, 1872}, {1604, 11399}, {1609, 2164}, {1611, 3290}, {1621, 4233}, {1633, 1721}, {1659, 3068}, {1698, 5142}, {1699, 9572}, {1707, 1719}, {1708, 1713}, {1709, 8558}, {1743, 1783}, {1745, 23619}, {1759, 21061}, {1773, 13161}, {1802, 2324}, {1830, 2246}, {1831, 2268}, {1836, 1901}, {1837, 1852}, {1843, 2876}, {1847, 10509}, {1857, 2357}, {1862, 6154}, {1864, 17810}, {1877, 2347}, {1878, 5183}, {1883, 17369}, {1944, 10446}, {1964, 2186}, {1966, 18832}, {1968, 16968}, {1969, 3403}, {1974, 16972}, {1986, 7724}, {1990, 2160}, {2128, 2129}, {2148, 2190}, {2156, 17871}, {2159, 3708}, {2166, 3376}, {2192, 7007}, {2194, 2219}, {2195, 2212}, {2203, 2214}, {2204, 2218}, {2207, 2281}, {2215, 8747}, {2255, 7151}, {2256, 3057}, {2258, 3192}, {2267, 30503}, {2273, 9620}, {2278, 17443}, {2287, 3869}, {2290, 2962}, {2305, 2652}, {2313, 9251}, {2319, 3186}, {2321, 17742}, {2326, 13739}, {2578, 2588}, {2579, 2589}, {2809, 8271}, {2822, 5667}, {2911, 21853}, {2960, 4877}, {3059, 7716}, {3060, 11445}, {3062, 10859}, {3069, 6203}, {3091, 9537}, {3144, 34920}, {3169, 34895}, {3174, 16550}, {3176, 10396}, {3189, 7718}, {3199, 21796}, {3207, 11363}, {3211, 24474}, {3219, 6994}, {3247, 4262}, {3306, 16706}, {3330, 7355}, {3333, 22088}, {3400, 3408}, {3401, 3409}, {3402, 3404}, {3462, 7344}, {3497, 3500}, {3567, 11460}, {3574, 32370}, {3575, 6253}, {3576, 7501}, {3589, 5834}, {3666, 15509}, {3683, 11323}, {3692, 5174}, {3694, 5687}, {3729, 20602}, {3731, 4222}, {3772, 16318}, {3811, 22021}, {3875, 31906}, {3877, 17519}, {3958, 12526}, {4063, 17924}, {4183, 4512}, {4194, 5296}, {4200, 5749}, {4209, 7131}, {4212, 17754}, {4228, 20243}, {4254, 11398}, {4295, 5746}, {4320, 17807}, {4327, 22769}, {4360, 31910}, {4361, 5792}, {4384, 16566}, {4394, 6591}, {4429, 5125}, {4466, 18634}, {5095, 32277}, {5130, 17275}, {5155, 17330}, {5200, 13427}, {5248, 25081}, {5282, 7102}, {5292, 34266}, {5322, 10829}, {5412, 5415}, {5413, 5416}, {5437, 17917}, {5521, 20623}, {5739, 8896}, {5742, 26066}, {5747, 12047}, {5776, 6001}, {5778, 5887}, {5802, 18391}, {5813, 28739}, {5829, 5880}, {5928, 13567}, {6059, 7083}, {6152, 6255}, {6180, 34371}, {6252, 6291}, {6404, 6406}, {6763, 31901}, {7017, 17787}, {7054, 11101}, {7190, 18162}, {7282, 8545}, {7350, 7351}, {7359, 12699}, {7412, 10268}, {7498, 31435}, {7534, 26921}, {7537, 8227}, {7559, 7989}, {7714, 34607}, {7982, 22356}, {7994, 14493}, {8148, 22147}, {8539, 8541}, {8743, 16470}, {8897, 18134}, {9310, 17452}, {9786, 12664}, {9817, 33849}, {10222, 23073}, {10311, 10315}, {10436, 15149}, {10476, 22065}, {10636, 10641}, {10637, 10642}, {10977, 33630}, {10985, 10988}, {11341, 20172}, {11433, 18921}, {12135, 17299}, {12329, 21867}, {13041, 13051}, {13042, 13052}, {14543, 17220}, {14557, 34048}, {14953, 17134}, {15344, 28847}, {16502, 20227}, {16551, 20367}, {16571, 17799}, {16572, 34498}, {16747, 32092}, {16814, 17516}, {17151, 31918}, {17465, 34080}, {17747, 28070}, {17861, 21364}, {17904, 20083}, {18163, 18677}, {18206, 31919}, {18344, 23351}, {18650, 24683}, {18679, 27659}, {18691, 20902}, {19213, 19214}, {19215, 19218}, {19216, 19217}, {19298, 19300}, {19299, 19301}, {19302, 21773}, {19432, 19446}, {19433, 19447}, {20258, 24334}, {20266, 21621}, {20291, 31015}, {20305, 24682}, {21368, 26665}, {21442, 24587}, {21770, 34434}, {22124, 34040}, {22127, 35631}, {22840, 22970}, {24315, 34830}, {24684, 25523}, {24701, 25365}, {25078, 25440}, {25590, 31925}, {26118, 34822}, {26273, 28023}, {26690, 35974}, {26704, 29068}, {26919, 26952}, {27472, 31346}, {31903, 32922}, {33790, 33793}, {36131, 36151}

X(19) = midpoint of X(4329) and X(20061)
X(19) = reflection of X(i) in X(j) for these {i,j}: {63, 34176}, {4319, 1486}, {4329, 18589}, {24701, 25365}, {30265, 3}, {31158, 2}
X(19) = isogonal conjugate of X(63)
X(19) = isotomic conjugate of X(304)
X(19) = complement of X(4329)
X(19) = anticomplement of X(18589)
X(19) = circumcircle-inverse of X(32756)
X(19) = polar-circle-inverse of X(5179)
X(19) = polar conjugate of X(75)
X(19) = complement of the isogonal conjugate of X(7169)
X(19) = complement of the isotomic conjugate of X(7219)
X(19) = isogonal conjugate of the anticomplement of X(226)
X(19) = isogonal conjugate of the complement of X(5905)
X(19) = isotomic conjugate of the anticomplement of X(16583)
X(19) = isotomic conjugate of the complement of X(21216)
X(19) = isotomic conjugate of the isogonal conjugate of X(1973)
X(19) = isogonal conjugate of the isotomic conjugate of X(92)
X(19) = isotomic conjugate of the polar conjugate of X(1096)
X(19) = isogonal conjugate of the polar conjugate of X(158)
X(19) = polar conjugate of the isotomic conjugate of X(1)
X(19) = polar conjugate of the isogonal conjugate of X(31)
X(19) = excentral-isogonal conjugate of X(2947)
X(19) = orthic-isogonal conjugate of X(33)
X(19) = perspector of circumconic centered at X(36103)
X(19) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1037, 2897}, {1041, 2893}, {7084, 3151}
X(19) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 36103}, {7097, 141}, {7169, 10}, {7219, 2887}
X(19) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 204}, {2, 36103}, {4, 33}, {27, 4}, {28, 25}, {29, 3192}, {40, 8802}, {57, 208}, {63, 1712}, {84, 7008}, {92, 1}, {108, 18344}, {158, 1096}, {196, 207}, {278, 34}, {281, 2331}, {653, 7649}, {823, 24006}, {1172, 6}, {1396, 4186}, {1748, 920}, {1783, 6591}, {2190, 31}, {2322, 1249}, {2580, 2588}, {2581, 2589}, {6336, 1870}, {7003, 7129}, {7012, 8750}, {7128, 108}, {8748, 393}, {8751, 2201}, {14493, 7071}, {20624, 2202}, {24000, 32676}, {24019, 661}, {32714, 513}, {36124, 2356}, {36126, 810}, {36127, 663}
X(19) = X(i)-cross conjugate of X(j) for these (i,j): {25, 34}, {31, 1}, {63, 2129}, {444, 7119}, {607, 33}, {608, 7129}, {649, 32674}, {661, 24019}, {663, 36127}, {774, 9258}, {798, 32676}, {810, 36126}, {1195, 284}, {1400, 6}, {1824, 4}, {1880, 393}, {1973, 1096}, {2083, 63}, {2170, 3064}, {2179, 31}, {2181, 158}, {2183, 913}, {2310, 513}, {2312, 1910}, {2333, 25}, {2354, 1474}, {2355, 28}, {2356, 36124}, {2357, 64}, {2578, 2576}, {2579, 2577}, {2624, 36131}, {2643, 24006}, {3209, 2331}, {3708, 661}, {6591, 1783}, {7083, 2191}, {7154, 7007}, {8020, 2207}, {12723, 7}, {16583, 2}, {17442, 92}, {17872, 75}, {18344, 108}
X(19) = X(i)-isoconjugate of X(j) for these (i,j): {1, 63}, {2, 3}, {4, 394}, {5, 97}, {6, 69}, {7, 219}, {8, 222}, {9, 77}, {10, 1790}, {19, 326}, {20, 1073}, {21, 1214}, {22, 14376}, {23, 34897}, {25, 3926}, {27, 3682}, {28, 3998}, {30, 14919}, {31, 304}, {32, 305}, {33, 7183}, {34, 3719}, {37, 1444}, {38, 34055}, {39, 1799}, {41, 7182}, {42, 17206}, {46, 6513}, {48, 75}, {49, 11140}, {50, 328}, {51, 34386}, {54, 343}, {55, 348}, {56, 345}, {57, 78}, {58, 306}, {59, 26932}, {60, 26942}, {65, 1812}, {66, 20806}, {67, 22151}, {68, 1993}, {71, 86}, {72, 81}, {73, 333}, {74, 11064}, {76, 184}, {80, 22128}, {83, 3917}, {85, 212}, {87, 22370}, {88, 5440}, {89, 3940}, {90, 6505}, {92, 255}, {94, 22115}, {95, 216}, {98, 36212}, {99, 647}, {100, 905}, {101, 4025}, {103, 26006}, {105, 25083}, {106, 3977}, {109, 6332}, {110, 525}, {111, 6390}, {112, 3265}, {125, 249}, {140, 31626}, {141, 1176}, {154, 34403}, {155, 6504}, {158, 6507}, {162, 24018}, {163, 14208}, {172, 7019}, {185, 801}, {187, 30786}, {189, 7078}, {190, 1459}, {192, 23086}, {193, 6391}, {194, 3504}, {200, 7177}, {201, 2185}, {217, 34384}, {220, 7056}, {223, 271}, {225, 6514}, {226, 283}, {228, 274}, {232, 6394}, {239, 295}, {248, 325}, {250, 15526}, {251, 3933}, {253, 15905}, {254, 6503}, {257, 3955}, {261, 2197}, {264, 577}, {265, 323}, {268, 347}, {269, 3692}, {273, 2289}, {275, 5562}, {276, 418}, {278, 1259}, {279, 1260}, {280, 7011}, {281, 1804}, {282, 7013}, {284, 307}, {286, 3990}, {287, 511}, {290, 3289}, {291, 20769}, {293, 1959}, {296, 1944}, {297, 17974}, {302, 32585}, {303, 32586}, {308, 20775}, {310, 2200}, {311, 14533}, {312, 603}, {314, 1409}, {318, 7125}, {321, 1437}, {324, 19210}, {329, 1433}, {330, 20760}, {331, 6056}, {332, 1400}, {335, 7193}, {336, 1755}, {337, 1914}, {339, 23357}, {341, 7099}, {346, 7053}, {350, 2196}, {371, 11090}, {372, 11091}, {385, 36214}, {393, 3964}, {401, 14941}, {427, 28724}, {441, 1297}, {459, 35602}, {476, 8552}, {480, 30682}, {485, 5408}, {486, 5409}, {487, 494}, {488, 493}, {491, 6414}, {492, 6413}, {512, 4563}, {513, 1332}, {514, 1331}, {516, 1815}, {518, 1814}, {519, 1797}, {520, 648}, {521, 651}, {522, 1813}, {523, 4558}, {524, 895}, {561, 9247}, {563, 20571}, {571, 20563}, {593, 3695}, {604, 3718}, {607, 7055}, {608, 1264}, {610, 19611}, {649, 4561}, {650, 6516}, {652, 664}, {656, 662}, {661, 4592}, {668, 22383}, {670, 3049}, {671, 3292}, {672, 31637}, {673, 1818}, {684, 2966}, {686, 18878}, {691, 14417}, {692, 15413}, {693, 906}, {694, 12215}, {757, 3949}, {765, 3942}, {775, 6508}, {799, 810}, {805, 24284}, {811, 822}, {827, 2525}, {850, 32661}, {858, 18876}, {878, 2396}, {879, 2421}, {894, 7015}, {903, 22356}, {908, 1795}, {912, 2990}, {914, 36052}, {916, 2989}, {932, 25098}, {940, 34259}, {943, 18607}, {999, 30680}, {1000, 22129}, {1002, 23151}, {1014, 3694}, {1016, 3937}, {1025, 23696}, {1029, 22136}, {1031, 22138}, {1032, 1498}, {1037, 27509}, {1038, 2339}, {1040, 7131}, {1068, 6512}, {1069, 5905}, {1088, 1802}, {1092, 2052}, {1096, 1102}, {1101, 20902}, {1105, 6509}, {1124, 13387}, {1125, 1796}, {1126, 4001}, {1147, 5392}, {1156, 6510}, {1178, 4019}, {1211, 1798}, {1220, 22097}, {1221, 22389}, {1231, 2194}, {1249, 15394}, {1252, 1565}, {1255, 3916}, {1262, 2968}, {1265, 1407}, {1267, 34121}, {1268, 22054}, {1270, 6415}, {1271, 6416}, {1273, 11077}, {1275, 3270}, {1292, 24562}, {1301, 20580}, {1306, 17431}, {1307, 17432}, {1310, 2522}, {1333, 20336}, {1335, 13386}, {1412, 3710}, {1414, 8611}, {1415, 35518}, {1425, 7058}, {1427, 1792}, {1434, 2318}, {1439, 2287}, {1441, 2193}, {1465, 1809}, {1472, 19799}, {1473, 30701}, {1494, 3284}, {1502, 14575}, {1509, 3690}, {1576, 3267}, {1577, 4575}, {1586, 26922}, {1619, 2139}, {1634, 4580}, {1636, 16077}, {1783, 4131}, {1789, 16577}, {1791, 3666}, {1793, 18593}, {1794, 5249}, {1803, 4847}, {1807, 3218}, {1808, 16609}, {1810, 3008}, {1811, 16610}, {1819, 8808}, {1822, 2582}, {1823, 2583}, {1897, 4091}, {1909, 7116}, {1937, 6518}, {1943, 7016}, {1946, 4554}, {1958, 9255}, {1976, 6393}, {1994, 3519}, {2113, 20742}, {2148, 18695}, {2149, 17880}, {2165, 9723}, {2169, 14213}, {2207, 4176}, {2286, 30479}, {2327, 3668}, {2351, 7763}, {2353, 34254}, {2359, 4357}, {2373, 14961}, {2407, 14380}, {2420, 34767}, {2435, 34211}, {2481, 20752}, {2482, 15398}, {2504, 29241}, {2510, 2858}, {2524, 3222}, {2574, 8115}, {2575, 8116}, {2580, 2584}, {2581, 2585}, {2715, 6333}, {2972, 23582}, {2983, 18650}, {2985, 23154}, {2986, 13754}, {2987, 3564}, {2991, 34381}, {2994, 3157}, {2995, 22134}, {2996, 3167}, {2998, 20794}, {3053, 6340}, {3064, 6517}, {3083, 6213}, {3084, 6212}, {3108, 7767}, {3112, 4020}, {3158, 27832}, {3219, 7100}, {3226, 20785}, {3229, 8858}, {3260, 18877}, {3261, 32656}, {3262, 14578}, {3263, 32658}, {3264, 32659}, {3266, 14908}, {3268, 32662}, {3269, 18020}, {3295, 30679}, {3316, 5406}, {3317, 5407}, {3346, 6617}, {3433, 28420}, {3569, 17932}, {3580, 5504}, {3618, 34817}, {3669, 4571}, {3676, 4587}, {3708, 24041}, {3781, 14621}, {3784, 17743}, {3796, 18840}, {3912, 36057}, {3927, 25417}, {3952, 7254}, {3978, 17970}, {4064, 4556}, {4143, 32713}, {4358, 36058}, {4373, 20818}, {4391, 36059}, {4466, 4570}, {4552, 23189}, {4555, 22086}, {4557, 15419}, {4560, 23067}, {4562, 22384}, {4564, 7004}, {4566, 23090}, {4567, 18210}, {4574, 7192}, {4590, 20975}, {4591, 14429}, {4598, 22090}, {4652, 25430}, {4846, 15066}, {4855, 8056}, {4998, 7117}, {5391, 34125}, {5456, 16841}, {5467, 14977}, {5468, 10097}, {5490, 10132}, {5491, 10133}, {5546, 17094}, {5976, 15391}, {6010, 24560}, {6061, 20618}, {6148, 11079}, {6334, 10420}, {6335, 23224}, {6337, 8770}, {6338, 15369}, {6339, 19588}, {6356, 7054}, {6368, 18315}, {6374, 15389}, {6376, 15373}, {6511, 7040}, {6515, 15316}, {6527, 28783}, {6528, 32320}, {6542, 17972}, {6553, 23089}, {6601, 23144}, {6625, 22139}, {6626, 15377}, {6630, 22148}, {6650, 17976}, {7017, 7335}, {7023, 30681}, {7045, 34591}, {7074, 34400}, {7105, 7364}, {7114, 34404}, {7123, 17170}, {7124, 8817}, {7128, 24031}, {7219, 22119}, {7224, 23150}, {7261, 20741}, {7309, 16840}, {7319, 23140}, {7357, 20739}, {7361, 20764}, {7372, 23084}, {7473, 35911}, {7578, 23039}, {8024, 10547}, {8044, 22133}, {8046, 22141}, {8047, 22144}, {8048, 22132}, {8049, 22126}, {8050, 22154}, {8222, 8948}, {8223, 8946}, {8606, 17095}, {8677, 13136}, {8709, 22092}, {8750, 30805}, {8779, 35140}, {8813, 13615}, {8911, 34391}, {9146, 30491}, {9289, 9306}, {9295, 22158}, {9517, 17708}, {10159, 22352}, {10217, 11131}, {10218, 11130}, {10316, 18018}, {10317, 18019}, {10405, 22117}, {10411, 14582}, {10607, 34208}, {10665, 13428}, {10666, 13439}, {11517, 15474}, {11547, 16391}, {12028, 34834}, {13388, 30556}, {13389, 30557}, {13485, 22146}, {13575, 23115}, {13577, 22131}, {14206, 35200}, {14210, 36060}, {14379, 15466}, {14534, 22076}, {14570, 23286}, {14585, 18022}, {14615, 14642}, {14999, 35909}, {15329, 15421}, {15407, 15595}, {15412, 23181}, {15455, 23226}, {15740, 17811}, {15958, 18314}, {17434, 18831}, {17790, 17971}, {17946, 17977}, {17947, 17975}, {17950, 17973}, {18023, 23200}, {18026, 36054}, {18027, 23606}, {18750, 19614}, {18890, 20477}, {19126, 31360}, {19188, 31504}, {19354, 34401}, {20568, 23202}, {20777, 32020}, {21739, 23071}, {22053, 32008}, {22060, 32009}, {22061, 32010}, {22066, 32011}, {22067, 32012}, {22068, 32013}, {22074, 31643}, {22079, 31618}, {22080, 32014}, {22091, 30610}, {22093, 27805}, {22096, 31625}, {22143, 35511}, {22341, 31623}, {22344, 32017}, {22345, 30710}, {22350, 34234}, {22381, 31008}, {22388, 31624}, {22458, 35058}, {23095, 27494}, {23201, 32018}, {23210, 31622}, {26920, 34392}, {28408, 34436}, {28419, 34207}, {28696, 34427}, {28754, 34435}, {30807, 36056}, {31617, 32078}, {32657, 35517}, {32660, 35519}, {32663, 35520}, {32679, 36061}, {34483, 34545}, {35910, 35912}
X(19) = cevapoint of X(i) and X(j) for these (i,j): {1, 1707}, {2, 21216}, {6, 2178}, {25, 607}, {31, 1973}, {42, 2198}, {44, 17465}, {63, 2128}, {444, 1829}, {512, 14936}, {513, 17463}, {608, 3209}, {649, 2170}, {661, 3708}, {672, 17464}, {798, 2643}, {896, 17466}, {1400, 1880}, {1575, 20600}, {1755, 17462}, {1824, 2333}, {1953, 2180}, {2082, 30677}, {2179, 2181}, {2501, 8735}, {6203, 6204}, {8020, 16583}, {8769, 19213}, {19215, 19216}
X(19) = crosspoint of X(i) and X(j) for these (i,j): {1, 2184}, {2, 7219}, {4, 278}, {6, 8761}, {27, 28}, {57, 84}, {92, 158}, {107, 23984}, {108, 7128}, {112, 7115}, {281, 7003}, {648, 15742}, {653, 7012}, {823, 24000}, {1172, 8748}, {24033, 32714}
X(19) = X(19) = crosssum of X(i) and X(j) for these (i,j): {1, 610}, {2, 6360}, {3, 219}, {6, 3556}, {9, 40}, {19, 1712}, {48, 255}, {71, 72}, {220, 12329}, {222, 7011}, {394, 6511}, {520, 35072}, {521, 34591}, {523, 6506}, {525, 26932}, {647, 3937}, {652, 7004}, {656, 3708}, {822, 2632}, {905, 3942}, {1400, 12089}, {3049, 22386}, {3916, 3958}
X(19) = X(i)-vertex conjugate of X(j) for these (i,j): {1, 3422}, {1946, 2202}, {2191, 3433}
X(19) = X(i)-line conjugate of X(j) for these (i,j): {1, 8766},{8768, 656}
X(19) = trilinear pole of line {661, 663}
X(19) = crossdifference of every pair of points on line {521, 656}
X(19) = X(i)-Hirst inverse of X(j) for these (i,j): (1,240), (4,242)
X(19) = X(i)-aleph conjugate of X(j) for these (i,j): (2,610), (92,19), (508,223), (648,163)
X(19) = X(i)-beth conjugate of X(j) for these (i,j): (9,198), (19,608), (112,604), (281,281), (648,273), (653,19)
X(19) = inverse-in-polar-circle of X(5179)
X(19) = inverse-in-circumconic-centered-at-X(9) of X(1861)
X(19) = Zosma transform of X(9)
X(19) = perspector of ABC and extraversion triangle of X(19) (which is also the anticevian triangle of X(19))
X(19) = intersection of tangents at X(9) and X(57) to Thomson cubic K002
X(19) = intersection of tangents at X(40) and X(84) to Darboux cubic K004
X(19) = trilinear product of PU(i) for these i: 4, 23, 157
X(19) = barycentric product of PU(15)
X(19) = vertex conjugate of PU(19)
X(19) = bicentric sum of PU(127)
X(19) = PU(127)-harmonic conjugate of X(656)
X(19) = perspector of ABC and unary cofactor triangle of hexyl triangle
X(19) = perspector of unary cofactor triangles of 2nd and 4th extouch triangles
X(19) = perspector of ABC and unary cofactor triangle of extraversion triangle of X(9)
X(19) = complement of X(4329)
X(19) = {X(48),X(1953)}-harmonic conjugate of X(1)
X(19) = {X(92),X(1748)}-harmonic conjugate of X(63)
X(19) = trilinear product X(2)*X(25)
X(19) = trilinear pole of line X(661)X(663) (the polar of X(75) wrt polar circle)
X(19) = pole wrt polar circle of trilinear polar of X(75) (line X(514)X(661))
X(19) = polar conjugate of X(75)
X(19) = X(i)-isoconjugate of X(j) for these {i,j}: {1,63}, {6,69}, {31,304}, {48,75}, {67, 22151}, {92,255}
X(19) = X(571)-of-excentral-triangle
X(19) = perspector, wrt excentral triangle, of polar circle
X(19) = barycentric product X(i)*X(j) for these {i,j}: {1, 4}, {3, 158}, {5, 2190}, {6, 92}, {7, 33}, {8, 34}, {9, 278}, {10, 28}, {11, 7012}, {12, 270}, {21, 225}, {24, 91}, {25, 75}, {27, 37}, {29, 65}, {30, 36119}, {31, 264}, {32, 1969}, {38, 32085}, {41, 331}, {42, 286}, {44, 6336}, {46, 7040}, {47, 847}, {48, 2052}, {53, 2167}, {55, 273}, {56, 318}, {57, 281}, {63, 393}, {64, 1895}, {69, 1096}, {72, 8747}, {73, 1896}, {74, 1784}, {76, 1973}, {77, 1857}, {78, 1118}, {79, 6198}, {80, 1870}, {81, 1826}, {82, 427}, {83, 17442}, {84, 7952}, {85, 607}, {86, 1824}, {88, 8756}, {90, 1068}, {93, 2964}, {95, 2181}, {98, 240}, {100, 7649}, {101, 17924}, {104, 1785}, {105, 1861}, {107, 656}, {108, 522}, {110, 24006}, {112, 1577}, {125, 24000}, {162, 523}, {163, 14618}, {185, 821}, {186, 2166}, {189, 2331}, {190, 6591}, {196, 282}, {200, 1119}, {204, 253}, {207, 1034}, {208, 280}, {220, 1847}, {221, 7020}, {223, 7003}, {226, 1172}, {232, 1821}, {235, 775}, {242, 291}, {243, 1937}, {244, 15742}, {250, 1109}, {251, 20883}, {254, 920}, {255, 1093}, {256, 7009}, {257, 7119}, {267, 451}, {269, 7046}, {274, 2333}, {275, 1953}, {276, 2179}, {277, 7719}, {279, 7079}, {293, 6530}, {294, 5236}, {297, 1910}, {304, 2207}, {306, 5317}, {309, 3195}, {312, 608}, {313, 2203}, {321, 1474}, {322, 7151}, {324, 2148}, {326, 6524}, {329, 7129}, {333, 1880}, {335, 2201}, {336, 34854}, {341, 1398}, {342, 2192}, {346, 1435}, {347, 7008}, {349, 2204}, {388, 1039}, {394, 6520}, {403, 36053}, {415, 2652}, {419, 1581}, {423, 9278}, {429, 2363}, {436, 9251}, {458, 2186}, {459, 610}, {460, 8773}, {467, 2168}, {468, 897}, {469, 2214}, {470, 2153}, {471, 2154}, {477, 36063}, {497, 1041}, {502, 2906}, {511, 36120}, {512, 811}, {513, 1897}, {514, 1783}, {515, 36121}, {516, 36122}, {517, 36123}, {518, 36124}, {519, 36125}, {520, 36126}, {521, 36127}, {524, 36128}, {525, 24019}, {526, 36129}, {560, 18022}, {561, 1974}, {577, 6521}, {596, 4222}, {604, 7017}, {647, 823}, {648, 661}, {649, 6335}, {650, 653}, {651, 3064}, {657, 13149}, {662, 2501}, {663, 18026}, {664, 18344}, {673, 5089}, {693, 8750}, {757, 7140}, {759, 860}, {765, 2969}, {774, 1105}, {798, 6331}, {799, 2489}, {810, 6528}, {822, 15352}, {849, 7141}, {850, 32676}, {862, 18827}, {896, 17983}, {915, 1737}, {917, 1736}, {921, 3542}, {933, 2618}, {941, 5307}, {943, 1838}, {969, 4207}, {977, 5090}, {994, 5136}, {1018, 17925}, {1020, 17926}, {1043, 1426}, {1061, 1478}, {1063, 1479}, {1065, 1905}, {1088, 7071}, {1097, 31942}, {1101, 2970}, {1110, 2973}, {1113, 2588}, {1114, 2589}, {1120, 1878}, {1123, 6212}, {1128, 8120}, {1146, 7128}, {1148, 3362}, {1156, 23710}, {1170, 1855}, {1214, 8748}, {1220, 1829}, {1222, 1828}, {1247, 3144}, {1249, 2184}, {1255, 1839}, {1257, 1842}, {1268, 2355}, {1300, 1725}, {1301, 17898}, {1304, 36035}, {1309, 1769}, {1320, 1877}, {1336, 6213}, {1390, 1890}, {1395, 3596}, {1396, 2321}, {1400, 31623}, {1407, 7101}, {1411, 5081}, {1427, 2322}, {1441, 2299}, {1446, 2332}, {1488, 8122}, {1490, 7149}, {1503, 8767}, {1594, 2216}, {1697, 11546}, {1707, 34208}, {1712, 3346}, {1733, 3563}, {1734, 26705}, {1735, 32706}, {1738, 15344}, {1745, 7049}, {1748, 2165}, {1755, 16081}, {1757, 17982}, {1760, 13854}, {1820, 11547}, {1825, 3615}, {1827, 21453}, {1835, 6740}, {1843, 3112}, {1848, 2298}, {1867, 5331}, {1876, 14942}, {1886, 36101}, {1929, 17927}, {1940, 7105}, {1945, 1948}, {1947, 7106}, {1952, 2202}, {1957, 9307}, {1959, 6531}, {1966, 17980}, {1967, 17984}, {1990, 2349}, {2074, 5620}, {2089, 8121}, {2129, 6392}, {2155, 15466}, {2156, 17907}, {2161, 17923}, {2169, 13450}, {2173, 16080}, {2183, 16082}, {2185, 8736}, {2189, 6358}, {2212, 6063}, {2217, 17555}, {2218, 5125}, {2312, 6330}, {2326, 6354}, {2334, 5342}, {2354, 30710}, {2356, 2481}, {2358, 27398}, {2362, 14121}, {2432, 24035}, {2433, 24001}, {2435, 24024}, {2517, 32691}, {2574, 2586}, {2575, 2587}, {2576, 2592}, {2577, 2593}, {2580, 8105}, {2581, 8106}, {2616, 35360}, {2631, 15459}, {2632, 32230}, {2643, 18020}, {2648, 17985}, {2766, 21180}, {2799, 36104}, {2804, 36110}, {2962, 3518}, {2968, 24033}, {2971, 24037}, {2972, 24021}, {2995, 3192}, {3068, 19218}, {3069, 19217}, {3120, 5379}, {3176, 3345}, {3186, 3223}, {3209, 34404}, {3239, 32714}, {3270, 24032}, {3377, 13429}, {3378, 13440}, {3424, 23052}, {3577, 34231}, {3668, 4183}, {3708, 23582}, {3718, 7337}, {3900, 36118}, {3912, 8751}, {3924, 34406}, {4185, 31359}, {4186, 34860}, {4213, 13610}, {4235, 23894}, {4336, 34398}, {4358, 8752}, {4391, 32674}, {4564, 8735}, {4858, 7115}, {5338, 5936}, {5627, 35201}, {5663, 36130}, {5932, 7007}, {6059, 7182}, {6149, 6344}, {6353, 8769}, {6525, 19611}, {6529, 24018}, {6590, 36099}, {6995, 23051}, {7073, 7282}, {7090, 16232}, {7094, 17902}, {7096, 17904}, {7097, 17903}, {7108, 7120}, {7133, 13390}, {7219, 36103}, {8119, 10215}, {8749, 14206}, {8753, 14210}, {8754, 24041}, {8755, 36100}, {8772, 35142}, {8791, 16568}, {8806, 8885}, {8882, 14213}, {9239, 11380}, {9247, 18027}, {9258, 9308}, {11019, 14493}, {11109, 34434}, {11325, 18832}, {13476, 14004}, {14208, 32713}, {14248, 18156}, {14249, 19614}, {14273, 36085}, {14304, 36067}, {14312, 36044}, {14377, 17916}, {14571, 34234}, {14581, 33805}, {14776, 36038}, {14975, 20565}, {15149, 18785}, {15369, 33787}, {16230, 36084}, {16263, 18477}, {17171, 18098}, {17763, 17981}, {17954, 17987}, {17994, 36036}, {18070, 35325}, {18486, 22455}, {18833, 27369}, {20879, 33631}, {20902, 23964}, {20975, 23999}, {21185, 26706}, {21189, 26704}, {21666, 24027}, {23290, 36134}, {23604, 30733}, {23984, 34591}, {27376, 34055}
X(19) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 69}, {2, 304}, {3, 326}, {4, 75}, {5, 18695}, {6, 63}, {7, 7182}, {8, 3718}, {9, 345}, {10, 20336}, {11, 17880}, {21, 332}, {25, 1}, {27, 274}, {28, 86}, {29, 314}, {31, 3}, {32, 48}, {33, 8}, {34, 7}, {37, 306}, {38, 3933}, {41, 219}, {42, 72}, {44, 3977}, {47, 9723}, {48, 394}, {53, 14213}, {55, 78}, {56, 77}, {57, 348}, {58, 1444}, {63, 3926}, {64, 19611}, {65, 307}, {71, 3998}, {75, 305}, {77, 7055}, {78, 1264}, {81, 17206}, {82, 1799}, {91, 20563}, {92, 76}, {98, 336}, {100, 4561}, {101, 1332}, {105, 31637}, {107, 811}, {108, 664}, {109, 6516}, {110, 4592}, {112, 662}, {115, 20902}, {125, 17879}, {132, 17875}, {136, 17881}, {158, 264}, {162, 99}, {163, 4558}, {169, 28420}, {181, 201}, {184, 255}, {200, 1265}, {204, 20}, {205, 22132}, {207, 5932}, {208, 347}, {210, 3710}, {212, 1259}, {213, 71}, {219, 3719}, {220, 3692}, {221, 7013}, {222, 7183}, {225, 1441}, {226, 1231}, {228, 3682}, {232, 1959}, {235, 17858}, {240, 325}, {242, 350}, {244, 1565}, {250, 24041}, {251, 34055}, {255, 3964}, {256, 7019}, {264, 561}, {269, 7056}, {270, 261}, {273, 6063}, {278, 85}, {281, 312}, {284, 1812}, {286, 310}, {291, 337}, {293, 6394}, {318, 3596}, {326, 4176}, {331, 20567}, {393, 92}, {394, 1102}, {407, 18698}, {419, 1966}, {422, 5209}, {427, 1930}, {429, 18697}, {430, 4647}, {431, 18692}, {451, 20932}, {458, 3403}, {460, 1733}, {461, 4673}, {468, 14210}, {469, 33935}, {512, 656}, {513, 4025}, {514, 15413}, {522, 35518}, {523, 14208}, {560, 184}, {577, 6507}, {603, 1804}, {604, 222}, {607, 9}, {608, 57}, {614, 17170}, {647, 24018}, {648, 799}, {649, 905}, {650, 6332}, {653, 4554}, {656, 3265}, {661, 525}, {662, 4563}, {663, 521}, {667, 1459}, {669, 810}, {672, 25083}, {685, 36036}, {692, 1331}, {728, 30681}, {738, 30682}, {756, 3695}, {798, 647}, {800, 6508}, {810, 520}, {811, 670}, {823, 6331}, {847, 20571}, {860, 35550}, {862, 740}, {869, 3781}, {872, 3690}, {884, 23696}, {896, 6390}, {897, 30786}, {902, 5440}, {904, 7015}, {905, 30805}, {910, 26006}, {911, 1815}, {913, 2990}, {922, 3292}, {923, 895}, {1015, 3942}, {1019, 15419}, {1021, 15411}, {1039, 30479}, {1041, 8817}, {1042, 1439}, {1055, 6510}, {1068, 20930}, {1096, 4}, {1100, 4001}, {1104, 18650}, {1106, 7053}, {1109, 339}, {1118, 273}, {1119, 1088}, {1148, 18749}, {1172, 333}, {1196, 18671}, {1197, 22065}, {1249, 18750}, {1253, 1260}, {1254, 6356}, {1333, 1790}, {1334, 3694}, {1395, 56}, {1396, 1434}, {1397, 603}, {1398, 269}, {1400, 1214}, {1402, 73}, {1407, 7177}, {1415, 1813}, {1422, 34400}, {1426, 3668}, {1430, 5088}, {1435, 279}, {1438, 1814}, {1459, 4131}, {1460, 1038}, {1474, 81}, {1500, 3949}, {1501, 9247}, {1576, 4575}, {1577, 3267}, {1580, 12215}, {1611, 2128}, {1707, 6337}, {1712, 6527}, {1748, 7763}, {1755, 36212}, {1760, 34254}, {1781, 28754}, {1783, 190}, {1784, 3260}, {1785, 3262}, {1824, 10}, {1826, 321}, {1827, 4847}, {1828, 3663}, {1829, 4357}, {1839, 4359}, {1840, 3963}, {1841, 5249}, {1842, 17863}, {1843, 38}, {1848, 20911}, {1851, 3673}, {1852, 18690}, {1855, 1229}, {1856, 23528}, {1857, 318}, {1859, 6734}, {1860, 17866}, {1861, 3263}, {1862, 4986}, {1870, 320}, {1875, 22464}, {1876, 9436}, {1878, 1266}, {1880, 226}, {1886, 30807}, {1889, 32092}, {1890, 26234}, {1895, 14615}, {1897, 668}, {1900, 4967}, {1910, 287}, {1911, 295}, {1914, 20769}, {1917, 14575}, {1918, 228}, {1919, 22383}, {1922, 2196}, {1923, 20775}, {1924, 3049}, {1927, 17970}, {1950, 7364}, {1951, 6518}, {1953, 343}, {1957, 1975}, {1959, 6393}, {1964, 3917}, {1967, 36214}, {1968, 1958}, {1969, 1502}, {1973, 6}, {1974, 31}, {1976, 293}, {1981, 15418}, {1990, 14206}, {2052, 1969}, {2082, 27509}, {2083, 6389}, {2085, 20819}, {2112, 20742}, {2128, 6338}, {2129, 6339}, {2148, 97}, {2155, 1073}, {2156, 14376}, {2157, 34897}, {2159, 14919}, {2164, 6513}, {2166, 328}, {2167, 34386}, {2170, 26932}, {2171, 26942}, {2172, 20806}, {2173, 11064}, {2175, 212}, {2176, 22370}, {2177, 3940}, {2178, 6505}, {2179, 216}, {2181, 5}, {2184, 34403}, {2187, 7078}, {2189, 2185}, {2190, 95}, {2192, 271}, {2193, 6514}, {2194, 283}, {2199, 7011}, {2200, 3990}, {2201, 239}, {2202, 1944}, {2203, 58}, {2204, 284}, {2205, 2200}, {2206, 1437}, {2208, 1433}, {2209, 20760}, {2210, 7193}, {2211, 1755}, {2212, 55}, {2223, 1818}, {2251, 22356}, {2258, 34259}, {2260, 18607}, {2280, 23151}, {2295, 4019}, {2299, 21}, {2300, 22097}, {2308, 3916}, {2310, 2968}, {2312, 441}, {2326, 7058}, {2328, 1792}, {2331, 329}, {2332, 2287}, {2333, 37}, {2342, 1809}, {2345, 19799}, {2354, 3666}, {2355, 1125}, {2356, 518}, {2358, 8808}, {2484, 2522}, {2489, 661}, {2501, 1577}, {2576, 8115}, {2577, 8116}, {2586, 15164}, {2587, 15165}, {2588, 22339}, {2589, 22340}, {2624, 8552}, {2642, 14417}, {2643, 125}, {2908, 22130}, {2969, 1111}, {2970, 23994}, {2971, 2643}, {2972, 24020}, {3049, 822}, {3051, 4020}, {3052, 4855}, {3063, 652}, {3064, 4391}, {3079, 1097}, {3122, 18210}, {3124, 3708}, {3125, 4466}, {3162, 18596}, {3172, 610}, {3176, 33672}, {3186, 17149}, {3192, 3869}, {3194, 8822}, {3195, 40}, {3199, 1953}, {3209, 223}, {3213, 18623}, {3239, 15416}, {3248, 3937}, {3270, 24031}, {3271, 7004}, {3377, 13441}, {3378, 13430}, {3542, 33808}, {3553, 26872}, {3554, 26871}, {3563, 8773}, {3575, 17859}, {3708, 15526}, {3709, 8611}, {3725, 22076}, {3914, 20235}, {3939, 4571}, {4017, 17094}, {4118, 4121}, {4183, 1043}, {4185, 10436}, {4186, 3875}, {4206, 1010}, {4212, 33943}, {4213, 17762}, {4214, 25590}, {4222, 4360}, {4235, 24039}, {4705, 4064}, {4730, 14429}, {5089, 3912}, {5095, 24038}, {5101, 33937}, {5139, 17876}, {5151, 20900}, {5190, 17878}, {5254, 21406}, {5307, 34284}, {5317, 27}, {5338, 3616}, {5379, 4600}, {5521, 17877}, {5523, 20884}, {6059, 33}, {6139, 14414}, {6186, 7100}, {6187, 1807}, {6198, 319}, {6212, 1267}, {6213, 5391}, {6331, 4602}, {6335, 1978}, {6336, 20568}, {6353, 18156}, {6423, 19215}, {6424, 19216}, {6520, 2052}, {6521, 18027}, {6524, 158}, {6525, 1895}, {6529, 823}, {6531, 1821}, {6591, 514}, {6620, 4008}, {6748, 20879}, {7003, 34404}, {7007, 1034}, {7008, 280}, {7009, 1909}, {7012, 4998}, {7017, 28659}, {7032, 3784}, {7040, 20570}, {7046, 341}, {7071, 200}, {7076, 7283}, {7079, 346}, {7083, 1040}, {7102, 4385}, {7104, 7116}, {7113, 22128}, {7115, 4564}, {7118, 268}, {7119, 894}, {7120, 1943}, {7121, 23086}, {7122, 3955}, {7128, 1275}, {7129, 189}, {7140, 1089}, {7147, 20618}, {7151, 84}, {7154, 282}, {7156, 27382}, {7337, 34}, {7649, 693}, {7713, 17321}, {7719, 344}, {7952, 322}, {8020, 16583}, {8022, 22364}, {8061, 2525}, {8105, 2582}, {8106, 2583}, {8557, 6350}, {8609, 914}, {8640, 22090}, {8678, 23874}, {8735, 4858}, {8736, 6358}, {8743, 1760}, {8744, 16568}, {8745, 1748}, {8747, 286}, {8748, 31623}, {8749, 2349}, {8750, 100}, {8751, 673}, {8752, 88}, {8753, 897}, {8754, 1109}, {8756, 4358}, {8767, 35140}, {8769, 6340}, {8772, 3564}, {8792, 21378}, {8879, 20931}, {8882, 2167}, {9247, 577}, {9258, 9289}, {9292, 9255}, {9406, 3284}, {9417, 3289}, {9454, 20752}, {9456, 1797}, {9459, 23202}, {10151, 18699}, {10312, 18042}, {11325, 1740}, {11363, 3879}, {11380, 1582}, {11383, 997}, {11396, 17272}, {11406, 936}, {12723, 34822}, {14004, 17143}, {14213, 28706}, {14248, 8769}, {14398, 2631}, {14560, 36061}, {14571, 908}, {14580, 18669}, {14581, 2173}, {14585, 4100}, {14593, 91}, {14618, 20948}, {14776, 36037}, {14827, 1802}, {14936, 34591}, {14975, 35}, {15148, 2669}, {15149, 18157}, {15487, 28409}, {15742, 7035}, {16080, 33805}, {16228, 17894}, {16229, 17893}, {16240, 1099}, {16502, 7289}, {16544, 28696}, {16545, 28408}, {16583, 18589}, {16584, 20727}, {16716, 18648}, {17171, 16703}, {17408, 21147}, {17409, 2172}, {17442, 141}, {17453, 10316}, {17469, 7767}, {17516, 17151}, {17562, 17394}, {17872, 1368}, {17902, 20926}, {17903, 20914}, {17904, 20444}, {17905, 20927}, {17906, 21580}, {17907, 20641}, {17910, 21585}, {17911, 18137}, {17912, 18050}, {17913, 18138}, {17914, 21604}, {17915, 18040}, {17916, 17233}, {17917, 21605}, {17920, 20923}, {17922, 20949}, {17923, 20924}, {17924, 3261}, {17925, 7199}, {17927, 20947}, {17980, 1581}, {17982, 18032}, {17984, 1926}, {18020, 24037}, {18022, 1928}, {18026, 4572}, {18191, 17219}, {18210, 17216}, {18266, 17976}, {18344, 522}, {18384, 2166}, {18596, 28419}, {18676, 21579}, {18677, 21581}, {18680, 21583}, {18681, 21584}, {18756, 23079}, {19118, 1707}, {19215, 8222}, {19216, 8223}, {19217, 5491}, {19218, 5490}, {19554, 20741}, {19614, 15394}, {20613, 28739}, {20624, 8777}, {20883, 8024}, {20970, 3958}, {20975, 2632}, {20978, 10167}, {20979, 25098}, {21148, 1763}, {21389, 28423}, {21750, 23620}, {21760, 20785}, {21832, 24459}, {21833, 21046}, {22363, 22057}, {22383, 4091}, {23050, 10327}, {23503, 2524}, {23566, 20736}, {23710, 30806}, {23894, 14977}, {23985, 7128}, {24000, 18020}, {24006, 850}, {24018, 4143}, {24019, 648}, {24022, 32230}, {27369, 1964}, {27376, 20883}, {28044, 3886}, {28615, 1796}, {31623, 28660}, {31900, 16709}, {31905, 30940}, {32085, 3112}, {32230, 23999}, {32664, 20739}, {32674, 651}, {32676, 110}, {32691, 1310}, {32696, 36084}, {32713, 162}, {32714, 658}, {32715, 36034}, {32739, 906}, {32740, 36060}, {33581, 19614}, {33781, 19583}, {34121, 3084}, {34125, 3083}, {34248, 3504}, {34397, 6149}, {34417, 18477}, {34591, 23983}, {34854, 240}, {34858, 1795}, {35201, 6148}, {36059, 6517}, {36063, 35520}, {36084, 17932}, {36103, 4329}, {36104, 2966}, {36114, 18878}, {36118, 4569}, {36119, 1494}, {36120, 290}, {36121, 34393}, {36122, 18025}, {36123, 18816}, {36124, 2481}, {36125, 903}, {36126, 6528}, {36127, 18026}, {36128, 671}, {36129, 35139}
X(19) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 240, 23052}, {1, 610, 48}, {1, 16545, 18596}, {1, 18594, 610}, {2, 3101, 10319}, {2, 4329, 18589}, {2, 9536, 3101}, {2, 18589, 31261}, {2, 20061, 4329}, {2, 26789, 27127}, {2, 26837, 27180}, {4, 40, 11471}, {4, 281, 1826}, {4, 6197, 40}, {4, 7719, 7079}, {5, 8141, 8251}, {6, 1841, 34}, {6, 2182, 2261}, {6, 3197, 19350}, {6, 14571, 2331}, {6, 21767, 1409}, {7, 7291, 7289}, {8, 4198, 1891}, {8, 5279, 5227}, {9, 40, 71}, {9, 281, 7079}, {9, 2270, 2183}, {9, 16547, 169}, {9, 16548, 1766}, {25, 1824, 33}, {25, 2355, 5338}, {25, 11406, 55}, {27, 92, 5307}, {28, 1172, 1474}, {31, 1096, 204}, {31, 2181, 1096}, {33, 5338, 25}, {37, 910, 198}, {41, 2171, 3553}, {46, 1723, 579}, {46, 1731, 1732}, {48, 1953, 1}, {48, 2173, 610}, {48, 18597, 18596}, {51, 3611, 11435}, {55, 1859, 33}, {57, 278, 1435}, {57, 2257, 2260}, {65, 2182, 19350}, {65, 2264, 6}, {75, 92, 20883}, {75, 1760, 63}, {75, 16568, 1760}, {75, 20915, 20884}, {75, 21582, 21406}, {92, 1748, 63}, {169, 1766, 9}, {225, 1452, 208}, {269, 18725, 3942}, {284, 1630, 48}, {326, 18713, 1959}, {346, 10327, 3610}, {573, 15830, 71}, {579, 1723, 1732}, {579, 1731, 1723}, {604, 2170, 3554}, {607, 608, 6}, {607, 1880, 2331}, {607, 4185, 7119}, {608, 1880, 34}, {610, 18594, 2173}, {610, 18595, 18596}, {1195, 1400, 1182}, {1610, 2303, 48}, {1707, 1957, 8765}, {1707, 8769, 33781}, {1726, 1730, 1708}, {1729, 1765, 1741}, {1733, 1747, 920}, {1755, 16567, 63}, {1762, 24310, 63}, {1820, 2180, 920}, {1824, 2355, 25}, {1826, 1839, 4}, {1826, 8756, 281}, {1827, 7071, 33}, {1829, 4185, 34}, {1839, 8756, 1826}, {1841, 14571, 1880}, {1842, 1869, 4}, {1857, 30223, 7008}, {1861, 1890, 4}, {1950, 1951, 577}, {1953, 2173, 48}, {1958, 1959, 326}, {1973, 17442, 1}, {2082, 2285, 6}, {2164, 2178, 1609}, {2182, 2262, 6}, {2207, 16583, 36103}, {2264, 3197, 2261}, {2358, 3209, 207}, {2358, 7154, 7129}, {2362, 16232, 34}, {3209, 7154, 1033}, {3377, 3378, 920}, {4329, 18589, 31158}, {5271, 21376, 63}, {5341, 7297, 6}, {5341, 7300, 5356}, {5356, 7297, 7300}, {5356, 7300, 6}, {6203, 7348, 3069}, {6204, 7347, 3068}, {6212, 6213, 4}, {7713, 7719, 2333}, {8735, 8736, 53}, {9816, 10319, 2}, {10536, 11428, 184}, {11190, 11435, 3611}, {12329, 21867, 28043}, {13438, 13460, 208}, {14206, 21406, 21582}, {16027, 16033, 5}, {16031, 16036, 54}, {16545, 18595, 18597}, {16547, 16548, 9}, {18650, 25935, 26130}, {19555, 19591, 63}, {21160, 30265, 3}, {24683, 26130, 18650}, {26998, 27059, 2}, {31158, 31261, 18589}, {34121, 34125, 25}

leftri

Centers 20-30,

rightri

2- 5, 140, 186, 199, 235, 237, 297, 376- 379,381- 384,
401- 475, 546- 550, 631, 632 (and others) lie on the Euler line.


X(20) = DE LONGCHAMPS POINT

Trilinears    cos A - cos B cos C : cos B - cos C cos A : cos C - cos A cos B
Trilinears    sec A - sec B sec C : sec B - sec C sec A : sec C - sec A sec B
Trilinears    2 cos A - sin B sin C : 2 cos B - sin C sin A : 2 cos C - sin A sin B
Trilinears    (csc A)(tan B + tan C - tan A) : :
Barycentrics    tan B + tan C - tan A : tan C + tan A - tan B : tan A + tan B - tan C
Barycentrics    [-3a4 + 2a2(b2 + c2) + (b2 - c2)2] : :
Barycentrics    S^2 - 2 SB SC : :

Tripolars    Sqrt[a^6-3 a^2 b^4+2 b^6+6 a^2 b^2 c^2-2 b^4 c^2-3 a^2 c^4-2 b^2 c^4+2 c^6] : :
X(20) = 2(r + 2R)*X(1) - (r +4R)*X(7) = 3X(2) - 4X(3)
X(20) = (1 - J) X(1113) + (1 + J) X(1114)

X(20) = 9 X(2) - 8 X(5) = 3 X(4) - 4 X(5) = 3 X(3) - 2 X(5) = 15 X(2) - 16 X(140) = 5 X(4) - 8 X(140) = 5 X(5) - 6 X(140) = 5 X(3) - 4 X(140) = 2 X(10) - 3 X(165) = 8 X(140) - 15 X(376) = 4 X(5) - 9 X(376) = 2 X(3) - 3 X(376) = X(4) - 3 X(376) = 10 X(5) - 9 X(381) = 5 X(4) - 6 X(381) = 5 X(2) - 4 X(381) = 5 X(3) - 3 X(381) = 4 X(140) - 3 X(381) = 5 X(376) - 2 X(381) = 12 X(140) - 5 X(382) = 9 X(381) - 5 X(382) = 9 X(2) - 4 X(382) = 3 X(4) - 2 X(382) = 9 X(376) - 2 X(382) = 3 X(3) - X(382) = 7 X(382) - 12 X(546)

As a point on the Euler line, X(20) has Shinagawa coefficients (1, -2).

Let La be the polar of X(4) wrt the circle centered at A and passing through X(3), and define Lb and Lc cyclically. (Note: X(4) is the perspector of any circle centered at a vertex of ABC.) Let A″ = Lb∩\Lc, and define B″ and C″ cyclically. Triangle A″B″C″ is homothetic to the anticomplementary triangle, and the center of homothety is X(20), which is also the orthocenter of A″B″C″. Also, let La be the line through the intersections of the B- and C-Soddy ellipses, and define Lb and Lc cyclically. Then La,Lb,Lc concur in X(20). Also, let A'B'C' be the cevian triangle of X(253). Let A″ be the orthocenter of AB'C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(20). (Randy Hutson, November 18, 2015)

Let L be the Brocard axis of the intouch triangle. Let La be the Brocard axis of the A-extouch triangle, and define Lb and Lc cyclically. The lines L, La, Lb, Lc concur in X(20). (Randy Hutson, September 14, 2016)

Let A' be the reflection in BC of the A-vertex of the anticevian triangle of X(4), and define B' and C' cyclically. The circumcircles of A'BC, B'CA, and C'AB concur at X(20). (Randy Hutson, December 10, 2016)

Let A'B'C' be the reflection of ABC in X(3) (i.e., the circumcevian triangle of X(3)). Let A″ be the cevapoint of B' and C', and define B″ and C″ cyclically. The lines AA″, BB″, and CC″ concur in X(20). (Randy Hutson, December 10, 2016)

Let A'B'C' be the hexyl triangle. Let A″ be the cevapoint of B' and C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(20).

Let A'B'C' be the half-altitude triangle. Let A″ be the trilinear pole of line B'C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(20).

Let A'B'C' be the hexyl triangle and A″B″C″ be the side-triangle of ABC and hexyl triangle. Let A* be the {B',C'}-harmonic conjugate of A″, and define B*, C* cyclically. The lines AA*, BB*, CC* concur in X(20). (Randy Hutson, June 27, 2018)

If you have The Geometer's Sketchpad, you can view De Longchamps point.
If you have GeoGebra, you can view De Longchamps point.

Let OA be the circle centered at the A-vertex of the Ara triangle and passing through A; define OB and OC cyclically. X(20) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

Let OA be the circle centered at the A-vertex of the Johnson triangle and passing through A; define OB, OC cyclically. X(20) is the radical center of OA and OB, OC. (Randy Hutson, August 30, 2020)

Consider the chain of (blue) right triangles T1, T2, T3, ... in this figure: "Pythagorean snail". The points X(20)-of-T1, X(20)-of-T2, X(20-of-T3, ... all lie on a single circle. The same sort of concyclicity hold for X(i) for these i: 20, 22, 110, 175, 253, 280, 347, 858, 401, 925* 1114, 1294, 1295, 1297, 1303* 1305*, 1370*,... where (*) menas that the point is on theline at infinity. The lines X(3)X(652) of the triangles converge to the center of the snail; it appears that, with repect to each of the triangles, X(625) lies on the hypotenuse. (Dan Reznik, October 30, 2021)

Let Ea be the ellipse through A having foci B and C, and define Eb and Ec cyclically. These ellipses meet in 6 points. The lines through pairs of opposite intersections concur on X(20). Figure. (Dan Reznik, December 12, 2021)

In the plane of a triangle ABC, let
A' = A-excircle of ABFC, and define B' and C' cyclically; A1 = orthopole of BC wrt B'C', and define B1 and C1 cyclically; Then X(2) = A1B1C1-to-ABC orthology center, and X(3062) = ABC-to-A1B1C1 orthology center. (Ivan Pavlov, August 21, 2022)

In the plane of a triangle ABC, let E(A) be the ellipse whose foci are B and C, and which passes through A, and define E(b) and E(C) cyclically. Then X(20) = radical center of E(A), E(B), E(C). See X(20) Ellipses. (Benjamin Lee Warren, January 28, 2024)

X(20) lies on the the following curves: Q046, Q063, Q070, Q073, Q115, K004, K007, K032, K041, K047, K071, K077, K080, K096, K099, K122, K169, K182, K236, K268, K270, K313, K329, K344, K364, K401, K425, K426, K443, K449, K462, K499, K522, K566, K609, K617, K648, K649, K650, K651, K652, K706, K753, K763, K778, K809, K814, K824, K825, K827, K850, K894. Euler-Gergonne-Soddy circle, GEOS circle, Steiner/Wallace rectangular hyperbola, anticomplement of Kiepert hyperbola, anticomplement of Feuerbach hyperbola, anticomplement of Jerabek hyperbola, and these lines: {1,7}, {2,3}, {6,6459}, {8,40}, {9,10429}, {10,165}, {11,5204}, {12,5217}, {13,5238}, {14,5237}, {15,3412}, {16,3411}, {17,5352}, {18,5351}, {32,2549}, {33,1038}, {34,1040}, {35,1478}, {36,1479}, {39,7737}, {46,10572}, {51,9729}, {52,5890}, {54,4846}, {55,388}, {56,497}, {57,938}, {58,387}, {61,10653}, {62,10654}, {64,69}, {65,3474}, {68,74}, {72,144}, {76,3424}, {78,329}, {81,5706}, {97,1217}, {98,148}, {99,147}, {100,153}, {101,152}, {103,150}, {104,149}, {107,3184}, {109,151}, {110,146}, {112,10316}, {113,10721}, {114,7912}, {115,5206}, {116,10725}, {117,10726}, {118,10727}, {119,10728}, {120,10729}, {121,10730}, {122,10152}, {123,10731}, {124,10732}, {125,10733}, {126,10734}, {127,10735}, {142,5436}, {145,517}, {154,2883}, {155,323}, {159,2139}, {172,9598}, {182,7787}, {184,9545}, {185,193}, {187,3767}, {190,1265}, {192,9962}, {200,5815}, {212,1935}, {216,3087}, {220,5781}, {222,3562}, {224,4511}, {226,3601}, {227,9371}, {230,5023}, {243,1118}, {254,1300}, {262,5395}, {265,11270}, {284,5746}, {298,5868}, {299,5869}, {316,7763}, {325,6337}, {333,5786}, {343,6247}, {345,7270}, {346,1766}, {348,4872}, {355,3579}, {371,1587}, {372,1588}, {385,6392}, {386,9535}, {389,3060}, {391,573}, {392,9856}, {393,577}, {394,1032}, {395,5339}, {396,5340}, {399,6188}, {476,2693}, {477,10420}, {484,10573}, {485,1131}, {486,1132}, {487,638}, {488,637}, {495,9655}, {496,9668}, {498,3585}, {499,3583}, {518,3189}, {519,5493}, {524,11148}, {527,11523}, {529,3913}, {535,5537}, {541,9143}, {542,8591}, {543,7751}, {551,11522}, {553,11518}, {568,10263}, {574,2548}, {576,5032}, {578,5012}, {579,5802}, {590,6409}, {597,10541}, {601,3072}, {602,3073}, {603,1936}, {610,8804}, {615,6410}, {616,633}, {617,634}, {620,7825}, {621,627}, {622,628}, {648,9530}, {650,8142}, {651,7078}, {653,3176}, {664,7973}, {671,11623}, {691,2697}, {754,7758}, {901,2734}, {908,4855}, {910,6554}, {936,1750}, {942,3488}, {946,3576}, {952,3621}, {956,5082}, {958,2550}, {960,5698}, {986,11031}, {999,1058}, {1001,8273}, {1007,7773}, {1056,3295}, {1060,6198}, {1062,1870}, {1074,1838}, {1075,5667}, {1076,1785}, {1078,7616}, {1104,4000}, {1124,9660}, {1125,1699}, {1141,11671}, {1147,1614}, {1151,3068}, {1152,3069}, {1154,11271}, {1155,1788}, {1160,10784}, {1161,10783}, {1176,10548}, {1181,1993}, {1204,1899}, {1210,3586}, {1212,5819}, {1216,4549}, {1249,3172}, {1290,2694}, {1293,2370}, {1296,2373}, {1320,10305}, {1327,3590}, {1328,3591}, {1330,3430}, {1335,9647}, {1340,2542}, {1341,2543}, {1342,2546}, {1343,2547}, {1351,7839}, {1352,2896}, {1376,2551}, {1384,5305}, {1385,3622}, {1394,5930}, {1420,9580}, {1440,1804}, {1445,5809}, {1453,5222}, {1482,3623}, {1483,8148}, {1499,6563}, {1511,7728}, {1519,4881}, {1568,11202}, {1578,3092}, {1579,3093}, {1610,1633}, {1619,9914}, {1621,11496}, {1632,2892}, {1689,2545}, {1690,2544}, {1697,10106}, {1698,10164}, {1706,5795}, {1729,5011}, {1743,10443}, {1764,10449}, {1768,9803}, {1834,4252}, {1836,2646}, {1853,6696}, {1857,1940}, {1891,10319}, {1902,7718}, {1914,9597}, {1992,8550}, {1994,7592}, {2077,5080}, {2128,3685}, {2130,2131}, {2287,5776}, {2420,6794}, {2456,10131}, {2482,7888}, {2781,6293}, {2782,5984}, {2797,9409}, {2800,6224}, {2801,5904}, {2822,2939}, {2823,4552}, {2888,3357}, {2893,10432}, {2894,2975}, {2899,5205}, {2917,2935}, {2944,3923}, {2947,3682}, {2979,5562}, {3047,5504}, {3053,5254}, {3054,5585}, {3057,3476}, {3058,3304}, {3062,5785}, {3095,7709}, {3180,5865}, {3181,5864}, {3182,3347}, {3183,3348}, {3218,5709}, {3219,3587}, {3241,5882}, {3244,11531}, {3278,3608}, {3303,5434}, {3311,7581}, {3312,7582}, {3313,5596}, {3316,6451}, {3317,6452}, {3333,10580}, {3334,3609}, {3339,6738}, {3353,3354}, {3355,3637}, {3359,5554}, {3361,11019}, {3398,10788}, {3419,3916}, {3421,5687}, {3431,3521}, {3452,5438}, {3472,3473}, {3475,10404}, {3564,7893}, {3567,5446}, {3598,3673}, {3618,5085}, {3619,10516}, {3624,3817}, {3634,7989}, {3635,11224}, {3648,5693}, {3655,10222}, {3666,5716}, {3667,5592}, {3697,9947}, {3734,7800}, {3788,7842}, {3796,11425}, {3812,10178}, {3813,11194}, {3849,7759}, {3869,6001}, {3870,6769}, {3871,10306}, {3872,9874}, {3876,5777}, {3911,5704}, {3917,5907}, {3933,10513}, {3935,5534}, {3972,7803}, {4257,5292}, {4385,7172}, {4640,5794}, {4652,5175}, {4678,5690}, {4848,5128}, {4857,10072}, {5007,7739}, {5013,7736}, {5044,5927}, {5126,11373}, {5174,6350}, {5208,10441}, {5223,6743}, {5226,9612}, {5247,9441}, {5250,9800}, {5270,10056}, {5303,11680}, {5316,9842}, {5318,11480}, {5321,11481}, {5328,6700}, {5418,6564}, {5420,6565}, {5422,10982}, {5432,10588}, {5433,10589}, {5439,5806}, {5440,5658}, {5441,5902}, {5447,5891}, {5450,10527}, {5462,9781}, {5550,8227}, {5587,6684}, {5601,9834}, {5602,9835}, {5640,10110}, {5654,7712}, {5663,6101}, {5714,11374}, {5720,5811}, {5730,10609}, {5749,10445}, {5758,5905}, {5766,8545}, {5841,10528}, {5853,6762}, {5876,10627}, {5893,10192}, {6102,6243}, {6146,6515}, {6193,6241}, {6197,9536}, {6214,10518}, {6215,10517}, {6221,7583}, {6249,9751}, {6264,9802}, {6326,9809}, {6390,7776}, {6398,7584}, {6449,8981}, {6455,8976}, {6462,6465}, {6463,6466}, {6526,11589}, {6680,7872}, {6737,7992}, {6744,10980}, {6765,7994}, {6766,9797}, {6767,10386}, {7074,9370}, {7596,10885}, {7618,7775}, {7620,8182}, {7694,7752}, {7730,11802}, {7731,11562}, {7749,8588}, {7755,11648}, {7761,7795}, {7768,11057}, {7774,7783}, {7784,7789}, {7785,9737}, {7797,9753}, {7799,7860}, {7801,7873}, {7818,7863}, {7820,7935}, {7832,7910}, {7835,7911}, {7836,7898}, {7864,9748}, {7885,7891}, {7921,10983}, {7971,11682}, {7998,11439}, {8069,10629}, {8081,9793}, {8082,9795}, {8111,9783}, {8112,9787}, {8117,8118}, {8119,8124}, {8120,8123}, {8164,9654}, {8234,9789}, {8235,9791}, {8726,9776}, {8861,9474}, {8983,9615}, {9529,9979}, {9786,11433}, {9927,11468}, {9957,11035}, {9993,10583}, {9996,10357}, {10246,10595}, {10267,10532}, {10269,10531}, {10282,11449}, {10359,10796}, {10453,10476}, {10470,10478}, {10525,10785}, {10526,10786}, {10543,11246}, {10584,10893}, {10585,10894}, {10601,11745}, {10679,10805}, {10680,10806}, {11180,11645}, {11451,11695}, {11470,11511}, {11472,11487}, {11473,11513}, {11474,11514}, {11475,11515}, {11476,11516}

X(20) = midpoint of X(i) and X(j) for these {i,j}: {3, 1657}, {4, 3529}, {376, 11001}, {944, 6361}, {1498, 5925}, {3146, 5059}, {3869, 9961}, {6241, 11412}, {10575, 10625}
X(20) = reflection of X(i) in X(j) for these (i,j): (1, 4297), (2, 376), (3, 550), (4, 3), (5, 548), (7, 5732), (8, 40), (23, 10295), (64, 5894), (65, 9943), (68, 7689), (69, 1350), (76, 5188), (107, 3184), (144, 5759), (145, 944), (146, 110), (147, 99), (148, 98), (149, 104), (150, 103), (151, 109), (152, 101), (153, 100), (176, 8984), (193, 6776), (194, 11257), (329, 6282), (355, 3579), (376, 3534), (381, 8703), (382, 5), (616, 5473), (617, 5474), (650, 8142), (938, 9841), (962, 1), (1330, 3430), (1352, 3098), (1375, 8153), (2475, 3651), (2550, 11495), (2888, 7691), (3091, 3522), (3146, 4), (3153, 2071), (3421, 6244), (3434, 3428), (3436, 10310), (3448, 74), (3529, 1657), (3543, 2), (3627, 140), (3830, 549), (3832, 3528), (3839, 10304), (3853, 3530), (3868, 1071), (4846, 8717), (5059, 3529), (5073, 3627), (5080, 2077), (5189, 7464), (5691, 10), (5768, 7171), (5876, 10627), (5878, 6759), (5881, 11362), (5889, 185), (5895, 2883), (5921, 69), (5984, 9862), (6223, 1490), (6225, 1498), (6241, 10575), (6243, 6102), (6256, 6796), (6515, 10605), (6655, 7470), (6764, 6762), (6839, 7411), (6840, 6909), (6895, 6906), (6925, 7580), (7379, 4229), (7391, 378), (7408, 3537), (7620, 8182), (7710, 8719), (7728, 1511), (7731, 11562), (7758, 7781), (7982, 5882), (7991, 5493), (8148, 1483), (9589, 4301), (9797, 9845), (9799, 84), (9802, 6264), (9803, 1768), (9809, 6326), (9812, 3576), (9863, 7750), (9965, 2096), (10152, 122), (10296, 858), (10431, 1012), (10446, 991), (10721, 113), (10722, 114), (10723, 115), (10724, 11), (10725, 116), (10726, 117), (10727, 118), (10728, 119), (10729, 120), (10730, 121), (10731, 123), (10732, 124), (10733, 125), (10734, 126), (10735, 127), (10736, 1313), (10737, 1312), (11185, 8722), (11381, 5907), (11412, 10625), (11455, 5891), (11477, 8550), (11531, 3244), (11541, 5073), (11671, 1141)
X(20) = isogonal conjugate of X(64)
X(20) = isotomic conjugate of X(253)
X(20) = cyclocevian conjugate of X(1032)
X(20) = complement of X(3146)
X(20) = anticomplement of X(4)
X(20) = anticomplementary conjugate of X(4)
X(20) = X(i)-Ceva conjugate of X(j) for these (i,j): (69,2), (489,487), (490,488), (801,6), (1043,1), (1350,6194), (1503,147), (1975,194), (5921,9742), (7750,2896), (8822,63)
X(20) = X(i)-cross conjugate of X(j) for these (i,j): (64,2131), (122,8057), (154,1249), (1249,2), (1498,6616), (3183,2060), (3198,610), (5895,4), (5930,1895), (6525,3344)
X(20) = crosspoint of X(1) and X(7038)
X(20) = crosssum of X(i) and X(j) for these (i,j): {1,1044}, {512,3269}, {649,3270}
X(20) = crossdifference of every pair of points on line X(647)X(657)
X(20) = trilinear pole of X(6587)X(8057)
X(20) = circumcircle-inverse of X(2071)
X(20) = orthocentroidal-circle-inverse of X(3091)
X(20) = Steiner-circle-inverse of X(858)
X(20) = polar-circle-inverse of X(10151)
X(20) = orthoptic-circle-of-Steiner-inellipse-inverse of X(5159)
X(20) = orthoptic-circle-of-Steiner-circumelipse-inverse of X(858)
X(20) = Grebe-circle-inverse of X(37944)
X(20) = Johnson-circle-inverse of anticomplement of X(37968)
X(20) = anticomplement-of-circumcircle-inverse of X(3153)
X(20) = X(i)-aleph conjugate of X(j) for these (i,j): (8,191), (9,1045), (21,3216), (29,1714), (188,1046), (333,2), (556,1762), (645,3882), (1043,20), (3699,4427), (4182,846), (6731,2938)
X(20) = X(i)-beth conjugate of X(j) for these (i,j): (8,5691), (20,1394), (21,4306), (643,1259), (664,20), (1043,280)
X(20) = X(i)-gimel conjugate of X(j) for these (i,j): (21,6848), (1792,20), (3900,20), (4397,20), (7253,20)
X(20) = X(i)-he conjugate of X(j) for these (i,j): (645,20), (799,20), (7256,20), (7258,20)
X(20) = X(i)-zayin conjugate of X(j) for these (i,j): (1,64), (200,7580), (1043,20), (2287,573), (4397,3667), (6737,40)
X(20) = antigonal conjugate of X(10152)
X(20) = syngonal conjugate of X(3184)
X(20) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (1,4), (3,8), (4,5906), (6,5905), (19,6515), (31,193), (47,6193), (48,2), (55,5942), (58,3868), (63,69), (69,6327), (71,2895), (72,1330), (73,2475), (75,11442), (77,3434), (78,3436), (82,3060), (91,68), (92,317), (101,4391), (102,5081), (109,521), (110,7253), (162,520), (163,525), (184,192), (212,144), (219,329), (222,7), (228,1654), (255,20), (268,189), (283,3869), (284,92), (293,511), (295,4645), (304,315), (326,1370), (394,4329), (577,6360), (603,145), (656,3448), (662,850), (810,148), (905,150), (906,514), (921,11411), (922,7665), (947,318), (951,5174), (1069,11415), (1101,110), (1110,3732), (1214,2893), (1262,4566), (1331,513), (1333,3187), (1433,962), (1437,1), (1459,149), (1496,11469), (1790,75), (1794,72), (1795,517), (1796,319), (1797,320), (1803,85), (1807,5080), (1813,693), (1815,4872), (1822,2574), (1823,2575), (1923,10340), (1964,8878), (1973,6392), (2148,1993), (2149,651), (2159,3580), (2164,2994), (2167,264), (2168,5392), (2169,3), (2190,5889), (2193,63), (2196,6542), (2200,1655), (2216,52), (2349,340), (2359,321), (2360,1895), (2576,2592), (2577,2593), (2964,11271), (3916,2891), (3990,3151), (4020,2896), (4303,2894), (4558,7192), (4575,523), (4587,4462), (4592,512), (6507,6527), (7011,5932), (7015,4388), (7078,6223), (7099,4452), (7116,6646), (7125,347), (7177,6604), (9247,194), (9255,1899)
X(20) = X(3532)-complementary conjugate of X(10)
X(20) = X(i)-vertex conjugate of X(j) for these (i,j): {3,3346}, {4,5879}, {523,2071}
X(20) = X(4)-of-anticomplementary triangle
X(20) = X(52)-of-hexyl-triangle
X(20) = reflection of X(10296) in the De Longchamps line
X(20) = perspector of anticomplementary triangle and polar triangle of de Longchamps circle
X(20) = isogonal conjugate of X(4) wrt anticevian triangle of X(4) (or 'anticevian-isogonal conjugate of X(4)')
X(20) = perspector of ABC and pedal triangle of X(1498)
X(20) = exsimilicenter of circumcircle and 1st Steiner circle (the insimilicenter is X(631))
X(20) = X(4)-of-circumcevian-triangle-of-X(30)
X(20) = anticomplementary isotomic conjugate of X(193)
X(20) = excentral isogonal conjugate of X(1046)
X(20) = excentral isotomic conjugate of X(1045)
X(20) = cevapoint of X(i) and X(j) for these {i,j}: {1,3182}, {3,1498}, {4,3183}, {6,1661}, {30,3184}, {40,1490}, {64,2130}, {84,3353}, {122,8057}, {577,1660}, {610,7070}, {1249,3079}, {3198,8804}, {3345,3472}, {3346,3355}
X(20) = radical center of power circles
X(20) = radical center of circles centered at the vertices of ABC with radius equal to opposite side
X(20) = intersection of tangents to conic {X(4),X(13),X(14),X(15),X(16)} at X(15) and X(16)
X(20) = trilinear pole wrt anticomplementary triangle of de Longchamps line
X(20) = trilinear pole of polar of X(459) wrt polar circle, which is also the perspectrix of ABC and the half-altitude triangle
X(20) = pole wrt polar circle of trilinear polar of X(459)
X(20) = isoconjugate of X(j) and X(j) for these (i,j): {1,64}, {2,2155}, {6,2184}, {19,1073}, {31,253}, {48,459}, {55,8809}, {255,6526}, {656,1301}, {1036,10375}, {1402,5931}, {2190,8798}
X(20) = circumcevian isogonal conjugate of X(3)
X(20) = perspector of ABC and circumcircle antipode of circumanticevian triangle of X(4)
X(20) = X(98)-of-6th-Brocard-triangle
X(20) = perspector of hexyl triangle and cross-triangle of ABC and hexyl triangle
X(20) = Thomson isogonal conjugate of X(3167)
X(20) = Lucas isogonal conjugate of X(2)
X(20) = inner-Conway-to-Conway similarity image of X(8)
X(20) = cyclocevian conjugate of X(2) wrt anticevian triangle of X(2)
X(20) = trilinear product of vertices of X(4)-anti-altimedial triangle
X(20) = homothetic center of X(20)-altimedial and X(2)-anti-altimedial triangles
X(20) = inverse-in-orthoptic-circle-of-Steiner-inellipse of X(5159)
X(20) = inverse-in-circumconic-centered-at-X(4) of X(1559)
X(20) = anticevian isogonal conjugate of X(4)
X(20) = X(5562)-of-excentral-triangle
X(20) = X(74)-of-X(3)-Fuhrmann-triangle
X(20) = Ehrmann-mid-to-Johnson similarity image of X(3)
X(20) = perspector of hexyl triangle and anticevian triangle of X(63)
X(20) = perspector of excentral triangle and tangential triangle wrt hexyl triangle of the excentral-hexyl ellipse
X(20) = perspector of excentral triangle and extraversion triangle of X(7)
X(20) = homothetic center of ABC and the reflection in X(3) of the pedal triangle of X(3) (medial triangle)
X(20) = homothetic center of ABC and the reflection in X(4) of the antipedal triangle of X(4) (anticomplementary triangle)
X(20) = orthic-isogonal conjugate of X(32605)
X(20) = pole of Brocard axis wrt conic {X(4),X(13),X(14),X(15),X(16)}}
X(20) = QA-P5 (Isotomic Center) of the incenter-excenters quadrangle (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/26-qa-p5.html)
X(20) = barycentric product X(i)*X(j) for these {i,j}: {63,1895}, {69,1249}, {75,610}, {76,154}, {85,7070}, {86,8804}, {99,6587}, {204,304}, {274,3198}, {305,3172}, {312,1394}, {333,5930}, {648,8057}, {801,2883}, {1032,6616}, {1097,2184}, {3213,3718}, {3344,6527}, {3926,6525}, {7156,7182}, {10152,11064}
X(20) = barycentric quotient X(i)/X(j) for these (i,j): (1,2184), (2,253), (3,1073), (4,459), (6,64), (31,2155), (57,8809), (112,1301), (154,6), (204,19), (216,8798), (333,5931), (393,6526), (610,1), (1249,4), (1394,57), (1498,3343), (1562,125), (1895,92), (2285,10375), (3079,1249), (3172,25), (3198,37), (3213,34), (3284,11589), (3344,3346), (5930,226), (6525,393), (6587,523), (7070,9), (7156,33), (8057,525), (8804,10)
X(20) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,7,11036), (1,1044,1042), (1,1742,4300), (1,1770,4295), (1,3100,9538), (1,4292,7), (1,4293,3600), (1,4294,390), (1,4297,5731), (1,4298,11037), (1,4299,4293), (1,4301,5734), (1,4302,4294), (1,4304,4313), (1,4311,4308), (1,4312,3671), (1,4316,4299), (1,4324,4302), (1,4325,4317), (1,4330,4309), (1,4333,1770), (1,4340,3945), (1,4355,5542), (1,5732,10884), (1,9589,4301), (1,10624,9785), (2,3,3523), (2,4,3091), (2,5,7486), (2,22,10565), (2,23,4232), (2,376,10304), (2,377,4208), (2,452,5129), (2,1370,7396), (2,2475,5177), (2,3091,5056), (2,3146,4), (2,3522,3), (2,3523,10303), (2,3543,3839), (2,3832,5), (2,4190,6904), (2,5046,6919), (2,5059,3146), (2,5068,3090), (2,6837,6884), (2,6838,6960), (2,6839,6993), (2,6847,6888), (2,6848,6979), (2,6872,452), (2,6890,6972), (2,6987,6992), (2,6995,7398), (2,7391,7378), (2,7408,6997), (2,7409,5133), (2,7500,6995), (2,11106,405), (3,4,2), (3,5,631), (3,22,7488), (3,26,186), (3,140,3524), (3,376,3522), (3,381,140), (3,382,5), (3,405,6986), (3,546,3525), (3,548,3528), (3,549,10299), (3,550,376), (3,1012,21), (3,1532,6921), (3,1593,7503), (3,1597,7395), (3,1656,549), (3,1885,6816), (3,2937,1658), (3,3079,2060), (3,3091,10303), (3,3146,3091), (3,3149,404), (3,3522,10304), (3,3526,3530), (3,3529,3146), (3,3534,550), (3,3543,5056), (3,3560,1006), (3,3575,6815), (3,3627,3090), (3,3830,1656), (3,3843,3526), (3,3845,3533), (3,3851,5054), (3,3853,5067), (3,5059,3543), (3,5073,381), (3,5076,3628), (3,6638,417), (3,6756,6803), (3,6823,7494), (3,6827,6926), (3,6831,6910), (3,6836,6890), (3,6840,6972), (3,6842,6954), (3,6850,6908), (3,6851,6847), (3,6868,6987), (3,6872,6992), (3,6882,6961), (3,6895,6888), (3,6905,4188), (3,6906,4189), (3,6907,6988), (3,6911,6940), (3,6914,6875), (3,6917,6889), (3,6923,6825), (3,6925,6838), (3,6928,6891), (3,6929,6967), (3,6934,4190), (3,6938,6872), (3,6985,6905), (3,7387,24), (3,7395,7485), (3,7488,10298), (3,7491,6827), (3,7517,6644), (3,7553,7401), (3,7580,411), (3,8727,6857), (3,9122,1817), (3,9818,7509), (3,9909,3515), (3,10323,6636), (3,10431,6837), (3,11001,5059), (3,11413,2071), (3,11414,22), (3,11479,7484), (3,11676,3552), (4,5,3832), (4,21,6837), (4,24,3089), (4,140,5068), (4,186,3542), (4,376,3), (4,377,6839), (4,378,3088), (4,404,6953), (4,411,6838), (4,443,6835), (4,550,3522), (4,631,5), (4,1006,6846), (4,1595,7409), (4,1656,3854), (4,1657,5059), (4,3079,6616), (4,3088,7378), (4,3090,381), (4,3091,3839), (4,3146,3543), (4,3147,403), (4,3520,3541), (4,3522,3523), (4,3523,5056), (4,3524,3090), (4,3525,3545), (4,3528,631), (4,3533,3851), (4,3537,6803), (4,3538,6804), (4,3542,6623), (4,3545,546), (4,3651,6908), (4,3855,3843), (4,4188,6979), (4,4189,6888), (4,5067,3855), (4,5084,6957), (4,6353,235), (4,6616,1559), (4,6756,7408), (4,6803,6997), (4,6815,7544), (4,6824,6870), (4,6825,6871), (4,6826,6894), (4,6827,5046), (4,6833,6844), (4,6836,6840), (4,6850,2475), (4,6851,6895), (4,6852,6866), (4,6853,6867), (4,6854,6849), (4,6856,7548), (4,6857,6828), (4,6865,2478), (4,6868,6872), (4,6875,6824), (4,6876,6825), (4,6878,6990), (4,6880,6941), (4,6889,6843), (4,6891,5187), (4,6897,6826), (4,6899,6827), (4,6902,6929), (4,6903,6928), (4,6905,6848), (4,6906,6847), (4,6908,5177), (4,6909,6890), (4,6916,377), (4,6926,6919), (4,6927,1532), (4,6935,6831), (4,6940,6964), (4,6942,6834), (4,6947,6893), (4,6948,4190), (4,6949,6968), (4,6950,6833), (4,6951,6917), (4,6954,5141), (4,6961,5154), (4,6977,6830), (4,6986,6886), (4,6987,452), (4,6988,2476), (4,7386,6816), (4,7401,7394), (4,7412,4194), (4,7487,6995), (4,7512,3547), (4,7714,5198), (4,8889,7507), (4,10299,1656), (4,10304,10303), (4,10323,7400), (4,10996,6815), (4,11001,3529), (4,11111,6912), (5,140,5070), (5,382,4), (5,548,3), (5,550,548), (5,631,2), (5,3526,5067), (5,3530,3526), (5,3627,3861), (5,3832,3091), (5,3843,3855), (5,3845,3859), (5,3853,3843), (5,3859,3851), (5,3861,381), (5,5070,3090), (5,7486,5056), (5,9715,7493), (7,3188,279), (7,4313,1), (8,9778,40), (8,10430,9799), (8,10538,280), (11,5204,7288), (12,5217,5218), (21,377,2), (21,6839,6884), (21,7411,3), (22,858,7493), (22,1370,2), (22,2071,10298), (22,11413,3), (24,3089,4232), (24,6643,2), (24,7387,23), (25,1885,4), (25,7386,2), (25,7667,7386), (27,464,2), (32,2549,5286), (32,5286,5304), (32,7756,2549), (32,7765,5319), (35,1478,3085), (35,3085,5281), (35,10483,1478), (36,1479,3086), (36,3086,5265), (40,84,63), (40,3101,9537), (40,5881,11362), (55,7354,388), (56,6284,497), (57,950,938), (65,5918,9943), (68,11457,3448), (69,6527,253), (76,5188,6194), (98,5171,7793), (99,315,3926), (99,7802,315), (100,3436,7080), (140,381,3090), (140,546,10109), (140,3090,2), (140,3627,381), (140,3861,5), (140,5073,4), (140,8703,3), (145,9965,3868), (154,5895,2883), (165,5691,10), (175,176,347), (187,7748,3767), (226,3601,5703), (235,3515,6353), (315,8721,147), (316,7782,7763), (355,3579,5657), (355,5657,3617), (371,1587,7585), (371,6560,1587), (372,1588,7586), (372,6561,1588), (376,631,3528), (376,1370,2071), (376,1657,3146), (376,3146,3523), (376,3524,8703), (376,3528,548), (376,3529,4), (376,5059,3091), (376,6240,7400), (376,6851,4189), (376,6869,4190), (376,6916,7411), (376,6938,6987), (376,11541,140), (377,1012,6837), (377,6837,6993), (377,10431,4), (378,6240,4), (378,10323,3), (381,3090,5068), (381,3524,2), (381,3627,4), (381,5068,3091), (381,5070,5), (381,5073,3627), (381,8703,3524), (381,10109,3545), (382,548,631), (382,550,3528), (382,631,3832), (382,3526,3843), (382,3528,2), (382,3530,3855), (382,3843,3853), (382,5070,3861), (384,7791,2), (384,7833,7791), (390,3600,1), (394,1498,11441), (404,2478,2), (404,11114,2478), (405,443,2), (405,6835,6886), (405,11111,11106), (405,11112,443), (411,6836,2), (411,6840,6960), (411,6909,3), (411,6943,6962), (427,7494,2), (428,7484,7392), (428,11403,4), (440,7490,2), (442,6857,2), (442,8727,6828), (443,6912,6886), (443,11111,405), (452,6904,2), (474,5084,2), (474,11113,5084), (485,6200,9540), (485,9540,8972), (487,638,1271), (488,637,1270), (489,490,69), (498,3585,10590), (499,3583,10591), (546,549,1656), (546,1656,3545), (546,3545,3854), (546,3830,4), (546,10299,2), (547,3858,5072), (548,3528,3522), (548,3853,3530), (549,1656,3525), (549,3545,2), (549,3830,3545), (550,1657,4), (550,3146,10304), (550,3529,2), (550,3627,8703), (550,5059,3523), (550,6240,6636), (550,11001,3146), (574,7747,2548), (578,10984,5012), (631,3528,3), (631,3832,7486), (631,3855,5067), (631,5067,3526), (632,3850,5055), (858,7493,2), (858,10296,3153), (944,2096,1071), (946,3576,3616), (958,11495,5584), (962,5731,1), (962,5734,4301), (962,10884,11036), (991,3332,3945), (1006,6826,2), (1011,6817,2), (1012,6916,2), (1042,3000,1044), (1092,6759,110), (1113,1114,2071), (1131,8972,485), (1147,1614,9544), (1151,3070,3068), (1152,3071,3069), (1155,1837,1788), (1352,3098,10519), (1352,10519,3620), (1368,6353,2), (1368,9909,6353), (1370,7493,858), (1385,5603,3622), (1478,3085,5261), (1479,3086,5274), (1482,7967,3623), (1490,6282,78), (1532,6922,4193), (1583,6805,2), (1584,6806,2), (1585,1589,2), (1586,1590,2), (1587,9541,371), (1593,3575,4), (1593,7503,7527), (1593,10996,2), (1595,7399,5133), (1597,3537,2), (1597,6756,4), (1598,3538,2), (1610,1633,3556), (1656,3525,2), (1656,3830,546), (1656,10109,3090), (1657,3534,3), (1657,8703,11541), (1658,2937,7556), (1699,7987,1125), (1764,10454,10449), (1764,10463,10461), (1836,2646,3485), (1853,8567,6696), (1975,7750,69), (2041,2042,4), (2043,2044,376), (2045,2046,3533), (2060,3146,1559), (2071,7488,3), (2071,10296,858), (2077,6256,5552), (2475,4189,2), (2475,6847,3091), (2475,6895,4), (2475,6906,6888), (2476,6910,2), (2478,3149,6953), (2479,2480,441), (2549,5319,7765), (3053,5254,7735), (3060,10574,389), (3088,6636,3523), (3088,7400,2), (3090,3524,140), (3090,3529,11541), (3090,11541,3627), (3091,3523,2), (3091,3543,4), (3091,4208,6993), (3091,6992,5129), (3091,7486,5), (3091,10304,3523), (3098,9873,2896), (3100,4296,1), (3146,3522,2), (3146,3523,3839), (3146,3528,7486), (3146,3854,3830), (3146,7411,4208), (3146,10304,5056), (3146,11413,7396), (3147,11585,2), (3149,6865,2), (3151,7560,2), (3152,7538,2), (3153,10298,2), (3474,3486,65), (3520,7512,3), (3522,3529,3543), (3522,3543,10303), (3522,3854,10299), (3522,5059,4), (3523,3543,3091), (3523,10304,3), (3524,3529,5073), (3524,3627,5068), (3524,11541,4), (3525,3545,1656), (3525,3830,3854), (3525,10299,549), (3526,3530,631), (3526,3843,5), (3526,3853,3855), (3526,5067,2), (3528,3529,382), (3528,3832,3523), (3528,3855,3530), (3529,3534,3522), (3529,11001,1657), (3530,3843,5067), (3530,3853,5), (3530,3855,2), (3533,3628,2), (3533,5071,3628), (3534,11001,2), (3541,3547,2), (3542,3546,2), (3543,7396,3153), (3543,10304,2), (3545,3854,3091), (3545,10299,3525), (3548,7505,2), (3552,6655,2), (3560,6826,6846), (3560,6897,2), (3567,5446,11002), (3575,7503,7544), (3583,7280,499), (3585,5010,498), (3601,9579,226), (3616,9812,946), (3627,8703,140), (3627,11541,3146), (3628,3845,3851), (3628,3851,5071), (3628,3859,5), (3628,5054,3533), (3651,6851,2), (3651,6906,3), (3734,7830,7800), (3839,5056,3091), (3839,10303,5056), (3843,3853,4), (3843,3855,3832), (3845,5054,5071), (3845,5076,4), (3851,5054,3628), (3851,5076,3845), (3853,5067,3832), (3855,5067,5), (3868,11220,1071), (3911,9581,5704), (3917,5907,11444), (3917,11381,5907), (3972,7847,7803), (4188,5046,2), (4188,6926,3523), (4189,6895,6847), (4189,6908,3523), (4190,6868,6992), (4190,6872,2), (4190,6987,3523), (4191,6818,2), (4193,6921,2), (4195,4201,2), (4197,10883,5), (4208,6837,5056), (4292,4297,10884), (4292,4304,1), (4292,4313,11036), (4293,4294,1), (4293,4302,390), (4294,4299,3600), (4295,4305,1), (4298,4314,1), (4299,4302,1), (4299,4309,4317), (4299,4317,4325), (4299,4324,4294), (4301,9589,962), (4302,4309,4330), (4302,4316,4293), (4302,4317,4309), (4304,5732,5731), (4308,9785,1), (4309,4317,1), (4309,4330,4294), (4311,10624,1), (4316,4324,1), (4316,4330,4325), (4317,4325,4293), (4319,4320,1), (4324,4325,4330), (4325,4330,1), (4345,6049,1), (4348,7221,1), (4351,4354,1), (4652,6734,5744), (5004,5005,25), (5013,7745,7736), (5046,6848,3091), (5046,6905,6979), (5054,5071,2), (5054,5076,3851), (5056,10303,2), (5059,5068,5073), (5068,11541,3543), (5073,8703,3090), (5077,7866,8357), (5085,5480,3618), (5128,5727,4848), (5175,5744,6734), (5177,6888,5056), (5189,7492,2), (5218,5229,12), (5225,7288,11), (5261,5281,3085), (5265,5274,3086), (5318,11480,11488), (5319,7765,5286), (5321,11481,11489), (5432,10895,10588), (5433,10896,10589), (5446,9730,3567), (5447,5891,7999), (5550,9779,8227), (5550,10248,9779), (5584,6253,2550), (5587,6684,9780), (5806,11227,5439), (5878,6759,5656), (5881,11362,8), (5882,7982,3241), (6143,6639,2), (6143,7552,6639), (6225,11206,1498), (6459,6460,6), (6560,9541,7585), (6636,7391,2), (6643,7387,3089), (6644,7517,3518), (6676,8889,2), (6756,7395,6997), (6781,7756,32), (6803,7395,2), (6815,7503,2), (6824,6889,2), (6824,6917,6843), (6825,6833,2), (6827,6848,6919), (6827,6905,2), (6827,6970,6963), (6827,6985,6848), (6830,6842,5141), (6830,6954,2), (6831,6907,2476), (6831,6988,2), (6832,6989,2), (6833,6923,6871), (6834,6891,2), (6834,6928,5187), (6835,6986,2), (6836,6925,4), (6836,6962,6943), (6836,7580,6838), (6837,6839,3091), (6838,6840,3091), (6838,6890,2), (6842,6977,2), (6843,6870,3091), (6844,6871,3091), (6845,6937,5), (6846,6894,3091), (6847,6850,5177), (6847,6908,2), (6848,6926,2), (6850,6851,4), (6850,6892,6937), (6850,6906,2), (6853,6862,2), (6854,6883,2), (6856,7483,2), (6863,6952,2), (6868,6869,4), (6868,6885,6936), (6868,6934,2), (6868,6948,3), (6869,6938,3146), (6869,6948,6934), (6872,6904,5129), (6875,6917,2), (6875,6951,6889), (6876,6950,3), (6878,6881,2), (6880,6882,2), (6882,6941,5154), (6884,6993,5056), (6885,6930,5), (6885,6936,2), (6892,6937,2), (6893,6911,6964), (6893,6940,2), (6895,6908,3091), (6899,6905,6926), (6899,6985,2), (6902,6924,2), (6903,6942,6891), (6904,6992,10303), (6905,6963,6970), (6905,7491,5046), (6906,6937,6892), (6907,6935,2), (6909,6925,2), (6909,6932,6966), (6911,6947,2), (6912,6986,405), (6914,6917,6824), (6914,6951,2), (6916,10431,6839), (6919,6979,5056), (6922,6927,2), (6923,6950,2), (6924,6929,6944), (6925,6966,6932), (6928,6942,2), (6930,6948,6955), (6930,6955,2), (6932,6943,5), (6932,6966,2), (6934,6936,6885), (6934,6938,4), (6934,6987,6904), (6935,6988,6910), (6936,6955,631), (6938,6948,2), (6938,6955,6930), (6941,6961,2), (6943,6962,2), (6944,6967,2), (6949,6958,2), (6960,6972,2), (6962,6966,631), (6963,6970,2), (6985,7491,4), (6997,7485,2), (7381,11340,2), (7383,7404,2), (7388,11291,2), (7389,11292,2), (7390,7406,3091), (7391,7400,3091), (7392,7484,2), (7395,7408,3091), (7396,7488,10303), (7396,10565,2), (7399,7409,3091), (7401,7509,2), (7411,10431,2), (7464,7512,3520), (7470,11676,3), (7484,11403,11479), (7502,11250,3), (7509,7553,7394), (7509,7576,7401), (7511,7549,7557), (7538,7560,7520), (7540,7550,7533), (7553,9818,4), (7555,7574,7552), (7576,9818,7394), (7689,11750,11457), (7714,10691,2), (7761,7816,7795), (7783,7823,7774), (8226,8728,6991), (8227,10165,5550), (8357,8369,7866), (8550,11477,1992), (8703,11541,5068), (9778,10430,63), (9825,11479,7392), (10267,10532,10587), (10269,10531,10586), (10304,10565,10298), (10310,11500,100), (11015,11220,944), (11111,11112,2), (11291,11292,7819), (11293,11294,2), (11413,11414,7488)


X(21) = SCHIFFLER POINT

Trilinears       1/(cos B + cos C) : 1/(cos C + cos A) : 1/(cos A + cos B)
                        = f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)/(b + c)
Barycentrics  a/(cos B + cos C) : b/(cos C + cos A) : c/(cos A + cos B)
X(21) = 3R*X(2) + 2r*X(3)

As a point on the Euler line, X(21) has Shinagawa coefficients ($aSA$, abc - $aSA$).

The name of this point honors Kurt Schiffler.

Let A'B'C' be the incentral triangle of ABC, and let LA be the reflection of line B'C' in line BC; define LB and LC cyclically. The triangle formed by the lines LA, LB, LC is perspective to ABC, and the perspector is X(21). (Randy Hutson, 9/23/2011)

Write I for the incenter; the Euler lines of the four triangles IBC, ICA, IAB, and ABC concur in X(21). This configuration extends to Kirikami-Schiffler points and generalizations found by Peter Moses, as introduced just before X(3648).

Let A'B'C' be the 2nd circumperp triangle. Let A″ be the cevapoint of B' and C', and define B″ and C″ cyclically. The lines AA″, BB″, and CC″ concur in X(21). (Randy Hutson, April 9, 2016)

Let A'B'C' be the Feuerbach triangle. Let La be the tangent to the nine-point circle at A', and define Lb and Lc cyclically. Let A″ be the isogonal conjugate of the trilinear pole of La, and define B″ and C″ cyclically. Let A*B*C* be the circumcevian triangle, wrt A″B″C″, of X(1). The lines AA*, BB*, CC* concur in X(21). (Randy Hutson, April 9, 2016)

Let A'B'C' be the 2nd circumperp triangle. Let A″ be the trilinear product B'*C', and define B″ and C″ cyclically. Then A″, B″, C″ are collinear on line X(36)X(238) (the trilinear polar of X(81)). The lines A'A″, B'B″, C'C″ concur in X(21). (Randy Hutson, April 9, 2016)

Let Oa be the reflection of the A-excircle in the perpendicular bisector of BC, and define Ob, Oc cyclically. Then X(21) is the radical center of Oa, Ob, Oc. (Randy Hutson, April 9, 2016)

Let Ab, Ac, Bc, Ba, Ca, Cb be as in the construction of the Conway circle (see http://mathworld.wolfram.com/ConwayCircle.html). Let Oa be the circumcircle of AAbAc, and define Ob, Oc cyclically. X(21) is the radical center of Oa, Ob, Oc; see also X(8) and X(274). (Randy Hutson, April 9, 2016)

Let A'B'C' be the excentral triangle. X(21) is the radical center of the circles O(3,4) of triangles A'BC, B'CA, C'AB. (Randy Hutson, July 31 2018)

If you have The Geometer's Sketchpad, you can view Schiffler point.
If you have GeoGebra, you can view Schiffler point.

Lev Emelyanov and Tatiana Emelyanova, A note on the Schiffler point, Forum Geometricorum 3 (2003) pages 113-116.

See Dasari Naga Vijay Krishna, On a Conic Through Twelve Notable Points, Int. J. Adv. Math. and Mech. 7(2) (2019) 1-15.

In the plane of a triangle ABC, let
A' = A-excenter, and define B' and C' cyclically
A'' = A'X(3)∩BC, and define B'' and C'' cyclically.
The lines AA'', BB'',CC'' concur in X(21). (Yuda Chen, April 13, 2022)

X(21) lies on these lines: {1, 31}, {2, 3}, {6, 941}, {7, 56}, {8, 55}, {9, 41}, {10, 35}, {11, 4996}, {12, 5080}, {15, 5362}, {16, 5367}, {19, 4288}, {32, 981}, {34, 17080}, {36, 79}, {37, 172}, {39, 33854}, {40, 3577}, {42, 4281}, {43, 37574}, {44, 4273}, {45, 3285}, {46, 17098}, {48, 15656}, {51, 970}, {57, 4652}, {60, 960}, {65, 4640}, {69, 10432}, {71, 4269}, {72, 943}, {73, 651}, {74, 34800}, {75, 272}, {76, 37670}, {77, 1394}, {84, 285}, {85, 3188}, {90, 224}, {92, 41227}, {99, 105}, {101, 3294}, {104, 110}, {106, 34594}, {107, 1295}, {108, 39435}, {109, 37558}, {112, 26703}, {119, 31659}, {125, 38612}, {141, 4265}, {142, 10123}, {144, 954}, {145, 956}, {149, 2894}, {162, 3194}, {177, 7587}, {183, 18135}, {184, 13323}, {187, 5277}, {193, 37492}, {194, 16998}, {198, 5296}, {200, 4866}, {210, 4420}, {214, 501}, {219, 2335}, {220, 40779}, {226, 37583}, {238, 256}, {243, 1896}, {261, 314}, {268, 280}, {270, 1172}, {286, 1441}, {294, 1212}, {307, 2062}, {321, 7283}, {323, 5453}, {329, 5703}, {332, 1036}, {355, 11491}, {385, 1655}, {386, 1724}, {387, 24597}, {388, 37579}, {390, 6601}, {391, 4254}, {394, 36746}, {476, 2687}, {484, 3754}, {495, 20060}, {497, 10527}, {498, 11681}, {499, 14793}, {511, 15988}, {514, 23775}, {515, 10902}, {516, 15909}, {517, 1389}, {518, 2346}, {519, 3746}, {527, 34917}, {529, 15888}, {535, 5270}, {551, 5557}, {572, 1765}, {593, 6051}, {603, 17074}, {612, 989}, {614, 988}, {643, 1320}, {644, 1334}, {645, 36798}, {662, 1156}, {667, 29150}, {672, 41239}, {691, 2752}, {740, 27368}, {741, 932}, {748, 978}, {750, 37603}, {756, 5293}, {757, 15569}, {849, 30576}, {884, 885}, {900, 42741}, {902, 5255}, {908, 12572}, {912, 24299}, {915, 925}, {930, 26707}, {934, 26702}, {936, 3305}, {938, 5744}, {940, 4252}, {942, 3218}, {944, 10267}, {946, 11012}, {950, 5745}, {952, 34352}, {961, 1402}, {962, 3428}, {965, 37504}, {966, 36744}, {976, 983}, {982, 28082}, {986, 3924}, {987, 2206}, {990, 9962}, {991, 37659}, {992, 5110}, {995, 15315}, {999, 3296}, {1019, 35355}, {1021, 23893}, {1030, 1213}, {1038, 1041}, {1039, 1040}, {1056, 10587}, {1058, 10529}, {1060, 1063}, {1061, 1062}, {1064, 3073}, {1078, 18140}, {1083, 3110}, {1104, 3666}, {1107, 1914}, {1149, 8421}, {1150, 10449}, {1152, 31473}, {1155, 3812}, {1183, 7058}, {1191, 40153}, {1201, 17187}, {1210, 41557}, {1211, 26064}, {1214, 1396}, {1220, 26115}, {1251, 5240}, {1254, 1758}, {1260, 20007}, {1261, 4723}, {1279, 16696}, {1290, 12030}, {1296, 9061}, {1304, 2694}, {1319, 1408}, {1329, 5432}, {1330, 3936}, {1335, 9678}, {1376, 5217}, {1392, 2098}, {1412, 1420}, {1414, 43736}, {1453, 5256}, {1466, 5435}, {1470, 5555}, {1475, 16503}, {1478, 10198}, {1479, 11680}, {1482, 14497}, {1500, 5291}, {1503, 26543}, {1610, 2217}, {1617, 3600}, {1626, 40462}, {1633, 24723}, {1682, 3271}, {1697, 3680}, {1698, 5010}, {1706, 35445}, {1709, 9961}, {1728, 41568}, {1761, 2294}, {1762, 18673}, {1764, 10451}, {1770, 12609}, {1781, 25081}, {1788, 11509}, {1798, 40454}, {1805, 30556}, {1806, 7133}, {1807, 35194}, {1808, 43748}, {1809, 36795}, {1834, 24883}, {1836, 28628}, {1837, 26066}, {1870, 37565}, {1936, 2654}, {1946, 4391}, {1975, 16992}, {1993, 36742}, {1994, 36750}, {2053, 7155}, {2077, 6684}, {2078, 10106}, {2096, 5553}, {2112, 39244}, {2136, 31509}, {2175, 35628}, {2183, 41263}, {2223, 16830}, {2238, 18755}, {2241, 16975}, {2269, 23640}, {2271, 37657}, {2274, 16690}, {2276, 4426}, {2280, 21384}, {2295, 17735}, {2310, 2648}, {2320, 5289}, {2341, 5549}, {2344, 3061}, {2417, 43737}, {2550, 7676}, {2551, 5218}, {2778, 12826}, {2782, 5985}, {2787, 16158}, {2802, 13143}, {2804, 14224}, {2886, 6284}, {2895, 41014}, {2906, 18455}, {2979, 37482}, {3006, 5015}, {3011, 13161}, {3035, 17100}, {3052, 5710}, {3053, 5275}, {3058, 3813}, {3060, 5752}, {3062, 5732}, {3074, 22350}, {3085, 3436}, {3086, 8071}, {3120, 24161}, {3178, 38456}, {3207, 5781}, {3208, 4390}, {3210, 19851}, {3216, 4256}, {3220, 4357}, {3241, 3303}, {3244, 5288}, {3254, 12053}, {3256, 4848}, {3270, 43746}, {3290, 16716}, {3293, 33771}, {3304, 5558}, {3306, 15803}, {3315, 3953}, {3333, 4666}, {3336, 5883}, {3337, 4973}, {3361, 10582}, {3419, 5791}, {3421, 10528}, {3423, 17206}, {3427, 5731}, {3434, 4294}, {3452, 27385}, {3453, 30115}, {3454, 25645}, {3467, 10176}, {3474, 28629}, {3476, 11510}, {3487, 5905}, {3488, 12649}, {3496, 5060}, {3501, 41423}, {3509, 21808}, {3551, 15485}, {3555, 3957}, {3565, 15344}, {3579, 3753}, {3583, 14794}, {3585, 3822}, {3586, 5705}, {3589, 5096}, {3614, 6668}, {3617, 5687}, {3618, 36741}, {3621, 7317}, {3623, 6767}, {3624, 5561}, {3632, 25439}, {3634, 9342}, {3635, 13602}, {3663, 17189}, {3670, 30117}, {3673, 16749}, {3678, 7161}, {3679, 8715}, {3681, 3811}, {3684, 3691}, {3689, 4662}, {3695, 32849}, {3701, 7081}, {3716, 8648}, {3720, 37607}, {3730, 14964}, {3731, 33628}, {3733, 23836}, {3737, 6615}, {3738, 35055}, {3742, 32636}, {3748, 34791}, {3757, 4968}, {3816, 5433}, {3825, 10090}, {3826, 26060}, {3827, 41582}, {3831, 32918}, {3833, 5131}, {3841, 4324}, {3870, 7160}, {3886, 4483}, {3895, 4853}, {3912, 24632}, {3920, 5266}, {3925, 15338}, {3928, 11518}, {3929, 3951}, {3931, 17016}, {3935, 34790}, {3948, 26243}, {3971, 8669}, {4005, 15481}, {4011, 25591}, {4026, 20872}, {4067, 41696}, {4084, 5425}, {4101, 4416}, {4129, 39577}, {4251, 16552}, {4253, 16783}, {4255, 4383}, {4257, 37522}, {4258, 37658}, {4293, 7742}, {4297, 12617}, {4302, 19854}, {4309, 31458}, {4314, 4847}, {4322, 9363}, {4326, 42015}, {4344, 21002}, {4359, 16817}, {4366, 26801}, {4385, 26227}, {4413, 19877}, {4418, 24850}, {4422, 30906}, {4423, 5204}, {4427, 17164}, {4436, 40625}, {4438, 36568}, {4482, 29699}, {4516, 4612}, {4518, 18265}, {4520, 6603}, {4558, 8759}, {4565, 9372}, {4567, 5377}, {4570, 24433}, {4646, 4689}, {4668, 4803}, {4679, 25681}, {4680, 30172}, {4857, 10707}, {4867, 5424}, {4881, 10308}, {4972, 34868}, {4995, 21031}, {5016, 33113}, {5044, 5440}, {5045, 29817}, {5082, 20075}, {5119, 14923}, {5124, 17398}, {5132, 17277}, {5138, 10477}, {5176, 10039}, {5179, 27068}, {5183, 10107}, {5211, 31108}, {5239, 33653}, {5263, 8053}, {5264, 30116}, {5278, 9534}, {5280, 25092}, {5281, 7080}, {5287, 37554}, {5294, 5314}, {5297, 37589}, {5300, 29641}, {5325, 12437}, {5414, 31453}, {5422, 36754}, {5438, 7308}, {5439, 27003}, {5444, 14800}, {5482, 33852}, {5484, 20999}, {5506, 13146}, {5535, 31870}, {5537, 43174}, {5554, 5657}, {5584, 9778}, {5587, 6796}, {5603, 11249}, {5686, 6600}, {5690, 11849}, {5692, 9275}, {5694, 37733}, {5697, 21398}, {5708, 23958}, {5709, 21165}, {5711, 17126}, {5719, 17484}, {5735, 11522}, {5777, 33597}, {5790, 32141}, {5795, 6735}, {5814, 33077}, {5818, 11499}, {5832, 12701}, {5836, 37568}, {5880, 30295}, {5882, 10031}, {5886, 16159}, {5901, 22765}, {5902, 15173}, {5903, 30147}, {5919, 11260}, {5943, 15489}, {6001, 23059}, {6043, 17015}, {6147, 17483}, {6211, 25024}, {6326, 20117}, {6361, 35239}, {6514, 30223}, {6516, 17095}, {6554, 32561}, {6595, 17643}, {6599, 21634}, {6626, 7261}, {6651, 27954}, {6667, 7294}, {6693, 25441}, {6713, 18861}, {6727, 15997}, {6737, 18249}, {6745, 18250}, {6762, 10389}, {7004, 40602}, {7049, 7361}, {7085, 26065}, {7100, 13486}, {7149, 8885}, {7173, 31260}, {7179, 25581}, {7191, 37592}, {7226, 36565}, {7253, 23189}, {7257, 8851}, {7259, 30618}, {7270, 33116}, {7284, 37618}, {7292, 37599}, {7330, 12528}, {7354, 25466}, {7373, 18490}, {7588, 8250}, {7595, 8225}, {7621, 32479}, {7680, 11827}, {7688, 31730}, {7713, 24611}, {7745, 37661}, {7750, 37664}, {7754, 17002}, {7783, 17000}, {7793, 16997}, {7967, 16202}, {8062, 23226}, {8109, 8391}, {8110, 8372}, {8185, 39582}, {8227, 16125}, {8273, 10429}, {8296, 16484}, {8568, 34867}, {8582, 10164}, {8686, 8690}, {8720, 24165}, {8760, 26641}, {8844, 33295}, {8847, 43747}, {8886, 41084}, {8932, 43751}, {8983, 19080}, {9579, 25525}, {9612, 31266}, {9656, 34739}, {9668, 31493}, {9670, 11235}, {9710, 34612}, {9956, 33862}, {9957, 38460}, {10085, 12669}, {10087, 12531}, {10157, 40262}, {10165, 12608}, {10167, 34862}, {10179, 20323}, {10197, 37719}, {10202, 26877}, {10246, 13465}, {10266, 12524}, {10305, 24558}, {10307, 38031}, {10309, 24556}, {10385, 43745}, {10390, 18164}, {10396, 41576}, {10435, 10444}, {10436, 18655}, {10454, 13478}, {10544, 40966}, {10571, 34027}, {10585, 10590}, {10595, 10680}, {10601, 36745}, {10679, 12245}, {10914, 24297}, {11231, 17619}, {11240, 42842}, {11374, 31053}, {11375, 18977}, {11376, 16142}, {11507, 18391}, {11544, 20084}, {11683, 25255}, {11752, 15788}, {11789, 15789}, {12388, 12390}, {12515, 13145}, {12522, 12538}, {12523, 12539}, {12527, 13405}, {12532, 12739}, {12607, 34606}, {12615, 15326}, {12641, 13278}, {12671, 37837}, {12699, 17173}, {12775, 37562}, {12913, 37737}, {12953, 31245}, {13100, 15325}, {13151, 13369}, {13205, 32157}, {13384, 15829}, {13397, 39439}, {13464, 34485}, {13887, 19014}, {13940, 19013}, {13971, 19079}, {14192, 37741}, {14496, 31663}, {14526, 16152}, {14547, 41243}, {14795, 37710}, {14803, 16154}, {14804, 37701}, {14882, 40663}, {15178, 37518}, {15179, 24928}, {15446, 17104}, {15654, 24552}, {15852, 25939}, {16020, 16752}, {16114, 27180}, {16153, 33593}, {16155, 30384}, {16466, 17127}, {16478, 17017}, {16502, 31449}, {16549, 24047}, {16566, 32118}, {16579, 33178}, {16678, 23383}, {16683, 16693}, {16684, 32922}, {16691, 23393}, {16712, 33955}, {16720, 24358}, {16780, 31429}, {16824, 32932}, {16826, 37609}, {16887, 17219}, {16974, 41269}, {16996, 20081}, {17019, 33774}, {17052, 25447}, {17054, 17595}, {17056, 24936}, {17123, 27627}, {17168, 41691}, {17171, 18589}, {17181, 27187}, {17202, 29097}, {17257, 24320}, {17349, 37502}, {17379, 37507}, {17496, 22160}, {17594, 25059}, {17596, 24443}, {17601, 24440}, {17609, 42819}, {17613, 31787}, {17733, 32915}, {17749, 37687}, {17756, 31448}, {17778, 20077}, {17811, 37501}, {18123, 34435}, {18228, 27383}, {18299, 31008}, {18357, 18524}, {18481, 22798}, {18623, 41402}, {18642, 26167}, {18645, 43177}, {18653, 24564}, {18990, 20067}, {19684, 19762}, {19701, 19759}, {19716, 19753}, {19732, 19760}, {19783, 44094}, {19784, 37557}, {19785, 19844}, {19786, 19841}, {19808, 19842}, {19822, 19845}, {19862, 25542}, {20018, 37652}, {20066, 31419}, {20470, 25508}, {20653, 33160}, {21044, 23907}, {21246, 27401}, {21321, 40605}, {22080, 35203}, {22344, 24627}, {22345, 38000}, {22369, 26045}, {22376, 27002}, {22753, 38306}, {22768, 31631}, {22935, 36865}, {23144, 34046}, {23181, 35097}, {23206, 26634}, {23369, 23843}, {23537, 33129}, {23864, 27527}, {24159, 33146}, {24392, 41864}, {24436, 24697}, {24467, 37615}, {24470, 26842}, {24512, 33863}, {24586, 29966}, {24602, 29968}, {24619, 26526}, {24700, 25371}, {24880, 31204}, {24931, 31247}, {25005, 26285}, {25055, 25056}, {25354, 34053}, {25500, 30949}, {26102, 37608}, {26128, 36505}, {26144, 39200}, {26241, 39581}, {26244, 27040}, {26321, 34773}, {26437, 41545}, {26487, 37821}, {26558, 26629}, {26728, 26729}, {26818, 42884}, {26878, 31837}, {26921, 37533}, {26932, 43735}, {27006, 34847}, {27025, 31020}, {27097, 27185}, {27804, 41813}, {28386, 41346}, {28612, 34886}, {28813, 30847}, {29632, 30984}, {30302, 30387}, {30303, 30388}, {30304, 30363}, {31157, 37722}, {31393, 36846}, {31546, 31549}, {32010, 40415}, {32456, 36812}, {32919, 35633}, {32937, 36507}, {33297, 34016}, {33668, 37535}, {33814, 34122}, {33860, 34123}, {33925, 34610}, {34124, 38619}, {34277, 39167}, {34545, 37509}, {35658, 35660}, {36607, 38249}, {37503, 37654}, {37605, 41695}, {38722, 38752}, {38859, 40719}, {40457, 41364}, {41592, 41728}, {41601, 41734}, {41604, 41739}, {41606, 41741}

X(21) is the {X(2),X(3)}-harmonic conjugate of X(404). For a list of other harmonic conjugates of X(21), click Tables at the top of this page.

X(21) = midpoint of X(1) and X(191)
X(21) = reflection of X(3651) in X(3)
X(21) = isogonal conjugate of X(65)
X(21) = isotomic conjugate of X(1441)
X(21) = circumcircle-inverse of X(1325)
X(21) = polar-circle-inverse of X(37982)
X(21) = orthoptic-circle-of-Steiner-inellipse-inverse of complement of X(37959)
X(21) = orthoptic-circle-of-Steiner-circumellipse-inverse of anticomplement of X(37959)
X(21) = complement of X(2475)
X(21) = anticomplement of X(442)
X(21) = X(i)-Ceva conjugate of X(j) for these (i,j): (86,81), (261,333)
X(21) = cevapoint of X(i) and X(j) for these (i,j): (1,3), (9,55), (1805,1806)
X(21) = X(i)-cross conjugate of X(j) for these (i,j): (1,29), (3,283), (9,333), (55,284), (58,285), (284,81), (522,100)
X(21) = crosspoint of X(i) and X(j) for these {i,j}: {86,333}, {1805,1806}
X(21) = crosssum of X(i) and X(j) for these (i,j): (1,1046), (42,1400), (1254,1425), (1402,1409)
X(21) = crossdifference of every pair of points on line X(647)X(661)
X(21) = X(i)-Hirst inverse of X(j) for these (i,j): (2,448), (3,416), (4,425)
X(21) = X(i)-beth conjugate of X(j) for these (i,j): (21,58), (99,21), (643,21), (1043,1043), (1098,21)
X(21) = intersection of tangents at X(1) and X(3) to the Stammler hyperbola
X(21) = X(54)-of-2nd-circumperp-triangle
X(21) = X(3574)-of-excentral-triangle
X(21) = crosspoint of X(1) and X(3) wrt the excentral triangle
X(21) = crosspoint of X(1) and X(3) wrt the tangential triangle
X(21) = trilinear pole of line X(521)X(650)
X(21) = similitude center of ABC and X(1)-Brocard triangle
X(21) = X(i)-isoconjugate of X(j) for these (i,j): (6,226), (75,1402)
X(21) = {X(1),X(63)}-harmonic conjugate of X(3868)
X(21) = perspector of 2nd circumperp triangle and cross-triangle of ABC and 2nd circumperp triangle
X(21) = perspector of ABC and cross-triangle of ABC and 1st Conway triangle
X(21) = perspector of Gemini triangles 1 and 8
X(21) = barycentric product of Feuerbach hyperbola intercepts of line X(2)X(6)


X(22) = EXETER POINT

Trilinears   a(b4 + c4 - a4) : b(c4 + a4 - b4) : c(a4 + b4 - c4)
Barycentrics  a2(b4 + c4 - a4) : b2(c4 + a4 - b4) : c2(a4 + b4 - c4)
Barycentrics   sin 2A - tan ω : sin 2B - tan ω : : (M. Iliev, 5/13/07)
Barycentrics    tan B + tan C - tan A + tan ω : : (R. Hutson, 10/13/15)
X(22) = 3 R^2 X(2) - SW X(3)

As a point on the Euler line, X(22) has Shinagawa coefficients (E + 2F, -2E - 2F).

X(22) is the perspector of the circummedial triangle and the tangential triangle; also X(22) = X(55)-of-the-tangential triangle if ABC is acute. See the note just before X(1601) for a generalization.

Let La be the polar of X(3) wrt the A-power circle, and define Lb, Lc cyclically. Let A' = Lb∩Lc, B' = Lc∩La, C' = La∩Lb. The triangle A'B'C' is homothetic to the anticomplementary triangle, and the center of homothety is X(22). (Randy Hutson, September 5, 2015)

For a generalization and related references, see Peter Csiba and László Nément, Mathematics 2021: "Some Properties of the Exeter Transformation". The Exeter transformation is closely related to TCC Perspectors, introducted in 2003 in the preamble just before X(1601). This subject is developed in I. Minevich and P. Morton, International Journal of Geometry 2017, "Synthetic foundations of cevian geometry, IV"

If you have The Geometer's Sketchpad, you can view Exeter point.

If you have GeoGebra, you can view Exeter point.

X(22) lies on these lines: {1, 5310}, {2, 3}, {6, 251}, {8, 8193}, {9, 5314}, {10, 8185}, {11, 9673}, {12, 9658}, {31, 5329}, {32, 1194}, {35, 612}, {36, 614}, {40, 9626}, {42, 37576}, {49, 16266}, {51, 182}, {52, 7592}, {54, 36747}, {55, 3100}, {56, 977}, {57, 7293}, {63, 3220}, {64, 11440}, {66, 34177}, {68, 32048}, {69, 159}, {74, 2931}, {75, 21407}, {76, 1799}, {81, 36740}, {83, 41928}, {97, 19189}, {98, 925}, {99, 305}, {100, 197}, {105, 13397}, {107, 15466}, {110, 154}, {111, 2079}, {112, 3162}, {114, 23217}, {125, 41674}, {127, 11605}, {132, 35969}, {141, 20987}, {143, 36753}, {145, 8192}, {146, 9919}, {147, 9861}, {148, 13175}, {149, 13222}, {153, 9913}, {155, 1614}, {156, 6101}, {157, 183}, {160, 325}, {161, 343}, {165, 9590}, {184, 511}, {187, 1196}, {193, 19119}, {194, 9917}, {195, 12226}, {198, 27396}, {206, 3313}, {216, 10311}, {220, 26911}, {221, 19367}, {230, 8553}, {232, 577}, {238, 27661}, {262, 40393}, {264, 1629}, {280, 7172}, {315, 23208}, {316, 14558}, {321, 23847}, {323, 3167}, {324, 33971}, {347, 1617}, {348, 39732}, {353, 11173}, {371, 9683}, {373, 17508}, {385, 3164}, {386, 9571}, {388, 10831}, {389, 10984}, {390, 16541}, {399, 12219}, {476, 2697}, {477, 16167}, {485, 35776}, {486, 35777}, {487, 9921}, {488, 9922}, {491, 26307}, {492, 26306}, {497, 10832}, {515, 15177}, {519, 37546}, {524, 35707}, {543, 3455}, {567, 39522}, {569, 5446}, {573, 9570}, {574, 9699}, {575, 15004}, {576, 13366}, {595, 2922}, {599, 19596}, {616, 9916}, {617, 9915}, {627, 22657}, {628, 22656}, {638, 8996}, {669, 6563}, {675, 1305}, {689, 40362}, {842, 10420}, {901, 10016}, {907, 40189}, {930, 15959}, {940, 4265}, {956, 33090}, {958, 9712}, {962, 9911}, {991, 1790}, {999, 17024}, {1001, 20988}, {1007, 44180}, {1030, 5275}, {1040, 24611}, {1069, 9638}, {1078, 40022}, {1092, 10282}, {1112, 19154}, {1147, 9707}, {1151, 9694}, {1152, 12224}, {1154, 18445}, {1181, 5889}, {1184, 1627}, {1192, 43601}, {1199, 37493}, {1216, 10539}, {1225, 2934}, {1269, 23365}, {1270, 5594}, {1271, 5595}, {1289, 39436}, {1294, 1302}, {1295, 9058}, {1296, 14657}, {1311, 41906}, {1324, 7081}, {1351, 1994}, {1352, 31383}, {1369, 11641}, {1376, 9713}, {1383, 5024}, {1384, 5354}, {1407, 26910}, {1437, 37482}, {1460, 17126}, {1473, 3218}, {1486, 1621}, {1495, 3098}, {1498, 2917}, {1602, 1626}, {1603, 2933}, {1605, 2925}, {1606, 2926}, {1609, 7735}, {1611, 5023}, {1612, 7742}, {1613, 2076}, {1615, 2919}, {1616, 2920}, {1620, 2929}, {1634, 6148}, {1637, 25644}, {1661, 38918}, {1670, 8881}, {1671, 8880}, {1691, 3981}, {1714, 5358}, {1760, 4123}, {1843, 9813}, {1853, 23293}, {1899, 3580}, {1915, 3094}, {1972, 40870}, {1974, 11574}, {1975, 8024}, {1992, 32621}, {2000, 21370}, {2056, 5104}, {2077, 36984}, {2172, 4456}, {2178, 26242}, {2192, 11446}, {2194, 4259}, {2370, 9059}, {2393, 27365}, {2493, 38872}, {2693, 9060}, {2770, 14729}, {2777, 22109}, {2782, 5986}, {2799, 42659}, {2888, 9920}, {2896, 9918}, {2923, 24303}, {2924, 24304}, {2930, 14682}, {2932, 35221}, {2967, 23606}, {2975, 22654}, {3006, 23361}, {3007, 18613}, {3011, 36152}, {3051, 5017}, {3052, 5078}, {3066, 11451}, {3085, 10037}, {3086, 10046}, {3124, 38880}, {3197, 11445}, {3219, 7085}, {3291, 5206}, {3292, 44110}, {3295, 9538}, {3410, 18440}, {3434, 10829}, {3436, 10830}, {3447, 5968}, {3448, 12310}, {3504, 8782}, {3556, 3869}, {3563, 13398}, {3567, 36752}, {3576, 9625}, {3592, 34516}, {3594, 34515}, {3616, 11365}, {3648, 16119}, {3681, 12329}, {3705, 15654}, {3721, 21771}, {3734, 8891}, {3736, 44118}, {3746, 9643}, {3757, 23850}, {3781, 26885}, {3784, 26884}, {3819, 5651}, {3868, 37547}, {3870, 40910}, {3871, 20020}, {3873, 22769}, {3926, 40123}, {3955, 26892}, {3964, 37668}, {4057, 20294}, {4252, 33774}, {4260, 5320}, {4383, 5096}, {4440, 24822}, {4549, 32111}, {4550, 16194}, {5010, 5268}, {5013, 9608}, {5050, 9777}, {5085, 5640}, {5092, 5943}, {5093, 16981}, {5134, 24054}, {5138, 40952}, {5157, 9969}, {5172, 29665}, {5188, 42671}, {5191, 38553}, {5201, 14614}, {5204, 7292}, {5217, 5297}, {5272, 7280}, {5276, 36744}, {5304, 8573}, {5324, 24597}, {5406, 12305}, {5407, 12306}, {5408, 11825}, {5409, 8989}, {5412, 11514}, {5413, 11513}, {5462, 13336}, {5480, 37649}, {5523, 13854}, {5552, 26309}, {5562, 6759}, {5601, 8190}, {5602, 8191}, {5621, 9140}, {5687, 33091}, {5695, 23848}, {5706, 41723}, {5858, 14179}, {5859, 14173}, {5864, 11126}, {5865, 11127}, {5866, 19583}, {5890, 37489}, {5897, 9064}, {5907, 26883}, {5921, 39879}, {5938, 6031}, {5966, 14656}, {5976, 14713}, {5987, 13188}, {6090, 8780}, {6193, 9908}, {6194, 22655}, {6198, 9645}, {6200, 8854}, {6221, 9695}, {6223, 9910}, {6224, 9912}, {6225, 9914}, {6241, 8718}, {6243, 12161}, {6284, 9672}, {6337, 40125}, {6360, 20999}, {6396, 8855}, {6462, 8194}, {6463, 8195}, {6467, 40318}, {6480, 32567}, {6481, 32574}, {6503, 7710}, {6515, 6776}, {6527, 15589}, {6560, 18289}, {6561, 18290}, {6688, 22112}, {6781, 9745}, {7071, 9539}, {7083, 17127}, {7193, 26893}, {7262, 24436}, {7354, 9659}, {7585, 19006}, {7586, 19005}, {7669, 8667}, {7689, 10575}, {7750, 15270}, {7754, 8267}, {7761, 21248}, {7774, 20775}, {7779, 20794}, {7781, 19568}, {7787, 10790}, {7802, 16275}, {7823, 8878}, {7842, 30747}, {7878, 42037}, {7893, 19597}, {7910, 30785}, {7998, 17811}, {7999, 43598}, {8053, 20291}, {8125, 8131}, {8126, 8132}, {8276, 9540}, {8277, 13935}, {8280, 35820}, {8281, 35821}, {8546, 8584}, {8588, 20481}, {8591, 9876}, {8680, 24321}, {8717, 14855}, {8743, 10316}, {8793, 19613}, {8879, 41361}, {8903, 8904}, {8911, 26875}, {8939, 19406}, {8943, 19407}, {8972, 13889}, {9056, 41904}, {9057, 41905}, {9070, 39435}, {9084, 20187}, {9123, 34519}, {9209, 39228}, {9536, 11406}, {9537, 10306}, {9732, 10132}, {9733, 10133}, {9744, 23195}, {9781, 43651}, {9786, 10574}, {9833, 14516}, {9865, 23173}, {9874, 12411}, {9924, 12272}, {9927, 11750}, {9934, 12825}, {9937, 11411}, {9967, 44077}, {10192, 11064}, {10203, 13423}, {10263, 32046}, {10314, 10979}, {10330, 25046}, {10519, 14826}, {10527, 26308}, {10528, 10834}, {10529, 10835}, {10540, 15068}, {10541, 12834}, {10545, 31860}, {10546, 41424}, {10602, 37784}, {10605, 15072}, {10606, 11454}, {10641, 11516}, {10642, 11515}, {10733, 19457}, {10982, 13434}, {11012, 36986}, {11061, 32262}, {11174, 41328}, {11202, 36987}, {11245, 37644}, {11363, 37613}, {11422, 11477}, {11424, 13598}, {11433, 25406}, {11439, 15811}, {11443, 17813}, {11444, 17814}, {11447, 17819}, {11448, 17820}, {11449, 17821}, {11452, 17826}, {11453, 17827}, {11455, 11472}, {11456, 13754}, {11457, 12359}, {11459, 14157}, {11464, 37483}, {11480, 37776}, {11481, 37775}, {11511, 44102}, {11550, 21243}, {11580, 15655}, {11610, 22075}, {11629, 14184}, {11630, 14183}, {11643, 33998}, {11671, 14652}, {11820, 35450}, {11898, 14683}, {12017, 15018}, {12118, 19908}, {12160, 19347}, {12164, 43605}, {12203, 40814}, {12221, 12978}, {12222, 12979}, {12256, 12972}, {12257, 12973}, {12270, 17835}, {12271, 17836}, {12273, 17838}, {12274, 17839}, {12275, 17842}, {12276, 17840}, {12277, 17843}, {12278, 17845}, {12280, 17846}, {12284, 15085}, {12289, 12293}, {12383, 12412}, {12384, 12413}, {12414, 12849}, {12429, 34799}, {12824, 15462}, {12827, 36201}, {12893, 16111}, {13009, 13055}, {13010, 13056}, {13015, 17841}, {13016, 17844}, {13289, 16163}, {13321, 15037}, {13330, 14153}, {13340, 22115}, {13346, 13367}, {13348, 43652}, {13352, 18475}, {13394, 23292}, {13421, 32136}, {13567, 18911}, {13630, 37490}, {13638, 44192}, {13678, 13680}, {13758, 44193}, {13798, 13800}, {13858, 36329}, {13859, 35751}, {13941, 13943}, {14370, 17042}, {14389, 31670}, {14547, 22390}, {14577, 15355}, {14602, 43183}, {14673, 34186}, {14793, 24239}, {14852, 25739}, {14927, 32064}, {15024, 15805}, {15033, 37506}, {15043, 37514}, {15053, 20791}, {15060, 33533}, {15069, 15581}, {15109, 31489}, {15241, 31842}, {15302, 15815}, {15360, 43273}, {15512, 33495}, {15513, 40350}, {15520, 44111}, {15578, 23332}, {15588, 35213}, {15801, 19468}, {15812, 26156}, {15931, 30265}, {16030, 43768}, {16102, 39346}, {16261, 32620}, {16318, 42459}, {16472, 31757}, {16681, 23339}, {16989, 40981}, {16990, 22062}, {16998, 18666}, {17018, 37580}, {17093, 38859}, {17150, 20247}, {17165, 20249}, {17824, 32338}, {17837, 22534}, {17907, 41375}, {18124, 34436}, {18287, 39653}, {18392, 18405}, {18436, 32139}, {18438, 34397}, {18616, 20911}, {18617, 33936}, {18912, 41587}, {19122, 19132}, {19131, 39588}, {19137, 44091}, {19153, 22151}, {19167, 19180}, {19357, 34148}, {19412, 19430}, {19413, 19431}, {19588, 20080}, {19724, 19759}, {19725, 19760}, {19785, 41230}, {19798, 19841}, {19799, 19842}, {19835, 19845}, {20045, 20222}, {20127, 32227}, {20676, 28395}, {20878, 23385}, {20998, 21001}, {21072, 29065}, {21167, 35283}, {21368, 24430}, {22089, 30474}, {22135, 34137}, {22241, 32817}, {22647, 22658}, {22676, 35278}, {22802, 23358}, {23061, 37672}, {23115, 39575}, {23128, 41480}, {23216, 36849}, {23368, 23374}, {23380, 26232}, {23381, 32929}, {23383, 26230}, {23843, 26227}, {23864, 26248}, {23958, 26866}, {24163, 30117}, {24686, 25343}, {25335, 34437}, {25524, 29666}, {26228, 37579}, {26275, 39478}, {26302, 26394}, {26303, 26418}, {26304, 26494}, {26305, 26503}, {26895, 26909}, {26912, 26953}, {26913, 26958}, {29680, 37564}, {30270, 36212}, {30435, 34482}, {32248, 32276}, {32354, 32357}, {32379, 41590}, {32458, 39644}, {32762, 38227}, {32911, 36741}, {33854, 36743}, {33974, 37667}, {34013, 36521}, {34247, 36559}, {34424, 36836}, {34425, 36843}, {34565, 39561}, {34809, 37689}, {34966, 41615}, {35260, 37669}, {35325, 38663}, {36988, 37813}, {37492, 37685}, {37511, 44080}, {37516, 44085}, {37517, 44109}, {37779, 39899}, {38738, 39854}, {38749, 39825}, {39172, 40358}, {39807, 39820}, {39836, 39849}, {40120, 44064}, {40643, 41262}, {41447, 41468}, {41448, 41469}, {41594, 41730}, {41602, 41736}, {41605, 41740}, {41612, 41743}, {43816, 43829}

X(22) is the {X(3),X(25)}-harmonic conjugate of X(2). For a list of other harmonic conjugates of X(22), click Tables at the top of this page.

X(22) = reflection of X(378) in X(3)
X(22) = isogonal conjugate of X(66)
X(22) = isotomic conjugate of X(18018)
X(22) = complement of X(7391)
X(22) = anticomplement of X(427)
X(22) = circumcircle-inverse of X(858)
X(22) = polar-circle-inverse of X(37981)
X(22) = X(76)-Ceva conjugate of X(6)
X(22) = cevapoint of X(3) and X(159)
X(22) = crosspoint of X(99) and X(250)
X(22) = crosssum of X(125) and X(512)
X(22) = crossdifference of every pair of points on the line X(647)X(826)
X(22) = X(i)-beth conjugate of X(j) for these (i,j): (643,345), (833,22)
X(22) = pole, with respect to circumcircle, of the de Longchamps line
X(22) = isotomic conjugate of the isogonal conjugate of X(206)
X(22) = tangential isogonal conjugate of X(6)
X(22) = crosspoint of X(3) and X(159) wrt both the excentral and tangential triangles
X(22) = homothetic center of the tangential triangle and the orthic triangle of the anticomplementary triangle
X(22) = insimilicenter of circumcircle and tangential circle when ABC is acute
X(22) = exsimilicenter of circumcircle and tangential circle when ABC is obtuse
X(22) = inverse-in-de-Longchamps-circle of X(5189)
X(22) = inverse-in-{circumcircle, nine-point circle}-inverter of X(2072)
X(22) = X(75)-isoconjugate of X(2353)
X(22) = trilinear pole of line X(2485)X(8673)
X(22) = homothetic center of anticomplementary and Ara triangles
X(22) = Thomson-isogonal conjugate of X(5654)
X(22) = Lucas-isogonal conjugate of X(11459)
X(22) = X(19)-isoconjugate of X(14376)


X(23) = FAR-OUT POINT

Trilinears    a[b4 + c4 - a4 - b2c2] : :
Barycentrics  2 sin 2A - 3 tan ω : 2 sin 2B - 3 tan ω : 2 sin 2C - 3 tan ω     (M. Iliev, 5/13/07)
Tripolars    Sqrt[2(b^2 + c^2) - a^2] : :

X(23) = 9 R^2 X(2) - 2 SW X(3)

As a point on the Euler line, X(23) has Shinagawa coefficients (E + 4F, -4E - 4F).

Let A'B'C' be the antipedal triangle of X(3) (the tangential triangle). The circumcircles of AA'X(3), BB'X(3), CC'X(3) concur in two points: X(3) and X(23). (Randy Hutson, October 13, 2015)

Let A'B'C' be the anti-orthocentroidal triangle. Let A″ be the reflection of A' in line BC, and define B″ and C″ cyclically. Then X(23) is the centroid of A″B″C″; see X(9140), X(11002). (Randy Hutson, December 10, 2016)

If you have The Geometer's Sketchpad, you can view Far-out point.
If you have GeoGebra, you can view Far-out point.

X(23) lies on the Parry circle, anti-Brocard circle, anti-McCay circumcircle, and these lines: {2, 3}, {6, 353}, {8, 8185}, {10, 9591}, {15, 11629}, {16, 11630}, {31, 5363}, {32, 5354}, {35, 5297}, {36, 5370}, {49, 10263}, {50, 2493}, {51, 575}, {52, 1614}, {54, 5446}, {55, 5160}, {56, 7286}, {61, 34395}, {62, 34394}, {67, 8262}, {69, 20987}, {74, 9060}, {75, 21408}, {76, 26233}, {94, 98}, {99, 2770}, {100, 2752}, {105, 1290}, {107, 2697}, {108, 37798}, {110, 323}, {111, 187}, {112, 14580}, {114, 40604}, {125, 29012}, {137, 14652}, {141, 32218}, {143, 1199}, {145, 9798}, {146, 2931}, {147, 39828}, {148, 19577}, {154, 1993}, {156, 6243}, {157, 17008}, {159, 193}, {160, 7777}, {161, 6515}, {182, 5640}, {183, 2453}, {184, 576}, {194, 15652}, {195, 14449}, {206, 18882}, {216, 10985}, {230, 11063}, {232, 250}, {238, 27679}, {251, 1194}, {262, 7578}, {316, 5099}, {324, 1629}, {325, 3447}, {343, 3410}, {351, 9213}, {352, 2502}, {373, 5092}, {385, 523}, {388, 9658}, {390, 10833}, {394, 35264}, {477, 1302}, {497, 9673}, {512, 9138}, {515, 9625}, {516, 9590}, {524, 2930}, {530, 13859}, {531, 13858}, {542, 15360}, {568, 15032}, {569, 9781}, {574, 15302}, {577, 15355}, {598, 11628}, {612, 7298}, {614, 5345}, {647, 13114}, {667, 9980}, {671, 3455}, {675, 2690}, {689, 14603}, {895, 1177}, {896, 24436}, {925, 40118}, {935, 2373}, {946, 9626}, {1030, 37675}, {1078, 26235}, {1112, 18449}, {1147, 26882}, {1151, 33502}, {1152, 33503}, {1154, 5609}, {1173, 11692}, {1176, 9969}, {1180, 7772}, {1184, 22331}, {1196, 1627}, {1204, 12279}, {1216, 43598}, {1283, 3724}, {1287, 9076}, {1291, 5966}, {1296, 10102}, {1297, 1304}, {1300, 16167}, {1311, 2689}, {1324, 37764}, {1350, 15066}, {1351, 11004}, {1379, 6141}, {1380, 6142}, {1384, 40126}, {1460, 30652}, {1473, 23958}, {1486, 20061}, {1493, 16982}, {1501, 3981}, {1503, 3448}, {1511, 37477}, {1531, 22109}, {1533, 2777}, {1587, 35776}, {1588, 35777}, {1609, 37689}, {1621, 20988}, {1634, 7840}, {1691, 3124}, {1692, 39024}, {1799, 3456}, {1843, 19121}, {1853, 15579}, {1915, 20859}, {1916, 17938}, {1974, 11511}, {1975, 9464}, {1976, 13137}, {2076, 3231}, {2079, 5913}, {2080, 5191}, {2353, 40232}, {2421, 33928}, {2452, 7766}, {2459, 7599}, {2460, 7598}, {2492, 10561}, {2548, 9700}, {2549, 9699}, {2550, 9713}, {2551, 9712}, {2687, 9058}, {2688, 9057}, {2691, 9061}, {2692, 9083}, {2693, 9064}, {2694, 9107}, {2695, 9056}, {2696, 9084}, {2758, 9059}, {2766, 26703}, {2780, 32222}, {2781, 15139}, {2782, 5987}, {2854, 12367}, {2888, 12134}, {2916, 3589}, {2917, 16252}, {2936, 8591}, {2967, 16978}, {2979, 9306}, {2981, 10613}, {3043, 20773}, {3047, 15647}, {3055, 15109}, {3066, 5085}, {3098, 5651}, {3218, 3220}, {3219, 5285}, {3240, 37576}, {3303, 10149}, {3304, 17024}, {3314, 16335}, {3457, 36759}, {3458, 36760}, {3563, 10420}, {3564, 12310}, {3565, 40119}, {3581, 5663}, {3600, 18954}, {3617, 8193}, {3620, 37485}, {3621, 12410}, {3622, 11365}, {3623, 8192}, {3704, 33091}, {3743, 3746}, {3767, 33802}, {3796, 5422}, {3849, 14682}, {3934, 10130}, {3935, 40910}, {4265, 37633}, {4442, 23848}, {4550, 16261}, {5017, 9463}, {5028, 44116}, {5029, 41185}, {5032, 32621}, {5038, 13410}, {5078, 16686}, {5096, 37680}, {5097, 44109}, {5106, 5162}, {5111, 20976}, {5134, 24055}, {5143, 6187}, {5158, 10311}, {5166, 32740}, {5168, 41183}, {5171, 38528}, {5205, 26262}, {5206, 39576}, {5210, 20481}, {5225, 9672}, {5229, 9659}, {5261, 10831}, {5274, 10832}, {5304, 16303}, {5314, 27065}, {5322, 5563}, {5329, 17127}, {5347, 32911}, {5358, 24883}, {5412, 11418}, {5413, 11417}, {5462, 38848}, {5467, 5968}, {5480, 13394}, {5520, 26231}, {5523, 8428}, {5607, 9163}, {5608, 9162}, {5642, 19924}, {5643, 5943}, {5650, 14810}, {5866, 14360}, {5888, 15082}, {5889, 6759}, {5907, 7691}, {5921, 37488}, {5938, 20099}, {5944, 37472}, {5965, 24981}, {5972, 29317}, {5978, 14368}, {5979, 14369}, {5984, 9861}, {5986, 38664}, {5990, 26249}, {5991, 26277}, {6000, 15054}, {6038, 33997}, {6054, 36829}, {6090, 33878}, {6101, 18350}, {6104, 6109}, {6105, 6108}, {6151, 10614}, {6153, 10203}, {6193, 32048}, {6403, 44080}, {6453, 8854}, {6454, 8855}, {6530, 37766}, {6566, 7602}, {6567, 7601}, {6593, 9019}, {6776, 37644}, {6781, 10418}, {6795, 16311}, {7083, 30653}, {7291, 37782}, {7293, 27003}, {7295, 17126}, {7665, 14712}, {7669, 22329}, {7684, 8838}, {7685, 8836}, {7689, 12290}, {7711, 9155}, {7728, 32227}, {7735, 16306}, {7736, 9609}, {7738, 9608}, {7767, 34992}, {7779, 16316}, {7782, 11059}, {7783, 31088}, {7785, 23208}, {7806, 33801}, {7816, 30749}, {7823, 15270}, {7825, 30747}, {7885, 31076}, {7911, 30785}, {7928, 31124}, {8024, 16276}, {8290, 38998}, {8538, 44077}, {8542, 11188}, {8585, 8588}, {8586, 39689}, {8593, 9966}, {8596, 9876}, {8644, 9137}, {8680, 24322}, {8717, 37470}, {8718, 40647}, {8744, 10317}, {8859, 16092}, {8996, 43133}, {9070, 12030}, {9123, 11616}, {9135, 9212}, {9140, 11645}, {9157, 15562}, {9185, 14270}, {9189, 39477}, {9301, 9999}, {9420, 22734}, {9534, 9571}, {9535, 9570}, {9538, 9645}, {9540, 9683}, {9541, 9682}, {9542, 9695}, {9543, 9694}, {9545, 9707}, {9780, 37557}, {9827, 11817}, {9833, 34799}, {9871, 9879}, {9911, 20070}, {9912, 20085}, {9917, 20081}, {9918, 20088}, {9921, 12221}, {9922, 12222}, {9924, 40318}, {9968, 41715}, {9971, 19127}, {9972, 41713}, {9979, 42659}, {10046, 14986}, {10095, 13353}, {10110, 13434}, {10282, 34148}, {10329, 20965}, {10330, 12215}, {10355, 13492}, {10528, 26309}, {10529, 26308}, {10539, 11412}, {10541, 10601}, {10564, 15035}, {10641, 11421}, {10642, 11420}, {10721, 12893}, {10722, 39825}, {10723, 39854}, {10733, 13289}, {10984, 15043}, {11064, 15448}, {11078, 41023}, {11092, 41022}, {11130, 14539}, {11131, 14538}, {11134, 36980}, {11137, 36978}, {11141, 11549}, {11142, 11537}, {11381, 11440}, {11402, 11482}, {11416, 44102}, {11438, 15072}, {11441, 17834}, {11442, 31383}, {11449, 13346}, {11451, 43650}, {11456, 37489}, {11459, 15052}, {11464, 13352}, {11472, 41398}, {11550, 23293}, {11557, 40640}, {11574, 41464}, {11643, 34320}, {11671, 15959}, {11793, 43614}, {11809, 26228}, {12111, 26883}, {12160, 14530}, {12254, 12370}, {12307, 31834}, {12319, 32123}, {12359, 16659}, {12364, 12380}, {12384, 14731}, {13175, 20094}, {13203, 32125}, {13222, 20095}, {13336, 15024}, {13339, 13363}, {13349, 41473}, {13350, 41472}, {13366, 21849}, {13367, 13598}, {13391, 22115}, {13445, 15021}, {13451, 15038}, {13470, 43821}, {13474, 15062}, {13754, 14094}, {13851, 15044}, {14173, 37786}, {14174, 25225}, {14179, 37785}, {14180, 25226}, {14262, 38338}, {14611, 25045}, {14669, 38679}, {14671, 15564}, {14703, 34549}, {14704, 15753}, {14705, 15754}, {14805, 34513}, {14906, 30541}, {14918, 20625}, {14927, 37643}, {14984, 40114}, {14996, 36740}, {14997, 36741}, {15004, 22234}, {15026, 37471}, {15028, 37515}, {15033, 18475}, {15034, 43574}, {15039, 40111}, {15059, 29323}, {15068, 37494}, {15077, 34438}, {15361, 20126}, {15516, 44107}, {15574, 15589}, {15577, 35260}, {15655, 21448}, {15786, 39371}, {15801, 32379}, {16119, 20084}, {16166, 29011}, {16186, 30510}, {16187, 33879}, {16272, 33925}, {16313, 40896}, {16321, 16990}, {16324, 16989}, {16332, 29831}, {16463, 36211}, {16464, 36210}, {16510, 41720}, {16760, 23217}, {16776, 32154}, {16823, 26261}, {16835, 43689}, {16836, 43584}, {16881, 43845}, {17100, 37762}, {17128, 31078}, {17497, 18617}, {17508, 22112}, {17984, 23962}, {18125, 34437}, {18487, 41358}, {18860, 38704}, {19128, 44084}, {19189, 43768}, {20079, 34207}, {20080, 37491}, {20185, 23096}, {21009, 23406}, {21401, 34424}, {21402, 34425}, {21659, 41482}, {21766, 31884}, {21969, 34986}, {21970, 26869}, {22113, 22657}, {22114, 22656}, {22239, 34168}, {22687, 25233}, {22689, 25234}, {23395, 23862}, {24164, 30117}, {24650, 44123}, {24651, 44124}, {24687, 25344}, {25328, 35218}, {25739, 36253}, {27866, 41671}, {30716, 30737}, {31606, 38337}, {31652, 38862}, {31817, 43609}, {32171, 37495}, {32235, 40291}, {32239, 32257}, {32428, 38552}, {32479, 42008}, {32609, 37496}, {32624, 37761}, {32625, 37763}, {32739, 41323}, {33155, 41230}, {33582, 33861}, {33873, 36213}, {34137, 38356}, {34224, 41587}, {35266, 40112}, {35356, 39099}, {35360, 41204}, {36201, 41603}, {36414, 36417}, {36849, 39644}, {36990, 37638}, {37538, 37685}, {37801, 38971}, {38225, 38611}, {38672, 38678}, {40130, 41413}, {40911, 40917}, {41583, 41721}, {41596, 41732}, {41607, 41742}, {41613, 41744}, {43829, 43838}

X(23) is the {X(22),X(25)}-harmonic conjugate of X(2). For a list of other harmonic conjugates of X(23), click Tables at the top of this page.

X(23) = reflection of X(i) in X(j) for these (i,j): (110,1495), (323,110), (691,187), (858,468)
X(23) = isogonal conjugate of X(67)
X(23) = isotomic conjugate of X(18019)
X(23) = inverse-in-circumcircle of X(2)
X(23) = anticomplement of X(858)
X(23) = anticomplementary conjugate of X(2892)
X(23) = crosspoint of X(111) and X(251)
X(23) = crosssum of X(i) and X(j) for these (i,j): (125,690), (141,524)
X(23) = crossdifference of every pair of points on line X(39)X(647)
X(23) = complement of X(5189)
X(23) = perspector of ABC and reflection of circummedial triangle in the Euler line
X(23) = antigonal image of X(316)
X(23) = trilinear pole of line X(2492)X(6593)
X(23) = reflection of X(858) in the orthic axis
X(23) = reflection of X(110) in the Lemoine axis
X(23) = polar conjugate of isotomic conjugate of X(22151)
X(23) = X(352)-of-circumsymmedial-triangle
X(23) = X(110)-of-1st-anti-Brocard-triangle
X(23) = crosspoint of X(3) and X(2930) wrt both the excentral and tangential triangles
X(23) = inverse-in-circumcircle of X(2)
X(23) = inverse-in-polar-circle of X(427)
X(23) = inverse-in-{circumcircle, nine-point circle}-inverter of X(5)
X(23) = inverse-in-de-Longchamps-circle of X(1370)
X(23) = X(75)-isoconjugate of X(3455)
X(23) = common radical trace of similitude circles of pairs of the Stammler circles
X(23) = one of two harmonic traces of Ehrmann circles; the other is X(6)
X(23) = X(111)-of-anti-McCay-triangle
X(23) = X(691)-of-1st-Parry-triangle
X(23) = X(842)-of-2nd-Parry-triangle
X(23) = X(1296)-of-3rd-Parry-triangle
X(23) = inverse-in-Parry-isodynamic-circle of X(352) (see X(2))
X(23) = X(111)-of-4th-anti-Brocard-triangle
X(23) = similitude center of anti-McCay and 4th anti-Brocard triangles
X(23) = anti-Artzt-to-4th-anti-Brocard similarity image of X(110)
X(23) = intersection of de Longchamps lines of 1st and 2nd Ehrmann circumscribing triangles
X(23) = intersection of orthic axes of antipedal triangles of PU(1)
X(23) = intersection of de Longchamps lines of anticevian triangles of PU(4)
X(23) = circumtangential isogonal conjugate of X(32305)
X(23) = inverse of X(33502) in the Lucas circles radical circle
X(23) = inverse of X(33503) in the Lucas(-1) circles radical circle
X(23) = trilinear pole, wrt 1st Parry triangle, of line X(110)X(1296)
X(23) = X(19)-isoconjugate of X(34897)
X(23) = X(63)-isoconjugate of X(8791)


X(24) = PERSPECTOR OF ABC AND ORTHIC-OF-ORTHIC TRIANGLE

Trilinears    sec A cos 2A : sec B cos 2B : sec C cos 2C
Tllrilinears    sec A - 2 cos A : sec B - 2 cos B : sec C - 2 cos C
Barycentrics    tan A cos 2A : tan B cos 2B : tan C cos 2C
Barycentrics    tan A - sin 2A : tan B - sin 2B : tan C - sin 2C
Barycentrics    a(sec A - 2 cos A) : :
Barycentrics    a^2 (a^4 + b^4 + c^4 - 2 a^2 b^2 - 2 a^2 c^2)/(a^2 - b^2 - c^2) : :
X(24) = 6 X(2) + (J^2 - 5) X(3)
X(24) = (a^2 - b^2 - c^2) (a^2 + b^2 - c^2) (a^2 - b^2 + c^2) X(3) + 2 a^2 b^2 c^2 X(4)

As a point on the Euler line, X(24) has Shinagawa coefficients (2F, -E - 2F).

Let A'B'C' be the orthic triangle. Let A″ = inverse-in-circumcircle of A', and define B'' and C'' cyclically. The lines AA″, BB″, CC″ concur in X(24). (Randy Hutson, September 5, 2015)

X(24) = homothetic center of the tangential triangle and the triangle obtained by reflecting X(4) in the sidelines of ABC.

If you have The Geometer's Sketchpad, you can view X(24).
If you have GeoGebra, you can view X(24).

X(24) lies on these lines: {1, 1061}, {2, 3}, {6, 54}, {10, 15177}, {11, 9672}, {12, 9659}, {15, 10642}, {16, 10641}, {19, 2337}, {32, 232}, {33, 35}, {34, 36}, {39, 10311}, {40, 9625}, {49, 568}, {51, 578}, {52, 1147}, {53, 8553}, {55, 6197}, {56, 1870}, {58, 3192}, {60, 36742}, {61, 8739}, {62, 8740}, {64, 74}, {66, 34439}, {68, 3580}, {69, 37488}, {70, 34438}, {96, 847}, {98, 1289}, {99, 39828}, {101, 41320}, {102, 41401}, {104, 3435}, {107, 1093}, {108, 915}, {110, 155}, {111, 8428}, {112, 2079}, {113, 12893}, {114, 39825}, {115, 39854}, {125, 12140}, {132, 34217}, {136, 5961}, {137, 23320}, {143, 32171}, {154, 1181}, {156, 6102}, {157, 31381}, {159, 6776}, {161, 6146}, {165, 9591}, {182, 1843}, {183, 1235}, {184, 389}, {185, 1495}, {187, 1968}, {195, 6242}, {197, 11491}, {206, 19161}, {208, 37583}, {220, 26915}, {221, 19368}, {225, 36152}, {230, 27376}, {242, 1602}, {250, 18879}, {254, 393}, {264, 1078}, {275, 19185}, {278, 7742}, {281, 15817}, {317, 7763}, {340, 7796}, {371, 5413}, {372, 5412}, {386, 9570}, {388, 10037}, {394, 11412}, {399, 7722}, {476, 15112}, {477, 22239}, {487, 12972}, {488, 12973}, {497, 10046}, {498, 11392}, {499, 11393}, {511, 1092}, {515, 8185}, {517, 11363}, {539, 41598}, {569, 5422}, {571, 8745}, {572, 44103}, {573, 1474}, {574, 9700}, {575, 8541}, {576, 44102}, {581, 2299}, {584, 44095}, {601, 2212}, {602, 1395}, {648, 6179}, {842, 10423}, {912, 41609}, {917, 26705}, {925, 39437}, {933, 2383}, {935, 14729}, {944, 1610}, {953, 10016}, {958, 9713}, {974, 15647}, {1033, 33630}, {1058, 16541}, {1063, 1775}, {1075, 3186}, {1112, 1511}, {1141, 15959}, {1151, 3092}, {1152, 3093}, {1172, 36744}, {1173, 14528}, {1192, 1498}, {1199, 11402}, {1204, 6000}, {1216, 15066}, {1249, 8573}, {1294, 30249}, {1297, 39417}, {1304, 14264}, {1324, 1603}, {1351, 19118}, {1376, 9712}, {1384, 3172}, {1385, 1829}, {1407, 26914}, {1473, 26877}, {1485, 6530}, {1503, 11457}, {1533, 22962}, {1560, 14655}, {1604, 18283}, {1619, 18913}, {1620, 10606}, {1627, 3162}, {1748, 42700}, {1753, 2077}, {1824, 32613}, {1828, 32612}, {1853, 23294}, {1857, 7040}, {1861, 25440}, {1862, 33814}, {1871, 2355}, {1872, 26285}, {1876, 37582}, {1878, 23961}, {1892, 11374}, {1899, 9833}, {1900, 33862}, {1902, 3579}, {1905, 2646}, {1971, 39643}, {1975, 28706}, {1994, 9545}, {2080, 11380}, {2192, 11461}, {2203, 5752}, {2211, 5017}, {2332, 4262}, {2356, 37576}, {2373, 30251}, {2374, 30247}, {2501, 5926}, {2698, 32542}, {2781, 38851}, {2914, 12316}, {2918, 11387}, {2930, 32234}, {2935, 10721}, {2936, 23235}, {2965, 8746}, {2967, 44162}, {2979, 37486}, {3043, 19504}, {3044, 39810}, {3047, 19456}, {3060, 11449}, {3068, 8276}, {3069, 8277}, {3085, 10831}, {3086, 10832}, {3087, 31400}, {3095, 44089}, {3098, 12294}, {3100, 9645}, {3167, 12160}, {3197, 11460}, {3259, 39479}, {3292, 14531}, {3311, 5411}, {3312, 5410}, {3357, 11381}, {3426, 11270}, {3431, 3527}, {3432, 25044}, {3438, 3442}, {3439, 3443}, {3447, 14385}, {3448, 12412}, {3455, 11623}, {3532, 16835}, {3564, 41584}, {3574, 23358}, {3576, 7713}, {3581, 15068}, {3767, 5523}, {3785, 15574}, {3796, 15045}, {3964, 32001}, {4293, 18954}, {4294, 10833}, {5012, 15043}, {5013, 9609}, {5023, 33885}, {5050, 12167}, {5085, 7716}, {5090, 26446}, {5092, 19124}, {5095, 12584}, {5185, 38599}, {5186, 33813}, {5307, 39578}, {5338, 15931}, {5347, 36745}, {5359, 40938}, {5398, 44113}, {5446, 12038}, {5449, 18474}, {5480, 35228}, {5562, 9306}, {5594, 10784}, {5595, 10783}, {5603, 11365}, {5609, 13148}, {5621, 32274}, {5622, 8549}, {5640, 13434}, {5651, 11793}, {5657, 7718}, {5667, 14673}, {5690, 12135}, {5706, 38879}, {5878, 32111}, {5879, 40082}, {5892, 13336}, {5938, 41377}, {5944, 5946}, {5962, 16391}, {5963, 32762}, {5972, 15473}, {6101, 37494}, {6104, 8737}, {6105, 8738}, {6193, 6515}, {6200, 9683}, {6221, 9694}, {6243, 16266}, {6247, 16655}, {6284, 9673}, {6291, 12974}, {6293, 15139}, {6361, 9911}, {6396, 11474}, {6406, 12975}, {6413, 8954}, {6414, 32589}, {6458, 26886}, {6560, 35776}, {6561, 35777}, {6593, 11477}, {6684, 37557}, {6696, 16621}, {6749, 9606}, {6770, 9916}, {6771, 12142}, {6773, 9915}, {6774, 12141}, {6799, 33643}, {6800, 9730}, {7085, 26878}, {7293, 37534}, {7354, 9658}, {7581, 19006}, {7582, 19005}, {7583, 13884}, {7584, 13937}, {7649, 39225}, {7687, 32607}, {7688, 11471}, {7689, 12162}, {7690, 12298}, {7691, 11444}, {7692, 12299}, {7709, 22655}, {7717, 21151}, {7731, 17847}, {7735, 41361}, {7749, 27371}, {7754, 41676}, {7786, 36794}, {7952, 8069}, {7967, 8192}, {7999, 17811}, {8071, 34231}, {8127, 8131}, {8128, 8132}, {8148, 31948}, {8190, 11843}, {8191, 11844}, {8194, 11846}, {8195, 11847}, {8262, 15069}, {8550, 15582}, {8718, 22948}, {8754, 34218}, {8780, 12164}, {8883, 14518}, {8939, 19424}, {8943, 19425}, {8982, 8996}, {9638, 19354}, {9704, 15087}, {9706, 11422}, {9729, 10984}, {9737, 44099}, {9744, 23208}, {9777, 11426}, {9781, 10982}, {9861, 9862}, {9876, 12243}, {9908, 11411}, {9910, 12246}, {9912, 12247}, {9913, 12248}, {9914, 12250}, {9917, 12251}, {9918, 12252}, {9919, 12244}, {9920, 12254}, {9921, 12256}, {9922, 12257}, {9924, 12283}, {9925, 11271}, {9934, 17854}, {9938, 12278}, {9967, 26206}, {10098, 40119}, {10110, 11424}, {10192, 12233}, {10246, 11396}, {10267, 11383}, {10269, 22479}, {10274, 32352}, {10310, 20872}, {10313, 23115}, {10519, 37485}, {10539, 11441}, {10540, 32139}, {10546, 15056}, {10564, 43898}, {10574, 15053}, {10601, 15024}, {10610, 11576}, {10645, 11475}, {10646, 11476}, {10722, 39841}, {10723, 39812}, {10733, 12302}, {10785, 10829}, {10786, 10830}, {10788, 10790}, {10805, 10834}, {10806, 10835}, {11204, 32062}, {11206, 18909}, {11245, 31804}, {11362, 37546}, {11386, 26316}, {11388, 26341}, {11389, 26348}, {11390, 26492}, {11391, 26487}, {11400, 16203}, {11401, 16202}, {11408, 11486}, {11409, 11485}, {11423, 17809}, {11431, 32621}, {11433, 18925}, {11439, 11454}, {11440, 15305}, {11442, 12134}, {11458, 17813}, {11459, 17814}, {11462, 17819}, {11463, 17820}, {11465, 17825}, {11466, 17826}, {11467, 17827}, {11496, 20988}, {11500, 20989}, {11508, 15500}, {11550, 13419}, {11572, 23325}, {11589, 39268}, {11616, 14273}, {11641, 13200}, {11695, 43650}, {11704, 32345}, {11745, 23292}, {11750, 43817}, {11898, 12325}, {11935, 15110}, {12022, 19467}, {12041, 12133}, {12042, 12131}, {12095, 34756}, {12111, 12163}, {12112, 12315}, {12115, 26309}, {12116, 26308}, {12118, 32048}, {12136, 34862}, {12137, 12619}, {12138, 38602}, {12145, 38624}, {12174, 32063}, {12228, 16222}, {12235, 27365}, {12236, 15317}, {12245, 12410}, {12249, 12411}, {12253, 12413}, {12255, 12414}, {12281, 17835}, {12282, 17836}, {12284, 17838}, {12285, 17839}, {12286, 17842}, {12287, 17840}, {12288, 17843}, {12289, 17845}, {12295, 12901}, {12296, 12984}, {12297, 12985}, {12310, 12383}, {12324, 18931}, {12509, 12978}, {12510, 12979}, {12675, 41611}, {12828, 30714}, {13017, 17841}, {13018, 17844}, {13019, 13061}, {13020, 13062}, {13035, 13055}, {13036, 13056}, {13049, 13051}, {13050, 13052}, {13093, 34469}, {13166, 38608}, {13172, 13175}, {13199, 13222}, {13202, 13293}, {13292, 37644}, {13321, 14627}, {13323, 44092}, {13346, 44079}, {13363, 34513}, {13417, 17701}, {13429, 44193}, {13440, 44192}, {13450, 22261}, {13452, 43719}, {13472, 43908}, {13561, 34514}, {13568, 15448}, {13674, 13680}, {13794, 13800}, {13851, 34786}, {13886, 13889}, {13939, 13943}, {14111, 34835}, {14216, 16659}, {14270, 16230}, {14581, 35007}, {14644, 18394}, {14649, 20410}, {14651, 39832}, {14708, 20773}, {14831, 43844}, {14900, 15562}, {14912, 19459}, {14915, 43604}, {14984, 41616}, {15004, 37505}, {15032, 19347}, {15035, 15472}, {15072, 43601}, {15080, 43584}, {15107, 37483}, {15135, 19362}, {15344, 26706}, {15460, 24650}, {15461, 24651}, {15475, 39606}, {15513, 33842}, {15581, 19596}, {15644, 43652}, {15873, 16657}, {16035, 19173}, {16116, 16119}, {16172, 32132}, {16229, 39537}, {16261, 43613}, {16473, 31760}, {16625, 34986}, {16654, 23328}, {16658, 43607}, {16836, 35268}, {17824, 32339}, {17837, 22535}, {17927, 38903}, {17984, 38907}, {18344, 39227}, {18374, 34117}, {18390, 21659}, {19123, 19132}, {19149, 43617}, {19168, 19180}, {19169, 19192}, {19414, 19430}, {19415, 19431}, {19416, 19454}, {19417, 19455}, {19440, 19446}, {19441, 19447}, {20477, 44131}, {21637, 23042}, {21639, 34788}, {22241, 32830}, {22352, 37515}, {22480, 40108}, {22531, 22656}, {22532, 22657}, {22533, 22658}, {22538, 22978}, {22661, 32123}, {23096, 38545}, {24206, 32348}, {24320, 28731}, {24817, 24822}, {25739, 26917}, {26286, 40985}, {26302, 26381}, {26303, 26405}, {26304, 26439}, {26305, 26440}, {26306, 26441}, {26371, 26398}, {26372, 26422}, {26373, 26498}, {26374, 26507}, {26375, 26516}, {26376, 26521}, {26704, 32706}, {26869, 43808}, {26896, 26909}, {26916, 26953}, {28724, 40801}, {30250, 39439}, {31860, 41448}, {32137, 32210}, {32217, 40929}, {32247, 32262}, {32249, 32276}, {32250, 32305}, {32337, 32357}, {32340, 32401}, {32649, 39265}, {32704, 40101}, {33582, 41425}, {33586, 35602}, {33802, 43976}, {33843, 37512}, {34146, 43896}, {34292, 35719}, {34382, 40318}, {35217, 36990}, {35265, 43605}, {36103, 37817}, {36750, 44097}, {36754, 44105}, {37472, 39522}, {37475, 43597}, {37499, 38852}, {37638, 41171}, {37672, 43572}, {38457, 38906}, {38461, 38900}, {38462, 38901}, {38850, 41503}, {39478, 39534}, {39808, 39820}, {39809, 39831}, {39837, 39849}, {39838, 39860}, {39874, 39879}, {40321, 41400}, {40825, 41363}, {40981, 41371}, {43394, 43823}, {43818, 43829}

X(24) is the {X(3),X(4)}-harmonic conjugate of X(378). For a list of other harmonic conjugates of X(24), click Tables at the top of this page.

X(24) = reflection of X(4) in X(235)
X(24) = isogonal conjugate of X(68)
X(24) = isotomic conjugate of X(20563)
X(24) = complement of X(37444)
X(24) = complement of complement of X(31304)
X(24) = anticomplement of X(11585)
X(24) = perspector of ABC and reflection of X(4) in orthic triangle
X(24) = {X(3),X(25)}-harmonic conjugate of X(4)
X(24) = trilinear pole of line X(924)X(6753)
X(24) = circumcircle-inverse of X(403)
X(24) = orthocentroidal circle-inverse of X(1594)
X(24) = orthoptic-circle-of-Steiner-inellipse-inverse of X(37981)
X(24) = orthoptic-circle-of-Steiner-circumellipse-inverse of anticomplement of X(37929)
X(24) = de-Longchamps-circle-inverse of anticomplement of X(37951)
X(24) = X(249)-Ceva conjugate of X(112)
X(24) = X(52)-cross conjugate of X(4)
X(24) = crosspoint of X(107) and X(250)
X(24) = crosssum of X(i) and X(j) for these (i,j): (6,161), (125,520), (637,638)
X(24) = X(4)-Hirst inverse of X(421)
X(24) = X(46)-of-orthic-triangle if ABC is acute
X(24) = X(56)-of-the-tangential triangle if ABC is acute
X(24) = tangential isogonal conjugate of X(1498)
X(24) = exsimilicenter of circumcircle and tangential circle when ABC is acute (Yuda Chen, November 7, 2021)
X(24) = insimilicenter of circumcircle and tangential circle when ABC is obtuse
X(24) = inverse-in-polar-circle of X(2072)
X(24) = homothetic center of tangential and circumorthic triangles
X(24) = homothetic center of orthic and Kosnita triangles
X(24) = X(i)-isoconjugate of X(j) for these (i,j): (75,2351), (91,3)


X(25) = HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES

Trilinears    sin A tan A : :
Trilinears    a/(b2 + c2 - a2) : :
Trilinears    cos A - sec A : :
Barycentrics    tan A - tan ω : :
Barycentrics    sec A sin(A - ω) : :
X(25) = 6 R^2 X(2) - SW X(3)

As a point on the Euler line, X(25) has Shinagawa coefficients (F, -E - F).

Constructed as indicated by the name; also X(25) is the pole of the orthic axis (the line having trilinear coefficients cos A : cos B : cos C) with respect to the circumcircle.

If you have The Geometer's Sketchpad, you can view X(25).
If you have GeoGebra, you can view X(25).

Let A' be the center of the conic through the contact points of the incircle and the A-excircle with the sidelines of ABC. Define B' and C' cyclically. Let A″ be the center of the conic through the contact points of the B- and C- excircles with the sidelines of ABC. Define B″ and C″ cyclically. Let A* be the midpoint of A' and A″, and define B* and C* cyclically. The triangle A*B*C* is perspective to ABC at X(25). See also X(6), X(218), X(222), X(940), X(1743). (Randy Hutson, July 23, 2015)

Let A' be the radical center of the nine-point circle and the B- and C-power circles. efine B' and C' cyclically. The triangle A'B'C' is homothetic with the orthic triangle, and the center of homothety is X(25). Also X(25) is the point of intersection of these two lines: isotomic conjugate of polar conjugate of van Aubel line (i.e., line X(2)X(3)), and polar conjugate of isotomic conjugate of van Aubel line (i.e., line X(25)X(393)). Also, X(25) is the trilinear pole of line X(512)X(1692), this line being the isogonal conjugate of the isotomic conjugate of the orthic axis; the line X(512)X(1692) is also the polar of X(76) wrt polar circle, and the line is also the radical axis of circumcircle and 2nd Lemoine circle. (Randy Hutson, September 5, 2015)

Let A'B'C' be the orthic triangle. Let A″ be the barycentric product of the (real or imaginary) circumcircle intercepts of line B'C'. Define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X(25). (Randy Hutson, October 27, 2015)

The 2nd Ehrmann triangle, defined in the preamble to X(8537), can be generalized as follows. Let P be a point in the plane of ABC and not on BC∪CA∪AB. Let Ab the the point of intersection of the circle {P,B,C}} and the line AB, and define Bc and Ca cyclically. Define Ac symmetrically, and define Ba and Cb cyclically. Let A' = BcBa∩CaCb, and define B' and C' cyclically. Triangle A'B'C', here introduced as the P-Ehrmann triangle, is homothetic to the orthic triangle. The X(1)-Ehrmann triangle is the intangents triangle, and the X(6)-Ehrmann triangle is the 2nd Ehrmann triangle. If P lies on the circumcircle, the P-Ehrmann triangle is the tangential triangle. If P is on the Brocard 2nd cubic K018 or the circumcircle, then the P-Ehrmann triangle is perspective to ABC. The homothetic center of the orthic triangle and the X(4)-Ehrmann triangle is X(25). (Randy Hutson, February 10, 2016)

Let H(X) denote hodpoint of a point X, as defined in the preamble just before X(40139). Then H(X(2)) = H(X(8115)) = H(X(8116)) = X(25). (Radosław Żak, October 29, 2020)

Let Ga = reflection of X(6) in line BC, and define Gb and Gc cyclically. Let T = tangential triangle and H = orthic triangle. Then each pair of the triangles GaGbGc, T, and H are perspective, and the perspector in all three cases is X(25). (Dasari Naga Vijay Krishna, March 14, 2021)

In the plane of a triangle ABC, let
H = X(4) = orthocenter
D = AH∩BC
K = intersection of BH and line through D perpendicular to AB
L = intersection of CH and line through D perpendicular to AC
A' = intersection of KL with line through D perpendicular to KL, and define B' and C' cyclically.
The lines AA', BB', CC' concur in X(25). The triangle A'B'C' is here named the Iran triangle. (Jeffrey Liu, May 10, 2024).

Continuing, the with notes from Peter Moses, May 13, 2024, the vertex A' is given by

A' = (b^4 + c^4 - a^2 b^2 - a^2 c^2)/(a^2 - b^2 - c^2)
         : (c^4 + a^4 - b^2 c^2 - b^2 a^2)/(b^2 - c^2 - a^2)
         : (a^4 + b^4 - c^2 a^2 - c^2 b^2)/)c^2 - a^2 - b^2) ,

or, using Conway notation,
A' = SB*SC*(S^2 + SB*SC) : S^2*SC*(SA + SC) : S^2*SB*(SA + SB).

A'B'C' is the half-altitude triangle of the orthic triangle of ABC.

X(25) lies on these lines: {1, 1036}, {2, 3}, {6, 51}, {7, 7717}, {8, 7718}, {9, 5285}, {10, 5090}, {11, 10829}, {12, 10830}, {13, 9916}, {14, 9915}, {17, 22482}, {18, 22481}, {19, 33}, {31, 608}, {32, 1184}, {34, 56}, {35, 1900}, {36, 1878}, {39, 39951}, {40, 1902}, {41, 42}, {43, 37576}, {48, 14547}, {49, 36749}, {52, 155}, {53, 157}, {54, 3527}, {57, 1473}, {58, 967}, {63, 24320}, {64, 1192}, {65, 1452}, {66, 34207}, {67, 32239}, {68, 9908}, {69, 8263}, {72, 37547}, {74, 3426}, {76, 1241}, {79, 16114}, {80, 9912}, {81, 37492}, {83, 9918}, {84, 9910}, {92, 242}, {97, 14489}, {98, 107}, {99, 2374}, {100, 1862}, {101, 3190}, {104, 3420}, {105, 108}, {106, 9088}, {110, 1112}, {111, 112}, {113, 2931}, {114, 135}, {115, 3455}, {125, 1853}, {132, 136}, {133, 14703}, {137, 15959}, {141, 37485}, {143, 156}, {146, 34796}, {160, 3815}, {162, 37128}, {171, 7295}, {181, 2175}, {182, 3066}, {183, 264}, {185, 1498}, {187, 5140}, {190, 24814}, {193, 6339}, {195, 2904}, {200, 4006}, {209, 2911}, {210, 12329}, {212, 2183}, {216, 10314}, {219, 26885}, {220, 3690}, {221, 1425}, {222, 20122}, {225, 1842}, {226, 1892}, {236, 8132}, {238, 5329}, {244, 36570}, {250, 5968}, {251, 5359}, {262, 275}, {263, 2211}, {265, 12140}, {269, 40223}, {273, 1447}, {281, 7102}, {286, 1218}, {305, 683}, {317, 325}, {318, 7081}, {324, 42354}, {339, 1289}, {340, 7788}, {343, 1352}, {351, 14998}, {353, 11226}, {354, 22769}, {371, 493}, {372, 494}, {373, 5085}, {385, 2998}, {387, 38879}, {389, 1181}, {393, 1033}, {394, 511}, {399, 1986}, {459, 3424}, {476, 38552}, {485, 9922}, {486, 9921}, {487, 12169}, {488, 12170}, {497, 16541}, {512, 2433}, {516, 21062}, {518, 41611}, {524, 41585}, {542, 12828}, {543, 2936}, {568, 10540}, {571, 39109}, {573, 2328}, {574, 15433}, {576, 21849}, {577, 26953}, {578, 10110}, {581, 2360}, {588, 3311}, {589, 3312}, {597, 35707}, {599, 34992}, {604, 2208}, {610, 10382}, {647, 34212}, {648, 3228}, {667, 18344}, {669, 878}, {671, 9876}, {675, 2973}, {685, 40820}, {691, 40119}, {692, 913}, {694, 1613}, {800, 15259}, {842, 1304}, {847, 1179}, {884, 6591}, {895, 41616}, {908, 27388}, {915, 9058}, {917, 9057}, {925, 2974}, {933, 5966}, {935, 2770}, {940, 18165}, {941, 1172}, {954, 21319}, {958, 1891}, {973, 32341}, {974, 9934}, {982, 36572}, {993, 5155}, {999, 1870}, {1001, 1848}, {1007, 9723}, {1030, 39982}, {1062, 9645}, {1068, 26228}, {1073, 1297}, {1084, 16098}, {1092, 37498}, {1093, 8794}, {1096, 1402}, {1108, 40970}, {1119, 3598}, {1141, 15960}, {1147, 5446}, {1151, 6291}, {1152, 6406}, {1154, 15068}, {1162, 8904}, {1163, 8903}, {1164, 8974}, {1165, 13950}, {1169, 2189}, {1173, 43908}, {1177, 32246}, {1180, 3108}, {1216, 37486}, {1235, 1239}, {1249, 5304}, {1261, 7046}, {1291, 23096}, {1298, 22551}, {1299, 34333}, {1300, 1302}, {1309, 2726}, {1311, 21666}, {1324, 1785}, {1327, 13668}, {1328, 13788}, {1329, 9712}, {1350, 3917}, {1376, 1861}, {1383, 5354}, {1396, 42290}, {1397, 3271}, {1403, 8852}, {1407, 3937}, {1436, 7008}, {1437, 36742}, {1441, 26260}, {1466, 22344}, {1470, 1877}, {1482, 41722}, {1501, 1976}, {1503, 1619}, {1506, 9700}, {1511, 15472}, {1560, 2079}, {1565, 39732}, {1576, 2493}, {1604, 1863}, {1605, 6108}, {1606, 6109}, {1610, 3486}, {1611, 1968}, {1614, 3567}, {1620, 8567}, {1622, 38870}, {1626, 20470}, {1631, 1826}, {1633, 3474}, {1634, 9766}, {1637, 42659}, {1659, 30386}, {1698, 9591}, {1699, 9590}, {1716, 7093}, {1726, 1736}, {1730, 1754}, {1753, 10310}, {1790, 37474}, {1825, 23844}, {1830, 23845}, {1831, 23846}, {1838, 7742}, {1839, 8053}, {1840, 23851}, {1841, 2178}, {1857, 42069}, {1864, 2182}, {1866, 26437}, {1867, 39585}, {1869, 37601}, {1871, 10267}, {1872, 11248}, {1887, 11509}, {1897, 8851}, {1946, 40134}, {1989, 6103}, {1990, 5306}, {1994, 5093}, {2004, 36759}, {2005, 36760}, {2051, 9570}, {2053, 4426}, {2056, 13330}, {2076, 21001}, {2155, 2357}, {2181, 6187}, {2192, 3270}, {2198, 14974}, {2217, 3435}, {2262, 18621}, {2270, 7070}, {2332, 4258}, {2350, 5021}, {2353, 3767}, {2489, 6041}, {2502, 11173}, {2548, 23208}, {2697, 22239}, {2752, 2766}, {2781, 15106}, {2793, 34519}, {2854, 6096}, {2881, 42665}, {2883, 13568}, {2886, 9713}, {2914, 5898}, {2916, 31521}, {2917, 3574}, {2929, 5895}, {2930, 5095}, {2932, 5151}, {2934, 2963}, {2935, 13202}, {2968, 26703}, {2979, 10546}, {2981, 10632}, {3052, 8750}, {3064, 23865}, {3068, 13884}, {3069, 13937}, {3087, 7736}, {3098, 3819}, {3168, 9755}, {3197, 3611}, {3202, 27375}, {3259, 10016}, {3286, 39984}, {3292, 11470}, {3295, 3920}, {3305, 5314}, {3306, 7293}, {3357, 13474}, {3407, 37892}, {3425, 6530}, {3431, 3531}, {3456, 7755}, {3506, 35431}, {3511, 23173}, {3519, 41598}, {3532, 22334}, {3564, 6515}, {3572, 43925}, {3580, 11442}, {3581, 18435}, {3589, 3867}, {3679, 37546}, {3695, 5687}, {3705, 5081}, {3772, 23847}, {3794, 26625}, {3818, 21243}, {3868, 42461}, {3924, 8615}, {3933, 40123}, {4057, 7649}, {4108, 14618}, {4265, 37674}, {4383, 5347}, {4650, 24436}, {5012, 5050}, {5024, 37808}, {5092, 6688}, {5096, 37679}, {5120, 33854}, {5121, 40293}, {5146, 29681}, {5152, 32527}, {5174, 29641}, {5201, 8667}, {5203, 5866}, {5204, 5370}, {5210, 8585}, {5217, 7302}, {5248, 39579}, {5254, 9608}, {5292, 5358}, {5305, 41361}, {5307, 16678}, {5311, 17442}, {5324, 37642}, {5342, 16823}, {5364, 20678}, {5406, 9739}, {5407, 9738}, {5408, 9733}, {5409, 9732}, {5418, 9683}, {5462, 36752}, {5476, 32267}, {5480, 10192}, {5504, 20771}, {5512, 14657}, {5521, 14667}, {5523, 5938}, {5544, 6030}, {5562, 17814}, {5584, 11471}, {5587, 9625}, {5593, 18130}, {5596, 26926}, {5597, 8190}, {5598, 8191}, {5622, 12099}, {5644, 7712}, {5647, 42445}, {5650, 31884}, {5707, 18180}, {5889, 11441}, {5890, 11456}, {5891, 37478}, {5913, 21397}, {5926, 39533}, {5965, 41599}, {5986, 12188}, {5989, 6331}, {6000, 10605}, {6054, 20774}, {6088, 10103}, {6094, 33900}, {6102, 32139}, {6114, 31688}, {6115, 31687}, {6145, 32332}, {6146, 9833}, {6151, 10633}, {6193, 12309}, {6197, 10306}, {6200, 41438}, {6217, 19352}, {6218, 19351}, {6239, 12313}, {6241, 12315}, {6242, 12316}, {6243, 18350}, {6247, 16621}, {6289, 12973}, {6290, 12972}, {6344, 31676}, {6391, 12272}, {6396, 41437}, {6400, 12314}, {6423, 8577}, {6424, 8576}, {6561, 9682}, {6564, 8280}, {6565, 8281}, {6696, 16656}, {6749, 9300}, {6750, 15512}, {6751, 17849}, {6752, 41373}, {6753, 8651}, {6767, 29815}, {6776, 11206}, {7017, 17987}, {7028, 8131}, {7079, 40175}, {7160, 12139}, {7179, 7282}, {7283, 19799}, {7373, 17024}, {7607, 39284}, {7612, 8796}, {7664, 34517}, {7665, 8878}, {7687, 13289}, {7691, 15056}, {7722, 12308}, {7745, 15270}, {7746, 27371}, {7762, 19597}, {7766, 38262}, {7773, 37804}, {7774, 20794}, {7779, 22152}, {7784, 21248}, {7792, 17907}, {7828, 33802}, {7952, 39600}, {8024, 22241}, {8071, 15654}, {8105, 42668}, {8106, 42667}, {8157, 10214}, {8227, 9626}, {8266, 15271}, {8267, 22253}, {8278, 32577}, {8537, 11422}, {8550, 15581}, {8584, 15471}, {8588, 33880}, {8735, 23402}, {8745, 8882}, {8746, 14577}, {8749, 20975}, {8756, 15621}, {8840, 17984}, {8887, 31381}, {8901, 19174}, {8911, 26868}, {8939, 19404}, {8943, 19405}, {9056, 32706}, {9059, 40101}, {9060, 32710}, {9061, 26706}, {9070, 39439}, {9083, 32704}, {9084, 30247}, {9135, 11631}, {9209, 39201}, {9475, 38867}, {9659, 10895}, {9672, 10896}, {9694, 43512}, {9695, 43509}, {9704, 14627}, {9708, 29667}, {9709, 29679}, {9748, 38918}, {9756, 42400}, {9792, 19170}, {9822, 19126}, {9927, 19908}, {9932, 22660}, {9935, 11577}, {9993, 14165}, {10095, 32046}, {10098, 10102}, {10111, 12419}, {10263, 16266}, {10266, 12146}, {10272, 11566}, {10274, 11808}, {10278, 41357}, {10313, 15355}, {10317, 36414}, {10478, 17188}, {10519, 33522}, {10535, 11436}, {10536, 11435}, {10545, 11451}, {10571, 41401}, {10606, 21663}, {10620, 12292}, {10643, 11516}, {10644, 11515}, {10961, 11514}, {10963, 11513}, {10974, 16471}, {10984, 37514}, {11003, 34545}, {11064, 31670}, {11174, 36794}, {11175, 20965}, {11179, 20192}, {11188, 41614}, {11202, 11430}, {11412, 43598}, {11424, 11425}, {11427, 14853}, {11439, 11440}, {11444, 43614}, {11457, 16659}, {11459, 33523}, {11464, 15033}, {11472, 16194}, {11475, 11480}, {11476, 11481}, {11487, 16543}, {11574, 19137}, {11580, 40103}, {11695, 37515}, {11743, 32391}, {11745, 12233}, {11746, 13198}, {11817, 15047}, {12007, 15580}, {12162, 12163}, {12220, 26206}, {12228, 20773}, {12235, 19458}, {12236, 19456}, {12237, 19461}, {12238, 19462}, {12239, 19463}, {12240, 19464}, {12241, 15873}, {12242, 19468}, {12279, 43601}, {12290, 13093}, {12293, 12301}, {12295, 12302}, {12296, 12303}, {12297, 12304}, {12298, 12305}, {12299, 12306}, {12300, 12307}, {12311, 12509}, {12312, 12510}, {12324, 18913}, {12335, 40953}, {12420, 12421}, {12429, 14516}, {13007, 13051}, {13008, 13052}, {13013, 19465}, {13014, 19466}, {13019, 13021}, {13020, 13022}, {13023, 13035}, {13024, 13036}, {13148, 14094}, {13233, 36523}, {13321, 15087}, {13336, 15805}, {13346, 13598}, {13390, 30385}, {13394, 14561}, {13403, 34785}, {13417, 17847}, {13419, 18381}, {13450, 34449}, {13507, 13597}, {13754, 18451}, {13851, 18405}, {13858, 36330}, {13859, 35752}, {14216, 16655}, {14264, 35372}, {14458, 16080}, {14490, 43713}, {14492, 43530}, {14529, 42450}, {14535, 32581}, {14683, 18947}, {14713, 14715}, {14810, 16187}, {14845, 37513}, {14852, 18474}, {14855, 35237}, {15053, 15072}, {15069, 41586}, {15121, 41603}, {15126, 15127}, {15135, 34117}, {15139, 37473}, {15141, 38851}, {15300, 33850}, {15302, 38862}, {15321, 34436}, {15462, 16165}, {15463, 32609}, {15475, 15551}, {15589, 32000}, {15591, 40321}, {15651, 40052}, {15655, 20481}, {15668, 17171}, {15740, 43690}, {15741, 32605}, {16010, 32250}, {16178, 16188}, {16221, 42426}, {16231, 39225}, {16263, 22455}, {16277, 43678}, {16317, 36878}, {16583, 18616}, {16776, 19127}, {16778, 39954}, {16817, 19798}, {16835, 43719}, {16974, 21010}, {17054, 24163}, {17808, 40124}, {17824, 32352}, {17830, 35711}, {17835, 21650}, {17836, 21651}, {17837, 21652}, {17838, 21649}, {17839, 21653}, {17840, 21655}, {17841, 21657}, {17842, 21654}, {17843, 21656}, {17844, 21658}, {17845, 21659}, {17846, 21660}, {17924, 26249}, {17983, 18818}, {18020, 31632}, {18390, 18396}, {18475, 37506}, {18613, 23710}, {18615, 22363}, {18619, 41015}, {18651, 24701}, {18755, 39967}, {18906, 37894}, {18907, 41370}, {18909, 34781}, {18912, 34224}, {18914, 18916}, {18928, 25406}, {18950, 39874}, {18997, 19039}, {18998, 19040}, {19140, 40291}, {19149, 19161}, {19169, 19172}, {19180, 21638}, {19349, 19366}, {19358, 19410}, {19359, 19411}, {19418, 19424}, {19419, 19425}, {19430, 21642}, {19431, 21643}, {19460, 22530}, {19583, 40324}, {19724, 19756}, {19725, 19763}, {20032, 20034}, {20266, 26933}, {20271, 21771}, {20423, 35266}, {20468, 42071}, {20791, 43584}, {21148, 40934}, {21461, 34394}, {21462, 34395}, {21661, 22552}, {21850, 37645}, {21851, 34779}, {22080, 37499}, {22109, 36518}, {22240, 23635}, {22331, 36616}, {22466, 22483}, {22538, 22549}, {22662, 22953}, {23039, 37494}, {23180, 36849}, {23224, 42772}, {23291, 32064}, {23359, 41011}, {23361, 26357}, {23675, 28037}, {23858, 42070}, {24682, 25343}, {24686, 25344}, {24855, 43618}, {26227, 41013}, {26235, 44142}, {26262, 38462}, {26266, 44143}, {26269, 41375}, {26275, 39200}, {26877, 26928}, {26878, 26938}, {26886, 26894}, {26907, 26909}, {26918, 26936}, {27365, 41615}, {27370, 40643}, {28476, 32691}, {28782, 28783}, {30249, 34168}, {30687, 31394}, {30737, 44131}, {31382, 35709}, {32001, 37668}, {32078, 40674}, {32137, 32138}, {32145, 32166}, {32234, 32254}, {32260, 32276}, {32264, 32285}, {32340, 32345}, {32359, 32377}, {32445, 40951}, {32654, 39072}, {32674, 34068}, {32676, 34079}, {32929, 42707}, {33801, 35222}, {33863, 39966}, {33873, 35458}, {34096, 41278}, {34382, 41619}, {34448, 41221}, {34482, 39955}, {34783, 37490}, {34803, 44180}, {35012, 36067}, {35219, 36851}, {35278, 39656}, {36743, 39798}, {36901, 44176}, {36983, 43616}, {37483, 43586}, {37502, 39971}, {37503, 39974}, {37644, 39899}, {37665, 40065}, {37667, 43981}, {37671, 44134}, {37778, 40102}, {38292, 41894}, {38920, 41414}, {38956, 40082}, {39111, 39112}, {39417, 40358}, {39530, 41244}, {39644, 39645}, {39646, 40814}, {39806, 39810}, {39809, 39812}, {39817, 39820}, {39835, 39839}, {39838, 39841}, {39846, 39849}, {40116, 43079}, {40169, 40184}, {40182, 40195}, {40185, 42484}, {40187, 40189}, {40190, 40219}, {40220, 40226}, {40285, 41725}, {40316, 40317}, {40454, 43742}, {41410, 41445}, {41411, 41444}, {42394, 44144}, {43460, 43462}, {43725, 43726}

X(25) is the {X(5),X(26)}-harmonic conjugate of X(3). For a list of other harmonic conjugates of X(25), click Tables at the top of this page.

X(25) = reflection of X(i) in X(j) for these (i,j): (4,1596), (1370,1368)
X(25) = isogonal conjugate of X(69)
X(25) = isotomic conjugate of X(305)
X(25) = circumcircle-inverse of X(468)
X(25) = nine-point-circle-inverse of X(37981)
X(25) = orthocentroidal-circle-inverse of X(427)
X(25) = complement of X(1370)
X(25) = anticomplement of X(1368)
X(25) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,6), (28,19), (250,112)
X(25) = X(32)-cross conjugate of X(6)
X(25) = crosspoint of X(i) and X(j) for these (i,j): (4,393), (6,64), (19,34), (112,250)
X(25) = crosssum of X(i) and X(j) for these (i,j): (2,20), (3,394), (6,159), (8,329), (63,78), (72,306), (125,525)
X(25) = crossdifference of every pair of points on line X(441)X(525)
X(25) = X(i)-Hirst inverse of X(j) for these (i,j): (4,419), (6,232)
X(25) = X(i)-beth conjugate of X(j) for these (i,j): (33,33), (108,25), (162,278)
X(25) = trilinear pole of line X(512)X(1692)
X(25) = de-Longchamps-circle-inverse of anticomplement of X(37777)
X(25) = cevapoint of X(i) and X(j) for these {i,j}: {6, 3053}, {32, 1974}
X(25) = crosspoint of PU(4)
X(25) = barycentric product of PU(i) for these i: 4,18,23,157
X(25) = barycentric product of vertices of half-altitude triangle
X(25) = barycentric product of vertices of orthocentroidal triangle
X(25) = perspector of circumconic centered at X(3162)
X(25) = center of circumconic that is locus of trilinear poles of lines passing through X(3162)
X(25) = X(2)-Ceva conjugate of X(3162)
X(25) = pole, wrt circumcircle, of orthic axis
X(25) = pole, wrt polar circle, of de Longchamps line
X(25) = X(i)-isoconjugate of X(j) for these (i,j): (6,304), (48,76), (75,3), (92,394), (1101,339)
X(25) = tangential isogonal conjugate of X(159)
X(25) = insimilicenter of nine-point circle and tangential circle
X(25) = orthic isogonal conjugate of X(6)
X(25) = homothetic center of ABC and the 2nd pedal triangle of X(4)
X(25) = homothetic center of ABC and the 2nd antipedal triangle of X(3)
X(25) = homothetic center of the medial triangle and the 3rd pedal triangle of X(4)
X(25) = homothetic center of the anticomplementary triangle and the 3rd antipedal triangle of X(3)
X(25) = homothetic center of reflection of orthic triangle in X(4) and reflection of tangential triangle in X(3)
X(25) = homothetic center of reflections of orthic and tangential triangles in their respective Euler lines
X(25) = inverse-in-polar-circle of X(858)
X(25) = inverse-in-{circumcircle, nine-point circle}-inverter of X(403)
X(25) = inverse-in-circumconic-centered-at-X(4) of X(450)
X(25) = Danneels point of X(4)
X(25) = Danneels point of X(1113)
X(25) = Danneels point of X(1114)
X(25) = X(2)-vertex conjugate of X(2)
X(25) = vertex conjugate of X(8105) and X(8106)
X(25) = vertex conjugate of foci of orthic inconic
X(25) = vertex conjugate of PU(112)
X(25) = Zosma transform of X(63)
X(25) = X(57)-of-the-tangential triangle if ABC is acute
X(25) = perspector of ABC and the (pedal triangle of X(4) in the orthic triangle)
X(25) = X(57) of orthic triangle if ABC is acute
X(25) = intersection of tangents at X(371) and X(372) to the orthocubic K006
X(25) = insimilicenter of circumcircle and incircle of orthic triangle if ABC is acute; the exsimilicenter is X(1593)
X(25) = perspector of ABC and circummedial tangential triangle
X(25) = homothetic center of ABC and orthocevian triangle of X(2)
X(25) = homothetic center of orthocevian triangle of X(2) and Ara triangle
X(25) = {X(8880),X(8881)}-harmonic conjugate of X(184)
X(25) = homothetic center of medial triangle and cross-triangle of ABC and Ara triangle
X(25) = perspector of ABC and cross-triangle of ABC and 4th Brocard triangle
X(25) = harmonic center of circumcircle and circle O(PU(4))
X(25) = Thomson-isogonal conjugate of X(5656)
X(25) = homothetic center of Aries and 2nd Hyacinth triangles
X(25) = intersection of tangents to Hatzipolakis-Lozada hyperbola at X(4) and X(193)
X(25) = crosspoint, wrt orthic triangle, of X(4) and X(193)
X(25) = barycentric product of (real or nonreal) circumcircle intercepts of orthic axis
X(25) = vertex conjugate of X(24007) and X(24008) (the Kiepert hyperbola intercepts of the orthic axis)
X(25) = excentral-to-ABC functional image of X(57)
X(25) = barycentric product of vertices of infinite altitude triangle
X(25) = intersection of tangents to Walsmith rectangular hyperbola at X(74) and X(110)
X(25) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,3,7484), (2,4,427), (2,5,7539), (3,4,1593), (3,5,7395), (4,5,7507), (4,24,3)


X(26) = CIRCUMCENTER OF THE TANGENTIAL TRIANGLE

Trilinears      a[b2cos 2B + c2cos 2C - a2cos 2A] : :
Trilinears    (J2 - 3) cos A + 4 cos B cos C : : , where J is as at X(1113)
Barycentrics    a2(b2cos 2B + c2cos 2C - a2cos 2A) : :
Barycentrics    a^2 (a^8 - 2 a^6 (b^2 + c^2) + 2 a^2 (b^6 + c^6) - (b^2 - c^2)^2 (b^4 + c^4)) : :
X(26) = 6 X(2) + (J^2 - 7) X(3)
X(26) = (J^2 - 3) X(3) + 2 X(4)

As a point on the Euler line, X(26) has Shinagawa coefficients (E + 4F, -3E - 4F).

If you have The Geometer's Sketchpad, you can view X(26).
If you have GeoGebra, you can view X(26).

Theorems involving X(26), published in 1889 by A. Gob, are discussed in
Roger A. Johnson, Advanced Euclidean Geometry, Dover, 1960, 259-260.

Let OA be the circle centered at the A-vertex of the circumorthic tangential triangle and passing through A; define OB and OC cyclically. X(26) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

X(26) lies on these lines: {1, 9625}, {2, 3}, {6, 143}, {10, 9712}, {39, 9608}, {40, 9590}, {49, 1993}, {51, 569}, {52, 184}, {54, 3060}, {55, 4354}, {56, 4351}, {64, 32138}, {68, 161}, {74, 12279}, {98, 1286}, {110, 7731}, {113, 22109}, {154, 155}, {157, 2934}, {159, 3564}, {182, 5462}, {195, 9704}, {197, 32141}, {206, 511}, {221, 32143}, {232, 10316}, {343, 12134}, {355, 8185}, {394, 6101}, {495, 10037}, {496, 10046}, {512, 39537}, {524, 9925}, {542, 15581}, {568, 6800}, {577, 14576}, {578, 5446}, {912, 40660}, {952, 9798}, {970, 9570}, {1092, 10625}, {1112, 12228}, {1151, 9682}, {1177, 14984}, {1181, 6102}, {1199, 11003}, {1204, 10575}, {1209, 32332}, {1216, 9306}, {1350, 10627}, {1351, 14449}, {1352, 20987}, {1353, 19459}, {1478, 9658}, {1479, 9672}, {1483, 8192}, {1493, 19468}, {1495, 5562}, {1498, 2931}, {1503, 12359}, {1511, 35602}, {1601, 25043}, {1602, 35220}, {1603, 35221}, {1605, 1607}, {1606, 1608}, {1609, 42459}, {1614, 5889}, {1829, 24301}, {1843, 19131}, {1853, 13561}, {1971, 23128}, {1974, 9967}, {2079, 5023}, {2165, 8553}, {2192, 32168}, {2351, 31381}, {2393, 8548}, {2777, 12893}, {2781, 15132}, {2782, 39828}, {2794, 39825}, {2916, 5085}, {2929, 35237}, {2935, 34584}, {3070, 35776}, {3071, 35777}, {3098, 5447}, {3197, 32158}, {3205, 36979}, {3206, 36981}, {3220, 24467}, {3357, 14915}, {3425, 10547}, {3527, 13451}, {3532, 34802}, {3556, 14988}, {3567, 5012}, {3580, 25738}, {3581, 11456}, {3796, 5946}, {5092, 11695}, {5285, 26921}, {5347, 36754}, {5412, 10898}, {5413, 10897}, {5422, 13353}, {5448, 23358}, {5449, 18381}, {5594, 5874}, {5595, 5875}, {5621, 20379}, {5640, 38848}, {5690, 8193}, {5844, 12410}, {5876, 18451}, {5890, 37490}, {5891, 44082}, {5892, 37515}, {5901, 11365}, {5907, 32237}, {5944, 10263}, {6000, 7689}, {6030, 15045}, {6146, 32269}, {6237, 10536}, {6238, 10535}, {6247, 44158}, {6403, 19121}, {6407, 9694}, {6515, 32358}, {6759, 9932}, {6776, 18951}, {7293, 37612}, {7352, 26888}, {7691, 11459}, {7712, 15032}, {7742, 14667}, {7767, 15574}, {8190, 32146}, {8191, 32147}, {8194, 32177}, {8195, 32178}, {8276, 8981}, {8277, 13966}, {8538, 44102}, {8550, 35707}, {8718, 15072}, {8743, 10317}, {8746, 36418}, {9143, 25714}, {9730, 10984}, {9781, 13434}, {9786, 13630}, {9820, 10192}, {9861, 9918}, {9911, 28174}, {9915, 22657}, {9916, 22656}, {9917, 22655}, {9920, 12310}, {9927, 18400}, {9938, 13289}, {10095, 17810}, {10113, 19457}, {10182, 29317}, {10264, 13171}, {10312, 22240}, {10313, 22120}, {10533, 10665}, {10534, 10666}, {10540, 11441}, {10601, 15026}, {10605, 13491}, {10606, 32210}, {10610, 10982}, {10628, 40276}, {10632, 11421}, {10633, 11420}, {10634, 10642}, {10635, 10641}, {10661, 30402}, {10662, 30403}, {10663, 10682}, {10664, 10681}, {10733, 40242}, {10790, 32134}, {10828, 32151}, {10829, 10943}, {10830, 10942}, {10833, 15171}, {10834, 32213}, {10835, 32214}, {10880, 11418}, {10881, 11417}, {11202, 12038}, {11206, 11411}, {11248, 20872}, {11399, 37729}, {11402, 37493}, {11430, 13598}, {11432, 16881}, {11438, 40647}, {11440, 12290}, {11444, 43598}, {11449, 43574}, {11455, 15062}, {11464, 15107}, {11468, 13445}, {11472, 15811}, {11477, 13421}, {11482, 43697}, {11499, 20989}, {11591, 17814}, {11645, 14864}, {11671, 34418}, {12006, 37514}, {12111, 14157}, {12160, 26864}, {12162, 26883}, {12164, 14530}, {12220, 19128}, {12236, 13198}, {12280, 12380}, {12289, 41482}, {12293, 17845}, {12295, 32607}, {12307, 41726}, {12370, 19467}, {12891, 13288}, {12892, 13287}, {13142, 43595}, {13292, 31804}, {13336, 22352}, {13352, 13367}, {13391, 17821}, {13419, 21243}, {13558, 15653}, {13562, 37485}, {13567, 18952}, {13889, 13925}, {13943, 13993}, {14128, 33533}, {14531, 43844}, {14641, 43604}, {14657, 33962}, {14693, 32762}, {15035, 25487}, {15043, 15080}, {15067, 35259}, {15069, 19596}, {15085, 17838}, {15172, 16541}, {15454, 16104}, {15462, 40949}, {15478, 34428}, {15912, 40947}, {15959, 25150}, {16010, 35218}, {16165, 25711}, {16252, 22660}, {16391, 23181}, {17809, 32136}, {17811, 32142}, {17813, 32155}, {17819, 32169}, {17820, 32170}, {17825, 32205}, {17826, 32207}, {17827, 32208}, {18350, 23039}, {18376, 32393}, {18379, 18405}, {18874, 31860}, {18954, 18990}, {19005, 19116}, {19006, 19117}, {19129, 39588}, {19132, 19155}, {19165, 20993}, {19180, 19211}, {19189, 19210}, {19194, 26887}, {19908, 32048}, {20191, 23329}, {20299, 29012}, {20424, 32333}, {20477, 44138}, {20771, 41673}, {20791, 43597}, {21651, 34750}, {21849, 37505}, {22115, 37484}, {22533, 22550}, {22654, 32153}, {23698, 39854}, {23709, 34292}, {26446, 37557}, {29181, 35228}, {32613, 39582}, {32620, 33537}, {32829, 44180}, {34116, 44078}, {34118, 34177}, {34380, 37491}, {34397, 35603}, {34417, 37513}, {34514, 34826}, {35219, 39879}, {35719, 41244}, {39805, 39835}, {39806, 39834}, {39823, 39853}, {39824, 39852}, {39829, 39859}, {39830, 39858}, {43575, 43829}

X(26) is the {X(154),X(155)}-harmonic conjugate of X(156). For a list of other harmonic conjugates of X(26), click Tables at the top of this page.

X(26) = reflection of X(155) in X(156)
X(26) = isogonal conjugate of X(70)
X(26) = isotomic conjugate of X(20564)
X(26) = tangential isogonal conjugate of X(155)
X(26) = inverse-in-circumcircle of X(2072)
X(26) = crosssum of X(125) and X(924)
X(26) = complement of X(14790)
X(26) = anticomplement of X(13371)
X(26) = intouch-to-ABC functional image of X(3)
X(26) = orthoptic-circle-of-Steiner-inellipse-inverse of complement of X(37978)
X(26) = orthoptic-circle-of-Steiner-circumellipse-inverse of anticomplement of X(37978)


X(27) = CEVAPOINT OF ORTHOCENTER AND CLAWSON CENTER

Trilinears       (sec A)/(b + c) : (sec B)/(c + a) : (sec C)/(a + b)
Barycentrics  (tan A)/(b + c) : (tan B)/(c + a) : (tan C)/(a + b)
X(27) = 3 SA SB SC X(2) - 2 S^2 s^2 X(3)

As a point on the Euler line, X(27) has Shinagawa coefficients (F, -E - F - $bc$).

If you have The Geometer's Sketchpad, you can view X(27).
If you have GeoGebra, you can view X(27).

X(27) lies on these lines: {2, 3}, {6, 1246}, {7, 81}, {8, 19848}, {19, 63}, {33, 5287}, {34, 5256}, {53, 37646}, {57, 273}, {58, 270}, {69, 19793}, {71, 40435}, {84, 1896}, {86, 1474}, {99, 9085}, {101, 22000}, {103, 107}, {110, 917}, {112, 675}, {116, 40076}, {162, 673}, {198, 27287}, {225, 2363}, {226, 284}, {239, 1829}, {240, 11031}, {242, 2355}, {243, 1859}, {264, 14829}, {272, 2189}, {274, 19798}, {275, 2051}, {281, 5235}, {295, 335}, {306, 1043}, {310, 17206}, {317, 4417}, {318, 10461}, {321, 5279}, {329, 2287}, {331, 15467}, {393, 967}, {394, 10446}, {516, 2328}, {579, 1751}, {607, 39741}, {648, 903}, {653, 18815}, {662, 913}, {811, 19801}, {823, 34234}, {871, 19800}, {908, 2327}, {933, 26708}, {946, 2360}, {1014, 1440}, {1019, 40213}, {1071, 1871}, {1086, 16099}, {1088, 1434}, {1119, 21454}, {1230, 44146}, {1240, 19810}, {1249, 37666}, {1259, 19845}, {1268, 1796}, {1301, 41905}, {1304, 2688}, {1333, 3772}, {1412, 17197}, {1427, 18603}, {1441, 3101}, {1625, 35096}, {1659, 2067}, {1699, 17188}, {1719, 1733}, {1730, 1746}, {1770, 1780}, {1785, 39595}, {1803, 2332}, {1810, 19815}, {1812, 17139}, {1836, 2194}, {1840, 40033}, {1841, 3666}, {1844, 10122}, {1851, 14024}, {1865, 18679}, {1870, 17011}, {1880, 25059}, {1882, 1940}, {1973, 2296}, {2052, 13478}, {2193, 37695}, {2203, 14621}, {2206, 3120}, {2221, 4000}, {2354, 28287}, {2400, 7192}, {2659, 26892}, {2969, 6650}, {3011, 19849}, {3187, 3868}, {3194, 5222}, {3306, 19802}, {3332, 11206}, {3423, 5324}, {3661, 5090}, {3687, 5081}, {3794, 17616}, {3914, 44119}, {4304, 4653}, {4373, 9965}, {4384, 5342}, {4393, 11396}, {4786, 7649}, {4921, 19819}, {5057, 6061}, {5088, 18607}, {5333, 17917}, {5523, 18686}, {5732, 17194}, {5905, 39695}, {6198, 17019}, {6331, 19816}, {6335, 40039}, {6358, 16548}, {6384, 19803}, {6502, 13390}, {6542, 12135}, {6548, 17925}, {6748, 37662}, {7017, 19807}, {7119, 40418}, {7140, 17927}, {7249, 40432}, {7283, 42706}, {7354, 40980}, {7718, 17316}, {8025, 19823}, {8044, 34440}, {8736, 37770}, {8756, 19797}, {9308, 37683}, {10444, 17185}, {10449, 19838}, {11363, 16826}, {13243, 35360}, {16077, 35161}, {16747, 20880}, {17182, 24556}, {17189, 23681}, {17277, 44103}, {17903, 41364}, {17921, 43925}, {18344, 24601}, {18742, 40010}, {19752, 19767}, {19799, 33932}, {19812, 25507}, {19820, 39710}, {19821, 29766}, {19824, 36606}, {19827, 28650}, {19830, 39707}, {20291, 38852}, {20527, 20751}, {21370, 44178}, {21621, 24019}, {23383, 34429}, {31424, 39585}, {32000, 37655}, {34255, 39749}, {39704, 42028}, {41342, 43729}

X(27) is the {X(2),X(4)}-harmonic conjugate of X(469). For a list of other harmonic conjugates of X(27), click Tables at the top of this page.

X(27) = isogonal conjugate of X(71)
X(27) = isotomic conjugate of X(306)
X(27) = inverse-in-circumcircle of X(2073)
X(27) = inverse-in-orthocentroidal-circle of X(469)
X(27) = complement of X(3151)
X(27) = anticomplement of X(440)
X(27) = X(286)-Ceva conjugate of X(29)
X(27) = cevapoint of X(i) and X(j) for these (i,j): (4,19), (57,278)
X(27) = X(i)-cross conjugate of X(j) for these (i,j): (4,286), (19,28), (57,81), (58,86)
X(27) = crossdifference of every pair of points on line X(647)X(810)
X(27) = X(i)-Hirst inverse of X(j) for these (i,j): (2,447), (4,423)
X(27) = X(i)-beth conjugate of X(j) for these (i,j): (648,27), (923,27)
X(27) = trilinear pole of line X(242)X(514) (the polar of X(10) wrt polar circle)
X(27) = pole wrt polar circle of trilinear polar of X(10) (line X(523)X(661))
X(27) = polar conjugate of X(10)
X(27) = X(6)-isoconjugate of X(72)
X(27) = X(19)-isoconjugate of X(3682)
X(27) = X(75)-isoconjugate of X(2200)
X(27) = crosspoint of X(4) and X(19) wrt excentral triangle
X(27) = trilinear product X(2)*X(28)


X(28) = CEVAPOINT OF X(19) AND X(25)

Trilinears       (tan A)/(b + c) : (tan B)/(c + a) : (tan C)/(a + b)
Barycentrics  (sin A tan A)/(b + c) : (sin B tan B)/(c + a) : (sin C tan C)/(a + b)
X(28) = 8 sa sb sc X(4) + a b c (3 + J^2) X(25)

As a point on the Euler line, X(28) has Shinagawa coefficients ($a$F, -$a$(E + F) - abc).

If you have The Geometer's Sketchpad, you can view X(28).
If you have GeoGebra, you can view X(28).

X(28) lies on these lines: 1,19   2,3   10,1891   11,1852   33,975   34,57   35,1869   36,1838   46,1780   54,1243   56,278   60,81   65,1175   72,1257   88,162   104,107   105,112   108,225   110,915   142,1890   228,943   242,261   272,273   279,1014   281,958   291,1783   501,1831   579,1724   580,1730   607,1002   608,959   614,1472   956,1219   957,1191   961,1169   1104,1333   1125,1848   1155,1888   1170,1876   1178,1432   1224,1826   1255,1824   1295,1301   1385,1871   1412,1422   1633,1770   1710,1725

X(28) is the {X(27),X(29)}-harmonic conjugate of X(4). For a list of other harmonic conjugates of X(28), click Tables at the top of this page.

X(28) = isogonal conjugate of X(72)
X(28) = isotomic conjugate of X(20336)
X(28) = anticomplement of X(21530)
X(28) = trilinear pole of line X(513)X(1430) (the polar of X(321) wrt polar circle)
X(28) = polar conjugate of X(321)
X(28) = X(6)-isoconjugate of X(306)
X(28) = X(75)-isoconjugate of X(228)
X(28) = circumcircle-inverse of X(2074)
X(28) = X(i)-Ceva conjugate of X(j) for these (i,j): (270,58), (286,81)
X(28) = cevapoint of X(i) and X(j) for these (i,j): (19,25), (34,56)
X(28) = X(i)-cross conjugate of X(j) for these (i,j): (19,27), (58,58)
X(28) = crossdifference of every pair of points on line X(647)X(656)
X(28) = X(4)-Hirst inverse of X(422)
X(28) = X(i)-beth conjugate of X(j) for these (i,j): (29,29), (107,28), (162,28), (270,28)


X(29) = CEVAPOINT OF INCENTER AND ORTHOCENTER

Trilinears       (sec A)/(cos B + cos C) : (sec B)/(cos C + cos A) : (sec C)/(cos A + cos B)
Barycentrics  (tan A)/(cos B + cos C) : (tan B)/(cos C + cos A) : (tan C)/(cos A + cos B)
Barycentrics    (a - b - c)/((b + c) (a^2 - b^2 - c^2)) : :

As a point on the Euler line, X(29) has Shinagawa coefficients (F*S2, $bcSBSC$ - F*S2).

If you have The Geometer's Sketchpad, you can view X(29).
If you have GeoGebra, you can view X(29) and X(29) cevapoint..

X(29) lies on these lines: 1,92   2,3   8,219   10,1794   33,78   34,77   58,162   65,296   81,189   102,107   112,1311   226,951   242,257   270,283   284,950   314,1039   388,1037   392,1871   497,1036   515,947   648,1121   662,1800   758,1844   894,1868   960,1859   1056,1059   1057,1058   1125,1838   1220,1474   1737,1780   1807,1897   1842,1848

X(29) is the {X(3),X(4)}-harmonic conjugate of X(412). For a list of other harmonic conjugates of X(29), click Tables at the top of this page.

X(29) = isogonal conjugate of X(73)
X(29) = isotomic conjugate of X(307)
X(29) = inverse-in-circumcircle of X(2075)
X(29) = complement of X(3152)
X(29) = anticomplement of X(18641)
X(29) = X(286)-Ceva conjugate of X(27)
X(29) = cevapoint of X(i) and X(j) for these (i,j): (1,4), (33,281)
X(29) = X(i)-cross conjugate of X(j) for these (i,j): (1,21), (284,333), (497,314)
X(29) = crosssum of X(i) and X(j) for these (i,j): (1,1047), (228,1409)
X(29) = crossdifference of every pair of points on line X(647)X(822)
X(29) = X(4)-Hirst inverse of X(415)
X(29) = X(i)-beth conjugate of X(j) for these (i,j): (29,28), (811,29)
X(29) = intersection of tangents at X(1) and X(4) to hyperbola passing through X(1), X(4) and the excenters
X(29) = crosspoint of X(1) and X(4) wrt the excentral triangle
X(29) = trilinear pole of line X(243)X(522) (the polar of X(226) wrt polar circle)
X(29) = pole wrt polar circle of trilinear polar of X(226) (line X(523)X(656))
X(29) = polar conjugate of X(226)
X(29) = X(6)-isoconjugate of X(1214)


X(30) = EULER INFINITY POINT

Trilinears    cos A - 2 cos B cos C : cos B - 2 cos C cos A : cos C - 2 cos A cosB
Trilinears    bc[2a4 - (b2 - c2)2 - a2(b2 + c2)] : :
Trilinears    2 sec A - sec B sec C : :
Trilinears    sin B sin C - 3 cos B cos C : :
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2a4 - (b2 - c2)2 - a2(b2 + c2)
Barycentrics    S^2 - 3 SB SC : :
X(30) = X(2) - X(3); if (i) and X(j) are on the Euler line, then X(30) = X(i) - X(j)

As a point on the Euler line, X(30) has Shinagawa coefficients (1, -3).

Let A'B'C' be the reflection triangle. Let A″ be the cevapoint of B' and C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ are parallel to the Euler line, and therefore concur in X(30). (Randy Hutson, December 10, 2016)

X(30) is the point of intersection of the Euler line and the line at infinity. Thus, each of the lines listed below is parallel to the Euler line.

If you have The Geometer's Sketchpad, you can view Euler Infinity Point.

X(30) lies on the Neuberg cubic, the Darboux quintic, and these (parallel) lines: {1, 79}, {2, 3}, {6, 2549}, {7, 3488}, {8, 3578}, {9, 3587}, {10, 3579}, {11, 36}, {12, 35}, {13, 15}, {14, 16}, {17, 5238}, {18, 5237}, {32, 5254}, {33, 1060}, {34, 1062}, {40, 191}, {46, 1837}, {49, 1614}, {50, 1989}, {51, 5946}, {52, 185}, {53, 577}, {54, 3521}, {55, 495}, {56, 496}, {57, 3586}, {58, 1834}, {61, 397}, {62, 398}, {63, 3419}, {64, 68}, {65, 1770}, {69, 3426}, {74, 265}, {80, 484}, {84, 3928}, {98, 671}, {99, 316}, {100, 2687}, {101, 2688}, {102, 2689}, {103, 2690}, {104, 1290}, {105, 2691}, {106, 2692}, {107, 2693}, {108, 2694}, {109, 2695}, {110, 477}, {111, 2696}, {112, 2697}, {113, 1495}, {114, 2482}, {115, 187}, {119, 2077}, {128, 6592}, {133, 3184}, {137, 6150}, {141, 3098}, {143, 389}, {146, 323}, {148, 385}, {154, 5654}, {155, 1498}, {156, 1147}, {165, 5587}, {182, 597}, {226, 4304}, {250, 6530}, {262, 598}, {284, 1901}, {298, 616}, {299, 617}, {315, 1975}, {329, 3940}, {340, 1494}, {371, 3070}, {372, 3071}, {388, 3295}, {390, 1056}, {485, 1151}, {486, 1152}, {489, 638}, {490, 637}, {497, 999}, {498, 5217}, {499, 5204}, {511, 512}, {551, 946}, {553, 942}, {567, 5012}, {568, 3060}, {574, 3815}, {582, 1724}, {590, 6200}, {599, 1350}, {615, 6396}, {618, 623}, {619, 624}, {620, 625}, {664, 5195}, {841, 1302}, {908, 5440}, {910, 5179}, {925, 5962}, {935, 1297}, {938, 5708}, {944, 962}, {956, 3434}, {993, 2886}, {1043, 1330}, {1058, 3600}, {1117, 5671}, {1125, 3824}, {1131, 6407}, {1132, 6408}, {1141, 1157}, {1145, 5176}, {1146, 5011}, {1155, 1737}, {1160, 5860}, {1161, 5861}, {1213, 4877}, {1216, 5907}, {1285, 5304}, {1292, 2752}, {1293, 2758}, {1294, 1304}, {1295, 2766}, {1296, 2770}, {1319, 1387}, {1337, 3479}, {1338, 3480}, {1351, 1353}, {1376, 3820}, {1465, 1877}, {1490, 5763}, {1565, 4872}, {1587, 3311}, {1588, 3312}, {1625, 3289}, {1691, 6034}, {1699, 3576}, {1750, 5720}, {1754, 5398}, {1765, 5755}, {1768, 5535}, {1807, 3465}, {1838, 1852}, {1865, 2193}, {1870, 3100}, {1990, 3163}, {2021, 2023}, {2093, 5727}, {2094, 2095}, {2132, 2133}, {2292, 5492}, {2456, 5182}, {2548, 5013}, {2646, 4870}, {2654, 4303}, {2895, 4720}, {2931, 2935}, {2968, 5081}, {3003, 6128}, {3023, 6023}, {3027, 6027}, {3035, 3814}, {3053, 3767}, {3068, 6221}, {3069, 6398}, {3085, 5229}, {3086, 5225}, {3167, 5656}, {3255, 3577}, {3260, 6148}, {3292, 5609}, {3303, 4309}, {3304, 4317}, {3357, 5894}, {3424, 5485}, {3429, 4052}, {3436, 5687}, {3481, 3482}, {3485, 4305}, {3486, 4295}, {3487, 4313}, {3565, 5203}, {3589, 4045}, {3665, 4056}, {3703, 4680}, {3746, 4330}, {3829, 5450}, {3911, 5122}, {3917, 5891}, {3925, 5251}, {4030, 4692}, {4252, 5292}, {4296, 6198}, {4298, 5045}, {4301, 5882}, {4325, 4857}, {4421, 6256}, {4424, 5724}, {4511, 5057}, {4669, 5493}, {4677, 5881}, {4999, 5267}, {5008, 5355}, {5010, 5432}, {5032, 5093}, {5103, 5149}, {5107, 5477}, {5119, 5252}, {5180, 6224}, {5188, 6248}, {5207, 6393}, {5418, 6409}, {5420, 6410}, {5424, 5561}, {5448, 5893}, {5459, 5478}, {5460, 5479}, {5461, 6036}, {5463, 5473}, {5464, 5474}, {5538, 6326}, {5562, 5876}, {5603, 5731}, {5657, 5790}, {5703, 5714}, {5732, 5805}, {5758, 6223}, {5759, 5779}, {5858, 5864}, {5859, 5865}, {5889, 6241}, {5892, 5943}, {6104, 6107}, {6105, 6106}, {6193, 6225}, {6237, 6254}, {6238, 6285}

X(30) = isogonal conjugate of X(74)
X(30) = isotomic conjugate of X(1494)
X(30) = anticomplementary conjugate of X(146)
X(30) = complementary conjugate of X(113)
X(30) = orthopoint of X(523)
X(30) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,113), (265,5), (476,523)
X(30) = cevapoint of X(3) and X(399)
X(30) = crosspoint of X(i) and X(j) for these (i,j): (13,14), (94,264)
X(30) = crosssum of X(i) and X(j) for these (i,j): (15,16), (50,184)
X(30) = crossdifference of every pair of points on line X(6)X(647)
X(30) = ideal point of PU(30)
X(30) = vertex conjugate of PU(87)
X(30) = perspector of circumconic centered at X(3163)
X(30) = center of circumconic that is locus of trilinear poles of lines passing through X(3163)
X(30) = X(2)-Ceva conjugate of X(3163)
X(30) = trilinear pole of line X(1636)X(1637) (the line that is the tripolar centroid of the Euler line)
X(30) = X(517)-of-orthic triangle if ABC is acute
X(30) = X(542)-of-1st Brocard triangle
X(30) = crosspoint of X(3) and X(399) wrt both the excentral and tangential triangles
X(30) = crosspoint of X(616) and X(617) wrt both the excentral and anticomplementary triangles
X(30) = cevapoint of X(616) and X(617)
X(30) = X(6)-isoconjugate of X(2349)
X(30) = perspector of 2nd isogonal triangle of X(4) and cross-triangle of ABC and 2nd isogonal triangle of X(4)
X(30) = Thomson isogonal conjugate of X(110)
X(30) = Lucas isogonal conjugate of X(110)
X(30) = homothetic center of X(20)-altimedial and X(140)-anti-altimedial triangles
X(30) = X(1154)-of-excentral-triangle
X(30) = homothetic center of Ehrmann vertex-triangle and Trinh triangle
X(30) = homothetic center of Ehrmann side-triangle and dual of orthic triangle
X(30) = homothetic center of Ehrmann mid-triangle and medial triangle
X(30) = excentral-to-ABC functional image of X(517)
X(30) = 1st-Brocard-isogonal conjugate of X(18332)
X(30) = polar conjugate of X(16080)
X(30) = X(63)-isoconjugate of X(8749)


X(31) = 2nd POWER POINT

Trilinears    a2 : b2 : c2
Trilinears    1 - cos 2A : 1 - cos 2B : 1 - cos 2C
Trilinears    cot B + cot C : :
Trilinears    SB + SC : :
Trilinears    d(a,b,c) : : , where d(a,b,c) = distance between A and de Longchamps line
Barycentrics    a3 : b3 : c3

X(31) = (r2 + s2)*X(1) - 6rR*X(2) - 4r2*X(3)   (Peter Moses, April 2, 2013)

Let A'B'C' be the circumsymmedial triangle. Let A″ be the trilinear product B'*C', and define B″ and C″ cyclically. Then A″, B″, C″ are collinear on line X(667)X(788) (the trilinear polar of X(31)). The lines AA″, BB″, CC″ concur in X(31). (Randy Hutson, February 10, 2016)

Let A'B'C' be the Apus triangle. Let A″ be the trilinear product B'*C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(31). (Randy Hutson, February 10, 2016)

Let A'B'C' be the Ara triangle. Let A″ be the trilinear product B'*C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(31). (Randy Hutson, February 10, 2016)

Define the 1st Kenmotu diagonals triangle as the triangle formed by the diagonals of the squares in the Kenmotu configuration with center X(371) that do not include X(371). Define the 2nd Kenmotu diagonals triangle as the triangle formed by the diagonals of the squares in the Kenmotu configuration with center X(372) that do not include X(372). (Randy Hutson, February 10, 2016)

Let A1B1C1 and A2B2C2 be the 1st and 2nd Kenmotu diagonals triangles. Let A' be the trilinear product A1*A2, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(31). (Randy Hutson, February 10, 2016)

X(31) is the Brianchon point (perspector) of the inellipse that is the trilinear square of the Lemoine axis. The center of the inellipse is X(16584). (Randy Hutson, October 15, 2018)

If you have The Geometer's Sketchpad, you can view X(31) (1), X(31) (2), X(31) (3).


If you have GeoGebra, you can view X(31).

X(31) lies on these lines: 1,21   2,171   3,601   4,3072   6,42   8,987   9,612   10,964   19,204   25,608   28,2282   32,41   33,2250   34,1254   35,386   36,995   37,2214   40,580   43,100   44,210   48,560   51,181   56,154   57,105   65,1104   72,976   75,82   76,734   86,2296   91,1087   92,162   99,715   101,609   106,2163   110,593   112,2249   158,2190   163,923   165,2999   172,1613   184,604   197,2183   198,2255   199,2277   200,1261   218,1260   222,1458   226,3011   237,904   240,1748   278,1430   284,2258   292,1915   354,1279   388,1935   404,978   497,1936   516,1754   561,722   582,3579   607,2357   649,884   663,2423   669,875   678,3158   692,2877   701,789   708,1502   740,3187   743,825   745,827   759,994   775,1097   872,2220   893,1691   899,1376   901,2382   937,1103   940,1001   982,3218   984,3219   990,1709   999,1149   1066,3157   1098,2363   1124,3076   1182,3192   1210,1771   1331,2991   1335,3077   1393,1454   1403,1428   1427,1456   1438,2279   1450,1470   1474,2215   1486,2260   1572,2170   1582,1740   1616,3304   1633,3123   1820,1953   1836,3120   1910,2186   1911,1922   1917,2085   1927,1967   1932,1973   1951,3010   1974,2281   1979,2107   2003,2078   2054,2248   2083,2156   2153,2154   2188,2638   2242,3230   2264,3198   2274,3286   2318,2911   3074,3085   3075,3086   3220,3415

X(31) is the {X(1),X(63)}-harmonic conjugate of X(38). For a list of other harmonic conjugates of X(31), click Tables at the top of this page.

X(31) = isogonal conjugate of X(75)
X(31) = isotomic conjugate of X(561)
X(31) = complement of X(6327)
X(31) = anticomplement of X(2887)
X(31) = anticomplementary conjugate of anticomplement of X(38813)
X(31) = circumcircle-inverse of X(5161)
X(31) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,48), (6,41), (9,205), (58,6), (82,1)
X(31) = X(213)-cross conjugate of X(6)
X(31) = crosspoint of X(i) and X(j) for these (i,j): (1,19), (6,56)
X(31) = crosssum of X(i) and X(j) for these (i,j): (1,63), (2,8), (7,347), (10,321), (239,1281), (244,514), (307,1441), (523,1086), (693,1111)
X(31) = crossdifference of every pair of points on line X(514)X(661)
X(31) = X(1403)-Hirst inverse of X(1428)
X(31) = X(i)-aleph conjugate of X(j) for these (i,j): (82,31), (83,75)
X(31) = X(i)-beth conjugate of X(j) for these (i,j): (21,993), (55,55), (109,31), (110,57), (643,31), (692,31)
X(31) = barycentric product of PU(8)
X(31) = vertex conjugate of PU(8)
X(31) = bicentric sum of PU(i) for these i: 23, 48
X(31) = PU(23)-harmonic conjugate of X(661)
X(31) = PU(48)-harmonic conjugate of X(649)
X(31) = trilinear product of PU(36)
X(31) = trilinear product X(55)*X(56)
X(31) = trilinear pole of line X(667)X(788)
X(31) = pole wrt polar circle of trilinear polar of X(1969)
X(31) = X(48)-isoconjugate (polar conjugate) of X(1969)
X(31) = X(6)-isoconjugate of X(76)
X(31) = X(92)-isoconjugate of X(63)
X(31) = trilinear square of X(6)
X(31) = trilinear cube root of X(1917)
X(31) = vertex conjugate of foci of incentral inellipse
X(31) = perspector of ABC and extraversion triangle of X(31) (which is also the anticevian triangle of X(31))
X(31) = {X(1),X(1707)}-harmonic conjugate of X(63)
X(31) = perspector of ABC and unary cofactor triangle of extraversion triangle of X(7)
X(31) = perspector of ABC and unary cofactor triangle of extraversion triangle of X(8) (2nd Conway triangle)
X(31) = perspector of ABC and unary cofactor triangle of 4th Conway triangle
X(31) = perspector of unary cofactor triangles of 2nd and 4th Conway triangles
X(31) = perspector of unary cofactor triangles of Gemini triangles 2 and 30
X(31) = perspector of ABC and cross-triangle of Gemini triangles 33 and 34
X(31) = perspector of ABC and cross-triangle of ABC and Gemini triangle 33
X(31) = perspector of ABC and cross-triangle of ABC and Gemini triangle 34
X(31) = barycentric product of vertices of Gemini triangle 33
X(31) = barycentric product of vertices of Gemini triangle 34
X(31) = barycentric product of (nonreal) circumcircle intercepts of the antiorthic axis
X(31) = center of circumconic locus of trilinear poles of lines passing through X(32664)
X(31) = perspector of circumconic centered at X(32664)
X(31) = X(2)-Ceva conjugate of X(32664)


X(32) = 3rd POWER POINT

Trilinears    a3 : b3 : c3
Trilinears    sin(A - ω) : sin(B - ω) : sin(C - ω)
Trilinears    sin A + sin(A - 2ω) : sin B + sin(B - 2ω) : sin C + sin(C - 2ω)
Trilinears    cos A - cos(A - 2ω) : cos B - cos(B - 2ω) : cos C - cos(C -2ω)
Trilinears    cos A - sin A cot ω : :
Trilinears    sin A - cos A tan ω : :
Trilinears    a - 2R cos A tan ω : :
Barycentrics    a4 : b4 : c4
X(32) = -(r2 + 4rR - s2)(r2 + 2rR + s2)*X(1) + 6rR(r2 + 4rR - s2)*X(2) + 2r2(r2 + 4rR - 3s2)*X(3)   (Peter Moses, April 2, 2013)

If you have The Geometer's Sketchpad, you can view X(32).

The 5th Brocard triangle is here introduced as the vertex triangle of the circumcevian triangles of the 1st and 2nd Brocard points. (Randy Hutson, December 26, 2015)

The 5th Brocard triangle is homothetic to ABC at X(32), homothetic to the medial triangle at X(3096), homothetic to the anticomplementary triangle at X(2896), perspective to the 1st Brocard triangle at X(2896), and perspective to the 3rd Brocard triangle at X(32).(Randy Hutson, December 26, 2015)

Let A'B'C' be the 1st Brocard triangle. Let A″, B″, C″ be inverse-in-circumcircle of A', B' and C' resp. AA″, BB″, CC″ concur in X(32). (Randy Hutson, July 20, 2016)

Let A'B'C' be the 1st Brocard triangle. Let A″ be the cevapoint, wrt A'B'C', of B and C, and define B″, C″ cyclically. A'A″, B'B″, C'C″ concur in X(32). (Randy Hutson, July 20, 2016)

X(32) is the Brianchon point (perspector) of the inellipse that is the barycentric square of the Lemoine axis. The center of this inellipse is X(8265). (Randy Hutson, October 15, 2018)

X(32) lies on these lines: 1,172   2,83   3,6   4,98   5,230   9,987   20,2549   21,981   22,1194   24,232   25,1184   31,41   35,2276   48,1472   51,2351   55,1500   56,1015   71,2273   75,746   76,384   81,980   99,194   100,713   101,595   110,729   111,1383   163,849   165,1571   184,211   218,906   220,3052   262,3406   263,1976   512,878   538,1003   560,1918   561,724   590,640   604,1106   615,639   632,3055   637,3069   638,3068   682,1974   695,3492   710,1502   731,825   733,827   902,1334   904,1933   910,1104   941,1169   958,1572   983,3495   988,1449   993,1107   1009,1724   1055,1201   1084,1576   1092,3289   1191,3207   1204,3269   1376,1574   1395,1402   1423,3500   1468,2280   1613,1915   1843,2353   1911,1932   1919,3249   1922,1923   1950,2285   1951,2082   1992,2482   1995,3291   2004,2005   2319,3494   2508,2881   2698,2715   3087,3088   3124,3457   3170,3171   3497,3512   3499,3511

X(32) is the {X(3),X(6)}-harmonic conjugate of X(39). For a list of other harmonic conjugates of X(32), click Tables at the top of this page.

X(32) = midpoint of X(371) and X(372)
X(32) = reflection of X(315) in X(626)
X(32) = isogonal conjugate of X(76)
X(32) = isotomic conjugate of X(1502)
X(32) = complement of X(315)
X(32) = anticomplement of X(626)
X(32) = circumcircle-inverse of X(1691)
X(32) = Brocard-circle-inverse of X(39)
X(32) = 1st-Lemoine-circle-inverse of X(1692)
X(32) = antigonal conjugate of X(37841)
X(32) = anticomplementary conjugate of anticomplement of X(38826)
X(32) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,206), (6,184), (112,512), (251,6)
X(32) = crosspoint of X(i) and X(j) for these (i,j): (2,66), (6,25)
X(32) = crosssum of X(i) and X(j) for these (i,j): (2,69), (6,22), (75,312), (115,826), (311,343), (313,321), (338,850), (339,525), (349,1231), (693,1086), (1229,1233), (1230,1269)
X(32) = crossdifference of every pair of points on line X(325)X(523)
X(32) = X(184)-Hirst inverse of X(237)
X(32) = X(i)-beth conjugate of X(j) for these (i,j): (41,41), (163,56), (919,32)
X(32) = external center of similitude of circumcircle and Moses circle
X(32) = radical trace of circumcircle and circle {X(1687),X(1688),PU(1),PU(2)}
X(32) = trilinear product of vertices of circumsymmedial triangle
X(32) = trilinear product of vertices of 3rd Brocard triangle
X(32) = insimilicenter of circles O(15,16) and O(61,62); the exsimilicenter is X(39)
X(32) = insimilicenter of circles {X(15),X(62),PU(1)} and {X(16),X(61),PU(1)}; the exsimilicenter is X(182)
X(32) = intersection of tangents at PU(1) to Brocard circle
X(32) = intersection of lines P(1)U(2) and U(1)P(2)
X(32) = vertex conjugate of PU(1)
X(32) = trilinear product of PU(9)
X(32) = barycentric product of PU(36)
X(32) = bicentric sum of PU(39)
X(32) = midpoint of PU(39)
X(32) = center of circle {X(371),X(372),PU(1),PU(39)}} (the circle orthogonal to the Brocard circle through the 1st and 2nd Brocard points)
X(32) = crosssum of polar conjugates of PU(4)
X(32) = perspector ABC and tangential triangle of 1st Brocard triangle
X(32) = trilinear cube of X(6)
X(32) = trilinear square root of X(1917)
X(32) = inverse-in-2nd-Brocard-circle of X(3094)
X(32) = perspector of circumconic centered at X(206)
X(32) = center of circumconic that is locus of trilinear poles of lines passing through X(206)
X(32) = trilinear pole of line X(669)X(688) (the isogonal conjugate of the isotomic conjugate of the Lemoine axis)
X(32) = perspector of ABC and 3rd Brocard triangle
X(32) = {X(61),X(62)}-harmonic conjugate of X(576)
X(32) = {X(1340),X(1341)}-harmonic conjugate of X(5116)
X(32) = {X(1687),X(1688)}-harmonic conjugate of X(3)
X(32) = reflection of X(5028) in X(6)
X(32) = X(32)-of-circumsymmedial-triangle
X(32) = X(75)-isoconjugate of X(2)
X(32) = X(92)-isoconjugate of X(69)
X(32) = X(1577)-isoconjugate of X(99)
X(32) = X(4048) of 1st anti-Brocard triangle
X(32) = homothetic center of circumnormal triangle and unary cofactor triangle of Stammler triangle
X(32) = perspector of ABC and cross-triangle of ABC and 1st Brocard triangle
X(32) = homothetic center of medial triangle and cross-triangle of ABC and 5th Brocard triangle
X(32) = homothetic center of medial triangle and cross-triangle of ABC and 5th anti-Brocard triangle
X(32) = Cundy-Parry Phi transform of X(511)
X(32) = Cundy-Parry Psi transform of X(98)
X(32) = X(169)-of-orthic-triangle if ABC is acute
X(32) = Steiner-circumellipse-inverse of X(16985)
X(32) = barycentric square of X(6)
X(32) = barycentric product of (nonreal) circumcircle intercepts of the Lemoine axis


X(33) = PERSPECTOR OF THE ORTHIC AND INTANGENTS TRIANGLES

Trilinears    1 + sec A : 1 + sec B : 1 + sec C
Trilinears    tan A cot(A/2) : tan B cot(B/2) : tan C cot(C/2)
Trilinears    (b + c - a)/(b2 + c2 - a2) : :
Trilinears    sec A cos2(A/2) : :
Barycentrics    sin A + tan A : sin B + tan B : sin C + tan C
Barycentrics    tan A cos2(A/2) : :

X(33) = (r + 2R - s)(r + 2R + s)*X(1) - 6rR*X(2) + 4rR*X(3)   (Peter Moses, April 2, 2013)

Let LA be the reflection of line BC in the internal angle bisector of angle A, and define LB and LC cyclically. Let DEF be the triangle formed by LA, LB, LC. Then DEF (the intangents triangle) is homothetic to the orthic triangle, and the homothetic center is X(33). (Randy Hutson, 9/23/2011)

If you have The Geometer's Sketchpad, you can view X(33).


If you have GeoGebra, you can view X(33).

X(33) lies on these lines: 1,4   2,1040   5,1062   6,204   7,1041   8,1039   9,212   10,406   11,427   12,235   19,25   20,1038   24,35   28,975   29,78   30,1060   36,378   40,201   42,393   47,90   56,963   57,103   63,1013   64,65   79,1063   80,1061   84,603   112,609   200,281   210,220   222,971   264,350

X(33) is the {X(1),X(4)}-harmonic conjugate of X(34). For a list of other harmonic conjugates, click Tables at the top of this page.

X(33) = isogonal conjugate of X(77)
X(33) = isotomic conjugate of X(7182)
X(33) = anticomplement of X(34822)
X(33) = trilinear pole of line X(657)X(4041) (the polar of X(85) wrt polar circle)
X(33) = pole wrt polar circle of trilinear polar of X(85) (line X(522)X(693))
X(33) = polar conjugate of X(85)
X(33) = perspector of ABC and extraversion triangle of X(34)
X(33) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,19), (29,281), (318,9)
X(33) = X(i)-cross conjugate of X(j) for these (i,j): (41,9), (42,55)
X(33) = crosspoint of X(i) and X(j) for these (i,j): (1,282), (4,281)
X(33) = crosssum of X(i) and X(j) for these (i,j): (1,223), (3,222), (57,1394), (73,1214)
X(33) = crossdifference of every pair of points on line X(652)X(905)
X(33) = X(33)-beth conjugate of X(25)
X(33) = homothetic center of anti-excenters-incenter reflections triangle and anti-tangential midarc triangle


X(34) = X(4)-BETH CONJUGATE OF X(4)

Trilinears    1 - sec A : 1 - sec B : 1 - sec C
Trilinears    tan A tan(A/2) : tan B tan(B/2) : tan C tan(C/2)
Trilinears    1/[(b + c - a)(b2 + c2 - a2)] : :
Trilinears    sec A sin2(A/2)
Barycentrics    sin A - tan A : sin B - tan B : sin C - tan C
Barycentrics    tan A sin2(A/2) : :
Tripolars    (pending)
X(34) = (r + 2R - s)(r + 2R + s)*X(1) + 6rR*X(2) - 4rR*X(3)   (Peter Moses, April 2, 2013)

X(34) is the center of perspective of the orthic triangle and the reflection in the incenter of the intangents triangle.

If you have The Geometer's Sketchpad, you can view X(34) (1) and X(34) (2).


If you have GeoGebra, you can view X(34).

X(34) lies on these lines: 1,4   2,1038   5,1060   6,19   7,1039   8,1041   9,201   10,475   11,235   12,427   20,1040   24,36   25,56   28,57   29,77   30,1062   35,378   40,212   46,47   55,227   79,1061   80,1063   87,242   106,108   196,937   207,1042   222,942   244,1106   331,870   347,452   860,997

X(34) is the {X(1),X(4)}-harmonic conjugate of X(33). For a list of other harmonic conjugates of X(34), click Tables at the top of this page.

X(34) = isogonal conjugate of X(78)
X(34) = isotomic conjugate of X(3718)
X(34) = anticomplement of X(34823)
X(34) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,207), (4,208), (28,56), (273,57), (278,19)
X(34) = X(25)-cross conjugate of X(19)
X(34) = crosssum of X(219) and X(1260)
X(34) = crossdifference of every pair of points on line X(521)X(652)
X(34) = circumcircle-inverse of X(32757)
X(34) = X(56)-Hirst inverse of X(1430)
X(34) = trilinear pole of polar of X(312) wrt polar circle (line X(649)X(4017))
X(34) = pole wrt polar circle of trilinear polar of X(312) (line X(522)X(3717))
X(34) = polar conjugate of X(312)
X(34) = perspector of ABC and extraversion triangle of X(33)
X(34) = homothetic center of intangents triangle and reflection of orthic triangle in X(4)
X(34) = homothetic center of orthic triangle and anti-tangential midarc triangle
X(34) = X(8078)-of-orthic-triangle if ABC is acute
X(34) = X(i)-beth conjugate of X(j) for these (i,j): (1,221), (4,4), (28,34), (29,1), (107,158), (108,34), (110,47), (162,34), (811,34)
X(34) = X(i)-isoconjugate of X(j) for these {i,j}: {1,78}, {31,3718}, {48,312}


X(35) = {X(1),X(3)}-HARMONIC CONJUGATE OF X(36)

Trilinears    1 + 2 cos A : 1 + 2 cos B : 1 + 2 cos C
Trilinears    a(b2 + c2 - a2 + bc) : :
Trilinears    sin(3A/2) csc(A/2) : :
Barycentrics    sin A + sin 2A : :
Barycentrics    a2(b2 + c2 - a2 + bc) : :

X(35) = 3 X[1] - 2 X[11011], 3 X[1] - X[11280], 3 X[2] + X[20066], 6 X[2] - 5 X[31262], 2 X[10] + X[11015], 2 X[12] - 3 X[3584], 2 X[12] + X[4324], X[12] - 3 X[4995], 3 X[2330] - X[19369], 4 X[2646] - X[11009], 2 X[2646] + X[11010], 3 X[2646] - X[11011], 6 X[2646] - X[11280], X[2975] - 3 X[17549], 3 X[3576] - X[11014], 3 X[3584] - X[3585], 3 X[3584] + X[4324], 3 X[3584] + 2 X[15338], X[3585] - 6 X[4995], X[3585] + 2 X[15338], X[3871] + 2 X[5267], 2 X[3871] + X[5288], X[3871] + 3 X[17549], X[4324] + 6 X[4995], 3 X[4995] + X[15338], 4 X[5267] - X[5288], 2 X[5267] - 3 X[17549], X[5288] - 6 X[17549], 3 X[5902] - 4 X[13750], 4 X[6668] - 3 X[17530], 3 X[7676] + X[8543], X[11009] + 2 X[11010], 3 X[11009] - 4 X[11011], 3 X[11009] - 2 X[11280], 3 X[11010] + 2 X[11011], 3 X[11010] + X[11280], X[11012] + 2 X[11849], 3 X[15015] - 2 X[33598], X[15908] - 3 X[21155], X[20066] + 2 X[25639], 2 X[20066] + 3 X[31159], 2 X[20066] + 5 X[31262], 4 X[25639] - 3 X[31159], 4 X[25639] - 5 X[31262], 3 X[31159] - 5 X[31262]

Let A' be the inverse-in-circumcircle of the A-excenter, and define B' and C' cyclically. Then the lines AA', BB', CC' concur in X(35).

Let A'B'C' be the orthic triangle. Let B'C'A″ be the triangle similar to ABC such that segment A'A″ crosses the line B'C', and define B″ and C″ cyclically. (Equivalently, A″ is the reflection of A in B'C'.) Let Ia be the incenter of B'C'A″, and define Ib and Ic cyclically. The lines AIa, BIb, CIc concur in X(35). (Randy Hutson, November 18, 2015)

If you have The Geometer's Sketchpad, you can view X(35).
If you have GeoGebra, you can view X(35).

X(35) lies on the cubics K073, K434, K679, K1056 and these lines: {1, 3}, {2, 1479}, {4, 498}, {5, 3583}, {6, 5312}, {8, 993}, {9, 90}, {10, 21}, {11, 140}, {12, 30}, {15, 1250}, {16, 5357}, {19, 14017}, {20, 1478}, {22, 612}, {23, 5297}, {24, 33}, {25, 1900}, {28, 1869}, {31, 386}, {32, 2276}, {34, 378}, {37, 267}, {38, 7293}, {39, 1914}, {41, 3730}, {42, 58}, {43, 1011}, {47, 212}, {54, 6286}, {60, 5127}, {61, 7127}, {63, 3811}, {71, 284}, {72, 191}, {73, 74}, {75, 21410}, {77, 7163}, {78, 3422}, {79, 226}, {81, 4278}, {84, 7162}, {86, 25599}, {98, 10086}, {99, 1909}, {101, 1334}, {103, 1803}, {104, 5559}, {106, 28218}, {110, 501}, {112, 13116}, {125, 12896}, {145, 8666}, {149, 24387}, {172, 187}, {181, 35203}, {182, 3056}, {183, 3760}, {186, 1825}, {192, 7793}, {197, 8185}, {198, 3731}, {199, 1961}, {200, 1259}, {201, 1725}, {202, 5237}, {203, 5238}, {210, 31445}, {213, 17735}, {214, 3884}, {218, 4258}, {221, 10076}, {225, 7414}, {228, 846}, {238, 3216}, {255, 991}, {259, 34201}, {270, 2073}, {291, 8671}, {306, 24632}, {319, 34016}, {350, 1078}, {355, 6914}, {371, 3301}, {372, 2066}, {376, 388}, {377, 10198}, {380, 1723}, {381, 12953}, {382, 10895}, {384, 27020}, {385, 25264}, {389, 11429}, {390, 3086}, {392, 2932}, {404, 1125}, {405, 1376}, {411, 516}, {442, 6690}, {474, 1001}, {495, 550}, {496, 549}, {497, 499}, {500, 1154}, {511, 2330}, {515, 6906}, {518, 3916}, {519, 2975}, {528, 4999}, {535, 20060}, {538, 4400}, {546, 3614}, {548, 4325}, {551, 5253}, {572, 1404}, {573, 1405}, {574, 2241}, {575, 8540}, {578, 11436}, {580, 14547}, {590, 31499}, {595, 902}, {596, 32923}, {603, 10623}, {609, 3053}, {611, 1350}, {613, 5085}, {614, 7485}, {632, 10593}, {650, 11247}, {672, 4251}, {674, 5135}, {692, 1437}, {741, 29151}, {748, 17749}, {750, 16451}, {758, 20612}, {813, 2711}, {849, 1326}, {851, 29640}, {910, 16601}, {920, 10393}, {936, 4512}, {944, 5450}, {946, 5443}, {947, 1795}, {950, 1006}, {953, 23153}, {954, 4312}, {956, 3632}, {958, 3679}, {960, 5440}, {968, 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13204}, {3011, 7465}, {3023, 33813}, {3027, 12042}, {3028, 12041}, {3035, 4187}, {3052, 4255}, {3065, 7161}, {3070, 9646}, {3071, 9660}, {3090, 5225}, {3099, 11494}, {3100, 7488}, {3101, 25080}, {3120, 24160}, {3146, 10590}, {3157, 12163}, {3159, 32936}, {3207, 32625}, {3218, 3874}, {3219, 3647}, {3230, 21008}, {3237, 8161}, {3238, 8160}, {3244, 5303}, {3270, 13367}, {3286, 4649}, {3293, 4267}, {3294, 35342}, {3297, 6410}, {3298, 6409}, {3311, 19037}, {3312, 19038}, {3357, 7355}, {3419, 26066}, {3434, 6910}, {3454, 29846}, {3474, 3487}, {3485, 6361}, {3486, 5657}, {3501, 16788}, {3509, 3970}, {3515, 7071}, {3522, 4293}, {3525, 10589}, {3526, 9669}, {3529, 5229}, {3530, 15172}, {3534, 9655}, {3541, 11393}, {3560, 5587}, {3600, 10304}, {3616, 4188}, {3626, 17574}, {3627, 10592}, {3628, 5326}, {3633, 12513}, {3634, 5047}, {3648, 17484}, {3649, 5719}, {3652, 12738}, {3683, 5044}, {3684, 16552}, {3688, 7193}, {3689, 34790}, {3693, 17744}, {3695, 3712}, {3697, 5302}, {3705, 4894}, {3720, 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{13061, 19475}, {13062, 19476}, {13080, 18244}, {13089, 13146}, {13173, 13174}, {13206, 13221}, {13253, 22775}, {13278, 26726}, {13293, 19505}, {13541, 34139}, {13588, 25526}, {13602, 15179}, {13665, 13897}, {13666, 13714}, {13675, 13679}, {13723, 29674}, {13743, 17663}, {13785, 13954}, {13786, 13837}, {13795, 13799}, {13887, 13888}, {13935, 13962}, {13940, 13942}, {13966, 19029}, {14100, 31658}, {14118, 24025}, {14377, 30949}, {14873, 33329}, {14986, 15717}, {15104, 21165}, {15462, 32290}, {15815, 16781}, {15908, 21155}, {15950, 22791}, {16058, 16569}, {16059, 25502}, {16061, 16818}, {16113, 16152}, {16117, 16118}, {16139, 33857}, {16154, 21077}, {16163, 18968}, {16286, 17123}, {16299, 33174}, {16342, 19858}, {16346, 19859}, {16405, 29825}, {16436, 29573}, {16453, 17122}, {16468, 20992}, {16469, 21002}, {16471, 19764}, {16475, 16688}, {16484, 20470}, {16496, 22769}, {16583, 34872}, {16783, 17754}, {16817, 28611}, {16819, 33047}, {16825, 23407}, {16826, 19308}, {16842, 19872}, {16845, 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{19513, 21321}, {19649, 24239}, {19864, 32942}, {20107, 31272}, {20108, 32944}, {20459, 23530}, {20475, 23370}, {20677, 21004}, {20831, 20989}, {20963, 33863}, {20999, 22344}, {21319, 33099}, {21477, 29598}, {21516, 29596}, {21518, 29602}, {21537, 26626}, {21616, 27385}, {21669, 31673}, {22060, 32913}, {22267, 27248}, {22556, 22650}, {22557, 22651}, {22558, 22652}, {22559, 22653}, {22676, 22729}, {22843, 22884}, {22890, 22929}, {22951, 22980}, {22954, 22962}, {23226, 35057}, {23358, 32378}, {24036, 33950}, {24046, 28082}, {24068, 32927}, {24161, 24715}, {24248, 24309}, {24424, 24684}, {24466, 31775}, {24813, 24845}, {24820, 24821}, {24953, 31419}, {25582, 27127}, {26298, 26493}, {26299, 26502}, {26300, 26512}, {26301, 26513}, {26321, 28204}, {26353, 26498}, {26354, 26507}, {26355, 26516}, {26356, 26521}, {26562, 30131}, {26686, 35297}, {27143, 28410}, {27324, 33816}, {27622, 33109}, {27802, 31320}, {28606, 30142}, {28761, 31058}, {29055, 29300}, {29574, 35276}, {30107, 33819}, {30110, 33830}, {30295, 30424}, {30296, 30425}, {30297, 30426}, {30385, 30431}, {30386, 30432}, {30435, 31461}, {30944, 33140}, {31140, 31493}, {32143, 32210}, {32168, 32171}, {32233, 32307}, {32256, 32261}, {32259, 32305}, {32330, 32403}, {32347, 32356}, {32350, 32401}, {33635, 33671}, {33866, 33949}, {34927, 34931}

X(35) = midpoint of X(i) and X(j) for these {i,j}: {1, 11010}, {3, 11849}, {12, 15338}, {2975, 3871}, {3585, 4324}, {5086, 11015}, {6906, 11491}
X(35) = reflection of X(i) in X(j) for these {i,j}: {1, 2646}, {79, 14526}, {2975, 5267}, {3584, 4995}, {3585, 12}, {4324, 15338}, {5086, 10}, {5288, 2975}, {6763, 3916}, {6842, 31659}, {11009, 1}, {11012, 3}, {11280, 11011}, {12047, 13411}, {24390, 4999}, {31159, 2}
X(35) = isogonal conjugate of X(79)
X(35) = isotomic conjugate of X(20565)
X(35) = isogonal conjugate of the anticomplement of X(3647)
X(35) = isogonal conjugate of the complement of X(3648)
X(35) = isogonal conjugate of the isotomic conjugate of X(319)
X(35) =Thomson-isogonal conjugate of X(5659)
X(35) =excentral-isogonal conjugate of X(2949)
X(35) = complement of isogonal conjugate of X(34441)
X(35) = anticomplement of X(25639)
X(35) = orthocenter of cross-triangle of ABC and inner Yff triangle
X(35) = insimilicenter of circumcircles of ABC and inner Yff triangle; the exsimilicenter is X(1)
X(35) = homothetic center of Trinh triangle and anti-tangential midarc triangle
X(35) = circumcircle-inverse of X(484)
X(35) = isogonal conjugate of the anticomplement of X(3647)
X(35) = isogonal conjugate of the complement of X(3648)
X(35) = isogonal conjugate of the isotomic conjugate of X(319)
X(35) = Thomson isogonal conjugate of X(5659)
X(35) = excentral isogonal conjugate of X(2949)
X(35) = complement of isotomic conjugate of isogonal conjugate of X(20988)
X(35) = complement of polar conjugate of isogonal conjugate of X(22122)
X(35) = complement of complement of X(20066)
X(35) = Cundy-Parry Psi transform of X(15175)
X(35) = X(34441)-complementary conjugate of X(10)
X(35) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 35197}, {21, 35194}, {943, 1}, {1255, 6}, {1442, 2003}, {5951, 484}, {11107, 6198}, {15168, 1757}, {18359, 2323}, {33670, 33669}
X(35) = X(i)-cross conjugate of X(j) for these (i,j): {500, 1}, {2174, 2003}, {2594, 6198}, {6149, 7343}
X(35) = X(i)-isoconjugate of X(j) for these (i,j): {1, 79}, {2, 2160}, {4, 7100}, {6, 30690}, {7, 7073}, {31, 20565}, {36, 2166}, {57, 7110}, {58, 6757}, {65, 3615}, {75, 6186}, {81, 8818}, {94, 7113}, {191, 30602}, {225, 1789}, {265, 1870}, {273, 8606}, {513, 6742}, {522, 26700}, {523, 13486}, {554, 1251}, {649, 15455}, {1081, 33653}, {1127, 10230}, {1989, 3218}, {3466, 34301}, {3468, 34303}, {4707, 32678}, {7004, 34922}, {11060, 20924}, {11076, 21739}, {13610, 14844}, {21044, 35049}, {21828, 32680}
X(35) = cevapoint of X(i) and X(j) for these (i,j): {55, 1030}, {2594, 22342}
X(35) = crosspoint of X(i) and X(j) for these (i,j): {1, 3467}, {21, 35196}, {59, 8701}, {100, 4570}, {1442, 3219}, {11107, 35193}
X(35) = crosssum of X(i) and X(j) for these (i,j): {1, 3336}, {6, 20988}, {11, 4977}, {481, 482}, {513, 3120}, {1086, 23729}, {2160, 7073}, {3122, 23751}, {4466, 23727}
X(35) = trilinear pole of line {2605, 9404}
X(35) = crossdifference of every pair of points on line {650, 4802}
X(35) = barycentric product X(i)*X(j) for these {i,j}: {1, 3219}, {6, 319}, {8, 2003}, {9, 1442}, {21, 16577}, {31, 33939}, {42, 34016}, {50, 20566}, {55, 17095}, {57, 4420}, {58, 3969}, {63, 6198}, {75, 2174}, {80, 323}, {81, 3678}, {100, 14838}, {101, 4467}, {110, 7265}, {190, 2605}, {219, 7282}, {226, 35193}, {249, 21054}, {261, 21794}, {304, 14975}, {312, 1399}, {314, 21741}, {321, 17104}, {333, 2594}, {445, 1794}, {593, 7206}, {651, 35057}, {664, 9404}, {692, 18160}, {765, 7202}, {943, 16585}, {1126, 3578}, {1214, 11107}, {1255, 3647}, {1268, 17454}, {1441, 35192}, {1812, 1825}, {2167, 35194}, {2611, 4567}, {2982, 31938}, {4557, 16755}, {4570, 8287}, {4600, 20982}, {6149, 18359}, {6187, 7799}, {6335, 23226}, {7110, 7279}, {7186, 17743}, {7343, 17484}, {8652, 23883}, {19620, 33670}, {21824, 24041}, {22342, 31623}
X(35) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 30690}, {2, 20565}, {6, 79}, {31, 2160}, {32, 6186}, {37, 6757}, {41, 7073}, {42, 8818}, {48, 7100}, {50, 36}, {55, 7110}, {80, 94}, {100, 15455}, {101, 6742}, {163, 13486}, {186, 17923}, {284, 3615}, {319, 76}, {323, 320}, {500, 5249}, {526, 4707}, {1399, 57}, {1415, 26700}, {1442, 85}, {2003, 7}, {2161, 2166}, {2174, 1}, {2193, 1789}, {2307, 554}, {2477, 2003}, {2594, 226}, {2605, 514}, {2611, 16732}, {3219, 75}, {3444, 30602}, {3578, 1269}, {3647, 4359}, {3678, 321}, {3969, 313}, {4420, 312}, {4467, 3261}, {6149, 3218}, {6187, 1989}, {6198, 92}, {7115, 34922}, {7186, 3662}, {7202, 1111}, {7206, 28654}, {7265, 850}, {7279, 17095}, {7282, 331}, {7343, 21739}, {8287, 21207}, {9404, 522}, {11107, 31623}, {14270, 21828}, {14838, 693}, {14975, 19}, {16577, 1441}, {17095, 6063}, {17104, 81}, {17190, 16709}, {17454, 1125}, {18755, 14844}, {20566, 20573}, {20982, 3120}, {21054, 338}, {21741, 65}, {21794, 12}, {21824, 1109}, {22094, 4466}, {22115, 22128}, {22342, 1214}, {23226, 905}, {33939, 561}, {34016, 310}, {35057, 4391}, {35192, 21}, {35193, 333}, {35194, 14213}, {35197, 17483}
X(35) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 3, 36}, {1, 36, 5563}, {1, 40, 5903}, {1, 46, 5902}, {1, 55, 3746}, {1, 57, 18398}, {1, 65, 5425}, {1, 165, 46}, {1, 484, 65}, {1, 3336, 942}, {1, 3337, 354}, {1, 3550, 5264}, {1, 3576, 21842}, {1, 5010, 3}, {1, 5119, 5697}, {1, 5131, 3337}, {1, 7280, 56}, {1, 7991, 25415}, {1, 10902, 14798}, {1, 11280, 11011}, {1, 14794, 11012}, {1, 15803, 3338}, {1, 16192, 15803}, {1, 17596, 3670}, {1, 17700, 30274}, {1, 30282, 3612}, {1, 32613, 14795}, {2, 1479, 7741}, {2, 4294, 1479}, {2, 5248, 5259}, {2, 5259, 25542}, {2, 25639, 31262}, {3, 55, 1}, {3, 56, 7280}, {3, 999, 5204}, {3, 1482, 26286}, {3, 3295, 56}, {3, 3579, 7688}, {3, 3612, 14803}, {3, 3746, 5563}, {3, 5217, 5010}, {3, 6244, 5584}, {3, 10246, 32612}, {3, 10267, 3576}, {3, 10306, 3428}, {3, 10310, 165}, {3, 10679, 11249}, {3, 10902, 15931}, {3, 11248, 40}, {3, 11507, 46}, {3, 14882, 484}, {3, 15931, 35202}, {3, 16202, 10269}, {3, 26285, 2077}, {3, 26357, 14793}, {3, 31393, 13370}, {3, 32613, 10902}, {3, 32760, 14798}, {3, 35000, 3579}, {3, 35238, 35242}, {3, 35251, 35238}, {4, 498, 7951}, {4, 5218, 498}, {5, 6284, 3583}, {8, 993, 5258}, {8, 4189, 993}, {10, 21, 5251}, {10, 4304, 10572}, {10, 10572, 80}, {11, 15171, 4857}, {15, 1250, 5353}, {15, 7006, 2307}, {16, 10638, 5357}, {20, 1478, 10483}, {20, 3085, 1478}, {20, 5281, 3085}, {21, 100, 10}, {31, 386, 1203}, {32, 2276, 5280}, {32, 31451, 2276}, {36, 3746, 1}, {36, 5537, 3245}, {36, 14795, 14798}, {36, 14799, 15931}, {36, 14803, 14800}, {39, 1914, 5299}, {40, 3601, 1}, {40, 5903, 3245}, {40, 11248, 5537}, {41, 3730, 5526}, {55, 56, 3295}, {55, 1470, 31393}, {55, 5010, 36}, {55, 5172, 24929}, {55, 5204, 3303}, {55, 5217, 3}, {55, 10310, 11507}, {55, 10966, 26358}, {55, 11492, 8186}, {55, 11493, 8187}, {55, 14794, 11009}, {55, 16678, 3750}, {55, 26357, 1697}, {55, 32613, 32760}, {55, 34879, 2078}, {56, 3295, 1}, {56, 7280, 36}, {58, 33771, 42}, {63, 3811, 5904}, {65, 3579, 484}, {65, 14882, 3256}, {65, 24929, 1}, {72, 4640, 191}, {73, 109, 34043}, {74, 10088, 19470}, {75, 21586, 21410}, {78, 12514, 5692}, {78, 35258, 12514}, {79, 15228, 1770}, {80, 5441, 10572}, {100, 10058, 80}, {104, 10087, 7972}, {110, 10065, 7727}, {140, 15171, 11}, {165, 8069, 36}, {172, 1500, 16785}, {187, 1500, 172}, {197, 13730, 8185}, {212, 601, 47}, {221, 10606, 10076}, {226, 1770, 79}, {226, 31730, 1770}, {226, 32610, 8606}, {267, 1717, 33642}, {371, 5414, 3301}, {372, 2066, 3299}, {376, 388, 4299}, {381, 12953, 18514}, {382, 10895, 18513}, {382, 31479, 10895}, {390, 3523, 3086}, {404, 1621, 1125}, {405, 1376, 1698}, {474, 1001, 3624}, {484, 5172, 36}, {484, 15932, 46}, {484, 24929, 5425}, {484, 34871, 3}, {495, 550, 7354}, {495, 7354, 5270}, {496, 549, 5433}, {496, 5433, 3582}, {496, 10386, 3058}, {497, 631, 499}, {498, 4302, 4}, {499, 4309, 497}, {548, 15888, 4325}, {548, 18990, 15326}, {549, 3058, 3582}, {549, 10386, 496}, {550, 7354, 4316}, {574, 2241, 2275}, {574, 10987, 16784}, {595, 1193, 5315}, {595, 4256, 1193}, {672, 4251, 17745}, {902, 1193, 595}, {902, 4256, 5315}, {942, 1155, 3336}, {942, 31663, 1155}, {943, 3651, 226}, {944, 6950, 5450}, {950, 6684, 1737}, {954, 11495, 4312}, {956, 3913, 3632}, {958, 4421, 5687}, {958, 5687, 3679}, {968, 975, 27785}, {988, 3749, 1}, {993, 8715, 8}, {999, 3303, 1}, {1011, 19763, 1724}, {1012, 11500, 5691}, {1058, 3524, 7288}, {1058, 7288, 10072}, {1125, 10624, 30384}, {1158, 18446, 15071}, {1250, 2307, 7006}, {1259, 20835, 31424}, {1319, 9957, 1}, {1381, 1382, 484}, {1385, 3057, 1}, {1385, 3579, 13145}, {1385, 14792, 36}, {1385, 26086, 3}, {1388, 34880, 5193}, {1399, 2594, 2003}, {1420, 1470, 13370}, {1420, 31393, 1}, {1466, 1617, 3361}, {1470, 11510, 1420}, {1478, 31452, 3085}, {1482, 34471, 1}, {1656, 9668, 10896}, {1657, 9654, 12943}, {1697, 3576, 1}, {1698, 3586, 10826}, {1737, 6684, 5445}, {1770, 31730, 15228}, {1837, 26446, 18395}, {2077, 10902, 3}, {2077, 32613, 15931}, {2077, 32760, 36}, {2078, 34879, 15931}, {2098, 10246, 1}, {2177, 4257, 16474}, {2241, 2275, 16784}, {2275, 10987, 2241}, {2307, 7006, 5353}, {2550, 6857, 19854}, {2594, 6149, 35197}, {2646, 11010, 11009}, {2646, 14794, 36}, {2915, 20872, 9591}, {2975, 17549, 5267}, {3052, 4255, 16466}, {3057, 26086, 14792}, {3058, 5433, 496}, {3085, 5281, 31452}, {3149, 11496, 1699}, {3219, 4420, 3678}, {3256, 7688, 484}, {3295, 35239, 3340}, {3303, 5204, 999}, {3304, 6767, 1}, {3333, 10389, 1}, {3428, 10306, 7991}, {3434, 6910, 26363}, {3486, 5657, 10573}, {3515, 7071, 11399}, {3524, 10385, 10072}, {3529, 8164, 5229}, {3530, 15172, 15325}, {3560, 11499, 5587}, {3576, 14793, 36}, {3579, 24929, 65}, {3583, 4330, 6284}, {3584, 3585, 12}, {3584, 4324, 3585}, {3612, 5119, 1}, {3612, 5697, 24926}, {3624, 9614, 23708}, {3647, 3678, 3219}, {3666, 5266, 1}, {3730, 4262, 41}, {3746, 14799, 14798}, {3746, 14803, 24926}, {3748, 5045, 1}, {3748, 32636, 5045}, {3869, 22836, 4867}, {3871, 5267, 5288}, {3871, 17549, 2975}, {3920, 6636, 5322}, {3976, 17715, 1}, {4189, 8715, 5258}, {4251, 24047, 672}, {4255, 16466, 5313}, {4261, 5301, 16470}, {4292, 13405, 13407}, {4299, 10056, 388}, {4302, 5218, 7951}, {4304, 10572, 5441}, {4314, 10164, 1210}, {4316, 5270, 7354}, {4326, 21153, 15299}, {4366, 7824, 26959}, {4421, 16370, 3679}, {4423, 16408, 34595}, {4428, 16371, 25055}, {4855, 5250, 997}, {4995, 15338, 12}, {5010, 11010, 14794}, {5010, 32613, 14799}, {5010, 32760, 15931}, {5015, 32851, 30171}, {5045, 5122, 32636}, {5046, 27529, 3814}, {5048, 15178, 1}, {5085, 10387, 613}, {5126, 31792, 20323}, {5127, 15792, 60}, {5132, 8053, 238}, {5132, 16287, 3216}, {5172, 14882, 65}, {5172, 35000, 484}, {5217, 11849, 14794}, {5248, 25440, 2}, {5263, 19270, 19863}, {5268, 7298, 25}, {5284, 17531, 19862}, {5300, 33113, 30172}, {5326, 7173, 3628}, {5432, 6284, 5}, {5563, 15931, 34890}, {5597, 26423, 1}, {5598, 26399, 1}, {5687, 16370, 958}, {5691, 31434, 10827}, {5697, 14803, 5563}, {5703, 9778, 4295}, {5711, 19765, 1}, {5919, 24928, 1}, {6398, 31474, 18995}, {6656, 26629, 30104}, {6769, 10383, 1}, {6914, 32141, 355}, {6938, 10786, 6256}, {7081, 7283, 1089}, {7288, 10385, 1058}, {7691, 10066, 7356}, {7742, 15803, 36}, {7807, 26590, 30103}, {7982, 13384, 1}, {7987, 8071, 36}, {8069, 10310, 46}, {8069, 11507, 1}, {8071, 11508, 1}, {8186, 8187, 57}, {8666, 25439, 145}, {9627, 18447, 1}, {9819, 30389, 1}, {9957, 13624, 1319}, {10088, 19470, 6126}, {10267, 14793, 21842}, {10267, 26357, 1}, {10269, 26358, 1}, {10306, 22766, 25415}, {10679, 11249, 7982}, {10679, 22768, 1}, {10902, 32760, 14795}, {10965, 16203, 1}, {10966, 16202, 1}, {10966, 26358, 7962}, {11011, 11280, 11009}, {11375, 12699, 18393}, {11461, 11464, 9638}, {11849, 33862, 11012}, {12100, 15170, 5298}, {12512, 13405, 4292}, {12515, 12739, 11571}, {13743, 18524, 18480}, {14795, 14799, 10902}, {14796, 14797, 1}, {14796, 14802, 36}, {14797, 14801, 36}, {14801, 14802, 3}, {15326, 15888, 18990}, {15326, 18990, 4325}, {15696, 31480, 9657}, {17637, 22937, 1749}, {17735, 18755, 213}, {18518, 28444, 18761}, {20323, 31792, 1}, {24929, 35000, 3256}, {24953, 34612, 31419}, {25414, 26287, 1}, {26285, 32613, 3}, {30282, 31508, 5119}, {30478, 34607, 5082}, {31159, 31262, 25639}, {32622, 32623, 15931}
X(35) = X(943)-aleph conjugate of X(35)
X(35) = X(i)-beth conjugate of X(j) for these (i,j): (100,35), (643,10)
X(35) = perspector of ABC and orthic triangle of incentral triangle
X(35) = X(2975) of X(1)-Brocard triangle
X(35) = crossdifference of every pair of points on line X(650)X(4802)
X(35) = homothetic center of intangents and Kosnita triangles
X(35) = perspector of ABC and extraversion triangle of X(36)
X(35) = Hofstadter 3/2 point
X(35) = homothetic center of 2nd isogonal triangle of X(1) and cevian triangle of X(3); see X(36)
X(35) = insimilicenter of circumcircle and circumcircle of reflection triangle of X(1); exsimilicenter is X(36)
X(35) = Cundy-Parry Phi transform of X(5902)


X(36) = INVERSE-IN-CIRCUMCIRCLE OF INCENTER

Trilinears     1 - 2 cos A : 1 - 2 cos B : 1 - 2 cos C
Trilinears     a(b2 + c2 - a2 - bc)
Trilinears     sec(A/2) cos(3A/2) : :
Barycentrics   sin A - sin 2A : :
Barycentrics   a2(b2 + c2 - a2 - bc) : :
Tripolars    Sqrt[b c (b + c - a)] : :
Tripolars    sec A' : :, where A'B'C' is the excentral triangle

If you have The Geometer's Sketchpad, you can view X(36).
If you have GeoGebra, you can view X(36).

Let A' be the isogonal conjugate of A with respect to BCX(1), and define B' and C' cyclically. Let A″ be the circumcenter of BCX(1), and define B″ and C″ cyclically. The lines A'A″, B'B″, C'C″ concur in X(36). Also, X(36) is the QA-P4 center (Isogonal Center) of quadrangle ABCX(1) (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/25-qa-p4.html)

Let P be a point in the plane of triangle ABC, not on a sideline of ABC. Let A1 be the isogonal conjugate of A with respect to triangle BCP, and define B1, C1 cyclically. Call triangle A1B1C1 the 1st isogonal triangle of P. A1B1C1 is also the reflection triangle of the isogonal conjugate of P. A1B1C1 is perspective to ABC iff P lies on the Neuberg cubic. The perspector lies on cubic K060 (pK(X1989, X265), O(X5) orthopivotal cubic). Let A2 be the isogonal conjugate of A1 with respect to triangle B1C1P, and define B2, C2 cyclically. Call triangle A2B2C2 the 2nd isogonal triangle of P. Continuing, let An be the isogonal conjugate of A(n-1) with respect to triangle B(n-1)C(n-1)P, and define B(n), C(n) cyclically. Call triangle AnBnCn the nth isogonal triangle of P. For n >= 2, all triangles AnBnCn are perspective to A(n-1)B(n-1)C(n-1). Call the perspector, Pn, the nth isogonal perspector of P. Pn is the orthocenter of A(n-1)B(n-1)C(n-1) and either the incenter or an excenter of AnBnCn. The triangles AnBnCn are all concyclic, with P as center. Call the circle the isogonal circle of P. For P = X(1), the 2nd isogonal triangle of X(1) is homothetic to ABC at X(36); see also X(35), X(1478), X(1479), X(3583), X(3585), X(5903), X(7741), X(7951). (Randy Hutson, November 18, 2015)

Let A'B'C' be the incentral triangle. Let A″ be the reflection of A in line B'C', and define B″, C″ cyclically. The lines A'A″, B'B″, C'C″ concur in X(36). (Randy Hutson, June 27, 2018)

X(36) is the {X(3),X(56)}-harmonic conjugate of X(1). For a list of other harmonic conjugates of X(36), click Tables at the top of this page.

X(36) lies on these lines: 1,3   2,535   4,499   6,609   10,404   11,30   12,140   15,202   16,203   21,79   22,614   24,34   31,995   33,378   39,172   47,602   48,579   54,73   58,60   59,1110   63,997   80,104   84,90   99,350   100,519   101,672   106,901   109,953   187,1015   191,960   214,758   226,1006   238,513   255,1106   376,497   388,498   474,958   495,549   496,550   573,604   1030,1100

X(36) = midpoint of X(1) and X(484)
X(36) = reflection of X(i) in X(j) for these (i,j): (1,1319), (484,1155) (2077,3)
X(36) = isogonal conjugate of X(80)
X(36) = isotomic conjugate of X(20566)
X(36) = complement of X(5080)
X(36) = anticomplement of X(3814)
X(36) = circumcircle-inverse of X(1)
X(36) = inccircle-inverse of X(942)
X(36) = Bevan-circle-inverse of X(46)
X(36) = polar conjugate of isotomic conjugate of X(22128)
X(36) = X(i)-Ceva conjugate of X(j) for these (i,j): (88,6), (104,1)
X(36) = crosspoint of X(58) and X(106)
X(36) = crosssum of X(i) and X(j) for these (i,j): (1,484), (10,519), (11,900)
X(36) = crossdifference of every pair of points on line X(37)X(650)
X(36) = X(104)-aleph conjugate of X(36)
X(36) = X(i)-beth conjugate of X(j) for these (i,j): (21,36), (100,36), (643,519)
X(36) = X(2070)-of-intouch-triangle
X(36) = X(186)-of-2nd circumperp-triangle
X(36) = {X(55),X(56)}-harmonic conjugate of X(999)
X(36) = reflection of X(484) in the antiorthic axis
X(36) = inverse-in-{circumcircle, nine-point circle}-inverter of X(354)
X(36) = perspector of ABC and extraversion triangle of X(35)
X(36) = homothetic center of intangents and Trinh triangles
X(36) = perspector of ABC and the reflection of the 2nd circumperp triangle in line X(1)X(3)
X(36) = X(186)-of-reflection-triangle-of-X(1)
X(36) = Cundy-Parry Psi transform of X(15446)
X(36) = exsimilicenter of circumcircle and circumcircle of reflection triangle of X(1); insimilicenter is X(35)
X(36) = homothetic center of medial triangle and cross-triangle of ABC and 2nd isogonal triangle of X(1)
X(36) = perspector of ABC and the reflection of the excentral triangle in the antiorthic axis (the reflection of the anticevian triangle of X(1) in the trilinear polar of X(1))
X(36) = Cundy-Parry Phi transform of X(5903)
X(36) = homothetic center of Kosnita triangle and anti-tangential midarc triangle
X(36) = orthocenter of cross-triangle of ABC and outer Yff triangle
X(36) = exsimilicenter of circumcircles of ABC and outer Yff triangle; the insimilicenter is X(1)
X(36) = outer-Yff-isogonal conjugate of X(34789)


X(37) = CROSSPOINT OF INCENTER AND CENTROID

Trilinears       b + c : c + a : a + b
Trilinears       ar - S : br - S : cr - S
Trilinears    semi-major axis of A-Soddy ellipse : :

Barycentrics  a(b + c) : b(c + a) : c(a + b)
Tripolars    (pending)
X(37) = (r2 + 2rR - s2)*X(1) - 6rR*X(2) - 2r2*X(3)   (Peter Moses, April 2, 2013)

Let A'B'C' be the cevian triangle of X(1). Let A″ be the centroid of triangle AB'C', and define B″ and C″ cyclically. Then the lines AA″, BB″, CC″ concur in X(37). (Eric Danneels, Hyacinthos 7892, 9/13/03)

A simple construction of X(37) as a crosspoint can be generalized as follows: let DEF be the medial triangle of ABC and let A'B'C' be the cevian triangle of a point U other than the centroid, X(2). The crosspoint of X(2) and U is then the point of concurrence of lines LD,ME,NF, where L,M,N are the respective midpoints of AA', BB', CC'. If U=u : v : w (trilinears), then crosspoint(X(2),U) = b/w+c/v : c/u+a/w : a/v+b/u, assuming that uvw is nonzero. In particular, if U=X(1), then the crosspoint is X(37). (Seiichi Kirikami, July 10, 2011)

X(37) = perspector of ABC and the medial triangle of the incentral triangle of ABC. (Randy Hutson, August 23, 2011)

X(37) = center of the Hofstadter ellipse E(1/2); see X(359). This is the incentral inellipse, which is the trilinear square of the antiorthic axis. (Randy Hutson, August 9, 2014)

Let A' be the trilinear pole of the tangent to the Apollonius circle where it meets the A-excircle, and define B' and C' cyclically. The triangle A'B'C' is homothetic to ABC at X(37). (Randy Hutson, April 9, 2016)

Let Oa be the A-extraversion of the Conway circle (the circle centered at the A-excenter and passing through A, with radius sqrt(ra^2 + s^2), where ra is the A-exradius). Let La be the radical axis of the circumcircle and Oa. Let A' = Lb∩Lc, B' = Lc∩La, C' = La∩Lb. The lines AA', BB', CC' concur in X(37). (Randy Hutson, April 9, 2016)

If you have The Geometer's Sketchpad, you can view X(37).
If you have GeoGebra, you can view X(37).

X(37) lies on these lines: 1,6   2,75   3,975   7,241   8,941   10,594  12,225   19,25   21,172   35,267   38,354   39,596   41,584   48,205   63,940   65,71   73,836   78,965   82,251   86,190   91,498   100,111   101,284   141,742   142,1086   145,391   158,281   171,846   226,440   256,694   347,948   513,876   517,573   537,551   579,942   626,746   665,900   971,991   1953,2183

X(37) is the {X(1),X(9)}-harmonic conjugate of X(6). For a list of other harmonic conjugates of X(37), click Tables at the top of this page.

X(37) = midpoint of X(i) and X(j) for these (i,j): (75,192), (190,335)
X(37) = isogonal conjugate of X(81)
X(37) = isotomic conjugate of X(274)
X(37) = complement of X(75)
X(37) = complementary conjugate of X(2887)
X(37) = anticomplement of X(3739)
X(37) = circumcircle-inverse of X(32758)
X(37) = X(i)-Ceva conjugate of X(j) for these (i,j):
(1,42), (2,10), (4,209), (9,71), (10,210), (190,513), (226,65), (321,72), (335,518)
X(37) = cevapoint of X(213) and X(228)
X(37) = X(i)-cross conjugate of X(j) for these (i,j): (42,65), (228,72)
X(37) = crosspoint of X(i) and X(j) for these (i,j): (1,2), (9,281), (10,226)
X(37) = X(1)-line conjugate of X(238)
X(37) = crosssum of X(i) and X(j) for these (i,j): (1,6), (57,222), (58,284), (1333,1437)
X(37) = crossdifference of every pair of points on line X(36)X(238)
X(37) = X(10)-Hirst inverse of X(740)
X(37) = X(1)-aleph conjugate of X(1051)
X(37) = X(i)-beth conjugate of X(j) for these (i,j): (9,37), (644,37), (645,894), (646,37), (1018,37)
X(37) = midpoint of PU(i), for these i: 6, 52, 53
X(37) = bicentric sum of PU(i), forthese i: 6, 52, 53
X(37) = trilinear product of PU(32)
X(37) = center of circumconic that is locus of trilinear poles of lines passing through X(10)
X(37) = perspector of circumconic centered at X(10)
X(37) = trilinear pole of line X(512)X(661) (polar of X(286) wrt polar circle)
X(37) = trilinear pole wrt medial triangle of Gergonne line
X(37) = pole wrt polar circle of trilinear polar of X(286) (line X(693)X(905))
X(37) = X(48)-isoconjugate (polar conjugate) of X(286)
X(37) = {X(6),X(9)}-harmonic conjugate of X(44)
X(37) = X(160)-of-intouch triangle
X(37) = perpector of ABC and n(Medial)*n(Incentral) triangle
X(37) = homothetic center of medial triangle and inverse of n(Medial)*n(Incentral) triangle
X(37) = perspector of incentral triangle and tangential triangle, wrt incentral triangle, of circumconic of incentral triangle centered at X(1) (the bicevian conic of X(1) and X(57))
X(37) = inverse-in-circumconic-centered-at-X(9) of X(1757)
X(37) = complement wrt incentral triangle of X(2667)
X(37) = perspector of ABC and unary cofactor triangle of 2nd circumperp triangle
X(37) = perspector of medial triangle and Gergonne line extraversion triangle
X(37) = trilinear pole, wrt Gergonne line extraversion triangle, of Gergonne line
X(37) = perspector of ABC and cross-triangle of Gemini triangles 3 and 4
X(37) = perspector of ABC and cross-triangle of ABC and Gemini triangle 3
X(37) = perspector of ABC and cross-triangle of ABC and Gemini triangle 4
X(37) = perspector of ABC and cross-triangle of ABC and Gemini triangle 16
X(37) = center of the {ABC, Gemini 17}-circumconic
X(37) = perspector of ABC and unary cofactor triangle of Gemini triangle 23
X(37) = incentral-to-ABC barycentric image of X(37)


X(38) = CROSSPOINT OF X(1) AND X(75)

Trilinears    b2 + c2 : :
Trilinears    csc A sin(A + ω) : csc B sin(B + ω) : csc C sin(C + ω)
Trilinears    SA + Sω : :
Barycentrics    a(b2 + c2) : :
Barycentrics    sin(A + ω) : sin(B + ω) : sin(C + ω)
X(38) = (3r2 + 8rR - s2)*X(1) - 6rR*X(2) - 4r2*X(3)   (Peter Moses, April 2, 2013)

X(38) lies on these lines: 1,21   2,244   3,976   8,986   9,614   10,596   37,354   42,518   56,201   57,612   75,310   78,988   92,240   99,745   210,899   321,726   869,980   912,1064   1038,1106

X(38) is the {X(1),X(63)}-harmonic conjugate of X(31). For a list of other harmonic conjugates of X(38), click Tables at the top of this page.

X(38) = isogonal conjugate of X(82)
X(38) = isotomic conjugate of X(3112)
X(38) = anticomplement of X(1215)
X(38) = crosspoint of X(1) and X(75)
X(38) = crosssum of X(1) and X(31)
X(38) = crossdifference of every pair of points on line X(661)X(830)
X(38) = X(643)-beth conjugate of X(38)
X(38) = bicentric sum of PU(35)
X(38) = PU(35)-harmonic conjugate of X(661)
X(38) = trilinear pole of line X(2084)X(2530)
X(38) = perspector of ABC and extraversion triangle of X(38) (which is also the anticevian triangle of X(38))
X(38) = barycentric square root of X(8041)


X(39) = BROCARD MIDPOINT

Trilinears    a(b2 + c2) : b(c2 + a2) : c(a2 + b2)
Trilinears    sin(A + ω) : sin(B + ω) : sin(C + ω)
Trilinears    sin A + sin(A + 2ω) : sin B + sin(B + 2ω) : sin C + sin(C + 2ω)
Trilinears    cos A - cos(A + 2ω) : cos B - cos(B + 2ω) : cos C - cos(C + 2ω)
Trilinears    sin A + cos A tan ω : :
Trilinears    cos A + sin A cot ω : :
Trilinears    a + 2R cos A tan ω : :
Barycentrics    a2(b2 + c2) : b2(c2 + a2) : c2(a2 + b2)
X(39) = (r2 + 4rR - s2)*(r2 + 2rR + s2)X(1) - 6rR(r2 + 4rR - s2)*X(2) - 2r2(r2 + 4rR + s2)*X(3)   (Peter Moses, April 2, 2013)
X(39) = P(1) + U(1)

X(39) is the midpoint of the 1st and 2nd Brocard points, given by trilinears c/b : a/c : b/a and b/c : c/a : a/b. The third and fourth trilinear representations were given by Peter J. C. Moses (10/3/03); cf. X(511), X(32), X(182).

The locus of the nine-point center in a Brocard porism (triangles sharing circumcircle and Brocard inellipse with ABC) is the circle through X(5) with center X(39). (Randy Hutson, August 29, 2018)

As illustrations of the Brocard porism mentioned just above, see
Brocard-Poncelet Porism, Stationary Brocard Points and Invariant Brocard Angle,
Brocard Porism, Steiner Ellipses, and the Homothetic Poncelet Pair, (Dan Reznik and Ronaldo Garcia, August 9, 2020),
The Poncelet Homothetic Pair Contains an Aspect-Ratio Invariant Brocard Inellipse. Here the orbit of X(39) is an ellipse. (Dan Reznik and Ronaldo Garcia, September 13, 2020)

Ross Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, Mathematical Association of America, 1995. Chapter 10: The Brocard Points.

X(39) lies on the bicevian conic of X(2) and X(99) and on these lines: 1,291   2,76   3,6   4,232   5,114   9,978   10,730   36,172   37,596   51,237   54,248   83,99   110,755   140,230   141,732   185,217   213,672   325,626   395,618   493,494   512,881   588,589   590,642   597,1084   615,641

X(39) is the {X(3),X(6)}-harmonic conjugate of X(32). For a list of other harmonic conjugates of X(39), click Tables at the top of this page.

X(39) = midpoint of X(76) and X(194)
X(39) = reflection of X(5052) in X(6)
X(39) = isogonal conjugate of X(83)
X(39) = isotomic conjugate of X(308)
X(39) = complement of X(76)
X(39) = complementary conjugate of X(626)
X(39) = circumcircle-inverse of X(2076)
X(39) = Brocard-circle-inverse of X(32)
X(39) = 1st-Lemoine-circle-inverse of X(2458)
X(39) = antitomic conjugate of anticomplement of X(39076)
X(39) = Steiner-circumellipse-inverse of anticomplement of X(3978)
X(39) = eigencenter of anticevian triangle of X(512)
X(39) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,141), (3,23208), (4,211), (99,512)
X(39) = crosspoint of X(i) and X(j) for these (i,j): (2,6), (141,427)
X(39) = crosssum of X(i) and X(j) for these (i,j): (2,6), (251,1176)
X(39) = crossdifference of every pair of points on line X(23)X(385)
X(39) = radical trace of 1st and 2nd Brocard circles
X(39) = exsimilicenter of circles O(15,16) and O(61,62); the insimilicenter is X(32)
X(39) = radical trace of circles {X(15),X(62),PU(1)} and {X(16),X(61),PU(1)}
X(39) = anticenter of cyclic quadrilateral PU(1)PU(39)
X(39) = bicentric sum of PU(i) for these i: 1, 67
X(39) = midpoint of PU(1)
X(39) = PU(67)-harmonic conjugate of X(351)
X(39) = X(5007) of 5th Brocard triangle
X(39) = X(5026) of 6th Brocard triangle
X(39) = center of Moses circle
X(39) = center of Gallatly circle
X(39) = inverse-in-2nd-Brocard-circle of X(6)
X(39) = inverse-in-Kiepert-hyperbola of X(5)
X(39) = {X(61),X(62)}-harmonic conjugate of X(575)
X(39) = {X(1687),X(1688)}-harmonic conjugate of X(3398)
X(39) = {X(2009),X(2010)}-harmonic conjugate of X(5)
X(39) = Brocard axis intercept of radical axis of nine-point circles of ABC and circumsymmedial triangle
X(39) = intersection of tangents to hyperbola {A,B,C,X(2),X(6)}} at X(2) and X(6)
X(39) = perspector of circumconic centered at X(141)
X(39) = center of circumconic that is locus of trilinear poles of lines passing through X(141)
X(39) = trilinear pole, wrt medial triangle, of orthic axis
X(39) = trilinear pole of line X(688)X(3005)
X(39) = perspector of medial triangle of ABC and medial triangle of 1st Brocard triangle
X(39) = X(2029)-of-2nd-Brocard triangle
X(39) = X(39)-of-circumsymmedial-triangle
X(39) = perspector, wrt symmedial triangle, of bicevian conic of X(6) and X(25)
X(39) = intersection of Brocard axes of ABC and 5th Euler triangle
X(39) = X(92)-isoconjugate of X(1176)
X(39) = X(1577)-isoconjugate of X(827)
X(39) = eigencenter of Steiner triangle
X(39) = perspector of ABC and unary cofactor triangle of circummedial triangle
X(39) = center of (equilateral) unary cofactor triangle of Stammler triangle
X(39) = X(7753)-of-4th-anti-Brocard-triangle
X(39) = X(11)-of-X(3)PU(1)
X(39) = X(115)-of-X(3)PU(1)
X(39) = X(125)-of-X(3)PU(1)
X(39) = homothetic center of Kosnita triangle and mid-triangle of 1st and 2nd Kenmotu diagonals triangles
X(39) = Cundy-Parry Phi transform of X(182)
X(39) = Cundy-Parry Psi transform of X(262)
X(39) = endo-similarity image of antipedal triangles of PU(1); the similitude center of these triangles is X(3)
X(39) = orthoptic-circle-of-Steiner-inellipse-inverse of X(32526)
X(39) = QA-P42 (QA-Orthopole Center) of quadrangle ABCX(2) (see http://www.chrisvantienhoven.nl/quadrangle-objects/index.php/15-mathematics/encyclopedia-of-quadri-figures/quadrangle-objects/artikelen-qa/228-qa-p42.html)


X(40) = BEVAN POINT

Trilinears     cos B + cos C - cos A - 1 : :
Trilinears    b/(c + a - b) + c/(a + b - c) - a/(b + c -a) : :
Trilinears    sin2(B/2) + sin2(C/2) - sin2(A/2) : :
Trilinears    a^3 + a^2(b + c) - a(b + c)^2 - (b + c)(b - c)^2 : :
Trilinears    b(tan B/2) + c(tan C/2) - a(tan A/2) : :
Trilinears    (r/R) - 2 cos A : :
X(40) = X(1) - 2X(3) = 2R*X(4) - (r + 4R)*X(9)
X(40) = X[1] - 3 X[165] = 2 X[3] - 3 X[165] = 3 X[376] - X[944] = 5 X[631] - 4 X[1125] = 3 X[1] - 4 X[1385] = 9 X[165] - 4 X[1385] = 3 X[3] - 2 X[1385] = 3 X[1] - 2 X[1482] = 9 X[165] - 2 X[1482] = 3 X[3] - X[1482] = 4 X[5] - 5 X[1698] = 4 X[5] - 3 X[1699] = 5 X[1698] - 3 X[1699] = 2 X[3095] - 3 X[3097] = X[145] - 5 X[3522] = 2 X[551] - 3 X[3524] = 2 X[3244] - 7 X[3528] = 8 X[1385] - 9 X[3576] = 4 X[1482] - 9 X[3576] = 2 X[1] - 3 X[3576] = 4 X[3] - 3 X[3576] = 3 X[3576] - 8 X[3579]

If you have GeoGebra, you can view X(40).

This point is mentioned in a problem proposal by Benjamin Bevan, published in Leybourn's Mathematical Repository, 1804, p. 18.

Constructions received from Randy Hutson, January 29, 2015:
(1) Let A'B'C' be the extangents triangle. Let A″ be the reflection of A' in BC, and define B″, C″ cyclically. A'A″, B'B″, C'C″ concur in X(40).
(2) Let A'B'C' be the extangents triangle. Let A″ be the cevapoint of B' and C', and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X(40).
(3) Let A'B'C' be the hexyl triangle and A″B″C″ be the side-triangle of ABC and hexyl triangle. Let A* be the {B,C}-harmonic conjugate of A″, and define B*, C* cyclically. The lines A'A*, B'B*, C'C* concur in X(40).
(4) Let Ia be the reflection of X(1) in the perpendicular bisector of BC, and define Ib, Ic cyclically. X(40) = X(104)-of-IaIbIc.
(5) Let A'B'C' be the cevian triangle of X(189). Let A″ be the orthocenter of AB'C', and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X(40). (6) Let A'B'C' be the mixtilinear incentral triangle. Let A″ be the trilinear product B'*C', and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X(40).

Let A'B'C' be the intouch triangle of the extangents triangle, if ABC is acute. Then A'B'C' is homothetic to the Aquila triangle at X(40). (Randy Hutson, December 2, 2017)

Let A'B'C' be the excentral triangle. Let A″ be the isogonal conjugate, wrt A'BC, of A. Define B″ and C″ cyclically. (A″ is also the reflection of A' in BC, and cyclically for B″ and C″). The lines A'A″, B'B″, C'C″ concur in X(40). (Randy Hutson, December 2, 2017)

Let OA be the circle centered at the A-vertex of the anti-Mandart-incircle triangle and passing through A; define OB and OC cyclically. X(40) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

Let OA be the circle centered at the A-vertex of the Gemini triangle 4 and passing through A; define OB and OC cyclically. X(40) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

Let I = X(1) and P = X(40), and let
DEF = pedal triangle of P,
Ab = AI∩PE, and define Bc and Ca cyclically,
Ac = AI∩PF, and define Ba and Cb cyclically.
The centers of the circles {Ab, Bc, Ca}} and {Ac, Ba, Cb}} are the bicentric pair PU(44). (Angel Montesdeoca, April 12, 2021)

X(40) lies on the following curves: Q010, Q122, K004, K033, K100, K133, K179, K199, K269, K308, K333, K338, K343, K384, K414, K619, K667, K679, K692, K710, K736, K750, K806, K807, K815, K826, the Jerabek hyhperbola of the excentral triangle, the Mandart hyperbola, and these lines: {1,3}, {2,946}, {4,9}, {5,1698}, {6,380}, {7,7160}, {8,20}, {11,6922}, {12,1836}, {15,10636}, {16,10637}, {21,3577}, {22,9626}, {24,9625}, {25,1902}, {26,9590}, {28,2328}, {30,191}, {31,580}, {32,9620}, {33,201}, {34,212}, {39,1571}, {41,2301}, {42,581}, {43,970}, {47,1774}, {49,9586}, {58,601}, {64,72}, {74,6011}, {75,10444}, {77,947}, {78,100}, {79,4338}, {80,90}, {92,412}, {98,6010}, {101,972}, {103,1292}, {104,1293}, {108,207}, {109,255}, {140,3624}, {144,5815}, {145,3218}, {149,10265}, {151,2816}, {153,3648}, {154,7973}, {164,188}, {168,9837}, {170,1282}, {173,8351}, {182,1700}, {184,9622}, {185,3611}, {187,10988}, {190,341}, {196,208}, {197,3556}, {198,2324}, {209,3293}, {210,1750}, {214,10698}, {219,610}, {220,910}, {221,223}, {226,3085}, {228,3191}, {238,1722}, {256,989}, {258,8092}, {269,8829}, {307,4329}, {322,8822}, {329,6260}, {347,7013}, {371,5415}, {372,5416}, {376,519}, {381,7989}, {382,5790}, {386,1064}, {387,579}, {388,3474}, {389,11435}, {390,938}, {392,474}, {402,12696}, {404,3877}, {405,1730}, {442,5715}, {483,3645}, {485,13893}, {486,13947}, {495,4312}, {497,1210}, {498,5219}, {499,6891}, {500,8274}, {511,1045}, {518,1071}, {542,9881}, {548,3633}, {549,3656}, {550,952}, {551,3524}, {572,1449}, {574,9619}, {576,8539}, {578,11428}, {595,602}, {612,2292}, {614,3915}, {631,1125}, {653,1895}, {659,2821}, {664,7183}, {672,2082}, {726,12251}, {728,1018}, {730,11257}, {738,1323}, {758,3158}, {774,4319}, {813,2724}, {846,9840}, {901,2716}, {902,3924}, {908,5552}, {912,5534}, {920,4302}, {936,960}, {943,5665}, {950,1708}, {953,2743}, {954,12560}, {956,3916}, {958,1012}, {971,2951}, {978,1050}, {979,9359}, {984,1721}, {990,7174}, {993,6906}, {997,3878}, {1000,4315}, {1001,3812}, {1006,3754}, {1043,7415}, {1054,11512}, {1056,4298}, {1058,11019}, {1066,4306}, {1104,3052}, {1118,1785}, {1130,6585}, {1145,2829}, {1147,9621}, {1151,7969}, {1152,7968}, {1154,6255}, {1160,5588}, {1161,5589}, {1181,2323}, {1191,3752}, {1253,1254}, {1256,9376}, {1320,11715}, {1329,1532}, {1330,2792}, {1334,7390}, {1386,5085}, {1419,3157}, {1421,1772}, {1423,13161}, {1448,7273}, {1473,8192}, {1475,11200}, {1478,1770}, {1479,1737}, {1480,5315}, {1483,3655}, {1496,4320}, {1503,1761}, {1519,6834}, {1537,3035}, {1587,13883}, {1588,13936}, {1593,1829}, {1621,6986}, {1630,2289}, {1633,2823}, {1656,7988}, {1657,4668}, {1707,5247}, {1712,3176}, {1723,2955}, {1724,3073}, {1725,1775}, {1726,2949}, {1727,4324}, {1728,1837}, {1736,4907}, {1739,12659}, {1743,2264}, {1745,2818}, {1746,4714}, {1748,5174}, {1765,3696}, {1769,9525}, {1777,1935}, {1783,7156}, {1790,3193}, {1817,1819}, {1834,2245}, {1859,1872}, {1870,4347}, {1888,3074}, {2066,2362}, {2123,3421}, {2130,3354}, {2131,3472}, {2177,2650}, {2218,4674}, {2222,2745}, {2254,2814}, {2266,4251}, {2269,2285}, {2294,3247}, {2331,3194}, {2717,2742}, {2771,5531}, {2777,10119}, {2778,2915}, {2782,9860}, {2784,4050}, {2794,4769}, {2796,12243}, {2801,5528}, {2807,5562}, {2835,3939}, {2886,5705}, {2900,10605}, {2939,2947}, {2940,2948}, {2945,2953}, {2946,2952}, {2956,9370}, {2957,14026}, {2975,3872}, {3008,7397}, {3062,4866}, {3065,12747}, {3068,13912}, {3069,13975}, {3070,13911}, {3071,13973}, {3086,3911}, {3090,3634}, {3091,3305}, {3095,3097}, {3098,9941}, {3099,9821}, {3100,9610}, {3146,3219}, {3160,7177}, {3169,5847}, {3182,3346}, {3185,7420}, {3207,6603}, {3208,3509}, {3220,9798}, {3241,10304}, {3243,3874}, {3244,3528}, {3306,3523}, {3309,4063}, {3348,3353}, {3355,3473}, {3358,5787}, {3398,10789}, {3434,6734}, {3436,6256}, {3452,6848}, {3467,5560}, {3476,4311}, {3485,5218}, {3486,4304}, {3487,3671}, {3488,4314}, {3515,11363}, {3516,11396}, {3526,11230}, {3529,3626}, {3534,4677}, {3545,3828}, {3555,10167}, {3560,5251}, {3575,5090}, {3583,6928}, {3585,6923}, {3622,5734}, {3636,10299}, {3640,11825}, {3641,11824}, {3647,11530}, {3653,12100}, {3661,6999}, {3663,10521}, {3681,3951}, {3683,3698}, {3689,3962}, {3690,11381}, {3692,5279}, {3697,5927}, {3710,10327}, {3711,4005}, {3714,5695}, {3715,3983}, {3719,7270}, {3729,4385}, {3781,5907}, {3784,13348}, {3814,6941}, {3822,6937}, {3825,6963}, {3827,12329}, {3839,10248}, {3841,6829}, {3844,10516}, {3868,3870}, {3873,12005}, {3880,12513}, {3884,6940}, {3886,10449}, {3890,5253}, {3914,5230}, {3918,6920}, {3947,5714}, {3955,13346}, {3980,12545}, {3984,4420}, {4026,5799}, {4047,5776}, {4084,12559}, {4187,7681}, {4293,10106}, {4299,6948}, {4326,5728}, {4333,5841}, {4384,6996}, {4413,6918}, {4414,10459}, {4421,12635}, {4450,5016}, {4511,4855}, {4654,10056}, {4662,5220}, {4663,11477}, {4669,11001}, {4678,5059}, {4816,12103}, {4847,5082}, {4880,13369}, {5013,9592}, {5044,8580}, {5056,9779}, {5057,6932}, {5067,10171}, {5084,7682}, {5088,9312}, {5171,11364}, {5180,6960}, {5234,9708}, {5252,7354}, {5259,6883}, {5260,6912}, {5261,8545}, {5267,6950}, {5274,5704}, {5281,5703}, {5295,5774}, {5312,5396}, {5314,7503}, {5316,6964}, {5330,13587}, {5426,5428}, {5432,11375}, {5433,11376}, {5435,9785}, {5439,10582}, {5440,5730}, {5442,6713}, {5445,6882}, {5550,10303}, {5554,6872}, {5559,7284}, {5561,7161}, {5688,5870}, {5689,5871}, {5692,5720}, {5722,10384}, {5726,9654}, {5729,9844}, {5735,5880}, {5744,6705}, {5745,6847}, {5763,11374}, {5768,5853}, {5791,8727}, {5804,8257}, {5805,8728}, {5806,11108}, {5905,10528}, {5909,10374}, {5918,12680}, {6043,11991}, {6048,9566}, {6068,6259}, {6198,9611}, {6200,9615}, {6221,9618}, {6237,9928}, {6241,11460}, {6246,10724}, {6265,13253}, {6407,9584}, {6700,6927}, {6759,10536}, {6764,12516}, {6842,7951}, {6889,10198}, {6890,10527}, {6897,10532}, {6899,10916}, {6943,11680}, {6947,10531}, {6949,11813}, {6967,10200}, {6990,12558}, {6998,9746}, {7082,12953}, {7387,8185}, {7413,12544}, {7589,7590}, {7596,8231}, {7672,7675}, {7673,7677}, {7970,11711}, {7978,11720}, {7983,11710}, {7984,11709}, {7993,12773}, {8075,8081}, {8076,8082}, {8078,8091}, {8089,8099}, {8090,8100}, {8107,8111}, {8108,8112}, {8140,12488}, {8144,9576}, {8188,10669}, {8189,10673}, {8197,9834}, {8204,9835}, {8214,9838}, {8215,9839}, {8224,8234}, {8244,12490}, {8245,9959}, {8423,12491}, {8616,13732}, {8981,13888}, {8983,9540}, {9521,10015}, {9751,12264}, {9786,10382}, {9857,9873}, {9896,12417}, {9906,12662}, {9907,12663}, {10087,11570}, {10090,12758}, {10197,11263}, {10373,11347}, {10386,12433}, {10404,11246}, {10436,10446}, {10437,10447}, {10578,11036}, {10606,12262}, {10695,11714}, {10696,11700}, {10697,11712}, {10703,11713}, {10705,12265}, {10738,12619}, {10791,12110}, {10899,10900}, {10912,11194}, {10915,12115}, {11251,11852}, {11445,12111}, {11722,13099}, {11754,11756}, {11763,11765}, {11772,11774}, {11781,11783}, {11828,12440}, {11829,12441}, {11900,12113}, {11919,12648}, {11920,12649}, {12059,12665}, {12247,13199}, {12387,12398}, {12407,12661}, {12408,13221}, {12517,12843}, {12518,12844}, {12519,12845}, {12530,12683}, {12556,12660}, {12653,12737}, {12670,12671}, {12756,12757}, {12786,13465}, {13935,13971}, {13942,13966}

X(40) = midpoint of X(i) and X(j) for these {i,j}: {1, 7991}, {3, 12702}, {4, 6361}, {8, 20}, {10, 5493}, {65, 7957}, {944, 12245}, {1768, 5541}, {2093, 7994}, {2100, 2101}, {2136, 6762}, {2448, 2449}, {2948, 9904}, {2951, 5223}, {3245, 5537}, {5531, 12767}, {6764, 12632}, {9860, 13174}, {9961, 12528}, {11826, 11827}, {12247, 13199}, {12408, 13221}, {12488, 12489}, {12526, 12565}, {12697, 12698}
X(40) = reflection of X(i) in X(j) for these {i,j}: {1, 3}, {3, 3579}, {4, 10}, {8, 11362}, {57, 3359}, {84, 1158}, {145, 5882}, {149, 10265}, {355, 5690}, {944, 4297}, {946, 6684}, {962, 946}, {1012, 4640}, {1071, 9943}, {1320, 11715}, {1482, 1385}, {1490, 11500}, {1537, 3035}, {1768, 12515}, {1836, 6907}, {2077, 13528}, {2948, 12778}, {3062, 5779}, {3555, 12675}, {3576, 165}, {3655, 8703}, {3656, 549}, {3679, 3654}, {3811, 8715}, {3868, 5884}, {4297, 12512}, {4301, 1125}, {5531, 12331}, {5535, 484}, {5587, 5657}, {5603, 10164}, {5691, 355}, {5693, 72}, {5732, 11495}, {5735, 5880}, {5881, 8}, {6210, 573}, {6261, 6796}, {6264, 104}, {6282, 6244}, {6326, 100}, {6361, 5493}, {6765, 3913}, {6769, 10306}, {7688, 7964}, {7701, 191}, {7970, 11711}, {7971, 6261}, {7978, 11720}, {7982, 1}, {7983, 11710}, {7984, 11709}, {7991, 12702}, {7993, 12773}, {8148, 10222}, {9579, 6850}, {9580, 6827}, {9589, 12699}, {9799, 9948}, {9812, 10175}, {9845, 9841}, {9856, 5044}, {10222, 13624}, {10695, 11714}, {10696, 11700}, {10697, 11712}, {10698, 214}, {10703, 11713}, {10705, 12265}, {10724, 6246}, {10738, 12619}, {10864, 84}, {10912, 11260}, {11014, 11012}, {11224, 10246}, {11372, 9}, {11477, 4663}, {11523, 3811}, {11531, 1482}, {12398, 12387}, {12407, 13211}, {12520, 12511}, {12629, 12513}, {12650, 12114}, {12651, 11496}, {12653, 12737}, {12665,14740}, {12672, 960}, {12688, 5777}, {12696, 402}, {12699, 5}, {12701, 6922}, {12703, 5119}, {12704, 46}, {12705, 12514}, {12717, 1766}, {12751, 1145}, {12842, 12516}, {12843, 12517}, {12844, 12518}, {12845, 12519}, {13099, 11722}, {13253, 6265}
X(40) = isogonal conjugate of X(84)
X(40) = isotomic conjugate of X(309)
X(40) = inverse-in-circumcircle of X(2077)
X(40) = complement of X(962)
X(40) = anticomplement of X(946)
X(40) = X(963)-complementary conjugate of X(10)
X(40) = X(947)-anticomplementary conjugate of X(8)
X(40) = X(i)-Ceva conjugate of X(j) for these (i,j): (4, 2910), (8, 1), (20, 1490), (63, 9), (329, 2324), (347, 223), (515, 6326), (1817, 198), (7080, 1103), (7128, 101), (8822, 329), (9369, 1050), (9778, 2951)
X(40) = X(i)-cross conjugate of X(j) for these (i,j): (64, 3354), (198, 223), (208, 3342), (221, 1), (227, 7952), (2187, 2331), (7074, 2324)
X(40) = crosspoint of X(i) and X(j) for these (i,j): (329,347)
X(40) = crosssum of X(i) and X(j) for these (i,j): {19, 7008}, {56, 1413}, {513, 7004}, {649, 2310}, {1436, 2192}, {1903, 2357}
X(40) = crossdifference of every pair of points on line X(650)X(1459)
X(40) = cevapoint of X(i) and X(j) for these (i,j): {1, 2956}, {19, 8802}, {55, 3197}, {65, 8803}, {71, 3198}, {198, 7074}
X(40) = crosspoint of X(i) and X(j) for these (i,j): {8, 7080}, {63, 7013}, {100, 7012}, {190, 7045}, {329, 347}, {1817, 8822}
X(40) = trilinear pole of line {6129, 10397}
X(40) = point of concurrence of the perpendiculars from the excenters to the respective sides
X(40) = circumcenter of the excentral triangle
X(40) = incenter of the extangents triangle if triangle ABC is acute
X(40) = perspector of the excentral and extangents triangles
X(40) = perspector of the excentral and extouch triangles
X(40) = X(4)-of-hexyl-triangle
X(40) = X(4)-of-1st-circumperp-triangle
X(40) = X(20)-of-2nd-circumperp-triangle
X(40) = Miquel associate of X(8)
X(40) = perspector of hexyl triangle and cevian triangle of X(63)
X(40) = perspector of hexyl triangle and anticevian triangle of X(9)
X(40) = perspector of hexyl triangle and antipedal triangle of X(84)
X(40) = perspector of ABC and the reflection in X(57) of the antipedal triangle of X(57)
X(40) = excentral isogonal conjugate of X(1)
X(40) = excentral isotomic conjugate of X(1742)
X(40) = hexyl isogonal conjugate of X(1)
X(40) = perspector of ABC and extraversion triangle of X(84)
X(40) = trilinear product of extraversions of X(84)
X(40) = homothetic center of extangents triangle and reflection of intangents triangle in X(3)
X(40) = trilinear product of centers of mixtilinear incircles
X(40) = intangents-to-extangents similarity image of X(1)
X(40) = X(26)-of-reflection-triangle of X(1)
X(40) = {X(56),X(3057)}-harmonic conjugate of X(1)
X(40) = perspector of extangents triangle and cross-triangle of ABC and extangents triangle
X(40) = perspector of ABC and cross-triangle of ABC and hexyl triangle
X(40) = circumcircle-inverse of X(2077)
X(40) = inverse-in-incircle-of-anticomplementary-triangle of X(10538)
X(40) = X(1)-Hirst inverse of X(9371)
X(40) = outer-Garcia-to-ABC similarity image of X(4)
X(40) = Cundy-Parry Phi transform of X(57)
X(40) = Cundy-Parry Psi transform of X(9)
X(40) = anticevian isogonal conjugate of X(1)
X(40) = X(i)-vertex conjugate of X(j) for these (i,j): {3, 3345}, {513, 2077}, {2077, 513}, {3345, 3}
X(40) = X(5)-of-6th-mixtilinear-triangle
X(40) = orthic-to-ABC functional image of X(3), if ABC is acute
X(40) = intouch-to-ABC barycentric image of X(4)
X(40) = intouch-to-excentral similarity image of X(1)
X(40) = ABC-to-excentral barycentric image of X(1)
X(40) = antipode of X(12665) in the Mandart hyperbola
X(40) = extouch-isogonal conjugate of X(14872)
X(40) = X(10306)-of-Mandart-incircle-triangle
X(40) = X(65)-of-anti-Mandart-incircle-triangle
X(40) = endo-homothetic center of Ehrmann side-triangle and anti-excenters-incenter reflections triangle; the homothetic center is X(382).
X(40) = X(i)-isoconjugate of X(j) for these (i,j): {1, 84}, {2, 1436}, {4, 1433}, {6, 189}, {7, 2192}, {8, 1413}, {9, 1422}, {31, 309}, {34, 271}, {40, 1256}, {55, 1440}, {56, 280}, {57, 282}, {63, 7129}, {65, 285}, {69, 7151}, {75, 2208}, {77, 7008}, {81, 1903}, {85, 7118}, {86, 2357}, {222, 7003}, {268, 278}, {273, 2188}, {279, 7367}, {284, 8808}, {346, 6612}, {348, 7154}, {513, 13138}, {522, 8059}, {603, 7020}, {1174, 13156}, {1812, 2358}, {3341, 3345}, {3346, 8886}, {7054, 13853}, {9375, 9376}
X(40) = X(i)-aleph conjugate of X(j) for these (i,j): {1, 978}, {2, 57}, {8, 40}, {9, 1742}, {100, 4551}, {188, 1}, {259, 1740}, {366, 1743}, {522, 2957}, {556, 63}, {4146, 1445}, {4182, 165}, {6728, 1052}, {6731, 2951}, {7025, 361}, {7043, 7150}
X(40) = X(i)-beth conjugate of X(j) for these (i,j): {8, 4}, {21, 1420}, {40, 221}, {100, 40}, {643, 78}, {644, 728}, {13138, 3341}
X(40) = X(i)-gimel conjugate of X(j) for these (i,j): {8, 84}, {78, 40}, {521, 10085}, {522, 40}, {3717, 40}, {4041, 40}, {4086, 40}, {4147, 40}, {4163, 40}, {4391, 40}, {4397, 40}, {4723, 40}, {4768, 40}, {4811, 40}, {4985, 40}, {6615, 40}, {6735, 40}, {7628, 40}, {7629, 40}, {7646, 40}, {7647, 40}
X(40) = X(i)-he conjugate of X(j) for these (i,j): {2, 516}, {190, 40}, {312, 6211}, {645, 40}, {646, 40}, {3699, 40}, {4518, 1766}, {4554, 40}, {4582, 40}, {4621, 40}, {4633, 40}, {4876, 165}, {4997, 40}, {6335, 40}, {6559, 10860}, {8707, 40}, {9365, 1}, {11609, 3}, {13136, 40}
X(40) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 84}, {8, 40}, {10, 1158}, {11, 1768}, {21, 7701}, {55, 1709}, {65, 7992}, {72, 1490}, {145, 10864}, {200, 10860}, {210, 165}, {390, 11372}, {497, 57}, {517, 6001}, {518, 971}, {519, 515}, {521, 513}, {522, 3667}, {950, 4}, {952, 2829}, {958, 7330}, {960, 3}, {1001, 3358}, {1145, 2950}, {1697, 12705}, {1837, 46}, {1864, 1750}, {2098, 10085}, {2321, 1766}, {2802, 2800}, {2804, 900}, {3036, 12515}, {3057, 1}, {3058, 1699}, {3059, 2951}, {3239, 649}, {3271, 9355}, {3678, 6796}, {3680, 6762}, {3686, 573}, {3688, 1742}, {3706, 1764}, {3717, 6211}, {3738, 2827}, {3877, 3576}, {3878, 6261}, {3880, 517}, {3883, 6210}, {3884, 5450}, {3885, 7982}, {3886, 12717}, {3893, 7991}, {3900, 3309}, {3907, 6002}, {4046, 2941}, {4111, 2938}, {4520, 3294}, {4534, 5540}, {4662, 3579}, {4673, 10476}, {4847, 63}, {5119, 12686}, {5245, 1277}, {5246, 1276}, {5289, 7171}, {5697, 7971}, {5795, 10}, {5837, 12514}, {5853, 516}, {5854, 952}, {5856, 5851}, {6366, 2826}, {6737, 20}, {6738, 9948}, {8058, 522}, {8275, 7966}, {9119, 5776}, {9785, 3333}, {9898, 7160}, {9957, 12114}, {10106, 12246}, {10866, 3361}, {10950, 5691}, {12448, 8158}, {12527, 6223}, {12541, 6766}, {12572, 6260}, {12575, 946}
X(40) = barycentric product X(i)*X(j) for these {i,j}: {1, 329}, {6, 322}, {7, 2324}, {8, 223}, {9, 347}, {10, 1817}, {37, 8822}, {57, 7080}, {63, 7952}, {69, 2331}, {75, 198}, {76, 2187}, {78, 196}, {85, 7074}, {92, 7078}, {189, 1103}, {190, 6129}, {208, 345}, {219, 342}, {221, 312}, {227, 333}, {281, 7013}, {304, 3195}, {306, 3194}, {318, 7011}, {321, 2360}, {341, 6611}, {651, 8058}, {1088, 7368}, {2199, 3596}, {3209, 3718}, {5514, 7045}, {7017, 7114}, {7128, 7358}
X(40) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 189}, {2, 309}, {6, 84}, {9, 280}, {25, 7129}, {31, 1436}, {32, 2208}, {33, 7003}, {41, 2192}, {42, 1903}, {48, 1433}, {55, 282}, {56, 1422}, {57, 1440}, {65, 8808}, {101, 13138}, {196, 273}, {198, 1}, {208, 278}, {212, 268}, {213, 2357}, {219, 271}, {221, 57}, {223, 7}, {227, 226}, {281, 7020}, {284, 285}, {322, 76}, {329, 75}, {342, 331}, {347, 85}, {354, 13156}, {604, 1413}, {607, 7008}, {1103, 329}, {1106, 6612}, {1253, 7367}, {1254, 13853}, {1415, 8059}, {1436, 1256}, {1817, 86}, {1819, 1812}, {1973, 7151}, {2175, 7118}, {2187, 6}, {2199, 56}, {2212, 7154}, {2324, 8}, {2331, 4}, {2360, 81}, {3194, 27}, {3195, 19}, {3197, 3341}, {3209, 34}, {6129, 514}, {6611, 269}, {7011, 77}, {7013, 348}, {7074, 9}, {7078, 63}, {7080, 312}, {7114, 222}, {7368, 200}, {7952, 92}, {8058, 4391}, {8822, 274}, {10397, 521}
X(40) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3, 3576), (1, 35, 3601), (1, 36, 1420), (1, 46, 57), (1, 57, 3333), (1, 65, 11529), (1, 165, 3), (1, 484, 46), (1, 1764, 10476), (1, 2093, 65), (1, 3336, 3338), (1, 3339, 942), (1, 3361, 999), (1, 3612, 13384), (1, 3670, 3677), (1, 3746, 10389), (1, 5010, 3612), (1, 5119, 1697), (1, 5264, 5269), (1, 5697, 7962), (1, 5709, 12704), (1, 5902, 11518), (1, 5903, 3340), (1, 7987, 1385), (1, 7994, 6769), (1, 9819, 9957), (1, 10268, 10902), (1, 10980, 5045), (1, 11010, 5119), (1, 11224, 10222), (1, 11531, 1482), (2, 946, 8227), (2, 962, 946), (3, 55, 10902), (3, 942, 8726), (3, 1385, 7987), (3, 1482, 1385), (3, 2095, 9940), (3, 3428, 11012), (3, 3579, 165), (3, 5584, 7688), (3, 5708, 11227), (3, 5709, 57), (3, 6244, 10310), (3, 7991, 7982), (3, 8148, 10246), (3, 8158, 999), (3, 8251, 10319), (3, 9940, 10857), (3, 10246, 13624), (3, 10306, 55), (3, 10310, 2077), (3, 10679, 10267), (3, 10680, 10269), (3, 11248, 35), (3, 11249, 36), (4, 10, 5587), (4, 5657, 10), (4, 6197, 19), (4, 12705, 11372), (5, 12699, 1699), (8, 9778, 20), (8, 10860, 10864), (9, 1706, 10), (10, 573, 9548), (10, 12514, 9), (10, 12572, 2551), (11, 12701, 9614), (12, 1836, 9612), (19, 71, 9), (19, 11471, 4), (20, 63, 84), (20, 9537, 3101), (20, 11362, 5881), (35, 3245, 5903), (35, 5537, 11248), (35, 5903, 1), (36, 5697, 1), (39, 1571, 9574), (39, 1572, 9575), (42, 4300, 581), (43, 1695, 970), (46, 484, 5128), (46, 1697, 3333), (46, 3338, 3336), (46, 5119, 1), (46, 5709, 5535), (46, 11010, 1697), (55, 65, 1), (55, 2093, 11529), (55, 5183, 2093), (55, 5584, 3), (55, 7957, 6769), (55, 7964, 165), (56, 3057, 1), (57, 1697, 1), (57, 5128, 46), (57, 7991, 6766), (63, 9778, 10860), (64, 3198, 1490), (65, 3579, 7688), (65, 6769, 7982), (65, 7964, 5584), (65, 10268, 3576), (72, 5687, 200), (72, 7580, 1490), (78, 6261, 6326), (100, 411, 6796), (100, 3869, 78), (140, 5886, 3624), (145, 3522, 5731), (145, 5731, 5882), (164, 505, 188), (165, 484, 3359), (165, 5538, 5010), (165, 6282, 2077), (165, 6769, 10902), (165, 7991, 1), (165, 7994, 55), (165, 11531, 7987), (165, 12702, 7982), (169, 3730, 9), (191, 2960, 1710), (191, 5691, 7330), (200, 12526, 72), (200, 12565, 1490), (210, 12688, 5777), (221, 227, 223), (221, 7074, 7078), (227, 7074, 1103), (354, 3303, 1), (355, 3654, 5690), (355, 5690, 3679), (376, 944, 4297), (376, 12245, 944), (381, 9956, 7989), (388, 3474, 4292), (392, 474, 8583), (411, 3869, 6261), (484, 5119, 57), (484, 7991, 5709), (484, 11010, 1), (497, 1788, 1210), (498, 12047, 5219), (573, 1766, 9), (595, 13329, 602), (631, 4301, 9624), (631, 5603, 1125), (942, 3295, 1), (946, 6684, 2), (950, 1708, 10396), (956, 10914, 4853), (958, 5836, 9623), (960, 1376, 936), (962, 6684, 8227), (986, 5255, 1), (999, 9957, 1), (1125, 4301, 5603), (1125, 5603, 9624), (1125, 10164, 631), (1151, 7969, 9583), (1155, 3057, 56), (1210, 10624, 497), (1276, 1277, 9), (1276, 6192, 6191), (1277, 6191, 6192), (1319, 2098, 1), (1381, 1382, 2077), (1385, 1482, 1), (1385, 7987, 3576), (1388, 5048, 1), (1402, 10480, 1), (1420, 7962, 1), (1467, 10388, 1), (1478, 1770, 9579), (1478, 10039, 9578), (1479, 1737, 9581), (1482, 11531, 7982), (1490, 12526, 5693), (1571, 1572, 39), (1656, 9955, 7988), (1697, 5128, 57), (1698, 1699, 5), (1698, 9589, 1699), (1699, 9589, 12699), (1700, 1701, 182), (1702, 1703, 6), (1704, 1705, 182), (1706, 12705, 5587), (1709, 7330, 7701), (1750, 7995, 12688), (1754, 5264, 3072), (1770, 10039, 1478), (1837, 6284, 3586), (2017, 2018, 39), (2077, 11012, 3), (2093, 10306, 7982), (2098, 5204, 1319), (2099, 2646, 1), (2099, 5217, 2646), (2136, 3928, 6762), (2136, 9841, 944), (2292, 4220, 8235), (2551, 5698, 12572), (2572, 2573, 3), (2975, 6909, 5450), (3057, 10270, 3576), (3085, 4295, 226), (3091, 9780, 10175), (3158, 11523, 3811), (3303, 5221, 354), (3304, 5919, 1), (3336, 3338, 57), (3340, 3601, 1), (3359, 3587, 165), (3359, 5119, 3576), (3359, 5709, 46), (3359, 7991, 12704), (3361, 9819, 1), (3428, 6244, 2077), (3428, 6282, 3576), (3428, 10310, 3), (3428, 13528, 165), (3436, 6925, 6256), (3485, 5218, 13411), (3496, 3501, 9), (3523, 3616, 10165), (3555, 10167, 12675), (3576, 5535, 57), (3576, 7982, 1), (3576, 12704, 3333), (3579, 7957, 10902), (3579, 7991, 3576), (3579, 10306, 10268), (3579, 12702, 1), (3587, 5709, 3), (3587, 12702, 1697), (3624, 11522, 5886), (3634, 3817, 3090), (3666, 5710, 1), (3671, 13405, 3487), (3679, 5691, 355), (3681, 9961, 12528), (3681, 11684, 3951), (3730, 5011, 169), (3746, 5902, 1), (3811, 8715, 3158), (3868, 3871, 3870), (3868, 7411, 10884), (3869, 6796, 6326), (3872, 4652, 2975), (3890, 9352, 5253), (3895, 5731, 7966), (3911, 12053, 3086), (3916, 10914, 956), (3931, 5711, 1), (4297, 6762, 9845), (4297, 12512, 376), (4301, 10164, 1125), (4302, 10573, 10572), (4314, 6738, 3488), (4424, 5264, 1), (4512, 12651, 11496), (4640, 5836, 958), (4853, 10914, 11525), (4855, 11682, 4511), (5044, 9709, 8580), (5045, 5708, 10980), (5045, 6767, 1), (5119, 5128, 3333), (5119, 5709, 7982), (5221, 8273, 9940), (5252, 7354, 9613), (5535, 7982, 12704), (5536, 7987, 3338), (5541, 6763, 3632), (5552, 11415, 908), (5584, 6769, 3576), (5584, 7957, 1), (5584, 7991, 11529), (5584, 10306, 10902), (5657, 6361, 4), (5687, 7580, 11500), (5708, 6767, 5045), (5709, 11010, 12703), (5714, 8164, 3947), (5758, 6908, 226), (5812, 6907, 9612), (5887, 11499, 5720), (6191, 6192, 9), (6210, 6211, 9), (6210, 12717, 11372), (6212, 6213, 9), (6252, 6404, 3779), (6736, 12527, 3421), (6769, 10268, 55), (6838, 11415, 12608), (6922, 12700, 9614), (6923, 10526, 3585), (6928, 10525, 3583), (7589, 12445, 7590), (7672, 7676, 7675), (7688, 10902, 3), (7742, 11508, 2078), (7957, 7964, 3), (7987, 7991, 11531), (7987, 11531, 1), (7991, 7994, 7957), (8075, 8093, 8081), (8076, 8094, 8082), (8107, 9805, 8111), (8108, 9806, 8112), (8148, 10222, 11224), (8148, 10246, 10222), (8148, 11224, 7982), (8148, 13624, 1), (8158, 9819, 7982), (8224, 9808, 8234), (9572, 9573, 8141), (9574, 9575, 39), (9576, 9577, 8144), (9578, 9579, 1478), (9580, 9581, 1479), (9582, 9583, 1151), (9584, 9585, 6407), (9586, 9587, 49), (9588, 9589, 5), (9590, 9591, 26), (9780, 9812, 3091), (9955, 11231, 1656), (10222, 10246, 1), (10222, 13624, 10246), (10267, 10679, 3746), (10269, 10680, 5563), (10306, 12702, 7957), (10389, 11518, 1), (10434, 12435, 1), (10470, 11521, 1), (10572, 10573, 5727), (10912, 11194, 11260), (11019, 12575, 1058), (11822, 11823, 55), (12000, 13373, 1), (12703, 12704, 7982)


X(41) = X(6)-CEVA CONJUGATE OF X(31)

Trilinears    a2(b + c - a) : b2(c + a - b) : c2(a + b - c)
Trilinears    a2cot(A/2) : b2cot(B/2) : c2cot(C/2)
Trilinears    a2(a - s) : b2(b - s) : c2(c - s)
Trilinears    a tan A' : : , where A'B'C' is the excentral triangle
Barycentrics    a3(b + c - a) : b3(c + a - b) : c3(a + b - c)
X(41) = (r + 2R)(r2 + 4rR + s2)*X(1) - 6rR(r + 4R)*X(2) -2r(2 + 4rR - s2)*X(3)   (Peter Moses, April 2, 2013)

For an artistic design generated by X(41), see X(244).

If you have The Geometer's Sketchpad, you can view X(41).

X(41) lies on these lines: 1,101   3,218   6,48   9,21   25,42   31,32   37,584   55,220   58,609   65,910   219,1036   226,379   560,872   601,906   603,911   663,884

X(41) is the {X(32),X(213)}-harmonic conjugate of X(31). For a list of other harmonic conjugates of X(41), click Tables at the top of this page.

X(41) = isogonal conjugate of X(85)
X(41) = isotomic conjugate of X(20567)
X(41) = complement of X(21285)
X(41) = anticomplement of X(17046)
X(41) = X(i)-Ceva conjugate of X(j) for these (i,j): (6,31), (9,212), (284,55)
X(41) = crosspoint of X(i) and X(j) for these (i,j): (6,55), (9,33)
X(41) = crosssum of X(i) and X(j) for these (i,j): (1,169), (2,7), (57,77), (92,342), (226,1441), (514,1111)
X(41) = crossdifference of every pair of points on line X(522)X(693)
X(41) = X(i)-beth conjugate of X(j) for these (i,j): (41,32), (101,41), (220,220)
X(41) = X(75)-isoconjugate of X(57)
X(41) = X(92)-isoconjugate of X(77)
X(41) = trilinear product of vertices of 4th mixtilinear triangle
X(41) = trilinear product of vertices of 5th mixtilinear triangle
X(41) = trilinear product of PU(93)
X(41) = barycentric product of PU(104)
X(41) = PU(93)-harmonic conjugate of X(663)
X(41) = bicentric sum of PU(93)
X(41) = perspector of unary cofactor triangles of Gemini triangles 1 and 13


X(42) = CROSSPOINT OF INCENTER AND SYMMEDIAN POINT

Trilinears    a(b + c) : b(c + a) : c(a + b)
Trilinears    (1 + cos A)(cos B + cos C) : (1 + cos B)(cos C + cos A) : (1 + cos C)(cos A + cos B)
Trilinears    a(ar - S) : b(br - S): c(cr - S)
Trilinears    csc B + csc C : :
Barycentrics    a2(b + c) : b2(c + a) : c2(a + b)
Tripolars    (pending)

If you have The Geometer's Sketchpad, you can view X(42).

Let A'B'C' be the extangents triangle. Let La be the trilinear polar of A', and define Lb and Lc cyclically. Let A″ = Lb∩Lc, B″ = Lc∩La, C″ = La∩Lb. The lines AA″, BB″, CC″ concur in X(42). (Randy Hutson, December 26, 2015)

Let A'B'C' be the extangents triangle. Let A″ be the trilinear product B'*C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(42).(Randy Hutson, December 26, 2015)

Let Ab, Ac, Bc, Ba, Ca, Cb be as defined at X(3588). Let A* be the intersection of the tangents to the Myakishev conic at Ba and Ca, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(42).(Randy Hutson, December 26, 2015)

X(42) lies on these lines: 1,2   3,967   6,31   9,941   25,41   33,393   35,58   37,210   38,518   40,581   48,197   57,1001   65,73   81,100   101,111   165,991   172,199   181,228   244,354   308,313   321,740   517,1064   560,584   649,788   694,893   748,1001   750,940   894,1045   942,1066

X(42) is the {X(1),X(43)}-harmonic conjugate of X(2). For a list of other harmonic conjugates of X(42), click Tables at the top of this page.

X(42) = reflection of X(321) in X(1215)
X(42) = isogonal conjugate of X(86)
X(42) = isotomic conjugate of X(310)
X(42) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,37), (6,213), (10,71), (55,228)
X(42) = crosspoint of X(i) and X(j) for these (i,j): (1,6), (33,55), (37,65)
X(42) = crosssum of X(i) and X(j) for these (i,j): (1,2), (7,77), (21,81)
X(42) = crossdifference of every pair of points on line X(239)X(514)
X(42) = circumcircle-inverse of X(32759)
X(42) = X(1)-line conjugate of X(239)
X(42) = X(i)-beth conjugate of X(j) for these (i,j): (21,551), (55,42), (100,226), (210,210), (643,171)
X(42) = bicentric sum of PU(8)
X(42) = PU(8)-harmonic conjugate of X(649)
X(42) = barycentric product of PU(32)
X(42) = trilinear product of PU(85)
X(42) = trilinear pole of line X(512)X(798)
X(42) = Danneels point of X(1)
X(42) = {X(1),X(2)}-harmonic conjugate of X(3720)
X(42) = X(75)-isoconjugate of X(58)
X(42) = X(92)-isoconjugate of X(1790)
X(42) = trilinear square root of X(872)
X(42) = perspector of ABC and unary cofactor triangle of 1st Conway triangle
X(42) = perspector of ABC and unary cofactor triangle of 5th Conway triangle
X(42) = perspector of unary cofactor triangles of 1st and 5th Conway triangles
X(42) = perspector of ABC and unary cofactor triangle of Gemini triangle 2
X(42) = barycentric product of vertices of Gemini triangle 15


X(43) = X(6)-CEVA CONJUGATE OF X(1)

Trilinears    ab + ac - bc : bc + ba - ca : ca + cb - ab
Trilinears    csc B + csc C - csc A : csc C + csc A - csc B : csc A + csc B - csc C
Barycentrics    a(ab + ac - bc) : b(bc + ba - ca) : c(ca + cb - ab)

Let A' be the inverse-in-excircles-radical-circle of the midpoint of BC, and define B' and C' cyclically. Triangle A'B'C' is inscribed in the Apollonius circle and homothetic to the excentral triangle at X(43). (Randy Hutson, November 30, 2018)

X(43) lies on the Kiepert hyperbola of the excentral triangle and these lines: 1,2   6,87   9,256   31,100   35,1011   40,970   46,851   55,238   57,181   58,979   72,986   75,872   81,750   165,573   170,218   210,984   312,740   518,982

X(43) is the {X(2),X(42)}-harmonic conjugate of X(1). For a list of other harmonic conjugates of X(43), click Tables at the top of this page. X(43) is the external center of similitude of the Bevan circle and Apollonius circle; the internal center is X(1695).

X(43) = reflection of X(1) in X(995)
X(43) = isogonal conjugate of X(87)
X(43) = isotomic conjugate of X(6384)
X(43) = anticomplement of X(3840)
X(43) = X(6)-Ceva conjugate of X(1)
X(43) = X(192)-cross conjugate of X(1)
X(43) = crosssum of X(2) and X(330)
X(43) = X(55)-Hirst inverse of X(238)
X(43) = inverse-in-excircles-radical-circle of X(5121)
X(43) = perspector of ABC and extraversion triangle of X(87)
X(43) = trilinear product of extraversions of X(87)
X(43) = excentral-isogonal conjugate of X(1766)
X(43) = polar conjugate of isotomic conjugate of X(22370)
X(43) = perspector of Gemini triangle 5 and cross-triangle of Gemini triangles 3 and 5
X(43) = X(i)-aleph conjugate of X(j) for these (i,j):
(1,9), (6,43), (55,170), (100,1018), (174,169), (259,165), (365,1), (366,63), (507,362), (509,57)

X(43) = X(660)-beth conjugate of X(43)
X(43) = {X(2),X(8)}-harmonic conjugate of X(3741)


X(44) = X(6)-LINE CONJUGATE OF X(1)

Trilinears    b + c - 2a : c + a - 2b : a + b - 2c
Barycentrics    a(b + c - 2a) : b(c + a - 2b) : c(a + b - 2c)
X(44) = (3r2 + 6rR + s2)*X(1) - 18rR*X(2) -6r2*X(3)    (Peter Moses, April 2, 2013)
X(44) = X[1] - 3 X[238], 2 X[1] - 3 X[1279], X[1] + 3 X[1757], 4 X[1] - 3 X[4864], 3 X[2] - 4 X[6687], 3 X[2] + X[20072], 6 X[2] - 5 X[31243], 3 X[190] + X[17160], 3 X[238] - 2 X[3246], 4 X[238] - X[4864], 3 X[239] - X[17160], X[320] - 4 X[6687] (and many more)

X(44) lies on the curves K137, K453, K454, K717, K752, K1050, Q043, Q087, and these lines: {1, 6}, {2, 89}, {3, 34877}, {7, 14564}, {8, 4217}, {10, 752}, {15, 11790}, {16, 11791}, {19, 1828}, {21, 4273}, {31, 210}, {32, 25066}, {36, 3196}, {39, 37599}, {40, 21896}, {41, 2267}, {42, 3683}, {43, 4640}, {48, 3204}, {51, 209}, {55, 4849}, {56, 1732}, {57, 16602}, {58, 5044}, {63, 3752}, {65, 374}, {69, 17231}, {71, 2264}, {75, 16816}, {81, 17021}, {86, 4698}, {88, 679}, {100, 2384}, {101, 2718}, {105, 6017}, {141, 4416}, {142, 4896}, {144, 4000}, {169, 16605}, {171, 3740}, {172, 5035}, {181, 375}, {190, 239}, {192, 3759}, {193, 344}, {198, 2261}, {200, 3052}, {212, 1864}, {214, 1017}, {241, 651}, {259, 17641}, {281, 3087}, {292, 660}, {294, 1156}, {312, 37652}, {319, 17229}, {321, 19742}, {329, 3772}, {333, 27064}, {346, 3621}, {350, 17029}, {354, 748}, {386, 31445}, {391, 2345}, {394, 39249}, {480, 21002}, {511, 15977}, {513, 649}, {517, 1168}, {519, 2325}, {524, 3912}, {527, 1086}, {545, 1266}, {572, 2174}, {573, 3579}, {579, 16415}, {580, 5777}, {583, 992}, {584, 2268}, {594, 3626}, {595, 34790}, {597, 4364}, {599, 17284}, {602, 14872}, {610, 37500}, {612, 3715}, {614, 21342}, {645, 19623}, {662, 1931}, {674, 3271}, {678, 902}, {726, 4974}, {742, 17755}, {751, 2276}, {756, 2308}, {758, 21331}, {765, 1252}, {799, 30938}, {872, 2309}, {894, 3739}, {897, 9278}, {908, 23593}, {936, 4252}, {940, 3305}, {941, 28625}, {948, 12848}, {966, 3823}, {971, 13329}, {976, 4005}, {990, 5779}, {991, 31658}, {993, 5114}, {1015, 8610}, {1046, 3812}, {1052, 3509}, {1125, 19898}, {1150, 30818}, {1151, 32555}, {1152, 32556}, {1193, 5109}, {1211, 5294}, {1213, 3454}, {1253, 14100}, {1319, 1404}, {1333, 1778}, {1334, 3780}, {1376, 1707}, {1418, 1445}, {1427, 1708}, {1438, 34893}, {1465, 19619}, {1468, 25917}, {1471, 8581}, {1573, 3997}, {1639, 22086}, {1643, 24457}, {1654, 17239}, {1698, 31151}, {1721, 16112}, {1726, 14557}, {1737, 6128}, {1738, 17768}, {1739, 17960}, {1754, 5927}, {1766, 12702}, {1776, 9371}, {1781, 5356}, {1783, 14571}, {1785, 1990}, {1834, 12572}, {1836, 21949}, {1842, 6047}, {1859, 7076}, {1877, 8756}, {1903, 7118}, {1918, 22271}, {1944, 34852}, {1963, 33766}, {1964, 22343}, {1992, 17316}, {1999, 35652}, {2082, 21872}, {2092, 3647}, {2164, 35251}, {2170, 17444}, {2171, 17443}, {2177, 21870}, {2178, 5120}, {2194, 26890}, {2220, 3965}, {2223, 4557}, {2241, 3991}, {2251, 3285}, {2260, 28352}, {2262, 14737}, {2270, 5128}, {2277, 5069}, {2285, 5221}, {2293, 4878}, {2295, 3691}, {2298, 28615}, {2321, 3625}, {2341, 5127}, {2342, 2361}, {2364, 37525}, {2609, 3709}, {2664, 9359}, {2895, 33157}, {2999, 3929}, {3003, 13006}, {3006, 4144}, {3009, 3248}, {3059, 21059}, {3068, 30412}, {3069, 30413}, {3070, 31562}, {3071, 31561}, {3122, 20456}, {3158, 21000}, {3161, 17314}, {3175, 3187}, {3216, 3916}, {3219, 3666}, {3220, 5096}, {3241, 31722}, {3244, 4029}, {3245, 5540}, {3264, 4506}, {3290, 3999}, {3306, 31197}, {3332, 5817}, {3452, 37646}, {3501, 21868}, {3589, 4357}, {3618, 4657}, {3629, 3879}, {3632, 4873}, {3635, 4982}, {3644, 25269}, {3661, 4690}, {3662, 17345}, {3663, 17334}, {3664, 6666}, {3678, 5007}, {3681, 3744}, {3684, 5524}, {3685, 28581}, {3694, 5301}, {3696, 3923}, {3697, 5264}, {3706, 32864}, {3717, 5846}, {3720, 4722}, {3729, 4361}, {3730, 4266}, {3742, 17123}, {3753, 21373}, {3763, 17272}, {3764, 4735}, {3765, 4377}, {3769, 27538}, {3775, 24295}, {3781, 37516}, {3782, 17781}, {3791, 3971}, {3792, 9037}, {3838, 33096}, {3842, 33682}, {3844, 33082}, {3875, 4718}, {3876, 7296}, {3880, 16561}, {3899, 21338}, {3924, 3962}, {3928, 23511}, {3932, 5847}, {3939, 15733}, {3940, 37817}, {3941, 34247}, {3946, 17246}, {3950, 15828}, {3951, 37549}, {3958, 4016}, {3966, 33163}, {3967, 4362}, {3975, 17790}, {3986, 15808}, {4003, 5282}, {4007, 4816}, {4009, 17763}, {4011, 32853}, {4033, 25298}, {4037, 17162}, {4113, 32945}, {4253, 16604}, {4255, 31424}, {4307, 38057}, {4353, 4989}, {4358, 16704}, {4360, 4681}, {4363, 4384}, {4371, 4461}, {4386, 5782}, {4387, 17156}, {4388, 33118}, {4389, 17333}, {4393, 4664}, {4399, 4431}, {4402, 4488}, {4410, 20913}, {4419, 5222}, {4434, 23552}, {4440, 4912}, {4445, 17286}, {4454, 24599}, {4471, 37586}, {4472, 24603}, {4473, 4725}, {4515, 14974}, {4530, 14584}, {4553, 9025}, {4646, 12514}, {4648, 18230}, {4650, 16569}, {4659, 16833}, {4662, 5255}, {4667, 17392}, {4679, 11269}, {4683, 29850}, {4687, 17379}, {4695, 5183}, {4703, 25453}, {4704, 17393}, {4706, 32845}, {4713, 17026}, {4716, 28484}, {4721, 29433}, {4726, 17117}, {4739, 17116}, {4755, 16826}, {4794, 9321}, {4850, 14997}, {4883, 5284}, {4888, 20195}, {4889, 17315}, {4899, 9053}, {4914, 33162}, {4957, 24209}, {4966, 34379}, {4967, 7227}, {5032, 29585}, {5036, 37572}, {5057, 33139}, {5087, 33140}, {5158, 17102}, {5179, 12019}, {5217, 36744}, {5219, 31187}, {5224, 17331}, {5228, 8545}, {5242, 23302}, {5243, 23303}, {5257, 17398}, {5271, 19723}, {5275, 9349}, {5276, 5297}, {5277, 25068}, {5278, 26223}, {5296, 5550}, {5341, 16547}, {5393, 32787}, {5405, 32788}, {5438, 8951}, {5530, 18253}, {5541, 21885}, {5546, 19622}, {5548, 9456}, {5704, 27382}, {5714, 5746}, {5723, 22464}, {5739, 32777}, {5745, 37662}, {5781, 8544}, {5838, 30332}, {5852, 24231}, {5880, 24695}, {5905, 24789}, {6144, 17311}, {6173, 31183}, {6181, 15855}, {6184, 6594}, {6329, 17045}, {6351, 7585}, {6352, 7586}, {6510, 16578}, {6541, 17772}, {6646, 16706}, {7074, 30223}, {7083, 12329}, {7090, 13911}, {7126, 7127}, {7133, 19038}, {7232, 17282}, {7238, 28333}, {7308, 37674}, {7321, 31300}, {7330, 36745}, {7772, 37592}, {7963, 33804}, {8287, 26012}, {8584, 29574}, {8607, 23980}, {9018, 20670}, {9300, 24239}, {9324, 9325}, {9330, 9347}, {9355, 9441}, {10436, 17259}, {10445, 31673}, {11063, 26744}, {11488, 30414}, {11489, 30415}, {12723, 21867}, {13883, 31595}, {13936, 31594}, {13973, 14121}, {14555, 26065}, {14578, 32641}, {14621, 25384}, {15534, 29573}, {15624, 20992}, {15988, 25099}, {16560, 18735}, {16583, 36283}, {16696, 27644}, {16700, 27643}, {16713, 27058}, {16726, 18198}, {16732, 17895}, {16738, 27078}, {16825, 32935}, {16834, 17318}, {17011, 33761}, {17013, 28606}, {17028, 21264}, {17228, 17343}, {17230, 17342}, {17232, 17341}, {17233, 17339}, {17234, 17338}, {17236, 17329}, {17238, 17328}, {17240, 17373}, {17241, 17375}, {17242, 17377}, {17244, 17378}, {17247, 17380}, {17248, 17381}, {17249, 17383}, {17251, 17308}, {17252, 17307}, {17253, 17306}, {17254, 17305}, {17255, 17304}, {17258, 17302}, {17263, 17300}, {17265, 17298}, {17266, 17297}, {17267, 17296}, {17268, 17295}, {17269, 17294}, {17270, 17293}, {17271, 17292}, {17273, 17291}, {17274, 17290}, {17283, 17288}, {17285, 17287}, {17317, 20090}, {17325, 29598}, {17387, 29572}, {17390, 32455}, {17394, 27268}, {17461, 21886}, {17483, 26724}, {17484, 33129}, {17495, 30579}, {17593, 17779}, {17605, 24892}, {17720, 24597}, {17754, 37673}, {17780, 36872}, {17786, 29542}, {18228, 37642}, {18644, 36949}, {18743, 37683}, {19717, 37869}, {19750, 22034}, {19872, 31252}, {20016, 28329}, {20530, 37686}, {20568, 32012}, {20662, 38989}, {20970, 25092}, {21026, 25621}, {21035, 23659}, {21281, 24735}, {21371, 27623}, {21746, 22277}, {22276, 23638}, {23073, 37618}, {23972, 35091}, {24320, 36741}, {24330, 24592}, {24352, 24600}, {24407, 27918}, {24512, 30950}, {24632, 30906}, {24692, 25351}, {24697, 29633}, {24703, 33137}, {24715, 28534}, {24890, 25651}, {25067, 37659}, {25971, 26671}, {26048, 26076}, {26070, 32851}, {26792, 33133}, {26799, 26971}, {26878, 37528}, {27003, 37687}, {27191, 29607}, {27253, 32088}, {28254, 28283}, {28309, 36522}, {28538, 32847}, {28582, 32922}, {28604, 28633}, {29610, 31144}, {30829, 37684}, {32005, 36857}, {32779, 37656}, {32843, 33115}, {32861, 33164}, {32914, 32938}, {33075, 33166}, {33099, 33132}, {35068, 35079}, {35069, 35090}, {35116, 35508}, {35242, 37499}, {35595, 37633}, {36289, 36294}, {37595, 37685}

X(44) = midpoint of X(i) and X(j) for these {i,j}: {190, 239}, {238, 1757}, {320, 20072}, {1266, 4480}, {2325, 4700}, {3218, 3257}, {3271, 20683}, {3943, 4969}, {4432, 4753}, {9355, 9441}, {39150, 39151}
X(44) = reflection of X(i) in X(j) for these {i,j}: {1, 3246}, {320, 3834}, {1086, 3008}, {1266, 4395}, {1279, 238}, {3834, 6687}, {3912, 4422}, {3943, 2325}, {4432, 4759}, {4645, 3823}, {4702, 4432}, {4727, 3943}, {4864, 1279}, {4887, 17067}, {4908, 4370}, {4969, 4700}, {17374, 3912}, {24692, 25351}, {31138, 2}
X(44) = isogonal conjugate of X(88)
X(44) = isotomic conjugate of X(20568)
X(44) = complement of X(320)
X(44) = anticomplement of X(3834)
X(44) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,214), (88,1), (104,55)
X(44) = crosspoint of X(i) and X(j) for these (i,j): (1,88), (2,80)
X(44) = crosssum of X(i) and X(j) for these (i,j): (1,44), (6,36), (57,1465)
X(44) = crossdifference of every pair of points on line X(1)X(513)
X(44) = crossdifference of PU(55)
X(44) = X(i)-line conjugate of X(j) for these (i,j): {6, 1} }, {517, 14190}, {649, 513}, {1643, 24457}, {1739, 17960}
X(44) = X(88)-cross conjugate of X(44)
X(44) = bicentric sum of PU(i) for these i: 33, 50
X(44) = midpoint of PU(i) for these i: 33, 50
X(44) = perspector of circumconic centered at X(214)
X(44) = center of circumconic that is locus of trilinear poles of lines passing through X(214)
X(44) = inverse-in-circumconic-centered-at-X(9) of X(1)
X(44) = polar conjugate of isotomic conjugate of X(5440)
X(44) = trilinear pole of line X(678)X(1635)
X(44) = polar conjugate of isogonal conjugate of X(23202)
X(44) = cevapoint of X(i) and X(j) for these (i,j): {1, 9324}, {6, 3196}, {1635, 2087}, {2251, 23202}, {4120, 4530}
X(44) = crosspoint of X(i) and X(j) for these (i,j): {1, 88}, {2, 80}, {57, 8686}, {100, 5376}, {214, 19618}, {519, 3911}, {765, 3257}, {1016, 6079}, {1252, 32641}, {4358, 38462}, {7126, 19551}
X(44) = crosssum of X(i) and X(j) for these (i,j): {1, 44}, {2, 17495}, {6, 36}, {9, 3880}, {37, 31855}, {57, 1465}, {106, 2316}, {244, 1635}, {513, 2087}, {1015, 6085}, {1086, 10015}, {9456, 36058}, {37772, 37773}
X(44) = trilinear pole of line {678, 1635}
X(44) = crossdifference of every pair of points on line {1, 513}
X(44) = X(32012)-anticomplementary conjugate of X(6327)
X(44) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 214}, {32, 16586}, {42, 31845}, {80, 2887}, {759, 3741}, {1168, 21241}, {1402, 6739}, {1411, 2886}, {1807, 1368}, {1918, 35069}, {1989, 21236}, {2006, 17046}, {2161, 141}, {2222, 17072}, {2341, 21246}, {6187, 10}, {11060, 5249}, {14975, 1511}, {18359, 626}, {18815, 17047}, {20566, 21235}, {24624, 21240}, {32671, 21196}, {32675, 4885}, {34079, 3739}, {34857, 3454}, {36804, 21262}, {36815, 20542}, {36910, 21244}
X(44) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 678}, {2, 214}, {19, 17465}, {57, 17460}, {88, 1}, {100, 3251}, {104, 55}, {519, 3689}, {1022, 3722}, {1023, 1635}, {2161, 37}, {2743, 4162}, {3218, 517}, {3257, 513}, {3911, 1319}, {4358, 5440}, {4585, 8674}, {5376, 100}, {16704, 519}, {17780, 1960}, {23703, 4895}, {30731, 14425}, {32012, 2}, {36037, 3900}
X(44) = X(i)-cross conjugate of X(j) for these (i,j): {678, 1}, {902, 1319}, {1635, 1023}, {1960, 17780}, {2087, 1635}, {3251, 100}, {4895, 23703}, {14408, 24004}, {20972, 6}, {21805, 519}, {23202, 5440}
X(44) = X(i)-isoconjugate of X(j) for these (i,j): {1, 88}, {2, 106}, {3, 6336}, {4, 1797}, {6, 903}, {7, 2316}, {31, 20568}, {44, 679}, {56, 4997}, {57, 1320}, {58, 4080}, {63, 36125}, {69, 8752}, {75, 9456}, {81, 4674}, {89, 4792}, {92, 36058}, {100, 1022}, {101, 6548}, {110, 4049}, {190, 23345}, {244, 5376}, {264, 32659}, {292, 27922}, {312, 1417}, {512, 4615}, {513, 3257}, {514, 901}, {519, 2226}, {523, 4591}, {593, 4013}, {649, 4555}, {651, 23838}, {661, 4622}, {673, 34230}, {693, 32665}, {798, 4634}, {900, 4638}, {908, 10428}, {1086, 9268}, {1168, 3218}, {1252, 6549}, {1293, 2403}, {1318, 3911}, {1635, 4618}, {2163, 4945}, {2291, 36887}, {2712, 17953}, {3261, 32719}, {3445, 31227}, {3676, 5548}, {4462, 36042}, {4510, 30650}, {4588, 23598}, {4604, 23352}, {6545, 6551}, {6635, 21143}, {8046, 39148}, {9325, 9326}, {14190, 37131}, {14260, 34234}, {16489, 36592}, {16944, 18359}, {17109, 36805}, {17969, 35153}, {20332, 36814}
X(44) = X(i)-aleph conjugate of X(j) for these (i,j): {1, 9324}, {2, 16561}, {366, 5541}, {903, 21372}, {1320, 17613}, {3257, X(44) = 1023}
X(44) = X(i)-beth conjugate of X(j) for these (i,j): {21, 3246}, {333, 24593}, {645, 239}, {3939, 2223}, {5546, 7113}
X(44) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 88}, {6, 4674}, {9, 106}, {19, 36058}, {43, 903}, {46, 1797}, {57, 2316}, {63, 9456}, {104, 1465}, {106, 16610}, {513, 1022}, {517, 57}, {610, 36125}, {649, 23345}, {650, 23838}, {672, 34230}, {978, 4997}, {1022, 1635}, {1023, 244}, {1052, 5376}, {1054, 3257}, {1575, 36814}, {1635, 513}, {1726, 32659}, {1731, 3}, {1739, 63}, {1740, 20568}, {1743, 1320}, {1745, 6336}, {1763, 8752}, {1766, 1417}, {1769, 905}, {2087, 1054}, {2161, 36}, {2183, 14260}, {2246, 14190}, {2316, 517}, {2640, 4622}, {2664, 27922}, {2827, 3669}, {3216, 4080}, {3218, 6}, {3257, 2087}, {3737, 4049}, {3960, 650}, {4040, 6548}, {4498, 2441}, {4585, 3125}, {4674, 3218}, {4893, 23352}, {5539, 4615}, {5540, 901}, {5541, 2226}, {6909, 1427}, {9324, 679}, {9359, 4555}, {9456, 1739}, {12034, 1319}, {16548, 16944}, {16554, 1168}, {16560, 32665}, {16576, 4582}, {16610, 9}, {16670, 4792}, {17613, 269}, {21214, 31227}, {21372, 31}, {21381, 4591}, {21382, 32719}, {21385, 649}, {21864, 3337}, {23345, 21385}, {23650, 4083}, {23838, 3960}, {24625, 1575}, {32486, 3911}, {35338, 6549}, {36814, 24625}, {37680, 37}, {39150, 37773}, {39151, 37772}
X(44) = barycentric product X(i)*X(j) for these {i,j}: {1, 519}, {3, 38462}, {4, 5440}, {6, 4358}, {7, 3689}, {8, 1319}, {9, 3911}, {19, 3977}, {31, 3264}, {37, 16704}, {42, 30939}, {56, 4723}, {57, 2325}, {58, 3992}, {63, 8756}, {72, 37168}, {75, 902}, {76, 2251}, {78, 1877}, {80, 214}, {81, 3943}, {86, 21805}, {88, 4370}, {89, 4908}, {92, 22356}, {99, 4730}, {100, 900}, {101, 3762}, {104, 1145}, {106, 4738}, {109, 4768}, {162, 14429}, {190, 1635}, {219, 37790}, {256, 4434}, {264, 23202}, {291, 4432}, {312, 1404}, {321, 3285}, {513, 17780}, {514, 1023}, {517, 36944}, {522, 23703}, {528, 14191}, {561, 9459}, {594, 30576}, {643, 30572}, {644, 30725}, {649, 24004}, {651, 1639}, {653, 14418}, {658, 14427}, {660, 4448}, {662, 4120}, {664, 4895}, {668, 1960}, {673, 14439}, {678, 903}, {679, 8028}, {693, 23344}, {741, 4783}, {765, 1647}, {799, 14407}, {898, 30583}, {899, 36872}, {934, 4528}, {985, 4439}, {1002, 4702}, {1016, 2087}, {1017, 20568}, {1019, 4169}, {1100, 31011}, {1120, 17460}, {1126, 4975}, {1156, 6174}, {1227, 6187}, {1255, 4969}, {1317, 1320}, {1809, 1846}, {1811, 5151}, {2171, 30606}, {2320, 36920}, {2323, 14628}, {2334, 4742}, {2415, 4394}, {2429, 4462}, {3251, 4555}, {3257, 6544}, {3445, 4487}, {3576, 36925}, {3669, 30731}, {3903, 4922}, {4506, 30650}, {4511, 14584}, {4530, 4564}, {4598, 14408}, {4606, 4773}, {4607, 14437}, {4618, 33922}, {4700, 25430}, {4727, 25417}, {4753, 30571}, {4958, 37211}, {4984, 37212}, {5298, 32635}, {5376, 35092}, {6335, 22086}, {7126, 36668}, {8851, 24816}, {9278, 31059}, {9456, 36791}, {13143, 33812}, {14425, 27834}, {14436, 37133}, {16670, 36915}, {17100, 38544}, {17455, 18359}, {19551, 36669}, {20972, 36805}, {23757, 36037}, {28602, 37135}
X(44) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 903}, {2, 20568}, {6, 88}, {9, 4997}, {19, 6336}, {25, 36125}, {31, 106}, {32, 9456}, {37, 4080}, {41, 2316}, {42, 4674}, {45, 4945}, {48, 1797}, {55, 1320}, {99, 4634}, {100, 4555}, {101, 3257}, {106, 679}, {110, 4622}, {163, 4591}, {184, 36058}, {214, 320}, {238, 27922}, {244, 6549}, {513, 6548}, {519, 75}, {644, 4582}, {649, 1022}, {661, 4049}, {662, 4615}, {663, 23838}, {667, 23345}, {678, 519}, {692, 901}, {750, 4510}, {756, 4013}, {900, 693}, {901, 4618}, {902, 1}, {1023, 190}, {1110, 9268}, {1145, 3262}, {1155, 36887}, {1252, 5376}, {1319, 7}, {1397, 1417}, {1404, 57}, {1635, 514}, {1639, 4391}, {1647, 1111}, {1743, 31227}, {1877, 273}, {1960, 513}, {1973, 8752}, {2087, 1086}, {2177, 4792}, {2223, 34230}, {2251, 6}, {2325, 312}, {2429, 27834}, {3009, 36814}, {3251, 900}, {3264, 561}, {3285, 81}, {3689, 8}, {3762, 3261}, {3911, 85}, {3943, 321}, {3977, 304}, {3992, 313}, {4120, 1577}, {4152, 4723}, {4169, 4033}, {4358, 76}, {4370, 4358}, {4394, 2403}, {4432, 350}, {4434, 1909}, {4439, 33931}, {4448, 3766}, {4528, 4397}, {4530, 4858}, {4543, 4768}, {4700, 19804}, {4702, 4441}, {4723, 3596}, {4727, 28605}, {4730, 523}, {4738, 3264}, {4759, 30963}, {4768, 35519}, {4773, 4801}, {4775, 23352}, {4783, 35544}, {4792, 36594}, {4893, 23598}, {4895, 522}, {4908, 4671}, {4922, 4374}, {4958, 4823}, {4969, 4359}, {4975, 1269}, {4984, 4978}, {5168, 17960}, {5440, 69}, {6174, 30806}, {6187, 1168}, {6544, 3762}, {8028, 4738}, {8661, 764}, {8756, 92}, {9247, 32659}, {9324, 9460}, {9456, 2226}, {9459, 31}, {14122, 17089}, {14191, 18821}, {14407, 661}, {14408, 3835}, {14418, 6332}, {14425, 4462}, {14427, 3239}, {14429, 14208}, {14436, 3250}, {14437, 4728}, {14439, 3912}, {14584, 18815}, {16704, 274}, {17455, 3218}, {17460, 1266}, {17780, 668}, {20972, 16610}, {21781, 9326}, {21805, 10}, {21821, 3943}, {22086, 905}, {22356, 63}, {22371, 5440}, {23202, 3}, {23214, 22067}, {23344, 100}, {23644, 1739}, {23703, 664}, {23757, 36038}, {24004, 1978}, {30572, 4077}, {30576, 1509}, {30725, 24002}, {30731, 646}, {30939, 310}, {31011, 32018}, {32665, 4638}, {32739, 32665}, {34858, 10428}, {36872, 31002}, {36944, 18816}, {37168, 286}, {37790, 331}, {38462, 264}, {39251, 17023}
X(44) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 6, 16666}, {1, 9, 45}, {1, 45, 37}, {1, 238, 3246}, {1, 1743, 16670}, {1, 3246, 1279}, {1, 16505, 16507}, {1, 16666, 1100}, {1, 16670, 6}, {1, 16676, 16672}, {2, 320, 3834}, {2, 3758, 4670}, {2, 3834, 31243}, {2, 4643, 17237}, {2, 4644, 4675}, {2, 4741, 17227}, {2, 17256, 4708}, {2, 20072, 320}, {2, 26768, 27106}, {2, 26816, 27159}, {6, 9, 37}, {6, 37, 1100}, {6, 45, 1}, {6, 1100, 16668}, {6, 1743, 16669}, {6, 3973, 15492}, {6, 15492, 16814}, {6, 16669, 16671}, {6, 16675, 16884}, {6, 16685, 20228}, {6, 16777, 1449}, {6, 16814, 3723}, {6, 16884, 16667}, {6, 16885, 9}, {6, 17796, 2323}, {7, 37650, 17278}, {9, 37, 16814}, {9, 1449, 3731}, {9, 1743, 6}, {9, 2324, 34524}, {9, 3973, 16885}, {9, 16572, 8557}, {9, 16667, 16675}, {9, 16669, 1100}, {9, 16670, 1}, {9, 16671, 3723}, {9, 16885, 15492}, {10, 3707, 17330}, {37, 1100, 3723}, {37, 1743, 16671}, {37, 15492, 9}, {37, 16666, 1}, {37, 16669, 6}, {37, 16671, 16668}, {41, 2267, 2278}, {43, 7262, 4640}, {45, 16670, 16666}, {45, 16672, 16676}, {48, 3217, 3204}, {57, 37679, 16602}, {63, 4383, 3752}, {69, 17279, 17231}, {69, 26685, 17279}, {71, 2347, 4271}, {72, 1724, 1104}, {75, 17349, 17348}, {75, 17350, 17351}, {75, 21591, 21417}, {86, 17260, 4698}, {88, 9326, 679}, {101, 5053, 7113}, {141, 4416, 17344}, {141, 17353, 17357}, {144, 4000, 17276}, {144, 37681, 4000}, {192, 3759, 4852}, {193, 344, 4851}, {213, 16552, 1107}, {219, 1723, 1108}, {220, 8557, 37}, {238, 4649, 16801}, {241, 651, 6610}, {319, 17280, 17229}, {320, 3834, 31138}, {320, 6687, 31243}, {320, 27637, 28362}, {346, 5839, 17299}, {391, 2345, 17275}, {597, 4364, 17023}, {651, 37787, 241}, {662, 1931, 16702}, {672, 899, 20331}, {672, 2183, 2245}, {672, 2238, 1575}, {672, 2246, 1155}, {672, 2348, 910}, {679, 3257, 9326}, {748, 32912, 354}, {756, 2308, 3745}, {894, 17277, 3739}, {896, 899, 1155}, {896, 2246, 2243}, {899, 20331, 1575}, {902, 21805, 3689}, {966, 5749, 17303}, {966, 26039, 9780}, {984, 16468, 1386}, {992, 1400, 28244}, {992, 28249, 27627}, {1100, 16671, 6}, {1100, 16814, 37}, {1155, 2246, 910}, {1155, 2348, 2246}, {1386, 15481, 984}, {1400, 27627, 28249}, {1404, 22356, 17455}, {1445, 6180, 1418}, {1449, 3731, 16777}, {1654, 17289, 17239}, {1708, 34048, 1427}, {1743, 3973, 9}, {1743, 15492, 1100}, {1743, 16885, 37}, {1778, 2287, 1333}, {2170, 21801, 17444}, {2176, 21384, 17448}, {2182, 2183, 910}, {2183, 2265, 2182}, {2238, 20331, 899}, {2243, 20331, 1155}, {2276, 37657, 21904}, {2316, 12034, 2161}, {2323, 5526, 17796}, {2325, 3943, 4908}, {2325, 4969, 4727}, {3008, 4887, 17067}, {3204, 4268, 48}, {3218, 37680, 16610}, {3219, 32911, 3666}, {3247, 16667, 16884}, {3247, 16675, 37}, {3554, 34524, 37}, {3589, 4357, 17384}, {3589, 17332, 4357}, {3618, 17257, 4657}, {3629, 17243, 3879}, {3661, 17346, 4690}, {3661, 17354, 17359}, {3662, 17347, 17345}, {3662, 17352, 17356}, {3664, 6666, 17245}, {3681, 17127, 3744}, {3686, 17355, 594}, {3723, 16668, 1100}, {3729, 4361, 4686}, {3731, 16777, 37}, {3758, 17335, 2}, {3759, 17336, 192}, {3834, 6687, 2}, {3875, 17262, 4718}, {3875, 25728, 17262}, {3879, 25101, 17243}, {3943, 4370, 2325}, {4360, 17261, 4681}, {4363, 4384, 4688}, {4370, 4700, 4727}, {4370, 4969, 3943}, {4389, 17367, 17382}, {4416, 17353, 141}, {4419, 5222, 17301}, {4440, 29590, 37756}, {4473, 6542, 17264}, {4659, 16833, 17119}, {4663, 15254, 1}, {4667, 29571, 17392}, {4687, 17379, 28639}, {4690, 17359, 3661}, {4727, 4908, 3943}, {4753, 4759, 4702}, {4887, 17067, 1086}, {5222, 6172, 4419}, {5223, 7290, 3242}, {5224, 17368, 17385}, {5278, 26223, 31993}, {5540, 16548, 7297}, {5749, 9780, 26039}, {6646, 16706, 17235}, {6687, 20072, 31138}, {7174, 16469, 38315}, {7277, 17245, 3664}, {8609, 17796, 6603}, {9780, 26039, 17303}, {10436, 17259, 31238}, {14439, 39251, 902}, {15492, 16669, 37}, {15569, 16801, 1279}, {16477, 16521, 16666}, {16505, 23343, 1}, {16669, 16814, 16668}, {16669, 16885, 16814}, {16671, 16814, 1100}, {16672, 16676, 37}, {16673, 16677, 37}, {16675, 16884, 3247}, {17120, 17260, 86}, {17121, 17261, 4360}, {17123, 32913, 3742}, {17233, 17363, 17372}, {17234, 17364, 17376}, {17248, 17381, 25498}, {17254, 29630, 17305}, {17328, 17371, 17238}, {17329, 17370, 17236}, {17330, 17369, 10}, {17331, 17368, 5224}, {17333, 17367, 4389}, {17334, 17366, 3663}, {17337, 17365, 142}, {17338, 17364, 17234}, {17339, 17363, 17233}, {17340, 17362, 2321}, {17341, 17361, 17232}, {17342, 17360, 17230}, {17343, 17358, 17228}, {17344, 17357, 141}, {17345, 17356, 3662}, {17346, 17354, 3661}, {17347, 17352, 3662}, {17348, 17351, 75}, {17349, 17350, 75}, {17781, 26723, 3782}, {20372, 21760, 20363}, {20568, 32091, 32012}, {24597, 31018, 17720}, {25454, 25658, 2}, {26975, 27036, 2}, {27265, 29978, 30823}, {27268, 37677, 17394}, {27627, 28249, 28244}, {30556, 30557, 5289}, {31138, 31243, 3834}, {32091, 32103, 20568}, {32864, 32930, 3706}, {33082, 33159, 3844}, {33096, 33138, 3838}, {34247, 36635, 3941}


X(45) = X(9)-BETH CONJUGATE OF X(1)

Trilinears       2b + 2c - a : 2c + 2a - b : 2a + 2b - c
Barycentrics  a(2b + 2c - a) : b(2c + 2a - b) : c(2a + 2b - c)

X(45) = (3r2 + 6rR - s2)*X(1) - 18rR*X(2) - 6r2*X(3)    (Peter Moses, April 2, 2013)

X(45) lies on these lines: 1,6   2,88   53,281   55,678   141,344   198,1030   210,968   346,594

X(45) is the {X(1),X(9)}-harmonic conjugate of X(44). For a list of other harmonic conjugates of X(45), click Tables at the top of this page.

X(45) = isogonal conjugate of X(89)
X(45) = isotomic conjugate of X(20569)
X(45) = anticomplement of X(34824)
X(45) = crosssum of X(6) and X(999)
X(45) = anticomplement of isotomic conjugate of X(32013)
X(45) = X(i)-beth conjugate of X(j) for these (i,j): (9,1), (644,45)
X(45) = complement of polar conjugate of isogonal conjugate of X(22129)
X(45) = anticomplement of anticomplement of X(31285)


X(46) = X(4)-CEVA CONJUGATE OF X(1)

Trilinears    cos B + cos C - cos A : :
Trilinears    a^3 + a^2(b + c) - a(b^2 + c^2) - (b - c)^2(b + c) : :
Trilinears    ra - R : :, where ra, rb, rc are the exradii

Barycentrics    a(cos B + cos C - cos A) : :
Tripolars    (pending)

Let Ja' be the reflection of the A-excenter in BC, and define Jb', Jc' cyclically. Let Oa be the circumcenter of AJb'Jc', and define Ob, Oc cyclically. OaObOc and ABC are perspective at X(46). (Randy Hutson, July 20, 2016)

Let A' be the inverse-in-Bevan-circle of the A-vertex of the hexyl triangle, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(46). (Randy Hutson, July 20, 2016)

Let JaJbJc be the excentral triangle. Let A″ be the inverse-in-Bevan-circle of A, and define B″, C″ cyclically. The lines JaA″, JbB″, JcC″ concur in X(46). (Randy Hutson, July 20, 2016)

X(46) lies on these lines: 1,3   4,90   9,79   10,63   19,579   34,47   43,851   58,998   78,758   80,84   100,224   158,412   169,672   200,1004   218,910   222,227   225,254   226,498   269,1103   404,997   474,960   499,946   595,614   750,975   978,1054

X(46) is the {X(3),X(65)}-harmonic conjugate of X(1). For a list of other harmonic conjugates of X(46), click Tables at the top of this page.

X(46) = reflection of X(i) in X(j) for these (i,j): (1,56), (1479,1210)
X(46) = isogonal conjugate of X(90)
X(46) = isotomic conjugate of X(20570)
X(46) = circumcircle-inverse of X(32760)
X(46) = Bevan-circle-inverse of X(36)
X(46) = X(4)-Ceva conjugate of X(1)
X(46) = crosssum of X(3) and X(1069)
X(46) = X(i)-aleph conjugate of X(j) for these (i,j): (4,46), (174,223), (188,1079), (366,610), (653, 1020)
X(46) = X(100)-beth conjugate of X(46)
X(46) = perspector of excentral and orthic triangles
X(46) = orthic isogonal conjugate of X(1)
X(46) = excentral isogonal conjugate of X(1490)
X(46) = X(24)-of-excentral-triangle
X(46) = {X(1),X(3)}-harmonic conjugate of X(3612)
X(46) = {X(1),X(40)}-harmonic conjugate of X(5119)
X(46) = perspector of ABC and extraversion triangle of X(90)
X(46) = trilinear product of extraversions of X(90)
X(46) = X(24) of reflection triangle of X(1)
X(46) = homothetic center of ABC and orthic triangle of reflection triangle of X(1)
X(46) = Cundy-Parry Phi transform of X(46)
X(46) = Cundy-Parry Psi transform of X(90)
X(46) = {X(1),X(57)}-harmonic conjugate of X(3338)


X(47) = X(110)-BETH CONJUGATE OF X(34)

Trilinears       cos 2A : cos 2B : cos 2C = f(a,b,c) : f(b,c,a) : f(c,a,b), where
                        f(a,b,c) = a2(a4 + b4 + c4 - 2a2b2 - 2a2c2)
Trilinears        a2 - 2R2 : b2 - 2R2 : c2 - 2R2
Barycentrics  a cos 2A : b cos 2B : c cos 2C
Trilinears    tan A cot 2A : :
Trilinears    cos^2 A - sin^2 A : :
Trilinears    1 - 2 sin^2 A : :
Trilinears    1 - 2 cos^2 A : :

X(47) = (r2 - R2 + s2)*X(1) - 6rR*X(2) - 4r2*X(3)    (Peter Moses, April 2, 2013)

Let A'B'C' be the Kosnita triangle. Let A″ be the trilinear product B'*C', and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X(47). (Randy Hutson, March 21, 2019)

Let A'B'C' and A″B″C″ be the Lucas and Lucas(-1) tangents triangles. Let A* be the trilinear product A'*A″, and define B*, C* cyclically. The lines AA*, BB*, CC* concur in X(47). (Randy Hutson, March 21, 2019)

X(47) lies on these lines: 1,21   19,921   33,90   34,46   35,212   36,602   91,92   158,162   171,498   238,499

X(47) is the {X(91),X(92)}-harmonic conjugate of X(564). For a list of other harmonic conjugates of X(47), click Tables at the top of this page.

X(47) = isogonal conjugate of X(91)
X(47) = isotomic conjugate of X(20571)
X(47) = anticomplement of X(34825)
X(47) = trilinear product X(371)*X(372)
X(47) = X(92)-isoconjugate of X(1820)
X(47) = perspector of ABC and extraversion triangle of X(47) (which is also the anticevian triangle of X(47))
X(47) = eigencenter of cevian triangle of X(92)
X(47) = eigencenter of anticevian triangle of X(48)
X(47) = X(92)-Ceva conjugate of X(48)
X(47) = crosssum of X(i) and X(j) for these (i,j): (656,1109)
X(47) = X(275)-aleph conjugate of X(92)
X(47) = X(i)-beth conjugate of X(j) for these (i,j): (110,34), (643,47)
X(47) = trilinear product of X(371) and X(372)


X(48) = CROSSPOINT OF X(1) AND X(63)

Trilinears    tan B + tan C : :
Trilinears    a2(b2 + c2 - a2) ::
Trilinears    SBSC - S2 : SCSA - S2 : SASB - S2
Trilinears    1 - cot B cot C : :
Trilinears    sin 2A : :

X(48) = (r2 + 4rR + 4R2 + s2)*X(1) - 6R(2R + r)*X(2) - 2(r2 + 2rR - s2)*X(3)    (Peter Moses, April 2, 2013)

Let A'B'C' be the hexyl triangle. Let A″ be the barycentric product of the circumcircle intercepts of line B'C'. Define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X(48). (Randy Hutson, July 31 2018)

X(48) lies on these lines: 1,19   3,71   6,41   9,101   31,560   36,579   37,205   42,197   55,154   63,326   75,336   163,1094   184,212   220,963   255,563   281,944   282,947   354,584   577,603   692,911   949,1037   958,965

X(48) is the {X(41),X(604)}-harmonic conjugate of X(6). For a list of other harmonic conjugates of X(48), click Tables at the top of this page.

X(48) = isogonal conjugate of X(92)
X(48) = isotomic conjugate of X(1969)
X(48) = complement of X(21270)
X(48) = anticomplement of X(20305)
X(48) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,31), (2,36033), (3,212), (63,255), (92,47), (284, 6)
X(48) = X(228)-cross conjugate of X(3)
X(48) = crosspoint of X(i) and X(j) for these (i,j): (1,63), (3,222), (91,92), (219,268)
X(48) = crosssum of X(i) and X(j) for these (i,j): (1,19), (4,281), (47,48), (196,278), (523, 1146), (661,1109)
X(48) = crossdifference of every pair of points on line X(240)X(522)
X(48) = X(1)-line conjugate of X(240)
X(48) = X(i)-beth conjugate of X(j) for these (i,j): (101,48), (219,219), (284,604), (906,48)
X(48) = barycentric product of PU(16)
X(48) = vertex conjugate of PU(18)
X(48) = bicentric sum of PU(22)
X(48) = PU(22)-harmonic conjugate of X(656)
X(48) = trilinear pole of line X(810)X(822)
X(48) = X(2)-isoconjugate of X(4)
X(48) = X(75)-isoconjugate of X(19)
X(48) = X(91)-isoconjugate of X(1748)
X(48) = perspector of ABC and extraversion triangle of X(48) (which is also the anticevian triangle of X(48))
X(48) = crosspoint of X(2066) and X(5414)
X(48) = {X(1),X(19)}-harmonic conjugate of X(1953)
X(48) = perspector of circumconic centered at X(36033)


X(49) = CENTER OF SINE-TRIPLE-ANGLE CIRCLE

Trilinears       cos 3A : cos 3B : cos 3C
Barycentrics  sin A cos 3A : sin B cos 3B : sin C cos 3C
Barycentrics    a^4 (a^2 - b^2 - c^2) (a^4 + b^4 + c^4 - 2 a^2 b^2 - 2 a^2 c^2 - b^2 c^2) : :
X(49) = (r2 + 2 r R+ s2)*X(1) - 6 R (R + r)*X(2) - 3(r2 + 4 r R + R2 - s2)*X(3)    (Peter Moses, April 2, 2013)

V. Thebault, "Sine-triple-angle-circle," Mathesis 65 (1956) 282-284.

X(49) lies on these lines: 1,215   3,155   4,156   5,54   24,568   52,195   93,94   381,578

Suppose that P and Q are distinct points in the plane of a triangle ABC . Let PA = reflection of P in line AQ, let QA = reflection of Q in line AP, and let MA = midpoint of segment PAQA. Define MB and MC cyclically. César Lozada found that if Q = isogonal conjugate of P, then the locus of P for which MAMBMC is perspective to ABC is the union of a cubic and 6 circles: specifically, the McCay cubic (K003), the circles {B,C,B',C'}}, {C,A,C',A'}}, {A,B,A',B'}}, and the circles {B,C,A'}}, {C,A,B'}}, {A,B,C'}}, where A',B',C' are the excenters of ABC. Moreover, if P = X(3) and Q = X(4), then MAMBMC is not only perspective, but homothetic, to ABC, and the center of homothety is X(49). See Hyacinthos 23265, June 1, 2015.

X(49) is the {X(54),X(110)}-harmonic conjugate of X(5). For a list of other harmonic conjugates of X(49), click Tables at the top of this page.

X(49) = isogonal conjugate of X(93)
X(49) = isotomic conjugate of X(20572)
X(49) = anticomplement of X(34826)
X(49) = X(4)-isoconjugate of X(2962)
X(49) = X(92)-isoconjugate of X(2963)
X(49) = eigencenter of cevian triangle of X(94)
X(49) = eigencenter of anticevian triangle of X(50)
X(49) = X(94)-Ceva conjugate of X(50)


X(50) = X(74)-CEVA CONJUGATE OF X(184)

Trilinears    sin 3A : :
Trilinears    cos A sin 2A + sin A cos 2A : :
Trilinears    sin A + cos A cot D/2 : : , where cot D/2 = (4*area)/(6R2 - a2 - b2 - c2), where R = abc/(4*area)       (Peter Moses, 12/19/2011; cf. X(568))
Trilinears    a(1 - 4 cos2A) : b(1 - 4 cos2B) : c(1 - 4 cos2C)
Trilinears         a(1 + 2 cos 2A) : b(1 + 2 cos 2B) : c(1 + 2 cos 2C)
Barycentrics    sin A sin 3A : sin B sin 3B : sin C sin 3C

Barycentrics    a^4 ((a^2 - b^2 - c^2)^2 - b^2 c^2) : :
Tripolars    (pending)
X(50) = -(r2 + 2rR + s2)(r2 + 4rR + 3R2 - s2)*X(1) + 6rR(r2 + 4rR + 3R2 - s2)*X(2) + 2r2(r2 + 4rR + 3R2 - 3s2)*X(3)    (Peter Moses, April 2, 2013)

Let DEF be any equilateral triangle inscribed in the circumcircle of ABC. Let D' be the barycentric product E*F, and define E', F' cyclically. Then D',E',F' all line on a line passing through X(50). In the special case that DEF is the circumtangential triangle, the points D',E',F' lie on the Brocard axis, and in case DEF is the circumnormal triangle, the points D',E'F' lie on the line X(50)X(647). See also X(6149). (Randy Hutson, January 29, 2015)

Let A'B'C' and A″B″C″ be the (equilateral) circumcevian triangles of X(15) and X(16). Let A* be the barycentric product A'*A″, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(50). See also X(6149). (Randy Hutson, January 29, 2015)

Let AA1A2, BB1B2, CC1C2 be the circumcircle-inscribed equilateral triangles used in the construction of the Trinh triangle. Let A' be the barycentric product A1*A2, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(50); see also X(6149). (Randy Hutson, October 13, 2015)

Let A1B1C1 and A2B2C2 be the 1st and 2nd Ehrmann circumscribing triangles. Let A' be the crossdifference of A1 and A2, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(50). (Randy Hutson, June 27, 2018)

X(50) lies on these lines: 3,6   67,248   112,477   115,231   230,858   338,401   647,654

X(50) is the {X(3),X(6)}-harmonic conjugate of X(566). For a list of other harmonic conjugates of X(40), click Tables at the top of this page.

X(50) = isogonal conjugate of X(94)
X(50) = isotomic conjugate of X(20573)
X(50) = anticomplement of X(34827)
X(50) = complement of isogonal conjugate of X(34448)
X(50) = circumcircle-inverse of X(32761)
X(50) = Brocard-circle-inverse of X(566)
X(50) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,215), (74,184), (94,49)
X(50) = crosspoint of X(i) and X(j) for these (i,j): (93,94), (186,323)
X(50) = crosssum of X(49) and X(50)
X(50) = crossdifference of every pair of points on line X(5)X(523)
X(50) = barycentric product of X(15) and X(16)
X(50) = X(i)-isoconjugate of X(j) for these (i,j): (92,265), (1577,476)
X(50) = Cundy-Parry Phi transform of X(568)
X(50) = perspector of circumconic passing through X(110) and the isogonal conjugates of PU(5)
X(50) = X(2)-Ceva conjugate of X(11597)
X(50) = perspector of ABC and unary cofactor triangle of Ehrmann vertex-triangle
X(50) = barycentric product X(35)*X(36)
X(50) = crossdifference of PU(173)


X(51) = CENTROID OF ORTHIC TRIANGLE

Trilinears    a2cos(B - C) : :
Trilinears    a[a2(b2 + c2) - (b2 - c2)2] : :
Trilinears    sin A (sin 2B + sin 2C) : :
Trilinears    sec A (csc 2B + csc 2C) : :
Barycentrics    a3cos(B - C) : b3cos(C - A) : c3cos(A - B)

X(51) = (r2 + 2rR + s2)*X(1) + 6R(R - r)*X(2) - (r2 + 4rR - s2)*X(3)    (Peter Moses, April 2, 2013)

Let A'B'C' be the anticomplementary triangle and let Ba and Ca be the orthogonal projections of B' and C' on BC, respectively. Define Cb and Ac cyclically, and define Ab and Bc cyclically. Then X(51) is the centroid of BaCaCbAbAcBc. (Randy Hutson, April 9, 2016)

Let L be the van Aubel line. Let U = X(6)X(25), the isogonal conjugate of polar conjugate of L; let V = X(4)X(51), the polar conjugate of the isogonal conjugate of L. Then X(51) = U∩V. (Randy Hutson, April 9, 2016)

Let A'B'C' be the orthic triangle. Let Oa be the A-McCay circle of triangle AB'C', and define Ob, Oc cyclically. X(51) is the radical center of circles Oa, Ob, Oc. (Randy Hutson, July 31 2018)

X(51) lies on these lines: 2,262   4,185   5,52   6,25   21,970   22,182   23,575   24,578   26,569   31,181   39,237   44,209   54,288   107,275   125,132   129,137   130,138   199,572   210,374   216,418   381,568   397,462   398,463   573,1011

X(51) is the {X(5),X(143)}-harmonic conjugate of X(52). For a list of other harmonic conjugates of X(51), click Tables at the top of this page.

X(51) = reflection of X(210) in X(375)
X(51) = isogonal conjugate of X(95)
X(51) = isotomic conjugate of X(34384)
X(51) = complement of X(2979)
X(51) = anticomplement of X(3819)
X(51) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,53), (5,216), (6,217)
X(51) = X(2)-Ceva conjugate of complementary conjugate of X(34845)
X(51) = X(217)-cross conjugate of X(216)
X(51) = crosspoint of X(i) and X(j) for these (i,j): (4,6), (5,53)
X(51) = crosssum of X(i) and X(j) for these (i,j): (2,3), (6,160), (54,97)
X(51) = crossdifference of every pair of points on line X(323)X(401)
X(51) = inverse-in-orthosymmedial-circle of X(125)
X(51) = X(2) of tangential triangle of Johnson circumconic
X(51) = trilinear pole of polar of X(276) wrt polar circle
X(51) = pole wrt polar circle of trilinear polar of X(276) (line X(340)X(520))
X(51) = X(48)-isoconjugate (polar conjugate) of X(276)
X(51) = X(92)-isoconjugate of X(97)
X(51) = Zosma transform of X(92)
X(51) = intersection of tangents to Moses-Jerabek conic at X(6) and X(1204)
X(51) = perspector of 1st & 2nd orthosymmedial triangles
X(51) = bicentric sum of PU(157)
X(51) = PU(157)-harmonic conjugate of X(647)
X(51) = perspector of orthic-of-orthocentroidal triangle and orthocentroidal-of-orthic triangle
X(51) = centroid of reflection triangle of X(125)
X(51) = excentral-to-ABC functional image of X(2)
X(51) = centroid of anticomplementary circle intercepts of sidelines of ABC
X(51) = {X(34221),X(34222)}-harmonic conjugate of X(5480)


X(52) = ORTHOCENTER OF ORTHIC TRIANGLE

Trilinears    cos 2A cos(B - C) : cos 2B cos(C - A) : cos 2C cos(A - B)
Trilinears    sec A (sec 2B + sec 2C) : :
Trilinears    cos(A - 2B) + cos(A - 2C) : :
Barycentrics    tan A (sec 2B + sec 2C) : tan B (sec 2C + sec 2A) : tan C (sec 2A + sec 2B)
Barycentrics    a^2(a^4 + b^4 + c^4 - 2a^2b^2 - 2a^2c^2)[a^2(b^2 + c^2) - (b^2 - c^2)^2] : :
Tripolars    (pending)
X(52) = (r2 + 2rR + s2)*X(1) - 6rR*X(2) - (r2 - 4rR - 2R2 + s2)*X(3)    (Peter Moses, April 2, 2013)

Let Ha be the foot of the A-altitude. Let Ba be the foot of the perpendicular from Ha to CA, and define Cb and Ac cyclically. Let Ca be the foot of the perpendicular from Ha to AB, and define Ab and Bc cyclically. Let A' be the orthocenter of HaBaCa, and define B' and C' cyclically. The lines HaA', HbB', HcC' concur in X(52). (Randy Hutson, December 10, 2016)

Let OA be the circle centered at the A-vertex of the anti-Ursa-minor triangle and passing through A; define OB and OC cyclically. X(52) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

X(52) lies on these lines: 3,6   4,68   5,51   25,155   26,184   30,185   49,195   113,135   114,211   128,134   129,139

X(52) is the {X(5),X(143)}-harmonic conjugate of X(51). For a list of other harmonic conjugates of X(52), click Tables at the top of this page.

X(52) = reflection of X(i) in X(j) for these (i,j): (3,389), (5,143), (113,1112), (1209,973)
X(52) = isogonal conjugate of X(96)
X(52) = isotomic conjugate of X(34385)
X(52) = anticomplement of X(1216)
X(52) = circumcircle-inverse of X(32762)
X(52) = Brocard-circle-inverse of X(569)
X(52) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,5), (317,467), (324,216)
X(52) = crosspoint of X(4) and X(24)
X(52) = crosssum of X(3) and X(68)
X(52) = {X(3),X(6)}-harmonic conjugate of X(569)
X(52) = orthic isogonal conjugate of X(5)
X(52) = X(20)-of-2nd Euler triangle
X(52) = perspector of ABC and cross-triangle of ABC and 2nd Euler triangle
X(52) = perspector of ABC and cross-triangle of ABC and Kosnita triangle
X(52) = antipode of X(113) in Hatzipolakis-Lozada hyperbola
X(52) = Cundy-Parry Phi transform of X(571)
X(52) = Cundy-Parry Psi transform of X(5392)
X(52) = X(1577)-isoconjugate of X(32692)
X(52) = excentral-to-ABC functional image of X(4)


X(53) = SYMMEDIAN POINT OF ORTHIC TRIANGLE

Trilinears    tan A cos(B - C) : tan B cos(C - A) : tan C cos(A - B)
Barycentrics  a tan A cos(B - C) : b tan B cos(C - A) : c tan C cos(A - B)
Barycentrics    (a^2 (b^2 + c^2) - (b^2 - c^2)^2)/(a^2 - b^2 - c^2) : :

Let A'B'C' be the Euler triangle. Let LA be the trilinear polar of A', and define LB and LC cyclically. Let A″ = LB∩LC, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(53). (Randy Hutson, June 7, 2019)

X(53) lies on these lines: 4,6   5,216   25,157   30,577   45,281   115,133   128,139   137,138   141,264   232,427   273,1086   275,288   311,324   317,524   318,594   395,472   396,473

X(53) is the {X(4),X(393)}-harmonic conjugate of X(6). For a list of other harmonic conjugates of X(53), click Tables at the top of this page.

X(53) = isogonal conjugate of X(97)
X(53) = isotomic conjugate of X(34386)
X(53) = anticomplement of X(34828)
X(53) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,51), (324,5)
X(53) = X(51)-cross conjugate of X(5)
X(53) = crosssum of X(3) and X(577)
X(53) = Kosnita(X(4),X(6)) point
X(53) = trilinear pole of line X(12077)X(15451) (the polar of X(95) wrt polar circle)
X(53) = pole wrt polar circle of trilinear polar of X(95) (line X(323)X(401))
X(53) = polar conjugate of X(95)
X(53) = excentral-to-ABC functional image of X(6)


X(54) = KOSNITA POINT

Trilinears    sec(B - C) : sec(C - A) : sec(A - B)
Trilinears    a/(b cos B + c cos C) : :
Barycentrics    sin A sec(B - C) : sin B sec(C - A) : sin C sec(A - B)
Barycentrics    a^2/(S^2 + SB SC) : :
Barycentrics    a^2/(b^4 + c^4 - a^2b^2 - a^2c^2 - 2b^2c^2) : :
Barycentrics    a^2*(a^4 - 2*a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2)*(a^4 - a^2*b^2 - 2*a^2*c^2 - b^2*c^2 + c^4) : :
X(54) = 3 X[2] - 4 X[6689], X[3] + 2 X[1493], 3 X[3] - X[12307], 3 X[3] + X[12316], 2 X[3] + X[15801], X[4] + 2 X[10619], X[4] - 4 X[12242], 3 X[4] - 2 X[32340], 5 X[5] - 4 X[20584], X[5] + 4 X[20585], 3 X[51] - 2 X[11808], 2 X[52] + X[12226], X[64] + 3 X[17824], X[64] - 3 X[32345], X[74] + 2 X[2914], X[74] + 4 X[32226], X[110] + 2 X[15089], 4 X[140] - X[3519], 2 X[195] + X[7691], X[195] + 2 X[10610], 3 X[195] + X[12307], 3 X[195] - X[12316], 3 X[381] - 2 X[22804], 4 X[389] - X[6242], 2 X[389] + X[21660], X[550] + 2 X[11803], 4 X[575] - X[9972], 5 X[631] + X[11271], 5 X[631] - X[12325], 5 X[631] - 4 X[32348], 5 X[632] - 3 X[21357], 4 X[973] - 5 X[3567], 4 X[973] - 3 X[7730], 4 X[973] + X[12291], 4 X[973] - X[13423], X[1141] + 2 X[27423], 3 X[1157] - 4 X[12060], 4 X[1493] + X[7691], 6 X[1493] + X[12307], 6 X[1493] - X[12316], 4 X[1493] - X[15801], 5 X[1656] - 4 X[13565], X[2888] - 4 X[6689], 3 X[2917] - 5 X[17821], 3 X[2917] - X[17846], 7 X[3090] - 8 X[32396], 7 X[3523] + 2 X[13431], 11 X[3525] - 4 X[15605], 5 X[3567] - 2 X[6152], 5 X[3567] - 3 X[7730], 5 X[3567] + 4 X[11577], 5 X[3567] + X[12291], 5 X[3567] - X[13423], 2 X[3574] + X[12254], 3 X[3574] - X[32340], X[3627] - 4 X[30531], 8 X[5462] - 3 X[41713], X[5889] + 2 X[12606], 3 X[5890] - X[32339], X[5898] - 3 X[32609], 3 X[5946] - X[13368], 3 X[5946] + X[15532], 3 X[6030] - 2 X[13564], 2 X[6102] + X[22815], 2 X[6152] - 3 X[7730], X[6152] + 2 X[11577], 2 X[6152] + X[12291], 4 X[6153] - 3 X[41713], X[6241] + 2 X[12300], X[6241] + 4 X[15739], X[6242] + 2 X[21660], X[6242] + 4 X[40632], X[6288] - 4 X[8254], 5 X[6288] - 8 X[20584], X[6288] + 8 X[20585], X[6288] + 2 X[36966], X[6759] - 3 X[10274], 2 X[6759] - 3 X[32379], X[7691] - 4 X[10610], 3 X[7691] - 2 X[12307], 3 X[7691] + 2 X[12316], 3 X[7730] + 4 X[11577], 3 X[7730] + X[12291], 3 X[7730] - X[13423], X[7979] + 2 X[9905], X[7979] - 4 X[12266], 5 X[8254] - 2 X[20584], X[8254] + 2 X[20585], 2 X[8254] + X[36966], X[8718] + 3 X[13482], X[8718] + 4 X[37472], 3 X[9730] - 2 X[11802], 7 X[9781] - 4 X[11576], 8 X[9827] - 11 X[15024], 4 X[9827] - 3 X[41578], X[9905] + 2 X[12266], X[9935] - 4 X[10282], 4 X[10115] + X[12226], 6 X[10610] - X[12307], 6 X[10610] + X[12316], 4 X[10610] + X[15801], X[10619] + 2 X[12242], 3 X[10619] + X[32340], X[11271] + 4 X[32348], 3 X[11402] + X[32333], 3 X[11402] - X[32341], X[11412] - 4 X[12363], 4 X[11577] - X[12291], 4 X[11577] + X[13423], 4 X[12007] + X[13622], 4 X[12242] + X[12254], 6 X[12242] - X[32340], 3 X[12254] + 2 X[32340], X[12280] - 7 X[15043], 2 X[12307] + 3 X[15801], 2 X[12316] - 3 X[15801], X[12325] - 4 X[32348], 4 X[13365] - 5 X[15026], X[13432] + 11 X[15720], 3 X[13482] - 4 X[37472], 2 X[14049] + X[33565], 4 X[14865] - X[16835], 11 X[15024] - 6 X[41578], 5 X[15034] - 2 X[25714], X[15800] - 4 X[22051], 5 X[17821] - X[17846], 5 X[17821] - 6 X[32391], X[17846] - 6 X[32391], X[18368] + 2 X[34564], X[20584] + 5 X[20585], 4 X[20584] + 5 X[36966], 4 X[20585] - X[36966], X[32352] + 2 X[40632], 2 X[32369] - 3 X[32395].

See John Rigby, "Brief notes on some forgotten geometrical theorems," Mathematics and Informatics Quarterly 7 (1997) 156-158.

Let O be the circumcenter of triangle ABC, and OA the circumcenter of triangle BOC. Define OB and OC cyclically. Then the lines AOA, BOB, COC concur in X(54). For details and generalization, see

Darij Grinberg, A New Circumcenter Question.

The above construction of X(54) generalizes. Suppose that P and Q are points (as functions of a,b,c). Let A' = Q-of-BCP, B' = Q-of-CAP, C' = Q-of-ABP. If the lines AA', BB', CC' concur, the perspector is called the Kosnita(P,Q) point, denoted by K(P,Q). (Randy Hutson, 9/23/2011)

X(3) = K(X(20),X(2)) X(4) = K(X(20,X(20) X(5) = K(X(4),X(2))
X(13) = K(X(13),X(1)) X(17) = K(X(13),X(3)) X(18) = K(X(14),X(3))
X(140) = K(X(3), X(2)) X(251) = K(X(6), X(6))
X(481) = K(X(175),X(1)) X(482) = K(X(176),X(1))

Let NANBNC be the reflection triangle of X(5). Let OA be the circumcenter of ANBNC, and define OB and OC cyclically. Triangle OAOBOC is perspective to ABC at X(54), homothetic to the orthic-of-orthocentroidal triangle at X(54), and orthologic to the reflection triangle at X(54). (Randy Hutson, June 7, 2019)

Let (Na) be the reflection of the nine-point circle in BC, and define (Nb) and (Nc) cyclically. X(54) is the radical center of (Na), (Nb), (Nc). The tangents at A, B, C to the Napoleon-Feuerbach cubic K005 concur in X(54). Let A'B'C' be the reflection triangle. Let La be the trilinear polar of A', and define Lb and Lc cyclically. Let A″ be Lb∩Lc, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(54). (Randy Hutson, July 23, 2015)

If you have GeoGebra, you can view X(54).

let Na = reflection of X(5) in the line BC, and define Nb and Nc cyclically. The medial triangle of NaNbNc is perspective to ABC, and the perspector is X(54). (Dasari Naga Vijay Krishna, June 8, 2021)

X(54) lies on the Jarabek circumhyperbola, the cubics K005, K073, K112, K316, K361, K364, K373, K388, K439, K464, K466, K467, K469, K471, K499, K500, K502, K526, K566, K569, K589, K590, K629, K633, K646, K668, K822, K919, K942, K947, K976, K1107, K1180, the curves Q023, Q029, Q089, Q110, Q141, and these lines: {1, 2599}, {2, 68}, {3, 97}, {4, 184}, {5, 49}, {6, 24}, {10, 9562}, {11, 2477}, {12, 215}, {13, 3206}, {14, 3205}, {15, 10678}, {16, 10677}, {17, 3201}, {18, 3200}, {19, 16031}, {20, 4846}, {22, 36747}, {23, 5446}, {25, 3527}, {26, 3060}, {28, 1243}, {30, 3521}, {32, 9985}, {33, 9638}, {35, 6286}, {36, 73}, {39, 248}, {51, 288}, {52, 1166}, {55, 9666}, {56, 9653}, {59, 5399}, {60, 5396}, {61, 3166}, {62, 3165}, {64, 378}, {65, 1870}, {66, 3541}, {67, 5622}, {69, 95}, {70, 1899}, {71, 572}, {72, 1006}, {74, 185}, {98, 3203}, {99, 39805}, {112, 217}, {113, 3047}, {114, 3044}, {115, 9697}, {118, 3046}, {119, 3045}, {125, 3043}, {136, 6801}, {137, 33545}, {140, 252}, {143, 2070}, {154, 10594}, {155, 7503}, {156, 381}, {186, 389}, {193, 19131}, {206, 14853}, {219, 26915}, {222, 26914}, {251, 37123}, {262, 3202}, {276, 290}, {287, 37125}, {311, 34385}, {323, 1216}, {324, 37127}, {371, 6414}, {372, 6413}, {376, 10984}, {382, 18550}, {394, 7509}, {397, 11134}, {398, 11137}, {399, 33539}, {402, 12797}, {403, 12241}, {411, 34800}, {418, 2055}, {427, 31804}, {436, 8794}, {476, 36159}, {477, 36179}, {493, 12998}, {494, 12999}, {496, 40450}, {511, 1176}, {523, 36161}, {526, 38897}, {542, 18125}, {546, 10540}, {548, 13623}, {549, 34483}, {550, 11803}, {568, 1658}, {575, 895}, {576, 7556}, {577, 26876}, {632, 21357}, {695, 14153}, {826, 879}, {930, 35720}, {953, 36078}, {970, 1798}, {1062, 9637}, {1075, 21449}, {1087, 2595}, {1113, 14374}, {1114, 14375}, {1192, 35472}, {1204, 11270}, {1246, 7554}, {1263, 6343}, {1291, 15907}, {1296, 9226}, {1329, 9702}, {1351, 9715}, {1352, 18124}, {1353, 19129}, {1437, 6905}, {1439, 1443}, {1495, 10110}, {1498, 11455}, {1503, 15321}, {1506, 9696}, {1511, 12006}, {1587, 13440}, {1588, 13429}, {1593, 3426}, {1594, 6145}, {1595, 16659}, {1598, 3531}, {1656, 9703}, {1698, 9586}, {1699, 9587}, {1853, 38433}, {1879, 9378}, {1907, 16658}, {1976, 37334}, {2051, 9563}, {2071, 40647}, {2393, 32367}, {2574, 14709}, {2575, 14710}, {2616, 3737}, {2620, 7136}, {2623, 10097}, {2781, 34437}, {2886, 9701}, {2904, 34438}, {2929, 42016}, {2937, 10263}, {2981, 14818}, {3048, 5512}, {3068, 8995}, {3069, 13986}, {3090, 9306}, {3091, 9544}, {3147, 11433}, {3167, 7395}, {3292, 7550}, {3311, 6416}, {3312, 6415}, {3336, 3468}, {3357, 13452}, {3398, 36214}, {3431, 11438}, {3432, 32409}, {3448, 10116}, {3470, 38933}, {3471, 38935}, {3515, 11432}, {3517, 9777}, {3523, 13336}, {3524, 37515}, {3525, 15605}, {3526, 11935}, {3528, 37480}, {3529, 31371}, {3530, 13339}, {3532, 10605}, {3542, 14457}, {3545, 15749}, {3548, 18911}, {3563, 32692}, {3575, 34397}, {3580, 7542}, {3613, 11816}, {3627, 30531}, {3628, 40111}, {3796, 10323}, {3815, 9603}, {5050, 6391}, {5067, 5651}, {5085, 34817}, {5092, 41435}, {5093, 16195}, {5133, 12134}, {5157, 10519}, {5198, 14530}, {5254, 9604}, {5418, 9676}, {5422, 6642}, {5447, 15246}, {5462, 6153}, {5486, 35486}, {5494, 10693}, {5498, 15061}, {5504, 9730}, {5562, 34986}, {5587, 9622}, {5597, 12480}, {5598, 12481}, {5640, 7506}, {5643, 12099}, {5663, 11559}, {5721, 38850}, {5870, 10262}, {5871, 10261}, {5891, 41597}, {5898, 13363}, {5900, 25563}, {5907, 35500}, {5946, 13368}, {5972, 19481}, {6000, 14865}, {6030, 13391}, {6151, 14819}, {6198, 11429}, {6239, 12231}, {6240, 12233}, {6243, 7502}, {6400, 12232}, {6515, 41594}, {6561, 9677}, {6636, 10625}, {6640, 18952}, {6643, 37645}, {6644, 8907}, {6696, 16623}, {6794, 7765}, {6800, 7387}, {6815, 12318}, {6853, 18123}, {6875, 13323}, {7393, 15066}, {7505, 39571}, {7507, 32402}, {7514, 11444}, {7516, 7998}, {7517, 26881}, {7525, 37484}, {7526, 12111}, {7527, 12162}, {7529, 35264}, {7544, 34116}, {7547, 7699}, {7549, 41608}, {7574, 13470}, {7575, 16881}, {7576, 34782}, {7577, 16000}, {7689, 39242}, {7728, 11805}, {7731, 19362}, {8227, 9621}, {8562, 14380}, {8743, 40823}, {8795, 41204}, {8889, 38442}, {8918, 8930}, {8919, 8929}, {9140, 13561}, {9418, 39283}, {9590, 31760}, {9625, 31757}, {9652, 10895}, {9667, 10896}, {9729, 22962}, {9786, 14528}, {9818, 11441}, {9932, 15045}, {9971, 15582}, {10018, 13567}, {10024, 12370}, {10095, 13621}, {10202, 28787}, {10205, 35729}, {10226, 15055}, {10295, 13568}, {10299, 13347}, {10575, 12086}, {10601, 11465}, {10602, 11458}, {10721, 11744}, {10950, 40437}, {11004, 37478}, {11077, 41335}, {11202, 13472}, {11263, 38535}, {11264, 34826}, {11381, 13596}, {11403, 32063}, {11416, 15074}, {11439, 31861}, {11440, 18570}, {11443, 38263}, {11460, 19350}, {11461, 19354}, {11462, 19355}, {11463, 19356}, {11466, 19363}, {11467, 19364}, {11477, 19127}, {11491, 20986}, {11591, 34864}, {12007, 13622}, {12023, 12024}, {12041, 35498}, {12084, 15072}, {12106, 15019}, {12110, 40643}, {12112, 13474}, {12229, 12509}, {12230, 12510}, {12281, 19457}, {12282, 19458}, {12283, 19459}, {12284, 19456}, {12285, 19461}, {12286, 19462}, {12287, 19463}, {12288, 19464}, {12359, 41730}, {12834, 13365}, {13011, 13035}, {13012, 13036}, {13017, 19465}, {13018, 19466}, {13351, 37813}, {13364, 18369}, {13382, 21663}, {13420, 18368}, {13432, 15720}, {13433, 34565}, {13445, 13491}, {13488, 32111}, {13598, 37925}, {13754, 14118}, {13856, 38618}, {14070, 37493}, {14071, 25150}, {14106, 14111}, {14152, 26897}, {14220, 34210}, {14371, 14379}, {14491, 34417}, {14518, 34756}, {14542, 18533}, {14587, 18114}, {14641, 37944}, {14788, 37649}, {14805, 15091}, {15053, 16867}, {15056, 15068}, {15059, 24572}, {15093, 32448}, {15121, 15124}, {15232, 32381}, {15305, 32139}, {15328, 38936}, {15340, 27371}, {15401, 15537}, {15644, 22352}, {15646, 16665}, {15712, 26861}, {15760, 18433}, {16252, 16657}, {16766, 31674}, {16837, 34449}, {16868, 18390}, {17702, 34007}, {17711, 23329}, {18324, 37490}, {18374, 22336}, {18376, 40276}, {18474, 34799}, {18559, 34785}, {18874, 21308}, {18945, 32393}, {19123, 19125}, {19124, 39874}, {19136, 38005}, {19142, 22829}, {19151, 37473}, {19186, 19408}, {19187, 19409}, {19212, 33971}, {19349, 19368}, {19358, 19414}, {19359, 19415}, {19440, 19502}, {19441, 19503}, {19460, 22535}, {20190, 32599}, {20193, 30551}, {20421, 23040}, {21394, 30504}, {21849, 37939}, {22233, 41448}, {22330, 37953}, {22533, 32375}, {22950, 22972}, {23128, 26216}, {23293, 25738}, {24385, 36837}, {26877, 26889}, {26896, 26898}, {26916, 26920}, {31376, 34837}, {32110, 38448}, {32248, 39562}, {32249, 32251}, {32284, 37784}, {32321, 41715}, {32357, 34207}, {32661, 41334}, {32713, 42873}, {32737, 38394}, {33543, 37483}, {33695, 35909}, {33992, 35728}, {34351, 41596}, {34384, 39287}, {34418, 35467}, {34664, 41615}, {35480, 40242}, {35602, 37514}, {35724, 35885}, {37489, 38444}, {39808, 39810}, {39837, 39839}

X(54) = midpoint of X(i) and X(j) for these {i,j}: {1, 9905}, {3, 195}, {4, 12254}, {5, 36966}, {125, 14049}, {389, 40632}, {973, 11577}, {1263, 6343}, {1493, 10610}, {2929, 42016}, {3574, 10619}, {5889, 32338}, {6276, 6277}, {7691, 15801}, {11271, 12325}, {11597, 15089}, {12026, 31675}, {12291, 13423}, {12307, 12316}, {13368, 15532}, {17824, 32345}, {21660, 32352}, {27196, 27423}, {32333, 32341}, {32346, 32354}
X(54) = reflection of X(i) in X(j) for these {i,j}: {1, 12266}, {3, 10610}, {4, 3574}, {5, 8254}, {52, 10115}, {110, 11597}, {195, 1493}, {265, 11804}, {1141, 27196}, {1209, 6689}, {2888, 1209}, {2914, 32226}, {2917, 32391}, {3519, 21230}, {3574, 12242}, {6145, 32351}, {6152, 973}, {6153, 5462}, {6242, 32352}, {6288, 5}, {7691, 3}, {7728, 11805}, {7979, 1}, {9972, 9977}, {9977, 575}, {11412, 41590}, {12254, 10619}, {12300, 15739}, {12785, 10}, {12797, 402}, {13121, 10066}, {13122, 10082}, {13423, 6152}, {15062, 14130}, {15800, 20424}, {15801, 195}, {20424, 22051}, {21230, 140}, {21660, 40632}, {23061, 15137}, {32196, 143}, {32338, 12606}, {32352, 389}, {32379, 10274}, {33565, 125}, {41590, 12363}
X(54) = isogonal conjugate of X(5)
X(54) = isotomic conjugate of X(311)
X(54) = inverse-in-circumcircle of X(1157)
X(54) = complement of X(2888)
X(54) = anticomplement of X(1209)
X(54) = circumcircle-inverse of X(1157)
X(54) = Brocard-circle-inverse of X(18335)
X(54) = polar conjugate of X(324)
X(54) = antigonal image of X(33565)
X(54) = symgonal image of X(11597)
X(54) = Thomson-isogonal conjugate of X(35885)
X(54) = psi-transform of X(14656)
X(54) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1166, 8}, {2216, 2888}
X(54) = X(i)-complementary conjugate of X(j) for these (i,j): {3432, 10}, {40140, 21231}
X(54) = X(i)-Ceva conjugate of X(j) for these (i,j): {5, 2120}, {95, 97}, {97, 33629}, {275, 8882}, {288, 6}, {933, 23286}, {1141, 1157}, {1166, 25044}, {8884, 8883}, {14533, 26887}, {14587, 110}, {18315, 2623}, {18831, 15412}, {20574, 1614}, {23286, 19208}, {35196, 2169}, {39287, 95}
X(54) = X(i)-cross conjugate of X(j) for these (i,j): {3, 96}, {6, 275}, {184, 14533}, {186, 74}, {389, 4}, {523, 110}, {570, 2}, {1199, 1173}, {2594, 1}, {2623, 18315}, {3269, 39181}, {8603, 2981}, {8604, 6151}, {13366, 6}, {13367, 3}, {14533, 97}, {16030, 95}, {16035, 8884}, {19189, 1298}, {21638, 8795}, {21660, 3519}, {23286, 933}, {30451, 4558}, {32352, 6145}, {39199, 109}, {39201, 112}, {39478, 901}, {40632, 13418}, {41218, 654}
X(54) = cevapoint of X(i) and X(j) for these (i,j): {3, 1147}, {6, 184}, {15, 3200}, {16, 3201}, {32, 3202}, {39, 3203}, {55, 3204}, {58, 9563}, {61, 3205}, {62, 3206}, {215, 2245}, {523, 8901}, {572, 9562}, {654, 41218}, {3270, 9404}
X(54) = crosspoint of X(i) and X(j) for these (i,j): {1, 3461}, {3, 3463}, {4, 3459}, {5, 2121}, {95, 275}, {3467, 3469}, {3489, 3490}
X(54) = crosssum of X(i) and X(j) for these (i,j): {1, 3460}, {2, 17035}, {3, 195}, {4, 3462}, {11, 8819}, {51, 216}, {54, 2120}, {61, 8839}, {62, 8837}, {233, 3078}, {288, 38816}, {523, 8902}, {627, 628}, {1953, 7069}, {2600, 41218}, {3336, 3468}, {3470, 38933}, {3471, 38935}, {6368, 35442}, {8918, 8930}, {8919, 8929}, {12077, 41221}, {17434, 41219}, {21011, 21807}
X(54) = trilinear pole of line {50, 647}
X(54) = crossdifference of every pair of points on line {2081, 2600}
X(54) = trilinear pole of line X(50)X(647) (the polar of X(324) wrt polar circle)
X(54) = pole wrt polar circle of trilinear polar of X(324)
X(54) = X(48)-isoconjugate (polar conjugate) of X(324)
X(54) = X(92)-isoconjugate of X(216)
X(54) = intersection of tangents to hyperbola {A,B,C,X(4),X(5)}} at X(4) and X(3459)
X(54) = {X(2595),X(2596)}-harmonic conjugate of X(1087)
X(54) = trilinear product of vertices of circumnormal triangle
X(54) = intersection of tangents at X(3) and X(4) to Neuberg cubic K001
X(54) = exsimilicenter of nine-point circle and sine-triple-angle circle
X(54) = homothetic center of orthocevian triangle of X(3) and circumorthic triangle
X(54) = perspector of ABC and unary cofactor triangle of reflection triangle
X(54) = X(3)-of-reflection-triangle-of-X(5)
X(54) = perspector of ABC and cross-triangle of ABC and circumorthic triangle
X(54) = perspector of ABC and Hatzipolakis-Moses triangle
X(54) = X(191)-of-orthic-triangle if ABC is acute
X(54) = trilinear product of vertices of X(4)-altimedial triangle
X(54) = X(i)-isoconjugate of X(j) for these (i,j): {1, 5}, {2, 1953}, {6, 14213}, {7, 7069}, {8, 1393}, {10, 18180}, {19, 343}, {25, 18695}, {31, 311}, {37, 17167}, {38, 17500}, {48, 324}, {51, 75}, {52, 91}, {53, 63}, {54, 1087}, {69, 2181}, {76, 2179}, {79, 35194}, {81, 21011}, {86, 21807}, {92, 216}, {94, 2290}, {100, 21102}, {110, 2618}, {143, 2962}, {158, 5562}, {162, 6368}, {163, 18314}, {217, 1969}, {255, 13450}, {293, 39569}, {304, 3199}, {318, 30493}, {326, 14569}, {467, 1820}, {523, 2617}, {561, 40981}, {610, 13157}, {655, 2600}, {656, 35360}, {661, 14570}, {662, 12077}, {811, 15451}, {823, 17434}, {897, 41586}, {920, 8800}, {921, 41587}, {1154, 2166}, {1209, 2216}, {1263, 1749}, {1474, 42698}, {1568, 36119}, {1577, 1625}, {1707, 27364}, {1826, 16697}, {1895, 8798}, {1956, 32428}, {1972, 2313}, {1973, 28706}, {2081, 32680}, {2153, 33529}, {2154, 33530}, {2167, 36412}, {2180, 5392}, {2184, 42459}, {2222, 6369}, {2595, 7135}, {2596, 2603}, {2599, 3615}, {2621, 18114}, {2964, 25043}, {4560, 35307}, {4575, 23290}, {8769, 41588}, {10015, 35321}, {17438, 31610}, {18070, 35319}, {18833, 27374}, {20577, 36148}, {23181, 24006}, {24000, 35442}, {24041, 41221}, {27371, 34055}, {32678, 41078}, {36035, 36831}
X(54) = barycentric product X(i)*X(j) for these {i,j}: {1, 2167}, {3, 275}, {4, 97}, {6, 95}, {25, 34386}, {32, 34384}, {39, 39287}, {48, 40440}, {63, 2190}, {69, 8882}, {75, 2148}, {83, 16030}, {92, 2169}, {96, 1993}, {99, 2623}, {110, 15412}, {140, 288}, {182, 42300}, {184, 276}, {226, 35196}, {249, 8901}, {252, 1994}, {253, 33629}, {264, 14533}, {287, 19189}, {290, 41270}, {323, 1141}, {338, 14587}, {340, 11077}, {371, 16032}, {372, 16037}, {394, 8884}, {401, 1298}, {520, 16813}, {523, 18315}, {525, 933}, {571, 34385}, {577, 8795}, {578, 37872}, {647, 18831}, {648, 23286}, {662, 2616}, {801, 16035}, {850, 14586}, {1073, 38808}, {1092, 8794}, {1105, 19180}, {1157, 13582}, {1166, 37636}, {1502, 14573}, {1577, 36134}, {1634, 39182}, {2052, 19210}, {2245, 39277}, {2888, 40140}, {2984, 11245}, {3051, 41488}, {3431, 4993}, {3904, 36078}, {4551, 39177}, {6504, 8883}, {6563, 32692}, {7763, 41271}, {11140, 25044}, {13366, 31617}, {14096, 39283}, {14528, 19188}, {14618, 15958}, {15351, 19208}, {15414, 32713}, {19166, 41890}, {19170, 40448}, {19174, 28724}, {20574, 40684}, {22052, 39286}, {35311, 39181}, {37225, 39274}, {39201, 42405}
X(54) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 14213}, {2, 311}, {3, 343}, {4, 324}, {6, 5}, {15, 33529}, {16, 33530}, {24, 467}, {25, 53}, {31, 1953}, {32, 51}, {41, 7069}, {42, 21011}, {50, 1154}, {51, 36412}, {58, 17167}, {63, 18695}, {64, 13157}, {69, 28706}, {72, 42698}, {95, 76}, {96, 5392}, {97, 69}, {110, 14570}, {112, 35360}, {154, 42459}, {160, 41480}, {163, 2617}, {184, 216}, {186, 14918}, {187, 41586}, {213, 21807}, {232, 39569}, {251, 17500}, {252, 11140}, {275, 264}, {276, 18022}, {288, 40410}, {323, 1273}, {389, 34836}, {393, 13450}, {512, 12077}, {523, 18314}, {526, 41078}, {560, 2179}, {570, 1209}, {571, 52}, {577, 5562}, {604, 1393}, {647, 6368}, {649, 21102}, {654, 6369}, {661, 2618}, {850, 15415}, {933, 648}, {1141, 94}, {1157, 37779}, {1166, 40393}, {1173, 31610}, {1298, 1972}, {1333, 18180}, {1437, 16697}, {1501, 40981}, {1510, 20577}, {1576, 1625}, {1609, 41587}, {1843, 27371}, {1953, 1087}, {1971, 32428}, {1973, 2181}, {1974, 3199}, {1993, 39113}, {2148, 1}, {2167, 75}, {2168, 91}, {2169, 63}, {2174, 35194}, {2190, 92}, {2207, 14569}, {2501, 23290}, {2616, 1577}, {2623, 523}, {2963, 25043}, {2965, 143}, {3049, 15451}, {3053, 41588}, {3124, 41221}, {3202, 40588}, {3269, 35442}, {3284, 1568}, {3518, 14129}, {5063, 5891}, {6748, 14978}, {8648, 2600}, {8739, 6117}, {8740, 6116}, {8770, 27364}, {8795, 18027}, {8882, 4}, {8883, 6515}, {8884, 2052}, {8901, 338}, {9409, 14391}, {10311, 39530}, {10312, 30506}, {11077, 265}, {13338, 13364}, {13342, 27355}, {13366, 233}, {14270, 2081}, {14533, 3}, {14573, 32}, {14575, 217}, {14579, 1263}, {14585, 418}, {14586, 110}, {14587, 249}, {14642, 8798}, {15109, 21230}, {15412, 850}, {15958, 4558}, {16029, 1591}, {16030, 141}, {16032, 34391}, {16034, 1592}, {16035, 13567}, {16037, 34392}, {16813, 6528}, {18315, 99}, {18353, 565}, {18831, 6331}, {19180, 41005}, {19189, 297}, {19208, 39352}, {19210, 394}, {19306, 1749}, {20574, 31626}, {21449, 9291}, {21461, 36300}, {21462, 36301}, {21741, 2599}, {23195, 42445}, {23286, 525}, {25044, 1994}, {26887, 3164}, {32445, 42453}, {32640, 36831}, {32661, 23181}, {32692, 925}, {33629, 20}, {33872, 14845}, {34384, 1502}, {34386, 305}, {34397, 11062}, {34756, 39114}, {35196, 333}, {36078, 655}, {36134, 662}, {37636, 1225}, {38808, 15466}, {39109, 41536}, {39177, 18155}, {39201, 17434}, {39287, 308}, {40440, 1969}, {40633, 13595}, {41213, 41222}, {41270, 511}, {41271, 2165}, {41331, 27374}, {41373, 41481}, {41488, 40016}, {42293, 34983}, {42300, 327}
X(54) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 3460, 2599}, {2, 2888, 1209}, {2, 9545, 1147}, {2, 18912, 26917}, {3, 1493, 15801}, {3, 1993, 11412}, {3, 7592, 5890}, {3, 11402, 7592}, {3, 12161, 5889}, {3, 12316, 12307}, {3, 15087, 6102}, {3, 16266, 2979}, {3, 19210, 97}, {3, 25044, 1157}, {3, 32046, 5012}, {4, 184, 1614}, {4, 275, 4994}, {4, 578, 15033}, {4, 1614, 14157}, {4, 19467, 12289}, {5, 49, 110}, {5, 567, 13434}, {5, 9706, 9705}, {5, 14516, 41171}, {6, 24, 3567}, {6, 2917, 973}, {6, 14533, 8882}, {6, 14585, 10312}, {6, 15073, 8537}, {6, 19189, 9792}, {6, 19357, 24}, {6, 19468, 6152}, {24, 12291, 12380}, {24, 19357, 11464}, {25, 9707, 26882}, {26, 36749, 3060}, {36, 35197, 7356}, {49, 110, 9705}, {49, 567, 5}, {51, 3518, 38848}, {51, 10282, 3518}, {52, 18475, 7488}, {110, 9706, 49}, {110, 13434, 5}, {140, 11245, 26879}, {140, 40631, 252}, {143, 5944, 2070}, {154, 10982, 10594}, {155, 7503, 11459}, {155, 37506, 7503}, {182, 1092, 631}, {184, 275, 26887}, {184, 578, 4}, {184, 3574, 32379}, {184, 11424, 6759}, {184, 15033, 14157}, {185, 3520, 74}, {185, 11430, 3520}, {186, 1199, 389}, {195, 5012, 10203}, {195, 10610, 7691}, {195, 12307, 12316}, {217, 1970, 112}, {275, 8884, 4}, {275, 38808, 8884}, {288, 20574, 1173}, {323, 37126, 1216}, {378, 1181, 6241}, {381, 9704, 156}, {389, 13366, 1199}, {389, 13367, 186}, {389, 21660, 6242}, {394, 7509, 7999}, {394, 37476, 7509}, {427, 31804, 34224}, {569, 1147, 2}, {578, 6759, 11424}, {578, 10274, 3574}, {627, 628, 1273}, {631, 14912, 18916}, {973, 6152, 7730}, {973, 32391, 24}, {1157, 25042, 3}, {1173, 38848, 51}, {1173, 39667, 288}, {1181, 11425, 378}, {1209, 6689, 2}, {1216, 37513, 37126}, {1263, 10285, 31392}, {1493, 12363, 1993}, {1495, 10110, 34484}, {1498, 35502, 11455}, {1511, 36153, 12006}, {1593, 11456, 12290}, {1593, 19347, 11456}, {1594, 6146, 25739}, {1614, 15033, 4}, {1899, 37119, 23294}, {1994, 7488, 52}, {2070, 14627, 143}, {2595, 2596, 1087}, {2917, 13423, 12380}, {2937, 10263, 15107}, {3091, 9544, 10539}, {3518, 37505, 1173}, {3520, 15032, 185}, {3541, 6776, 11457}, {3567, 7730, 973}, {3567, 11464, 24}, {3567, 12291, 6152}, {3567, 13423, 7730}, {3567, 19468, 12380}, {3574, 21659, 32365}, {3796, 37498, 10323}, {5012, 34148, 3}, {5422, 6642, 15024}, {5622, 32245, 32234}, {5889, 11422, 12161}, {5889, 19167, 19194}, {5890, 11423, 7592}, {5946, 15532, 13368}, {6102, 32136, 15087}, {6143, 33565, 14076}, {6146, 23292, 1594}, {6150, 18016, 3}, {6152, 11577, 12291}, {6640, 18952, 26913}, {6644, 36753, 15043}, {6750, 35717, 4}, {6759, 11424, 4}, {7526, 18445, 12111}, {7542, 13292, 3580}, {7547, 18396, 18394}, {7592, 11402, 11423}, {7592, 16030, 19168}, {7592, 32333, 32339}, {7699, 18394, 7547}, {7722, 32607, 74}, {7730, 12291, 13423}, {7730, 13423, 6152}, {8254, 20585, 36966}, {8254, 36966, 6288}, {8882, 14533, 33629}, {9706, 13434, 110}, {9707, 11426, 9781}, {9730, 12038, 22467}, {9781, 26882, 25}, {9818, 11441, 15058}, {9905, 12266, 7979}, {10066, 10082, 1}, {10274, 12254, 1614}, {10282, 37505, 51}, {10605, 35477, 11468}, {10619, 12242, 4}, {10984, 13346, 376}, {11402, 16030, 19170}, {11402, 32333, 32341}, {11425, 17809, 1181}, {11427, 18925, 4}, {11427, 32354, 3574}, {11430, 15032, 74}, {11449, 15043, 6644}, {11464, 13423, 2917}, {12006, 36153, 15037}, {12038, 22467, 15035}, {12161, 12606, 15801}, {12227, 32607, 7722}, {12234, 23358, 32352}, {13121, 13122, 7979}, {13198, 15463, 74}, {13353, 22115, 140}, {13366, 13367, 389}, {13366, 21660, 12234}, {13367, 21638, 19185}, {13367, 32352, 23358}, {13621, 15038, 10095}, {13630, 43394, 3}, {14389, 14516, 5}, {16029, 16034, 6}, {16030, 16035, 3}, {16030, 19170, 19209}, {16031, 16036, 19}, {16032, 16037, 2}, {16035, 19210, 8883}, {17821, 17846, 2917}, {17928, 36752, 15045}, {18388, 21659, 4}, {18570, 34783, 11440}, {18925, 32346, 12254}, {19095, 19096, 6}, {19172, 19180, 19206}, {19459, 39588, 12283}, {21638, 21660, 19207}, {37481, 37814, 15053}


X(55) = INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE)

Trilinears    a(b + c - a) : b(c + a - b) : c(a + b - c)
Trilinears    1 + cos A : 1 + cos B : 1 + cos C
Trilinears    cos2(A/2) : cos2(B/2) : cos2(B/2)
Trilinears    tan(B/2) + tan(C/2) : tan(C/2) + tan(A/2) : tan(A/2) + tan(B/2)
Trilinears    a(a - s) : b(b - s) : c(c - s)
Trilinears    a(cot A/2) : :
Trilinears    a2/(1 - cos A) : :
Trilinears    a(2ar - S) : :
Barycentrics   a2(b + c - a) : b2(c + a - b) : c2(a + b - c) : :
Barycentrics   area(A'BC) : : , where A'B'C' = 1st circumperp triangle
X(55) = R*X(1) + r*X(3)
X(55) = (Ra+Rb+Rc)*X(1) + r*Ja + r*Jb + r*Jc, where Ja, Jb, Jc are excenters, and Ra, Rb, Rc are the exradii

X(55) = center of homothety of three triangles:   tangential, intangents, and extangents. Also, X(55) is the pole-with-respect-to-the-circumcircle of the trilinear polar of X(1). These properties and others are given in

O. Bottema and J. T. Groenman, "De gemene raaklijnen van de vier raakcirkels van een driehoek, twee aan twee," Nieuw Tijdschrift voor Wiskunde 67 (1979-80) 177-182.

Let A', B', C' be the second points of intersection of the angle bisectors of triangle ABC with its incircle. Let Oa be the circle externally tangent to the incircle at A', and internally tangent to the circumcircle; define Ob and Oc cyclically. Then X(55) is the radical center of circles Oa, Ob, Oc. Let A″ be the touchpoint of Oa and the circumcircle, and define B″ and C″ cyclically. The lines A'A″, B'B″, C'C″ concur in X(55). Let Ba, Ca be the intersections of lines CA, AB, respectively, and the antiparallel to BC through a point P. Define Cb, Ab, Ac, Bc cyclically. Triangles ABaCa, AbBCb, AcBcC are congruent only when P = X(55) or one of its 3 extraversions. Let A*B*C* be the incentral triangle. Let La be the reflection of line BC in line AA*, and define Lb and Lc cyclically. Let A''' = Lb∩Lc, and define B''' and C'''. The lines A*A''', B*B''', C*C''' concur in X(55). (Randy Hutson, November 18, 2015)

Let A'B'C' be the extouch triangle. Let A″ be the barycentric product of the circumcircle intercepts of line B'C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(55). (Randy Hutson, July 31 2018)

Let (Oa) be the circumcircle of BCX(1). Let Pa be the perspector of (Oa). Let La be the polar of Pa wrt (Oa). Define Lb and Lc cyclically. The lines La, Lb, Lc concur in X(55). (Randy Hutson, July 31 2018)

X(55) lies on these lines: 1,3   2,11   4,12   5,498   6,31   7,2346   8,21   9,200   10,405   15,203   16,202   19,25   20,388   30,495   34,227   41,220   43,238   45,678   48,154   63,518   64,73   77,1037   78,960   81,1002   92,243   103,109   104,1000   108,196   140,496   181,573   182,613  183,350   184,215   192,385   199,1030   201,774   204,1033   219,284   223,1456   226,516   255,601   256,983   329,1005   376,1056   386,595   392,997   411,962   511,611   515,1012   519,956   574,1015   603,963   631,1058   650,884   654,926   748,899   840,901   846,984   869,893   1026,1083   1070,1076   1072,1074   2195,5452  

X(55) is the {X(1),X(3)}-harmonic conjugate of X(56). For a list of other harmonic conjugates of X(55), click Tables at the top of this page.

X(55) = reflection of X(i) in X(j) for these (i,j): (1478,495), (2099,1)
X(55) = isogonal conjugate of X(7)
X(55) = isotomic conjugate of X(6063)
X(55) = complement of X(3434)
X(55) = anticomplement of X(2886)
X(55) = centroid of curvatures of circumcircle and excircles
X(55) = circumcircle-inverse of X(1155)
X(55) = antigonal conjugate of polar conjugate of X(37767)
X(55) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,6), (3,198), (7,218), (8,219), (9,220), (21,9), (59,101), (104,44), (260,259), (284,41)
X(55) = cevapoint of X(42) and X(228) for these (i,j)
X(55) = X(i)-cross conjugate of X(j) for these (i,j): (41,6), (42,33), (228,212)
X(55) = crosspoint of X(i) and X(j) for these (i,j): (1,9), (3,268), (7,277), (8,281), (21,284), (59,101)
X(55) = crosssum of X(i) and X(j) for these (i,j): (1,57), (2,145), (4,196), (11,514), (55,218), (56,222), (63,224), (65,226), (81,229), (177,234), (241,1362), (513,1086), (905,1364), (1361,1465)
X(55) = crossdifference of every pair of points on line X(241)X(514)
X(55) = X(i)-Hirst inverse of X(j) for these (i,j): (6,672), (43,241)
X(55) = X(1)-line conjugate of X(241)
X(55) = X(i)-beth conjugate of X(j) for these (i,j): (21,999), (55,31), (100,55), (200,200), (643,2), (1021,1024)
X(55) = insimilicenter of the intangents and extangents circles
X(55) = insimilicenter of the intangents and tangential circles
X(55) = exsimilicenter of then extangents and tangential circles
X(55) = X(22)-of-intouch-triangle
X(55) = trilinear pole of line X(657)X(663) (polar of X(331) wrt polar circle)
X(55) = pole wrt polar circle of trilinear polar of X(331)
X(55) = X(48)-isoconjugate (polar conjugate) of X(331)
X(55) = homothetic center of ABC and Mandart-incircle triangle
X(55) = inverse-in-Feuerbach-hyperbola of X(1001)
X(55) = inverse-in-circumconic-centered-at-X(1) of X(1936)
X(55) = {X(1),X(40)}-harmonic conjugate of X(65)
X(55) = trilinear square of X(259)
X(55) = Danneels point of X(100)
X(55) = vertex conjugate of PU(48)
X(55) = vertex conjugate of foci of Mandart inellipse
X(55) = excentral isotomic conjugate of X(2942)
X(55) = homothetic center of the reflections of the intangents and extangents triangles in their respective Euler lines
X(55) = perspector of ABC and extraversion triangle of X(56)
X(55) = trilinear product of PU(104)
X(55) = barycentric product of PU(112)
X(55) = bicentric sum of PU(112)
X(55) = PU(112)-harmonic conjugate of X(650)
X(55) = perspector of ABC and unary cofactor triangle of 7th mixtilinear triangle
X(55) = perspector of 4th mixtilinear triangle and unary cofactor triangle of 7th mixtilinear triangle
X(55) = perspector of unary cofactor triangles of 3rd, 4th and 5th extouch triangles
X(55) = {X(3513),X(3514)}-harmonic conjugate of X(56)
X(55) = perspector of ABC and cross-triangle of ABC and extangents triangle
X(55) = perspector of ABC and cross-triangle of ABC and Hutson extouch triangle
X(55) = homothetic center of ABC and cross-triangle of ABC and 2nd Johnson-Yff triangle
X(55) = Thomson-isogonal conjugate of X(5657)
X(55) = homothetic center of midarc triangle and 2nd-circumperp-of-1st-circumperp triangle (which is also 1st-circumperp-of-2nd-circumperp triangle)
X(55) = homothetic center of 2nd midarc triangle and 1st-circumperp-of-1st-circumperp triangle (which is also 2nd-circumperp-of-2nd-circumperp triangle)
X(55) = Cundy-Parry Phi transform of X(942)
X(55) = Cundy-Parry Psi transform of X(943)
X(55) = X(4)-of-1st-Johnson-Yff-triangle
X(55) = homothetic center of anti-Hutson intouch triangle and anti-tangential midarc triangle
X(55) = barycentric product of circumcircle intercepts of excircles radical circle


X(56) = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE)

Trilinears    a/(b + c - a) : b/(c + a - b) : c/(a + b - c)
Trilinears    1 - cos A : 1 - cos B : 1 - cos C
Trilinears    sin2(A/2) : sin2(B/2) : sin2(C/2)
Trilinears    a(tan A/2) : :
Trilinears    Ra - r : Rb - r : Rc - r, where Ra, Rb, Rc are the exradii
Trilinears    a*Ra : b*Rb : c*Rc, where Ra, Rb, Rc are the exradii
Trilinears    a cos A - (c + a) cos B - (a + b) cos C : :
Barycentrics    a2/(b + c - a) : b2/(c + a - b) : c2/(a + b - c)
Barycentrics    area(A'BC) : : , where A'B'C' = 2nd circumperp triangle
X(56) = R*X(1) - r*X(3)

X(56) is the perspector of the tangential triangle and the reflection of the intangents triangle in X(1).

Let A'B'C' be the Fuhrmann triangle. Let La be the line through A' parallel to BC, and define Lb and Lc cyclically. Let A″ = Lb∩Lc, B″ = Lc∩La, C″ = La∩Lb. The lines AA″, BB″, CC″ concur in X(56). Also, let AaBaCa be the poristic triangle (i.e., a triangle with common circumcircle and incircle as ABC) such that BaCa is parallel to BC. Define AbBbCb and AcBcCc cyclically. The lines AAa, BBb, CCc concur in X(56). (Randy Hutson, November 18, 2015)

Let A'B'C' be the intouch triangle. Let A″ be the barycentric product of the circumcircle intercepts of line B'C', and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X(56). (Randy Hutson, June 27, 2018)

See Dasari Naga Vijay Krishna, On a Conic Through Twelve Notable Points, Int. J. Adv. Math. and Mech. 7(2) (2019) 1-15.

If you have Geometer's Sketchpad, X(56).
If you have GeoGebra, you can view X(56).

In the plane of a triangle ABC, let
A'B'C' = circumcevian triangle of X(1);
Ta = line tangent to circumcircle at A', and define Tb and Tc cyclically;
Va = Tb∩Tc, and define Vb and Vc cyclically.
The triangle VaVbVc is perspective to ABC, and the perspector is X(56).
(Dasari Naga Vijay Krishna, June 19, 2021)

In the plane of a triangle ABC, let (Oa) be the circle tangent internally to the incircle and tangent internally to the circumcircle at A. Define (Ob) and (Oc) cyclically. The radical center of (Oa), (Ob), (Oc) is X(56). (Ivan Pavlov, February 24, 2022)

X(56) lies on these lines: 1,3   2,12   4,11   5,499   6,41   7,21   8,404   10,474   19,207   20,497   22,977   25,34   28,278   30,496   31,154   32,1015   33,963   38,201   58,222   61,202   62,203   63,960   72,997   77,1036   78,480   81,959   85,870   87,238   100,145   101,218   105,279   106,109   140,495   181,386   182,611   197,227   212,939   219,579   220,672   223,937   226,405   255,602   266,289   269,738   330,385   376,1058   411,938   511,613   551,553   607,911   631,1056   667,764   946,1012   978,979   1025,1083   1070,1074   1072,1076   1345,2464

X(56) is the {X(1),X(3)}-harmonic conjugate of X(55). For a list of other harmonic conjugates of X(56), click Tables at the top of this page.

X(56) = midpoint of X(1) and X(46)
X(56) = reflection of X(i) in X(j) for these (i,j): (1479,496), (2098,1)
X(56) = isogonal conjugate of X(8)
X(56) = isotomic conjugate of X(3596)
X(56) = complement of X(3436)
X(56) = anticomplement of X(1329)
X(56) = circumcircle-inverse of X(1319)
X(56) = antigonal conjugate of X(17101)
X(56) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,221), (7,222), (28,34), (57,6), (59,109), (108,513)
X(56) = X(31)-cross conjugate of X(6)
X(56) = crosspoint of X(i) and X(j) for these (i,j): (1,84), (7,278), (28,58), (57,269), (59,109)
X(56) = crosssum of X(i) and X(j) for these (i,j): (1,40), (2,144), (6,197), (9,200), (10,72), (11,522), (55,219), (175,176), (519,1145)
X(56) = crossdifference of every pair of points on line X(522)X(650)
X(56) = X(i)-Hirst inverse of X(j) for these (i,j): (6,1458), (34,1430), (57,1429), (604,1428), (1416,1438)
X(56) = X(266)-aleph conjugate of X(1050)
X(56) = X(i)-beth conjugate of X(j) for these (i,j): (1,1), (21,3), (56,1106), (58,56), and (P,57) for all P on the circumcircle
X(56) = homothetic center of the intouch triangle and the circumcevian triangle of X(1)
X(56) = trilinear pole of line X(649)X(854) (the isogonal conjugate of the isotomic conjugate of the Gergonne line)
X(56) = homothetic center of ABC and the reflection of the Mandart-incircle triangle in X(1)
X(56) = {X(1),X(40)}-harmonic conjugate of X(3057)
X(56) = {X(1),X(57)}-harmonic conjugate of X(65)
X(56) = trilinear square of X(266)
X(56) = trilinear square root of X(1106)
X(56) = X(92)-isoconjugate of X(219)
X(56) = vertex conjugate of PU(93)
X(56) = inverse-in-{circumcircle, incircle}-inverter of X(3660)
X(56) = perspector of ABC and extraversion triangle of X(55)
X(56) = perspector of ABC and unary cofactor triangle of Gemini triangle 15
X(56) = homothetic center of intangents triangle and reflection of tangential triangle in X(3)
X(56) = homothetic center of tangential triangle and reflection of intangents triangle in X(1)
X(56) = Brianchon point (perspector) of inellipse that is isogonal conjugate of isotomic conjugate of incircle
X(56) = pole wrt polar circle of trilinear polar of X(7017) (line X(2804)X(4397))
X(56) = X(48)-isoconjugate (polar conjugate) of X(7017)
X(56) = barycentric product of PU(46)
X(56) = bicentric sum of PU(60)
X(56) = PU(60)-harmonic conjugate of X(650)
X(56) = trilinear product of PU(92)
X(56) = perspector of ABC and cross-triangle of ABC and Apus triangle
X(56) = perspector of ABC and cross-triangle of ABC and Hutson intouch triangle
X(56) = homothetic center of ABC and cross-triangle of ABC and 1st Johnson-Yff triangle
X(56) = homothetic center of midarc triangle and 1st-circumperp-of-1st-circumperp triangle (which is also 2nd-circumperp-of-2nd-circumperp triangle)
X(56) = homothetic center of 2nd midarc triangle and 2nd-circumperp-of-1st-circumperp triangle (which is also 1st-circumperp-of-2nd-circumperp triangle)
X(56) = Cundy-Parry Phi transform of X(517)
X(56) = Cundy-Parry Psi transform of X(104)
X(56) = {X(3513),X(3514)}-harmonic conjugate of X(55)
X(56) = X(4)-of-2nd-Johnson-Yff-triangle
X(56) = homothetic center of tangential triangle and anti-tangential midarc triangle
X(56) = Ursa-major-to-Ursa-minor similarity image of X(4)
X(56) = barycentric product of (nonreal) circumcircle intercepts of the Gergonne line


X(57) = ISOGONAL CONJUGATE OF X(9)

Trilinears    1/(b + c - a) : 1/(c + a - b) : 1/(a + b - c)
Trilinears    tan(A/2) : tan(B/2) : tan(C/2)
Trilinears    1 + cos B + cos C - cos A
Trilinears    1 + sin(A/2)csc(B/2)csc(C/2) : :
Trilinears    cos2(B/2) + cos2(C/2) - cos2(A/2) ::
Trilinears    SA - bc : SB - ca : SC - ab : :
Trilinears    csc A - cot A : :
Trilinears    (1 - cos A) csc A : :
Trilinears    b(cot B/2) + c(cot C/2) - a(cot A/2) : :
Trilinears    cot A' : :, where A'B'C' is the excentral triangle
Trilinears    |AA'|/|AX(1)| : |BB'|/|BX(1)| : |CC'|/|CX(1)|, where A'B'C' is the excentral triangle
Trilinears    Ra : Rb : Rc, where Ra, Rb, Rc are the exradii
Barycentrics: Ra - r : Rb - r : Rc - r, where Ra, Rb, Rc are the exradii
Barycentrics    a/(b + c - a) : b/(c + a - b) : c/(a + b - c)
Barycentrics    1 - cos A : 1 - cos B : 1 - cos C
Barycentrics    area(A'BC) : : , where A'B'C' = excentral triangle
X(57) = (Ra + Rb + Rc)*X(1) - r*Ja - r*Jb - r*Jc, where Ja, Jb, Jc are excenters, and Ra, Rb, Rc are the exradii

Let Ja, Jb, Jc be the excenters and I the incenter of ABC. Let Ka be the symmedian point of JbJcI, and define Kb, Kc cyclically. Then KaKbKc is perspective to ABC at X(57). (Randy Hutson, September 14, 2016)

Let A' be the perspector of the circumconic centered at the A-excenter, and define B' and C'cyclically. The lines AA', BB', CC' concur in X(57). (Randy Hutson, September 14, 2016)

Let A'B'C' be the mixtilinear incentral triangle. Let A″ be the trilinear pole of line B'C', and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X(57). (Randy Hutson, September 14, 2016)

Let A' be the perspector of the A-mixtilinear incircle, and define B' and C'cyclically. The lines AA', BB', CC' concur in X(57). (Randy Hutson, September 14, 2016)

Let A', B' and C'be the inverse-in-{circumcircle, incircle}-inverter of A, B, C. Let A″B″C″ be the tangential triangle of A'B'C'. A″B″C″ is perspective to the intouch triangle at X(57). (Randy Hutson, September 14, 2016)

Let A'B'C' be the orthic triangle. Let La be the reflection of line B'C' in the internal angle bisector of A, and define Lb and Lc cyclically. Let A″ = Lb∩Lc, B″ = Lc∩La, C″ = La∩Lb. Triangle A″B″C″ is homothetic to ABC, with center of homothety X(57). (Randy Hutson, September 14, 2016)

Let Oa be the circle passing through B and C, and tangent to the incircle. Define Ob and Oc cyclically. Let A' be the point of tangency of Oa and the incircle, and define B' and C' cyclically. Triangle A'B'C' is perspective to the intouch triangle at X(57). Also, X(57) is the radical center of circles Oa, Ob, Oc. (Randy Hutson, July 31 2018)

Let A'B'C' be the intouch triangle. Let A″ be the trilinear product of the circumcircle intercepts of line B'C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(57). (Randy Hutson, July 31 2018)

Let A1B1C1 be Gemini triangle 1. Let A' be the perspector of conic {A,B,C,B1,C1}}, and define B' and C' cyclically. Triangle A'B'C' is the tangential of excentral triangle. The lines AA', BB', CC' concur in X(57). (Randy Hutson, January 15, 2019)

Let Va, Vb, Vc be the antipodes of V=X(40) in the circles (VBC), (VCA), (VAB), respectively. The lines AVa, BVb, CVc concur in X(57). (Angel Montesdeoca, October 14, 2019)

Let DEF be the intouch triangle. Let Ia be the internal bisector of angle BAC, and let D' be the point, other than D, where the line through D parallel to Ia meets the incircle. Let A' be the point, other than A, where AD' meets the incircle. Let La be the radical axis of the circumcircles of triangles A'BF and A'CE, and define Lb and Lc cyclically. The lines La, Lb, Lc concur in X(57). See X(57). (Angel Montesdeoca, December 21, 2019)

In the plane of a triangle ABC, let
A'B'C' = excentral triangle;
Ba = reflection of B in A', and define Cb and Ac cyclically;
Ca = reflection in C in A', and define Ab and Bc cyclically;
Va = AcBc∩CbAb, and define Vb and Vc cyclically.
The triangle VaVbVc is perspective to ABC, and the perspector is X(57).
(Dasari Naga Vijay Krishna, June 23, 2021)

X(57) is the perspector of the intouch triangle and excentral triangle.

X(57) lies on the Thomson cubic and these lines: 1,3   2,7   4,84   6,222   10,388   19,196   20,938   27,273   28,34   31,105   33,103   38,612   42,1002   43,181   72,474   73,386   77,81   78,404   79,90   85,274   88,651   92,653   164,177   169,277   173,174   200,518   201,975   234,362   239,330   255,580  279,479   282,3343   345,728   497,516   499,920   649,1024   658,673   748,896   758,997   955,991   957,995   959,1042   961,1106   978,1046   1020,1086   1073,3351   3342,3350

X(57) is the {X(2),X(7)}-harmonic conjugate of X(226). For a list of other harmonic conjugates of X(57), click Tables at the top of this page.

X(57) = midpoint of X(497) and X(3474)
X(57) = reflection of X(i) in X(j) for these (i,j): (1,999), (200,1376)
X(57) = isogonal conjugate of X(9)
X(57) = isotomic conjugate of X(312)
X(57) = complement of X(329)
X(57) = anticomplement of X(3452)
X(57) = circumcircle-inverse of X(2078)
X(57) = incircle-inverse of X(3660)
X(57) = Bevan-circle-inverse of X(1155)
X(57) = trilinear product of PU(46)
X(57) = antigonal conjugate of polar conjugate of X(37769)
X(57) = trilinear pole of PU(96) (line X(513)X(663), the polar of X(318) wrt polar circle, and the Monge line of the mixtilinear incircles)
X(57) = barycentric product of PU(94)
X(57) = pole wrt polar circle of trilinear polar of X(318)
X(57) = X(48)-isoconjugate (polar conjugate) of X(318)
X(57) = X(6)-isoconjugate of X(8)
X(57) = X(75)-isoconjugate of X(41)
X(57) = X(92)-isoconjugate of X(212)
X(57) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,223), (7,1), (27,278), (81,222), (85,77), (273,34), (279,269)
X(57) = cevapoint of X(i) and X(j) for these (i,j): (6,56), (19,208)
X(57) = X(i)-cross conjugate of X(j) for these (i,j): (6,1), (19,84), (56,269), (65,7)
X(57) = crosspoint of X(i) and X(j) for these (i,j): (1,3062), (2,189), (7,279), (27,81), (85,273), (1014,1434), (1659,13390)
X(57) = crosssum of X(i) and X(j) for these (i,j): (1,165), (6,198), (31,205), (37,71), (41,212), (55,220), (210,1334), (2066,5414)
X(57) = crossdifference of every pair of points on line X(650)X(663)
X(57) = X(i)-Hirst inverse of X(j) for these (i,j): (1,241), (7,1447), (56,1429), (105,1462), (910,1419)
X(57) = perspector of ABC and unary cofactor triangle of 6th mixtilinear triangle
X(57) = perspector of ABC and antipedal triangle of X(40)
X(57) = homothetic center of: ABC; orthic triangle of intouch triangle; tangential triangle of excentral triangle
X(57) = X(25)-of-excentral-triangle
X(57) = X(25)-of-intouch-triangle
X(57) = pole wrt Bevan circle of antiorthic axis
X(57) = perspector of Bevan circle
X(57) = perspector of circumconic centered at X(223)
X(57) = center of circumconic that is locus of trilinear poles of lines passing through X(223)
X(57) = perspector of pedal and antipedal (or anticevian) triangles of X(1)
X(57) = perspector of ABC and medial triangle of pedal triangle of X(84)
X(57) = inverse-in-circumconic-centered-at-X(9) of X(3911)
X(57) = orthocorrespondent of X(1)
X(57) = Danneels point of X(7)
X(57) = vertex conjugate of X(55) and X(57)
X(57) = perspector of ABC and extraversion triangle of X(9)
X(57) = trilinear product of extraversions of X(9)
X(57) = SS(A→A') of X(63), where A'B'C' is the excentral triangle
X(57) = Cundy-Parry Phi transform of X(40)
X(57) = Cundy-Parry Psi transform of X(84)
X(57) = perspector of ABC and cross-triangle of Gemini triangles 9 and 10
X(57) = perspector of ABC and cross-triangle of ABC and Gemini triangle 9
X(57) = perspector of ABC and cross-triangle of ABC and Gemini triangle 10
X(57) = barycentric product of vertices of Gemini triangle 9
X(57) = barycentric product of vertices of Gemini triangle 10
X(57) = perspector of ABC and tangential triangle, wrt Gemini triangle 2, of {ABC, Gemini 2}-circumconic
X(57) = perspector of Gemini triangle 36 and cross-triangle of ABC and Gemini triangle 36
X(57) = perspector of ABC and unary cofactor triangle of Gemini triangle 36
X(57) = X(i)-aleph conjugate of X(j) for these (i,j): (2,40), (7,57), (57,978), (174,1), (366,165), (507,503), (508,9), (509,43)
X(57) = X(i)-beth conjugate of X(j) for these (i,j): (2,2), (81,57), (88,57), (100,57), (110,31), (162,57), (190,57), (333,63), (648,92), (651,57), (653,57), (655,57), (658,57), (660,57), (662,57), (673,57), (771,57), (799,57), (823,57), (897, 57)
X(57) = {X(1),X(3)}-harmonic conjugate of X(3601)
X(57) = {X(1),X(40)}-harmonic conjugate of X(1697)
X(57) = {X(2),X(63)}-harmonic conjugate of X(9)
X(57) = {X(55),X(56)}-harmonic conjugate of X(1617)
X(57) = {X(56),X(65)}-harmonic conjugate of X(1)
X(57) = {X(3513),X(3514)}-harmonic conjugate of X(1)


X(58) = ISOGONAL CONJUGATE OF X(10)

Trilinears    a/(b + c) : b/(c + a) : c/(a + b)
Trilinears    (1 - cos A)/(cos B + cos C) : :
Trilinears    sa2 + SR : sb2 + SR : sc2 + SR
Trilinears    r cos A - s sin A : : , where s = semiperimeter and r = inradius
Trilinears    sin(A - U) : : , U as at X(572) and X(573)
Trilinears    (R/r) - 1/(cos B + cos C) : :
Trilinears    (r/R) - cos 2A + 1 : :
Trilinears    eccentricity of A-Soddy ellipse : :
Barycentrics     a2/(b + c) : b2/(c + a) : c2/(a + b)

X(58) is the point of concurrence of the Brocard axes of triangles BIC, CIA, AIB, ABC, (where I denotes the incenter, X(1)), as proved in Antreas P. Hatzipolakis, Floor van Lamoen, Barry Wolk, and Paul Yiu, Concurrency of Four Euler Lines, Forum Geometricorum 1 (2001) 59-68.

Let (Sa) be the reflection of the Spieker circle in BC, and define (Sb), (Sc) cyclically. X(58) is the radical center of (Sa), (Sb), (Sc). (Randy Hutson, July 20, 2016)

Let A'B'C' be the anticomplement of the Feuerbach triangle. Let A″B″C″ be the circumcevian triangle, wrt A'B'C', of X(1). The lines AA″, BB″, CC″ concur in X(58). (Randy Hutson, July 20, 2016)

Let A'B'C' be the Feuerbach triangle. Let La be the tangent to the nine-point circle at A', and define Lb, Lc cyclically. Let A″ be the isogonal conjugate of the trilinear pole of La, and define B″, C″ cyclically. Let A* = BB″∩CC″, B* = CC″∩AA″, C* = AA″∩BB″. The lines AA*, BB*, CC* concur in X(58). (Randy Hutson, July 20, 2016)

Let A'B'C' be the 2nd circumperp triangle. Let A″ be the trilinear product B'*C', and define B″, C″ cyclically. A″, B″, C″ are collinear on line X(36)X(238) (the trilinear polar of X(81)). The lines AA″, BB″, CC″ concur in X(58). (Randy Hutson, July 20, 2016)

For an artistic design generated by X(58), see X(244).

X(58) lies on these lines: 1,21   2,540   3,6   7,272   8,996   9,975   10,171   20,387   25,967   27,270   28,34   29,162   35,42   36,60   40,601   41,609   43,979   46,998   56,222   65,109   82,596   84,990   86,238   87,978   99,727   101,172   103,112   106,110   229,244   269,1014   274,870   314,987   405,940   519,1043   942,1104   977,982   1019,1027

X(58) is the {X(3),X(6)}-harmonic conjugate of X(386). For a list of other harmonic conjugates of X(58), click Tables at the top of this page.

X(58) = isogonal conjugate of X(10)
X(58) = isotomic conjugate of X(313)
X(58) = complement of X(1330)
X(58) = anticomplement of X(3454)
X(58) = circumcircle-inverse of X(1326)
X(58) = Brocard-circle-inverse of X(386)
X(58) = X(i)-Ceva conjugate of X(j) for these (i,j): (81,284), (267,501), (270,28)
X(58) = cevapoint of X(6) and X(31)
X(58) = X(i)-cross conjugate of X(j) for these (i,j): (6,81), (36,106), (56,28), (513,109)
X(58) = crosspoint of X(i) and X(j) for these (i,j): (1,267), (21,285), (27,86), (60,270)
X(58) = crosssum of X(i) and X(j) for these (i,j): (1,191), (6,199), (12,201), (37,210), (42,71), (65,227), (594, 756)
X(58) = crossdifference of every pair of points on line X(523)X(661)
X(58) = X(6)-Hirst inverse of X(1326)
X(58) = antigonal conjugate of isogonal conjugate of X(1324)
X(58) = antigonal conjugate of isotomic conjugate of X(21277)
X(58) = antigonal conjugate of polar conjugate of X(37770)
X(58) = X(i)-beth conjugate of X(j) for these (i,j): (21,21), (60,58), (110,58), (162,58), (643,58), (1098,283)
X(58) = barycentric product of PU(31)
X(58) = trilinear pole of line X(649)X(834)
X(58) = {X(1),X(31)}-harmonic conjugate of X(595)
X(58) = {X(21),X(283)}-harmonic conjugate of X(2328)
X(58) = X(42)-isoconjugate of X(75)
X(58) = X(71)-isoconjugate of X(92)
X(58) = X(101)-isoconjugate of X(1577)
X(58) = homothetic center of 2nd circumperp triangle and 'Hatzipolakis-Brocard triangle' (A'B'C' as defined at X(5429))
X(58) = trilinear product of vertices of 2nd circumperp triangle
X(58) = perspector of 2nd circumperp triangle and unary cofactor triangle of 1st circumperp triangle
X(58) = perspector of ABC and cross-triangle of ABC and 2nd circumperp triangle
X(58) = Cundy-Parry Phi transform of X(573)
X(58) = Cundy-Parry Psi transform of X(13478)
X(58) = perspector of ABC and unary cofactor triangle of Gemini triangle 11
X(58) = {X(1),X(21)}-harmonic conjugate of X(4653)


X(59) = ISOGONAL CONJUGATE OF X(11)

Trilinears       1/[1 - cos(B - C)] : 1/[1 - cos(C - A)] : 1/[1 - cos(A - B)]
Trilinears    csc^2 (B/2 - C/2) : :
Barycentrics  a/[1 - cos(B - C)] : b/[1 - cos(C - A)] : c/[1 - cos(A - B)]

X(59) lies on these lines: 36,1110   60,1101   100,521   101,657   109,901   513,651   518,765   523,655

X(59) = isogonal conjugate of X(11)
X(59) = isotomic conjugate of X(34387)
X(59) = cevapoint of X(i) and X(j) for these (i,j): (55,101), (56,109), (1381,1382)
X(59) = X(i)-cross conjugate of X(j) for these (i,j): (1,110), (3,100), (55,101), (56,109), (182,1492)
X(59) = crossdifference of every pair of points on line X(4530)X(14393)
X(59) = X(765)-beth conjugate of X(765)
X(59) = trilinear pole of line X(101)X(109)
X(59) = perspector of ABC and the reflection of the intouch triangle in line X(1)X(3)
X(59) = perspector of ABC and extraversion triangle of X(60)
X(59) = X(75)-isoconjugate of X(3271)
X(59) = trilinear square of X(6733)
X(59) = complement of isogonal conjugate of X(36902)


X(60) = ISOGONAL CONJUGATE OF X(12)

Trilinears       1/[1 + cos(B - C)] : 1/[1 + cos(C - A)] : 1/[1 + cos(A - B)]
Trilinears    sec^2 (B/2 - C/2) : :
Barycentrics  a/[1 + cos(B - C)] : b/[1 + cos(C - A)] : c/[1+ cos(A - B)]

Let A'B'C' be the cevian triangle of X(21). Let A″, B″, C″ be the inverse-in-circumcircle of A', B', C'. The lines AA″, BB″, CC″ concur in X(60). (Randy Hutson, October 15, 2018)

X(60) lies on these lines: 1,110   21,960   28,81   36,58   59,1101   86,272   283,284   404,662   757,1014

X(60) = isogonal conjugate of X(12)
X(60) = isotomic conjugate of X(34388)
X(60) = anticomplement of X(34829)
X(60) = X(58)-cross conjugate of X(270)
X(60) = X(i)-beth conjugate of X(j) for these (i,j): (60,849), (1098,1098)
X(60) = crossdifference of every pair of points on line X(2610)X(4024)
X(60) = perspector of ABC and extraversion triangle of X(59)
X(60) = X(75)-isoconjugate of X(181)
X(60) = trilinear square of X(6727)
X(60) = complement of isogonal conjugate of X(36903)


X(61) = ISOGONAL CONJUGATE OF X(17)

Trilinears    sin(A + π/6) : sin(B + π/6) : sin(C + π/6)
Trilinears    cos(A - π/3) : cos(B - π/3) : cos(C - π/3)
Trilinears    cos A + sqrt(3) sin A : :
Barycentrics   sin A sin(A + π/6) : sin B sin(B + π/6) : sin C sin(C + π/6)

X(61) lies on the Napoleon cubic and these lines: 1,203   2,18   3,6   4,13   5,14   30,397   56,202   140,395   299,636   302,629   618,627

X(61) is the {X(3),X(6)}-harmonic conjugate of X(62). For a list of other harmonic conjugates of X(61), click Tables at the top of this page.

X(61) = reflection of X(62) in X(5007)
X(61) = reflection of X(633) in X(635)
X(61) = isogonal conjugate of X(17)
X(61) = isotomic conjugate of X(34389)
X(61) = complement of X(633)
X(61) = anticomplement of X(635)
X(61) = Brocard-circle-inverse of X(62)
X(61) = eigencenter of cevian triangle of X(14)
X(61) = eigencenter of anticevian triangle of X(16)
X(61) = X(14)-Ceva conjugate of X(16)
X(61) = crossdifference of every pair of points on line X(523)X(14446)
X(61) = crosspoint of X(302) and X(473)
X(61) = point of concurrence of Brocard axes of BCX(15), CAX(15), ABX(15)
X(61) = perspector of ABC and centers of circles used in construction of X(1337)
X(61) = X(61)-of-circumsymmedial-triangle
X(61) = orthocorrespondent of X(16)
X(61) = {X(15),X(62)}-harmonic conjugate of X(3)
X(61) = {X(371),X(372)}-harmonic conjugate of X(15)
X(61) = perspector of inner Napoleon triangle and orthocentroidal triangle
X(61) = Cundy-Parry Phi transform of X(15)
X(61) = Cundy-Parry Psi transform of X(13)
X(61) = Kosnita(X(15),X(3)) point
X(61) = Kosnita(X(15),X(15)) point
X(61) = antigonal conjugate of X(34219)


X(62) = ISOGONAL CONJUGATE OF X(18)

Trilinears    sin(A - π/6) : sin(B - π/6) : sin(C - π/6)
Trilinears    cos(A + π/3) : cos(B + π/3) : cos(C + π/3)
Trilinears    cos A - sqrt(3) sin A : :
Barycentrics    sin A sin(A - π/6) : sin B sin(B - π/6) : sin C sin(C - π/6)

X(62) lies the Napoleon cubic and these lines: 1,202   2,17   3,6   4,14   5,13   30,398   56,203   140,396   298,635   303,630   619,628

X(62) is the {X(3),X(6)}-harmonic conjugate of X(61). For a list of other harmonic conjugates of X(62), click Tables at the top of this page.

X(62) = reflection of X(61) in X(5007)
X(62) = reflection of X(634) in X(636)
X(62) = isogonal conjugate of X(18)
X(62) = isotomic conjugate of X(34390)
X(62) = complement of X(634)
X(62) = anticomplement of X(636)
X(62) = Brocard-circle-inverse of X(61)
X(62) = eigencenter of cevian triangle of X(13)
X(62) = eigencenter of anticevian triangle of X(15)
X(62) = X(13)-Ceva conjugate of X(15)
X(62) = crosspoint of X(303) and X(472)
X(62) = crossdifference of every pair of points on line X(523)X(14447)
X(62) = point of concurrence of Brocard axes of BCX(16), CAX(16), ABX(16)
X(62) = perspector of ABC and centers of circles used in construction of X(1338)
X(62) = X(62)-of-circumsymmedial-triangle
X(62) = orthocorrespondent of X(15)
X(62) = {X(16),X(61)}-harmonic conjugate of X(3)
X(62) = {X(371),X(372)}-harmonic conjugate of X(16)
X(62) = perspector of outer Napoleon triangle and orthocentroidal triangle
X(62) = Cundy-Parry Phi transform of X(16)
X(62) = Cundy-Parry Psi transform of X(14)
X(62) = Kosnita(X(16),X(3)) point
X(62) = Kosnita(X(16),X(16)) point
X(62) = antigonal conjugate of X(34220)


X(63) = ISOGONAL CONJUGATE OF X(19)

Trilinears    cot A : cot B : cot C
Trilinears    b2 + c2 - a2 : c2 + a2 - b2 : a2 + b2 - c2
Trilinears    SA : SB : SC
Trilinears    csc A - tan(A/2) : :
Trilinears    csc A - cot(A/2) : :
Trilinears    tan(A/2) - cot(A/2) : :
Trilinears    d(a,b,c) : : , where d(a,b,c) = directed distance from A to the orthic axis
Trilinears    2 csc 2A - tan A : :
Barycentrics    cos A : cos B : cos C
X(63) = (r + 2R)*X(1) - 3R*X(2) - 2r*X(3)    (Peter Moses, April 2, 2013)
X(63) = (cos A)*[A] + (cos B)*[B] + (cos C)*[C], where A, B, C denote angles and [A], [B], [C] denote vertices

Let Oa be the A-extraversion of the Conway circle (the circle centered at the A-excenter and passing through A, with radius sqrt(r_a^2 + s^2), where r_a is the A-exradius). Let Pa be the perspector of Oa, and La the polar of Pa wrt Oa. Define Lb and Lc cyclically. Let A' = Lb∩Lc, B' = Lc∩La, C' = La∩Lb. Triangle A'B'C' is perspective to the excentral triangle at X(63). (Randy Hutson, February 10, 2016)

Let A'B'C' be the 2nd Brocard triangle. Let A″ be the trilinear product B'*C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(63). (Randy Hutson, February 10, 2016)

Let A'B'C' be the hexyl triangle. Let A″ be the trilinear pole of line B'C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(63). (Randy Hutson, February 10, 2016)

Let A'B'C' be the side-triangle of ABC and hexyl triangle. Let A″ be the {B,C}-harmonic conjugate of A', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(63). (Randy Hutson, February 10, 2016)

Let A'B'C' be the excentral triangle. Let A″ be the isotomic conjugate, wrt triangle A'BC, of X(1). Define B″ and C″ cyclically. The lines A'A″, B'B″, C'C″ concur in X(63). (Randy Hutson, July 31 2018)

X(63) lies on these lines: 1,21   2,7   3,72   6,2221   8,20   10,46   19,27   33,1013   36,997   37,940   48,326   55,518   56,960   65,958   69,71   77,219   91,921   100,103   162,204   169,379   171,612   190,312   194,239   201,603   210,1004   212,1040   213,980   220,241   223,651   238,614   240,1096   244,748   304,1102   318,412   354,1001   392,999   404,936   405,942   452,938   484,535   517,956   544,1018   561,799   654,918   750,756

X(63) is the {X(9),X(57)}-harmonic conjugate of X(2). For a list of other harmonic conjugates of X(63), click Tables at the top of this page.

X(63) = reflection of X(i) in X(j) for these (i,j): (1,993), (1478,10)
X(63) = isogonal conjugate of X(19)
X(63) = isotomic conjugate of X(92)
X(63) = complement of X(5905)
X(63) = anticomplement of X(226)
X(63) = anticomplementary conjugate of X(2893)
X(63) = X(i)-Ceva conjugate of X(j) for these (i,j): (7,224), (69,78), (75,1), (304,326), (333,2), (348,77)
X(63) = cevapoint of X(i) and X(j) for these (i,j): (3,219), (9,40), (48,255), (71,72)
X(63) = X(i)-cross conjugate of X(j) for these (i,j): (3,77), (9,271), (48,1), (71,3), (72,69), (219,78), (255,326)
X(63) = crosspoint of X(i) and X(j) for these (i,j): (69,348), (75,304)
X(63) = crosssum of X(25) and X(607)
X(63) = crossdifference of every pair of points on line X(661)X(663)
X(63) = trilinear product X(2)*X(3)
X(63) = trilinear product of PU(22)
X(63) = bicentric sum of PU(i) for these i: 128, 129
X(63) = PU(128)-harmonic conjugate of X(661)
X(63) = midpoint of PU(129)
X(63) = {X(1),X(1707)}-harmonic conjugate of X(31)
X(63) = {X(2),X(9)}-harmonic conjugate of X(3305)
X(63) = {X(2),X(57)}-harmonic conjugate of X(3306)
X(63) = {X(92),X(1748)}-harmonic conjugate of X(19)
X(63) = trilinear pole of line X(521)X(656)
X(63) = pole wrt polar circle of trilinear polar of X(158)
X(63) = X(48)-isoconjugate (polar conjugate) of X(158)
X(63) = X(i)-isoconjugate of X(j) for these {i,j}: {4,6}, {31,92}, {75,1973}
X(63) = excentral isogonal conjugate of X(1742)
X(63) = homothetic center of excentral triangle and anticomplement of the intouch triangle
X(63) = X(161)-of-intouch-triangle
X(63) = X(184)-of-excentral-triangle
X(63) = inverse-in-circumconic-centered-at-X(9) of X(908)
X(63) = trilinear square of X(5374)
X(63) = perspector of excentral triangle and Gemini triangle 2
X(63) = homothetic center of excentral triangle and Gemini triangle 30
X(63) = perspector of ABC and cross-triangle of Gemini triangles 35 and 36
X(63) = perspector of ABC and cross-triangle of ABC and Gemini triangle 35
X(63) = perspector of ABC and cross-triangle of ABC and Gemini triangle 36
X(63) = barycentric product of vertices of Gemini triangle 35
X(63) = barycentric product of vertices of Gemini triangle 36
X(63) = X(i)-aleph conjugate of X(j) for these (i,j):
(2,1), (75,63), (92,920), (99,662), (174,978), (190,100), (333,411), (366,43), (514,1052), (556,40), (648,162), (664,651), (668,190), (670,799), (671,897), (903,88)
X(63) = X(i)-beth conjugate of X(j) for these (i,j):
(63,222), (190,63), (333,57), (345,345), (643,63), (645,312), (662,223)
X(63) = perspector of ABC and extraversion triangle of X(63), which is also the anticevian triangle of X(63)


X(64) = ISOGONAL CONJUGATE OF X(20)

Trilinears    1/(cos A - cos B cos C) : 1/(cos B - cos C cos A) : 1/(cos C - cos A cos B)
Barycentrics    a/(cos A - cos B cos C) : b/(cos B - cos C cos A) : c/(cos C - cos A cos B)
Barycentrics    a^2/[3a^4 - 2a^2(b^2 + c^2) - (b^2 - c^2)^2] : :

A construction of X(64) appears in Lemoine's 1886 paper cited at X(19).

Let A'B'C' be the half-altitude triangle. Let La be the trilinear polar of A', and define Lb and Lc cyclically. Let A″ = Lb∩Lc, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(64). (Randy Hutson, November 18, 2015)

Let Oa be the circle with segment BC as diameter. Let A' be the perspector of Oa. Let La be the polar of A' wrt Oa. Define Lb and Lc cyclically. Let A″ = Lb∩Lc, B″ = Lc∩La, C″ = La∩Lb. The lines AA″, BB″, CC″ concur in X(64). (Randy Hutson, November 18, 2015)

Let A'B'C' be the cevian triangle of X(69). Let A″ be the orthocenter of AB'C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(64). (Randy Hutson, November 18, 2015)

Let A'B'C' be the reflection of the orthic triangle in X(4). Let A''B''C'' be the trangential triangle, with respect ot the orthic triangle, of the circumconic of the orthic triangle with center X(4); i.e., the bicevian conic of X(4) and X(459). Then X(64) is the perspector of A'B'C' and A''B''C''. (Randy Hutson, November 18, 2015)

The tangents at A, B, C to the Darboux cubic K004 concur in X(64). (Randy Hutson, November 18, 2015)

X(64) lies on the Darboux cubic and these lines: 1,3182   3,154   4,3183   6,185   20,69   24,74   30,68   33,65   40,72   54,378   55,73   71,198   84,3353   265,382   3345,3472   3346,3355

X(64) = reflection of X(1498) in X(3)
X(64) = isogonal conjugate of X(20)
X(64) = isotomic conjugate of X(14615)
X(64) = complement of X(6225)
X(64) = anticomplement of X(2883)
X(64) = circumcircle-inverse of X(11589)
X(64) = X(25)-cross conjugate of X(6)
X(64) = X(1)-beth conjugate of X(207)
X(64) = crosspoint of X(4) and X(3346)
X(64) = crosssum of X(3) and X(1498)
X(64) = perspector of hexyl triangle and anticevian triangle of X(2184)
X(64) = trilinear pole of line X(647)X(657)
X(64) = concurrence of normals to MacBeath circumconic at A, B, C
X(64) = isogonal conjugate, wrt tangential triangle of MacBeath circumconic (or anticevian triangle of X(3)), of X(1498)
X(64) = orthocenter of x(3)X(6)X(2435)
X(64) = orthology center of ABC and half-altitude triangle
X(64) = intersection of tangents at X(3) and X(4) to Thomson cubic K002
X(64) = intersection of tangents at X(20) and X(64) to Darboux cubic K004
X(64) = perspector of ABC and the reflection in X(3) of the antipedal triangle of X(3) (tangential triangle)
X(64) = perspector of ABC and circumcircle antipode of circumanticevian triangle of X(3)
X(64) = perspector of ABC and unary cofactor triangle of half-altitude triangle
X(64) = X(2136)-of-orthic-triangle if ABC is acute
X(64) = X(8905)-of-excentral-triangle
X(64) = X(3)-vertex conjugate of X(3)


X(65) = ORTHOCENTER OF THE INTOUCH TRIANGLE

Trilinears    cos B + cos C : cos C + cos A : cos A + cos B
Trilinears    (b + c)/(b + c - a) : (c + a)/(c + a - b) : (a + b)/(a + b - c)
Trilinears    sin(A/2) cos(B/2 - C/2) : sin(B/2) cos(C/2 - A/2) : sin(C/2) cos(A/2 - B/2)
Trilinears    Ra + r : Rb + r : Rc + r, where Ra, Rb, Rc are the exradii
Barycentrics   a(b + c)/(b + c - a) : b(c + a)/(c + a - b) : c(a + b)/(a + b - c)

Let A' be the intersections of the tangents to the Yiu conic at the points where they meet the A-excircle. Define B' and C' similarly. The lines AA', BB', CC' concur in X(65). (Randy Hutson, July 20, 2016)

Let Ab, Ac be the points where the A-excircle touches lines CA and AB resp., and define Bc, Ba, Ca, Cb cyclically. Let Ta be the intersection of the tangents to the Yiu conic (defined at X(478)) at Bc and Ca, and define Tb, Tc cyclically. Let Ta' be the intersection of the tangents to the Yiu conic at Ba and Cb, and define Tb', Tc' cyclically. Let Sa = TbTc∩Tb'Tc', Sb = TcTa∩Tc'Ta', Sc = TaTb∩Ta'Tb'. The lines ASa, BSb, CSc concur in X(65). (See also X(1903).) (Randy Hutson, July 20, 2016)

Let A' be the homothetic center of the orthic triangles of the intouch and A-extouch triangles, and define B' and C' cyclically. The triangle A'B'C' is perspective to the extouch triangle at X(65). (Randy Hutson, July 20, 2016)

Let A'B'C' be the orthic triangle. Let B'C'A″ be the triangle similar to ABC such that segment A'A″ crosses the line B'C'. Define B″ and C″ cyclically. Equivalently, A″ is the reflection of A in B'C', and cyclically for B″ and C″. Let Ia be the incenter of B'C'A″, and define Ib and Ic cyclically. The circumcenter of triangle IaIbIc is X(65). Let A* be the intersection of lines A″Ia and B'C', and define B* and C* cyclically. The lines A'A*, B'B*, C'C* concur in X(65). (Randy Hutson, July 20, 2016)

Let Ab, Ac be the points where the A-excircle touches lines CA and AB resp., and define Bc, Ba, Ca, Cb cyclically. Let IaIbIc be the intouch triangle. Let Oa be the circle through Ab, Ac, Ib, Ic, and define Ob, Oc cyclically. X(65) is the radical center of Oa, Ob, Oc. (Randy Hutson, July 20, 2016)

Let A'B'C' be the intouch triangle of the extangents triangle, if ABC is acute. Then A'B'C' is perspective to the intouch triangle and 4th and 5th extouch triangles at X(65). (Randy Hutson, December 2 2017)

Let OA be the circle centered at the A-vertex of the Wasat triangle and passing through A; define OB and OC cyclically. X(65) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

Let A' be the isogonal conjugate of A wrt the A-extouch triangle. Define B' and C' cyclically. The lines AA', BB', CC' concur in X(65). (Randy Hutson, August 30, 2020)

In the plane of a triangle ABC, let
Ba = reflection of A in the external angular bisector of angle B, and define Cb and Ac cyclically;
Ca = reflection of A in the external angular bisector of angle C, and define Ab and Bc cyclically;
Pa = AcBc∩AbCb, and define Pb and Pc cyclically;
Ka = AbBa∩AcCa, and define Kb and Kc cyclically.
Then PaPbPc and ABC, and also KaKbKc and ABC, are perspective, and the perspector is X(65).
(Dasari Naga Vijay Krishna, June 19, 2021)

X(65) lies on these lines: 1,3   2,959   4,158   6,19   7,8   10,12   11,117   29,296   31,1104   33,64   37,71   41,910   42,73   44,374   58,109   63,958   68,91   74,108   77,969   79,80   81,961   110,229   169,218   172,248   224,1004   225,407   243,412   257,894   278,387   279,1002   386,994   409,1098   474,997   497,938   516,950   519,553   604,1100   651,895   1039,1041   1061,1063

X(65) is the {X(1),X(40)}-harmonic conjugate of X(55). For a list of other harmonic conjugates of X(65), click Tables at the top of this page.

X(65) = reflection of X(i) in X(j) for these (i,j): (1,942), (72,10)
X(65) = isogonal conjugate of X(21)
X(65) = isotomic conjugate of X(314)
X(65) = complement of X(3869)
X(65) = anticomplement of X(960)
X(65) = circumcircle-inverse of X(5172)
X(65) = incircle-inverse of X(1319)
X(65) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,73), (4,225), (7,226), (10,227), (109,513), (226,37)
X(65) = X(42)-cross conjugate of X(37)
X(65) = crosspoint of X(i) and X(j) for these (i,j): (1,4), (7,57)
X(65) = crosssum of X(i) and X(j) for these (i,j): (1,3), (9,55), (56,1394), (1805,1806)
X(65) = crossdifference of every pair of points on line X(521)X(650)
X(65) = X(1284)-Hirst inverse of X(1400)
X(65) = X(i)-beth conjugate of X(j) for these (i,j): (1,65), (8,72), (10,10), (65,1042), (80,65), (100,65), (101,213), (291,65), (668,65), (1018, 65)
X(65) = bicentric sum of PU(15)
X(65) = PU(15)-harmonic conjugate of X(650)
X(65) = trilinear product of PU(81)
X(65) = trilinear pole of line X(647)X(661)
X(65) = perspector of intouch triangle and inverse(n(hexyl triangle))
X(65) = orthologic center of inverse(n(hexyl triangle)) to hexyl triangle; the reciprocal orthologic center is X(84)
X(65) = perspector of ABC and the extangents triangle
X(65) = X(1986)-of-Fuhrmann-triangle
X(65) = X(40) of Mandart-incircle triangle
X(65) = homothetic center of intangents triangle and reflection of extangents triangle in X(40)
X(65) = homothetic center of extangents triangle and reflection of intangents triangle in X(1)
X(65) = reflection of X(3057) in X(1)
X(65) = {X(1),X(3)}-harmonic conjugate of X(2646)
X(65) = {X(1),X(57)}-harmonic conjugate of X(56)
X(65) = {P,Q}-harmonic conjugate of X(1463), where P and Q are the intersections of the incircle and line X(7)X(8)
X(65) = pairwise perspector of: intouch triangle, 4th extouch triangle, 5th extouch triangle
X(65) = perspector of [reflection of incentral triangle in X(1)] and tangential triangle, wrt incentral triangle, of circumconic of incentral triangle centered at X(1) (bicevian conic of X(1) and X(57))
X(65) = inverse-in-{incircle, circumcircle}-inverter of X(2078)
X(65) = pedal-isogonal conjugate of X(1)
X(65) = X(5) of reflection triangle of X(1)
X(65) = radical trace of circumcircle and circumcircle of reflection triangle of X(1)
X(65) = X(188)-of-orthic-triangle if ABC is acute
X(65) = perspector of ABC and cross-triangle of ABC and 4th extouch triangle
X(65) = perspector of ABC and cross-triangle of ABC and 5th extouch triangle
X(65) = polar conjugate of X(31623)
X(65) = pole wrt polar circle of trilinear polar of X(31623) (line X(521)X(1948))
X(65) = perspector of ABC and anti-tangential midarc triangle
X(65) = homothetic center of extangents triangle and anti-tangential midarc triangle
X(65) = excentral-to-intouch similarity image of X(1)


X(66) = ISOGONAL CONJUGATE OF X(22)

Trilinears    bc/(b4 + c4 - a4) : :
Barycentrics    1/(b4 + c4 - a4) : :

Let P be a point on the circumcircle, and let LP be its Steiner line. The locus of the orthopole of LP, as P varies, is an ellipse with center X(4) and perspector X(66). (Randy Hutson, March 29, 2020)

In the plane of a triangle ABC, let
A'B'C' = anticomplementary triangle;
Oa = circle with diameter BC, and define Ob and Oc cyclically;
Ab = A'BC'∩Oa, and define Bc and Ca cyclically;
Ac = B'CA'∩Oa, and define Ba and Cb cyclically;
A″= BcBa∩CaCb, and define B″ and C″ cyclically.
The triangle A″B″C″ is perspective to ABC, and the perspector is X(66).
(Dasari Naga Vijay Krishna, April 15, 2021)

X(66) lies on these lines: 2,206   3,141   6,427   68,511   73,976   193,895   248,571   290,317   879,924

X(66) = midpoint of X(2892) and X(3448)
X(66) = reflection of X(i) in X(j) for these (i,j): (159,141), (1177,125)
X(66) = isogonal conjugate of X(22)
X(66) = isotomic conjugate of X(315)
X(66) = complement of X(5596)
X(66) = anticomplement of X(206)
X(66) = cyclocevian conjugate of X(2998)
X(66) = cevapoint of X(125) and X(512)
X(66) = X(32)-cross conjugate of X(2)
X(66) = crosssum of X(3) and X(159)
X(66) = trilinear pole of line X(647)X(826) (radical axis of Brocard and polar circles)
X(66) = antigonal image of X(1177)
X(66) = orthocenter of X(3)X(4)X(2435)
X(66) = X(3174)-of-orthic-triangle if ABC is acute
X(66) = polar conjugate of isotomic conjugate of X(14376)
X(66) = X(63)-isoconjugate of X(8743)


X(67) = ISOGONAL CONJUGATE OF X(23)

Trilinears    bc/(b4 + c4 - a4 - b2c2) : :
Barycentrics    1/(b4 + c4 - a4 - b2c2) : :

Let A' be the reflection in BC of the A-vertex of the antipedal triangle of X(6), and define B' and C' cyclically. The circumcircles of A'BC, B'CA, C'AB concur at X(67). Also, let A' be the reflection of X(6) in BC, and define B' and C' cyclically. The circumcircles of A'BC, B'CA, C'AB concur in X(67). Note: the above 2 sets of circumcircles are identical. (Randy Hutson, November 18, 2015)

X(67) lies on these lines: 3,542   4,338   6,125   50,248   74,935   110,141   265,511   290,340   524,858   526,879

X(67) = midpoint of X(69) and X(3448)
X(67) = reflection of X(i) in X(j) for these (i,j): (6,125), (110,141)
X(67) = isogonal conjugate of X(23)
X(67) = isotomic conjugate of X(316)
X(67) = circumcircle-inverse of X(3455)
X(67) = cevapoint of X(141) and X(524)
X(67) = X(187)-cross conjugate of X(2)
X(67) = antigonal image of X(6)
X(67) = trilinear pole of line X(39)X(647)
X(67) = polar conjugate of X(37765)
X(67) = pole wrt polar circle of trilinear polar of X(37765) (line X(9517)X(9979))
X(67) = X(63)-isoconjugate of X(8744)
X(67) = orthocenter of X(3)X(74)X(879)
X(67) = perspector of ABC and X(2)-Ehrmann triangle; see X(25)
X(67) = X(19)-isoconjugate of X(22151)


X(68) = PRASOLOV POINT

Trilinears        cos A sec 2A : cos B sec 2B : cos C sec 2C
Barycentrics   tan 2A : tan 2B : tan 2C
Barycentrics    (b^2 + c^2 - a^2)/(a^4 + b^4 + c^4 - 2a^2b^2 - 2a^2c^2) : :

Let A'B'C' be the 2nd Euler triangle. The lines AA', BB', CC' concur in X(68), as proved in V. V. Prasolov, Zadachi po planimetrii, Moscow, 4th edition, 2001.

Coordinates for X(68) can be obtained easily from the Ceva ratios in Prasolov's proof of concurrence.

Let Oa be the circle centered at the A-vertex of the orthic triangle and passing through A; define Ob and Oc cyclically. Then X(68) is the radical center of Oa, Ob, Oc. (Randy Hutson, November 2, 2017)

The X(3)-Fuhrmann triangle is inversely similar to ABC, with similitude center X(3), and perspective to ABC at X(68). (Randy Hutson, November 3, 2017)

X(68) lies on these lines: 2,54   3,343   4,52   5,6   11,1069   20,74   26,161   30,64   65,91   66,511   73,1060   136,254   290,315   568,973

X(68) = reflection of X(155) in X(5)
X(68) = isogonal conjugate of X(24)
X(68) = isotomic conjugate of X(317)
X(68) = anticomplement of X(1147)
X(68) = X(96)-Ceva conjugate of X(3)
X(68) = cevapoint of X(i) and X(j) for these (i,j): (6,161), (125,520)
X(68) = X(115)-cross conjugate of X(525)
X(68) = pedal antipodal perspector of X(4)
X(68) = pedal antipodal perspector of X(186)
X(68) = X(63)-isoconjugate of X(8745)
X(68) = crossdifference of every pair of points on line X(924)X(6753)
X(68) = trilinear product of vertices of X(3)-anti-altimedial triangle
X(68) = orthic-to-ABC barycentric image of X(52)
X(68) = cyclocevian conjugate of X(34287)


X(69) = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE

Trilinears    (cos A)/a2 : (cos B)/b2 : (cos C)/c2
Trilinears    bc(b2 + c2 - a2) : ca(c2 + a2 - b2) : ab(a2 + b2 - c2)
Trilinears    sec2(A/2) - csc2(A/2) : :
Barycentrics    cot A : cot B : cot C
Barycentrics    b2 + c2 - a2 : c2 + a2 - b2 : a2 + b2 - c2
Barycentrics    cot B + cot C - cot ω : :
Barycentrics    cot B + cot C - cot A - cot ω : :
Barycentrics    SA : SB : SC
X(69) = 2(r2 + 2rR + s2)*X(1) + 3(r2 - s2)*X(2) - 4r2*X(3)    (Peter Moses, April 2, 2013)
X(69) = (cot A)*[A] + (cot B)*[B] + (cot C)*[C], where A, B, C denote angles and [A], [B], [C] denote vertices

Let A'B'C' be the anticomplementary triangle. Let A″ be the inverse-in-anticomplementary-circle of A, and define B″ and C″ cyclically. The lines A'A″, B'B″, C'C″ concur in X(69). (Randy Hutson, February 10, 2016)

Let A'B'C' be the anticomplementary triangle. Let A″ be the orthogonal projection of A' on line BC, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(69). (Randy Hutson, February 10, 2016)

Let A'B'C' be the half-altitude triangle. Let A″ be the trilinear pole of line B'C', and define B″ and C″ cyclically. Let A* be the trilinear pole of line B″C″, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(69). (Randy Hutson, February 10, 2016)

Let A2B2C2 be the 2nd Conway triangle. Let A' be the cevapoint of B2 and C2, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(69). (Randy Hutson, December 10, 2016)

X(69) is the barycentric multiplier for the MacBeath circumconic. (The barycentric product of X(69) and the circumcircle is the MacBeath circumconic.) (Randy Hutson, August 19, 2019)

Let OA be the circle centered at the A-vertex of the 1st Ehrmann triangle and passing through A; define OB and OC cyclically. X(69) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

X(69) lies on the Lucas cubic and these lines: 2,6   3,332   4,76   7,8   9,344   10,969   20,64   22,159   54,95   63,71   72,304   73,77   74,99   110,206   125,895   144,190   150,668   189,309   192,742   194,695   200,269   219,1332   248,287   263,308   265,328   274,443   290,670   297,393   347,664   350,497   404,1014   478,651   485,639   486,640   520,879   1225,2888   1369,3410  

X(69) is the {X(7),X(8)}-harmonic conjugate of X(75). For a list of other harmonic conjugates of X(69), click Tables at the top of this page.

If you have The Geometer's Sketchpad, you can view X(69).

X(69) = reflection of X(i) in X(j) for these (i,j): (2,599), (4,1352), (6,141), (20,1350), (193,6), (895,125), (1351,5), (1353,140)
X(69) = isogonal conjugate of X(25)
X(69) = isotomic conjugate of X(4)
X(69) = complement of X(193)
X(69) = anticomplement of X(6)
X(69) = anticomplementary conjugate of X(2)
X(69) = circumcircle-inverse of X(5866)
X(69) = cyclocevian conjugate of X(253)
X(69) = X(i)-Ceva conjugate of X(j) for these (i,j): (76,2), (304,345), (314,75), (332,326)
X(69) = cevapoint of X(i) and X(j) for these (i,j): (2,20), (3,394), (6,159), (8,329), (63,78), (72,306), (125,525)
X(69) = X(i)-cross conjugate of X(j) for these (i,j): (3,2), (63,348), (72,63), (78,345), (125,525), (306,304), (307,75), (343,76)
X(69) = crosspoint of X(i) and X(j) for these (i,j): (2,2996), (76,305), (314,332)
X(69) = X(2)-Hirst inverse of X(325)
X(69) = X(i)-beth conjugate of X(j) for these (i,j): (69,77), (99,347), (314,7), (332,69), (645,69), (668,69)
X(69) = barycentric product of PU(37)
X(69) = bicentric sum of PU(132)
X(69) = midpoint of PU(132)
X(69) = perspector of the orthic-of-medial triangle and the reference triangle
X(69) = perspector of ABC and the pedal triangle of X(20)
X(69) = perspector of ABC and (reflection in X(2) of the pedal triangle of X(2))
X(69) = intersection of extended sides P(11)U(45) and U(11)P(45) of the trapezoid PU(11)PU(45)
X(69) = perspector of ABC and 4th extouch triangle
X(69) = antipode of X(287) in hyperbola {A,B,C,X(2),X(69)}}
X(69) = trilinear pole of line X(441)X(525)
X(69) = pole wrt polar circle of trilinear polar of X(393) (line X(460)X(512))
X(69) = X(48)-isoconjugate (polar conjugate) of X(393)
X(69) = X(6)-isoconjugate of X(19)
X(69) = X(92)-isoconjugate of X(32)
X(69) = antigonal image of X(895)
X(69) = crosssum of X(i) and X(j) for these (i,j): (6,3053), (32,1974)
X(69) = perspector of ABC and the 2nd pedal triangle of X(3)
X(69) = crosspoint of X(6) and X(159) wrt both the excentral and tangential triangles
X(69) = crosspoint of X(2) and X(20) wrt both the excentral and anticomplementary triangles
X(69) = homothetic center of anticomplementary triangle and 2nd antipedal triangle of X(4) (i.e., of 1st and 2nd antipedal triangles of X(4))
X(69) = perspector of the complement of the polar circle
X(69) = perspector of the inconic with center X(3)
X(69) = pole, wrt de Longchamps circle, of trilinear polar of X(95)
X(69) = perspector of the extraversion triangles of X(7) and X(8)
X(69) = {X(2),X(6)}-harmonic conjugate of X(3618)
X(69) = perspector of ABC and anticomplement of submedial triangle
X(69) = perspector of ABC and mid-triangle of orthic and dual of orthic triangles
X(69) = perspector of ABC and cross-triangle of ABC and 2nd Brocard triangle
X(69) = perspector of 2nd Conway triangle and cross-triangle of ABC and 2nd Conway triangle
X(69) = Lucas-isogonal conjugate of X(376)
X(69) = anticevian-isogonal conjugate of X(2)
X(69) = inverse-in-MacBeath-circumconic of X(22151)
X(69) = {X(7),X(8)}-harmonic conjugate of X(75)
X(69) = intersection of van Aubel lines of outer and inner Vecten triangles
X(69) = orthic-isogonal conjugate of X(19583)
X(69) = X(4)-Ceva conjugate of X(19583)


X(70) = ISOGONAL CONJUGATE OF X(26)

Trilinears    bc/[b2 cos 2B + c2 cos 2C - a2 cos 2A] : :
Barycentrics    1/(a^8 - 2 a^6 (b^2 + c^2) + 2 a^2 (b^6 + c^6) - (b^2 - c^2)^2 (b^4 + c^4)) : :
X(70) = (S cot ω- 2R2)/(S cot ω-3 R2)*X(3) - X(8907)      (Peter Moses, December 28, 2015)

X(70) lies on the Jerabek circumhyperbola and these lines: {3,8907}, {6,1594}, {54,1899}, {64,6240}, {66,6403}, {71,2158}, {74,1288}, {265,6243}, {1176,1352}, {1177,3542}, {3448,5504}, {3527,7507}, {4846,6241}, {6145,6152}

X(70) = isogonal conjugate of X(26)
X(70) = anticomplement of X(34116)
X(70) = antigonal conjugate of X(38534)
X(70) = X(571)-crossconjugate of X(2)
X(70) = X(i)-isoconjugate of X(j) for these {i,j}: {1,26}, {63,8746}
X(70) = reflection of the isogonal conjugate of X(2072) in X(125)
X(70) = X(125)-cevapoint of X(924)
X(70) = X(161)-crosssum of X(8553)
X(70) = barycentric product X(525) X(1288)


X(71) = ISOGONAL CONJUGATE OF X(27)

Trilinears       (b + c) cos A : (c + a) cos B : (a + b) cos C
Barycentrics  (b + c) sin 2A : (c + a) sin 2B : (a + b) sin 2C

X(71) lies on these lines: 1,579   3,48   4,9   6,31   35,284   37,65   54,572   63,69   64,198   74,101   165,610   190,290   583,1100

X(71) is the intersection of the isotomic conjugate of the polar conjugate of the Nagel line (i.e., line X(63)X(69)), and the polar conjugate of the isotomic conjugate of the Nagel line (i.e., line X(4)X(9)). (Randy Hutson, July 11, 2019)

X(71) is the {X(9),X(40)}-harmonic conjugate of X(19). For a list of other harmonic conjugates of X(71), click Tables at the top of this page.

X(71) = isogonal conjugate of X(27)
X(71) = anticomplement of X(34830)
X(71) = X(i)-Ceva conjugate of X(j) for these (i,j): (3,228), (9, 37), (10,42), (63,72)
X(71) = X(228)-cross conjugate of X(73)
X(71) = crosspoint of X(i) and X(j) for these (i,j): (3,63), (9,219), (10,306)
X(71) = crosssum of X(i) and X(j) for these (i,j): (1,579), (4,19), (28,1127), (57,278), (58,1474)
X(71) = crossdifference of every pair of points on line X(242)X(514)
X(71) = X(4)-line conjugate of X(242)
X(71) = X(i)-beth conjugate of X(j) for these (i,j): (219,71), (1018,71)
X(71) = trilinear pole of line X(647)X(810)
X(71) = X(92)-isoconjugate of X(58)
X(71) = barycentric product of Jerabek hyperbola intercepts of Nagel line
X(71) = polar conjugate of isotomic conjugate of X(3682)
X(71) = antigonal conjugate of X(38535)
X(71) = X(63)-isoconjugate of X(8747)


X(72) = ISOGONAL CONJUGATE OF X(28)

Trilinears    (b + c) cot A : (c + a) cot B : (a + b) cot C
Trilinears    (b + c)(b2 + c2 - a2) : (c + a)(c2 + a2 - b2) : (a + b)(a2 + b2 - c2)
Barycentrics    (b + c) cos A : (c + a) cos B : (a + b) cos C
X(72) = (r + 2R)*X(1) - 3R*X(2) - r*X(3)    (Peter Moses, April 2, 2013)

Let Ia be the reflection of X(1) in the perpendicular bisector of BC, and define Ib and Ic cyclically. X(72) = X(11) of IaIbIc. (Randy Hutson, September 14, 2016)

X(72) lies on the Mandart hyperbola and these lines: 1,6   2,942   3,63   4,8   5,908   7,443   10,12   20,144   21,943   31,976   35,191   40,64   43,986   54,1006   56,997   57,474   69,304   73,201   74,100   145,452   171,1046   185,916   190,1043   222,1038   248,293   290,668   295,337   306,440   394,1060   519,950   672,1009   894,1010   940,975   978,982

X(72) is the {X(1),X(9)}-harmonic conjugate of X(405). For a list of other harmonic conjugates of X(72), click Tables at the top of this page.

X(72) is the perspector of the 1st and 2nd extouch triangles. X(72) is also the orthocenter of the 2nd extouch triangle. (Randy Hutson, August 23, 2011)

X(72) = reflection of X(i) in X(j) for these (i,j): (1,960), (65,10), (1145,14740), (3555,1)
X(72) = isogonal conjugate of X(28)
X(72) = isotomic conjugate of X(286)
X(72) = inverse-in-Fuhrmann circle of X(3419)
X(72) = anticomplement of X(942)
X(72) = X(i)-Ceva conjugate of X(j) for these (i,j): (8,10), (63,71), (69,306), (321,37)
X(72) = X(i)-cross conjugate of X(j) for these (i,j): (201,10), (228,37)
X(72) = crosspoint of X(i) and X(j) for these (i,j): (8,78), (63,69), (306,307)
X(72) = crosssum of X(i) and X(j) for these (i,j): (19,25), (34,56)
X(72) = crossdifference of every pair of points on line X(513)X(1430)
X(72) = X(i)-beth conjugate of X(j) for these (i,j): (8,65), (72,73), (78,72), (100,227), (644,72)
X(72) = trilinear pole of line X(647)X(656)
X(72) = complement of X(3868)
X(72) = X(149) of X(1)-Brocard triangle
X(72) = X(6)-isoconjugate of X(27)
X(72) = X(75)-isoconjugate of X(2203)
X(72) = X(92)-isoconjugate of X(1333)
X(72) = inverse-in-Fuhrmann-circle of X(3419)
X(72) = X(6146)-of-excentral-triangle
X(72) = perspector of ABC and cross-triangle of ABC and 2nd extouch triangle
X(72) = trilinear product of Jerabek hyperbola intercepts of Nagel line
X(72) = excentral-to-ABC barycentric image of X(4)
X(72) = antipode of X(1145) in the Mandart hyperbola
X(72) = extouch-isogonal conjugate of X(5687)


X(73) = CROSSPOINT OF INCENTER AND CIRCUMCENTER

Trilinears    sec B + sec C : sec C + sec A : sec A + sec B
Trilinears    a(b + c)(a^2 - b^2 - c^2)/(a - b - c) : :
Barycentrics    (cos B + cos C) sin 2A : (cos C + cos A) sin 2B : (cos A + cos B) sin 2C
X(73) = (r2 - 4R2 + s2)*X(1) - 6rR*X(2) + 4rR*X(3)    (Peter Moses, April 2, 2013)

X(73) lies on these lines: 1,4   3,212   6,41   21,651   35,74   36,54   37,836   42,65   55,64   57,386   66,976   68,1060   69,77   72,201   102,947   228,408   284,951   290,336   1036,1037   1057,1059

X(73) is the {X(1064),X(1066)}-harmonic conjugate of X(1). For a list of other harmonic conjugates of X(73), click Tables at the top of this page.

X(73) = isogonal conjugate of X(29)
X(73) = anticomplement of X(34381)
X(73) = X(1)-Ceva conjugate of X(65)
X(73) = X(228)-cross conjugate of X(71)
X(73) = crosspoint of X(i) and X(j) for these (i,j): (1,3), (77,222), (226,307)
X(73) = crosssum of X(i) and X(j) for these (i,j): (1,4), (33,281)
X(73) = crossdifference of every pair of points on line X(243)X(522)
X(73) = X(i)-Hirst inverse of X(j) for these (i,j): (1,243), (65,851)
X(73) = X(i)-beth conjugate of X(j) for these (i,j): (1,1042), (3,73), (21,946), (72,72), (100,10), (101,73), (295,73)
X(73) = bicentric sum of PU(16)
X(73) = PU(16)-harmonic conjugate of X(652)
X(73) = trilinear product of PU(83)
X(73) = trilinear pole of line X(647)X(822)
X(73) = X(63)-isoconjugate of X(8748)
X(73) = X(92)-isoconjugate of X(284)
X(73) = {X(1),X(1745)}-harmonic conjugate of X(4)


X(74) = ISOGONAL CONJUGATE OF EULER INFINITY POINT

Trilinears    1/(cos A - 2 cos B cos C) : 1/(cos B - 2 cos C cos A) : 1/(cos C - 2 cos A cos B)
Trilinears    1/(3 cos A - 2 sin B sin C) : 1/(3 cos B - 2 sin C sin A) : 1/(3 cos C - 2 sin A sin B)
Trilinears    a/[2a4 - (b2 - c2)2 - a2(b2 + c2)] : :
Barycentrics   a/(cos A - 2 cos B cos C) : b/(cos B - 2 cos C cos A) : c/(cos C - 2 cos A cos B)
Barycentrics   a^2/(3 SBSC - S2) : :
X(74) = (r2 + 2rR + s2)*X(1) - R(6r + 9R)*X(2) + (r2 + 12rR + 18R2 - 3s2)*X(3)    (Peter Moses, April 2, 2013)
X(74) = 3 X[2] - 4 X[6699], 9 X[2] - 8 X[12900], 3 X[3] - X[399], 3 X[3] - 2 X[1511], 5 X[3] - 2 X[5609], 5 X[3] - X[12308], 4 X[3] - X[14094], 10 X[3] - 7 X[15020], 2 X[3] - 5 X[15021], 22 X[3] - 19 X[15023], 8 X[3] - 5 X[15034], 4 X[3] - 3 X[15035], 8 X[3] - 7 X[15036], 13 X[3] - 7 X[15039], 7 X[3] - 5 X[15040], X[3] - 3 X[15041], 15 X[3] - 13 X[15042], 6 X[3] - 5 X[15051], 2 X[3] + X[15054], 2 X[3] - 3 X[15055], 5 X[3] - 3 X[32609], 5 X[3] + 4 X[38626], 13 X[3] - 4 X[38632], 5 X[3] - 9 X[38633], 13 X[3] - 9 X[38638], 3 X[4] - 4 X[7687], X[4] + 2 X[10990], 3 X[4] - 2 X[13202], 2 X[4] - 3 X[14644], 3 X[4] - 5 X[15081], X[4] - 4 X[20417], 4 X[5] - 7 X[15057], 4 X[5] - 5 X[15059], 2 X[5] - 3 X[15061], 3 X[5] - 4 X[40685], X[6] - 3 X[5621], 2 X[6] - 3 X[5622], X[20] + 2 X[16003], 3 X[51] - 2 X[11807], 3 X[54] - 2 X[2914], 5 X[54] - 4 X[32226], 3 X[64] + X[17812], 3 X[110] - 2 X[399], 3 X[110] - 4 X[1511], 5 X[110] - 4 X[5609], X[110] + 2 X[10620], X[110] - 4 X[12041], 5 X[110] - 2 X[12308], 5 X[110] - 7 X[15020], X[110] - 5 X[15021], 11 X[110] - 19 X[15023], 4 X[110] - 5 X[15034], 2 X[110] - 3 X[15035], 4 X[110] - 7 X[15036], 13 X[110] - 14 X[15039], 7 X[110] - 10 X[15040], X[110] - 6 X[15041], 15 X[110] - 26 X[15042], 3 X[110] - 5 X[15051]

Let T be the triangle formed by reflecting the orthic axis in the sidelines of ABC; then T is perspective to ABC, and the perspector is X(74). Let A' be the point of intersection of the orthic axis and line BC, and define B' and C' cyclically. Let OA be the circumcenter of AB'C', and define Let OB and OC cyclically; then the lines AOA, BOB, COC concur in X(74). (Randy Hutson, August 26, 2014)

Let A'B'C' be the anticomplementary triangle. Let L be the line through A' parallel to the Euler line, and define M and N cyclically. Let L' be the reflection of L in sideline BC, and define M' and N' cyclically. The lines L', M', N' concur in X(74). (Randy Hutson, August 26, 2014)

Let A'B'C' be the X(3)-Fuhrmann triangle. Let A'' be the reflection of A in line B'C', and define B'' and C'' cyclically. Then A''B''C'' is inversely similar to ABC, with similitude center X(265), and A''B''C'' is perspective to ABC at X(74), which is also the orthocenter of A''B''C''. (Randy Hutson, August 26, 2014)

In Hyacinthos 8129 (10/4/03), Floor van Lamoen noted that if X(74) is denoted by J, then each of the points A,B,C,J is J of the other three, in analogy with the well known property of orthocentric systems (that is, each of the points A,B,C,H is the orthocenter of the other three).

Let A'B'C' be the orthocentroidal triangle and A″B″C″ the anti-orthocentroidal triangle. Let A* be the reflection of A″ in B'C', and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(74). (Randy Hutson, December 10, 2016)

Let A'B'C' be the anti-orthocentroidal triangle. Let A″ be the reflection of A in line B'C', and define B″ and C″ cyclically. The lines A'A″, B'B″, C'C″ concur in X(74). (Randy Hutson, January 15, 2019)

The tangents at A, B, C to the Neuberg cubic K001 concur in X(74)

Let Ea be the ellipse through X(74) having foci B and C, and define Eb and Ec cyclically. The 6 minor vertices of these three ellipses lie on a pair of lines. Figure. (Dan Reznik, December 13, 2021)

Let T be a family of Poncelet triangles inscribed in a circle and circumscribing some conic. The line X(3)X(74) remains stationary iff the family contains an equilateral triangle, and X(74) and X(110) are stationary on the circumcircle. See video. (Dan Reznik, August 17, 2024)

X(74) lies on the circumcircle, Walsmith rectangular hyperbola, Jerabek hyperbola, Moses-Jerabek conic, 2nd Evans circle, the cubics K001, K039, K073, K114, K130, K187, K223, K255, K279, K374, K446, K447, K448, K488, K489, K496, K499, K505, K513, K523, K524, K536, K564, K595, K596, K597, K614, K639, K668, K669, K695, K698, K724, K802, K803, K811, K816, K818, K819, K854, K905, K923, K929, K930, K1106, K1107, K1166, K1169, K1170, K1172, the curves Q001, Q030, Q125, Q138, and these lines: {1, 3464}, {2, 113}, {3, 110}, {4, 107}, {5, 3521}, {6, 112}, {10, 12368}, {11, 10767}, {12, 12372}, {13, 5618}, {14, 5619}, {15, 5668}, {16, 5669}, {20, 68}, {21, 34800}, {22, 2931}, {23, 9060}, {24, 64}, {25, 3426}, {26, 12279}, {30, 265}, {32, 9984}, {35, 73}, {36, 7343}, {40, 6011}, {49, 10226}, {50, 11079}, {51, 11807}, {52, 11806}, {54, 185}, {55, 3028}, {56, 3024}, {65, 108}, {66, 1289}, {67, 935}, {69, 99}, {70, 1288}, {71, 101}, {72, 100}, {94, 39375}, {97, 19193}, {98, 690}, {102, 2773}, {103, 2774}, {104, 7429}, {105, 2775}, {106, 2776}, {111, 2433}, {140, 14643}, {141, 14982}, {154, 35472}, {165, 2948}, {182, 9970}, {184, 3043}, {186, 1304}, {187, 248}, {195, 22949}, {287, 4235}, {290, 16077}, {323, 2071}, {325, 2855}, {352, 35188}, {371, 19111}, {372, 19110}, {381, 1539}, {382, 10113}, {386, 34453}, {389, 1173}, {394, 17838}, {402, 12369}, {403, 1514}, {468, 10293}, {477, 523}, {484, 2222}, {493, 12377}, {494, 12378}, {511, 691}, {512, 842}, {513, 2687}, {514, 2688}, {515, 2689}, {516, 2690}, {517, 1290}, {518, 2691}, {519, 2692}, {520, 2693}, {521, 2694}, {522, 2695}, {524, 2696}, {525, 2697}, {526, 9142}, {546, 15025}, {548, 20189}, {549, 5655}, {550, 930}, {573, 3031}, {574, 15920}, {578, 13472}, {631, 5972}, {675, 7433}, {689, 7470}, {695, 35476}, {759, 14127}, {789, 33805}, {805, 35002}, {827, 1176}, {841, 8675}, {858, 16167}, {901, 35000}, {907, 34817}, {915, 3657}, {924, 15453}, {927, 5195}, {931, 6876}, {934, 1439}, {952, 36158}, {1064, 29038}, {1099, 9405}, {1112, 1593}, {1113, 2575}, {1114, 2574}, {1138, 5670}, {1147, 3047}, {1151, 11462}, {1152, 11463}, {1154, 1291}, {1157, 3484}, {1177, 10423}, {1181, 8567}, {1192, 10594}, {1199, 13382}, {1243, 4219}, {1245, 32691}, {1250, 36073}, {1263, 5671}, {1276, 5672}, {1277, 5673}, {1286, 18124}, {1287, 18125}, {1292, 2836}, {1293, 2842}, {1294, 9033}, {1295, 2850}, {1296, 1350}, {1297, 2435}, {1300, 15328}, {1303, 32438}, {1305, 20291}, {1309, 38955}, {1311, 7441}, {1337, 5674}, {1338, 5675}, {1352, 41737}, {1370, 12319}, {1385, 5606}, {1428, 32290}, {1464, 36064}, {1498, 3532}, {1499, 2770}, {1510, 14979}, {1533, 32223}, {1553, 22104}, {1587, 19002}, {1588, 19001}, {1594, 6696}, {1597, 3531}, {1620, 35479}, {1650, 10745}, {1656, 34128}, {1657, 11999}, {1764, 38482}, {1853, 18434}, {1870, 19505}, {1885, 10816}, {1899, 35481}, {1903, 40117}, {1942, 2713}, {1987, 3331}, {1989, 11070}, {1993, 19456}, {1994, 13482}, {1995, 11472}, {2070, 11559}, {2080, 32694}, {2088, 11060}, {2132, 39376}, {2133, 5676}, {2330, 32289}, {2372, 22037}, {2373, 32122}, {2374, 32121}, {2393, 5505}, {2646, 11670}, {2706, 22089}, {2709, 18860}, {2720, 5172}, {2752, 3309}, {2758, 3667}, {2764, 34109}, {2766, 6001}, {2790, 15111}, {2794, 11005}, {2883, 10018}, {2930, 31884}, {2940, 16143}, {2979, 12273}, {2981, 14816}, {3003, 32681}, {3060, 12236}, {3065, 5677}, {3068, 8994}, {3069, 13969}, {3081, 38246}, {3090, 6723}, {3091, 20397}, {3100, 12888}, {3101, 12661}, {3134, 39985}, {3146, 12295}, {3147, 6225}, {3153, 19479}, {3165, 10646}, {3166, 10645}, {3184, 15526}, {3231, 9091}, {3258, 32417}, {3260, 40423}, {3428, 22586}, {3440, 5678}, {3441, 5679}, {3455, 40080}, {3466, 5680}, {3479, 5681}, {3480, 5682}, {3481, 5683}, {3482, 5684}, {3483, 5685}, {3515, 13093}, {3516, 7592}, {3518, 11381}, {3522, 14683}, {3523, 16534}, {3524, 5642}, {3528, 24981}, {3529, 15077}, {3533, 38792}, {3541, 14542}, {3542, 12250}, {3563, 35364}, {3564, 32244}, {3565, 6391}, {3566, 40118}, {3574, 35482}, {3576, 11720}, {3616, 11723}, {3627, 11801}, {3628, 15029}, {3827, 10100}, {3832, 38725}, {3843, 20396}, {3851, 15088}, {4220, 9058}, {4231, 9107}, {4296, 19469}, {4299, 18968}, {4302, 12896}, {4549, 16063}, {4558, 15919}, {5012, 12228}, {5024, 9475}, {5070, 15046}, {5085, 6593}, {5094, 7699}, {5095, 14912}, {5158, 15816}, {5169, 7706}, {5181, 10519}, {5189, 36853}, {5318, 11139}, {5321, 11138}, {5480, 22336}, {5486, 6776}, {5562, 22978}, {5584, 11460}, {5597, 12365}, {5598, 12366}, {5603, 11735}, {5640, 31861}, {5656, 35486}, {5840, 10778}, {5866, 10425}, {5870, 10815}, {5871, 10814}, {5878, 7505}, {5889, 12084}, {5893, 35487}, {5894, 18560}, {5895, 11704}, {5900, 35489}, {5916, 39424}, {5917, 39425}, {5921, 32275}, {5925, 35490}, {5961, 15469}, {6003, 12030}, {6010, 30269}, {6037, 11676}, {6055, 9144}, {6091, 35191}, {6102, 15002}, {6103, 6794}, {6143, 25563}, {6145, 6240}, {6146, 35491}, {6151, 14817}, {6197, 10119}, {6200, 6413}, {6221, 6415}, {6233, 8722}, {6236, 43273}, {6284, 12904}, {6321, 14734}, {6325, 32228}, {6353, 35512}, {6396, 6414}, {6398, 6416}, {6403, 39382}, {6515, 18932}, {6570, 10979}, {6584, 22765}, {6636, 14855}, {6644, 10546}, {6698, 10516}, {6759, 11270}, {6799, 32339}, {6811, 13654}, {6813, 13774}, {6998, 9057}, {7059, 8459}, {7060, 8449}, {7164, 8432}, {7165, 8485}, {7280, 9638}, {7325, 8476}, {7326, 8468}, {7327, 8503}, {7328, 8527}, {7329, 8504}, {7354, 12903}, {7413, 9056}, {7417, 9084}, {7423, 9061}, {7434, 9083}, {7471, 38700}, {7488, 8718}, {7492, 8717}, {7503, 25711}, {7506, 11439}, {7512, 22109}, {7514, 20791}, {7526, 10574}, {7527, 9730}, {7547, 40686}, {7550, 16836}, {7556, 40291}, {7574, 19402}, {7575, 32124}, {7576, 15321}, {7577, 23329}, {7712, 10298}, {7732, 11824}, {7733, 11825}, {7953, 41435}, {8059, 37583}, {8172, 8174}, {8173, 8175}, {8431, 8440}, {8433, 8435}, {8434, 8436}, {8437, 8441}, {8438, 8442}, {8439, 8443}, {8444, 8464}, {8445, 8465}, {8446, 8466}, {8447, 8467}, {8448, 8462}, {8450, 8470}, {8451, 8469}, {8452, 8458}, {8453, 8471}, {8454, 8472}, {8455, 8473}, {8456, 8474}, {8457, 8475}, {8460, 8478}, {8461, 8477}, {8463, 8479}, {8480, 8505}, {8481, 8508}, {8482, 8509}, {8483, 8506}, {8484, 8507}, {8486, 8510}, {8487, 8511}, {8488, 8512}, {8489, 8515}, {8490, 8516}, {8491, 8513}, {8492, 8514}, {8493, 8517}, {8494, 8518}, {8495, 8519}, {8496, 8520}, {8497, 8521}, {8498, 8522}, {8499, 8525}, {8500, 8526}, {8501, 8523}, {8502, 8524}, {8528, 8530}, {8529, 8532}, {8531, 8533}, {8537, 13248}, {8550, 35492}, {8705, 32229}, {8744, 32687}, {8998, 9540}, {9069, 14605}, {9070, 41455}, {9100, 9759}, {9129, 38698}, {9143, 10304}, {9160, 14060}, {9181, 38702}, {9202, 14538}, {9203, 14539}, {9218, 38611}, {9544, 35493}, {9545, 35494}, {9703, 35495}, {9704, 35496}, {9705, 12038}, {9706, 35498}, {9716, 35499}, {9729, 16223}, {9777, 35501}, {9781, 9786}, {9818, 9826}, {9833, 35503}, {9938, 11412}, {10102, 30230}, {10111, 18917}, {10263, 13358}, {10282, 17506}, {10303, 38795}, {10310, 13204}, {10311, 41414}, {10409, 14369}, {10410, 14368}, {10421, 12380}, {10540, 15646}, {10610, 18364}, {10619, 18368}, {10625, 12226}, {10632, 10681}, {10633, 10682}, {10638, 36072}, {10663, 11420}, {10664, 11421}, {10698, 31525}, {10880, 13287}, {10881, 13288}, {11003, 39242}, {11004, 13352}, {11012, 39633}, {11017, 22462}, {11061, 25406}, {11081, 39380}, {11086, 39381}, {11179, 41720}, {11202, 20421}, {11248, 13217}, {11249, 13218}, {11250, 12092}, {11403, 15465}, {11410, 12165}, {11411, 30552}, {11414, 12310}, {11416, 12596}, {11417, 12891}, {11418, 12892}, {11423, 11425}, {11458, 11477}, {11465, 11479}, {11466, 11480}, {11467, 11481}, {11550, 18559}, {11562, 14118}, {11589, 15404}, {11634, 38873}, {11693, 15705}, {11694, 17504}, {11699, 13624}, {11804, 15800}, {11822, 13208}, {11823, 13209}, {11826, 13213}, {11827, 13214}, {11828, 13215}, {11829, 13216}, {12017, 26206}, {12042, 18332}, {12074, 12584}, {12082, 33534}, {12085, 38260}, {12100, 13392}, {12113, 13494}, {12162, 15052}, {12225, 15133}, {12254, 13418}, {12261, 12699}, {12262, 41722}, {12278, 32140}, {12282, 12301}, {12285, 12303}, {12286, 12304}, {12287, 12305}, {12288, 12306}, {12291, 12307}, {12315, 15750}, {12505, 32311}, {12898, 34773}, {13017, 13021}, {13018, 13022}, {13145, 26711}, {13391, 32608}, {13397, 28787}, {13403, 32325}, {13414, 41519}, {13415, 41518}, {13434, 13630}, {13452, 26883}, {13474, 34484}, {13568, 15559}, {13595, 16194}, {13603, 32062}, {13621, 32137}, {13665, 13915}, {13785, 13979}, {13868, 15626}, {13935, 13990}, {14110, 30238}, {14374, 14710}, {14375, 14709}, {14457, 18912}, {14458, 41443}, {14480, 14934}, {14490, 31860}, {14536, 41522}, {14540, 39636}, {14541, 39637}, {14561, 32271}, {14639, 15359}, {14651, 16278}, {14685, 14703}, {14809, 16169}, {14830, 20404}, {14833, 19905}, {14853, 15118}, {15030, 35904}, {15043, 16222}, {15058, 17928}, {15078, 18451}, {15085, 37486}, {15089, 15801}, {15131, 23328}, {15232, 26704}, {15320, 26705}, {15322, 41456}, {15329, 39987}, {15342, 34473}, {15459, 41204}, {15478, 40047}, {15545, 38741}, {15578, 19151}, {15579, 37473}, {15664, 32692}, {15749, 33703}, {16000, 18381}, {16164, 21161}, {16186, 34210}, {16340, 20957}, {16620, 16621}, {16658, 37458}, {16868, 22802}, {16936, 35446}, {18317, 20123}, {18324, 20773}, {18358, 26156}, {18363, 34563}, {18439, 37814}, {18445, 25487}, {18551, 21308}, {18916, 18947}, {19051, 42215}, {19052, 42216}, {19121, 19138}, {19168, 19172}, {19361, 32329}, {19376, 26283}, {19406, 19482}, {19407, 19483}, {19424, 19507}, {19425, 19508}, {19454, 19484}, {19455, 19485}, {20186, 40119}, {22115, 34152}, {22329, 38894}, {22535, 22549}, {23061, 37477}, {23240, 35442}, {25328, 29181}, {25641, 36172}, {25738, 34350}, {26861, 33923}, {26914, 26927}, {26915, 26935}, {26916, 26936}, {28788, 30268}, {29299, 37620}, {29317, 32273}, {30250, 34935}, {30257, 41454}, {30270, 39639}, {31074, 34796}, {31133, 40909}, {31384, 40097}, {31724, 34798}, {32235, 35268}, {32237, 37953}, {32251, 39588}, {32274, 36990}, {32349, 37970}, {32581, 42299}, {32618, 40894}, {32619, 40895}, {32620, 40916}, {33962, 35447}, {34007, 43577}, {34207, 39417}, {34298, 34310}, {34435, 38850}, {34437, 38851}, {34440, 38852}, {34568, 41433}, {34594, 37403}, {35265, 37952}, {35373, 40390}, {35465, 39373}, {35834, 42267}, {35835, 42266}, {36034, 36069}, {36071, 36131}, {36193, 38609}, {37426, 43356}, {37475, 41670}, {37948, 43572}, {38263, 39562}, {38323, 41171}, {38936, 39372}, {39808, 39812}, {39837, 39841}

X(74) = midpoint of X(i) and X(j) for these {i,j}: {1, 9904}, {3, 10620}, {4, 12244}, {20, 3448}, {40, 33535}, {64, 10117}, {110, 15054}, {125, 10990}, {265, 20127}, {476, 14508}, {1350, 16010}, {1657, 12902}, {2935, 17835}, {5889, 13201}, {6241, 12281}, {6776, 32247}, {7725, 7726}, {9862, 18331}, {10264, 14677}, {11412, 12284}, {12163, 12302}, {12270, 15100}, {12283, 32249}, {12317, 12383}, {13491, 15101}, {15545, 38741}, {16003, 16111}, {32608, 35452}
X(74) = reflection of X(i) in X(j) for these {i,j}: {1, 11709}, {3, 12041}, {4, 125}, {20, 16111}, {23, 32110}, {52, 11806}, {110, 3}, {113, 6699}, {125, 20417}, {146, 113}, {185, 17855}, {186, 21663}, {265, 10264}, {323, 10564}, {382, 10113}, {399, 1511}, {477, 36164}, {895, 11579}, {974, 15151}, {1112, 16270}, {1199, 34468}, {1498, 15647}, {1533, 32223}, {1539, 20304}, {1553, 22104}, {1986, 974}, {2930, 33851}, {2935, 11598}, {3146, 12295}, {3448, 16003}, {3627, 11801}, {5504, 12901}, {5622, 5621}, {5627, 40630}, {5655, 549}, {5921, 32275}, {6241, 17854}, {6321, 15535}, {7722, 185}, {7728, 5}, {7731, 1986}, {7978, 1}, {9138, 19902}, {9140, 20126}, {9144, 6055}, {9934, 13289}, {9970, 182}, {10113, 20379}, {10263, 13358}, {10540, 15646}, {10698, 31525}, {10706, 2}, {10721, 4}, {10733, 265}, {10752, 6}, {10767, 11}, {11005, 15357}, {11562, 40647}, {11579, 32305}, {11699, 13624}, {12111, 7723}, {12112, 1495}, {12121, 550}, {12244, 10990}, {12290, 12292}, {12292, 15738}, {12295, 36253}, {12308, 5609}, {12368, 10}, {12369, 402}, {12381, 10065}, {12382, 10081}, {12383, 16163}, {12505, 32311}, {12584, 14810}, {12699, 12261}, {12778, 3579}, {12825, 12358}, {12898, 34773}, {13202, 7687}, {13417, 389}, {14094, 110}, {14157, 186}, {14480, 14934}, {14683, 30714}, {14833, 19905}, {14982, 141}, {14989, 34150}, {15035, 15055}, {15054, 10620}, {15055, 15041}, {15063, 5972}, {15107, 3581}, {15131, 23328}, {15329, 39987}, {15463, 32607}, {15472, 19457}, {15800, 11804}, {15801, 15089}, {16105, 11746}, {16163, 37853}, {18332, 12042}, {18781, 18780}, {19140, 5092}, {19506, 20299}, {20127, 14677}, {20957, 16340}, {22115, 34152}, {22265, 98}, {23061, 37477}, {23236, 34153}, {23315, 6696}, {30714, 38726}, {32111, 468}, {32234, 6776}, {34150, 12079}, {34153, 548}, {36172, 25641}, {36193, 38609}, {36990, 32274}, {37477, 37950}, {38520, 38641}, {38789, 34128}, {38790, 1539}, {38791, 6723}, {38898, 13630}, {39985, 3134}, {41720, 11179}, {41737, 1352}, {43572, 37948}, {43574, 2071}, {43576, 7464}
X(74) = isogonal conjugate of X(30)
X(74) = isotomic conjugate of X(3260)
X(74) = complement of X(146)
X(74) = anticomplement of X(113)
X(74) = X(1)-Ceva conjugate of X(17149)
X(74) = cevapoint of X(i) and X(j) for these (i,j): (15,16), (50,184)
X(74) = crosssum of X(i) and X(j) for these (i,j): (3,399), (616),617)
X(74) = X(i)-cross conjugate of X(j) for these (i,j): (186,54), (526,110)
X(74) = circumcircle-antipode of X(110)
X(74) = polar-circle-inverse of X(133)
X(74) = 2nd-Droz-Farney-circle-inverse of X(17854)
X(74) = Schoutte-circle-inverse of X(2715)
X(74) = 2nd-Brocard-circle-inverse of X(38520)
X(74) = inverse-in-O(15,16) of X(2715), where O(15,16) is the circle having segment X(15)X(16) as diameter
X(74) = trilinear pole of line X(6)X(647)
X(74) = Ψ(X(6),X(647))
X(74) = antipode of X(1199) in Moses-Jerabek conic
X(74) = reflection of X(477) in the Euler line
X(74) = reflection of X(842) in the Brocard axis
X(74) = reflection of X(2687) in the line X(1)X(3)
X(74) = reflection of X(1296) in the line X(3)X(351)
X(74) = {X(3),X(399)}-harmonic conjugate of X(1511)
X(74) = X(128)-of-excentral-triangle
X(74) = X(137)-of-hexyl-triangle
X(74) = X(1296)-of-circumsymmedial
X(74) = trilinear pole wrt circumorthic triangle of van Aubel line
X(74) = X(1577)-isoconjugate of X(2420)
X(74) = orthocentroidal-to-ABC similarity image of X(4)
X(74) = 4th-Brocard-to-circumsymmedial similarity image of X(4)
X(74) = perspector of ABC and the reflection of the Kosnita triangle in X(3)
X(74) = orthocenter of X(3)X(67)X(879)
X(74) = intersection of tangents at X(3) and X(4) to Napoleon-Feuerbach cubic, K005
X(74) = X(1317)-of-tangential-triangle is ABC is acute
X(74) = 2nd-Parry-to-ABC similarity image of X(110)
X(74) = X(80)-of-Trinh-triangle if ABC is acute
X(74) = Trinh-isogonal conjugate of X(2071)
X(74) = trilinear product of PU(86)
X(74) = perspector of ABC and the (degenerate) side-triangle of the (equilateral) circumcevian triangles of X(15) and X(16)
X(74) = homothetic center of X(15)- and X(16)-Ehrmann triangles; see X(25)
X(74) = perspector of ABC and X(15)-Ehrmann triangle
X(74) = perspector of ABC and X(16)-Ehrmann triangle
X(74) = 3rd-Parry-to-circumsymmedial similarity image of X(23)
X(74) = perspector of ABC and unary cofactor triangle of orthocentroidal triangle
X(74) = endo-homothetic center of X(4)-altimedial and X(4)-anti-altimedial triangles
X(74) = Thomson isogonal conjugate of X(523)
X(74) = Lucas isogonal conjugate of X(523)
X(74) = X(100)-of-circumorthic-triangle if ABC is acute
X(74) = perspector of ABC and 2nd anti-Parry triangle
X(74) = X(110)-of-2nd-anti-Parry-triangle
X(74) = X(9138)-of-1st-anti-Parry-triangle
X(74) = excentral-to-ABC functional image of X(5541)
X(74) = orthic-to-ABC functional image of X(128)
X(74) = trilinear pole wrt, Thomson triangle, of line X(3)X(5646)
X(74) = trilinear pole, wrt Lucas triangle, of line X(4)X(15066)
X(74) = BSS(a→a^2) of X(1156)
X(74) = areal center of pedal triangles of PU(4)
X(74) = areal center of pedal triangles of PU(5)
X(74) = areal center of pedal triangles of PU(11)
X(74) = antipode of X(32111) in Walsmith rectangular hyperbola
X(74) = orthocenter of X(6)X(110)X(3569)
X(74) = X(5541)-of-orthic-triangle if ABC is acute
X(74) = perspector of circumconic centered at X(36896)
X(74) = trilinear pole of line X(6)X(647)
X(74) = crossdifference of every pair of points on line {1636, 1637}
X(74) = psi-transform of X(14685)
X(74) = Collings transform of X(i) for these i: {125, 2088, 3134, 39174, 39987}
X(74) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {10419, 8}, {36053, 146}, {40388, 5905}, {40423, 6327}
X(74) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 36896}, {34178, 10}
X(74) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 36896}, {30, 2132}, {1304, 14380}, {1494, 14919}, {2349, 15627}, {5627, 3470}, {9139, 9717}, {10419, 14385}, {14919, 15291}, {15395, 110}, {16077, 2394}, {16080, 8749}, {34568, 647}, {40384, 6}, {40423, 2}
X(74) = X(i)-cross conjugate of X(j) for these (i,j): {3, 10419}, {6, 40384}, {184, 11079}, {186, 54}, {526, 110}, {647, 34568}, {686, 4558}, {1464, 1}, {1495, 6}, {3003, 2}, {6000, 4}, {9142, 111}, {9409, 112}, {9717, 9139}, {11081, 2981}, {11086, 6151}, {12112, 14483}, {13289, 38534}, {13754, 14264}, {14157, 1173}, {14264, 5627}, {14380, 1304}, {16186, 523}, {18877, 14919}, {21650, 265}, {21663, 3}, {40352, 8749}
X(74) = cevapoint of X(i) and X(j) for these (i,j): {3, 13754}, {6, 1495}, {15, 16}, {50, 184}, {55, 2245}, {512, 2088}, {523, 3134}, {654, 3270}, {3269, 9409}, {3581, 4550}, {5663, 39987}, {6000, 39174}, {11074, 40355}, {18877, 40352}, {42789, 42790}
X(74) = crosspoint of X(i) and X(j) for these (i,j): {1, 7164}, {3, 8431}, {4, 1138}, {13, 8462}, {14, 8452}, {15, 8445}, {16, 8455}, {30, 2133}, {399, 8486}, {484, 7327}, {616, 8535}, {617, 8536}, {1157, 8487}, {1263, 8439}, {1337, 8489}, {1338, 8490}, {1494, 16080}, {2132, 8534}, {3065, 3466}, {3440, 3441}, {3464, 8488}, {3465, 7328}, {3479, 8491}, {3480, 8492}, {3481, 8494}, {5667, 8493}, {7059, 7325}, {7060, 7326}, {7165, 7329}, {8446, 8471}, {8456, 8479}, {8495, 8529}, {8496, 8531}, {8499, 8501}, {8500, 8502}, {18878, 39295}
X(74) = crosssum of X(i) and X(j) for these (i,j): {1, 3464}, {2, 39358}, {3, 399}, {4, 5667}, {13, 5623}, {14, 5624}, {15, 5668}, {16, 5669}, {74, 2132}, {484, 3465}, {616, 617}, {1138, 5670}, {1157, 3484}, {1263, 5671}, {1276, 5672}, {1277, 5673}, {1337, 5674}, {1338, 5675}, {1495, 3284}, {1650, 9033}, {2088, 21731}, {2133, 5676}, {3065, 5677}, {3081, 3163}, {3440, 5678}, {3441, 5679}, {3466, 5680}, {3479, 5681}, {3480, 5682}, {3481, 5683}, {3482, 5684}, {3483, 5685}, {7059, 8459}, {7060, 8449}, {7164, 8432}, {7165, 8485}, {7325, 8476}, {7326, 8468}, {7327, 8503}, {7328, 8527}, {7329, 8504}, {8172, 8174}, {8173, 8175}, {8431, 8440}, {8433, 8435}, {8434, 8436}, {8437, 8441}, {8438, 8442}, {8439, 8443}, {8444, 8464}, {8445, 8465}, {8446, 8466}, {8447, 8467}, {8448, 8462}, {8450, 8470}, {8451, 8469}, {8452, 8458}, {8453, 8471}, {8454, 8472}, {8455, 8473}, {8456, 8474}, {8457, 8475}, {8460, 8478}, {8461, 8477}, {8463, 8479}, {8480, 8505}, {8481, 8508}, {8482, 8509}, {8483, 8506}, {8484, 8507}, {8486, 8510}, {8487, 8511}, {8488, 8512}, {8489, 8515}, {8490, 8516}, {8491, 8513}, {8492, 8514}, {8493, 8517}, {8494, 8518}, {8495, 8519}, {8496, 8520}, {8497, 8521}, {8498, 8522}, {8499, 8525}, {8500, 8526}, {8501, 8523}, {8502, 8524}, {8528, 8530}, {8529, 8532}, {8531, 8533}, {12113, 12369}
X(74) = X(i)-isoconjugate of X(j) for these (i,j): {1, 30}, {2, 2173}, {3, 1784}, {6, 14206}, {9, 6357}, {19, 11064}, {31, 3260}, {37, 18653}, {57, 7359}, {63, 1990}, {74, 1099}, {75, 1495}, {76, 9406}, {92, 3284}, {100, 11125}, {110, 36035}, {113, 36053}, {162, 9033}, {163, 41079}, {190, 14399}, {240, 35912}, {265, 35201}, {304, 14581}, {402, 9390}, {561, 9407}, {647, 24001}, {648, 2631}, {649, 42716}, {651, 14400}, {653, 14395}, {656, 4240}, {661, 2407}, {662, 1637}, {759, 6739}, {799, 14398}, {811, 9409}, {823, 1636}, {896, 9214}, {897, 5642}, {1494, 42074}, {1511, 2166}, {1568, 2190}, {1577, 2420}, {1650, 24000}, {1725, 15454}, {1749, 3471}, {1895, 11589}, {1959, 35906}, {2153, 41887}, {2154, 41888}, {2159, 36789}, {2349, 3163}, {3431, 18486}, {5620, 16164}, {5664, 32678}, {6149, 14254}, {8772, 36891}, {9408, 33805}, {11251, 36062}, {14208, 23347}, {16163, 36119}, {24019, 41077}, {32679, 41392}, {34334, 35200}, {36037, 42750}
X(74) = barycentric product X(i)*X(j) for these {i,j}: {1, 2349}, {3, 16080}, {4, 14919}, {6, 1494}, {7, 15627}, {15, 36308}, {16, 36311}, {30, 40384}, {31, 33805}, {63, 36119}, {69, 8749}, {75, 2159}, {76, 40352}, {92, 35200}, {94, 14385}, {98, 35910}, {99, 2433}, {110, 2394}, {111, 36890}, {112, 34767}, {249, 12079}, {253, 15291}, {264, 18877}, {287, 35908}, {305, 40354}, {323, 5627}, {340, 11079}, {470, 39377}, {471, 39378}, {520, 15459}, {524, 9139}, {525, 1304}, {526, 39290}, {647, 16077}, {648, 14380}, {671, 9717}, {850, 32640}, {1073, 10152}, {1495, 31621}, {1577, 36034}, {2867, 15292}, {2966, 32112}, {2986, 14264}, {2987, 36875}, {3003, 40423}, {3260, 40353}, {3265, 32695}, {3267, 32715}, {3269, 42308}, {3470, 13582}, {3580, 10419}, {4558, 18808}, {7799, 40355}, {9033, 34568}, {14208, 36131}, {15412, 36831}, {20123, 40391}, {22455, 37638}, {30474, 32681}, {40050, 40351}
X(74) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 14206}, {2, 3260}, {3, 11064}, {6, 30}, {15, 41887}, {16, 41888}, {19, 1784}, {25, 1990}, {30, 36789}, {31, 2173}, {32, 1495}, {50, 1511}, {55, 7359}, {56, 6357}, {58, 18653}, {100, 42716}, {110, 2407}, {111, 9214}, {112, 4240}, {162, 24001}, {184, 3284}, {186, 14920}, {187, 5642}, {216, 1568}, {248, 35912}, {323, 6148}, {512, 1637}, {520, 41077}, {523, 41079}, {526, 5664}, {560, 9406}, {574, 13857}, {647, 9033}, {649, 11125}, {661, 36035}, {663, 14400}, {667, 14399}, {669, 14398}, {810, 2631}, {1304, 648}, {1384, 35266}, {1494, 76}, {1495, 3163}, {1501, 9407}, {1576, 2420}, {1946, 14395}, {1974, 14581}, {1976, 35906}, {1989, 14254}, {1990, 34334}, {2088, 3258}, {2159, 1}, {2173, 1099}, {2245, 6739}, {2349, 75}, {2394, 850}, {2420, 3233}, {2433, 523}, {2987, 36891}, {3003, 113}, {3049, 9409}, {3163, 23097}, {3269, 1650}, {3284, 16163}, {3310, 42750}, {3457, 36299}, {3458, 36298}, {3470, 37779}, {5063, 10564}, {5158, 1531}, {5627, 94}, {8739, 6110}, {8740, 6111}, {8749, 4}, {9139, 671}, {9406, 42074}, {9407, 9408}, {9408, 3081}, {9409, 14401}, {9412, 34582}, {9717, 524}, {10152, 15466}, {10317, 16165}, {10419, 2986}, {11060, 14583}, {11063, 10272}, {11074, 14993}, {11079, 265}, {12079, 338}, {14264, 3580}, {14380, 525}, {14385, 323}, {14560, 41392}, {14579, 3471}, {14581, 16240}, {14642, 11589}, {14910, 15454}, {14919, 69}, {15166, 14499}, {15167, 14500}, {15291, 20}, {15395, 39295}, {15451, 14391}, {15459, 6528}, {15627, 8}, {16077, 6331}, {16080, 264}, {18320, 38610}, {18808, 14618}, {18877, 3}, {19622, 16164}, {21906, 2682}, {22455, 43530}, {32112, 2799}, {32640, 110}, {32681, 1302}, {32695, 107}, {32715, 112}, {33805, 561}, {34397, 39176}, {34417, 18487}, {34568, 16077}, {34767, 3267}, {34952, 14397}, {35200, 63}, {35908, 297}, {35910, 325}, {36034, 662}, {36064, 38340}, {36119, 92}, {36131, 162}, {36308, 300}, {36311, 301}, {36430, 18484}, {36831, 14570}, {36890, 3266}, {36896, 146}, {39201, 1636}, {39290, 35139}, {39377, 40709}, {39378, 40710}, {39380, 10217}, {39381, 10218}, {40135, 13202}, {40351, 1974}, {40352, 6}, {40354, 25}, {40355, 1989}, {40384, 1494}, {40385, 39263}, {40388, 1300}, {40423, 40832}, {41336, 20772}, {42658, 14345}, {42671, 6793}, {43083, 18557}
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 146, 113}, {3, 110, 15035}, {3, 399, 1511}, {3, 1511, 15051}, {3, 5609, 15020}, {3, 6241, 1614}, {3, 11456, 11464}, {3, 12041, 15055}, {3, 12174, 9707}, {3, 12308, 32609}, {3, 14094, 15034}, {3, 14264, 14385}, {3, 15035, 15036}, {3, 15041, 12041}, {3, 15054, 14094}, {3, 32138, 11440}, {3, 32139, 11449}, {3, 33533, 41462}, {3, 38497, 38555}, {4, 125, 14644}, {4, 15081, 7687}, {4, 26937, 26917}, {5, 15061, 15059}, {6, 9412, 9408}, {20, 11457, 12289}, {24, 64, 12290}, {35, 19470, 10088}, {36, 7727, 10091}, {110, 15020, 32609}, {110, 15021, 15055}, {110, 15035, 15034}, {110, 15051, 1511}, {110, 15055, 3}, {113, 146, 10706}, {113, 6699, 2}, {125, 7687, 15081}, {125, 12244, 10721}, {125, 13202, 7687}, {184, 11204, 35473}, {185, 3520, 54}, {185, 11430, 15032}, {185, 13293, 15463}, {185, 32607, 13198}, {186, 12112, 1495}, {265, 10264, 9140}, {265, 20126, 10264}, {323, 2071, 10564}, {323, 10564, 43574}, {376, 12317, 12383}, {376, 12383, 16163}, {378, 1986, 15472}, {378, 5890, 15033}, {378, 10605, 5890}, {378, 17835, 7731}, {381, 38790, 1539}, {382, 38724, 10113}, {399, 1511, 110}, {399, 15042, 32609}, {548, 34153, 38723}, {549, 10272, 38794}, {616, 617, 6148}, {974, 1986, 5890}, {974, 2935, 15472}, {974, 11598, 378}, {974, 19457, 5622}, {1181, 8567, 35477}, {1204, 3357, 4}, {1495, 12112, 14157}, {1498, 32534, 26882}, {1511, 12358, 15066}, {1511, 15051, 15035}, {1539, 20304, 381}, {1995, 11472, 16261}, {2914, 15032, 12227}, {2930, 31884, 33851}, {2935, 5621, 19457}, {2935, 10605, 1986}, {2935, 10606, 11598}, {2935, 15151, 5622}, {2935, 19457, 378}, {3003, 40353, 36896}, {3357, 39174, 38937}, {3520, 7722, 15463}, {3520, 15032, 11430}, {3627, 15027, 15044}, {4550, 37470, 2}, {5092, 19140, 15462}, {5609, 32609, 110}, {5621, 17835, 974}, {5622, 10752, 6}, {5627, 14989, 34150}, {5642, 6053, 20125}, {5655, 38794, 10272}, {5890, 7731, 1986}, {5972, 38727, 631}, {6200, 12375, 10819}, {6241, 11464, 11456}, {6241, 11468, 3}, {6396, 12376, 10820}, {6723, 36518, 3090}, {6723, 38791, 36518}, {7488, 10575, 8718}, {7687, 13202, 4}, {7687, 15081, 14644}, {7722, 32607, 54}, {7725, 19059, 10752}, {7726, 19060, 10752}, {7728, 15061, 5}, {7731, 19457, 15033}, {8749, 18877, 15291}, {9140, 10733, 265}, {9408, 9412, 112}, {9717, 14264, 39239}, {9717, 39239, 3470}, {9786, 35502, 9781}, {9904, 11709, 7978}, {10065, 10081, 1}, {10113, 20379, 38724}, {10264, 20127, 10733}, {10605, 10606, 378}, {10605, 11598, 15472}, {10605, 19457, 974}, {10606, 17835, 2935}, {10620, 11454, 43578}, {10620, 12041, 110}, {10620, 13171, 17854}, {10620, 15021, 15035}, {10620, 15041, 3}, {10620, 15055, 14094}, {10620, 38633, 5609}, {10721, 14644, 4}, {10990, 20417, 4}, {11250, 34783, 34148}, {11413, 12163, 11412}, {11430, 15032, 54}, {11454, 15072, 3}, {11456, 11464, 1614}, {11598, 15151, 19457}, {12041, 15041, 15021}, {12041, 15054, 15035}, {12041, 38626, 32609}, {12079, 34150, 5627}, {12121, 38788, 550}, {12219, 12901, 43574}, {12244, 20417, 14644}, {12308, 32609, 5609}, {12308, 38633, 3}, {12358, 12412, 110}, {12358, 12825, 11459}, {12371, 12374, 10767}, {12381, 12382, 7978}, {13198, 15463, 54}, {13293, 17855, 13198}, {13293, 32607, 3520}, {13491, 32210, 3}, {13621, 33541, 32137}, {13630, 14130, 13434}, {14094, 15035, 110}, {14094, 15055, 15036}, {14264, 14385, 3470}, {14264, 15468, 110}, {14385, 39239, 9717}, {14480, 38701, 14934}, {14643, 38728, 140}, {14677, 20126, 10733}, {15021, 15054, 3}, {15021, 15055, 12041}, {15034, 15036, 15035}, {15041, 32609, 38633}, {15041, 34469, 17854}, {15054, 15055, 110}, {15057, 15059, 15061}, {15063, 38727, 5972}, {15072, 15100, 12270}, {15080, 15100, 399}, {16163, 37853, 376}, {17835, 19457, 1986}, {18933, 37643, 15081}, {19059, 19060, 6}, {20126, 20127, 265}, {20427, 26937, 4}, {22462, 33539, 11017}, {23236, 38723, 34153}, {31954, 31955, 5890}, {32616, 32617, 6241}, {34150, 40630, 12079}, {36518, 38729, 6723}, {38626, 38633, 110}, {38632, 38638, 110}, {38641, 38653, 110}, {38650, 38661, 110}, {38729, 38791, 3090}, {38937, 39174, 38933}


X(75) = ISOTOMIC CONJUGATE OF INCENTER

Trilinears    1/a2 : 1/b2 : 1/c2
Trilinears    1/(1 - cos 2A) : 1/(1 - cos 2B) : 1/(1 - cos 2C)
Trilinears    SA + S2 : SB + S2 : SC + S2
Trilinears    sec2(A/2) + csc2(A/2) : :
Trilinears    d(a,b,c) : : , where d(a,b,c) = distance from A to Lemoine axis
Trilinears    h(a,b,c) : : , where h(a,b,c) = (distance from A to antiorthic axis)2
Barycentrics    1/a : 1/b : 1/c
Barycentrics   csc A : csc B : csc C
X(75) = (csc A)*[A] + (csc B)*[B] + (csc C)*[C], where A, B, C denote angles and [A], [B], [C] denote vertices
X(75) = 3 X[2] + X[1278], 6 X[2] - X[3644], 3 X[2] - 4 X[3739], 9 X[2] - 4 X[4681], 3 X[2] + 2 X[4686], 6 X[2] - 5 X[4687], 9 X[2] - 8 X[4698], 3 X[2] - 5 X[4699], 9 X[2] - 5 X[4704], 9 X[2] - 2 X[4718], 3 X[2] + 4 X[4726], 3 X[2] - 8 X[4739], 6 X[2] - 7 X[4751], 5 X[2] - 4 X[4755], 6 X[2] + X[4764], 3 X[2] - 7 X[4772], 9 X[2] - X[4788], 3 X[2] + 5 X[4821], 9 X[2] - 7 X[27268], 9 X[2] - 10 X[31238], 2 X[37] + X[1278], 4 X[37] - X[3644], 4 X[37] - 3 X[4664], 3 X[37] - 2 X[4681], 4 X[37] - 5 X[4687], X[37] - 3 X[4688], 3 X[37] - 4 X[4698], 2 X[37] - 5 X[4699], 6 X[37] - 5 X[4704], 3 X[37] - X[4718], X[37] + 2 X[4726], X[37] - 4 X[4739], 2 X[37] + 3 X[4740], 4 X[37] - 7 X[4751], 5 X[37] - 6 X[4755], 4 X[37] + X[4764], 2 X[37] - 7 X[4772], 6 X[37] - X[4788], 2 X[37] + 5 X[4821], 6 X[37] - 7 X[27268], 3 X[37] - 5 X[31238], 4 X[142] - 3 X[27475], X[144] - 3 X[27484], X[192] - 4 X[3739], 2 X[192] - 3 X[4664], 3 X[192] - 4 X[4681], X[192] + 2 X[4686], 2 X[192] - 5 X[4687], X[192] - 6 X[4688], 3 X[192] - 8 X[4698], X[192] - 5 X[4699], 3 X[192] - 5 X[4704], 3 X[192] - 2 X[4718], X[192] + 4 X[4726], X[192] - 8 X[4739], X[192] + 3 X[4740], 2 X[192] - 7 X[4751], 5 X[192] - 12 X[4755], 2 X[192] + X[4764], X[192] - 7 X[4772]

Let A2B2C2 be the 2nd Conway triangle. Let A' be the trilinear pole of line B2C2, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(75). (Randy Hutson, December 10, 2016)

Let A4B4C4 be the 4th Conway triangle. Let A' be the trilinear pole of line B4C4, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(75). (Randy Hutson, December 10, 2016)

Let AaBaCa, AbBbCb, AcBcCc be the A-, B-, and C-anti-altimedial triangles, resp. Let A' be the trilinear product Ba*Ca, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(75). (Randy Hutson, November 2, 2017)

Let A23B23C23 be Gemini triangle 23. Let A' be the perspector of conic {A,B,C,B23,C23}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(75). (Randy Hutson, January 15, 2019)

Let A40B40C40 be Gemini triangle 40. Let A' be the perspector of conic {A,B,C,B40,C40}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(75). (Randy Hutson, January 15, 2019)

If you have Geometer's Sketchpad, X(75).
If you have GeoGebra, you can view X(75).

X(75) lies on the circumconic {A,B,C,X(2),X(7)}}, the cubics K014, K034, K132, K183, K184, K254, K276, K286, K323, K366, K432, K507, K605, K697, K738, K743, K766, K767, K768, K862, K863, K865, K868, K968, K970, K985, K986, K990, K992, K994, K995, K996, K999, K1015, K1020, K1022, K1031, K1032, K1038, the curve Q124, and these lines: {1, 86}, {2, 37}, {3, 17864}, {4, 12689}, {5, 19839}, {6, 239}, {7, 8}, {9, 190}, {10, 76}, {11, 16067}, {12, 21405}, {19, 27}, {20, 30271}, {21, 272}, {22, 21407}, {23, 21408}, {31, 82}, {32, 746}, {34, 37087}, {35, 21410}, {36, 21411}, {38, 310}, {39, 14622}, {40, 10444}, {41, 21414}, {42, 1218}, {43, 872}, {44, 16816}, {45, 16815}, {47, 2216}, {48, 336}, {55, 3757}, {56, 17887}, {57, 4032}, {58, 21421}, {66, 21423}, {71, 28287}, {72, 1246}, {77, 664}, {78, 7190}, {81, 2214}, {83, 14619}, {87, 3226}, {88, 21427}, {89, 21428}, {91, 20571}, {99, 261}, {100, 675}, {101, 767}, {105, 20628}, {110, 21430}, {115, 21431}, {141, 334}, {142, 2321}, {144, 391}, {145, 3945}, {149, 2805}, {150, 2893}, {158, 240}, {162, 37220}, {171, 3769}, {172, 4372}, {183, 1376}, {187, 21434}, {193, 4371}, {194, 1107}, {200, 4328}, {210, 18142}, {219, 1944}, {220, 27420}, {222, 1943}, {223, 20238}, {225, 264}, {226, 3687}, {234, 556}, {236, 40893}, {238, 3923}, {242, 24320}, {244, 1978}, {255, 2190}, {256, 3764}, {257, 698}, {267, 6763}, {269, 1222}, {277, 30701}, {279, 1219}, {280, 309}, {281, 27509}, {282, 20239}, {291, 9230}, {292, 20630}, {298, 1081}, {299, 554}, {305, 3914}, {306, 5249}, {308, 40093}, {311, 18815}, {315, 4911}, {317, 5081}, {325, 2886}, {329, 14555}, {330, 7187}, {339, 23674}, {342, 18026}, {354, 3706}, {355, 21277}, {365, 20631}, {366, 20434}, {379, 5279}, {384, 4426}, {385, 4386}, {404, 19850}, {405, 7283}, {480, 28058}, {489, 31550}, {490, 31549}, {491, 1659}, {492, 13390}, {497, 11997}, {511, 15978}, {513, 17159}, {514, 4406}, {516, 3883}, {517, 10446}, {519, 3664}, {522, 3261}, {523, 876}, {524, 4399}, {525, 17899}, {527, 3686}, {537, 668}, {538, 1573}, {545, 4957}, {551, 4717}, {560, 1580}, {573, 29069}, {584, 40744}, {596, 16887}, {597, 17225}, {599, 4445}, {612, 32926}, {614, 32942}, {646, 4859}, {647, 21437}, {649, 20909}, {650, 21438}, {651, 28968}, {656, 17893}, {659, 21439}, {667, 21440}, {669, 21441}, {670, 18827}, {671, 35181}, {672, 20632}, {689, 745}, {693, 4411}, {700, 871}, {712, 3735}, {716, 16829}, {723, 9063}, {728, 1223}, {730, 24293}, {735, 9065}, {748, 32930}, {749, 20456}, {750, 17763}, {751, 23659}, {753, 789}, {756, 18152}, {757, 1468}, {758, 994}, {765, 39293}, {775, 1496}, {798, 20910}, {799, 897}, {811, 1099}, {812, 20908}, {825, 40371}, {826, 18077}, {846, 32934}, {850, 4467}, {873, 40438}, {889, 16495}, {896, 20904}, {899, 32931}, {900, 3766}, {901, 2863}, {905, 23685}, {908, 4054}, {918, 24141}, {927, 2751}, {934, 2370}, {936, 40424}, {940, 1999}, {942, 5295}, {956, 5088}, {958, 1975}, {964, 5262}, {966, 3975}, {969, 17156}, {980, 10472}, {982, 1920}, {990, 13727}, {991, 29016}, {997, 4561}, {1001, 3685}, {1030, 27788}, {1073, 20240}, {1078, 25440}, {1088, 3668}, {1089, 1268}, {1100, 4393}, {1104, 4195}, {1108, 27334}, {1119, 7046}, {1125, 3993}, {1150, 3218}, {1211, 3782}, {1212, 25242}, {1213, 3948}, {1214, 27339}, {1228, 5051}, {1230, 41809}, {1233, 25006}, {1234, 41501}, {1237, 1240}, {1247, 6626}, {1249, 20241}, {1271, 38236}, {1272, 40716}, {1281, 5989}, {1330, 5814}, {1332, 2989}, {1369, 5014}, {1370, 3434}, {1423, 16609}, {1429, 24334}, {1442, 4861}, {1444, 2217}, {1446, 43533}, {1449, 16834}, {1486, 26241}, {1500, 27255}, {1501, 33733}, {1574, 3934}, {1577, 23894}, {1581, 1934}, {1621, 32929}, {1631, 7087}, {1654, 3765}, {1655, 26045}, {1697, 10889}, {1734, 35035}, {1742, 28850}, {1743, 16833}, {1746, 21375}, {1757, 32935}, {1766, 6996}, {1799, 17873}, {1812, 2219}, {1826, 18747}, {1836, 3966}, {1847, 40445}, {1890, 5342}, {1895, 5931}, {1897, 2000}, {1914, 4376}, {1916, 40099}, {1917, 33807}, {1925, 18276}, {1928, 2085}, {1931, 20634}, {1953, 1959}, {1962, 39737}, {1973, 34065}, {1992, 35578}, {2053, 14199}, {2054, 20636}, {2064, 40940}, {2092, 27042}, {2112, 20638}, {2140, 22011}, {2166, 23994}, {2167, 2168}, {2170, 33946}, {2172, 20931}, {2175, 24264}, {2176, 16827}, {2178, 11329}, {2210, 4381}, {2238, 24330}, {2262, 20348}, {2275, 16720}, {2285, 41245}, {2292, 28660}, {2295, 17033}, {2303, 26643}, {2309, 21352}, {2319, 20438}, {2324, 27384}, {2325, 6666}, {2329, 16822}, {2339, 41260}, {2352, 13588}, {2400, 4397}, {2475, 5016}, {2530, 18081}, {2640, 16556}, {2652, 30988}, {2783, 5977}, {2870, 21293}, {2887, 17889}, {2894, 2897}, {2895, 17483}, {2908, 30878}, {2968, 6356}, {2998, 16606}, {3006, 30632}, {3008, 17352}, {3009, 20637}, {3035, 37688}, {3056, 4459}, {3061, 17760}, {3086, 25583}, {3120, 25760}, {3121, 21224}, {3122, 17065}, {3123, 6386}, {3161, 18230}, {3164, 18606}, {3177, 4875}, {3185, 11688}, {3219, 5278}, {3227, 7208}, {3230, 35274}, {3241, 3902}, {3244, 4464}, {3247, 16831}, {3252, 4562}, {3260, 20565}, {3265, 23683}, {3266, 4442}, {3314, 20541}, {3405, 33793}, {3501, 27626}, {3509, 24586}, {3570, 9318}, {3573, 24346}, {3578, 30690}, {3589, 4395}, {3598, 7172}, {3616, 3702}, {3617, 4346}, {3618, 4402}, {3619, 29611}, {3622, 32105}, {3624, 6533}, {3625, 4896}, {3626, 4887}, {3629, 4405}, {3631, 4478}, {3632, 4888}, {3633, 7278}, {3634, 4066}, {3635, 4909}, {3670, 10468}, {3681, 4651}, {3684, 24333}, {3688, 14839}, {3694, 25521}, {3695, 8728}, {3700, 24622}, {3701, 5936}, {3703, 3925}, {3704, 25466}, {3707, 4480}, {3713, 5228}, {3714, 3812}, {3720, 4365}, {3723, 28639}, {3726, 30945}, {3731, 16832}, {3740, 3967}, {3747, 16690}, {3750, 29651}, {3763, 17290}, {3771, 33130}, {3773, 3836}, {3774, 25538}, {3775, 4710}, {3776, 29739}, {3780, 4754}, {3789, 17794}, {3790, 3826}, {3791, 4697}, {3807, 30997}, {3828, 4125}, {3831, 24172}, {3834, 17229}, {3835, 27485}, {3840, 17063}, {3841, 30172}, {3846, 3944}, {3848, 26103}, {3869, 17139}, {3870, 3996}, {3873, 13476}, {3874, 33297}, {3877, 17183}, {3882, 29382}, {3888, 9016}, {3889, 17169}, {3891, 3920}, {3896, 17018}, {3909, 25049}, {3926, 17869}, {3930, 30949}, {3933, 17867}, {3936, 27476}, {3938, 32923}, {3943, 17242}, {3946, 5750}, {3949, 30985}, {3950, 29571}, {3952, 40607}, {3961, 32920}, {3965, 26125}, {3969, 18139}, {3974, 26040}, {3981, 21954}, {3992, 18145}, {4007, 6173}, {4010, 25759}, {4011, 17123}, {4016, 4469}, {4024, 18154}, {4025, 17894}, {4030, 34612}, {4036, 18158}, {4044, 5257}, {4051, 9311}, {4053, 24063}, {4056, 4894}, {4058, 21255}, {4072, 29600}, {4073, 18033}, {4081, 6067}, {4082, 18153}, {4083, 23807}, {4085, 29659}, {4086, 18160}, {4094, 21254}, {4099, 32009}, {4124, 24482}, {4132, 17217}, {4136, 17062}, {4150, 18744}, {4234, 37817}, {4319, 14942}, {4353, 19868}, {4366, 17000}, {4377, 17235}, {4383, 27064}, {4387, 4423}, {4390, 9317}, {4391, 28898}, {4392, 35543}, {4396, 16997}, {4403, 33908}, {4408, 4926}, {4410, 4690}, {4412, 7122}, {4413, 5205}, {4414, 32845}, {4415, 5743}, {4422, 17337}, {4425, 33154}, {4432, 15485}, {4436, 8053}, {4438, 33138}, {4470, 26626}, {4471, 7246}, {4472, 17045}, {4475, 18168}, {4484, 7241}, {4488, 6172}, {4494, 17304}, {4495, 36480}, {4497, 7236}, {4500, 29808}, {4505, 25351}, {4511, 7269}, {4513, 25878}, {4515, 6706}, {4517, 20694}, {4519, 30947}, {4552, 17077}, {4553, 25279}, {4554, 5231}, {4555, 35175}, {4563, 23673}, {4568, 17761}, {4569, 18025}, {4572, 20567}, {4586, 43099}, {4592, 17881}, {4595, 21232}, {4597, 36818}, {4599, 37221}, {4641, 37652}, {4648, 17314}, {4649, 4716}, {4655, 32857}, {4660, 24715}, {4661, 25286}, {4668, 4902}, {4672, 4974}, {4675, 4851}, {4678, 25278}, {4679, 17777}, {4680, 7272}, {4683, 33098}, {4684, 5542}, {4693, 16484}, {4702, 42819}, {4703, 33099}, {4708, 28633}, {4712, 18031}, {4713, 37673}, {4723, 36588}, {4742, 38314}, {4748, 28635}, {4760, 10987}, {4766, 27477}, {4795, 40891}, {4797, 21793}, {4798, 29586}, {4854, 25597}, {4865, 32866}, {4869, 29616}, {4873, 20195}, {4882, 7274}, {4886, 5739}, {4889, 28329}, {4897, 23835}, {4898, 29602}, {4899, 24393}, {4901, 38200}, {4915, 7271}, {4941, 20340}, {4966, 25557}, {4970, 17592}, {4971, 17388}, {4972, 8024}, {4975, 25055}, {4981, 7226}, {4996, 7279}, {5082, 17170}, {5086, 21270}, {5176, 21286}, {5178, 21285}, {5211, 17721}, {5254, 26558}, {5267, 7782}, {5277, 30167}, {5283, 16819}, {5287, 34064}, {5294, 26723}, {5296, 28809}, {5301, 24335}, {5311, 32928}, {5341, 24587}, {5372, 23958}, {5437, 30567}, {5515, 37842}, {5540, 33951}, {5550, 25585}, {5620, 21207}, {5698, 24280}, {5736, 34772}, {5737, 38000}, {5738, 12649}, {5741, 31053}, {5744, 27472}, {5774, 36279}, {5791, 25446}, {5827, 9654}, {5902, 10452}, {6007, 17049}, {6147, 41014}, {6180, 40862}, {6210, 29057}, {6327, 20292}, {6335, 7101}, {6337, 30478}, {6360, 18607}, {6375, 6377}, {6379, 22184}, {6390, 23679}, {6535, 25961}, {6539, 40013}, {6547, 36230}, {6682, 17591}, {6703, 29841}, {6707, 29612}, {7009, 37581}, {7013, 37278}, {7017, 15466}, {7032, 18170}, {7034, 10010}, {7035, 9458}, {7146, 30097}, {7155, 24451}, {7176, 12513}, {7191, 24552}, {7193, 24332}, {7196, 24477}, {7200, 9263}, {7209, 7233}, {7225, 18048}, {7243, 18043}, {7244, 17598}, {7308, 30568}, {7318, 10527}, {7336, 24250}, {7377, 12610}, {7752, 25639}, {7758, 31416}, {7760, 30133}, {7763, 26363}, {7764, 31488}, {7781, 31456}, {7991, 10442}, {8033, 13610}, {8055, 41926}, {8056, 32017}, {8058, 30805}, {8061, 18070}, {8632, 24354}, {8769, 17890}, {8773, 17876}, {9466, 27076}, {9508, 14296}, {9776, 18141}, {9965, 14552}, {10025, 37658}, {10176, 33948}, {10327, 41916}, {10400, 16091}, {10448, 40430}, {10538, 20477}, {10589, 30740}, {10980, 35613}, {11104, 19849}, {11108, 19852}, {11680, 33864}, {12530, 20556}, {12618, 36652}, {14018, 41013}, {14548, 36845}, {14624, 39957}, {14923, 21271}, {14997, 41241}, {15149, 18721}, {15523, 18052}, {15668, 16777}, {16496, 24841}, {16504, 36236}, {16525, 17475}, {16547, 20602}, {16548, 16566}, {16549, 29433}, {16551, 40476}, {16552, 20605}, {16560, 24591}, {16569, 25106}, {16574, 20367}, {16583, 27299}, {16600, 30107}, {16604, 20363}, {16605, 25994}, {16608, 37796}, {16666, 37677}, {16672, 29578}, {16678, 23339}, {16696, 16738}, {16707, 17150}, {16710, 16726}, {16711, 16714}, {16713, 16728}, {16722, 36857}, {16727, 17154}, {16729, 30564}, {16737, 17166}, {16741, 17162}, {16747, 18656}, {16750, 17884}, {16751, 31296}, {16814, 25269}, {16876, 18610}, {16884, 29584}, {16886, 33841}, {16888, 36482}, {16892, 18071}, {16969, 25129}, {16973, 32029}, {16974, 17688}, {17011, 19684}, {17017, 32772}, {17026, 17754}, {17027, 24512}, {17028, 20331}, {17034, 17750}, {17038, 27798}, {17050, 29960}, {17053, 26979}, {17061, 29634}, {17064, 30761}, {17067, 29596}, {17073, 28755}, {17084, 30543}, {17122, 29649}, {17133, 29574}, {17136, 17221}, {17138, 17153}, {17171, 18720}, {17181, 24390}, {17184, 32782}, {17197, 18177}, {17205, 39697}, {17206, 23555}, {17251, 17252}, {17265, 17266}, {17309, 17310}, {17323, 17324}, {17372, 17373}, {17442, 18717}, {17443, 30052}, {17449, 31136}, {17451, 30036}, {17458, 21191}, {17484, 37656}, {17486, 23632}, {17489, 26035}, {17595, 24627}, {17596, 32916}, {17600, 29644}, {17671, 21073}, {17682, 17742}, {17717, 25385}, {17733, 37607}, {17736, 29473}, {17741, 17743}, {17751, 20247}, {17758, 40006}, {17793, 25120}, {17797, 17798}, {17888, 29857}, {17896, 35518}, {17903, 40015}, {18022, 22069}, {18054, 21026}, {18055, 20706}, {18066, 31079}, {18067, 28595}, {18073, 29591}, {18080, 21123}, {18136, 28654}, {18162, 20769}, {18206, 29767}, {18207, 23669}, {18208, 20274}, {18297, 20527}, {18359, 30608}, {18601, 27163}, {18661, 26734}, {18739, 30713}, {18745, 21094}, {18811, 25719}, {18821, 35171}, {18826, 36873}, {18834, 34054}, {18835, 24211}, {19582, 25917}, {19701, 20182}, {19857, 37039}, {19863, 25599}, {19974, 39786}, {19975, 19976}, {20016, 20090}, {20045, 39743}, {20072, 31300}, {20081, 41838}, {20133, 20164}, {20134, 20175}, {20138, 27949}, {20139, 20167}, {20146, 20168}, {20147, 20180}, {20151, 20178}, {20161, 20166}, {20176, 20177}, {20254, 20256}, {20255, 20271}, {20259, 20260}, {20262, 40880}, {20267, 30103}, {20276, 20545}, {20332, 20639}, {20335, 21101}, {20337, 34528}, {20351, 20538}, {20352, 21278}, {20358, 24351}, {20535, 30082}, {20598, 24458}, {20992, 32117}, {21008, 27954}, {21021, 21897}, {21033, 30961}, {21071, 29968}, {21085, 33064}, {21178, 21186}, {21180, 21205}, {21196, 25667}, {21208, 27808}, {21210, 24207}, {21212, 29427}, {21216, 41015}, {21226, 40908}, {21231, 22370}, {21240, 24190}, {21242, 29676}, {21244, 38406}, {21299, 24717}, {21345, 23488}, {21351, 22226}, {21362, 29698}, {21368, 24595}, {21454, 37655}, {21511, 38871}, {21769, 28365}, {21801, 29965}, {21805, 31161}, {21808, 29966}, {21816, 25661}, {21834, 42327}, {21868, 25102}, {21871, 41828}, {21872, 30011}, {21956, 26590}, {22047, 24050}, {22218, 26974}, {22279, 22289}, {22413, 23440}, {23478, 23498}, {23481, 23502}, {23482, 23483}, {23484, 23500}, {23485, 23493}, {23486, 23495}, {23518, 28706}, {23556, 34254}, {23557, 37804}, {23664, 40050}, {23681, 25527}, {23682, 24445}, {23690, 24248}, {23897, 27966}, {23903, 27706}, {23978, 25000}, {24003, 36863}, {24005, 25023}, {24046, 24166}, {24058, 24224}, {24162, 24178}, {24169, 33174}, {24180, 39714}, {24183, 24184}, {24188, 24233}, {24202, 30144}, {24204, 33833}, {24214, 31327}, {24241, 29655}, {24254, 40859}, {24319, 27691}, {24324, 27950}, {24338, 24517}, {24343, 39914}, {24378, 24439}, {24386, 32023}, {24411, 36278}, {24430, 40717}, {24450, 39717}, {24487, 25382}, {24505, 36225}, {24510, 39362}, {24599, 37681}, {24621, 37596}, {24663, 26106}, {24693, 32847}, {24725, 32843}, {24779, 24781}, {24880, 25471}, {24892, 33119}, {24922, 25469}, {24943, 33123}, {24957, 25687}, {24986, 25005}, {24988, 26235}, {24995, 36568}, {24999, 26541}, {25002, 25964}, {25019, 26001}, {25072, 31211}, {25082, 27109}, {25083, 25252}, {25122, 33789}, {25130, 32095}, {25237, 26770}, {25244, 26690}, {25271, 26049}, {25296, 33800}, {25345, 31090}, {25453, 32780}, {25496, 29821}, {25512, 27785}, {25525, 41878}, {25591, 27627}, {25741, 25758}, {26034, 33068}, {26061, 29850}, {26098, 33071}, {26100, 27026}, {26128, 32783}, {26223, 32911}, {26229, 26263}, {26265, 38869}, {26364, 32832}, {26580, 33151}, {26627, 37633}, {26685, 37650}, {26688, 37687}, {26738, 31030}, {26772, 26976}, {26819, 35058}, {26840, 37653}, {26842, 32863}, {26963, 27809}, {26978, 28598}, {27017, 27107}, {27035, 27285}, {27044, 27095}, {27164, 40773}, {27340, 40133}, {27482, 31344}, {27493, 30566}, {27508, 28827}, {27549, 38057}, {27757, 30834}, {27793, 31037}, {27797, 39994}, {28301, 41848}, {28470, 30183}, {28640, 29592}, {28739, 37800}, {28780, 37771}, {28965, 40863}, {29036, 41430}, {29379, 29511}, {29381, 29405}, {29456, 29559}, {29561, 34017}, {29585, 31342}, {29609, 31319}, {29613, 34573}, {29629, 40480}, {29631, 33128}, {29632, 33156}, {29635, 33135}, {29642, 33158}, {29643, 32848}, {29653, 33092}, {29671, 32855}, {29673, 32865}, {29679, 39998}, {29709, 29711}, {29840, 30660}, {29846, 33127}, {29849, 33105}, {29967, 34830}, {29979, 29985}, {30092, 37598}, {30099, 39244}, {30603, 40603}, {30964, 30970}, {31145, 32093}, {31270, 40780}, {31350, 31351}, {31418, 32816}, {31449, 31859}, {31625, 40095}, {31627, 34019}, {32019, 42326}, {32020, 40533}, {32033, 40881}, {32101, 39710}, {32775, 33143}, {32776, 33145}, {32781, 33125}, {32842, 33070}, {32844, 33104}, {32852, 32949}, {32854, 33072}, {32856, 33065}, {32861, 32946}, {32864, 32912}, {32947, 33094}, {32948, 33074}, {32950, 33083}, {33067, 33080}, {33069, 33081}, {33073, 33088}, {33091, 39723}, {33114, 33139}, {33115, 33161}, {33117, 33162}, {33118, 33163}, {33120, 33136}, {33121, 33137}, {33122, 33148}, {33124, 33171}, {33126, 33144}, {33676, 35026}, {33678, 33679}, {33776, 38831}, {33947, 34860}, {34018, 39959}, {34021, 34022}, {34261, 41251}, {35119, 36232}, {35157, 35164}, {35527, 40362}, {35616, 39773}, {36625, 38254}, {36805, 39963}, {37520, 37684}, {37680, 41242}, {38247, 39740}, {39570, 40333}, {39693, 39748}, {39698, 39706}, {39722, 39724}, {39729, 39730}, {39736, 39738}, {39742, 42055}, {39775, 42079}, {40024, 40094}, {41236, 41247}, {41777, 43040}, {42015, 42311}

X(75) = midpoint of X(i) and X(j) for these {i,j}: {2, 4740}, {8, 24349}, {31, 37003}, {37, 4686}, {192, 1278}, {3644, 4764}, {3739, 4726}, {4399, 7228}, {4440, 33888}, {4699, 4821}, {17362, 17365}, {17363, 17364}, {18830, 40844}
X(75) = reflection of X(i) in X(j) for these {i,j}: {1, 24325}, {2, 4688}, {8, 3696}, {20, 30271}, {31, 18805}, {37, 3739}, {76, 21443}, {190, 17755}, {192, 37}, {335, 1086}, {693, 4411}, {984, 10}, {1278, 4686}, {2667, 25124}, {3261, 20907}, {3644, 192}, {3739, 4739}, {3758, 31317}, {3879, 3664}, {3993, 1125}, {4043, 20891}, {4094, 21254}, {4416, 3686}, {4664, 2}, {4681, 4698}, {4686, 4726}, {4687, 4699}, {4704, 31238}, {4718, 4681}, {4751, 4772}, {4764, 1278}, {4788, 4718}, {6376, 10009}, {7199, 4374}, {17333, 17330}, {17334, 17332}, {17347, 4416}, {17362, 4399}, {17363, 17362}, {17364, 17365}, {17365, 7228}, {17377, 3879}, {17388, 17390}, {17389, 17392}, {17458, 21191}, {18080, 21123}, {18137, 20892}, {20430, 5}, {20448, 20435}, {20924, 20893}, {20949, 20906}, {20950, 20908}, {20954, 3261}, {20956, 20912}, {21143, 21211}, {21606, 21433}, {21746, 17049}, {21834, 42327}, {23794, 4408}, {24505, 36225}, {30273, 3}, {32453, 39}, {35959, 40878}, {36494, 27478}, {37842, 5515}, {39467, 6374}, {41683, 244}, {42083, 24003}
X(75) = isogonal conjugate of X(31)
X(75) = isotomic conjugate of X(1)
X(75) = complement of X(192)
X(75) = anticomplement of X(37)
X(75) = anticomplementary conjugate of X(2895)
X(75) = complementary conjugate of X(21250)
X(75) = antigonal conjugate of X(37842)
X(75) = antitomic image of X(335)
X(75) = cyclocevian conjugate of X(8044)
X(75) = trilinear pole of line X(514)X(661)
X(75) = crossdifference of every pair of points on line {667, 788}
X(75) = medial-isogonal conjugate of X(21250)
X(75) = anticomplementary-isogonal conjugate of X(2895)
X(75) = X(i)-Hirst inverse of X(j) for these (i,j): (2,350), (334,335)
X(75) = X(83)-aleph conjugate of X(31)
X(75) = X(i)-beth conjugate of X(j) for these (i,j): (8,984), (75,7), (99,77), (314,75), (522,876), (645,9), (646,75), (668,75), (811,342)
X(75) = X(37)-of-anticomplementary triangle.
X(75) = trilinear product of PU(i) for these i: 3, 35
X(75) = barycentric product of PU(10)
X(75) = trilinear product of PU(75)
X(75) = X(6752)-of-excentral-triangle
X(75) = trilinear pole of line X(514)X(661)
X(75) = pole wrt polar circle of trilinear polar of X(19) (line X(661)X(663))
X(75) = X(48)-isoconjugate (polar conjugate) of X(19)
X(75) = X(6)-isoconjugate of X(6)
X(75) = crosspoint of X(1) and X(63) with respect to the excentral triangle
X(75) = crosspoint of X(1) and X(63) with respect to the anticomplementary triangle
X(75) = trilinear square of X(2)
X(75) = trilinear square root of X(561)
X(75) = trilinear product of the four CPCC points; http://bernard-gibert.fr/Tables/table11.html
X(75) = perspector of ABC and extraversion triangle of X(75) (which is also the anticevian triangle of X(75))
X(75) = perspector of ABC and cross-triangle of Gemini triangles 3 and 6
X(75) = perspector of Gemini triangle 13 and cross-triangle of ABC and Gemini triangle 13
X(75) = perspector of ABC and cross-triangle of ABC and Gemini triangle 21
X(75) = perspector of ABC and cross-triangle of ABC and Gemini triangle 22
X(75) = perspector of ABC and cross-triangle of Gemini triangles 21 and 22
X(75) = barycentric product of vertices of Gemini triangle 21
X(75) = barycentric product of vertices of Gemini triangle 22
X(75) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 2895}, {2, 1330}, {3, 3151}, {6, 1654}, {7, 2893}, {21, 329}, {27, 4}, {28, 5905}, {31, 1655}, {48, 18666}, {56, 17778}, {57, 2475}, {58, 2}, {59, 3882}, {60, 63}, {75, 21287}, {77, 2897}, {81, 8}, {82, 3770}, {86, 69}, {99, 20295}, {101, 31290}, {110, 514}, {112, 25259}, {162, 4391}, {163, 17494}, {222, 3152}, {249, 4427}, {250, 14543}, {251, 17499}, {261, 20245}, {267, 1029}, {270, 92}, {274, 6327}, {284, 144}, {285, 189}, {286, 21270}, {310, 315}, {314, 21286}, {333, 3436}, {513, 21221}, {514, 3448}, {552, 20244}, {593, 1}, {603, 18667}, {643, 4462}, {648, 20293}, {649, 148}, {662, 513}, {667, 21220}, {670, 21304}, {693, 21294}, {741, 6542}, {757, 75}, {759, 17484}, {763, 17140}, {799, 21301}, {849, 17147}, {873, 17137}, {967, 26051}, {1014, 7}, {1019, 149}, {1098, 18750}, {1169, 894}, {1171, 10}, {1172, 5942}, {1175, 3219}, {1178, 6646}, {1333, 192}, {1396, 12649}, {1408, 3210}, {1412, 145}, {1414, 693}, {1434, 3434}, {1437, 6360}, {1444, 4329}, {1459, 39352}, {1474, 193}, {1509, 17135}, {1576, 21225}, {1790, 20}, {1798, 3101}, {1817, 6223}, {1914, 39367}, {1919, 25054}, {1929, 20349}, {2163, 37635}, {2185, 3869}, {2194, 3177}, {2203, 21216}, {2206, 194}, {2221, 26117}, {2248, 6625}, {2299, 30694}, {2328, 30695}, {2360, 20211}, {2363, 321}, {3285, 17487}, {3286, 20533}, {3733, 4440}, {3737, 37781}, {4025, 13219}, {4091, 34186}, {4184, 17732}, {4273, 17488}, {4556, 523}, {4558, 20294}, {4560, 33650}, {4564, 3909}, {4565, 522}, {4567, 3952}, {4570, 190}, {4573, 21302}, {4589, 21303}, {4591, 900}, {4600, 668}, {4610, 512}, {4620, 3888}, {4622, 21297}, {4623, 17217}, {4627, 4778}, {4629, 4977}, {4637, 3900}, {5009, 33888}, {5235, 21291}, {5331, 5739}, {5546, 4468}, {6385, 21275}, {6578, 4608}, {6628, 17143}, {6629, 14360}, {6650, 20558}, {6727, 16017}, {7192, 150}, {7199, 21293}, {7252, 39351}, {7303, 17152}, {7341, 3875}, {8025, 2891}, {8747, 6515}, {10566, 25051}, {13486, 1577}, {14534, 17751}, {14616, 21277}, {14953, 152}, {15376, 20017}, {16696, 21289}, {16704, 21290}, {16887, 1369}, {16948, 8055}, {17103, 30660}, {17167, 2888}, {17168, 2889}, {17169, 2890}, {17172, 2892}, {17187, 2896}, {17200, 40002}, {17206, 1370}, {17209, 147}, {17940, 2786}, {17962, 20536}, {18206, 20344}, {18268, 17759}, {18653, 146}, {18792, 20355}, {18827, 20553}, {21123, 39346}, {24624, 5080}, {30940, 20554}, {30941, 20552}, {32014, 20290}, {33295, 20345}, {33774, 24051}, {34079, 20072}, {36085, 30709}, {37128, 4645}, {38813, 17350}, {38832, 21219}, {39179, 25048}, {39949, 32863}, {39950, 33110}, {40142, 20077}, {40143, 14450}, {40153, 5484}, {40214, 3648}, {40398, 33091}, {40403, 10327}, {40408, 4651}, {40430, 33066}, {40432, 4388}, {40438, 319}, {40746, 40721}, {41629, 42020}, {42302, 2550}, {43076, 26824}, {43359, 4813}
X(75) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 21250}, {6, 34832}, {7, 20547}, {31, 6376}, {56, 20528}, {57, 20338}, {87, 141}, {330, 2887}, {604, 41886}, {649, 5518}, {932, 3835}, {1919, 40610}, {2053, 3452}, {2162, 10}, {2319, 1329}, {4598, 21260}, {6383, 21235}, {6384, 626}, {7121, 2}, {7153, 2886}, {7155, 21244}, {7209, 17047}, {15373, 3}, {16606, 3454}, {18830, 21262}, {21759, 1213}, {22381, 440}, {23086, 18589}, {23493, 1211}, {34071, 513}, {34252, 20333}, {39914, 20542}, {40881, 20551}, {42027, 21245}
X(75) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 17149}, {2, 6376}, {8, 20935}, {31, 33788}, {69, 18749}, {76, 312}, {81, 18148}, {82, 18058}, {83, 18057}, {85, 18743}, {86, 18133}, {92, 18156}, {99, 18155}, {158, 33787}, {190, 20954}, {192, 39467}, {256, 24732}, {264, 20930}, {274, 2}, {286, 18147}, {291, 19567}, {304, 18750}, {305, 20914}, {308, 18040}, {310, 76}, {312, 16284}, {313, 20932}, {314, 69}, {321, 17762}, {330, 6384}, {333, 18738}, {334, 20947}, {348, 18751}, {514, 18149}, {561, 304}, {668, 693}, {670, 7199}, {689, 18077}, {693, 18159}, {799, 1577}, {811, 14208}, {850, 20951}, {870, 30963}, {871, 21615}, {873, 18140}, {903, 39995}, {1088, 33780}, {1218, 4687}, {1221, 37}, {1240, 313}, {1434, 18739}, {1502, 20444}, {1509, 18136}, {1577, 20939}, {1821, 1966}, {1928, 20641}, {1930, 20934}, {1967, 18271}, {1969, 92}, {1978, 514}, {2481, 350}, {3112, 1}, {3223, 18832}, {3261, 18151}, {3264, 20937}, {3596, 322}, {3718, 33672}, {4554, 4391}, {4569, 15413}, {4572, 3261}, {4583, 3766}, {4602, 661}, {4620, 18740}, {4623, 18158}, {4625, 18160}, {6063, 85}, {6382, 20936}, {6383, 20923}, {6384, 20943}, {6385, 18137}, {6386, 20949}, {6650, 24731}, {7018, 20955}, {7033, 24524}, {7035, 668}, {7182, 40702}, {7199, 18150}, {7260, 25667}, {8024, 20933}, {14616, 30939}, {17143, 17135}, {17206, 18736}, {18021, 21581}, {18022, 20926}, {18023, 20956}, {18031, 3912}, {18070, 18060}, {18145, 39997}, {18152, 40006}, {18811, 39126}, {18816, 320}, {18830, 513}, {18832, 20945}, {18833, 561}, {18891, 20446}, {18895, 17789}, {20566, 17791}, {20567, 20927}, {20568, 4358}, {20569, 30829}, {20641, 20931}, {20948, 20941}, {23999, 811}, {24037, 799}, {27801, 20929}, {28659, 20928}, {28660, 4417}, {30710, 1909}, {30939, 39996}, {31002, 6381}, {31618, 344}, {31625, 4033}, {32018, 321}, {33805, 14206}, {34384, 21579}, {34537, 21604}, {36796, 33677}, {36863, 29226}, {37204, 18070}, {37220, 16568}, {38810, 6}, {38840, 33801}, {38847, 31}, {40014, 20942}, {40015, 38298}, {40016, 18050}, {40017, 3948}, {40023, 42034}, {40024, 3661}, {40072, 21596}, {40073, 21597}, {40087, 40034}, {40162, 21608}, {40216, 33943}, {40362, 21585}, {40363, 21594}, {40364, 21582}, {40421, 21583}, {40422, 319}, {40495, 20940}, {40827, 27792}, {41283, 20922}
X(75) = X(i)-cross conjugate of X(j) for these (i,j): {1, 92}, {2, 85}, {4, 40015}, {7, 309}, {8, 312}, {10, 2}, {19, 39733}, {37, 39735}, {38, 1}, {63, 304}, {69, 20570}, {76, 6384}, {82, 39727}, {83, 40037}, {192, 40025}, {239, 40845}, {244, 514}, {306, 40011}, {307, 69}, {310, 18298}, {312, 40014}, {313, 40010}, {318, 34404}, {320, 40716}, {321, 76}, {347, 322}, {350, 18032}, {514, 1978}, {519, 18359}, {522, 190}, {523, 4033}, {561, 18832}, {596, 35058}, {656, 662}, {661, 4602}, {693, 668}, {714, 41683}, {726, 335}, {756, 17758}, {774, 2184}, {982, 9311}, {984, 27475}, {1086, 7199}, {1089, 40013}, {1099, 14206}, {1109, 1577}, {1111, 693}, {1125, 30690}, {1210, 189}, {1227, 30939}, {1266, 903}, {1281, 17789}, {1441, 264}, {1577, 799}, {1631, 20444}, {1725, 2349}, {1733, 1821}, {1734, 100}, {1735, 36100}, {1736, 36101}, {1737, 34234}, {1738, 673}, {1739, 88}, {1769, 3257}, {1895, 18750}, {1930, 561}, {2085, 31}, {2172, 20641}, {2227, 1581}, {2254, 4562}, {2292, 226}, {2517, 37218}, {2588, 2580}, {2589, 2581}, {2643, 18070}, {2786, 4639}, {3112, 18834}, {3116, 2186}, {3123, 513}, {3262, 20566}, {3263, 334}, {3264, 40039}, {3578, 33939}, {3663, 7}, {3668, 253}, {3670, 81}, {3673, 1088}, {3687, 28660}, {3701, 40012}, {3705, 27424}, {3717, 36807}, {3728, 37}, {3739, 40004}, {3741, 257}, {3766, 4583}, {3778, 6}, {3821, 14621}, {3912, 18031}, {3914, 4}, {3948, 40017}, {3963, 308}, {3992, 39994}, {4025, 664}, {4041, 27805}, {4044, 40030}, {4086, 15455}, {4125, 40021}, {4151, 3952}, {4357, 86}, {4358, 20568}, {4359, 274}, {4374, 670}, {4389, 39704}, {4391, 4554}, {4397, 6335}, {4398, 39707}, {4425, 6625}, {4441, 40028}, {4442, 671}, {4443, 751}, {4446, 749}, {4453, 4555}, {4458, 35148}, {4467, 99}, {4642, 2051}, {4647, 321}, {4671, 40029}, {4695, 14554}, {4696, 32017}, {4712, 3912}, {4723, 36805}, {4736, 3936}, {4738, 4358}, {4768, 36804}, {4793, 4671}, {4847, 8}, {4858, 3261}, {4967, 1268}, {4968, 30710}, {4972, 83}, {4980, 32018}, {4981, 32009}, {4986, 7035}, {5051, 14534}, {5224, 30598}, {6376, 40027}, {6381, 31002}, {6382, 6376}, {6734, 333}, {6735, 4997}, {6736, 6557}, {9436, 18025}, {10916, 2994}, {11019, 10405}, {14206, 33805}, {14208, 811}, {14213, 1969}, {14837, 658}, {17072, 4598}, {17155, 34860}, {17205, 4608}, {17289, 40044}, {17446, 82}, {17755, 18157}, {17758, 40008}, {17797, 17788}, {17859, 40440}, {17860, 318}, {17861, 273}, {17862, 331}, {17863, 286}, {17864, 18022}, {17869, 2052}, {17871, 158}, {17872, 19}, {17873, 1235}, {17874, 41013}, {17875, 40703}, {17876, 24006}, {17879, 14208}, {17880, 35519}, {17883, 23999}, {17886, 850}, {17888, 17924}, {17896, 18026}, {17898, 823}, {17899, 6331}, {17903, 20914}, {18695, 20571}, {18697, 313}, {18698, 1441}, {18743, 40026}, {18827, 39718}, {18837, 17149}, {19962, 19975}, {20234, 1502}, {20235, 305}, {20236, 20567}, {20237, 28659}, {20322, 1895}, {20432, 18895}, {20433, 18891}, {20443, 35538}, {20517, 43190}, {20627, 1928}, {20880, 6063}, {20888, 310}, {20889, 18833}, {20890, 41283}, {20891, 6385}, {20892, 6383}, {20895, 3596}, {20896, 27801}, {20898, 1930}, {20899, 6382}, {20900, 3264}, {20901, 40495}, {20902, 20948}, {20903, 24037}, {20906, 6386}, {20907, 4572}, {20911, 7018}, {20912, 18023}, {21020, 10}, {21120, 21580}, {21147, 20928}, {21174, 36118}, {21184, 1305}, {21185, 37206}, {21186, 653}, {21189, 651}, {21192, 4610}, {21196, 7260}, {21208, 7192}, {21210, 10566}, {21403, 34384}, {21405, 18021}, {21406, 40364}, {21407, 40421}, {21409, 40362}, {21412, 40016}, {21420, 40363}, {21422, 40072}, {21423, 40073}, {21425, 8024}, {21431, 34537}, {21435, 40162}, {21443, 871}, {21935, 13478}, {22069, 3}, {22130, 20926}, {22271, 18137}, {22464, 34393}, {23518, 275}, {23528, 7017}, {23529, 281}, {23536, 278}, {23537, 27}, {23661, 31623}, {23662, 17907}, {23663, 393}, {23664, 25}, {23665, 20883}, {23666, 427}, {23667, 33}, {23668, 1826}, {23670, 38462}, {23672, 2501}, {23673, 14618}, {23674, 18020}, {23676, 7649}, {23681, 1847}, {23683, 6528}, {23686, 6591}, {23687, 1783}, {23928, 8818}, {23996, 1959}, {24010, 3239}, {24014, 30807}, {24026, 4391}, {24028, 908}, {24031, 6332}, {24038, 14210}, {24165, 330}, {24169, 39724}, {24174, 42304}, {24175, 27818}, {24176, 39747}, {24177, 279}, {24209, 18815}, {24211, 7224}, {24213, 1440}, {24443, 57}, {24462, 660}, {24589, 20569}, {24982, 40420}, {24993, 31643}, {24995, 40415}, {25001, 31618}, {25006, 32008}, {25015, 40414}, {25019, 30705}, {26013, 1952}, {26015, 1121}, {26234, 870}, {26563, 32023}, {27474, 21615}, {27966, 8033}, {28605, 40023}, {29062, 835}, {29673, 17743}, {31330, 31359}, {31627, 16284}, {33136, 80}, {33145, 79}, {33890, 30545}, {33940, 40038}, {33941, 40033}, {34387, 18155}, {36035, 32680}, {36038, 35174}, {36250, 1029}, {38407, 30022}, {38822, 21595}, {38995, 4481}, {39697, 39699}, {39994, 40041}, {40216, 40005}, {40973, 36907}, {40999, 95}, {41804, 1494}, {41809, 32014}, {42027, 2998}, {42285, 39705}
X(i)-isoconjugate of X(j) for these (i,j): {1, 31}, {2, 32}, {3, 25}, {4, 184}, {7, 2175}, {8, 1397}, {9, 604}, {10, 2206}, {13, 34394}, {14, 34395}, {15, 3457}, {16, 3458}, {19, 48}, {20, 33581}, {21, 1402}, {22, 2353}, {23, 3455}, {24, 2351}, {27, 2200}, {28, 228}, {30, 40352}, {33, 603}, {34, 212}, {35, 6186}, {36, 6187}, {37, 1333}, {39, 251}, {40, 2208}, {41, 57}, {42, 58}, {43, 7121}, {44, 9456}, {45, 28607}, {50, 1989}, {51, 54}, {52, 41271}, {53, 14533}, {55, 56}, {59, 3271}, {60, 181}, {61, 21461}, {62, 21462}, {63, 1973}, {64, 154}, {65, 2194}, {66, 206}, {67, 18374}, {69, 1974}, {71, 1474}, {72, 2203}, {73, 2299}, {74, 1495}, {75, 560}, {76, 1501}, {77, 2212}, {78, 1395}, {81, 213}, {82, 1964}, {83, 3051}, {84, 2187}, {85, 9447}, {86, 1918}, {87, 2209}, {88, 2251}, {92, 9247}, {94, 19627}, {95, 40981}, {97, 3199}, {98, 237}, {99, 669}, {100, 667}, {101, 649}, {105, 2223}, {106, 902}, {107, 39201}, {108, 1946}, {109, 663}, {110, 512}, {111, 187}, {112, 647}, {115, 23357}, {155, 39109}, {157, 1485}, {159, 34207}, {160, 2980}, {161, 34438}, {162, 810}, {163, 661}, {171, 904}, {172, 893}, {182, 263}, {190, 1919}, {197, 3435}, {198, 1436}, {199, 3437}, {200, 1106}, {204, 19614}, {205, 42467}, {208, 2188}, {210, 1408}, {216, 8882}, {217, 275}, {219, 608}, {220, 1407}, {221, 2192}, {222, 607}, {223, 7118}, {230, 32654}, {232, 248}, {238, 1911}, {239, 1922}, {244, 1110}, {249, 3124}, {250, 20975}, {255, 1096}, {256, 7122}, {262, 34396}, {264, 14575}, {265, 34397}, {268, 3209}, {269, 1253}, {274, 2205}, {279, 14827}, {282, 2199}, {284, 1400}, {287, 2211}, {290, 9418}, {291, 2210}, {292, 1914}, {297, 14600}, {308, 41331}, {311, 14573}, {315, 40146}, {323, 11060}, {325, 14601}, {334, 18892}, {335, 14599}, {338, 23963}, {350, 14598}, {351, 691}, {365, 18753}, {371, 8577}, {372, 8576}, {385, 9468}, {393, 577}, {394, 2207}, {418, 8884}, {419, 17970}, {427, 10547}, {460, 42065}, {468, 14908}, {476, 14270}, {480, 7023}, {493, 6423}, {494, 6424}, {511, 1976}, {513, 692}, {514, 32739}, {517, 34858}, {520, 32713}, {523, 1576}, {524, 32740}, {526, 14560}, {561, 1917}, {571, 2165}, {574, 1383}, {588, 5062}, {589, 5058}, {593, 1500}, {595, 40148}, {610, 2155}, {612, 1472}, {614, 7084}, {648, 3049}, {650, 1415}, {651, 3063}, {652, 32674}, {654, 32675}, {656, 32676}, {657, 1461}, {659, 34067}, {662, 798}, {665, 919}, {668, 1980}, {670, 9426}, {671, 14567}, {672, 1438}, {673, 9454}, {675, 8618}, {676, 32642}, {682, 40413}, {684, 32696}, {685, 39469}, {686, 32708}, {688, 4577}, {689, 9494}, {690, 32729}, {694, 1691}, {695, 1915}, {697, 8619}, {699, 3229}, {713, 8620}, {717, 8621}, {727, 3009}, {728, 7366}, {729, 3231}, {731, 8622}, {733, 8623}, {738, 6602}, {739, 3230}, {741, 3747}, {743, 8624}, {745, 8625}, {753, 8626}, {755, 8627}, {756, 849}, {757, 872}, {759, 3724}, {761, 8628}, {765, 3248}, {767, 8629}, {785, 2978}, {788, 1492}, {789, 8630}, {799, 1924}, {800, 41890}, {803, 8631}, {804, 17938}, {805, 5027}, {813, 8632}, {815, 8633}, {817, 8634}, {822, 24019}, {825, 3250}, {826, 4630}, {827, 3005}, {831, 8635}, {833, 8636}, {835, 8637}, {842, 5191}, {843, 2502}, {846, 18757}, {850, 14574}, {869, 985}, {870, 18900}, {875, 3573}, {878, 4230}, {881, 17941}, {884, 2283}, {887, 9150}, {888, 32717}, {890, 898}, {891, 32718}, {894, 7104}, {896, 923}, {897, 922}, {900, 32719}, {901, 1960}, {903, 9459}, {906, 6591}, {907, 3804}, {909, 2183}, {910, 911}, {913, 2252}, {924, 32734}, {925, 34952}, {926, 32735}, {927, 8638}, {931, 8639}, {932, 8640}, {933, 15451}, {934, 8641}, {935, 42659}, {941, 5019}, {943, 40956}, {961, 20967}, {963, 20991}, {967, 2271}, {983, 7032}, {999, 34446}, {1015, 1252}, {1016, 1977}, {1017, 2226}, {1030, 3444}, {1033, 28783}, {1035, 7037}, {1036, 1460}, {1037, 7083}, {1042, 2328}, {1055, 2291}, {1073, 3172}, {1084, 4590}, {1086, 23990}, {1092, 6524}, {1093, 23606}, {1100, 28615}, {1101, 2643}, {1109, 23995}, {1113, 42668}, {1114, 42667}, {1118, 6056}, {1126, 2308}, {1146, 23979}, {1147, 14593}, {1155, 34068}, {1167, 40958}, {1169, 2092}, {1170, 20229}, {1171, 20970}, {1172, 1409}, {1173, 13366}, {1174, 1475}, {1175, 40952}, {1176, 1843}, {1177, 2393}, {1178, 20964}, {1179, 23195}, {1191, 7050}, {1197, 1258}, {1214, 2204}, {1249, 14642}, {1259, 7337}, {1260, 1398}, {1262, 14936}, {1290, 42670}, {1291, 6140}, {1292, 8642}, {1293, 8643}, {1296, 8644}, {1297, 42671}, {1301, 42658}, {1302, 42660}, {1304, 9409}, {1308, 8645}, {1309, 23220}, {1310, 8646}, {1326, 2054}, {1334, 1412}, {1356, 6064}, {1357, 6065}, {1358, 6066}, {1379, 5638}, {1380, 5639}, {1384, 21448}, {1399, 7073}, {1403, 2053}, {1404, 2316}, {1405, 2364}, {1406, 7072}, {1410, 4183}, {1411, 2361}, {1413, 7074}, {1416, 2340}, {1417, 3689}, {1428, 7077}, {1431, 2330}, {1433, 3195}, {1435, 1802}, {1437, 1824}, {1447, 18265}, {1457, 2342}, {1458, 2195}, {1459, 8750}, {1468, 2258}, {1477, 8647}, {1486, 3433}, {1491, 34069}, {1494, 9407}, {1502, 9233}, {1509, 7109}, {1510, 32737}, {1511, 40355}, {1575, 34077}, {1580, 1967}, {1581, 1933}, {1582, 9288}, {1611, 40322}, {1613, 3224}, {1625, 2623}, {1631, 7087}, {1634, 18105}, {1635, 32665}, {1636, 32695}, {1637, 32640}, {1661, 34426}, {1676, 41378}, {1677, 41379}, {1692, 2987}, {1707, 38252}, {1716, 15370}, {1726, 7139}, {1740, 34248}, {1743, 38266}, {1755, 1910}, {1759, 40145}, {1783, 22383}, {1790, 2333}, {1799, 27369}, {1804, 6059}, {1821, 9417}, {1857, 7335}, {1880, 2193}, {1886, 32657}, {1916, 14602}, {1921, 18897}, {1923, 3112}, {1927, 1966}, {1929, 18266}, {1932, 9285}, {1945, 1951}, {1949, 2202}, {1950, 7106}, {1953, 2148}, {1965, 9236}, {1971, 1987}, {1979, 9265}, {1988, 32445}, {1990, 18877}, {1992, 39238}, {2052, 14585}, {2056, 8601}, {2070, 34448}, {2084, 4599}, {2112, 18783}, {2149, 2170}, {2150, 2171}, {2151, 2153}, {2152, 2154}, {2156, 2172}, {2159, 2173}, {2160, 2174}, {2161, 7113}, {2162, 2176}, {2163, 2177}, {2164, 2178}, {2167, 2179}, {2168, 2180}, {2169, 2181}, {2182, 32677}, {2189, 2197}, {2191, 21059}, {2196, 2201}, {2217, 3185}, {2218, 2352}, {2220, 39798}, {2222, 8648}, {2224, 2225}, {2238, 18268}, {2241, 30651}, {2242, 30650}, {2245, 34079}, {2248, 18755}, {2249, 42669}, {2254, 32666}, {2255, 2256}, {2259, 2260}, {2276, 40746}, {2279, 2280}, {2281, 2303}, {2282, 2304}, {2298, 2300}, {2310, 24027}, {2319, 41526}, {2321, 16947}, {2347, 3451}, {2349, 9406}, {2350, 4251}, {2354, 2359}, {2356, 36057}, {2357, 2360}, {2363, 3725}, {2384, 8649}, {2395, 14966}, {2420, 2433}, {2421, 2422}, {2423, 2427}, {2424, 2426}, {2425, 2432}, {2428, 2440}, {2429, 2441}, {2430, 2442}, {2431, 2443}, {2434, 2444}, {2435, 2445}, {2436, 2437}, {2481, 9455}, {2482, 41936}, {2489, 4558}, {2491, 2966}, {2501, 32661}, {2576, 2578}, {2577, 2579}, {2610, 32671}, {2624, 32678}, {2631, 36131}, {2635, 34078}, {2637, 36140}, {2638, 24033}, {2642, 36142}, {2701, 5075}, {2702, 5029}, {2703, 5040}, {2709, 9135}, {2712, 5168}, {2715, 3569}, {2748, 8650}, {2854, 32741}, {2908, 7094}, {2930, 22259}, {2963, 2965}, {3003, 14910}, {3016, 32730}, {3022, 7339}, {3043, 14595}, {3052, 3445}, {3053, 8770}, {3064, 32660}, {3068, 26454}, {3069, 26461}, {3094, 18898}, {3108, 5007}, {3114, 18899}, {3117, 3407}, {3121, 4567}, {3122, 4570}, {3129, 3438}, {3130, 3439}, {3131, 3442}, {3132, 3443}, {3135, 34449}, {3148, 3425}, {3158, 16945}, {3163, 40353}, {3167, 14248}, {3186, 15389}, {3197, 7152}, {3203, 30505}, {3207, 11051}, {3222, 9491}, {3225, 32748}, {3228, 33875}, {3249, 6632}, {3266, 19626}, {3269, 23964}, {3284, 8749}, {3288, 26714}, {3289, 6531}, {3292, 8753}, {3310, 32641}, {3331, 26717}, {3415, 37586}, {3423, 37580}, {3426, 26864}, {3431, 34417}, {3446, 16686}, {3447, 7669}, {3449, 21746}, {3450, 23638}, {3453, 20966}, {3456, 6636}, {3504, 11325}, {3506, 34214}, {3512, 19554}, {3527, 11402}, {3556, 7169}, {3565, 8651}, {3596, 41280}, {3709, 4565}, {3733, 4557}, {3768, 34075}, {3778, 38813}, {3926, 36417}, {3978, 8789}, {3990, 5317}, {4055, 8747}, {4057, 40519}, {4079, 4556}, {4100, 6520}, {4105, 6614}, {4117, 24037}, {4273, 28658}, {4370, 41935}, {4394, 34080}, {4471, 7236}, {4497, 7246}, {4559, 7252}, {4588, 4775}, {4591, 14407}, {4627, 4832}, {4628, 21123}, {4790, 34074}, {4834, 8652}, {4893, 34073}, {5008, 39389}, {5012, 27375}, {5023, 36616}, {5035, 39974}, {5039, 11175}, {5041, 34572}, {5052, 30535}, {5063, 34288}, {5089, 32658}, {5106, 5970}, {5147, 12031}, {5163, 35107}, {5285, 8615}, {5291, 17961}, {5383, 21762}, {5410, 6416}, {5411, 6415}, {5412, 6414}, {5413, 6413}, {5467, 9178}, {5486, 19136}, {5545, 8653}, {5546, 7180}, {5596, 22262}, {5649, 6041}, {5994, 6138}, {5995, 6137}, {6012, 8654}, {6013, 8655}, {6014, 8656}, {6016, 8657}, {6017, 8658}, {6037, 9420}, {6057, 7342}, {6061, 7143}, {6063, 9448}, {6078, 8659}, {6079, 8660}, {6129, 32652}, {6139, 14733}, {6184, 41934}, {6185, 39686}, {6233, 11186}, {6353, 40319}, {6366, 32728}, {6371, 32736}, {6391, 19118}, {6525, 14379}, {6529, 32320}, {6551, 8661}, {6574, 8662}, {6578, 8663}, {6589, 32653}, {6610, 18889}, {6611, 7367}, {6612, 7368}, {6651, 18263}, {6660, 41533}, {6759, 32319}, {6800, 14906}, {7008, 7114}, {7011, 7154}, {7053, 7071}, {7063, 7340}, {7064, 7341}, {7078, 7151}, {7079, 7099}, {7096, 32664}, {7100, 14975}, {7107, 7120}, {7115, 7117}, {7116, 7119}, {7123, 16502}, {7132, 20665}, {7261, 18262}, {7357, 40370}, {7649, 32656}, {7712, 14479}, {7735, 40799}, {7772, 39955}, {7953, 8664}, {7954, 8665}, {8053, 34444}, {8061, 34072}, {8265, 40416}, {8301, 41528}, {8574, 40173}, {8578, 40150}, {8603, 11083}, {8604, 11088}, {8609, 32655}, {8675, 32738}, {8677, 14776}, {8739, 36296}, {8740, 36297}, {8751, 20752}, {8752, 22356}, {8756, 32659}, {8761, 21767}, {8772, 36051}, {8791, 10317}, {8852, 17798}, {8911, 41515}, {8946, 10133}, {8948, 10132}, {9023, 9192}, {9033, 32715}, {9066, 9489}, {9125, 32648}, {9136, 9486}, {9186, 9188}, {9208, 32694}, {9217, 20998}, {9262, 41405}, {9266, 9299}, {9292, 9306}, {9310, 9315}, {9316, 9439}, {9408, 40384}, {9419, 34536}, {9427, 34537}, {9465, 9515}, {9470, 18264}, {9500, 20672}, {9697, 11538}, {9971, 19151}, {10117, 34190}, {10293, 40114}, {10316, 13854}, {10329, 14370}, {10355, 38533}, {10420, 21731}, {10423, 42665}, {10425, 42663}, {10630, 39689}, {10641, 32586}, {10642, 32585}, {11062, 11077}, {11063, 14579}, {11064, 40354}, {11079, 39176}, {11081, 11086}, {11082, 11136}, {11084, 19294}, {11087, 11135}, {11089, 19295}, {11134, 11138}, {11137, 11139}, {11422, 34154}, {11636, 17414}, {11672, 41932}, {12077, 14586}, {13338, 30537}, {13472, 15004}, {14085, 14088}, {14096, 42288}, {14251, 40820}, {14318, 43357}, {14376, 17409}, {14380, 23347}, {14385, 14583}, {14425, 32645}, {14528, 17810}, {14553, 37504}, {14569, 19210}, {14571, 14578}, {14580, 18876}, {14581, 14919}, {14582, 14591}, {14587, 41221}, {14603, 14604}, {14609, 41309}, {14621, 40728}, {14885, 21355}, {14946, 16985}, {15080, 41443}, {15166, 41941}, {15167, 41942}, {15369, 19588}, {15378, 20974}, {15382, 20455}, {15383, 23644}, {15388, 38356}, {15494, 34447}, {15526, 41937}, {15591, 41619}, {15742, 22096}, {15905, 41489}, {16277, 23208}, {16463, 37848}, {16464, 37850}, {16468, 40735}, {16685, 39964}, {16777, 34819}, {16813, 42293}, {16944, 40172}, {16946, 39956}, {16969, 36614}, {17407, 39172}, {17408, 39167}, {17415, 33514}, {17735, 17962}, {17939, 17989}, {17940, 17990}, {17942, 18000}, {17943, 18001}, {17944, 18002}, {17963, 17966}, {17974, 34854}, {17983, 23200}, {18018, 20968}, {18022, 40373}, {18038, 24479}, {18267, 39044}, {18334, 23588}, {18344, 36059}, {18384, 22115}, {18591, 40570}, {18756, 40737}, {18771, 20958}, {18772, 20959}, {18797, 40368}, {18881, 19220}, {18891, 18893}, {18894, 18895}, {18896, 18902}, {18901, 18903}, {19297, 19302}, {19561, 30648}, {19596, 34437}, {20228, 23617}, {20332, 21760}, {20468, 34183}, {20775, 32085}, {20859, 38826}, {20965, 42346}, {20979, 34071}, {20986, 34434}, {20987, 34436}, {20988, 34441}, {20989, 34442}, {20990, 34443}, {20992, 34445}, {20993, 34427}, {20999, 34179}, {21001, 36615}, {21747, 41434}, {21750, 40403}, {21751, 38810}, {21753, 40408}, {21759, 27644}, {21779, 40770}, {22052, 33631}, {22388, 26705}, {23099, 31614}, {23115, 40144}, {23202, 36125}, {23343, 23349}, {23344, 23345}, {23346, 23351}, {23383, 34429}, {23493, 38832}, {23590, 35071}, {23626, 38827}, {23858, 34184}, {23868, 34250}, {23971, 35508}, {23980, 41933}, {23984, 39687}, {23985, 35072}, {24012, 24013}, {24021, 42080}, {24576, 30634}, {26920, 41516}, {27374, 39287}, {28781, 28785}, {28784, 34167}, {30435, 39951}, {32230, 34980}, {32651, 33525}, {33882, 39981}, {34121, 34125}, {34238, 36213}, {34565, 34567}, {34570, 40135}, {34818, 36748}, {34921, 42657}, {36069, 42666}, {36617, 38297}, {37128, 41333}, {37183, 39644}, {39110, 39111}, {39625, 39627}, {40320, 40323}, {40321, 40324}, {40347, 41336}, {40363, 41281}, {40372, 40421}, {40415, 40935}, {40802, 40825}, {40814, 40823}, {41196, 41200}, {41197, 41201}, {41521, 41615}, {41534, 41882}, {41880, 41881}, {43112, 43113}
X(75) = cevapoint of X(i) and X(j) for these (i,j): {1, 63}, {2, 8}, {3, 22130}, {4, 17903}, {6, 1631}, {7, 347}, {10, 321}, {11, 21120}, {31, 2172}, {37, 22271}, {38, 1930}, {57, 21147}, {58, 38822}, {76, 6382}, {85, 31627}, {92, 1895}, {239, 1281}, {244, 514}, {307, 1441}, {323, 4996}, {512, 6377}, {522, 4858}, {523, 1086}, {525, 2968}, {561, 18837}, {656, 20902}, {693, 1111}, {740, 17755}, {758, 16586}, {850, 34387}, {894, 17797}, {908, 24028}, {984, 27474}, {1099, 14206}, {1109, 1577}, {1125, 3578}, {1227, 35550}, {1266, 20900}, {1580, 19572}, {1647, 21129}, {1734, 20901}, {1738, 20431}, {1739, 21427}, {1959, 23996}, {1962, 17746}, {2085, 20627}, {2292, 3687}, {2643, 8061}, {3120, 21124}, {3123, 20906}, {3239, 24010}, {3663, 20895}, {3670, 20896}, {3672, 28616}, {3705, 33890}, {3728, 20891}, {3778, 20234}, {3912, 4712}, {3914, 20235}, {3936, 4736}, {4000, 11677}, {4025, 17880}, {4151, 40619}, {4357, 18697}, {4358, 4738}, {4359, 4647}, {4391, 24026}, {4467, 17886}, {4793, 24589}, {4847, 20880}, {4972, 21425}, {5515, 23879}, {6332, 24031}, {7081, 17741}, {14208, 17879}, {14210, 24038}, {16732, 30591}, {17446, 21424}, {17864, 22069}, {17872, 21406}, {20237, 24443}, {20888, 21020}, {20899, 24165}, {24014, 30807}, {33679, 40844}
X(75) = crosspoint of X(i) and X(j) for these (i,j): {1, 3223}, {2, 330}, {76, 6063}, {99, 4998}, {274, 310}, {291, 24576}, {561, 1969}, {668, 7035}, {670, 31625}, {799, 24037}, {811, 23999}, {1916, 40098}, {3112, 18833}
X(75) = crosssum of X(i) and X(j) for these (i,j): {1, 1740}, {2, 17486}, {3, 23075}, {6, 2176}, {9, 32468}, {32, 2175}, {75, 33788}, {213, 1918}, {238, 19580}, {512, 3271}, {560, 9247}, {649, 38346}, {667, 3248}, {669, 1977}, {1923, 1964}, {1966, 18270}, {3708, 8061}, {3747, 20663}, {8022, 16584}, {38810, 38820}
X(75) = barycentric product X(i)*X(j) for these {i,j}: {1, 76}, {3, 1969}, {4, 304}, {6, 561}, {7, 312}, {8, 85}, {9, 6063}, {10, 274}, {19, 305}, {21, 349}, {25, 40364}, {27, 20336}, {28, 40071}, {29, 1231}, {30, 33805}, {31, 1502}, {32, 1928}, {37, 310}, {38, 308}, {39, 18833}, {41, 41283}, {42, 6385}, {43, 6383}, {48, 18022}, {55, 20567}, {56, 28659}, {57, 3596}, {58, 27801}, {63, 264}, {65, 28660}, {66, 20641}, {67, 20944}, {69, 92}, {77, 7017}, {78, 331}, {79, 33939}, {80, 20924}, {81, 313}, {82, 8024}, {83, 1930}, {86, 321}, {87, 6382}, {88, 3264}, {91, 7763}, {95, 14213}, {99, 1577}, {100, 3261}, {101, 40495}, {110, 20948}, {115, 24037}, {141, 3112}, {145, 40014}, {158, 3926}, {162, 3267}, {189, 322}, {190, 693}, {192, 6384}, {194, 18832}, {226, 314}, {238, 18895}, {239, 334}, {244, 31625}, {249, 23994}, {253, 18750}, {255, 18027}, {256, 1920}, {257, 1909}, {261, 6358}, {262, 3403}, {271, 40701}, {273, 345}, {275, 18695}, {278, 3718}, {279, 341}, {280, 40702}, {281, 7182}, {286, 306}, {287, 40703}, {290, 1959}, {291, 1921}, {292, 18891}, {297, 336}, {307, 31623}, {309, 329}, {311, 2167}, {318, 348}, {319, 30690}, {320, 18359}, {325, 1821}, {326, 2052}, {330, 6376}, {332, 40149}, {333, 1441}, {335, 350}, {338, 24041}, {343, 40440}, {346, 1088}, {347, 34404}, {384, 9239}, {385, 1934}, {479, 30693}, {512, 4602}, {513, 1978}, {514, 668}, {518, 18031}, {519, 20568}, {522, 4554}, {523, 799}, {525, 811}, {536, 31002}, {555, 7027}, {556, 4146}, {560, 40362}, {594, 873}, {596, 18140}, {604, 40363}, {610, 41530}, {626, 38847}, {645, 4077}, {646, 3676}, {648, 14208}, {649, 6386}, {650, 4572}, {651, 35519}, {653, 35518}, {656, 6331}, {658, 4397}, {661, 670}, {662, 850}, {664, 4391}, {671, 14210}, {673, 3263}, {679, 36791}, {683, 18671}, {689, 8061}, {694, 1926}, {695, 1925}, {717, 30875}, {726, 32020}, {740, 40017}, {753, 30874}, {757, 28654}, {765, 23989}, {774, 40830}, {789, 824}, {798, 4609}, {801, 17858}, {812, 4583}, {823, 3265}, {825, 30870}, {826, 4593}, {858, 37220}, {870, 3661}, {871, 2276}, {874, 4444}, {876, 27853}, {889, 4728}, {894, 7018}, {896, 18023}, {897, 3266}, {903, 4358}, {908, 18816}, {996, 33934}, {1000, 20925}, {1002, 21615}, {1014, 30713}, {1016, 1111}, {1019, 27808}, {1029, 20932}, {1031, 20934}, {1043, 1446}, {1086, 7035}, {1089, 1509}, {1093, 1102}, {1099, 31621}, {1101, 23962}, {1109, 4590}, {1121, 30806}, {1125, 32018}, {1218, 31330}, {1220, 20911}, {1221, 3741}, {1222, 26563}, {1226, 40399}, {1228, 2363}, {1229, 21453}, {1230, 40438}, {1233, 2346}, {1235, 34055}, {1237, 40432}, {1240, 3666}, {1241, 17446}, {1255, 1269}, {1265, 1847}, {1266, 36805}, {1268, 4359}, {1275, 24026}, {1278, 40027}, {1369, 39727}, {1370, 39733}, {1400, 40072}, {1434, 3701}, {1468, 40828}, {1491, 37133}, {1494, 14206}, {1580, 18896}, {1581, 3978}, {1655, 18298}, {1725, 40832}, {1733, 8781}, {1734, 31624}, {1740, 40162}, {1748, 20563}, {1755, 18024}, {1760, 18018}, {1799, 20883}, {1895, 34403}, {1897, 15413}, {1916, 1966}, {1917, 40359}, {1927, 18901}, {1953, 34384}, {1964, 40016}, {1965, 9229}, {1967, 14603}, {1973, 40050}, {1993, 20571}, {2064, 15314}, {2084, 42371}, {2113, 20446}, {2156, 40073}, {2157, 40074}, {2161, 40075}, {2166, 7799}, {2171, 18021}, {2172, 40421}, {2184, 14615}, {2185, 34388}, {2186, 20023}, {2190, 28706}, {2234, 34087}, {2254, 36803}, {2275, 7034}, {2292, 40827}, {2349, 3260}, {2350, 40088}, {2373, 20884}, {2400, 42719}, {2481, 3912}, {2517, 37215}, {2533, 7260}, {2580, 22339}, {2581, 22340}, {2582, 15164}, {2583, 15165}, {2643, 34537}, {2799, 36036}, {2887, 38810}, {2896, 18834}, {2962, 7769}, {2994, 20930}, {2995, 4417}, {2996, 18156}, {2997, 18134}, {2998, 17149}, {3005, 37204}, {3006, 37130}, {3113, 3314}, {3120, 4601}, {3212, 27424}, {3218, 20566}, {3219, 20565}, {3222, 20910}, {3223, 6374}, {3224, 18837}, {3227, 6381}, {3239, 4569}, {3241, 40029}, {3262, 34234}, {3268, 32680}, {3496, 18836}, {3497, 18835}, {3509, 18036}, {3613, 33764}, {3616, 40023}, {3621, 40026}, {3662, 7033}, {3663, 32017}, {3673, 30701}, {3679, 20569}, {3687, 31643}, {3699, 24002}, {3700, 4625}, {3717, 34018}, {3729, 32023}, {3758, 30635}, {3759, 30636}, {3762, 4555}, {3766, 4562}, {3835, 18830}, {3868, 40011}, {3875, 40012}, {3904, 35174}, {3936, 14616}, {3948, 18827}, {3952, 7199}, {3963, 32010}, {3964, 6521}, {3975, 7233}, {4008, 40824}, {4010, 4639}, {4024, 4623}, {4025, 6335}, {4033, 7192}, {4036, 4610}, {4043, 39734}, {4080, 30939}, {4086, 4573}, {4118, 38830}, {4120, 4634}, {4143, 36126}, {4163, 36838}, {4176, 6520}, {4329, 40015}, {4357, 30710}, {4360, 40013}, {4373, 18743}, {4374, 27805}, {4431, 32021}, {4441, 27475}, {4451, 7196}, {4453, 36804}, {4467, 15455}, {4475, 5388}, {4486, 41072}, {4505, 4817}, {4509, 8707}, {4518, 10030}, {4552, 18155}, {4561, 17924}, {4563, 24006}, {4564, 34387}, {4567, 21207}, {4576, 18070}, {4592, 14618}, {4597, 4791}, {4598, 20906}, {4599, 23285}, {4600, 16732}, {4624, 4811}, {4632, 30591}, {4633, 4815}, {4645, 40845}, {4647, 32014}, {4651, 40004}, {4671, 39704}, {4766, 43099}, {4789, 35181}, {4823, 32042}, {4847, 31618}, {4858, 4998}, {4862, 34523}, {4876, 18033}, {4978, 6540}, {5209, 11611}, {5249, 40422}, {5423, 23062}, {5466, 24039}, {5905, 20570}, {5936, 19804}, {6149, 20573}, {6332, 18026}, {6339, 33787}, {6393, 36120}, {6504, 33808}, {6528, 24018}, {6539, 16709}, {6542, 18032}, {6548, 24004}, {6553, 33780}, {6557, 39126}, {6601, 21609}, {6625, 17762}, {6630, 18159}, {6650, 20947}, {6664, 18064}, {6742, 18160}, {6757, 34016}, {7045, 23978}, {7056, 7101}, {7096, 40365}, {7121, 40367}, {7155, 30545}, {7168, 18275}, {7178, 7257}, {7209, 27538}, {7219, 20914}, {7224, 17788}, {7237, 7307}, {7249, 17787}, {7261, 17789}, {7319, 21605}, {7357, 20444}, {7361, 18749}, {8044, 20929}, {8046, 20937}, {8047, 18151}, {8048, 20928}, {8049, 18137}, {8050, 20949}, {8709, 20908}, {9230, 9285}, {9295, 18149}, {9436, 36796}, {9447, 41287}, {9476, 17875}, {9505, 18035}, {10405, 16284}, {10436, 34258}, {10453, 40025}, {13136, 36038}, {13476, 18152}, {13485, 20941}, {13575, 21582}, {13576, 18157}, {13577, 20927}, {14080, 14087}, {14207, 35179}, {14295, 37134}, {14377, 33932}, {14534, 18697}, {14621, 33931}, {14623, 30149}, {14624, 16739}, {14942, 40704}, {15224, 15225}, {15415, 36134}, {15416, 36118}, {15466, 19611}, {15467, 27396}, {15526, 23999}, {16099, 42709}, {16552, 40005}, {16568, 18019}, {16703, 18082}, {17135, 39735}, {17143, 17758}, {17147, 40010}, {17160, 39994}, {17206, 41013}, {17277, 40216}, {17280, 40038}, {17302, 40033}, {17316, 40028}, {17445, 31630}, {17484, 40716}, {17495, 40039}, {17739, 18760}, {17743, 33930}, {17760, 18299}, {17791, 21739}, {17862, 40424}, {17871, 42407}, {17879, 23582}, {17893, 43188}, {18020, 20902}, {18025, 30807}, {18062, 31065}, {18075, 25322}, {18133, 35058}, {18135, 34860}, {18143, 27807}, {18145, 39697}, {18147, 39700}, {18154, 42363}, {18276, 39932}, {18277, 24576}, {18297, 18297}, {18596, 40009}, {18797, 33790}, {18811, 30827}, {18815, 32851}, {18840, 39731}, {19579, 30633}, {20234, 40415}, {20235, 40411}, {20236, 40419}, {20332, 35538}, {20627, 40416}, {20879, 40410}, {20880, 32008}, {20888, 32009}, {20889, 39968}, {20891, 40418}, {20892, 32011}, {20893, 32012}, {20894, 32013}, {20895, 40420}, {20898, 40425}, {20903, 40429}, {20907, 30610}, {20913, 39717}, {20917, 41527}, {20923, 39741}, {20933, 39726}, {20939, 35511}, {20942, 36606}, {20943, 38247}, {20945, 38262}, {20946, 42361}, {21289, 40037}, {21378, 40036}, {21406, 40413}, {21598, 41513}, {22464, 36795}, {23051, 40022}, {23626, 38812}, {23970, 24011}, {23974, 24021}, {23983, 24032}, {24000, 36793}, {24001, 34767}, {24020, 34538}, {24325, 40024}, {24624, 35550}, {25417, 30596}, {27375, 33778}, {27447, 41318}, {27494, 30963}, {27569, 40164}, {27795, 41830}, {28605, 30598}, {28626, 42029}, {30565, 35171}, {30566, 35175}, {30579, 40040}, {30712, 42034}, {30736, 37132}, {30758, 39721}, {30829, 36588}, {30940, 43534}, {31008, 42027}, {31359, 34284}, {32679, 35139}, {33090, 40044}, {33672, 41514}, {33788, 42486}, {33809, 36955}, {33935, 43531}, {33938, 39724}, {33944, 39722}, {35516, 36100}, {35517, 36101}, {35520, 36102}, {35522, 36085}, {35543, 37129}, {35544, 37128}, {35551, 37208}, {36263, 40826}, {39044, 40098}, {39698, 39995}, {39699, 39996}, {39723, 40001}, {39725, 40035}, {39748, 40034}, {39798, 40087}, {39981, 40089}, {40327, 40339}
X(75) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 6}, {2, 1}, {3, 48}, {4, 19}, {5, 1953}, {6, 31}, {7, 57}, {8, 9}, {9, 55}, {10, 37}, {11, 2170}, {12, 2171}, {13, 2153}, {14, 2154}, {15, 2151}, {16, 2152}, {19, 25}, {20, 610}, {21, 284}, {22, 2172}, {25, 1973}, {27, 28}, {28, 1474}, {29, 1172}, {30, 2173}, {31, 32}, {32, 560}, {33, 607}, {34, 608}, {35, 2174}, {36, 7113}, {37, 42}, {38, 39}, {39, 1964}, {40, 198}, {41, 2175}, {42, 213}, {43, 2176}, {44, 902}, {45, 2177}, {46, 2178}, {47, 571}, {48, 184}, {51, 2179}, {52, 2180}, {53, 2181}, {54, 2148}, {55, 41}, {56, 604}, {57, 56}, {58, 1333}, {59, 2149}, {60, 2150}, {63, 3}, {64, 2155}, {65, 1400}, {66, 2156}, {67, 2157}, {68, 1820}, {69, 63}, {70, 2158}, {71, 228}, {72, 71}, {73, 1409}, {74, 2159}, {77, 222}, {78, 219}, {79, 2160}, {80, 2161}, {81, 58}, {82, 251}, {83, 82}, {84, 1436}, {85, 7}, {86, 81}, {87, 2162}, {88, 106}, {89, 2163}, {90, 2164}, {91, 2165}, {92, 4}, {94, 2166}, {95, 2167}, {96, 2168}, {97, 2169}, {98, 1910}, {99, 662}, {100, 101}, {101, 692}, {102, 32677}, {103, 911}, {104, 909}, {105, 1438}, {106, 9456}, {107, 24019}, {108, 32674}, {109, 1415}, {110, 163}, {111, 923}, {112, 32676}, {114, 17462}, {115, 2643}, {116, 17463}, {120, 17464}, {121, 17465}, {124, 38345}, {125, 3708}, {126, 17466}, {140, 17438}, {141, 38}, {142, 354}, {144, 165}, {145, 1743}, {147, 16559}, {148, 2640}, {149, 5540}, {150, 16560}, {158, 393}, {162, 112}, {163, 1576}, {165, 3207}, {169, 1486}, {171, 172}, {172, 7122}, {173, 42622}, {174, 266}, {177, 18888}, {178, 7707}, {184, 9247}, {187, 922}, {188, 259}, {189, 84}, {190, 100}, {191, 1030}, {192, 43}, {193, 1707}, {194, 1740}, {196, 208}, {197, 205}, {198, 2187}, {200, 220}, {201, 2197}, {204, 3172}, {206, 17453}, {208, 3209}, {209, 2198}, {210, 1334}, {213, 1918}, {214, 17455}, {218, 21059}, {219, 212}, {220, 1253}, {221, 2199}, {222, 603}, {223, 221}, {224, 3211}, {225, 1880}, {226, 65}, {228, 2200}, {230, 8772}, {234, 10490}, {237, 9417}, {238, 1914}, {239, 238}, {240, 232}, {241, 1458}, {242, 2201}, {243, 2202}, {244, 1015}, {249, 1101}, {253, 2184}, {255, 577}, {256, 893}, {257, 256}, {261, 2185}, {262, 2186}, {263, 3402}, {264, 92}, {267, 3444}, {268, 2188}, {269, 1407}, {270, 2189}, {271, 268}, {273, 278}, {274, 86}, {275, 2190}, {276, 40440}, {277, 2191}, {278, 34}, {279, 269}, {280, 282}, {281, 33}, {282, 2192}, {283, 2193}, {284, 2194}, {286, 27}, {287, 293}, {290, 1821}, {291, 292}, {292, 1911}, {293, 248}, {294, 2195}, {295, 2196}, {296, 1949}, {297, 240}, {304, 69}, {305, 304}, {306, 72}, {307, 1214}, {308, 3112}, {309, 189}, {310, 274}, {311, 14213}, {312, 8}, {313, 321}, {314, 333}, {315, 1760}, {316, 16568}, {317, 1748}, {318, 281}, {319, 3219}, {320, 3218}, {321, 10}, {322, 329}, {323, 6149}, {325, 1959}, {326, 394}, {329, 40}, {330, 87}, {331, 273}, {332, 1812}, {333, 21}, {334, 335}, {335, 291}, {336, 287}, {338, 1109}, {339, 20902}, {341, 346}, {342, 196}, {344, 3870}, {345, 78}, {346, 200}, {347, 223}, {348, 77}, {349, 1441}, {350, 239}, {354, 1475}, {360, 1049}, {364, 20673}, {365, 18753}, {366, 365}, {367, 20664}, {384, 1582}, {385, 1580}, {388, 2285}, {391, 4512}, {393, 1096}, {394, 255}, {399, 19303}, {401, 1955}, {411, 1630}, {412, 38860}, {427, 17442}, {440, 18673}, {441, 8766}, {442, 2294}, {445, 1844}, {448, 23692}, {452, 380}, {476, 32678}, {477, 36151}, {479, 738}, {480, 6602}, {483, 7014}, {484, 19297}, {487, 19216}, {488, 19215}, {497, 2082}, {502, 21353}, {508, 509}, {510, 20469}, {511, 1755}, {512, 798}, {513, 649}, {514, 513}, {515, 2182}, {516, 910}, {517, 2183}, {518, 672}, {519, 44}, {520, 822}, {521, 652}, {522, 650}, {523, 661}, {524, 896}, {525, 656}, {526, 2624}, {527, 1155}, {528, 2246}, {536, 899}, {537, 20331}, {538, 2234}, {542, 2247}, {551, 16666}, {553, 32636}, {554, 33654}, {555, 7371}, {556, 188}, {560, 1501}, {561, 76}, {564, 1879}, {572, 20986}, {573, 3185}, {579, 2352}, {593, 849}, {594, 756}, {595, 2220}, {596, 39798}, {599, 36263}, {604, 1397}, {607, 2212}, {608, 1395}, {610, 154}, {614, 16502}, {616, 19298}, {617, 19299}, {620, 17467}, {625, 17472}, {626, 4118}, {643, 5546}, {644, 3939}, {645, 643}, {646, 3699}, {647, 810}, {648, 162}, {649, 667}, {650, 663}, {651, 109}, {652, 1946}, {653, 108}, {654, 8648}, {655, 2222}, {656, 647}, {657, 8641}, {658, 934}, {659, 8632}, {660, 813}, {661, 512}, {662, 110}, {663, 3063}, {664, 651}, {666, 36086}, {667, 1919}, {668, 190}, {669, 1924}, {670, 799}, {671, 897}, {672, 2223}, {673, 105}, {674, 2225}, {675, 2224}, {677, 36039}, {678, 1017}, {679, 2226}, {685, 36104}, {687, 36114}, {689, 4593}, {690, 2642}, {691, 36142}, {692, 32739}, {693, 514}, {694, 1967}, {695, 9288}, {698, 2227}, {712, 2228}, {714, 2229}, {716, 2230}, {718, 2231}, {720, 2232}, {722, 2233}, {726, 1575}, {727, 34077}, {728, 480}, {730, 2235}, {732, 2236}, {734, 2237}, {738, 7023}, {740, 2238}, {741, 18268}, {742, 2239}, {744, 2240}, {748, 2241}, {749, 30651}, {750, 2242}, {751, 30650}, {752, 2243}, {754, 2244}, {756, 1500}, {757, 593}, {758, 2245}, {759, 34079}, {764, 21143}, {765, 1252}, {774, 800}, {775, 41890}, {789, 4586}, {798, 669}, {799, 99}, {801, 775}, {810, 3049}, {811, 648}, {812, 659}, {813, 34067}, {822, 39201}, {823, 107}, {824, 1491}, {825, 34069}, {826, 8061}, {827, 34072}, {830, 2483}, {846, 18755}, {850, 1577}, {851, 42669}, {858, 18669}, {869, 40728}, {870, 14621}, {872, 7109}, {873, 1509}, {874, 3570}, {876, 3572}, {883, 1025}, {885, 1024}, {889, 4607}, {891, 3768}, {892, 36085}, {893, 904}, {894, 171}, {895, 36060}, {896, 187}, {897, 111}, {898, 34075}, {899, 3230}, {900, 1635}, {901, 32665}, {902, 2251}, {903, 88}, {904, 7104}, {905, 1459}, {906, 32656}, {908, 517}, {909, 34858}, {912, 2252}, {914, 912}, {915, 913}, {916, 2253}, {918, 2254}, {919, 32666}, {920, 1609}, {922, 14567}, {923, 32740}, {925, 36145}, {927, 36146}, {930, 36148}, {932, 34071}, {934, 1461}, {936, 2256}, {937, 2255}, {938, 2257}, {940, 1468}, {941, 2258}, {942, 2260}, {943, 2259}, {944, 2261}, {946, 2262}, {948, 2263}, {950, 2264}, {952, 2265}, {954, 2266}, {956, 2267}, {958, 2268}, {960, 2269}, {962, 2270}, {966, 968}, {968, 2271}, {969, 967}, {971, 2272}, {976, 2273}, {978, 21769}, {980, 2274}, {982, 2275}, {984, 2276}, {985, 40746}, {986, 2277}, {993, 2278}, {996, 40401}, {1001, 2280}, {1002, 2279}, {1005, 2301}, {1006, 2302}, {1010, 2303}, {1011, 2304}, {1014, 1412}, {1015, 3248}, {1016, 765}, {1018, 4557}, {1019, 3733}, {1021, 21789}, {1022, 23345}, {1023, 23344}, {1024, 884}, {1025, 2283}, {1026, 2284}, {1029, 267}, {1031, 39725}, {1038, 2286}, {1040, 7124}, {1043, 2287}, {1045, 21779}, {1046, 2305}, {1052, 41395}, {1054, 9259}, {1064, 2288}, {1073, 19614}, {1077, 359}, {1078, 18042}, {1081, 2306}, {1082, 2307}, {1084, 4117}, {1086, 244}, {1087, 36412}, {1088, 279}, {1089, 594}, {1092, 4100}, {1093, 6520}, {1096, 2207}, {1097, 36413}, {1098, 7054}, {1099, 3163}, {1100, 2308}, {1101, 23357}, {1102, 3964}, {1107, 2309}, {1108, 40958}, {1109, 115}, {1110, 23990}, {1111, 1086}, {1113, 2576}, {1114, 2577}, {1115, 7039}, {1119, 1435}, {1120, 40400}, {1121, 1156}, {1124, 605}, {1125, 1100}, {1126, 28615}, {1143, 7010}, {1146, 2310}, {1147, 563}, {1150, 993}, {1154, 2290}, {1155, 1055}, {1156, 2291}, {1157, 19306}, {1172, 2299}, {1193, 2300}, {1201, 20228}, {1210, 1108}, {1211, 2292}, {1212, 2293}, {1213, 1962}, {1214, 73}, {1215, 2295}, {1218, 2296}, {1219, 2297}, {1220, 2298}, {1221, 40418}, {1222, 23617}, {1226, 17862}, {1228, 18697}, {1229, 4847}, {1230, 4647}, {1231, 307}, {1232, 20879}, {1233, 20880}, {1235, 20883}, {1236, 20884}, {1237, 3963}, {1240, 30710}, {1245, 2281}, {1246, 2282}, {1249, 204}, {1252, 1110}, {1253, 14827}, {1255, 1126}, {1257, 2983}, {1259, 2289}, {1260, 1802}, {1262, 24027}, {1264, 3719}, {1265, 3692}, {1266, 16610}, {1267, 3083}, {1268, 1255}, {1269, 4359}, {1274, 7001}, {1275, 7045}, {1276, 19304}, {1277, 19305}, {1278, 16569}, {1281, 19557}, {1282, 20672}, {1293, 34080}, {1302, 36149}, {1304, 36131}, {1319, 1404}, {1320, 2316}, {1323, 6610}, {1329, 17452}, {1330, 1761}, {1331, 906}, {1332, 1331}, {1333, 2206}, {1335, 606}, {1337, 19300}, {1338, 19301}, {1352, 16567}, {1368, 18671}, {1369, 21378}, {1370, 18596}, {1376, 9310}, {1385, 2317}, {1386, 21764}, {1400, 1402}, {1403, 41526}, {1407, 1106}, {1408, 16947}, {1412, 1408}, {1414, 4565}, {1422, 1413}, {1423, 1403}, {1427, 1042}, {1429, 1428}, {1432, 1431}, {1434, 1014}, {1435, 1398}, {1436, 2208}, {1440, 1422}, {1441, 226}, {1442, 2003}, {1444, 1790}, {1445, 1617}, {1446, 3668}, {1447, 1429}, {1459, 22383}, {1462, 1416}, {1465, 1457}, {1468, 5019}, {1474, 2203}, {1476, 3451}, {1488, 289}, {1490, 3197}, {1491, 3250}, {1492, 825}, {1494, 2349}, {1495, 9406}, {1496, 5065}, {1500, 872}, {1501, 1917}, {1502, 561}, {1503, 2312}, {1509, 757}, {1565, 3942}, {1575, 3009}, {1577, 523}, {1580, 1691}, {1581, 694}, {1582, 1915}, {1621, 4251}, {1631, 32664}, {1635, 1960}, {1638, 14413}, {1639, 4895}, {1647, 2087}, {1654, 846}, {1655, 1045}, {1659, 2362}, {1691, 1933}, {1698, 16777}, {1706, 1696}, {1707, 3053}, {1708, 37579}, {1712, 1033}, {1716, 21775}, {1724, 5301}, {1725, 3003}, {1726, 23843}, {1730, 23383}, {1733, 230}, {1734, 6586}, {1735, 8607}, {1736, 8608}, {1737, 8609}, {1738, 3290}, {1739, 8610}, {1740, 1613}, {1742, 20995}, {1743, 3052}, {1745, 21767}, {1748, 24}, {1749, 11063}, {1751, 2218}, {1755, 237}, {1757, 17735}, {1758, 17966}, {1759, 1631}, {1760, 22}, {1761, 199}, {1762, 3145}, {1763, 3556}, {1764, 23361}, {1766, 197}, {1769, 3310}, {1780, 41332}, {1783, 8750}, {1784, 1990}, {1785, 14571}, {1790, 1437}, {1791, 2359}, {1792, 2327}, {1795, 14578}, {1797, 36058}, {1799, 34055}, {1804, 7125}, {1812, 283}, {1813, 36059}, {1814, 36057}, {1815, 36056}, {1817, 2360}, {1818, 20752}, {1820, 2351}, {1821, 98}, {1824, 2333}, {1826, 1824}, {1829, 2354}, {1834, 40977}, {1837, 40968}, {1838, 1841}, {1839, 2355}, {1847, 1119}, {1848, 1829}, {1855, 1827}, {1858, 1195}, {1861, 5089}, {1895, 1249}, {1896, 8748}, {1897, 1783}, {1899, 2083}, {1903, 2357}, {1909, 894}, {1910, 1976}, {1911, 1922}, {1914, 2210}, {1915, 1932}, {1916, 1581}, {1917, 9233}, {1918, 2205}, {1919, 1980}, {1920, 1909}, {1921, 350}, {1922, 14598}, {1923, 41331}, {1924, 9426}, {1925, 9230}, {1926, 3978}, {1927, 8789}, {1928, 1502}, {1929, 17962}, {1930, 141}, {1931, 1326}, {1933, 14602}, {1934, 1916}, {1935, 1950}, {1936, 1951}, {1937, 1945}, {1940, 7120}, {1943, 1935}, {1944, 1936}, {1947, 1940}, {1948, 243}, {1952, 1937}, {1953, 51}, {1954, 1970}, {1955, 1971}, {1956, 1987}, {1957, 1968}, {1958, 9306}, {1959, 511}, {1962, 20970}, {1964, 3051}, {1965, 384}, {1966, 385}, {1967, 9468}, {1969, 264}, {1972, 1956}, {1973, 1974}, {1975, 1958}, {1978, 668}, {1981, 23353}, {1992, 36277}, {1993, 47}, {1994, 2964}, {1997, 36846}, {1999, 5247}, {2003, 1399}, {2006, 1411}, {2051, 34434}, {2052, 158}, {2064, 7270}, {2082, 7083}, {2083, 40947}, {2084, 688}, {2085, 8265}, {2090, 15997}, {2092, 3725}, {2099, 1405}, {2113, 18783}, {2128, 19588}, {2129, 15369}, {2151, 34394}, {2152, 34395}, {2153, 3457}, {2154, 3458}, {2155, 33581}, {2156, 2353}, {2157, 3455}, {2159, 40352}, {2160, 6186}, {2161, 6187}, {2162, 7121}, {2163, 28607}, {2166, 1989}, {2167, 54}, {2168, 41271}, {2169, 14533}, {2170, 3271}, {2171, 181}, {2172, 206}, {2173, 1495}, {2175, 9447}, {2176, 2209}, {2179, 40981}, {2181, 3199}, {2184, 64}, {2185, 60}, {2186, 263}, {2190, 8882}, {2192, 7118}, {2210, 14599}, {2221, 1472}, {2222, 32675}, {2223, 9454}, {2225, 8618}, {2227, 3229}, {2228, 8620}, {2230, 8621}, {2234, 3231}, {2235, 8622}, {2236, 8623}, {2238, 3747}, {2239, 8624}, {2240, 8625}, {2243, 8626}, {2244, 8627}, {2245, 3724}, {2247, 5191}, {2248, 18757}, {2251, 9459}, {2254, 665}, {2260, 40956}, {2269, 20967}, {2270, 20991}, {2275, 7032}, {2276, 869}, {2285, 1460}, {2287, 2328}, {2289, 6056}, {2291, 34068}, {2292, 2092}, {2293, 20229}, {2294, 40952}, {2295, 20964}, {2297, 7050}, {2299, 2204}, {2309, 1197}, {2310, 14936}, {2312, 42671}, {2319, 2053}, {2320, 2364}, {2321, 210}, {2322, 4183}, {2323, 2361}, {2324, 7074}, {2325, 3689}, {2329, 2330}, {2331, 3195}, {2339, 1036}, {2345, 612}, {2346, 1174}, {2348, 8647}, {2349, 74}, {2363, 1169}, {2421, 23997}, {2475, 1781}, {2481, 673}, {2482, 42081}, {2483, 8635}, {2484, 8646}, {2517, 6590}, {2530, 21123}, {2550, 40131}, {2574, 2578}, {2575, 2579}, {2578, 42668}, {2579, 42667}, {2580, 1113}, {2581, 1114}, {2582, 2574}, {2583, 2575}, {2588, 8105}, {2589, 8106}, {2592, 2588}, {2593, 2589}, {2594, 21741}, {2610, 42666}, {2611, 20982}, {2616, 2623}, {2617, 1625}, {2618, 12077}, {2624, 14270}, {2631, 9409}, {2632, 3269}, {2638, 39687}, {2640, 20998}, {2642, 351}, {2643, 3124}, {2644, 9218}, {2646, 21748}, {2648, 17963}, {2651, 5060}, {2664, 21788}, {2667, 21753}, {2669, 2106}, {2720, 32669}, {2770, 36150}, {2786, 9508}, {2886, 17451}, {2887, 3721}, {2893, 1762}, {2895, 191}, {2896, 16556}, {2951, 1615}, {2962, 2963}, {2964, 2965}, {2966, 36084}, {2968, 34591}, {2972, 37754}, {2975, 572}, {2986, 36053}, {2987, 36051}, {2990, 36052}, {2994, 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{25098, 22090}, {25259, 1734}, {25278, 25728}, {25280, 17261}, {25303, 17120}, {25430, 2334}, {25453, 16974}, {25527, 37549}, {25590, 37674}, {25639, 17443}, {25666, 4879}, {25728, 4421}, {25760, 3735}, {25900, 39541}, {25935, 5728}, {26015, 43065}, {26065, 37552}, {26166, 17859}, {26223, 5264}, {26227, 16788}, {26234, 17023}, {26538, 24239}, {26543, 11031}, {26544, 24802}, {26546, 21185}, {26563, 3663}, {26580, 4424}, {26591, 31397}, {26611, 24028}, {26626, 16475}, {26685, 3749}, {26792, 11010}, {26842, 3337}, {26932, 7004}, {26942, 201}, {26959, 18170}, {27003, 5563}, {27064, 5255}, {27065, 3746}, {27131, 5697}, {27164, 10458}, {27184, 986}, {27186, 18398}, {27191, 3315}, {27268, 42042}, {27287, 37529}, {27288, 4335}, {27318, 24661}, {27339, 37523}, {27382, 7070}, {27396, 3190}, {27424, 7155}, {27474, 3789}, {27475, 1002}, {27478, 28600}, {27481, 3795}, {27483, 30571}, {27495, 40774}, {27509, 1040}, {27538, 3208}, {27558, 24048}, {27569, 21085}, {27644, 38832}, {27647, 39548}, {27757, 4867}, {27792, 17164}, {27801, 313}, {27805, 3903}, {27808, 4033}, {27809, 18793}, {27818, 19604}, {27834, 1293}, {27853, 874}, {27855, 4375}, {27918, 27846}, {27929, 38348}, {28391, 41350}, {28604, 1961}, {28605, 1698}, {28606, 386}, {28609, 37567}, {28626, 39948}, {28650, 27789}, {28653, 17019}, {28654, 1089}, {28659, 3596}, {28660, 314}, {28706, 18695}, {28739, 8270}, {28740, 8271}, {28808, 3872}, {28809, 3886}, {28840, 4784}, {28878, 7659}, {28956, 13279}, {29007, 3256}, {29433, 16684}, {29477, 17221}, {29574, 4663}, {29579, 16496}, {29584, 16477}, {29598, 38315}, {29611, 7174}, {29616, 5223}, {29627, 3243}, {29965, 35645}, {29966, 35892}, {29982, 42057}, {30007, 35631}, {30019, 38484}, {30035, 37521}, {30225, 36278}, {30379, 3660}, {30473, 32925}, {30476, 17478}, {30479, 2339}, {30545, 3212}, {30547, 24579}, {30556, 2066}, {30557, 5414}, {30565, 3887}, {30566, 2802}, {30567, 12513}, {30568, 3913}, {30571, 25426}, {30578, 5541}, {30583, 14437}, {30591, 4988}, {30593, 30581}, {30594, 30582}, {30596, 28605}, {30598, 25417}, {30599, 25526}, {30608, 2320}, {30625, 11495}, {30660, 17739}, {30661, 18754}, {30667, 18794}, {30669, 18787}, {30687, 1905}, {30690, 79}, {30693, 5423}, {30694, 1721}, {30695, 2951}, {30699, 1722}, {30710, 1220}, {30712, 39980}, {30713, 3701}, {30730, 4069}, {30758, 17316}, {30805, 4131}, {30806, 527}, {30807, 516}, {30827, 2098}, {30829, 3241}, {30852, 1482}, {30854, 390}, {30855, 10700}, {30857, 10695}, {30866, 24841}, {30874, 35548}, {30875, 35533}, {30892, 26230}, {30939, 16704}, {30940, 33295}, {30941, 18206}, {30942, 16975}, {30946, 37555}, {30950, 16971}, {30963, 4393}, {30966, 40773}, {31002, 3227}, {31008, 33296}, {31018, 5119}, {31019, 5902}, {31053, 5903}, {31130, 17284}, {31134, 36283}, {31150, 4794}, {31164, 36279}, {31231, 1388}, {31330, 5283}, {31359, 941}, {31424, 37504}, {31526, 34497}, {31527, 2124}, {31598, 28039}, {31600, 34052}, {31605, 43049}, {31618, 21453}, {31623, 29}, {31625, 7035}, {31627, 3160}, {31631, 3193}, {31637, 1814}, {31995, 5437}, {31997, 17379}, {31998, 2644}, {32005, 36646}, {32008, 2346}, {32009, 40433}, {32010, 40432}, {32014, 40438}, {32017, 1222}, {32018, 1268}, {32020, 3226}, {32023, 9311}, {32024, 7676}, {32025, 33761}, {32033, 33784}, {32041, 37138}, {32042, 37211}, {32087, 7308}, {32092, 15668}, {32099, 3929}, {32104, 17259}, {32428, 2313}, {32462, 21778}, {32577, 34543}, {32635, 33635}, {32664, 40370}, {32665, 32719}, {32678, 14560}, {32679, 526}, {32680, 476}, {32726, 34078}, {32771, 17750}, {32777, 976}, {32779, 30115}, {32783, 16519}, {32784, 41269}, {32816, 18713}, {32851, 4511}, {32858, 5904}, {32863, 6763}, {32911, 595}, {32913, 33863}, {32922, 33854}, {32933, 25440}, {32937, 3501}, {32939, 404}, {33077, 5692}, {33091, 17744}, {33113, 22836}, {33116, 34772}, {33129, 30117}, {33137, 16968}, {33146, 24046}, {33296, 27644}, {33299, 3688}, {33651, 34065}, {33673, 18623}, {33677, 10025}, {33760, 1627}, {33761, 33771}, {33762, 33772}, {33763, 33773}, {33764, 1078}, {33765, 38859}, {33766, 33774}, {33767, 33762}, {33768, 33763}, {33769, 33764}, {33770, 33766}, {33775, 32025}, {33776, 33767}, {33777, 33768}, {33778, 33769}, {33779, 33770}, {33780, 4452}, {33781, 1611}, {33782, 33786}, {33784, 41396}, {33787, 6392}, {33788, 8264}, {33789, 32033}, {33790, 33801}, {33791, 33802}, {33792, 23374}, {33793, 35222}, {33794, 35224}, {33795, 33804}, {33796, 33790}, {33797, 33791}, {33798, 33793}, {33800, 33795}, {33805, 1494}, {33806, 33796}, {33807, 33797}, {33808, 6515}, {33809, 33799}, {33888, 2108}, {33889, 3507}, {33890, 41886}, {33930, 3662}, {33931, 3661}, {33932, 17233}, {33933, 17234}, {33934, 4389}, {33935, 5224}, {33936, 4643}, {33937, 17279}, {33938, 17280}, {33939, 319}, {33940, 16706}, {33941, 17289}, {33942, 4851}, {33943, 17300}, {33944, 17302}, {33945, 4657}, {33946, 3888}, {34020, 34063}, {34021, 39915}, {34050, 1455}, {34055, 1176}, {34060, 34488}, {34065, 19121}, {34072, 4630}, {34075, 32718}, {34085, 927}, {34086, 34020}, {34234, 104}, {34258, 31359}, {34284, 10436}, {34387, 4858}, {34388, 6358}, {34393, 36100}, {34403, 19611}, {34404, 280}, {34528, 42066}, {34537, 24037}, {34538, 24021}, {34544, 215}, {34589, 11998}, {34591, 3270}, {34824, 17450}, {34832, 17459}, {34860, 39956}, {34892, 34893}, {34914, 34916}, {35008, 35009}, {35025, 25820}, {35026, 36258}, {35027, 36256}, {35028, 36261}, {35029, 36257}, {35057, 9404}, {35058, 39748}, {35068, 4094}, {35071, 42080}, {35072, 2638}, {35092, 42084}, {35110, 42082}, {35139, 32680}, {35145, 37142}, {35148, 37135}, {35157, 37139}, {35171, 37143}, {35174, 655}, {35175, 37222}, {35179, 37216}, {35181, 37210}, {35193, 35192}, {35198, 11136}, {35199, 11135}, {35200, 18877}, {35201, 39176}, {35338, 35326}, {35342, 35327}, {35508, 24012}, {35511, 9395}, {35517, 30807}, {35518, 6332}, {35519, 4391}, {35543, 6381}, {35544, 3948}, {35550, 3936}, {35551, 4766}, {35956, 36269}, {35957, 24482}, {35958, 36268}, {35959, 36273}, {35960, 24504}, {35962, 36275}, {35963, 36265}, {35964, 36262}, {35965, 36264}, {36034, 32640}, {36035, 1637}, {36036, 2966}, {36037, 32641}, {36038, 10015}, {36039, 32642}, {36040, 32643}, {36041, 32644}, {36042, 32645}, {36043, 32646}, {36044, 32647}, {36045, 32648}, {36046, 32649}, {36047, 32650}, {36048, 32651}, {36049, 32652}, {36050, 32653}, {36051, 32654}, {36052, 32655}, {36053, 14910}, {36056, 32657}, {36057, 32658}, {36058, 32659}, {36059, 32660}, {36060, 14908}, {36061, 32662}, {36062, 32663}, {36067, 32667}, {36068, 32670}, {36069, 32671}, {36070, 32672}, {36071, 32673}, {36083, 32681}, {36084, 2715}, {36085, 691}, {36086, 919}, {36087, 32682}, {36088, 32683}, {36089, 32684}, {36090, 32685}, {36091, 32686}, {36092, 32687}, {36093, 32688}, {36094, 32689}, {36095, 10423}, {36096, 23969}, {36097, 32690}, {36098, 8687}, {36099, 32691}, {36100, 102}, {36101, 103}, {36102, 477}, {36103, 21148}, {36104, 32696}, {36105, 32697}, {36106, 32698}, {36107, 32699}, {36108, 32700}, {36109, 32701}, {36110, 32702}, {36111, 32703}, {36112, 32705}, {36113, 32707}, {36114, 32708}, {36115, 32709}, {36116, 32711}, {36117, 32712}, {36118, 32714}, {36119, 8749}, {36120, 6531}, {36124, 8751}, {36125, 8752}, {36126, 6529}, {36128, 8753}, {36131, 32715}, {36132, 32716}, {36133, 32717}, {36134, 14586}, {36135, 32720}, {36136, 32721}, {36137, 32723}, {36138, 32724}, {36139, 32725}, {36140, 32727}, {36141, 32728}, {36142, 32729}, {36143, 32731}, {36144, 32733}, {36145, 32734}, {36146, 32735}, {36147, 32736}, {36148, 32737}, {36149, 32738}, {36150, 32741}, {36215, 36276}, {36217, 36281}, {36263, 574}, {36277, 1384}, {36289, 9463}, {36479, 36404}, {36588, 39963}, {36598, 36614}, {36602, 36619}, {36606, 36603}, {36624, 36629}, {36625, 36627}, {36634, 36647}, {36640, 36636}, {36789, 1099}, {36790, 23996}, {36791, 4738}, {36792, 24038}, {36793, 17879}, {36796, 14942}, {36799, 8851}, {36805, 1120}, {36807, 1280}, {36834, 14969}, {36838, 4626}, {36845, 16572}, {36863, 4595}, {36908, 40933}, {37086, 41230}, {37128, 741}, {37129, 739}, {37130, 675}, {37131, 840}, {37132, 729}, {37133, 789}, {37134, 805}, {37135, 2702}, {37136, 2720}, {37137, 29055}, {37138, 8693}, {37139, 14733}, {37140, 36069}, {37141, 8059}, {37142, 2249}, {37143, 1308}, {37202, 26702}, {37203, 915}, {37204, 689}, {37205, 34594}, {37206, 1292}, {37207, 30664}, {37208, 761}, {37209, 29351}, {37210, 8691}, {37211, 8652}, {37212, 8701}, {37214, 43363}, {37215, 1310}, {37216, 1296}, {37217, 30247}, {37218, 835}, {37220, 2373}, {37221, 9076}, {37222, 2718}, {37279, 41227}, {37444, 18595}, {37543, 1451}, {37638, 18477}, {37680, 40091}, {37732, 21770}, {37754, 34980}, {37755, 1425}, {37756, 7292}, {37757, 38459}, {37758, 38460}, {37771, 1421}, {37772, 19373}, {37773, 7051}, {37774, 3100}, {37779, 1749}, {37780, 1323}, {37781, 1768}, {37783, 5127}, {37787, 2078}, {37788, 26015}, {37789, 5193}, {37790, 1877}, {37800, 34036}, {37805, 23710}, {37870, 5331}, {38247, 36598}, {38262, 38275}, {38275, 36615}, {38298, 20061}, {38340, 26700}, {38347, 38365}, {38417, 1133}, {38462, 8756}, {38468, 30379}, {38470, 34076}, {38810, 40415}, {38830, 38847}, {38847, 40416}, {38906, 30878}, {38930, 42446}, {38955, 2250}, {38986, 21762}, {39028, 39916}, {39044, 4366}, {39060, 2639}, {39062, 2633}, {39126, 5435}, {39130, 1903}, {39273, 3423}, {39337, 9431}, {39345, 39335}, {39346, 39336}, {39347, 39338}, {39348, 39340}, {39349, 39343}, {39350, 39341}, {39351, 9355}, {39352, 2629}, {39353, 39344}, {39355, 39342}, {39467, 41840}, {39694, 979}, {39695, 39947}, {39696, 39946}, {39697, 39981}, {39702, 39975}, {39703, 39969}, {39704, 89}, {39707, 26745}, {39708, 39982}, {39710, 39962}, {39712, 39957}, {39714, 39979}, {39716, 39958}, {39717, 39971}, {39718, 9510}, {39721, 39954}, {39722, 39977}, {39724, 7194}, {39725, 14370}, {39727, 41513}, {39731, 3618}, {39732, 40188}, {39733, 13575}, {39734, 39950}, {39735, 8049}, {39737, 39961}, {39738, 39972}, {39739, 39965}, {39741, 39970}, {39742, 39966}, {39747, 39949}, {39748, 39964}, {39749, 39959}, {39775, 34253}, {39797, 34444}, {39798, 40148}, {39914, 34252}, {39925, 2665}, {39947, 34430}, {39963, 41436}, {39970, 34445}, {39994, 39697}, {39995, 17495}, {39996, 30579}, {40001, 33090}, {40004, 39734}, {40005, 39735}, {40009, 39733}, {40010, 35058}, {40011, 2997}, {40012, 34860}, {40013, 596}, {40014, 4373}, {40015, 7219}, {40016, 18833}, {40017, 18827}, {40022, 39731}, {40023, 5936}, {40024, 39717}, {40025, 39741}, {40026, 36606}, {40027, 38247}, {40028, 39721}, {40029, 36588}, {40033, 39722}, {40034, 18133}, {40035, 20934}, {40036, 39727}, {40037, 39726}, {40038, 39724}, {40039, 39698}, {40040, 39699}, {40044, 39723}, {40050, 40364}, {40071, 20336}, {40072, 28660}, {40073, 20641}, {40074, 20944}, {40075, 20924}, {40087, 18140}, {40088, 18152}, {40089, 18145}, {40091, 33882}, {40098, 30663}, {40131, 37580}, {40133, 20978}, {40149, 225}, {40151, 16945}, {40152, 22341}, {40154, 17107}, {40162, 18832}, {40165, 7049}, {40166, 21132}, {40214, 17104}, {40215, 16944}, {40216, 17758}, {40217, 22116}, {40300, 40302}, {40301, 40300}, {40327, 40338}, {40339, 40301}, {40361, 33806}, {40362, 1928}, {40363, 28659}, {40364, 305}, {40365, 20444}, {40370, 40371}, {40382, 33783}, {40383, 40375}, {40393, 2216}, {40399, 1167}, {40414, 40431}, {40418, 1258}, {40420, 1476}, {40422, 40435}, {40424, 40399}, {40432, 1178}, {40434, 41434}, {40435, 943}, {40438, 1171}, {40439, 40408}, {40440, 275}, {40443, 1803}, {40471, 14991}, {40493, 36854}, {40495, 3261}, {40504, 40147}, {40515, 40504}, {40571, 1780}, {40592, 501}, {40593, 31526}, {40594, 39148}, {40603, 3159}, {40612, 6126}, {40619, 17761}, {40624, 34589}, {40626, 34588}, {40650, 38003}, {40699, 15891}, {40700, 15892}, {40701, 342}, {40702, 347}, {40703, 297}, {40704, 9436}, {40716, 21739}, {40718, 40747}, {40719, 5228}, {40720, 40753}, {40721, 40749}, {40722, 40751}, {40723, 40765}, {40724, 40754}, {40725, 40767}, {40728, 18900}, {40737, 40770}, {40738, 40763}, {40739, 40757}, {40740, 40766}, {40741, 40768}, {40742, 40772}, {40743, 40752}, {40773, 3736}, {40814, 4008}, {40836, 7129}, {40837, 207}, {40838, 7007}, {40843, 296}, {40845, 7261}, {40846, 7061}, {40848, 41531}, {40850, 34054}, {40862, 9364}, {40863, 23693}, {40864, 14189}, {40869, 41339}, {40873, 41532}, {40874, 2669}, {40875, 5205}, {40880, 9371}, {40882, 2651}, {40934, 21750}, {40935, 21751}, {40936, 21752}, {40937, 14547}, {40940, 1104}, {40952, 40978}, {40962, 40969}, {40963, 40970}, {40977, 40984}, {40999, 16577}, {41004, 26934}, {41005, 6508}, {41006, 14100}, {41013, 1826}, {41014, 3958}, {41072, 37207}, {41079, 36035}, {41081, 1433}, {41083, 3194}, {41140, 3246}, {41239, 19133}, {41283, 20567}, {41314, 23891}, {41318, 17752}, {41497, 41368}, {41514, 3345}, {41532, 41882}, {41535, 40874}, {41629, 16948}, {41760, 17871}, {41777, 7248}, {41798, 4845}, {41804, 18593}, {41809, 3743}, {41821, 5506}, {41847, 14996}, {41875, 20090}, {41878, 34195}, {41886, 20284}, {41915, 3646}, {42005, 5949}, {42012, 32561}, {42014, 32578}, {42027, 16606}, {42029, 9780}, {42033, 4420}, {42034, 3617}, {42051, 27627}, {42066, 9560}, {42074, 9408}, {42075, 9419}, {42079, 39686}, {42081, 39689}, {42285, 39974}, {42303, 42301}, {42311, 10509}, {42371, 37204}, {42455, 42462}, {42456, 21854}, {42467, 3435}, {42483, 8917}, {42554, 20898}, {42555, 6164}, {42696, 3305}, {42697, 3306}, {42708, 21674}, {42709, 16086}, {42710, 21081}, {42712, 4061}, {42713, 4062}, {42719, 2398}, {42720, 1026}, {42754, 42753}, {43035, 1456}, {43040, 1463}, {43071, 43070}, {43093, 37130}, {43099, 37208}, {43187, 36036}, {43531, 2214}
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 86, 17394}, {1, 274, 31997}, {1, 304, 18156}, {1, 1733, 4008}, {1, 1740, 1964}, {1, 1930, 304}, {1, 2234, 36289}, {1, 3403, 1966}, {1, 3875, 4360}, {1, 4360, 17393}, {1, 10436, 86}, {1, 10447, 314}, {1, 16571, 1740}, {1, 17143, 17144}, {1, 17151, 3875}, {1, 18691, 17858}, {1, 18695, 33808}, {1, 25528, 24661}, {1, 25590, 10436}, {1, 32092, 274}, {1, 32104, 17143}, {1, 33933, 33943}, {1, 33935, 17762}, {1, 33943, 41875}, {2, 37, 4687}, {2, 192, 37}, {2, 312, 18743}, {2, 321, 312}, {2, 344, 17263}, {2, 345, 33116}, {2, 346, 344}, {2, 350, 30963}, {2, 1229, 20946}, {2, 1267, 32791}, {2, 1278, 192}, {2, 2345, 17289}, {2, 3210, 3666}, {2, 3263, 30758}, {2, 3672, 17321}, {2, 3739, 4751}, {2, 4000, 16706}, {2, 4358, 30829}, {2, 4359, 19804}, {2, 4441, 350}, {2, 4452, 3672}, {2, 4461, 346}, {2, 4657, 17400}, {2, 4671, 4358}, {2, 4686, 3644}, {2, 4699, 3739}, {2, 4704, 27268}, {2, 4726, 4764}, {2, 4772, 4699}, {2, 4788, 4704}, {2, 4821, 1278}, {2, 4980, 42029}, {2, 5391, 32792}, {2, 16610, 31233}, {2, 16706, 17370}, {2, 17147, 28606}, {2, 17279, 17341}, {2, 17280, 17279}, {2, 17281, 17342}, {2, 17289, 17371}, {2, 17301, 17399}, {2, 17302, 4657}, {2, 17321, 17322}, {2, 17358, 17357}, {2, 17383, 17384}, {2, 17490, 3752}, {2, 17495, 4850}, {2, 17740, 32851}, {2, 17759, 2276}, {2, 19785, 19786}, {2, 19786, 19812}, {2, 19787, 19803}, {2, 19788, 19802}, {2, 19789, 19785}, {2, 19808, 19827}, {2, 19810, 19806}, {2, 19818, 19801}, {2, 19819, 19796}, {2, 19820, 19830}, {2, 19822, 19808}, {2, 19824, 19823}, {2, 19825, 19822}, {2, 19826, 19789}, {2, 19835, 19799}, {2, 20891, 20923}, {2, 20892, 30090}, {2, 24620, 16610}, {2, 25243, 26669}, {2, 26143, 25535}, {2, 26274, 3290}, {2, 26797, 27073}, {2, 26850, 27011}, {2, 26971, 25505}, {2, 27268, 4698}, {2, 28604, 17303}, {2, 28605, 321}, {2, 28808, 37758}, {2, 30998, 20530}, {2, 31130, 3263}, {2, 32791, 32803}, {2, 32792, 32804}, {2, 32793, 1267}, {2, 32794, 5391}, {2, 32797, 32801}, {2, 32798, 32802}, {2, 32799, 32795}, {2, 32800, 32796}, {2, 33150, 32774}, {2, 33168, 33113}, {2, 33931, 20947}, {2, 37759, 17720}, {2, 42029, 42034}, {2, 42034, 20942}, {3, 17864, 20926}, {3, 19844, 19841}, {3, 19845, 19842}, {3, 38906, 39556}, {4, 20235, 20914}, {4, 20914, 38298}, {5, 21403, 21579}, {6, 239, 3759}, {6, 894, 3758}, {6, 4361, 239}, {6, 4363, 894}, {6, 17118, 4363}, {6, 17119, 4361}, {6, 20172, 20179}, {6, 20181, 20172}, {6, 20234, 20444}, {6, 21776, 21751}, {6, 38301, 21776}, {7, 8, 69}, {7, 69, 320}, {7, 319, 17361}, {7, 1441, 85}, {7, 3212, 24471}, {7, 3262, 20930}, {7, 5564, 17360}, {7, 20895, 322}, {7, 31598, 7195}, {7, 31995, 42697}, {7, 32087, 8}, {7, 32099, 21296}, {7, 42696, 319}, {7, 42697, 7321}, {8, 69, 319}, {8, 85, 16284}, {8, 320, 17360}, {8, 377, 7270}, {8, 1441, 322}, {8, 1909, 24524}, {8, 2550, 32850}, {8, 4645, 3416}, {8, 7321, 17361}, {8, 20880, 85}, {8, 21296, 32099}, {8, 26135, 25311}, {8, 31995, 7}, {8, 32087, 42696}, {8, 33930, 20955}, {8, 34284, 1909}, {8, 42696, 5564}, {8, 42697, 320}, {9, 190, 17336}, {9, 3729, 190}, {9, 4384, 17277}, {9, 4659, 3729}, {9, 17277, 17335}, {9, 20236, 20927}, {10, 76, 6376}, {10, 1266, 4389}, {10, 1269, 18133}, {10, 1738, 4429}, {10, 3663, 4357}, {10, 3821, 32784}, {10, 4357, 5224}, {10, 4385, 341}, {10, 4389, 17250}, {10, 4398, 17249}, {10, 19950, 19951}, {10, 19951, 19969}, {10, 19957, 20006}, {10, 19958, 19957}, {10, 19962, 19950}, {10, 19963, 20001}, {10, 19967, 19971}, {10, 19968, 19954}, {10, 19995, 20005}, {10, 19996, 19968}, {10, 19999, 20000}, {10, 20000, 19963}, {10, 20004, 19962}, {10, 20888, 76}, {10, 21443, 1921}, {10, 23537, 16062}, {10, 33940, 33944}, {10, 33941, 33938}, {10, 34832, 25121}, {11, 21404, 21580}, {12, 21405, 21581}, {19, 63, 1760}, {19, 1760, 16568}, {19, 20883, 92}, {19, 20884, 20915}, {19, 21406, 21582}, {22, 21407, 21583}, {23, 21408, 21584}, {31, 20627, 20641}, {32, 21409, 21585}, {35, 21410, 21586}, {36, 21411, 21587}, {37, 192, 4664}, {37, 321, 4043}, {37, 1278, 3644}, {37, 3739, 2}, {37, 4681, 4704}, {37, 4688, 3739}, {37, 4698, 27268}, {37, 4699, 4751}, {37, 4718, 4681}, {37, 4726, 1278}, {37, 4739, 4699}, {37, 4740, 4764}, {37, 20891, 18137}, {37, 20923, 18743}, {37, 31238, 4698}, {38, 561, 17149}, {38, 2227, 3116}, {38, 17872, 17446}, {38, 20889, 561}, {38, 21020, 31330}, {39, 21412, 18050}, {40, 21413, 21588}, {41, 21414, 21589}, {42, 21415, 18138}, {43, 21416, 21590}, {44, 17348, 17349}, {44, 17351, 17350}, {44, 21417, 21591}, {45, 17259, 17260}, {45, 17262, 17261}, {45, 21418, 21592}, {48, 1958, 662}, {48, 17865, 21593}, {55, 20890, 20922}, {56, 21420, 21594}, {57, 11679, 14829}, {57, 20237, 20928}, {58, 21421, 21595}, {63, 92, 18750}, {63, 5271, 333}, {63, 14213, 92}, {63, 18655, 8822}, {63, 19591, 1755}, {63, 20883, 21582}, {65, 21422, 21596}, {66, 21423, 21597}, {69, 319, 17360}, {69, 320, 17361}, {69, 322, 16284}, {69, 1441, 20930}, {69, 3262, 322}, {69, 31995, 7321}, {69, 32087, 5564}, {69, 42696, 8}, {69, 42697, 7}, {76, 313, 30596}, {76, 1921, 21615}, {76, 3596, 313}, {76, 3821, 24731}, {76, 4389, 39995}, {76, 5224, 18133}, {76, 6376, 20943}, {76, 10009, 1921}, {76, 33937, 33938}, {76, 33940, 3673}, {76, 33941, 4385}, {76, 33945, 33944}, {81, 20896, 20929}, {82, 21424, 21598}, {82, 33760, 31}, {83, 21425, 20933}, {85, 322, 20930}, {85, 39126, 7}, {86, 274, 16709}, {86, 314, 30939}, {86, 3875, 17393}, {86, 4360, 1}, {86, 10436, 41847}, {86, 17160, 4360}, {86, 18697, 20932}, {87, 18194, 3248}, {87, 21426, 21599}, {88, 21427, 21600}, {89, 21428, 21601}, {92, 21582, 20915}, {99, 17886, 20951}, {100, 20901, 20940}, {101, 21429, 21602}, {105, 20628, 20642}, {110, 21430, 21603}, {115, 21431, 21604}, {141, 594, 3661}, {141, 1086, 3662}, {141, 3661, 17228}, {141, 3662, 17227}, {141, 3963, 18040}, {141, 4665, 594}, {141, 7263, 1086}, {141, 18143, 18150}, {141, 20913, 18143}, {142, 2321, 3912}, {142, 3912, 17234}, {142, 4431, 17233}, {142, 17233, 17241}, {145, 21432, 21605}, {150, 2893, 21276}, {171, 4362, 3769}, {187, 21434, 21607}, {190, 4384, 17335}, {190, 4858, 18151}, {190, 17277, 9}, {192, 3739, 4687}, {192, 4686, 4764}, {192, 4688, 4751}, {192, 4699, 2}, {192, 4704, 4681}, {192, 4740, 1278}, {192, 4772, 3739}, {192, 4788, 4718}, {192, 4821, 4686}, {192, 20171, 20173}, {192, 20891, 312}, {192, 20892, 20923}, {192, 27268, 4704}, {192, 30090, 18743}, {194, 21435, 21608}, {200, 21436, 21609}, {226, 3687, 4417}, {238, 3923, 4676}, {238, 20629, 20643}, {239, 894, 6}, {239, 4363, 3758}, {239, 17116, 894}, {239, 17117, 4361}, {239, 17120, 17121}, {239, 20234, 17788}, {239, 20432, 17789}, {244, 1978, 18149}, {244, 22167, 21330}, {256, 24478, 3764}, {257, 27447, 3863}, {264, 307, 18749}, {273, 307, 40702}, {273, 318, 264}, {274, 314, 86}, {274, 1930, 33943}, {274, 4647, 17762}, {274, 17143, 1}, {274, 17762, 41875}, {274, 32104, 17144}, {274, 33935, 304}, {280, 347, 6527}, {281, 27509, 37774}, {291, 20433, 20446}, {292, 20630, 20644}, {304, 4008, 1966}, {304, 17143, 4673}, {304, 31997, 41875}, {304, 39731, 1}, {306, 5249, 18134}, {307, 318, 33672}, {310, 31330, 17149}, {310, 40087, 561}, {312, 321, 42034}, {312, 18743, 20942}, {312, 19792, 19814}, {312, 19798, 19812}, {312, 19804, 2}, {312, 20923, 18137}, {312, 30090, 20923}, {312, 30758, 20947}, {312, 30829, 4358}, {312, 42029, 321}, {313, 1269, 76}, {313, 3264, 3596}, {313, 3663, 39995}, {313, 4357, 18133}, {313, 5224, 6376}, {314, 3875, 17144}, {314, 18697, 17762}, {314, 35550, 20932}, {319, 320, 69}, {319, 1441, 17791}, {319, 5564, 8}, {319, 7321, 320}, {319, 20930, 16284}, {320, 3262, 17791}, {320, 5564, 319}, {320, 7321, 7}, {321, 3263, 33931}, {321, 3739, 18137}, {321, 4358, 4671}, {321, 4359, 2}, {321, 4699, 20923}, {321, 4772, 30090}, {321, 4980, 28605}, {321, 16706, 40001}, {321, 19788, 19814}, {321, 19804, 18743}, {321, 19805, 19803}, {321, 19822, 19799}, {321, 20336, 42714}, {321, 20892, 20891}, {321, 20905, 1229}, {321, 21264, 20453}, {321, 24589, 4358}, {321, 25001, 346}, {321, 26234, 350}, {321, 26665, 17787}, {321, 27605, 25480}, {321, 27705, 27569}, {321, 28605, 42029}, {321, 32779, 42709}, {322, 20930, 17791}, {322, 39126, 320}, {326, 17858, 33808}, {326, 18695, 304}, {330, 20899, 20936}, {333, 32939, 63}, {335, 40848, 3862}, {341, 3673, 33780}, {344, 346, 17264}, {344, 20946, 18743}, {344, 37788, 20946}, {346, 1229, 312}, {346, 20905, 20946}, {349, 6734, 18738}, {350, 3263, 20947}, {350, 4441, 4479}, {350, 30758, 18743}, {350, 33931, 312}, {354, 3706, 10453}, {365, 20631, 20645}, {366, 20434, 20447}, {391, 4454, 144}, {561, 17149, 20945}, {594, 1086, 141}, {594, 3662, 17228}, {594, 3963, 4033}, {594, 7263, 3662}, {594, 20913, 18040}, {599, 4445, 17287}, {599, 7232, 17288}, {647, 21437, 21610}, {649, 20909, 20952}, {650, 21438, 21611}, {656, 17893, 20948}, {659, 21439, 21612}, {662, 18042, 48}, {662, 20902, 20941}, {667, 21440, 21613}, {668, 1111, 18159}, {669, 21441, 21614}, {672, 20632, 20646}, {673, 20431, 20445}, {693, 4411, 4828}, {798, 20910, 20953}, {799, 1109, 20939}, {850, 4467, 18155}, {894, 4361, 3759}, {894, 17116, 4363}, {894, 17117, 239}, {894, 17121, 17120}, {894, 20234, 17789}, {894, 21442, 17788}, {896, 20904, 20944}, {903, 17271, 17274}, {903, 20900, 20937}, {903, 32025, 17273}, {942, 5295, 10449}, {966, 4419, 17257}, {966, 17257, 17256}, {982, 1920, 6384}, {984, 1921, 6376}, {984, 21443, 21615}, {1001, 5695, 3685}, {1086, 3661, 17227}, {1086, 3963, 18143}, {1086, 4033, 18150}, {1086, 4665, 3661}, {1089, 7264, 3760}, {1089, 28611, 1698}, {1100, 4670, 17379}, {1100, 4852, 4393}, {1109, 20903, 799}, {1111, 3761, 20925}, {1111, 4692, 3761}, {1111, 4986, 668}, {1119, 7046, 32000}, {1211, 3782, 27184}, {1213, 4364, 17248}, {1213, 17246, 4364}, {1229, 20905, 37788}, {1229, 25001, 344}, {1229, 28974, 17264}, {1266, 3264, 39995}, {1266, 3663, 4398}, {1266, 4357, 3663}, {1266, 4967, 4357}, {1267, 5391, 2}, {1267, 32792, 32803}, {1267, 32793, 32801}, {1267, 32794, 32792}, {1267, 32795, 32799}, {1267, 32796, 32795}, {1267, 32797, 32793}, {1267, 32798, 5391}, {1267, 32802, 32804}, {1269, 3264, 313}, {1269, 3596, 30596}, {1269, 4357, 39995}, {1278, 3739, 4664}, {1278, 4688, 4687}, {1278, 4699, 37}, {1278, 4704, 4788}, {1278, 4739, 4751}, {1278, 4740, 4686}, {1278, 4772, 2}, {1278, 4821, 4740}, {1278, 20892, 312}, {1278, 27268, 4718}, {1281, 7061, 5989}, {1423, 20633, 20647}, {1441, 20895, 3262}, {1447, 7081, 183}, {1502, 4443, 24732}, {1502, 4446, 19567}, {1574, 3934, 27091}, {1575, 20440, 20453}, {1575, 21264, 2}, {1575, 28358, 27633}, {1580, 1582, 560}, {1631, 7087, 33801}, {1654, 4440, 6646}, {1654, 4643, 17328}, {1654, 6646, 4643}, {1654, 17276, 17329}, {1698, 3760, 18140}, {1733, 1930, 3403}, {1740, 1964, 36289}, {1740, 16571, 2234}, {1760, 20883, 20915}, {1760, 20884, 21582}, {1760, 21582, 18750}, {1836, 3966, 4388}, {1909, 20911, 20955}, {1909, 33930, 85}, {1920, 4087, 4485}, {1926, 3116, 17149}, {1928, 2085, 33788}, {1930, 4360, 20932}, {1930, 4647, 33935}, {1930, 17143, 17762}, {1930, 17445, 18051}, {1930, 17858, 18695}, {1930, 17859, 326}, {1930, 24325, 18157}, {1930, 39731, 18156}, {1931, 20634, 20648}, {1953, 1959, 18041}, {1958, 20902, 21593}, {1959, 17868, 1953}, {1964, 2234, 1740}, {1964, 17445, 1}, {2053, 20635, 20649}, {2054, 20636, 20650}, {2112, 20638, 20652}, {2234, 17445, 1964}, {2238, 24330, 24514}, {2275, 16720, 25918}, {2292, 31339, 31359}, {2319, 20438, 20451}, {2321, 3912, 17233}, {2321, 17234, 17240}, {2321, 24199, 17234}, {2325, 6666, 25101}, {2345, 4000, 2}, {2345, 16706, 17371}, {2345, 37756, 17370}, {2397, 28748, 28978}, {2580, 2581, 16568}, {2895, 17483, 32859}, {2968, 6356, 41005}, {3008, 17353, 17352}, {3008, 17355, 17353}, {3009, 20637, 20651}, {3112, 20898, 20934}, {3112, 33764, 31}, {3212, 20436, 20449}, {3218, 20887, 20920}, {3219, 20886, 20919}, {3262, 42697, 85}, {3263, 4441, 312}, {3263, 26234, 2}, {3264, 3663, 18133}, {3264, 4389, 6376}, {3290, 30748, 2}, {3416, 5880, 4645}, {3419, 41004, 2893}, {3589, 4395, 17366}, {3589, 7227, 17369}, {3589, 17366, 17367}, {3589, 17369, 17368}, {3596, 3718, 341}, {3596, 4357, 6376}, {3596, 4389, 18133}, {3596, 4398, 39995}, {3598, 7172, 15589}, {3644, 4664, 192}, {3644, 4687, 4664}, {3644, 4751, 37}, {3661, 3662, 141}, {3661, 3963, 17786}, {3661, 20913, 20917}, {3662, 3963, 20917}, {3662, 18040, 18150}, {3663, 4357, 4389}, {3663, 4967, 5224}, {3663, 5224, 17249}, {3663, 17861, 3673}, {3663, 24209, 17861}, {3663, 33937, 3718}, {3664, 3879, 17378}, {3666, 31993, 2}, {3666, 42051, 3210}, {3670, 27801, 18148}, {3672, 17321, 17320}, {3672, 31130, 20336}, {3673, 4385, 76}, {3673, 33937, 341}, {3673, 33938, 20943}, {3679, 3761, 668}, {3679, 4692, 4737}, {3679, 4862, 17272}, {3679, 17270, 32025}, {3679, 17272, 17270}, {3679, 17274, 17271}, {3679, 20894, 20925}, {3685, 16823, 1001}, {3686, 4416, 17346}, {3696, 20880, 20448}, {3703, 3925, 29641}, {3705, 7179, 325}, {3720, 4365, 32915}, {3723, 28639, 29570}, {3728, 20889, 40088}, {3729, 4384, 9}, {3729, 4858, 20927}, {3729, 17277, 17336}, {3739, 4681, 4698}, {3739, 4686, 192}, {3739, 4688, 4699}, {3739, 4698, 31238}, {3739, 4718, 27268}, {3739, 4739, 4688}, {3739, 4740, 3644}, {3739, 4821, 4764}, {3739, 25384, 17303}, {3740, 3967, 27538}, {3741, 24165, 982}, {3757, 32932, 55}, {3758, 3759, 6}, {3761, 4986, 4737}, {3761, 33934, 18159}, {3763, 17290, 17291}, {3763, 17293, 17292}, {3773, 3836, 29674}, {3778, 18891, 24732}, {3797, 24357, 4664}, {3834, 17229, 17231}, {3834, 17231, 17232}, {3875, 10436, 1}, {3875, 17151, 17160}, {3875, 25590, 86}, {3912, 4431, 2321}, {3912, 17233, 17240}, {3912, 17234, 17241}, {3912, 24199, 142}, {3923, 16825, 238}, {3943, 17243, 17242}, {3943, 17245, 17243}, {3943, 34824, 17244}, {3946, 5750, 17023}, {3946, 17023, 17380}, {3969, 18139, 32858}, {3980, 4362, 171}, {4000, 17289, 17370}, {4007, 6173, 17296}, {4007, 17296, 17294}, {4007, 17298, 17295}, {4025, 17894, 35519}, {4033, 18040, 17786}, {4033, 18143, 18040}, {4043, 4751, 18743}, {4043, 18137, 312}, {4044, 24603, 30830}, {4058, 21255, 29594}, {4110, 17786, 4033}, {4110, 20917, 17786}, {4118, 18053, 18060}, {4195, 19851, 1104}, {4329, 21279, 4872}, {4357, 4389, 17249}, {4357, 4967, 10}, {4357, 5224, 17250}, {4357, 20888, 1269}, {4358, 4359, 24589}, {4358, 4671, 312}, {4358, 24589, 2}, {4358, 30044, 29982}, {4358, 30829, 18743}, {4359, 4980, 321}, {4359, 19785, 19798}, {4359, 19835, 19808}, {4359, 20440, 21264}, {4359, 20891, 3739}, {4359, 20892, 4699}, {4359, 27569, 25457}, {4359, 28605, 312}, {4359, 42029, 18743}, {4360, 10436, 17394}, {4360, 17160, 3875}, {4360, 25590, 41847}, {4361, 4363, 6}, {4361, 17116, 3758}, {4361, 17118, 894}, {4361, 17119, 17117}, {4361, 20432, 20444}, {4363, 17117, 3759}, {4363, 17118, 17116}, {4363, 17119, 239}, {4363, 21442, 20444}, {4364, 17246, 17247}, {4371, 4644, 5839}, {4371, 7222, 193}, {4377, 17235, 18144}, {4377, 17239, 30473}, {4384, 4659, 190}, {4385, 33944, 20943}, {4389, 4398, 3663}, {4389, 5224, 4357}, {4393, 17379, 1100}, {4395, 7227, 3589}, {4395, 17369, 17367}, {4398, 4967, 17250}, {4398, 5224, 4389}, {4399, 17362, 29617}, {4399, 17365, 17363}, {4402, 5749, 5222}, {4402, 7229, 3618}, {4405, 7231, 3629}, {4418, 32914, 31}, {4419, 17257, 17258}, {4422, 17337, 17338}, {4422, 17340, 17339}, {4431, 24199, 3912}, {4436, 16684, 8053}, {4440, 6646, 17276}, {4440, 17275, 17329}, {4440, 28634, 17328}, {4441, 31130, 33931}, {4443, 4446, 3778}, {4445, 7232, 599}, {4461, 20905, 312}, {4472, 17045, 17398}, {4472, 17395, 17397}, {4478, 7238, 3631}, {4479, 30963, 350}, {4494, 17304, 18044}, {4568, 17761, 18061}, {4643, 6646, 17329}, {4643, 17275, 1654}, {4643, 17276, 6646}, {4643, 28634, 17275}, {4644, 5839, 193}, {4647, 18698, 18697}, {4647, 32092, 304}, {4648, 17314, 17316}, {4648, 17316, 17317}, {4651, 17165, 3681}, {4657, 17301, 17302}, {4657, 17302, 17399}, {4657, 17303, 2}, {4664, 4687, 37}, {4664, 4751, 4687}, {4664, 4764, 3644}, {4665, 7263, 141}, {4665, 20913, 4033}, {4670, 4852, 1100}, {4671, 4704, 22016}, {4671, 24589, 30829}, {4672, 4974, 16468}, {4673, 17158, 17144}, {4675, 4851, 17300}, {4675, 6542, 17387}, {4675, 17299, 4851}, {4681, 4698, 37}, {4681, 4704, 4664}, {4681, 4718, 192}, {4681, 31238, 27268}, {4686, 4688, 37}, {4686, 4698, 4788}, {4686, 4699, 4664}, {4686, 4726, 4740}, {4686, 4739, 2}, {4686, 4772, 4687}, {4686, 20892, 4043}, {4686, 31238, 4718}, {4687, 4751, 2}, {4687, 4764, 192}, {4687, 18137, 18743}, {4688, 4718, 31238}, {4688, 4726, 192}, {4688, 4739, 4772}, {4688, 4740, 4664}, {4688, 4821, 3644}, {4690, 17344, 17343}, {4690, 17345, 17344}, {4692, 4714, 3679}, {4698, 4718, 4704}, {4698, 4788, 4664}, {4698, 27268, 4687}, {4698, 31238, 2}, {4699, 4704, 31238}, {4699, 4726, 3644}, {4699, 4740, 192}, {4699, 4772, 4688}, {4699, 4788, 4698}, {4699, 20891, 30090}, {4704, 4788, 192}, {4704, 27268, 37}, {4704, 30044, 30829}, {4704, 31238, 4687}, {4714, 20894, 33934}, {4718, 4788, 3644}, {4718, 27268, 4664}, {4718, 31238, 37}, {4726, 4739, 37}, {4726, 4772, 4664}, {4739, 4740, 4687}, {4739, 4821, 4664}, {4740, 4772, 37}, {4740, 4821, 4726}, {4741, 17343, 17344}, {4751, 4764, 4664}, {4772, 4821, 192}, {4772, 28605, 20891}, {4788, 27268, 4681}, {4812, 20891, 20171}, {4847, 6063, 20935}, {4851, 6542, 17386}, {4851, 17299, 6542}, {4851, 17300, 17387}, {4858, 20881, 190}, {4859, 17282, 27191}, {4859, 17284, 17282}, {4859, 17286, 17283}, {4862, 17270, 17273}, {4862, 17272, 17274}, {4862, 17885, 1111}, {4886, 33066, 5739}, {4911, 5015, 315}, {4967, 20888, 313}, {4968, 20880, 34284}, {4968, 20911, 1909}, {4969, 7277, 3629}, {4970, 43223, 17592}, {4972, 8024, 18057}, {4986, 20894, 3761}, {5081, 7282, 317}, {5222, 5749, 3618}, {5222, 7229, 5749}, {5263, 32922, 1}, {5278, 32933, 3219}, {5391, 32791, 32804}, {5391, 32793, 32791}, {5391, 32794, 32802}, {5391, 32795, 32796}, {5391, 32796, 32800}, {5391, 32797, 1267}, {5391, 32798, 32794}, {5391, 32801, 32803}, {5564, 7321, 69}, {5564, 42697, 17361}, {5739, 5905, 33066}, {5750, 17023, 17381}, {5839, 7222, 4644}, {6063, 7182, 1088}, {6173, 17294, 17297}, {6173, 17296, 17298}, {6358, 11679, 20928}, {6358, 20882, 14829}, {6376, 33938, 341}, {6377, 6378, 6375}, {6382, 6383, 6374}, {6382, 24165, 6384}, {6382, 42027, 39467}, {6383, 42027, 6384}, {6385, 40088, 561}, {6542, 17300, 4851}, {6542, 26806, 17300}, {6646, 17275, 17328}, {7227, 17366, 17368}, {7228, 17362, 17364}, {7283, 16817, 405}, {7321, 20895, 17791}, {7321, 42696, 17360}, {8053, 16684, 23407}, {8301, 8424, 23868}, {8301, 20441, 20454}, {8621, 21751, 21776}, {9278, 20439, 20452}, {9776, 34255, 18141}, {10436, 17151, 4360}, {10436, 17160, 17393}, {10436, 18697, 304}, {10447, 17151, 17143}, {10447, 18698, 304}, {10447, 32092, 10436}, {10453, 30962, 41851}, {13476, 40216, 40004}, {14213, 20879, 63}, {14213, 21406, 20883}, {15668, 16777, 16826}, {15668, 17318, 16777}, {16062, 40071, 6376}, {16512, 16513, 63}, {16568, 20884, 20916}, {16571, 17445, 36289}, {16610, 30818, 2}, {16703, 16748, 16708}, {16706, 17289, 2}, {16706, 37756, 4000}, {16706, 40875, 29423}, {16709, 20932, 41875}, {16709, 30939, 86}, {16710, 17178, 16726}, {16748, 17163, 18059}, {16777, 17318, 17319}, {16815, 17260, 17259}, {16815, 17261, 17260}, {16816, 17349, 17348}, {16816, 17350, 17349}, {16819, 25264, 5283}, {16826, 17319, 16777}, {17026, 17754, 37686}, {17045, 17395, 17396}, {17045, 17398, 17397}, {17116, 17117, 6}, {17116, 17119, 3759}, {17116, 21442, 17789}, {17117, 17118, 3758}, {17117, 20432, 17788}, {17118, 17119, 6}, {17120, 17121, 6}, {17135, 17140, 3873}, {17135, 17146, 17145}, {17136, 18654, 17221}, {17137, 17141, 3868}, {17140, 17145, 17146}, {17140, 17163, 17135}, {17140, 40216, 16708}, {17143, 32092, 31997}, {17143, 33933, 304}, {17143, 39731, 17158}, {17144, 17762, 4673}, {17144, 31997, 1}, {17144, 33943, 18156}, {17145, 17146, 3873}, {17151, 25590, 1}, {17155, 31330, 38}, {17155, 40087, 17149}, {17160, 25590, 17394}, {17220, 20245, 17139}, {17227, 17228, 141}, {17227, 20917, 18150}, {17229, 17231, 17230}, {17230, 17232, 17231}, {17233, 17234, 3912}, {17235, 17237, 17236}, {17235, 17239, 17237}, {17236, 17238, 17237}, {17236, 29593, 17238}, {17237, 17239, 17238}, {17238, 29593, 17239}, {17240, 17241, 3912}, {17242, 17244, 17243}, {17242, 27147, 17244}, {17243, 17245, 17244}, {17243, 34824, 17245}, {17244, 27147, 17245}, {17245, 34824, 27147}, {17247, 17248, 4364}, {17247, 29576, 17248}, {17248, 29576, 1213}, {17249, 17250, 4357}, {17250, 30596, 18133}, {17251, 17253, 17252}, {17251, 17255, 17253}, {17252, 17254, 17253}, {17253, 17255, 17254}, {17256, 17258, 17257}, {17259, 17262, 45}, {17260, 17261, 45}, {17263, 17264, 344}, {17265, 17267, 17266}, {17265, 17269, 17267}, {17266, 17268, 17267}, {17267, 17269, 17268}, {17270, 17272, 17271}, {17270, 17274, 17272}, {17271, 17273, 17272}, {17271, 32025, 17270}, {17272, 17274, 17273}, {17273, 32025, 17271}, {17275, 17276, 4643}, {17276, 28634, 1654}, {17277, 20236, 18151}, {17277, 20927, 30854}, {17278, 17279, 2}, {17278, 17280, 17341}, {17278, 17281, 17279}, {17279, 17280, 17342}, {17279, 17281, 17280}, {17282, 17284, 17283}, {17282, 17286, 17284}, {17283, 17285, 17284}, {17283, 27191, 17282}, {17284, 17286, 17285}, {17285, 27191, 17283}, {17287, 17288, 599}, {17287, 29615, 4445}, {17288, 29615, 17287}, {17289, 17787, 29423}, {17289, 37756, 16706}, {17289, 40875, 29705}, {17290, 17293, 3763}, {17291, 17292, 3763}, {17294, 17296, 17295}, {17294, 17298, 17296}, {17295, 17297, 17296}, {17296, 17298, 17297}, {17299, 17300, 17386}, {17299, 26806, 17387}, {17300, 26806, 4675}, {17301, 17303, 4657}, {17301, 28604, 17400}, {17302, 17303, 17400}, {17302, 17790, 18046}, {17302, 25384, 4687}, {17302, 28604, 2}, {17304, 17306, 17305}, {17304, 17308, 17306}, {17305, 17307, 17306}, {17306, 17308, 17307}, {17309, 17311, 17310}, {17309, 17313, 17311}, {17310, 17312, 17311}, {17311, 17313, 17312}, {17314, 17316, 17315}, {17315, 17317, 17316}, {17320, 17322, 17321}, {17320, 28653, 17322}, {17322, 18147, 30963}, {17322, 28653, 2}, {17323, 17325, 17324}, {17323, 17327, 17325}, {17324, 17326, 17325}, {17324, 29610, 17326}, {17325, 17327, 17326}, {17326, 29610, 17327}, {17328, 17329, 4643}, {17330, 17332, 17331}, {17330, 17334, 17332}, {17331, 17333, 17332}, {17332, 17334, 17333}, {17335, 17336, 9}, {17335, 18151, 30854}, {17337, 17340, 4422}, {17338, 17339, 4422}, {17338, 29628, 17337}, {17339, 29628, 17338}, {17341, 17342, 17279}, {17344, 17345, 4741}, {17346, 17347, 4416}, {17348, 17351, 44}, {17349, 17350, 44}, {17352, 17354, 17353}, {17353, 17355, 17354}, {17356, 17357, 2}, {17356, 17359, 17357}, {17357, 17359, 17358}, {17360, 17361, 69}, {17360, 17791, 16284}, {17363, 29617, 17362}, {17364, 29617, 17363}, {17366, 17369, 3589}, {17367, 17368, 3589}, {17370, 17371, 2}, {17371, 40001, 18743}, {17372, 17374, 17373}, {17372, 17376, 17374}, {17373, 17375, 17374}, {17373, 20055, 17372}, {17374, 17376, 17375}, {17375, 20055, 17373}, {17377, 17378, 3879}, {17380, 17381, 17023}, {17382, 17384, 17383}, {17382, 17385, 17384}, {17384, 17385, 2}, {17386, 17387, 4851}, {17388, 17390, 17389}, {17388, 17392, 17390}, {17389, 17391, 17390}, {17390, 17392, 17391}, {17393, 17394, 1}, {17393, 41847, 17394}, {17394, 20932, 18156}, {17394, 41847, 86}, {17395, 17398, 17045}, {17396, 17397, 17045}, {17399, 17400, 4657}, {17446, 40364, 17149}, {17495, 31025, 2}, {17595, 37660, 24627}, {17731, 20437, 20450}, {17760, 30038, 3061}, {17762, 31997, 18156}, {17762, 33943, 304}, {17786, 20917, 18040}, {17787, 37756, 29484}, {17788, 17789, 20444}, {17790, 28604, 29388}, {17858, 17859, 1}, {17858, 18699, 18691}, {17859, 18699, 17858}, {17861, 33937, 3596}, {17863, 20336, 18147}, {17863, 24547, 17321}, {17865, 17882, 1958}, {17865, 18042, 20941}, {17879, 17883, 811}, {17880, 18689, 664}, {17882, 20902, 662}, {17889, 32778, 2887}, {18040, 18143, 20917}, {18133, 30596, 20943}, {18147, 42714, 312}, {18697, 18698, 35550}, {18743, 42034, 312}, {19785, 19788, 19790}, {19785, 19789, 19796}, {19785, 19792, 19821}, {19785, 19796, 19830}, {19785, 19797, 19827}, {19785, 19808, 19812}, {19785, 19810, 19803}, {19785, 19811, 19806}, {19785, 19819, 19789}, {19785, 19820, 19831}, {19785, 19822, 2}, {19785, 19823, 19829}, {19785, 19824, 19828}, {19785, 19825, 19808}, {19785, 19826, 19820}, {19785, 19835, 312}, {19786, 19787, 19821}, {19786, 19789, 19830}, {19786, 19792, 19803}, {19786, 19796, 19785}, {19786, 19797, 19808}, {19786, 19799, 18743}, {19786, 19807, 19806}, {19786, 19808, 2}, {19786, 19819, 19831}, {19786, 19820, 19796}, {19786, 19822, 19827}, {19786, 19828, 19829}, {19786, 19829, 19823}, {19786, 19833, 19822}, {19787, 19798, 19790}, {19787, 19799, 19814}, {19787, 19801, 19818}, {19787, 19809, 19807}, {19787, 19810, 19792}, {19787, 19811, 19810}, {19787, 19816, 19800}, {19787, 19822, 19806}, {19788, 19799, 19806}, {19789, 19796, 19831}, {19789, 19797, 19812}, {19789, 19807, 19821}, {19789, 19809, 19806}, {19789, 19811, 19803}, {19789, 19819, 19820}, {19789, 19822, 19786}, {19789, 19823, 19824}, {19789, 19825, 2}, {19789, 19826, 19819}, {19789, 19833, 19827}, {19790, 19814, 19803}, {19791, 19800, 19803}, {19792, 19804, 19802}, {19792, 19807, 19810}, {19792, 19808, 19806}, {19793, 19822, 33116}, {19794, 19795, 2}, {19796, 19797, 2}, {19796, 19807, 19803}, {19796, 19808, 19786}, {19796, 19810, 19821}, {19796, 19820, 19789}, {19796, 19822, 19812}, {19796, 19825, 19827}, {19796, 19828, 19824}, {19796, 19829, 19828}, {19796, 19832, 19829}, {19796, 19833, 19808}, {19797, 19808, 19822}, {19797, 19820, 19786}, {19797, 19826, 19830}, {19797, 19833, 19825}, {19798, 19799, 2}, {19798, 19810, 19802}, {19799, 19804, 19827}, {19799, 19805, 19802}, {19801, 19818, 19803}, {19804, 20923, 4751}, {19804, 28605, 42034}, {19804, 30090, 3739}, {19804, 42029, 312}, {19805, 19810, 19788}, {19805, 19835, 19806}, {19808, 19819, 19830}, {19808, 19820, 19785}, {19808, 19826, 19831}, {19808, 19829, 19832}, {19808, 19833, 19797}, {19809, 19810, 19811}, {19809, 19819, 19803}, {19810, 19811, 19807}, {19811, 19820, 19821}, {19812, 19827, 2}, {19812, 19831, 19785}, {19818, 19823, 19817}, {19819, 19822, 19785}, {19819, 19825, 19786}, {19819, 19833, 19812}, {19820, 19825, 19812}, {19820, 19832, 19824}, {19820, 19833, 2}, {19820, 19837, 19829}, {19822, 19824, 19832}, {19822, 19825, 19797}, {19822, 19826, 19796}, {19823, 19824, 19785}, {19824, 19828, 19830}, {19824, 19837, 19812}, {19825, 19826, 19785}, {19825, 28605, 19835}, {19827, 19830, 19786}, {19828, 19829, 19785}, {19828, 19832, 19823}, {19828, 19837, 19832}, {19829, 19832, 19786}, {19829, 19833, 19837}, {19829, 19837, 2}, {19830, 19831, 19796}, {19841, 19842, 3}, {19844, 19845, 3}, {19950, 19951, 19954}, {19950, 19953, 19952}, {19950, 19957, 19966}, {19950, 19962, 19968}, {19950, 19963, 10}, {19950, 19969, 19970}, {19950, 19972, 19967}, {19950, 19996, 19962}, {19950, 20000, 20001}, {19950, 20001, 19969}, {19950, 20003, 19996}, {19950, 20006, 19957}, {19951, 19954, 19970}, {19951, 19966, 19957}, {19951, 19968, 19950}, {19951, 20001, 10}, {19952, 19961, 19953}, {19954, 19969, 19951}, {19955, 19973, 19951}, {19957, 19958, 19950}, {19957, 19962, 19958}, {19957, 19966, 19954}, {19957, 20000, 20010}, {19957, 20006, 19951}, {19957, 20010, 10}, {19958, 20006, 19966}, {19958, 20010, 20006}, {19959, 19960, 19950}, {19962, 19963, 19951}, {19962, 19999, 19963}, {19962, 20000, 10}, {19962, 20001, 19954}, {19962, 20004, 19996}, {19962, 20010, 19957}, {19963, 19968, 19969}, {19963, 19996, 19950}, {19963, 20003, 19962}, {19963, 20004, 19968}, {19967, 19971, 19972}, {19968, 20001, 19951}, {19971, 19972, 19951}, {19996, 19999, 20001}, {19996, 20000, 19951}, {19996, 20003, 20004}, {19999, 20003, 19951}, {19999, 20004, 10}, {20000, 20003, 19968}, {20000, 20004, 19950}, {20236, 20881, 3729}, {20237, 20882, 11679}, {20244, 20245, 17220}, {20245, 21273, 3869}, {20292, 33075, 6327}, {20336, 26234, 17322}, {20367, 21061, 16574}, {20432, 21442, 20234}, {20435, 24349, 85}, {20627, 33760, 21598}, {20627, 33764, 20934}, {20706, 30028, 18055}, {20880, 20895, 1441}, {20880, 20911, 33930}, {20880, 32087, 322}, {20880, 42696, 20930}, {20883, 21406, 20884}, {20888, 33937, 4385}, {20888, 33945, 3673}, {20889, 21020, 40087}, {20895, 31995, 85}, {20895, 39126, 16284}, {20895, 42697, 20930}, {20898, 23665, 21424}, {20905, 25001, 2}, {20905, 28974, 17263}, {20911, 34284, 85}, {20915, 21582, 20916}, {20947, 30963, 18743}, {20955, 24524, 16284}, {21085, 33064, 33084}, {21296, 32099, 69}, {21776, 38301, 8621}, {22016, 24589, 4698}, {22016, 29982, 4358}, {23482, 23483, 23492}, {23502, 23507, 23481}, {24004, 29396, 29712}, {24004, 29446, 29396}, {24166, 24170, 24046}, {24463, 24478, 256}, {24547, 24993, 2}, {24589, 29982, 31238}, {24620, 30818, 31233}, {24715, 33076, 4660}, {24789, 32777, 2}, {24919, 24958, 2}, {25001, 37788, 17263}, {25242, 27304, 1212}, {25382, 27846, 24487}, {25457, 25660, 2}, {25590, 32104, 314}, {25753, 25811, 2}, {25757, 27918, 30963}, {25780, 25850, 2}, {25887, 25971, 2}, {25939, 26011, 2}, {26037, 32925, 756}, {26038, 27538, 3740}, {26163, 26213, 2}, {26234, 31130, 30758}, {26234, 33931, 30963}, {26538, 26665, 2}, {26587, 26632, 2}, {26591, 28930, 37758}, {26724, 33157, 2}, {26764, 26812, 2}, {26971, 27102, 2}, {27032, 27154, 2}, {27107, 27145, 27017}, {27136, 27192, 2}, {27186, 32858, 18139}, {27261, 27311, 2}, {27268, 29982, 30829}, {27474, 27478, 27475}, {27479, 27488, 27471}, {27480, 31347, 31306}, {27481, 27483, 31322}, {27481, 29576, 31323}, {27483, 31323, 29576}, {27487, 36494, 27475}, {27488, 27492, 27489}, {27494, 31329, 27495}, {27569, 27705, 42713}, {28653, 42714, 20947}, {28974, 37788, 344}, {29576, 31323, 31322}, {29667, 33131, 4972}, {30998, 41836, 2}, {31019, 33077, 3936}, {31993, 42051, 3666}, {31995, 32087, 69}, {31995, 42696, 320}, {32087, 42697, 319}, {32092, 32104, 1}, {32092, 33935, 33943}, {32104, 33933, 17762}, {32771, 32860, 42}, {32772, 32924, 17017}, {32779, 33129, 2}, {32780, 33132, 25453}, {32782, 33146, 17184}, {32783, 33147, 26128}, {32784, 33149, 3821}, {32791, 32792, 2}, {32791, 32801, 1267}, {32791, 32802, 32792}, {32792, 32801, 32791}, {32792, 32802, 5391}, {32793, 32794, 2}, {32793, 32798, 32792}, {32793, 32800, 32799}, {32793, 32802, 32803}, {32794, 32797, 32791}, {32794, 32799, 32800}, {32794, 32801, 32804}, {32795, 32796, 2}, {32795, 32799, 32791}, {32796, 32800, 32792}, {32797, 32798, 2}, {32799, 32800, 2}, {32801, 32802, 2}, {32803, 32804, 2}, {32842, 33112, 33070}, {32845, 32917, 4414}, {32855, 33111, 29671}, {32857, 33082, 4655}, {32861, 33097, 32946}, {32864, 32940, 32912}, {32865, 33169, 29673}, {32866, 33109, 4865}, {32923, 32945, 3938}, {33083, 33102, 32950}, {33084, 33103, 33064}, {33089, 33108, 3006}, {33090, 33110, 5014}, {33130, 33160, 3771}, {33138, 33167, 4438}, {33139, 33170, 33114}, {33148, 33175, 33122}, {33672, 40702, 18749}, {33772, 38813, 560}, {33933, 33935, 1930}, {33935, 39731, 4673}, {33937, 33940, 6376}, {33937, 33945, 10}, {33938, 33944, 6376}, {33940, 33941, 76}, {33941, 33945, 6376}, {34021, 40874, 34022}, {34387, 34388, 311}, {34884, 39556, 3}, {36928, 36929, 5252}, {38810, 38840, 38813}, {38813, 39671, 38810}, {42696, 42697, 69}


X(76) = 3rd BROCARD POINT

Trilinears    1/a3 : 1/b3 : 1/c3
Trilinears    csc(A - ω) : csc(B - ω) : csc(C - ω)
Barycentrics    1/a2 : 1/b2 : 1/c2
X(76) = 3*X(2) - 2*X(39) = 3*X(2) - P(1) - U(1)

Let A' be the perspector of the A-McCay circle, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(76). (Randy Hutson, April 9, 2016)

X(76) is the vertex conjugate of the foci of the inellipse that is the barycentric square of the de Longchamps line. The center of the inellipse is X(626) and its Brianchon point (perspector) is X(1502). (Randy Hutson, October 15, 2018)

Let A'B'C' be the obverse triangle of X(1). Let A″B″C″ be the N-obverse triangle of X(1). Let A* be the barycentric product A'*A″, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(76). (Randy Hutson, October 15, 2018)

X(76) is the barycentric multiplier for the Steiner circumellipse. (The barycentric product of X(76) and the circumcircle is the Steiner circumellipse.) (Randy Hutson, August 19, 2019)

For an artistic design generated by X(76), see X(244).

X(76) lies on these lines: 1,350   2,39   3,98   4,69   5,262   6,83   7,1240   8,668   10,75   13,299   14,298   17,303   18,302   20,3424   22,1799   25,1241   31,734   32,384   37,1218   85,226   95,96   100,767   107,2366   110,2367   115,626   141,698   148,2896   182,3406   187,3552   192,1221   251,1239   257,1926   275,276   297,343   321,561   330,1015   331,1231   333,1751   334,1089   335,871   338,599  485,491   486,492   524,598   620,1569   689,755   691,2868   693,764   761,789   799,1150   826,882   940,1509   1003,3053   1007,3090   1131,1271   1132,1270   1229,1446   1423,3403   1501,3115   1670,1677   1671,1676   1698,3097   2001,2909   2319,3500   2394,3267   3224,3225   3492,3506   3496,3512   3497,3509

X(76) is the {X(2),X(194)}-harmonic conjugate of X(39). For a list of other harmonic conjugates of X(76), click Tables at the top of this page.

X(76) = reflection of X(194) in X(39)
X(76) = isogonal conjugate of X(32)
X(76) = isotomic conjugate of X(6)
X(76) = complement of X(194)
X(76) = anticomplement of X(39)
X(76) = circumcircle-inverse of X(5152)
X(76) = 2nd-Brocard circle-inverse of X(99)
X(76) = anticomplementary conjugate of X(2896)
X(76) = cyclocevian conjugate of isogonal conjugate of X(2916)
X(76) = cyclocevian conjugate of isotomic conjugate of X(1369)
X(76) = X(i)-Ceva conjugate of X(j) for these (i,j): (308,2), (310,75)
X(76) = cevapoint of X(i) and X(j) for these (i,j): (2,69), (6,22), (75,312), (311,343), (313,321), (339,525)
X(76) = X(i)-cross conjugate of X(j) for these (i,j): (2,264), (69,305), (141,2), (321,75), (343,69), (525,99)
X(76) = crosssum of X(669) and X(1084)
X(76) = crossdifference of every pair of points on line X(669)X(688)
X(76) = X(i)-beth conjugate of X(j) for these (i,j): (76,85), (799,348)
X(76) = pole wrt polar circle of trilinear polar of X(25) (line X(512)X(1692))
X(76) = X(48)-isoconjugate (polar conjugate) of X(25)
X(76) = X(6)-isoconjugate of X(31)
X(76) = trilinear product of PU(i) for these i: 10, 86
X(76) = barycentric product of PU(11)
X(76) = antigonal image of X(1916)
X(76) = cevapoint of polar conjugates of PU(4)
X(76) = trilinear product of vertices of 1st Brocard triangle
X(76) = trilinear product of vertices of 1st anti-Brocard triangle
X(76) = X(2)-Ceva conjugate of X(6374)
X(76) = X(384)-of-5th-Brocard-triangle
X(76) = X(6)-of-6th-Brocard-triangle
X(76) = perspector of ABC and 1st Brocard triangle
X(76) = trilinear pole of de Longchamps line
X(76) = bicentric sum of PU(159)
X(76) = PU(159)-harmonic conjugate of X(9494)
X(76) = perspector of conic {A,B,C,X(670),X(689),X(1978)}} (isotomic conjugate of Lemoine axis.)
X(76) = X(1916) of 1st Brocard triangle
X(76) = crosspoint of X(6) and X(22) wrt both the anticomplementary and tangential triangles
X(76) = X(3094)-of-1st anti-Brocard-triangle
X(76) = trilinear product of vertices of mid-triangle of 1st Brocard and 1st anti-Brocard triangles
X(76) = perspector of ABC and cross-triangle of ABC and 3rd Brocard triangle
X(76) = trilinear product of vertices of the three anti-altimedial triangles
X(76) = Cundy-Parry Phi transform of X(98)
X(76) = Cundy-Parry Psi transform of X(511)
X(76) = barycentric product X(99)*X(850)
X(76) = intersection, other than X(4), of P(1)- and U(1)-Fuhrmann circles (aka -Hagge circles)
X(76) = {X(7737),X(14023)}-harmonic conjugate of X(20065)
X(76) = intersection of lines PU(1) of 1st and 2nd Ehrmann circumscribing triangles
X(76) = trilinear cube of X(2)
X(76) = barycentric square of X(75)
X(76) = trilinear product of vertices of Gemini triangle 19


X(77) = ISOGONAL CONJUGATE OF X(33)

Trilinears     1/(1 + sec A) : 1/(1 + sec B) : 1/(1 + sec C)
Trilinears     cos A sec2(A/2) : cos B sec2(B/2) : cos C sec2(C/2)
Trilinears     (b2 + c2 - a2)/(b + c - a) : :
Trilinears     SA(SA - bc) : :
Barycentrics  a/(1 + sec A) : b/(1 + sec B) : c/(1 + sec C)
Barycentrics    cot A (1 - cos A) : :

In the plane of a triangle ABC, let
Ia = line through X(7) parallel to BC, and define Ib and Ic cyclically.
Ac = Ia∩AB, and define Ba and Cb cyclically.
Ab = Ia∩AC, and define Bc and Ca cyclically.
Oa = circumcircle of A, Bc, Cb, and define Ob and Oc cyclically.
Then X(77) = radical center of Oa, Ob, Oc. See also X(77). (Ivan Pavlov, April 1, 2022)

X(77) lies on these lines: 1,7   2,189   6,241   9,651   29,34   40,947   55,1037   56,1036   57,81   63,219   65,969   69,73   75,664   102,934   283,603   309,318   738,951   988,1106   999,1057

X(77) = isogonal conjugate of X(33)
X(77) = isotomic conjugate of X(318)
X(77) = X(i)-Ceva conjugate of X(j) for these (i,j): (85,57), (86,7), (348,63)
X(77) = cevapoint of X(i) and X(j) for these (i,j): (1,223), (3,222)
X(77) = X(i)-cross conjugate of X(j) for these (i,j): (3,63), (73,222)
X(77) = trilinear pole of line X(652)X(905)
X(77) = {X(175),X(176)}-harmonic conjugate of X(962)
X(77) = X(92)-isoconjugate of X(41)
X(77) = perspector of ABC and extraversion triangle of X(78)
X(77) = X(i)-beth conjugate of X(j) for these (i,j): (21,990), (69,69), (86,269), (99,75), (332,326), (336,77), (662,77), (664,77), (811,77)


X(78) = ISOGONAL CONJUGATE OF X(34)

Trilinears    1/(1 - sec A) : 1/(1 - sec B) : 1/(1 - sec C)
Trilinears    cos A csc2(A/2) : :
Trilinears    (b + c - a)(b2 + c2 - a2) : :
Trilinears    SA(SA + bc) : :
Trilinears    (b + c - a) cot A : :
Trilinears    cot A cot(A/2) : :
Barycentrics  a/(1 - sec A) : b/(1 - sec B) : c/(1 - sec C)

If you have The Geometer's Sketchpad, you can view X(78).


If you have GeoGebra, you can view X(78).

X(78) lies on these lines: 1,2   3,63   4,908   9,21   20,329   29,33   37,965   38,988   40,100   46,758   55,960   56,480   57,404   69,73   101,205   207,653   210,958   212,283   220,949   226,377   271,394   273,322   280,282   345,1040   392,1057   474,942   517,945   644,728   999,1059

X(78) = isogonal conjugate of X(34)
X(78) = isotomic conjugate of X(273)
X(78) = X(i)-Ceva conjugate of X(j) for these (i,j): (69,63), (312,9), (332,345)
X(78) = X(i)-cross conjugate of X(j) for these (i,j): (3,271), (72,8), (212,9), (219,63)
X(78) = crosspoint of X(69) and X(345)
X(78) = crosssum of X(i) and X(j) for these (i,j): (25,608), (56,1406), (604,1395), (1042,1426)
X(78) = X(i)-beth conjugate of X(j) for these (i,j): (78,3), (643,40), (1043,1)
X(78) = trilinear pole of line X(521)X(652)
X(78) = {X(1),X(8)}-harmonic conjugate of X(3872)
X(78) = {X(2),X(145)}-harmonic conjugate of X(938)
X(78) = X(92)-isoconjugate of X(604)
X(78) = homothetic center of anticomplementary triangle and tangential triangle of the hexyl triangle
X(78) = perspector of ABC and extraversion triangle of X(77)


X(79) = ISOGONAL CONJUGATE OF X(35)

Trilinears    1/(1 + 2 cos A) : :
Trilinears    bc/(b2 + c2 - a2 + bc)
Trilinears    (sin A/2)(sin 3B/2)(sin 3C/2) : :
Trilinears    sin(A/2) csc(3A/2) : :
Barycentrics    1/(b2 + c2 - a2 + bc) : :
Barycentrics    1/(b c + 2 SA) : :

X(79) = (2r + 3R)*X(1) + 6r*X(2) - 6r*X(3)    (Peter Moses, April 2, 2013)

Let A' be the reflection of X(1) in sideline BC, and define B' and C' cyclically. Then the lines AA', BB', CC' concur in X(79). (Eric Danneels, Hyacinthos 7892, 9/13/03)

A'B'C' is also the reflection triangle of X(1). The lines AA', BB', CC' concur in X(79). (Randy Hutson, July 20, 2016)

Let P and Q be the intersections of line BC and circle {X(1),2r}. Let X = X(1). Let A' be the circumcenter of triangle PQX, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(79). (Compare to X(592), where the circle is the 1st Lemoine circle) (Randy Hutson, July 20, 2016)

Let A25B25C25 be Gemini triangle 25. Let A' be the perspector of conic {A,B,C,B25,C25}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(79). (Randy Hutson, January 15, 2019)

A construction for X(79) appears in Dasari Naga Vijay Krishna, On the Feuerbach Triangle.

In the plane of a triangle ABC, let
D = the point on line AC such that angle CAB = angle DAB and |AD| = |AB|
E = the point on line AB such that angle CAB = angle CAE and |AE| = |AC|
F = the point on line BC such that angle CBA = angle FBA and |BF| = |AB|
G = the point on line AB such that angle CBA = angle CBG and |BG| = |BC|
H = the point on line BC such that angle ACB = angle ACH and |CH| = |AC|
I = the point on line AC such that angle ACB = angle ICB and |CI| = |BC|
S = the nine point center of AFH
T = the nine point center of BDI
U = the nine point center of CEG
ABC and STU are homothetic about X(79) (labled V on the figure). See figure. (Benjamin Warren, October 24, 2024)

X(79) lies on these lines: 1,30   2,3647   8,758   9,46   12,484   21,36   33,1063   34,1061   35,226   57,90   65,80   104,946   314,320   388,1000

X(79) = reflection of X(191) in X(442)
X(79) = isogonal conjugate of X(35)
X(79) = isotomic conjugate of X(319)
X(79) = cevapoint of X(481) and X(482)
X(79) = crosssum of X(55) and X(1030)
X(79) = anticomplement of X(3647)
X(79) = X(2914) of Fuhrmann triangle
X(79) = antigonal image of X(3065)
X(79) = trilinear pole of line X(650)X(4802)
X(79) = perspector of ABC and extraversion triangle of X(80)
X(79) = Hofstadter -1/2 point
X(79) = trilinear pole of line X(650)X(4802)
X(79) = trilinear product of vertices of reflection triangle of X(1)
X(79) = X(6152)-of-excentral-triangle


X(80)  REFLECTION OF INCENTER IN FEUERBACH POINT

Trilinears    1/(1 - 2 cos A) : 1/(1 - 2 cos B) : 1/(1 - 2 cos C)
Trilinears    bc/(b2 + c2 - a2 - bc) : :
Trilinears    cos(A/2) sec(3A/2) : :
Barycentrics    1/(b2 + c2 - a2 - bc) : :
Barycentrics = 1/(bc - 2 SA) : :
X(80) = (2r + R)*X(1)- 6r*X(2) + 2r*X(3)    (Peter Moses, April 2, 2013)

Let A' be the reflection in BC of the A-vertex of the excentral triangle, and define B' and C' cyclically. The circumcircles of A'BC, B'CA, and C'AB concur in X(80). Also, the lines AA', BB', CC' concur in X(80). (Randy Hutson, December 10, 2016)

Let A'B'C' be the Fuhrmann triangle. Let La be the line through A parallel to B'C', and define Lb and Lc cyclically. Let A″ = Lb∩Lc, B″ = Lc∩La, C″ = La∩Lb. The lines A'A″, B'B″, C'C″ concur in X(80). (Randy Hutson, December 10, 2016)

Let A'B'C' be the orthic triangle. Let La be the antiorthic axis of AB'C', and define Lb and Lc cyclically. Let A″ = Lb∩Lc. B″ = Lc∩La, C″ = La∩Lb. The triangle A″B″C″ is inversely similar to ABC, with similitude center X(9). The incenter of triangle A″B″C″ is X(80). Also, the lines AA″, BB″, CC″ concur in X(80).(Randy Hutson, December 10, 2016)

Let A'B'C' be the excentral triangle. Let A″ be the isogonal conjugate, wrt A'BC, of A. Define B″, C″ cyclically. (A″ is also the reflection of A' in BC, and cyclically for B″ and C″). The lines AA″, BB″, CC″ concur in X(80). (Randy Hutson, January 29, 2018)

Let A'B'C' be the excentral triangle. Let Oa be the A'-Johnson circle of triangle A'BC, and define Ob and Oc cyclically. X(80) is the radical center of Oa, Ob, Oc. (Randy Hutson, June 27, 2018)

X(80) lies on these lines: 1,5   2,214   7,150   8,149   9,528   10,21   30,484   33,1061   34,1063   36,104   40,90   46,84   65,79   313,314   497,1000   499,944   516,655   519,908   943,950

X(80) = midpoint of X(8) and X(149)
X(80) = reflection of X(i) in X(j) for these (i,j): (1,11), (100,10), (1317,1387)
X(80) = isogonal conjugate of X(36)
X(80) = isotomic conjugate of X(320)
X(80) = circumcircle-inverse of X(10260)
X(80) = incircle-inverse of X(1387)
X(80) = Fuhrmann-circle-inverse of X(1)
X(80) = complement of X(6224)
X(80) = anticomplement of X(214)
X(80) = cevapoint of X(10) and X(519)
X(80) = X(i)-cross conjugate of X(j) for these (i,j): (44,2), (517,1)
X(80) = X(8)-beth conjugate of X(100)
X(80) = antigonal image of X(1)
X(80) = syngonal conjugate of X(10)
X(80) = X(186)-of-Fuhrmann triangle
X(80) = orthology center of ABC and Fuhrmann triangle
X(80) = reflection of any vertex of ABC in the corresponding side of the Fuhrmann triangle
X(80) = perspector of ABC and reflection of Fuhrmann triangle in X(11)
X(80) = trilinear pole of line X(37)X(650)
X(80) = inverse-in-circumconic-centered-at-X(1)-of-X(1807)
X(80) = perspector of ABC and extraversion triangle of X(79)
X(80) = X(1986)-of-excentral triangle
X(80) = perspector of ABC and mid-triangle of 1st and 2nd extouch triangles
X(80) = inner-Garcia-to-outer-Garcia similarity image of X(1)
X(80) = X(100)-of-outer-Garcia-triangle
X(80) = Conway-circle-inverse of X(35638)


X(81) = CEVAPOINT OF INCENTER AND SYMMEDIAN POINT

Trilinears       1/(b + c) : 1/(c + a) : 1/(a + b)
Barycentrics  a/(b + c) : b/(c + a) : c/(a + b)
Barycentrics    eccentricity of A-Soddy ellipse : :

X(81) = (r2 + 2rR + s2)*X(1) - 3rR*X(2) - 2r2*X(3)    (Peter Moses, April 2, 2013)

Let A'B'C' be the cevian triangle of X(1). Let A″ be the symmedian point of triangle AB'C', and define B″ and C″ cyclically. Then the lines AA″, BB″, CC″ concur in X(81). (Eric Danneels, Hyacinthos 7892, 9/13/03)

Let A'B'C' be the incentral triangle. Let LA be the reflection of B'C' in the internal angle bisector of vertex angle A, and define LB and LC cyclically. Let A'' = LB∩LC, B'' = LC∩LA, C'' = LA∩LB. The lines AA'', BB'', CC'' concur in X(81). (Randy Hutson, 9/23/2011)

Let H* be the Stammler hyperbola. Let A'B'C' be the tangential triangle and A″B″C″ be the excentral triangle. Let A* be the intersection of the tangents to H* at A' and A″, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(81). (Randy Hutson, February 10, 2016)

Let A'B'C' be the 2nd circumperp triangle. Let A″ be the trilinear pole of line B'C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(81). (Randy Hutson, February 10, 2016)

Let A'B'C' be the anticomplement of the Feuerbach triangle. Let A″ be BB'∩CC', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(81). (Randy Hutson, February 10, 2016)

Let Ba, Ca be the intersections of lines CA, AB, resp., and the antiparallel to BC through X(1). Define Cb, Ab, Ac, Bc cyclically. Triangles ABaCa, AbBCb, AcBcC are similar to each other and inversely similar to ABC. Let Sa be the similitude center of triangles AbBCb and AcBcC. Define Sb and Sc cyclically. The lines ASa, BSb, CSc concur in X(81). (Randy Hutson, February 10, 2016)

Let A10B10C10 be Gemini triangle 10. Let A' be the perspector of conic {A,B,C,B10,C10}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(81). (Randy Hutson, January 15, 2019)

Let A11B11C11 be Gemini triangle 11. Let A' be the perspector of conic {A,B,C,B11,C11}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(81). (Randy Hutson, January 15, 2019)

If you have The Geometer's Sketchpad, you can view X(81).


If you have GeoGebra, you can view X(81).

X(81) lies on these lines: 1,21   2,6   7,27   8,1010   19,969   28,60   29,189   32,980   42,100   43,750   55,1002   56,959   57,77   65,961   88,662   99,739   105,110   145,1043   226,651   239,274   314,321   377,387   386,404   411,581   593,757   715,932   859,957   941,967   982,985   1019,1022   1051,1054   1098,1104

X(81) = isogonal conjugate of X(37)
X(81) = isotomic conjugate of X(321)
X(81) = anticomplement of X(1211)
X(81) = circumcircle-inverse of X(5867)
X(81) = X(i)-Ceva conjugate of X(j) for these (i,j): (7,229), (86,21), (286,28)
X(81) = cevapoint of X(i) and X(j) for these (i,j): (1,6), (57,222), (58,284)
X(81) = X(i)-cross conjugate of X(j) for these (i,j): (1,86), (3,272), (6,58), (57,27), (284,21)
X(81) = crosspoint of X(274) and X(286)
X(81) = crosssum of X(i) and X(j) for these (i,j): (1,846), (6,1030), (42,1334), (213,228)
X(81) = crossdifference of every pair of points on line X(512)X(661)
X(81) = X(i)-beth conjugate of X(j) for these (i,j): (333,333), (643,81), (645,81), (648,81), (662,81), (931,81)
X(81) = trilinear product of PU(31)
X(81) = intersection of tangents at X(1) and X(6) to the Stammler hyperbola
X(81) = crosspoint of X(1) and X(6) wrt both the excentral and tangential triangles
X(81) = trilinear pole of line X(36)X(238) (the polar of X(1) wrt the circumcircle)
X(81) = {X(1),X(31)}-harmonic conjugate of X(1621)
X(81) = X(6)-isoconjugate of X(10)
X(81) = X(92)-isoconjugate of X(228)
X(81) = perspector of ABC and cross-triangle of Gemini triangles 1 and 2
X(81) = barycentric product of vertices of Gemini triangle 1
X(81) = barycentric product of vertices of Gemini triangle 2
X(81) = barycentric product of vertices of Gemini triangle 3
X(81) = barycentric product of vertices of Gemini triangle 4
X(81) = perspector of Gemini triangles 2 and 7
X(81) = perspector of ABC and cross-triangle of ABC and Gemini triangle 1
X(81) = perspector of ABC and cross-triangle of ABC and Gemini triangle 2
X(81) = perspector of Gemini triangle 24 and cross-triangle of ABC and Gemini triangle 24
X(81) = perspector of Gemini triangle 28 and cross-triangle of ABC and Gemini triangle 28


X(82) = ISOGONAL CONJUGATE OF X(38)

Trilinears    1/(b2 + c2) : 1/(c2 + a2) : 1/(a2 + b2)
Trilinears    sin A csc(A + ω) : sin B csc(B + ω) : sin C csc(C + ω)

Barycentrics  a/(b2 + c2) : b/(c2 + a2) : c/(a2 + b2)

Let A'B'C' be the circummedial triangle. Let A″ be the trilinear product B'*C', and define B″, C″ cyclically. A″, B″, C″ are collinear on line X(798)X(812) (the trilinear polar of X(3112)). The lines AA″, BB″, CC″ concur in X(82). (Randy Hutson, October 15, 2018)

X(82) lies on these lines: 1,560   10,83   31,75   37,251   58,596   689,715   759,827

X(82) = isogonal conjugate of X(38)
X(82) = isotomic conjugate of X(1930)
X(82) = anticomplement of X(21249)
X(82) = cevapoint of X(1) and X(31)
X(82) = trilinear pole of line X(661)X(830)
X(82) = crossdifference of every pair of points on line X(2084)X(2530)
X(82) = perspector of ABC and extraversion triangle of X(82) (which is also the anticevian triangle of X(82))
X(82) = crosspoint of X(1) and X(31) wrt the excentral triangle


X(83 = CEVAPOINT OF CENTROID AND SYMMEDIAN POINT

Trilinears       bc/(b2 + c2) : ca/(c2 + a2) : ab/(a2 + b2)
                        = csc(A + ω) : csc(B + ω) : csc(C + ω)

Barycentrics  1/(b2 + c2) : 1/(c2 + a2) : 1/(a2 + b2)

Let K denote the symmedian point, X(6). Let A'B'C' be the cevian triangle of K. Let KA be K of the triangle AB'C'; let KB be K of A'BC' and let KC be K of A'B'C. The lines AKA, BKB, CKC concur in X(83). (Randy Hutson, 9/23/2011)

Let A'B'C' be the 1st Brocard triangle. Let A″ be the reflection of A' in BC, and define B″ and C″ cyclically. AA″, BB″, CC″ concur in X(83). (Randy Hutson, December 26, 2015)

Let Ba, Ca be the intersections of lines CA, AB, resp., and the antiparallel to BC through X(2). Define Cb, Ab, Ac, Bc cyclically. Triangles ABaCa, AbBCb, AcBcC are similar to each other and inversely similar to ABC. Let Sa be the similitude center of triangles AbBCb and AcBcC. Define Sb and Sc cyclically. The lines ASa, BSb, CSc concur in X(83). (Randy Hutson, December 26, 2015)

Let (Oa) be the circle whose diameter is the orthogonal projections of PU(1) on line BC. Define (Ob) and (Oc) cyclically. X(83) is the radical center of circles (Oa), (Ob), (Oc). (Randy Hutson, December 26, 2015)

Let A'B'C' be the circummedial triangle. Let A″ be the trilinear pole of line B'C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(83). (Randy Hutson, December 26, 2015)

X(83) lies on these lines: 2,32   3,262   4,182   5,98   6,76   10,82   17,624   18,623   39,99   213,239   217,287   275,297   597,671   689,729

X(83) = isogonal conjugate of X(39)
X(83) = isotomic conjugate of X(141)
X(83) = complement of X(2896)
X(83) = cevapoint of X(2) and X(6)
X(83) = X(i)-cross conjugate of X(j) for these (i,j): (2,308), (6,251), (512,99)
X(83) = trilinear pole of line X(23)X(385) (line is the polar of X(2) wrt the circumcircle, and also the anticomplement of the de Longchamps line, and also the polar of X(5) wrt {circumcircle, nine-point circle}-inverter)
X(83) = crossdifference of every pair of points on line X(688)X(3005)
X(83) = pole wrt polar circle of trilinear polar of X(427)
X(83) = X(48)-isoconjugate (polar conjugate) of X(427)
X(83) = perspector of ABC and medial triangle of 1st Brocard triangle
X(83) = crosspoint of X(2) and X(6) wrt both the anticomplementary and tangential triangles
X(83) = trilinear product of vertices of circummedial triangle
X(83) = midpoint of PU(137)
X(83) = bicentric sum of PU(i) for these i: 137, 141
X(83) = homothetic center of 5th anti-Brocard triangle and medial triangle
X(83) = X(8290)-of-1st-Brocard-triangle
X(83) = perspector of ABC and 1st Brocard triangle of medial triangle
X(83) = perspector of ABC and 1st Brocard triangle of 5th anti-Brocard triangle
X(83) = homothetic center of ABC and cross-triangle of ABC and 5th anti-Brocard triangle
X(83) = Cundy-Parry Phi transform of X(262)
X(83) = Cundy-Parry Psi transform of X(182)
X(83) = barycentric product of circumcircle intercepts of line X(316)X(512)


X(84) = ISOGONAL CONJUGATE OF X(40)

Trilinears    1/(cos B + cos C - cos A - 1) : :
Trilinears    a^2[a^2 - (b - c)^2]^2 - (b - c)^2[a^2 - (b + c)^2]^2 : :
X(84) = (r + 2R)*X(1) - 6R*X(2) + (4R - 2r)*X(3)    (Peter Moses, April 2, 2013)

Let A',B',C' be the excenters. The perpendiculars from B' to AB and from C' to AC meet in a point A″. Points B″ and C″ are determined cyclically. The hexyl triangle, A″B″C″, is perspective to ABC, and X(84) is the perspector.

Let A'B'C' be the extouch triangle. Let A″ be the orthocenter of AB'C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(84). (Randy Hutson, September 14, 2016)

Let A1B1C1 be the 1st Conway triangle. Let A' be the crosspoint of B1 and C1, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(84). (Randy Hutson, December 2, 2017)

See Dasari Naga Vijay Krishna, On a Conic Through Twelve Notable Points, Int. J. Adv. Math. and Mech. 7(2) (2019) 1-15.

X(84) lies on the Darboux cubic, the circumellipse with center X(9), and these lines: 1,221   3,9   4,57   7,946   8,20   21,285   33,603   36,90   46,80   58,990   64,3353   171,989   256,988   294,580   309,314   581,941   944,1000   2130,3345   3346,3472   3347,3355

X(84) = reflection of X(i) in X(j) for these (i,j): (40,1158), (1490,3)
X(84) = isogonal conjugate of X(40)
X(84) = isotomic conjugate of X(322)
X(84) = X(i)-Ceva conjugate of X(j) for these (i,j): (189,282), (280,1)
X(84) = X(i)-cross conjugate of X(j) for these (i,j): (19,57), (56,1)
X(84) = X(280)-aleph conjugate of X(84)
X(84) = X(i)-beth conjugate of X(j) for these (i,j): (271,3), (280,280), (285,84)
X(84) = X(68)-of-the-hexyl-triangle
X(84) = trilinear pole of line X(650)X(1459)
X(84) = perspector of ABC and the reflection in X(9) of the antipedal triangle of X(9)
X(84) = Danneels point of X(110)
X(84) = trilinear product of vertices of hexyl triangle (i.e., the extraversions of X(40))
X(84) = hexyl-isotomic conjugate of X(12717)
X(84) = orthologic center of hexyl triangle to inverse(n(hexyl triangle)); the reciprocal orthologic center is X(65)
X(84) = perspector of ABC and cross-triangle of extouch and Hutson-extouch triangles
X(84) = Cundy-Parry Phi transform of X(9)
X(84) = Cundy-Parry Psi transform of X(57)
X(84) = intouch-to-excentral similarity image of X(4)


X(85) = ISOTOMIC CONJUGATE OF X(9)

Trilinears       b2c2/(b + c - a) : c2a2/(c + a - b) : a2b2/(a + b - c)
                        = tan(A/2) csc2A : tan(B/2) csc2B : tan(C/2) csc2C
Barycentrics  bc/(b + c - a) : ca/(c + a - b) : ab/(a + b - c)
Barycentrics    cot A' : :, where A'B'C' is the excentral triangle

Let A38B38C38 be Gemini triangle 38. Let A' be the perspector of conic {A,B,C,B38,C38}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(85). (Randy Hutson, January 15, 2019)

X(85) lies on these lines: 1,664   2,241   7,8   12,120   29,34   56,870   57,274   76,226   92,331   109,767   150,355   264,309

X(85) = isogonal conjugate of X(41)
X(85) = isotomic conjugate of X(9)
X(85) = complement of X(3177)
X(85) = anticomplement of X(1212)
X(85) = X(274)-Ceva conjugate of X(348)
X(85) = cevapoint of X(i) and X(j) for these (i,j): (1,169), (2,7), (57,77), (92,342)
X(85) = X(i)-cross conjugate of X(j) for these (i,j): (2,75), (57,273), (92,309), (142,2), (226,7)
X(85) = X(i)-beth conjugate of X(j) for these (i,j): (76,76), (85,279), (99,1), (274,85), (668,85), (789,85), (799,85), (811,85)
X(85) = trilinear pole of line X(522)X(693) (the isotomic conjugate of the circumconic centered at X(1), conic {A,B,C,X(100),X(664),X(1120),X(1320)}; also the polar of X(33) wrt polar circle)
X(85) = pole wrt polar circle of trilinear polar of X(33) (line X(657)X(4041))
X(85) = polar conjugate of X(33)
X(85) = {X(7),X(8)}-harmonic conjugate of X(6604)
X(85) = trilinear square of X(508)
X(85) = trilinear product of vertices of Gemini triangle 9
X(85) = trilinear product of vertices of Gemini triangle 10


X(86) = CEVAPOINT OF INCENTER AND CENTROID

Trilinears    bc/(b + c) : ca/(c + a) : ab/(a + b)
Barycentrics   1/(b + c) : 1/(c + a) : 1/(a + b)

X(86) = 2(r2 + 2rR + s2)*X(1) + 3(r2 + s2)*X(2) - 4r2*X(3)    (Peter Moses, April 2, 2013)

Let A'B'C' be the anticomplement of the Feuerbach triangle. Let La be the tangent to the circumcircle at A', and define Lb and Lc cyclically. Let A″ be the point where La is tangent to the Steiner circumellipse, and define B″ and C″ cyclically. Let A* = BB″∩CC″, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(86). (Randy Hutson, December 10, 2016)

Let A1B1C1 be the 1st Conway triangle. Let A' be the trilinear pole of line B1C1, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(86). (Randy Hutson, December 10, 2016)

Let A5B5C5 be the 5th Conway triangle. Let A' be the trilinear pole of line B5C5, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(86). (Randy Hutson, December 10, 2016)

Let A3B3C3 and A4B4C4 be Gemini triangles 3 and 4, resp. Let LA be the tangent at A to conic {A,B3,C3,B4,C4}}, and define LB, LC cyclically. Let A' = LB∩LC, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(86). (Randy Hutson, January 15, 2019)

Let A21B21C21 be Gemini triangle 21. Let A' be the perspector of conic {A,B,C,B21,C21}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(86). (Randy Hutson, January 15, 2019)

X(86) lies on these lines: 1,75   2,6   7,21   10,319   29,34   37,190   58,238   60,272   99,106   110,675   142,284   239,1100   269,1088   283,307   310,350   741,789   870,871

X(86) = isogonal conjugate of X(42)
X(86) = isotomic conjugate of X(10)
X(86) = complement of X(1654)
X(86) = anticomplement of X(1213)
X(86) = circumcircle-inverse of X(5937)
X(86) = X(274)-Ceva conjugate of X(333)
X(86) = cevapoint of X(i) and X(j) for these (i,j): (1,2), (7,77), (21,81)
X(86) = crosssum of X(1) and X(1045)
X(86) = crossdifference of every pair of points on line X(512)X(798)
X(86) = X(i)-cross conjugate of X(j) for these (i,j): (1,81), (2,274), (7,286), (21,333), (58,27), (513,190)
X(86) = X(i)-beth conjugate of X(j) for these (i,j): (86,1014), (99,86), (261,86), (314,314), (645,86), (811,86)
X(86) = X(2)-Ceva conjugate of X(6626)
X(86) = intersection of tangents at X(1) and X(2) to the bianticevian conic of X(1) and X(2); see X(99)
X(86) = crosspoint of X(1) and X(2) wrt both the excentral and anticomplementary triangles
X(86) = trilinear pole of line X(239)X(514) (Lemoine axis of excentral triangle)
X(86) = pole wrt polar circle of trilinear polar of X(1826)
X(86) = X(48)-isoconjugate (polar conjugate)-of-X(1826)
X(86) = perspector of Gemini triangle 1 and cross-triangle of ABC and Gemini triangle 1
X(86) = perspector of ABC and cross-triangle of ABC and Gemini triangle 23
X(86) = perspector of ABC and cross-triangle of ABC and Gemini triangle 24
X(86) = perspector of ABC and cross-triangle of Gemini triangles 23 and 24
X(86) = perspector of ABC and Gemini triangle 25
X(86) = barycentric product of vertices of Gemini triangle 23
X(86) = barycentric product of vertices of Gemini triangle 24
X(86) = barycentric product of vertices of Gemini triangle 25
X(86) = {X(2),X(6)}-harmonic conjugate of X(17277)
X(86) = {X(2),X(69)}-harmonic conjugate of X(5224)
X(86) = {X(2),X(141)}-harmonic conjugate of X(17307)


X(87) = X(2)-CROSS CONJUGATE OF X(1)

Trilinears       1/(ab + ac - bc) : 1/(bc + ba - ca) : 1/(ca + cb - ab)
Barycentrics  a/(ab + ac - bc) : b/(bc + ba - ca) : c/(ca + cb - ab)

X(87) lies on these lines: 1,192   6,43   9,292   10,979   34,242   56,238   58,978   106,932

X(87) = isogonal conjugate of X(43)
X(87) = isotomic conjugate of X(6376)
X(87) = anticomplement of X(34832)
X(87) = perspector of ABC and extraversion triangle of X(43)
X(87) = trilinear product of extraversions of X(43)
X(87) = cevapoint of X(2) and X(330)
X(87) = X(2)-cross conjugate of X(1)
X(87) = X(932)-beth conjugate of X(87)


X(88) = ISOGONAL CONJUGATE OF X(44)

Trilinears    1/(b + c - 2a) : 1/(c + a - 2b) : 1/(a + b - 2c)
Barycentrics  a/(b + c - 2a) : b/(c + a - 2b) : c/(a + b - 2c)
X(88) = (3r2 + 6rR - s2)*X(1) + 9rR*X(2) - 6r2*X(3)    (Peter Moses, April 2, 2013)

Let A9B9C9 be Gemini triangle 9. Let A' be the perspector of conic {A,B,C,B9,C9}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(88). (Randy Hutson, January 15, 2019)

See (and hear) Dan Reznik's Dance of the Swans: X(88) and X(162) and their Never-Touching Motion Over the Elliptic Billiard (March 4, 2020)

X(88) lies on these lines: 1,100   2,45   6,89   28,162   44,679   57,651   81,662   105,901   274,799   278,653   279,658   291,660

X(88) = isogonal conjugate of X(44)
X(88) = isotomic conjugate of X(4358)
X(88) = complement of X(30578)
X(88) = cevapoint of X(i) and X(j) for these (i,j): (1,44), (6,36)
X(88) = X(i)-cross conjugate of X(j) for these (i,j): (44,1), (517,7)
X(88) = X(i)-aleph conjugate of X(j) for these (i,j): (88,1), (679,88), (903,63), (1022,1052)
X(88) = X(333)-beth conjugate of X(190)
X(88) = trilinear product of PU(50)
X(88) = perspector of conic {A,B,C,PU(50)}}
X(88) = trilinear pole of PU(55); the line X(1)X(513), the line through X(1) parallel to its trilinear polar; also normal to Feuerbach hyperbola at X(1)
X(88) = crossdifference of every pair of points on line X(678)X(1635)
X(88) = X(6)-isoconjugate of X(519)
X(88) = BSS(a^2→a) of X(111)
X(88) = polar conjugate of X(38462)
X(88) = X(19)-isoconjugate of X(5440)
X(88) = X(48)-isoconjugate of X(38462)
X(88) = X(92)-isoconjugate of X(23202)


X(89) = ISOGONAL CONJUGATE OF X(45)

Trilinears       1/(2b + 2c - a) : 1/(2c + 2a - b) : 1/(2a + 2b - c)
Barycentrics  a/(2b + 2c - a) : b/(2c + 2a - b) : c/(2a + 2b - c)

Let A9B9C9 be the Gemini triangle 9. Let LA be the line through A9 parallel to BC, and define LB and LC cyclically. Let A'9 = LB∩LC, and define B'9, C'9 cyclically. Triangle A'9B'9C'9 is homothetic to ABC at X(89). (Randy Hutson, November 30, 2018)

X(89) lies on these lines: 1,902   2,44   6,88   649,1022

X(89) = isogonal conjugate of X(45)
X(89) = isotomic conjugate of X(4671)
X(89) = anticomplement of isotomic conjugate of isogonal conjugate of X(20973)
X(89) = anticomplement of polar conjugate of isogonal conjugate of X(22083)
X(89) = anticomplement of complementary conjugate of X(34824)


X(90) = X(3)-CROSS CONJUGATE OF X(1)

Trilinears    1/(cos B + cos C - cos A) : 1/(cos C + cos A - cos B) : 1/(cos A + cos B - cos C)
Trilinears    1/(a^3 + a^2 (b + c) - a (b^2 + c^2) - (b - c)^2 (b + c)) : :
Barycentrics  a/(cos B + cos C - cos A) : b/(cos C + cos A - cos B) : c/(cos A + cos B - cos C)

X(90) = (r + R)2*X(1) - 6rR*X(2) - 2r(r - R)*X(3)    (Peter Moses, April 2, 2013)

X(90) lies on these lines: 1,155   4,46   9,35   21,224   33,47   36,84   40,80   57,79

X(90) = isogonal conjugate of X(46)
X(90) = isotomic conjugate of X(20930)
X(90) = X(3)-cross conjugate of X(1)
X(90) = perspector of ABC and extraversion triangle of X(46)
X(90) = trilinear product of the extraversions of X(46), which is also the cross-triangle of the orthic and excentral triangles
X(90) = trilinear product of PU(125)
X(90) = Cundy-Parry Phi transform of X(90)
X(90) = Cundy-Parry Psi transform of X(46)


X(91) = ISOGONAL CONJUGATE OF X(47)

Trilinears       sec 2A : sec 2B : sec 2C
Barycentrics  sin A sec 2A : sin B sec 2B : sin C sec 2C
Trilinears    cot A tan 2A : :

X(91) lies on these lines: 19,920   31,1087   37,498   47,92   63,921   65,68   225,847   255,1109   759,925

X(91) = isogonal conjugate of X(47)
X(91) = X(48)-cross conjugate of X(92)
X(91) = trilinear product of X(485) and X(486)
X(91) = polar conjugate of X(1748)
X(91) = X(92)-isoconjugate of X(563)
X(91) = trilinear pole of line X(661)X(2618)
X(91) = perspector of ABC and extraversion triangle of X(91) (which is also the anticevian triangle of X(91))
X(91) = trilinear product of vertices of outer and inner Vecten triangles


X(92) = CEVAPOINT OF INCENTER AND CLAWSON POINT

Trilinears    csc 2A : csc 2B : csc 2C
Trilinears    cot A + tan A : :
Barycentrics    sec A : sec B : sec C
X(92) = 2r(r+2R)*X(4) - ((r + 2R)^2 - s^2)*X(8)
X(92) = (sec A)*[A] + (sec B)*[B] + (sec C)*[C], where A, B, C denote angles and [A], [B], [C] denote vertices

Let LA be the line through X(4) parallel to the internal bisector of angle A, and let
A' = BC∩LA. Define B' and C' cyclically.

Alexei Myakishev, "The M-Configuration of a Triangle," Forum Geometricorum 3 (2003) 135-144,

proves that the lines AA', BB', CC' concur in X(92). He notes that another construction follows from Proposition 2 of the article: let A1 be the midpoint of the arc BC of the circumcircle that passes through A, and let A2 be the point, other than A, in which the A-altitude meets the circumcircle. Let A″ = A1A2∩BC. Define B″ and C″ cyclically. Then the lines AA″, BB″, CC″ concur in X(92).

Suppose that T = A'B'C' is a central triangle. Let A'' be the pole with respect to the polar circle of the line B'C', and define B'' and C'' cyclically. The appearance of T in the following list means that the lines AA'', BB'', CC'' concur in X(92): Feurerbach, incentral, excentral, extangents, Apollonius, mixtilinear excentral. (Randy Hutson, December 26, 2015)

X(92) lies on these lines: 1,29   2,273   4,8   7,189   10,1838   19,27   25,242   28,2975   31,162   33,1897   34,1220   38,240   40,412   47,91   48,2167   53,4415   55,243   56,1940   57,653   81,2995   85,331   100,917   108,1311   171,1430   226,342   239,607   255,1087   257,297   264,306   304,561   345,3262   388,1118   394,1943   406,1068   427,2969   429,3948   429,3948   459,1446   497,1857   518,1859   608,894   651,2988   823,2349   938,3176   942,1148   960,1882   984,1860   994,1845   1146,1952   1172,2997   1211,1865   1309,2717   1435,3306   1585,1659   1621,4183   1707,1733   1726,1746   1731,1751   1785,4656   1842,1891   1844,3874   1870,5136   1947,2994   1954,1955   1956,2632   1973,3112   2064,3596      2331,5256   2399,4391   3064,4468   4198,4968

X(92) = isogonal conjugate of X(48)
X(92) = isotomic conjugate of X(63)
X(92) = anticomplement of X(1214)
X(92) = anticomplementary conjugate of X(2897)
X(92) = Fuhrmann-circle-inverse of X(5174)
X(92) = X(i)-Ceva conjugate of X(j) for these (i,j): (85, 342), (264,318), (286,4), (331,273)
X(92) = cevapoint of X(i) and X(j) for these (i,j): (1,19), (4,281), (47,48), (196,278)
X(92) = X(i)-cross conjugate of X(j) for these (i,j): (1,75), (4,273), (19,158), (48,91), (226,2), (281,318)
X(92) = crosspoint of X(i) and X(j) for these (i,j): (85,309), (264,331)
X(92) = crossdifference of every pair of points on line X(810)X(822)
X(92) = X(275)-aleph conjugate of X(47)
X(92) = X(i)-beth conjugate of X(j) for these (i,j): (92,278), (312,329), (648,57)
X(92) = {X(19),X(63)}-harmonic conjugate of X(1748)
X(92) = barycentric product of PU(20)
X(92) = trilinear product of PU(i) for these i: 21, 45
X(92) = bicentric sum of PU(130)
X(92) = midpoint of PU(130)
X(92) = trilinear product X(2)*X(4)
X(92) = trilinear pole of line X(240)X(522) (polar of X(1) wrt polar circle)
X(92) = pole of antiorthic axis wrt polar circle
X(92) = X(6)-isoconjugate of X(3)
X(92) = X(48)-isoconjugate (polar conjugate) of X(1)
X(92) = X(88)-isoconjugate of X(23202)
X(92) = X(91)-isoconjugate of X(563)
X(92) = perspector of ABC and extraversion triangle of X(92) (which is also the anticevian triangle of X(92))
X(92) = crosspoint of X(1) and X(19) wrt excentral triangle
X(92) = crosspoint of X(47) and X(48) wrt excentral triangle
X(92) = perspector of ABC and cross-triangle of Gemini triangles 37 and 38
X(92) = perspector of ABC and cross-triangle of ABC and Gemini triangle 37
X(92) = perspector of ABC and cross-triangle of ABC and Gemini triangle 38
X(92) = barycentric product of vertices of Gemini triangle 37
X(92) = barycentric product of vertices of Gemini triangle 38


X(93) = ISOGONAL CONJUGATE OF X(49)

Trilinears       sec 3A : sec 3B : sec 3C
Barycentrics  sin A sec 3A : sin B sec 3B : sin C sec 3C
Barycentrics    1/(a^2 (a^2 - b^2 - c^2) (a^4 + b^4 + c^4 - 2 a^2 b^2 - 2 a^2 c^2 - b^2 c^2)) : :

Let OAOBOC be the Kosnita triangle. Let A' be the pole wrt polar circle of line OBOC, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(93). (Randy Hutson, June 7, 2019)

X(93) lies on these lines: 4,562   49,94   186,252

X(93) = isogonal conjugate of X(49)
X(93) = anticomplement of X(34833)
X(93) = X(50)-cross conjugate of X(94)
X(93) = polar conjugate of X(1994)
X(93) = X(2964)-isoconjugate of X(3)


X(94) = ISOGONAL CONJUGATE OF X(50)

Trilinears        csc 3A : csc 3B : csc 3C
Barycentrics   sin A csc 3A : sin B csc 3B : sin C csc 3C
Barycentrics    b^2 c^2 / ((a^2 - b^2 - c^2)^2 - b^2 c^2) : :

Let A1B1C1 and A2B2C2 be the 1st and 2nd Ehrmann circumscribing triangles. Let A' be the trilinear pole of line A1A2, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(94). Let A1'B1'C1' and A2'B2'C2' be the 1st and 2nd Ehrmann inscribed triangles. Then X(94) is the radical center of nine-point circles of AA1'A2', BB1'B2', CC1'C2'. (Randy Hutson, June 27, 2018)

Let A'B'C' be the medial or orthic triangle of ABC and A″, B″, C″ the centers of circles {A', X(3), X(4)}}, {B', X(3), X(4)}} and {C', X(3), X(4)}}, respectively. Then A″, B″, C″ are collinear on the trilinear polar of X(94). (César Lozada, March 9, 2021).

X(94) lies on the Kiepert hyperbola and these lines: 2,300   4,143   23,98   49,93   96,925   275,324

X(94) = isogonal conjugate of X(50)
X(94) = isotomic conjugate of X(323)
X(94) = anticomplement of X(34834)
X(94) = cevapoint of X(49) and X(50)
X(94) = X(i)-cross conjugate of X(j) for these (i,j): (30,264), (50,93), (265,328)
X(94) = X(300)-Hirst inverse of X(301)
X(94) = trilinear pole of PU(5) (line X(5)X(523))
X(94) = pole wrt polar circle of trilinear polar of X(186)
X(94) = X(48)-isoconjugate (polar conjugate) of X(186)
X(94) = barycentric product X(476)*X(850)
X(94) = trilinear pole of PU(173)
X(94) = trilinear product X(74)*X(107) (circumcircle-X(4) antipodes)
X(94) = barycentric quotient X(476)/X(110)
X(94) = intersection of the tangent to hyperbola {A,B,C,X(6),X(13),X(16)}} at X(13) and the tangent to hyperbola {A,B,C,X(6),X(14),X(15)}} at X(14)


X(95) = CEVAPOINT OF CENTROID AND CIRCUMCENTER

Trilinears       b2c2sec(B - C) : :

Let A'B'C' be the symmedial triangle. Let La be the reflection of line B'C' in line BC, and define Lb and Lc cyclically. Let A″ = Lb∩Lc, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(95). (Randy Hutson, August 19, 2015)

Let A' be the intersection, other than A, of the circumcircle and the branch of the Lucas cubic that contains A, and define B' and C' cyclically. The triangle A'B'C' is here introduced as the Lucas triangle (not to be confused with the Lucas central triangle). The vertices A', B', C' lie on the rectangular hyperbola {X(2),X(20),X(54),X(69),X(110),X(2574),X(2575),X(2979)}}. (See http://bernard-gibert.fr/Exemples/k007.html.) Also, X(95) is the trilinear product of the vertices of the Lucas triangle. (Randy Hutson, August 19, 2015)

X(95) lies on these lines: 2,97   3,264   54,69   76,96   99,311   140,340   141,287   160,327   183,305   216,648   307,320

X(95) = isogonal conjugate of X(51)
X(95) = isotomic conjugate of X(5)
X(95) = anticomplement of X(233)
X(95) = X(276)-Ceva conjugate of X(275)
X(95) = cevapoint of X(i) and X(j) for these (i,j): (2,3), (6,160), (54,97)
X(95) = X(i)-cross conjugate of X(j) for these (i,j): (2,276), (3,97), (54,275), (140,2), (340,1494)
X(95) = intersection of tangents at X(2) and X(3) to bianticevian conic of X(2) and X(3)
X(95) = crosspoint of X(2) and X(3) wrt both the anticomplementary triangle and anticevian triangle of X(3)
X(95) = trilinear pole of line X(323)X(401) (polar of X(53) wrt polar circle, and polar of X(69) wrt de Longchamps circle)
X(95) = pole wrt polar circle of trilinear polar of X(53)
X(95) = X(48)-isoconjugate (polar conjugate) of X(53)
X(95) = X(92)-isoconjugate of X(217)


X(96) = ISOGONAL CONJUGATE OF X(52)

Trilinears    sec 2A sec(B - C) : sec 2B sec(C - A) : sec 2C sec(A - B)
Barycentrics  a sec 2A sec(B - C) : b sec 2B sec(C - A) : c sec 2C sec(A - B)

Let A'B'C' be the reflection triangle. Let BA and CA be the orthogonal projections of B' and C' on line BC, resp. Let (OA) be the circle with segment BACA as diameter. Define (OB) and (OC) cyclically. X(96) is the radical center of circles (OA), (OB), (OC). (Randy Hutson, June 7, 2019)

X(96) lies on these lines: 2,54   4,231   24,847   76,95   94,925

X(96) = isogonal conjugate of X(52)
X(96) = isotomic conjugate of X(39113)
X(96) = anticomplement of X(34835)
X(96) = cevapoint of X(3) and X(68)
X(96) = X(3)-cross conjugate of X(54)
X(96) = polar conjugate of X(467)
X(96) = Cundy-Parry Phi transform of X(5392)
X(96) = Cundy-Parry Psi transform of X(571)


X(97) = ISOGONAL CONJUGATE OF X(53)

Trilinears       cot A sec(B - C) : cot B sec(C - A) : cot C sec(A - B)
Barycentrics  cos A sec(B - C) : cos B sec(C - A) : cos C sec(A - B)

In the plane of a triangle ABC, let
Ia = line through X(3) parallel to BC, and define Ib and Ic cyclically.
Ac = Ia∩AB, and define Ba and Cb cyclically.
Ab = Ia∩AC, and define Bc and Ca cyclically.
Oa = circumcircle of A, Bc, Cb, and define Ob and Oc cyclically.
Then X(97) = radical center of Oa, Ob, Oc. See also X(77). (Ivan Pavlov, April 1, 2022)

X(97) lies on these lines: 2,95   3,54   110,418   216,288   276,401

X(97) = isogonal conjugate of X(53)
X(97) = isotomic conjugate of X(324)
X(97) = anticomplement of X(34836)
X(97) = complement of isogonal conjugate of X(34433)
X(97) = X(95)-Ceva conjugate of X(54)
X(97) = X(3)-cross conjugate of X(95)
X(97) = cevapoint of X(3) and X(577)
X(97) = X(51)-isoconjugate of X(92)
X(97) = Cundy-Parry Phi transform of X(7592)

leftri

Centers on the circumcircle: X(74), X(98)-X(112)

rightri

Centers X(74), X(98)-X(112), and many others in ETC lie on the circumcircle. Mappings Λ and Ψ derived from such a point P for application to points X, are defined here:

Λ(P,X) = isogonal conjugate of the point where line PX meets the line at infinity.

Let Y = Λ(P,X), let Q = isogonal conjugate of P, and let Y and Z be the points where line YQ meets the circumcircle;
then Ψ(P,X) = Z.


X(98) = TARRY POINT

Trilinears    sec(A + ω) : sec(B + ω) : sec(C + ω)
Trilinears    bc/(b4 + c4 - a2b2 - a2c2) : :
Barycentrics    1/(b4 + c4 - a2b2 - a2c2) : :
Tripolars    |cos(A + ω)| : :
Tripolars    |a (b^4 + c^4 - a^2 b^2 - a^2 c^2)| : :
X(98) = X(98) = 2(r2 + 4rR - s2)(r2 + 2rR + s2)*X(1) + 3(r4 + 4Rr3 + 2r2s2 - 4rRs2 + s4)*X(2) - 2(r2 + 4rR - s2)(3r2 + 4rR - s2)*X(3)    (Peter Moses, April 2, 2013)

Let A' be the apex of the isosceles triangle BA'C constructed inward on BC such that ∠A'BC = ∠A'CB = ω. Define B' nad C' cyclically. Let Ha be the orthocenter of BA'C, and define Hb and Hc cyclically. The lines AHa, BHb, CHc concur in X(98). (Randy Hutson, July 20, 2016)

Let A' be the apex of the isosceles triangle BA'C constructed outward on BC such that ∠A'BC = ∠A'CB = ω/2. Define B' and C' cyclically. Let Oa be the circumcenter of BA'C, and define Ob and Oc cyclically. The lines AOa, BOb, COc concur in X(98). (Randy Hutson, July 20, 2016)

If you have The Geometer's Sketchpad, you can view X(98).

J. W. Clawson, "Points on the circumcircle," American Mathematical Monthly 32 (1925) 169-174.

Let A'B'C' be the 1st Brocard triangle. Let A″ be the reflection of A in B'C', and define B″ and C″ cyclically. X(98) is the radical center of the circumcircles of AA'A″, BB'B″, CC'C″. (Randy Hutson, November 18, 2015)

Let A'B'C' be the orthic triangle. Let La be the Lemoine axis of triangle AB'C', and define Lb and Lc cyclically. Let A″ = Lb∩Lc, and define B″ and C″ cyclically. Triangle A″B″C″ is inversely similar to ABC, with similicenter X(2). The lines AA″, BB″, CC″ concur in X(98), which is X(4)-of-A″B″C″. (Randy Hutson, July 31 2018)

Let P be a point on the Steiner circumellipse. Let A' be the orthocenter of BCP, and define B' and C' cyclically. Let Q be the centroid of A'B'C'. The locus of Q as P varies is an ellipse similar and orthogonal to the Steiner circumellipse, and also centered at X(2). When P = X(671), Q = X(98). See also X(6054) and X(24808). (Randy Hutson, October 15, 2018)

X(98) lies on these lines: {1, 7970}, {2, 110}, {3, 76}, {4, 32}, {5, 83}, {6, 262}, {10, 101}, {11, 10768}, {12, 10799}, {13, 1080}, {14, 383}, {15, 33388}, {16, 33389}, {17, 6115}, {18, 6114}, {20, 148}, {22, 925}, {23, 94}, {24, 1289}, {25, 107}, {26, 1286}, {30, 671}, {35, 10086}, {36, 10089}, {39, 3399}, {40, 6010}, {54, 3203}, {55, 3027}, {56, 3023}, {57, 24469}, {61, 22691}, {62, 22692}, {69, 13355}, {74, 690}, {95, 20775}, {96, 32692}, {100, 228}, {102, 2785}, {103, 2786}, {104, 2787}, {105, 2788}, {106, 2789}, {108, 1402}, {109, 171}, {111, 1637}, {113, 13193}, {119, 13194}, {127, 12207}, {140, 7832}, {141, 19120}, {165, 13174}, {185, 1303}, {186, 935}, {187, 11676}, {194, 9737}, {230, 1503}, {235, 11380}, {237, 6037}, {250, 34978}, {251, 30505}, {264, 3425}, {265, 14849}, {275, 427}, {316, 15980}, {325, 2065}, {336, 1310}, {338, 1632}, {355, 12195}, {371, 13885}, {372, 13938}, {376, 543}, {378, 30247}, {381, 598}, {382, 22515}, {384, 6248}, {385, 511}, {386, 34454}, {402, 12181}, {403, 10423}, {446, 18858}, {459, 1301}, {468, 685}, {485, 6811}, {486, 6813}, {493, 12186}, {494, 12187}, {512, 2698}, {513, 2699}, {514, 2700}, {515, 2701}, {516, 2702}, {517, 2703}, {518, 2704}, {519, 2705}, {520, 2706}, {521, 2707}, {522, 2708}, {523, 842}, {524, 2709}, {525, 2710}, {526, 9161}, {530, 9203}, {531, 9202}, {538, 9888}, {541, 9144}, {546, 18502}, {548, 38731}, {549, 8724}, {550, 38730}, {573, 3029}, {574, 1569}, {575, 3329}, {576, 7766}, {618, 21157}, {619, 21156}, {620, 631}, {635, 32552}, {636, 32553}, {637, 1306}, {638, 1307}, {648, 2452}, {675, 7432}, {689, 1799}, {693, 2856}, {729, 2422}, {732, 36517}, {733, 11620}, {739, 35353}, {741, 4444}, {759, 7427}, {783, 3505}, {801, 1368}, {804, 878}, {813, 3404}, {841, 36164}, {843, 1499}, {850, 2857}, {858, 2986}, {868, 34174}, {895, 2407}, {901, 4080}, {919, 13576}, {927, 5991}, {929, 1324}, {930, 6636}, {931, 19544}, {932, 22381}, {934, 1410}, {946, 12194}, {1029, 37456}, {1113, 2593}, {1114, 2592}, {1125, 21636}, {1131, 7374}, {1132, 7000}, {1147, 3044}, {1181, 6570}, {1196, 37892}, {1284, 19637}, {1287, 2070}, {1288, 21213}, {1290, 37959}, {1291, 5189}, {1292, 2795}, {1293, 2796}, {1294, 2797}, {1295, 2798}, {1297, 2799}, {1302, 1995}, {1309, 17987}, {1311, 7439}, {1316, 5191}, {1342, 5404}, {1343, 5403}, {1350, 5969}, {1351, 14614}, {1370, 6504}, {1379, 3414}, {1380, 3413}, {1385, 29299}, {1423, 7350}, {1447, 7061}, {1478, 10802}, {1479, 10801}, {1482, 29269}, {1498, 13380}, {1576, 7668}, {1587, 18993}, {1588, 18994}, {1593, 5186}, {1598, 10790}, {1656, 7942}, {1657, 38733}, {1687, 2009}, {1688, 2010}, {1698, 29117}, {1699, 10789}, {1756, 7095}, {1764, 38481}, {1989, 23969}, {2051, 4279}, {2086, 8789}, {2222, 5143}, {2319, 6210}, {2378, 5917}, {2379, 5916}, {2387, 11674}, {2482, 3524}, {2558, 6177}, {2559, 6178}, {2691, 36001}, {2696, 7464}, {2711, 3309}, {2712, 3667}, {2713, 6000}, {2714, 6001}, {2720, 5061}, {2770, 4108}, {2777, 16278}, {2855, 3260}, {2858, 3266}, {3068, 8980}, {3069, 13967}, {3070, 14245}, {3071, 14231}, {3090, 6722}, {3091, 5395}, {3095, 7760}, {3098, 6194}, {3146, 38259}, {3153, 11635}, {3222, 3504}, {3225, 6234}, {3229, 8841}, {3269, 18338}, {3291, 9091}, {3314, 34507}, {3316, 10783}, {3317, 10784}, {3406, 7746}, {3407, 3818}, {3424, 9752}, {3428, 22514}, {3430, 8669}, {3456, 3518}, {3507, 23605}, {3510, 8923}, {3512, 7351}, {3522, 10992}, {3523, 38748}, {3525, 20399}, {3526, 7930}, {3533, 38746}, {3534, 33638}, {3543, 9880}, {3545, 5461}, {3563, 16230}, {3566, 23700}, {3574, 12208}, {3576, 11711}, {3579, 29151}, {3597, 5706}, {3616, 11724}, {3655, 9884}, {3734, 35925}, {3757, 38470}, {3815, 5038}, {3830, 17503}, {3832, 18845}, {3843, 18501}, {3849, 13241}, {3851, 15092}, {3855, 18843}, {3934, 5149}, {3972, 35930}, {4232, 9064}, {4239, 9058}, {4297, 28483}, {4558, 25051}, {4588, 30588}, {4995, 12350}, {5000, 34240}, {5001, 34239}, {5008, 22521}, {5020, 37874}, {5025, 9863}, {5026, 5085}, {5033, 7612}, {5034, 5477}, {5039, 5304}, {5050, 11174}, {5055, 22566}, {5071, 14971}, {5092, 8289}, {5169, 7578}, {5188, 6308}, {5210, 8719}, {5217, 15452}, {5254, 34870}, {5286, 13356}, {5293, 29052}, {5298, 12351}, {5306, 5480}, {5465, 10706}, {5469, 5479}, {5470, 5478}, {5475, 11170}, {5487, 22532}, {5488, 22531}, {5569, 19911}, {5587, 10791}, {5597, 12179}, {5598, 12180}, {5603, 10800}, {5657, 29127}, {5663, 9160}, {5840, 10769}, {5870, 14229}, {5871, 14244}, {5886, 29111}, {5913, 35188}, {5914, 9136}, {5965, 7779}, {5968, 34810}, {5970, 32472}, {5971, 6082}, {5978, 5982}, {5979, 5983}, {5996, 32730}, {6002, 12031}, {6011, 30273}, {6072, 22103}, {6199, 10839}, {6200, 10848}, {6201, 10793}, {6202, 10792}, {6221, 10841}, {6222, 8304}, {6233, 7610}, {6245, 12196}, {6246, 12198}, {6247, 12202}, {6249, 12206}, {6250, 12211}, {6251, 12210}, {6284, 13183}, {6295, 14539}, {6312, 6320}, {6316, 6319}, {6395, 10840}, {6396, 10847}, {6398, 10842}, {6399, 8305}, {6433, 10843}, {6434, 10844}, {6435, 10849}, {6436, 10850}, {6530, 16318}, {6539, 8701}, {6564, 35767}, {6565, 35766}, {6578, 32014}, {6582, 14538}, {6625, 7379}, {6655, 32152}, {6670, 36765}, {6684, 29119}, {6784, 6785}, {6795, 9832}, {6995, 8796}, {7166, 8926}, {7168, 8927}, {7354, 13182}, {7391, 13579}, {7414, 26706}, {7426, 9060}, {7438, 9107}, {7448, 9083}, {7449, 9056}, {7453, 9057}, {7458, 9061}, {7472, 38702}, {7494, 26870}, {7505, 26269}, {7607, 37637}, {7697, 10000}, {7751, 8178}, {7778, 15069}, {7781, 18768}, {7788, 11898}, {7790, 37242}, {7797, 37336}, {7818, 34623}, {7824, 13334}, {7827, 37345}, {7857, 37466}, {7875, 38317}, {7887, 10349}, {7901, 10333}, {7907, 10131}, {7911, 32151}, {7919, 9996}, {7932, 10345}, {8059, 8808}, {8150, 12252}, {8196, 11837}, {8203, 11838}, {8212, 11840}, {8213, 11841}, {8229, 24624}, {8295, 9865}, {8296, 10853}, {8297, 10854}, {8375, 10837}, {8376, 10838}, {8587, 8590}, {8591, 10304}, {8600, 11184}, {8690, 16434}, {8693, 37675}, {8704, 9831}, {8770, 34808}, {8784, 11606}, {8859, 11645}, {8997, 9540}, {9059, 26266}, {9066, 26233}, {9069, 26235}, {9070, 26227}, {9079, 26278}, {9080, 26276}, {9087, 9465}, {9123, 9215}, {9150, 34087}, {9167, 15702}, {9184, 9191}, {9221, 15032}, {9300, 12007}, {9307, 9308}, {9381, 21284}, {9413, 9414}, {9479, 14718}, {9738, 33371}, {9739, 33370}, {9746, 28841}, {9875, 34628}, {9927, 12193}, {9955, 29293}, {9956, 29155}, {10098, 10295}, {10101, 37979}, {10102, 36168}, {10113, 12201}, {10166, 31839}, {10264, 15545}, {10290, 32515}, {10299, 35022}, {10303, 38751}, {10310, 13173}, {10313, 35098}, {10486, 17005}, {10519, 15589}, {10531, 10803}, {10532, 10804}, {10733, 12066}, {10794, 10893}, {10795, 10894}, {10797, 10895}, {10798, 10896}, {10810, 22564}, {10997, 35375}, {11121, 36515}, {11122, 36514}, {11152, 11155}, {11172, 23055}, {11231, 29129}, {11248, 13189}, {11249, 13190}, {11414, 13175}, {11490, 11496}, {11538, 37349}, {11602, 23004}, {11603, 23005}, {11669, 31489}, {11822, 13176}, {11823, 13177}, {11826, 13180}, {11827, 13181}, {11828, 13184}, {11829, 13185}, {11839, 11897}, {12092, 31074}, {12112, 32732}, {12151, 22110}, {12200, 12599}, {12209, 12600}, {12245, 28532}, {12258, 31162}, {12355, 15681}, {12699, 29091}, {12702, 29177}, {13449, 14041}, {13580, 14808}, {13581, 14807}, {13672, 13687}, {13748, 14234}, {13749, 14238}, {13792, 13807}, {13935, 13989}, {14060, 14570}, {14236, 23261}, {14240, 23251}, {14458, 36990}, {14485, 18907}, {14534, 37360}, {14540, 22914}, {14541, 22869}, {14561, 16989}, {14575, 36794}, {14630, 14633}, {14631, 14632}, {14644, 15359}, {14719, 34783}, {14850, 38728}, {15054, 31854}, {15182, 19379}, {15246, 20189}, {15300, 19708}, {15357, 18331}, {15407, 34138}, {15483, 21163}, {15526, 34841}, {15598, 21167}, {15682, 32532}, {15687, 33698}, {15694, 26614}, {15698, 36521}, {15709, 22247}, {16083, 16089}, {16085, 35574}, {16086, 29241}, {16115, 16125}, {16166, 18366}, {16167, 37980}, {16188, 36173}, {16984, 18553}, {17008, 37182}, {17734, 32682}, {17758, 21554}, {17985, 36067}, {18440, 37071}, {18446, 36516}, {18806, 35385}, {18906, 35387}, {18931, 36893}, {19179, 31804}, {19642, 37443}, {19649, 34594}, {20126, 36826}, {20187, 21312}, {20190, 36811}, {20404, 32305}, {22143, 36841}, {22239, 37777}, {22467, 26179}, {22498, 35422}, {22520, 22753}, {22522, 22831}, {22523, 22832}, {22524, 22833}, {22791, 29265}, {23098, 36214}, {24815, 24828}, {24975, 25328}, {25213, 38403}, {25216, 38404}, {25406, 34229}, {25446, 32022}, {26273, 28476}, {26326, 26379}, {26327, 26403}, {26328, 26427}, {26329, 26428}, {26330, 26429}, {26331, 26430}, {26332, 26431}, {26333, 26432}, {26446, 29030}, {26613, 37461}, {27797, 28210}, {28474, 35103}, {28475, 35107}, {29067, 30115}, {30241, 37620}, {30670, 31394}, {30716, 38552}, {32111, 32681}, {32242, 32274}, {32335, 32369}, {32522, 33004}, {33512, 38793}, {33640, 37453}, {34133, 34134}, {34310, 35189}, {34396, 37988}, {34475, 37508}, {35705, 37451}, {36036, 36066}, {36071, 36104}, {36495, 36511}, {36526, 36548}, {36557, 36580}, {38253, 38282}

X(98) is the {X(2),X(147)}-harmonic conjugate of X(114). For a list of harmonic conjugates, click Tables at the top of this page.

X(98) = midpoint between X(20) and X(148)
X(98) = reflection of X(i) in X(j) for these (i,j): (4,115), (99,3), (147,114), (1513,230)
X(98) = isogonal conjugate of X(511)
X(98) = isotomic conjugate of X(325)
X(98) = complement of X(147)
X(98) = anticomplement of X(114)
X(98) = circumcircle antipode of X(99)
X(98) = X(290)-Ceva conjugate of X(287)
X(98) = cevapoint of X(i) and X(j) for these (i,j): (2,385), (6,237)
X(98) = X(i)-cross conjugate of X(j) for these (i,j): (230,2), (237,6), (248,287), (446,511)
X(98) = crosssum of X(385) and X(401)
X(98) = X(2)-Hirst inverse of X(287)
X(98) = perspector of ABC and triangle formed by Lemoine axis (or PU(1) or PU(2)) reflected in sides of ABC
X(98) = Λ(X(4), X(69)) (the line that is the isotomic conjugate of the Jerabek hyperbola)
X(98) = trilinear pole of line X(6)X(523) (polar of X(297) wrt polar circle, and the radical axis of circles with segments X(13)X(16) and X(14)X(15) as diameters)
X(98) = pole wrt polar circle of trilinear polar of X(297) (line X(114)X(132))
X(98) = pole wrt {circumcircle, nine-point circle}-inverter of line X(115)X(125)
X(98) = X(48)-isoconjugate (polar conjugate) of X(297)
X(98) = X(6)-isoconjugate of X(1959)
X(98) = inverse-in-polar-circle of X(132)
X(98) = inverse-in-{circumcircle, nine-point circle}-inverter of X(125)
X(98) = inverse-in-circle-{X(1687),X(1688),PU(1),PU(2)}} of X(2715)
X(98) = Ψ(X(6), X(523))
X(98) = Ψ(X(190), X(71))
X(98) = Kiepert-hyperbola antipode of X(4)
X(98) = reflection of X(842) in the Euler line
X(98) = reflection of X(2698) in the Brocard axis
X(98) = reflection of X(2699) in line X(1)X(3)
X(98) = X(129)-of-excentral-triangle
X(98) = X(130)-of-hexyl-triangle
X(98) = X(3)-of-1st-anti-Brocard-triangle
X(98) = perspector of ABC and 1st Neuberg triangle
X(98) = trilinear product of vertices of 1st Neuberg triangle
X(98) = orthocenter of X(13)X(14)X(2394)
X(98) = 2nd-Parry-to-ABC similarity image of X(2)
X(98) = trilinear product of PU(88)
X(98) = X(2456) of 6th Brocard triangle
X(98) = midpoint of PU(135)
X(98) = bicentric sum of PU(135)
X(98) = perspector of ABC and circumsymmedial triangle of Artzt triangle
X(98) = McCay-to-Artzt similarity image of X(381)
X(98) = circumcircle-antipode of X(99)
X(98) = the point of intersection, other than A, B, and C, of the circumcircle and Kiepert hyperbola
X(98) = Ψ(X(101), X(100)
X(98) = Λ(X(3), X(6))
X(98) = homothetic center of 5th anti-Brocard triangle and Euler triangle
X(98) = Thomson-isogonal conjugate of X(512)
X(98) = Lucas-isogonal conjugate of X(512)
X(98) = X(1380)-of-circummedial-triangle
X(98) = X(6233)-of-circumsymmedial-triangle
X(98) = Cundy-Parry Phi transform of X(76)
X(98) = Cundy-Parry Psi transform of X(32)
X(98) = perspector of ABC and vertex-triangle of reflection triangles of PU(1)
X(98) = X(2)-of-2nd-anti-Parry-triangle
X(98) = excentral-to-ABC functional image of X(1282)
X(98) = orthic-to-ABC functional image of X(129)
X(98) = Ψ(X(1), X(810))
X(98) = Ψ(X(811), X(1))
X(98) = barycentric product of circumcircle intercepts of line X(2)X(647)
X(98) = trilinear pole, wrt Thomson triangle, of line X(2)X(1350)
X(98) = areal center of pedal triangles of PU(1)
X(98) = orthocenter of X(98)X(99)X(31953)
X(98) = orthocenter of X(114)X(115)X(31953)
X(98) = X(2)-Ceva conjugate of X(36899)
X(98) = perspector of circumconic centered at X(36899)
X(98) = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {A,B,C,X(3),X(25),X(32)}} (isogonal conjugate of line X(4)X(69))
X(98) = BSS(a→a^2) of X(673)


X(99) = STEINER POINT

Trilinears    bc/(b2 - c2) : ca/(c2 - a2) : ab/(a2 - b2)
Trilinears    b2c2 csc(B - C) : c2a2 csc(C - A) : a2b2 csc(A - B)
Barycentrics    1/(b2 - c2) : 1/(c2 - a2) : 1/(a2 - b2)
Barycentrics    d(A,L) : : , where d = directed distance and L = X(115)X(125)
Tripolars    |a (b^2 - c^2)| : :
X(99) = 2(r2 + 4rR - s2)(r2 + 2rR + s2)*X(1) - 3(r4 + 4Rr3 + 2r2s2 - 4rRs2 + s4)*X(2) - 4r2(r2 + 4rR + 5s2)*X(3)    (Peter Moses, April 2, 2013)

Let LA be the reflection of the line X(3)X(6) in line BC, and define LB and LC cyclically. Let A' = LB∩LC, B' = LC∩LA, C' = LA∩LB. The lines AA', BB', CC' concur in X(99). (Randy Hutson, 9/23/2011)

X(99) is the only point on the circumcircle whose isotomic conjugate lies on the line at infinity. (Randy Hutson, 9/23/2011)

X(99) is the center of the bianticevian conic of X(1) and X(2), which is the rectangular hyperbola H that passes through these points: X(1), X(2), X(20), X(63), X(147), X(194), X(487), X(488), X(616), X(617), X(627), X(628), X(1764), X(2896), the excenters, the vertices of the anticomplementary triangle, and the extraversions of X(63). Also, H is the anticomplementary conjugate of line X(4)X(69), the anticomplementary isotomic conjugate of line X(2)X(6), the excentral isogonal conjugate of line X(40)X(511), and the excentral isotomic conjugate of line X(1045)X(2951); also, H is tangent to line X(1)X(75) at X(1), to line X(2)X(6) at X(2), and meets the line at infinity (and the Kiepert hyperbola, other than at X(2)) at X(3413) and X(3414). (Randy Hutson, December 26, 2015)

Let A'B'C' be the anticomplement of the Feuerbach triangle. Let A″B″C″ be the tangential triangle of A'B'C'. Let A* be the cevapoint of B″ and C″, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(99). (Randy Hutson, February 10, 2016)

Let La, Lb, Lc be the lines through A, B, C, resp. parallel to the Brocard axis. Let Ma, Mb, Mc be the reflections of BC, CA, AB in La, Lb, Lc, resp. Let A' = Mb∩Mc, and define B' and C' cyclically. Triangle A'B'C' is inversely similar to ABC with similarity ratio 3. Let A″B″C″ be the reflection of A'B'C' in the Brocard axis. The triangle A″B″C″ is homothetic to ABC, with center of homothety X(115) and centroid X(99). See Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, February 10, 2016)

Let A', B' and C' be the intersections of the de Longchamps line and lines BC, CA, AB, resp. The circumcircles of AB'C', BC'A', CA'B' concur in X(99). (Randy Hutson, February 10, 2016)

Let A', B', C' be the intersections of the Brocard axis and lines BC, CA, AB, resp. Let Oa, Ob, Oc be the circumcenters of AB'C', BC'A', CA'B', resp. The lines AOa, BOb, COc concur in X(99). (Randy Hutson, February 10, 2016)

Let A'B'C' be the 1st Brocard triangle. Let La be the line through A parallel to B'C', and define Lb and Lc cyclically. La, Lb, Lc concur in X(99). (Randy Hutson, February 10, 2016)

Let A' be the trilinear pole of the perpendicular bisector of BC, and define B' and C' cyclically. A'B'C' is also the anticomplement of the anticomplement of the midheight triangle. X(99) is the trilinear product A'*B'*C'. (Randy Hutson, January 29, 2018)

Let A'B'C' be the 1st Brocard triangle. Let A″ be the reflection of A in line B'C', and define B″, C″ cyclically. Let A″' be the reflection of A' in line BC, and define B″', C″' cyclically. Let A* = B″B″'∩\C″C″', and define B*, C* cyclically. Triangle A*B*C* is homothetic to A'B'C' at X(99). (Randy Hutson, June 27, 2018)

See a construction: Ercole Suppa, Hyacinthos 29064.

If you have The Geometer's Sketchpad, you can view the following dynamic sketches:
X(99) and Steiner Circum-ellipse (showing X(99) and an area-ratio property)

Let NANBNC and N'AN'BN'C be the inner and outer Napoleon triangles, resp. Let A' be the reflection of NA in line N'BN'C, and define B' and C' cyclically. Let A″ be the reflection of N'A in line NBNC, and define B″ and C″ cyclically. Triangles A'B'C' and A″B″C″ are equilateral and inversely similar, with similitude center X(99). (Randy Hutson, January 17, 2020)

For more about the Steiner circumellipse, visit MathWorld.

Let OA be the circle centered at the A-vertex of the Moses-Steiner osculatory triangle and passing through A; define OB and OC cyclically. X(99) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

Let LA be the tangent at A to circle {A,PU(1)}}, and define LB and LC cyclically. The lines LA, LB, LC concur in X(99). (Randy Hutson, August 30, 2020)

The osculating circle of the Steiner circumellipse at X(99) intersects the Steiner circumellipse in two points: X(99) and X(892). See Osculating circle of Steiner circumellipse at X(99). (Randy Hutson, November 5, 2021)

Let O be a point (not necessarily X(3). Let Q1,Q2,...Qn be a regular n-gon with center O. Let P be a point, and define n triangles T1=Q1Q2P, T2=Q2Q3P, ..., Tn=QnQ1P.
Claim 1: for all n, the points X(2)-of-Ti are concyclic on a first circle whose center O2 is collinear with O,P.
Claim 2: for all n, the lpoints X(99)-of-Ti are concyclic with P on a second circle whose center O99 is also collinear with O,P. Figure. (Dan Reznik, December 10, 2021)

For a construction, see Antreas Hatzipolakis and Peter Moses, euclid 7178.

X(99) lies on these lines: {1, 741}, {2, 111}, {3, 76}, {4, 114}, {5, 5966}, {6, 729}, {8, 28471}, {9, 35106}, {10, 6626}, {11, 10769}, {12, 13181}, {13, 303}, {14, 302}, {15, 22687}, {16, 22689}, {20, 147}, {21, 105}, {22, 305}, {23, 2770}, {25, 2374}, {27, 9085}, {28, 15344}, {30, 316}, {31, 715}, {32, 194}, {35, 1909}, {36, 350}, {37, 2375}, {38, 745}, {39, 83}, {40, 24469}, {51, 35919}, {53, 35920}, {55, 3023}, {56, 3027}, {57, 35176}, {58, 727}, {61, 22685}, {62, 22683}, {63, 2249}, {69, 74}, {75, 261}, {81, 739}, {86, 106}, {95, 311}, {100, 668}, {101, 190}, {102, 332}, {103, 1043}, {104, 314}, {107, 2797}, {108, 811}, {109, 643}, {110, 690}, {112, 648}, {113, 33512}, {116, 31001}, {125, 35922}, {127, 35923}, {140, 38224}, {141, 755}, {145, 28531}, {159, 2366}, {160, 1502}, {162, 32691}, {163, 825}, {165, 9860}, {172, 25264}, {182, 7709}, {184, 3044}, {186, 5866}, {187, 385}, {192, 2242}, {193, 5477}, {216, 35928}, {230, 14568}, {232, 15014}, {237, 3978}, {238, 2669}, {239, 1931}, {249, 525}, {250, 10423}, {251, 35929}, {253, 5896}, {262, 35930}, {264, 378}, {265, 14850}, {286, 915}, {287, 3269}, {298, 531}, {299, 530}, {304, 26702}, {310, 675}, {313, 2372}, {317, 1299}, {323, 32730}, {326, 26701}, {327, 35934}, {330, 2241}, {333, 1121}, {338, 34866}, {340, 10295}, {343, 35937}, {371, 19056}, {372, 19055}, {381, 22515}, {382, 7773}, {386, 3029}, {393, 35940}, {394, 26717}, {395, 35942}, {396, 35943}, {401, 36212}, {402, 13179}, {404, 18140}, {419, 19599}, {439, 6392}, {448, 25083}, {468, 37803}, {476, 850}, {477, 3260}, {487, 8982}, {488, 26441}, {489, 12123}, {490, 12124}, {491, 6560}, {492, 6561}, {493, 13184}, {494, 13185}, {511, 2698}, {512, 805}, {513, 2703}, {514, 2702}, {515, 2708}, {516, 2700}, {517, 2699}, {518, 2711}, {519, 2712}, {520, 2713}, {521, 2714}, {522, 2701}, {523, 691}, {524, 843}, {526, 9160}, {532, 6779}, {533, 6780}, {536, 16702}, {548, 7767}, {549, 11632}, {550, 3933}, {573, 34454}, {575, 35950}, {576, 10788}, {577, 35952}, {590, 35953}, {593, 17147}, {595, 34063}, {597, 35954}, {598, 5503}, {599, 6323}, {621, 5617}, {622, 5613}, {623, 23004}, {624, 23005}, {625, 7925}, {626, 6655}, {627, 25560}, {628, 25559}, {629, 11602}, {630, 11603}, {631, 6036}, {633, 22507}, {634, 22509}, {637, 6231}, {638, 6230}, {644, 8693}, {647, 9091}, {651, 32038}, {666, 919}, {669, 886}, {670, 804}, {689, 4609}, {692, 785}, {693, 1290}, {694, 11229}, {695, 711}, {697, 2206}, {698, 1691}, {703, 3051}, {707, 3117}, {712, 5006}, {713, 1333}, {719, 1964}, {723, 3116}, {726, 1326}, {731, 3736}, {732, 2076}, {736, 5162}, {737, 3094}, {740, 12031}, {753, 30966}, {754, 6781}, {757, 4360}, {767, 6385}, {769, 32739}, {783, 2531}, {789, 4602}, {801, 6509}, {813, 1016}, {815, 3888}, {826, 9218}, {827, 1576}, {835, 1978}, {839, 6386}, {840, 18821}, {858, 37804}, {859, 2726}, {873, 1621}, {883, 6606}, {889, 898}, {894, 38814}, {895, 13479}, {901, 4555}, {917, 7431}, {931, 4631}, {932, 18830}, {933, 18831}, {934, 4569}, {935, 3267}, {953, 17139}, {972, 1792}, {1010, 5988}, {1014, 8686}, {1015, 4366}, {1030, 3770}, {1046, 25607}, {1086, 25536}, {1113, 15164}, {1114, 15165}, {1125, 11599}, {1180, 9101}, {1194, 16951}, {1232, 13597}, {1235, 3520}, {1236, 2071}, {1270, 26617}, {1271, 26618}, {1273, 14979}, {1285, 1992}, {1287, 23285}, {1292, 2788}, {1293, 2789}, {1294, 2790}, {1295, 2791}, {1296, 2418}, {1298, 5562}, {1301, 2409}, {1302, 15329}, {1304, 3233}, {1305, 4572}, {1308, 7199}, {1310, 1633}, {1311, 4225}, {1316, 9155}, {1325, 2752}, {1331, 28624}, {1332, 36080}, {1350, 8719}, {1370, 34254}, {1379, 3413}, {1380, 3414}, {1384, 11055}, {1415, 14612}, {1434, 1477}, {1499, 2709}, {1500, 6645}, {1503, 2710}, {1506, 16044}, {1511, 18332}, {1562, 30227}, {1593, 12131}, {1605, 2926}, {1606, 2925}, {1613, 6380}, {1625, 2421}, {1627, 8267}, {1649, 9170}, {1655, 5277}, {1656, 38732}, {1657, 7776}, {1789, 21586}, {1799, 3455}, {1812, 32726}, {1921, 14195}, {1963, 17319}, {1995, 9084}, {2023, 5013}, {2052, 6503}, {2080, 32469}, {2106, 3230}, {2108, 24294}, {2134, 27954}, {2142, 14824}, {2185, 32939}, {2207, 6461}, {2222, 4998}, {2311, 24578}, {2368, 8053}, {2382, 18822}, {2383, 7576}, {2384, 16704}, {2502, 5108}, {2525, 9514}, {2548, 14035}, {2644, 21196}, {2668, 3750}, {2679, 31513}, {2687, 3262}, {2688, 35517}, {2689, 35519}, {2690, 3261}, {2695, 35516}, {2704, 3309}, {2705, 3667}, {2706, 6000}, {2707, 6001}, {2717, 35164}, {2718, 30939}, {2719, 4131}, {2721, 16741}, {2722, 15413}, {2725, 18157}, {2727, 30805}, {2748, 4076}, {2758, 3264}, {2764, 4143}, {2766, 7476}, {2857, 37183}, {2862, 16876}, {2868, 16084}, {2896, 7794}, {2930, 36792}, {2971, 36898}, {2975, 17143}, {2986, 34834}, {2996, 32989}, {3006, 5196}, {3008, 24378}, {3053, 6179}, {3058, 12351}, {3068, 8997}, {3069, 13989}, {3090, 6721}, {3091, 20399}, {3095, 12110}, {3096, 7791}, {3097, 10791}, {3108, 38278}, {3110, 14839}, {3146, 32816}, {3180, 9117}, {3181, 9115}, {3244, 32004}, {3286, 14665}, {3290, 16756}, {3314, 7761}, {3360, 33786}, {3363, 12040}, {3398, 32448}, {3407, 10290}, {3428, 22504}, {3448, 15357}, {3491, 14135}, {3522, 3785}, {3523, 11623}, {3524, 6055}, {3525, 20398}, {3526, 34127}, {3529, 32006}, {3533, 38735}, {3534, 7788}, {3543, 32827}, {3545, 9880}, {3564, 23700}, {3565, 35136}, {3566, 10425}, {3569, 14960}, {3570, 28841}, {3571, 7170}, {3576, 11710}, {3579, 29300}, {3582, 10070}, {3584, 10054}, {3589, 6034}, {3616, 11725}, {3618, 14039}, {3628, 38229}, {3632, 28547}, {3658, 9058}, {3666, 11611}, {3679, 28559}, {3699, 4614}, {3701, 37294}, {3729, 27958}, {3732, 28847}, {3746, 25303}, {3760, 7280}, {3761, 5010}, {3767, 7857}, {3788, 5025}, {3799, 29363}, {3815, 8370}, {3830, 22566}, {3832, 32835}, {3849, 7840}, {3854, 32873}, {3882, 6010}, {3886, 28842}, {3906, 9181}, {3912, 35163}, {3934, 7824}, {3948, 19308}, {3952, 4596}, {3977, 18653}, {4025, 35169}, {4037, 7267}, {4074, 10329}, {4093, 34996}, {4094, 7207}, {4108, 9080}, {4176, 11206}, {4188, 18135}, {4189, 5985}, {4203, 34020}, {4210, 18152}, {4215, 9074}, {4218, 30893}, {4228, 9061}, {4240, 9064}, {4243, 9057}, {4246, 9107}, {4249, 26705}, {4256, 37678}, {4357, 38453}, {4367, 24037}, {4391, 9090}, {4393, 9346}, {4467, 24041}, {4482, 4595}, {4499, 29325}, {4552, 4565}, {4553, 29071}, {4557, 8708}, {4570, 32682}, {4575, 15440}, {4582, 4591}, {4583, 4593}, {4588, 4597}, {4594, 29055}, {4613, 4639}, {4620, 7253}, {4658, 28523}, {4720, 28535}, {4736, 7266}, {4756, 28196}, {4760, 7200}, {4840, 29341}, {4897, 6064}, {4921, 8696}, {4996, 19628}, {5007, 7839}, {5008, 33694}, {5017, 32451}, {5024, 11174}, {5030, 37686}, {5055, 12355}, {5056, 32839}, {5059, 32825}, {5064, 12132}, {5088, 20924}, {5097, 22521}, {5099, 36174}, {5111, 13196}, {5113, 14959}, {5116, 24256}, {5171, 12251}, {5189, 30718}, {5201, 34000}, {5206, 7751}, {5207, 29012}, {5210, 8667}, {5235, 28317}, {5248, 31997}, {5254, 7807}, {5267, 20888}, {5283, 16915}, {5284, 33779}, {5286, 7856}, {5291, 17759}, {5309, 7806}, {5333, 28326}, {5434, 12350}, {5467, 11636}, {5469, 6670}, {5470, 6669}, {5473, 36776}, {5475, 7777}, {5479, 36765}, {5485, 11147}, {5490, 6568}, {5491, 6569}, {5597, 13176}, {5598, 13177}, {5603, 11724}, {5642, 9144}, {5648, 36883}, {5649, 5664}, {5663, 9161}, {5745, 11608}, {5840, 10768}, {5860, 33342}, {5861, 33343}, {5897, 6527}, {5921, 10008}, {5972, 16278}, {5987, 6031}, {5990, 9081}, {5992, 11115}, {5994, 17402}, {5995, 17403}, {5999, 9772}, {6071, 22103}, {6093, 9966}, {6114, 11304}, {6115, 11303}, {6194, 8722}, {6195, 9463}, {6200, 22716}, {6226, 11825}, {6227, 11824}, {6248, 37334}, {6284, 12185}, {6292, 10159}, {6295, 10645}, {6340, 38282}, {6353, 19583}, {6376, 25440}, {6396, 22718}, {6551, 6635}, {6563, 10420}, {6574, 7258}, {6582, 10646}, {6628, 27804}, {6646, 25435}, {6656, 7789}, {6658, 7747}, {6671, 22510}, {6672, 22511}, {6680, 7765}, {6683, 31652}, {6776, 35387}, {6782, 23013}, {6783, 23006}, {6786, 6787}, {7061, 33941}, {7255, 8684}, {7260, 30670}, {7279, 34388}, {7283, 15168}, {7354, 12184}, {7417, 9775}, {7419, 9083}, {7424, 33864}, {7426, 10102}, {7436, 32706}, {7450, 9056}, {7463, 26704}, {7471, 9060}, {7488, 28706}, {7495, 11056}, {7496, 13233}, {7502, 21395}, {7603, 17005}, {7610, 11151}, {7735, 32985}, {7736, 14033}, {7737, 7774}, {7738, 7803}, {7739, 16989}, {7745, 7858}, {7746, 7907}, {7749, 33259}, {7753, 19686}, {7758, 7877}, {7759, 7823}, {7762, 7905}, {7772, 7787}, {7778, 7841}, {7780, 15513}, {7784, 7881}, {7792, 7827}, {7800, 32965}, {7805, 35007}, {7815, 15515}, {7817, 16984}, {7818, 7897}, {7819, 7859}, {7821, 7842}, {7822, 7876}, {7825, 7888}, {7829, 10583}, {7834, 7864}, {7838, 13571}, {7843, 7941}, {7849, 7928}, {7851, 7942}, {7852, 7923}, {7853, 7880}, {7854, 7904}, {7855, 7893}, {7861, 7874}, {7862, 32966}, {7865, 9878}, {7866, 7918}, {7867, 7872}, {7868, 7937}, {7869, 7935}, {7873, 7895}, {7875, 14036}, {7878, 9605}, {7879, 7936}, {7884, 33220}, {7886, 33245}, {7887, 7940}, {7889, 19689}, {7894, 30435}, {7896, 7929}, {7900, 7903}, {7902, 7932}, {7915, 7948}, {7916, 7946}, {7926, 9766}, {7943, 33217}, {7953, 35137}, {8025, 8700}, {8029, 14728}, {8041, 10328}, {8149, 34999}, {8266, 33769}, {8287, 25469}, {8295, 10998}, {8299, 19635}, {8352, 22110}, {8362, 9478}, {8586, 12151}, {8588, 17131}, {8589, 9466}, {8597, 31173}, {8652, 32042}, {8666, 17144}, {8691, 24039}, {8703, 14830}, {8704, 13241}, {8707, 27808}, {8709, 16695}, {8715, 24524}, {8783, 23208}, {8787, 15534}, {8856, 28677}, {8860, 11149}, {8884, 23233}, {8980, 9540}, {9070, 13589}, {9078, 17521}, {9079, 17587}, {9082, 13588}, {9097, 16711}, {9108, 17524}, {9109, 17539}, {9110, 16748}, {9111, 35172}, {9112, 12155}, {9113, 12154}, {9136, 9487}, {9143, 9184}, {9185, 9216}, {9202, 27550}, {9203, 27551}, {9230, 20775}, {9293, 10190}, {9308, 10607}, {9428, 9494}, {9509, 35068}, {9512, 22085}, {9861, 11414}, {9870, 11580}, {9875, 19875}, {10007, 24273}, {10130, 31078}, {10278, 37879}, {10299, 35021}, {10302, 15810}, {10303, 32838}, {10304, 11177}, {10310, 12178}, {10350, 32452}, {10461, 29053}, {10488, 15533}, {10684, 11672}, {10796, 32447}, {11005, 17702}, {11060, 14972}, {11101, 33933}, {11112, 37664}, {11116, 30758}, {11121, 22848}, {11122, 22892}, {11123, 31614}, {11145, 34376}, {11146, 34374}, {11155, 11168}, {11176, 23356}, {11184, 11317}, {11237, 18969}, {11238, 12354}, {11248, 12189}, {11249, 12190}, {11285, 15815}, {11307, 36252}, {11308, 36251}, {11319, 27162}, {11320, 24598}, {11322, 30964}, {11328, 32524}, {11329, 30830}, {11353, 31234}, {11413, 34168}, {11593, 27779}, {11596, 22143}, {11612, 13858}, {11613, 13859}, {11822, 12179}, {11823, 12180}, {11826, 12182}, {11827, 12183}, {11828, 12186}, {11829, 12187}, {11842, 32519}, {12030, 35550}, {12054, 32516}, {12074, 24976}, {12093, 33962}, {12156, 18907}, {12195, 12782}, {12212, 32449}, {12258, 25055}, {12348, 34612}, {12349, 34606}, {12941, 18974}, {12942, 18975}, {12951, 13076}, {12952, 13075}, {13587, 18145}, {13637, 13642}, {13638, 35306}, {13712, 32808}, {13738, 30022}, {13757, 13761}, {13758, 35305}, {13835, 32809}, {13881, 33233}, {13935, 13967}, {14023, 33254}, {14060, 25051}, {14096, 22735}, {14118, 26166}, {14558, 18570}, {14574, 33514}, {14658, 19570}, {14720, 18436}, {14727, 21789}, {14734, 23105}, {14810, 14994}, {14849, 38728}, {14867, 31839}, {14927, 15428}, {14953, 35158}, {14961, 15013}, {15022, 32871}, {15034, 31854}, {15035, 22265}, {15059, 15359}, {15061, 15535}, {15075, 28405}, {15545, 32423}, {15574, 22241}, {15588, 35965}, {15696, 29322}, {15697, 32896}, {15701, 26614}, {15705, 32874}, {15708, 32885}, {15717, 32834}, {15814, 17503}, {16041, 37690}, {16049, 26703}, {16370, 16992}, {16371, 18146}, {16589, 16917}, {16592, 24962}, {16678, 18021}, {16806, 32036}, {16807, 32037}, {16887, 28485}, {16912, 36812}, {16914, 27318}, {16921, 31455}, {16924, 31401}, {16948, 28583}, {16956, 21838}, {16990, 33008}, {17004, 33274}, {17008, 21843}, {17034, 33863}, {17058, 26147}, {17132, 37792}, {17150, 33774}, {17183, 38452}, {17210, 28499}, {17280, 25433}, {17475, 18268}, {17499, 18755}, {17549, 37670}, {17596, 24291}, {17677, 30761}, {17729, 30109}, {17738, 18061}, {17777, 25533}, {17780, 28210}, {17944, 29313}, {18153, 37309}, {18206, 35167}, {18315, 32692}, {18481, 29096}, {19057, 32788}, {19058, 32787}, {19598, 36955}, {20023, 37184}, {20172, 31449}, {20580, 23582}, {20624, 31623}, {20982, 24504}, {21004, 21431}, {21163, 37455}, {21220, 21341}, {21359, 25166}, {21360, 25156}, {21937, 26244}, {22089, 22456}, {22239, 37937}, {22401, 28723}, {22407, 36511}, {22489, 22577}, {22490, 22578}, {22561, 22564}, {22691, 33409}, {22692, 33408}, {23096, 37943}, {23334, 25486}, {23610, 38017}, {23889, 35180}, {23999, 36068}, {24264, 24282}, {24345, 24714}, {24348, 24711}, {24630, 32851}, {25332, 30226}, {25526, 28491}, {25532, 30997}, {26243, 35276}, {26257, 30749}, {26615, 32810}, {26616, 32811}, {26619, 32805}, {26620, 32806}, {26716, 32040}, {26860, 28310}, {27082, 34403}, {27269, 33062}, {27656, 29983}, {27705, 34053}, {28348, 30092}, {28574, 33947}, {29039, 31730}, {29052, 35338}, {30528, 32690}, {30736, 37927}, {30785, 31107}, {31400, 32971}, {31404, 32979}, {31415, 33016}, {31450, 33269}, {31644, 36953}, {32465, 36759}, {32466, 36760}, {32483, 32485}, {32522, 37479}, {32823, 33703}, {33376, 33458}, {33377, 33459}, {33511, 38793}, {33705, 37465}, {33801, 35226}, {33932, 37405}, {33943, 37816}, {33972, 35298}, {34283, 36744}, {34389, 37848}, {34390, 37850}, {34505, 37637}, {35178, 35324}, {35287, 37667}, {35540, 37896}, {35549, 37903}, {36036, 36065}, {36070, 36085}, {36166, 38704}, {36329, 36769}, {36330, 36768}, {37205, 37212}, {37459, 38227}

X(99) is the {X(39),X(384)}-harmonic conjugate of X(83). For a list of other harmonic conjugates of X(99), click Tables at the top of this page.

X(99) = midpoint of X(i) and X(j) for these (i,j): (20,147), (616,617)
X(99) = reflection of X(i) in X(j) for these (i,j): (4,114), (13,619), (14,618), (98,3), (115,620), (148,115), (316,325), (385,187), (671,2)
X(99) = isogonal conjugate of X(512)
X(99) = isotomic conjugate of X(523)
X(99) = complement of X(148)
X(99) = anticomplement of X(115)
X(99) = cevapoint of X(i) and X(j) for these (i,j): (2,523), (3,525), (39,512), (100,190)
X(99) = X(1019)-cross conjugate of X(1509)
X(99) = crossdifference of every pair of points on line X(351)X(865)
X(99) = X(i)-cross conjugate of X(j) for these (i,j): (3,249), (22,250), (512,83), (523,2), (525,76)
X(99) = X(21)-beth conjugate of X(741)
X(99) = X(6)-of-1st-anti-Brocard-triangle
X(99) = X(381)-of-anti-McCay-triangle
X(99) = circumcircle-antipode of X(98)
X(99) = point of intersection, other than A, B, and C, of the circumcircle and Steiner ellipse
X(99) = Ψ(X(i), X(j) for these (i,j): (1,75), (2,39), (3,69), (4,69), (37,2), (51,5), (351,690)
X(99) = point of intersection, other than A, B, C, of the circumcircle and hyperbola {A,B,C,PU(1)}}
X(99) = point of intersection, other than A, B, C, of the circumcircle and hyperbola {A,B,C,PU(37)}}
X(99) = trilinear product of PU(90)
X(99) = similitude center of (equilateral) antipedal triangles of X(13) and X(14)
X(99) = Steiner-circumellipse-antipode of X(671)
X(99) = projection from Steiner inellipse to Steiner circumellipse of X(2482)
X(99) = trilinear pole of line X(2)X(6)
X(99) = pole wrt polar circle of trilinear polar of X(2501) (line X(115)X(2971))
X(99) = X(48)-isoconjugate (polar conjugate) of X(2501)
X(99) = X(6)-isoconjugate of X(661)
X(99) = X(1577)-isoconjugate of X(32)
X(99) = concurrence of reflections in sides of ABC of line X(4)X(69)
X(99) = Λ(X(1), X(512))
X(99) = isotomic conjugate wrt 1st Brocard triangle of X(76)
X(99) = perspector of ABC and the tangential triangle, wrt the anticomplementary triangle, of the bianticevian conic of X(1) and X(2)
X(99) = perspector of ABC and the tangential triangle, wrt the tangential triangle, of the Stammler hyperbola
X(99) = reflection of X(691) in the Euler line
X(99) = reflection of X(805) in the Brocard axis
X(99) = reflection of X(2703) in line X(1)X(3)
X(99) = reflection of X(316) in the de Longchamps line
X(99) = X(130)-of-excentral-triangle
X(99) = X(129)-of-hexyl-triangle
X(99) = inverse-in-polar-circle of X(5139)
X(99) = inverse-in-{circumcircle, nine-point circle}-inverter of X(126)
X(99) = inverse-in-2nd-Brocard-circle of X(76)
X(99) = trilinear product of vertices of circumcircle antipode of circumorthic triangle
X(99) = 1st-Parry-to-ABC similarity image of X(2)
X(99) = crossdifference of PU(105)
X(99) = X(1691) of 6th Brocard triangle
X(99) = eigencenter of circummedial triangle
X(99) = eigencenter of circumsymmedial triangle
X(99) = perspector of ABC and cross-triangle of circumcevian triangles of PU(1)
X(99) = X(98)-of-anti-Artzt-triangle
X(99) = X(2)-of-1st-anti-Parry-triangle
X(99) = Thomson-isogonal conjugate of X(511)
X(99) = Lucas-isogonal conjugate of X(511)
X(99) = X(1379)-of-circummedial-triangle
X(99) = X(6323)-of-circumsymmedial-triangle
X(99) = Kiepert image of X(2)
X(99) = Cundy-Parry Phi transform of X(14265)
X(99) = intersection of antipedal lines of X(1379) and X(1380)
X(99) = homothetic center of anticomplementary triangle and mid-triangle of antipedal triangles of X(13) and X(14)
X(99) = barycentric square root of X(4590)
X(99) = barycentric product of circumcircle intercepts of line X(2)X(39)
X(99) = perspector of ABC and vertex triangle of 1st and 2nd isodynamic-Dao triangles
X(99) = orthic-to-ABC functional image of X(130)
X(99) = tangential-isogonal conjugate of X(33704)
X(99) = trilinear pole, wrt circumtangential triangle, of Brocard axis
X(99) = Vu circlecevian point of PU(1)
X(99) = Vu circlecevian point of PU(37)
X(99) = areal center of pedal triangles of X(15) and X(16)
X(99) = areal center of pedal triangles of PU(2)
X(99) = areal center of cevian triangles of PU(40)
X(99) = X(2)-Ceva conjugate of X(31998)
X(99) = perspector of circumconic centered at X(31998)
X(99) = Cundy-Parry Psi transform of X(34157)
X(99) = X(39156)-of-orthic-triangle if ABC is acute
X(99) = Conway-circle-inverse of X(38477)
X(99) = Steiner-circumellipse-X(1)-antipode of X(18827)
X(99) = Steiner-circumellipse-X(3)-antipode of X(290)
X(99) = Steiner-circumellipse-X(4)-antipode of X(35142)
X(99) = Steiner-circumellipse-X(6)-antipode of X(3228)


X(100) = ANTICOMPLEMENT OF FEUERBACH POINT

Trilinears    1/(b - c) : :
Trilinears    (a - b)(a - c) : :
Barycentrics   a*(a - b)*(a - c) : :
Barycentrics  a/(b - c) : :
Tripolars    |b - c| : :
X(100) = 2R*X(1) - 3R*X(2) + 2r*X(3), 3 X[1] - X[12653], X[1] - 3 X[15015], 4 X[214] - X[1320], 2 X[214] + X[5541], 6 X[214] - X[12653], 2 X[214] - 3 X[15015], 2 X[1054] - 3 X[14193], X[1320] + 2 X[5541], 3 X[1320] - 2 X[12653], X[1320] - 6 X[15015], 3 X[5541] + X[12653], X[5541] + 3 X[15015], 3 X[10087] - 2 X[25439], X[12653] - 9 X[15015], 3 X[13278] - 4 X[25439], X[13279] - 4 X[25440], 3 X[2] - 4 X[3035], 3 X[2] + 2 X[6154], 9 X[2] - 8 X[6667], 3 X[2] + X[20095], 9 X[2] - 10 X[31235], 6 X[2] - 5 X[31272], 3 X[2] - 8 X[35023], 5 X[2] - 4 X[45310], X[11] - 3 X[6174], 3 X[11] - 4 X[6667], 4 X[11] - 3 X[10707], 2 X[11] + X[20095], 3 X[11] - 5 X[31235], 4 X[11] - 5 X[31272], X[11] - 4 X[35023], 5 X[11] - 6 X[45310], X[149] - 4 X[3035], X[149] + 2 X[6154], X[149] - 6 X[6174], 3 X[149] - 8 X[6667], 2 X[149] - 3 X[10707], 3 X[149] - 10 X[31235], 2 X[149] - 5 X[31272], X[149] - 8 X[35023], 5 X[149] - 12 X[45310], 2 X[3035] + X[6154], 2 X[3035] - 3 X[6174], 3 X[3035] - 2 X[6667], 8 X[3035] - 3 X[10707], 4 X[3035] + X[20095], 6 X[3035] - 5 X[31235], 8 X[3035] - 5 X[31272], 5 X[3035] - 3 X[45310], and many more

Let LA be the reflection of the line X(1)X(3) in line BC, and define LB and LC cyclically. Let A' = LB∩LC, B' = LC∩LA, C' = LA∩LB. The lines AA', BB', CC' concur in X(100). (Randy Hutson, 9/23/2011)

Let Ia be the reflection of X(1) in the perpendicular bisector of BC, and define Ib and Ic cyclically; then X(100) = X(36)-of-IaIbIc. Also, let P be a point on line X(4)X(8) other than X(4). Let A' be the reflection of P in BC, and define B' and C' cyclically. The circumcircles of AB'C', BC'A', and CA'B' concur at X(100). (Randy Hutson, 9/5/2015)

Let IaIbIc be the excentral triangle. The Euler lines of triangles BCIa, CAIb, ABIc concur in X(100). (Randy Hutson, June 27, 2018)

X(100) lies on the circumcircle, the circumcopnic with center X(9), the incircle of anticomplementary triangle, the cubics K299, K603, K661, K662, K817, K889, K1122, and these lines: {1, 88}, {2, 11}, {3, 8}, {4, 119}, {5, 10738}, {6, 739}, {7, 1004}, {9, 1005}, {10, 21}, {12, 2475}, {19, 7466}, {20, 153}, {22, 197}, {23, 2752}, {25, 1862}, {28, 5174}, {30, 2687}, {31, 43}, {32, 713}, {33, 35973}, {34, 35974}, {36, 519}, {37, 111}, {38, 3961}, {39, 32454}, {40, 78}, {41, 3501}, {42, 81}, {44, 2384}, {45, 9330}, {46, 224}, {56, 145}, {57, 1280}, {58, 3293}, {59, 521}, {63, 103}, {65, 12739}, {69, 5848}, {71, 2249}, {72, 74}, {75, 675}, {76, 767}, {79, 35982}, {85, 2369}, {86, 29822}, {89, 39428}, {92, 917}, {98, 228}, {99, 668}, {101, 644}, {107, 823}, {108, 653}, {109, 651}, {110, 643}, {112, 162}, {113, 10767}, {114, 10768}, {115, 10769}, {116, 10770}, {117, 10771}, {118, 10772}, {121, 10774}, {122, 10775}, {123, 10776}, {124, 10777}, {125, 10778}, {126, 10779}, {127, 10780}, {140, 1484}, {141, 33086}, {142, 2346}, {144, 480}, {146, 12327}, {147, 12178}, {148, 13173}, {169, 25082}, {172, 20691}, {182, 12199}, {183, 4441}, {184, 3045}, {187, 5291}, {189, 35987}, {190, 659}, {191, 3678}, {192, 9082}, {194, 12338}, {198, 346}, {199, 3969}, {209, 40571}, {210, 3219}, {212, 25308}, {213, 729}, {219, 23707}, {226, 3256}, {227, 4296}, {229, 3178}, {230, 17737}, {238, 899}, {239, 2223}, {242, 35993}, {260, 7588}, {274, 2368}, {278, 2376}, {279, 2377}, {281, 1013}, {283, 35995}, {286, 39438}, {292, 3121}, {294, 6184}, {306, 1817}, {312, 1311}, {313, 37219}, {314, 40600}, {318, 7412}, {319, 1444}, {322, 17134}, {325, 2856}, {329, 972}, {333, 4184}, {341, 2370}, {344, 1486}, {350, 9073}, {354, 3957}, {355, 6906}, {371, 19082}, {372, 19081}, {376, 3421}, {377, 3085}, {381, 22938}, {382, 22799}, {384, 26752}, {385, 17759}, {386, 3032}, {387, 36000}, {388, 4190}, {394, 7074}, {402, 13268}, {405, 5175}, {409, 21674}, {442, 943}, {474, 1387}, {476, 4036}, {477, 36001}, {484, 758}, {485, 35772}, {486, 35773}, {487, 12343}, {488, 12344}, {491, 26513}, {492, 26512}, {493, 13275}, {494, 13276}, {495, 11112}, {496, 13747}, {498, 2476}, {499, 5533}, {511, 2699}, {512, 2703}, {513, 765}, {514, 1308}, {515, 2077}, {516, 908}, {517, 953}, {518, 840}, {520, 2719}, {522, 655}, {523, 1290}, {524, 2721}, {525, 2722}, {527, 30295}, {529, 15326}, {535, 4316}, {536, 4396}, {545, 38530}, {548, 38754}, {550, 38753}, {551, 36006}, {560, 697}, {573, 29068}, {574, 16975}, {593, 6043}, {594, 1030}, {595, 3216}, {601, 37699}, {604, 3169}, {610, 3692}, {612, 17594}, {614, 3749}, {616, 12337}, {617, 12336}, {627, 22558}, {628, 22557}, {631, 5082}, {645, 931}, {646, 8707}, {648, 36077}, {649, 660}, {650, 919}, {656, 39026}, {658, 664}, {661, 2702}, {666, 31150}, {667, 898}, {669, 9362}, {672, 3684}, {677, 4131}, {689, 6386}, {691, 4567}, {693, 927}, {699, 2205}, {701, 18900}, {715, 1918}, {717, 40728}, {719, 41267}, {726, 32845}, {728, 2371}, {731, 869}, {733, 893}, {735, 14620}, {740, 3724}, {743, 2276}, {745, 21035}, {748, 8616}, {751, 20973}, {752, 32843}, {753, 984}, {755, 3954}, {756, 846}, {761, 3661}, {788, 35009}, {789, 874}, {805, 4603}, {825, 1492}, {827, 4599}, {831, 4568}, {835, 4033}, {839, 27808}, {842, 37959}, {843, 21839}, {850, 2864}, {851, 3936}, {855, 36926}, {872, 39441}, {883, 6183}, {891, 39443}, {894, 6015}, {896, 1757}, {903, 24405}, {905, 35350}, {910, 3693}, {912, 43078}, {929, 4391}, {932, 4595}, {935, 7476}, {936, 5250}, {938, 26062}, {940, 17018}, {941, 34261}, {942, 45395}, {946, 6915}, {950, 24982}, {954, 33993}, {958, 3036}, {960, 17638}, {962, 1537}, {964, 19763}, {968, 5268}, {971, 17613}, {976, 986}, {978, 3915}, {982, 3938}, {983, 41886}, {985, 3795}, {993, 3679}, {995, 37610}, {997, 3877}, {999, 3241}, {1006, 3419}, {1010, 26115}, {1011, 5278}, {1014, 3879}, {1037, 30619}, {1043, 4225}, {1045, 20964}, {1052, 45233}, {1055, 35291}, {1056, 11239}, {1058, 17567}, {1078, 17143}, {1086, 17724}, {1089, 2372}, {1100, 8700}, {1110, 1734}, {1113, 2580}, {1114, 2581}, {1125, 3746}, {1149, 12029}, {1158, 12528}, {1191, 28280}, {1193, 5255}, {1201, 37588}, {1211, 33083}, {1215, 4418}, {1220, 4267}, {1224, 27787}, {1229, 26268}, {1250, 5362}, {1253, 1958}, {1255, 1929}, {1257, 18673}, {1261, 7293}, {1265, 3556}, {1267, 9098}, {1268, 46896}, {1270, 11498}, {1271, 11497}, {1276, 44689}, {1277, 44688}, {1279, 7292}, {1284, 4442}, {1289, 4244}, {1292, 2414}, {1293, 2827}, {1294, 2828}, {1296, 2830}, {1297, 2831}, {1299, 31384}, {1300, 7414}, {1301, 7435}, {1302, 42716}, {1304, 5379}, {1309, 4397}, {1310, 4561}, {1312, 10782}, {1313, 10781}, {1319, 3880}, {1324, 16086}, {1325, 12030}, {1326, 30576}, {1329, 5046}, {1333, 21858}, {1334, 35106}, {1335, 9679}, {1375, 28757}, {1381, 3307}, {1382, 3308}, {1385, 4861}, {1386, 17012}, {1388, 10912}, {1389, 19920}, {1402, 1999}, {1403, 3210}, {1414, 4614}, {1415, 8687}, {1420, 2136}, {1421, 43048}, {1429, 19589}, {1445, 1998}, {1449, 17223}, {1458, 9364}, {1462, 43063}, {1465, 4318}, {1466, 3600}, {1468, 37603}, {1470, 3476}, {1476, 12640}, {1478, 17579}, {1479, 4193}, {1482, 6924}, {1483, 37535}, {1490, 2057}, {1499, 2746}, {1500, 2375}, {1503, 2747}, {1575, 1914}, {1577, 2690}, {1580, 3507}, {1593, 12138}, {1610, 2933}, {1612, 1714}, {1613, 21780}, {1615, 38876}, {1616, 28370}, {1617, 5435}, {1618, 6099}, {1624, 4243}, {1631, 17233}, {1634, 8050}, {1644, 31171}, {1657, 38756}, {1697, 3890}, {1698, 5047}, {1699, 15017}, {1706, 3601}, {1707, 28527}, {1708, 2900}, {1726, 29306}, {1737, 10073}, {1738, 3011}, {1739, 30117}, {1740, 2209}, {1743, 17222}, {1754, 3190}, {1761, 3949}, {1764, 35614}, {1766, 12530}, {1769, 46119}, {1770, 21077}, {1771, 3562}, {1788, 3189}, {1791, 3704}, {1792, 7270}, {1812, 22276}, {1813, 8059}, {1814, 24499}, {1816, 3682}, {1818, 1936}, {1824, 3563}, {1836, 31053}, {1837, 12743}, {1908, 3774}, {1931, 12031}, {1959, 2700}, {1983, 29044}, {2071, 2694}, {2078, 3911}, {2082, 26690}, {2092, 2298}, {2149, 32689}, {2161, 40988}, {2178, 17314}, {2220, 33760}, {2229, 41333}, {2238, 17735}, {2239, 3783}, {2265, 36278}, {2280, 17754}, {2284, 8693}, {2292, 5293}, {2295, 18755}, {2308, 28517}, {2320, 24297}, {2330, 15988}, {2340, 9441}, {2344, 19584}, {2345, 36744}, {2347, 23617}, {2352, 3187}, {2365, 3719}, {2367, 27801}, {2373, 20336}, {2374, 7438}, {2380, 42680}, {2381, 42677}, {2397, 9058}, {2427, 32722}, {2478, 4294}, {2481, 40619}, {2551, 6872}, {2611, 16598}, {2635, 23693}, {2646, 5836}, {2651, 37783}, {2664, 3747}, {2688, 14206}, {2689, 4086}, {2692, 4404}, {2698, 5360}, {2701, 4041}, {2705, 4729}, {2708, 6211}, {2711, 20683}, {2715, 36084}, {2723, 7360}, {2724, 28058}, {2725, 3912}, {2726, 3685}, {2727, 24018}, {2728, 6332}, {2729, 14210}, {2730, 44448}, {2734, 10538}, {2736, 4468}, {2737, 4462}, {2740, 14207}, {2742, 3309}, {2743, 3667}, {2744, 6000}, {2745, 6001}, {2748, 4401}, {2751, 3220}, {2757, 4723}, {2758, 3992}, {2766, 37964}, {2770, 42713}, {2796, 21093}, {2810, 3937}, {2841, 38512}, {2857, 42703}, {2859, 3267}, {2860, 3261}, {2861, 3262}, {2862, 3263}, {2863, 3264}, {2887, 29846}, {2888, 12341}, {2893, 40999}, {2895, 41811}, {2896, 11494}, {2899, 4186}, {2915, 3695}, {2991, 20455}, {3006, 32850}, {3025, 31877}, {3030, 3271}, {3051, 21792}, {3052, 4383}, {3057, 12740}, {3059, 38451}, {3065, 3647}, {3068, 13922}, {3069, 13991}, {3086, 6921}, {3090, 23513}, {3091, 11496}, {3098, 12499}, {3100, 9371}, {3120, 17719}, {3126, 5377}, {3145, 38903}, {3146, 27525}, {3193, 14868}, {3196, 4370}, {3197, 38875}, {3207, 4513}, {3208, 9310}, {3214, 5247}, {3222, 36860}, {3231, 21788}, {3234, 3239}, {3242, 4392}, {3244, 5563}, {3245, 4867}, {3251, 5376}, {3259, 31512}, {3286, 16704}, {3303, 3622}, {3304, 3623}, {3305, 4512}, {3333, 8000}, {3336, 3874}, {3337, 3881}, {3338, 3889}, {3339, 11520}, {3359, 18446}, {3361, 4917}, {3413, 36735}, {3414, 36736}, {3416, 33077}, {3428, 22775}, {3448, 13204}, {3474, 5905}, {3486, 5554}, {3488, 37249}, {3496, 33299}, {3509, 3930}, {3522, 38759}, {3523, 12777}, {3526, 34126}, {3555, 37582}, {3560, 5818}, {3565, 22280}, {3576, 3872}, {3582, 6681}, {3583, 3814}, {3584, 3822}, {3612, 3897}, {3621, 5204}, {3624, 17535}, {3625, 5288}, {3626, 5258}, {3632, 7280}, {3634, 5259}, {3648, 5951}, {3660, 37789}, {3662, 33122}, {3666, 3920}, {3675, 10699}, {3680, 45036}, {3683, 3740}, {3687, 5314}, {3694, 5279}, {3697, 31445}, {3700, 9090}, {3701, 7283}, {3703, 6636}, {3705, 5014}, {3711, 5220}, {3713, 37499}, {3715, 33519}, {3720, 3750}, {3723, 28338}, {3729, 24309}, {3739, 9110}, {3741, 32918}, {3742, 3748}, {3744, 3752}, {3745, 17011}, {3753, 6797}, {3756, 43055}, {3757, 4359}, {3759, 3941}, {3762, 39444}, {3771, 25957}, {3772, 29665}, {3780, 33863}, {3782, 33102}, {3784, 23155}, {3797, 8628}, {3802, 25800}, {3812, 37080}, {3813, 5433}, {3820, 11113}, {3821, 32775}, {3825, 4857}, {3828, 16861}, {3831, 35206}, {3832, 38758}, {3836, 29632}, {3840, 32943}, {3841, 31254}, {3846, 32947}, {3851, 38141}, {3876, 12514}, {3878, 11010}, {3884, 37563}, {3893, 11256}, {3900, 4564}, {3903, 29055}, {3914, 33133}, {3916, 28173}, {3918, 35016}, {3919, 5425}, {3923, 32931}, {3924, 24440}, {3927, 28145}, {3940, 28159}, {3943, 19297}, {3944, 33094}, {3948, 46501}, {3951, 28149}, {3968, 5426}, {3971, 32936}, {3980, 29670}, {3983, 5302}, {3984, 12511}, {3989, 28499}, {3990, 26717}, {3996, 4210}, {3999, 4864}, {4000, 26228}, {4023, 37656}, {4030, 15246}, {4042, 5361}, {4043, 26266}, {4062, 32846}, {4068, 27811}, {4069, 4606}, {4076, 6079}, {4080, 19636}, {4083, 43362}, {4084, 41696}, {4085, 29631}, {4090, 32938}, {4094, 5147}, {4105, 9358}, {4115, 15322}, {4123, 20243}, {4187, 15171}, {4191, 10453}, {4192, 4388}, {4197, 10198}, {4199, 41809}, {4200, 11398}, {4222, 40101}, {4224, 29641}, {4228, 33116}, {4240, 11848}, {4250, 26705}, {4251, 16549}, {4254, 5749}, {4255, 5710}, {4261, 9079}, {4262, 16788}, {4265, 33170}, {4293, 34605}, {4297, 6736}, {4302, 11114}, {4308, 8278}, {4312, 31164}, {4313, 37248}, {4314, 8582}, {4319, 26669}, {4342, 34639}, {4360, 20990}, {4362, 32860}, {4363, 24344}, {4367, 6631}, {4378, 4555}, {4384, 23407}, {4385, 45136}, {4393, 21010}, {4394, 6078}, {4415, 33100}, {4416, 41430}, {4417, 6327}, {4422, 16686}, {4424, 30115}, {4430, 23958}, {4432, 9458}, {4433, 4447}, {4437, 20468}, {4438, 33117}, {4440, 21320}, {4454, 24328}, {4458, 6742}, {4467, 4477}, {4471, 17354}, {4482, 23891}, {4487, 23205}, {4497, 17377}, {4498, 28520}, {4559, 32693}, {4562, 4586}, {4569, 6606}, {4570, 4636}, {4572, 34083}, {4574, 36080}, {4575, 36050}, {4585, 4588}, {4589, 4600}, {4596, 6578}, {4601, 9150}, {4617, 41353}, {4618, 14421}, {4621, 8684}, {4641, 4849}, {4646, 17016}, {4650, 28488}, {4652, 28193}, {4655, 33065}, {4660, 25760}, {4661, 27778}, {4663, 21870}, {4666, 5437}, {4671, 5695}, {4678, 17548}, {4682, 9507}, {4685, 4921}, {4687, 9094}, {4696, 22345}, {4722, 9340}, {4730, 39155}, {4737, 23206}, {4738, 39445}, {4757, 16126}, {4760, 24358}, {4763, 8645}, {4807, 39577}, {4825, 5385}, {4834, 29341}, {4847, 5659}, {4848, 12437}, {4853, 7987}, {4860, 42871}, {4865, 29849}, {4872, 33864}, {4876, 19557}, {4882, 16192}, {4893, 28875}, {4904, 26140}, {4919, 17439}, {4970, 32928}, {4973, 5131}, {4981, 38000}, {4999, 37291}, {5011, 38884}, {5012, 20986}, {5015, 37431}, {5016, 37399}, {5030, 45751}, {5061, 35104}, {5067, 38319}, {5078, 5096}, {5081, 37305}, {5088, 30806}, {5090, 12137}, {5091, 14839}, {5124, 17362}, {5125, 41227}, {5126, 35271}, {5128, 11523}, {5154, 10896}, {5172, 5427}, {5177, 10585}, {5178, 6684}, {5180, 28174}, {5183, 44663}, {5187, 5225}, {5219, 8543}, {5222, 21477}, {5227, 38883}, {5229, 31295}, {5230, 37030}, {5249, 13405}, {5252, 18976}, {5256, 5269}, {5261, 37435}, {5262, 5266}, {5265, 12632}, {5273, 20835}, {5275, 31477}, {5283, 31451}, {5287, 37553}, {5300, 37231}, {5308, 16412}, {5310, 33157}, {5311, 17592}, {5324, 33118}, {5328, 30332}, {5329, 33088}, {5330, 5697}, {5333, 43223}, {5347, 33093}, {5367, 10638}, {5384, 5386}, {5391, 9099}, {5445, 14795}, {5450, 5881}, {5520, 36175}, {5540, 24036}, {5550, 16408}, {5558, 12631}, {5584, 20007}, {5587, 6246}, {5597, 13228}, {5598, 13230}, {5601, 11492}, {5602, 11493}, {5603, 6911}, {5638, 11651}, {5639, 11652}, {5688, 6262}, {5689, 6263}, {5693, 40256}, {5698, 31018}, {5701, 38347}, {5703, 37229}, {5709, 9946}, {5711, 19767}, {5718, 33112}, {5727, 34701}, {5730, 12702}, {5739, 7085}, {5745, 25006}, {5748, 9812}, {5790, 6914}, {5815, 37426}, {5839, 36743}, {5842, 6840}, {5844, 22765}, {5880, 17718}, {5882, 37561}, {5886, 6946}, {5901, 45976}, {5903, 22836}, {5904, 37572}, {5921, 39877}, {5966, 21807}, {6011, 22003}, {6012, 33951}, {6049, 41426}, {6060, 6617}, {6075, 22102}, {6126, 46819}, {6161, 6551}, {6193, 12328}, {6194, 22556}, {6200, 35856}, {6223, 12330}, {6225, 12335}, {6253, 6895}, {6260, 46435}, {6361, 6985}, {6396, 35857}, {6462, 11503}, {6463, 11504}, {6540, 32042}, {6558, 6574}, {6584, 8702}, {6595, 7161}, {6596, 17097}, {6599, 12639}, {6604, 38859}, {6605, 8012}, {6645, 17693}, {6648, 32038}, {6666, 46916}, {6679, 29850}, {6685, 32772}, {6691, 37722}, {6700, 10624}, {6733, 45874}, {6737, 43174}, {6765, 15803}, {6767, 16417}, {6789, 34587}, {6827, 32554}, {6830, 37820}, {6839, 7680}, {6845, 18517}, {6850, 10522}, {6853, 31659}, {6871, 10588}, {6875, 38128}, {6876, 35239}, {6885, 10532}, {6891, 12116}, {6910, 19843}, {6918, 38038}, {6920, 9956}, {6931, 10591}, {6933, 31418}, {6934, 40245}, {6937, 26487}, {6941, 10525}, {6942, 11249}, {6944, 10531}, {6945, 26333}, {6948, 12115}, {6950, 22758}, {6952, 26470}, {6960, 15908}, {6961, 10785}, {6979, 7681}, {6981, 10598}, {6996, 20556}, {6999, 27526}, {7012, 36067}, {7017, 39429}, {7035, 8640}, {7046, 37441}, {7077, 14200}, {7083, 11345}, {7095, 40214}, {7098, 41538}, {7109, 21779}, {7110, 21065}, {7115, 32688}, {7123, 30706}, {7226, 28567}, {7262, 28502}, {7287, 29345}, {7288, 8668}, {7291, 25083}, {7320, 22754}, {7354, 12607}, {7373, 17573}, {7377, 28789}, {7427, 24808}, {7450, 23181}, {7461, 26704}, {7483, 31419}, {7489, 38042}, {7493, 28420}, {7504, 25639}, {7508, 38112}, {7538, 27410}, {7549, 14679}, {7585, 19000}, {7586, 18999}, {7587, 12748}, {7589, 8126}, {7671, 8257}, {7674, 8730}, {7705, 10826}, {7783, 21226}, {7787, 11490}, {7824, 26801}, {7951, 17577}, {7967, 10269}, {7982, 25485}, {7984, 31525}, {7989, 38161}, {7991, 11682}, {8021, 40435}, {8025, 18185}, {8027, 38018}, {8052, 21295}, {8053, 17277}, {8054, 46126}, {8069, 18391}, {8075, 8103}, {8076, 8104}, {8077, 8097}, {8107, 11685}, {8108, 11686}, {8109, 12733}, {8110, 12734}, {8168, 11194}, {8193, 9912}, {8197, 12460}, {8204, 12461}, {8214, 12741}, {8215, 12742}, {8224, 11687}, {8225, 12744}, {8227, 16174}, {8256, 10950}, {8270, 17080}, {8271, 17092}, {8273, 15717}, {8545, 47375}, {8583, 9951}, {8591, 12326}, {8641, 10006}, {8649, 9283}, {8652, 35327}, {8691, 37210}, {8696, 16885}, {8701, 37212}, {8728, 26060}, {8750, 32691}, {8758, 37782}, {8817, 13577}, {8844, 33889}, {8852, 18235}, {8935, 21085}, {8972, 13887}, {8988, 13893}, {9056, 42718}, {9057, 42719}, {9059, 24004}, {9067, 41314}, {9075, 33931}, {9076, 37221}, {9077, 17289}, {9078, 29679}, {9083, 18743}, {9084, 42724}, {9086, 21580}, {9095, 30829}, {9103, 28605}, {9105, 19804}, {9271, 9272}, {9305, 30694}, {9317, 21232}, {9318, 24685}, {9369, 22344}, {9471, 17755}, {9509, 20998}, {9534, 16452}, {9540, 13913}, {9580, 30827}, {9623, 30282}, {9668, 17556}, {9670, 31246}, {9708, 16370}, {9710, 24953}, {9776, 10578}, {9841, 12125}, {9857, 12498}, {9859, 41229}, {9874, 12333}, {9913, 11414}, {9957, 17614}, {10025, 39421}, {10032, 17781}, {10039, 10057}, {10165, 34486}, {10167, 17658}, {10174, 10536}, {10199, 34719}, {10222, 45977}, {10315, 40129}, {10386, 17527}, {10423, 36095}, {10434, 11679}, {10441, 45394}, {10448, 37574}, {10449, 16451}, {10595, 37622}, {10647, 37794}, {10648, 37795}, {10791, 12198}, {10860, 11678}, {10861, 15298}, {10882, 10890}, {10884, 37560}, {10915, 12749}, {10916, 12750}, {11012, 11362}, {11061, 32256}, {11101, 27690}, {11108, 19877}, {11110, 19874}, {11124, 30613}, {11236, 12943}, {11246, 17483}, {11263, 37731}, {11274, 37587}, {11329, 17316}, {11343, 29611}, {11350, 34255}, {11358, 19684}, {11512, 28011}, {11683, 45744}, {11689, 15323}, {11691, 12518}, {11814, 24709}, {11822, 12462}, {11823, 12463}, {11824, 12753}, {11825, 12754}, {11826, 12761}, {11828, 12765}, {11829, 12766}, {11900, 12729}, {12334, 12383}, {12340, 12384}, {12342, 12849}, {12387, 12389}, {12410, 37257}, {12432, 15932}, {12512, 12527}, {12516, 12533}, {12517, 12534}, {12519, 12535}, {12609, 33593}, {12611, 12699}, {12635, 37567}, {12647, 14793}, {12680, 46677}, {12701, 25681}, {12780, 40714}, {12781, 40713}, {12848, 18801}, {13145, 33858}, {13206, 13219}, {13245, 28162}, {13256, 36082}, {13595, 20988}, {13624, 32634}, {13675, 13678}, {13740, 26030}, {13743, 18357}, {13744, 38950}, {13795, 13798}, {13883, 19078}, {13935, 13977}, {13936, 19077}, {13940, 13941}, {13947, 13976}, {14008, 44411}, {14074, 45695}, {14189, 37780}, {14204, 18359}, {14213, 26708}, {14459, 17772}, {14511, 38707}, {14621, 40732}, {14667, 14686}, {14716, 34409}, {14969, 14996}, {14997, 30653}, {15175, 17057}, {15253, 37771}, {15254, 35595}, {15338, 15680}, {15440, 32656}, {15485, 17125}, {15507, 17777}, {15519, 23089}, {15523, 33079}, {15556, 20612}, {15569, 17021}, {15587, 15837}, {15625, 23361}, {15702, 38069}, {15703, 38084}, {15726, 43080}, {15733, 37787}, {16056, 18139}, {16058, 26038}, {16061, 26965}, {16113, 21075}, {16202, 38032}, {16342, 19853}, {16378, 17794}, {16395, 37507}, {16405, 37502}, {16409, 26103}, {16465, 41539}, {16477, 21747}, {16484, 30950}, {16552, 24047}, {16574, 33847}, {16592, 21341}, {16613, 28282}, {16666, 28310}, {16667, 28314}, {16669, 28298}, {16670, 28302}, {16687, 17150}, {16693, 20475}, {16706, 26230}, {16777, 28326}, {16778, 17156}, {16814, 28334}, {16823, 24589}, {16826, 25946}, {16828, 17557}, {16858, 19875}, {16859, 46932}, {16865, 46933}, {16872, 21278}, {16884, 28330}, {16920, 26687}, {16968, 39255}, {17017, 17716}, {17025, 38315}, {17061, 33150}, {17063, 17715}, {17124, 17782}, {17142, 18048}, {17147, 32926}, {17154, 24841}, {17155, 32920}, {17165, 32939}, {17184, 33068}, {17234, 29830}, {17263, 26261}, {17264, 26262}, {17280, 23868}, {17292, 21516}, {17349, 20992}, {17353, 35263}, {17367, 21540}, {17388, 21773}, {17449, 18201}, {17469, 29821}, {17475, 25804}, {17484, 17768}, {17494, 43986}, {17495, 20045}, {17532, 31479}, {17541, 27091}, {17546, 25542}, {17553, 19870}, {17570, 46931}, {17574, 38213}, {17593, 46901}, {17599, 29815}, {17600, 29816}, {17602, 33155}, {17603, 30284}, {17654, 31788}, {17668, 29007}, {17682, 28742}, {17686, 27020}, {17697, 26029}, {17717, 33104}, {17720, 33134}, {17725, 33143}, {17752, 18758}, {17761, 25532}, {17766, 32844}, {17889, 33127}, {17943, 40501}, {18064, 34020}, {18108, 36081}, {18166, 40433}, {18191, 22313}, {18265, 40848}, {18480, 21669}, {18481, 37403}, {18518, 35251}, {18519, 34627}, {18621, 35260}, {18642, 43735}, {18750, 41905}, {19314, 39581}, {19537, 20050}, {19582, 28077}, {19785, 37099}, {19822, 37090}, {19862, 33709}, {19876, 38104}, {19916, 44805}, {20011, 37639}, {20012, 37683}, {20103, 40998}, {20104, 31262}, {20118, 24914}, {20173, 26267}, {20323, 32427}, {20352, 20878}, {20470, 29824}, {20670, 24504}, {20777, 25298}, {20794, 25311}, {20846, 37601}, {20887, 29010}, {20967, 27064}, {20972, 40400}, {20974, 24484}, {21000, 37679}, {21002, 37681}, {21004, 21711}, {21026, 29862}, {21061, 37508}, {21081, 37294}, {21105, 24126}, {21221, 26075}, {21231, 24435}, {21285, 33298}, {21302, 36030}, {21553, 31546}, {21740, 37562}, {21842, 22837}, {21856, 28055}, {21891, 22311}, {22060, 29308}, {22149, 29228}, {22277, 41610}, {22300, 41723}, {22329, 37857}, {22559, 22647}, {22791, 37251}, {23363, 43350}, {23374, 29437}, {23600, 37419}, {23622, 24578}, {23844, 25253}, {23865, 27013}, {23969, 36096}, {24052, 28841}, {24165, 32923}, {24169, 29656}, {24170, 33953}, {24174, 28082}, {24178, 28027}, {24248, 33151}, {24318, 24712}, {24320, 27549}, {24392, 31231}, {24403, 26273}, {24447, 30997}, {24477, 37578}, {24612, 37416}, {24627, 37575}, {24635, 28043}, {24703, 27131}, {24723, 26580}, {24789, 29681}, {24850, 24852}, {24892, 32865}, {24943, 33174}, {25066, 33950}, {25312, 43360}, {25557, 37703}, {25737, 30236}, {25768, 25872}, {25940, 37555}, {25959, 30811}, {25961, 29642}, {26034, 32782}, {26128, 29848}, {26321, 37705}, {26393, 26394}, {26417, 26418}, {26493, 26494}, {26502, 26503}, {26611, 38357}, {26641, 35185}, {26733, 36074}, {26744, 36910}, {26892, 29326}, {26893, 29009}, {27097, 33828}, {27184, 32950}, {27248, 33830}, {27299, 33819}, {27518, 36855}, {27558, 37405}, {27622, 30029}, {27804, 34064}, {27805, 30670}, {28070, 41795}, {28125, 28869}, {28444, 38074}, {28523, 42043}, {28563, 36263}, {28795, 36698}, {29349, 38389}, {29473, 40006}, {29511, 46502}, {29615, 35276}, {29627, 37272}, {29634, 32774}, {29649, 32915}, {29658, 33128}, {29662, 33141}, {29671, 33072}, {29673, 33119}, {29674, 33156}, {29678, 33111}, {29683, 33135}, {29687, 33158}, {29814, 37674}, {29840, 41346}, {30147, 37571}, {30247, 37217}, {30626, 32735}, {30725, 46116}, {30733, 41507}, {30913, 32666}, {30942, 32941}, {30996, 40546}, {31141, 34626}, {31143, 33082}, {31204, 33138}, {31330, 32916}, {31385, 39437}, {31423, 38133}, {31544, 31547}, {31545, 31548}, {31798, 40262}, {31847, 38569}, {32076, 42014}, {32157, 32198}, {32347, 32354}, {32468, 41526}, {32612, 37727}, {32636, 34791}, {32641, 32685}, {32665, 32686}, {32681, 36083}, {32683, 36088}, {32684, 36039}, {32687, 36092}, {32690, 36097}, {32710, 37979}, {32778, 33074}, {32781, 32783}, {32847, 32848}, {32854, 32855}, {32856, 32857}, {32925, 32934}, {33064, 33067}, {33080, 33084}, {33081, 33085}, {33098, 33101}, {33105, 33109}, {33107, 37662}, {33136, 33140}, {33139, 35466}, {33142, 37646}, {33144, 33146}, {33145, 33152}, {33161, 33165}, {33162, 33167}, {33166, 44416}, {33171, 33172}, {33667, 41697}, {33956, 44784}, {33994, 42884}, {34140, 37758}, {34168, 42699}, {34404, 39451}, {34606, 37299}, {34880, 37738}, {34921, 35057}, {34927, 34932}, {35004, 37733}, {35249, 37430}, {35616, 35638}, {35657, 35659}, {35788, 35852}, {35789, 35853}, {36154, 38570}, {36508, 38286}, {36565, 37549}, {36977, 40293}, {37254, 39570}, {37262, 37581}, {37264, 37547}, {37308, 37730}, {37371, 37799}, {37442, 37573}, {37557, 39582}, {37604, 42042}, {37658, 42316}, {38711, 46636}, {40097, 46588}, {40216, 40419}, {40300, 40302}, {43816, 43847}, {43974, 43991}, {45269, 45272}, {45508, 45520}, {45509, 45521}

X(100) = midpoint of X(i) and X(j) for these {i,j}: {1, 5541}, {3, 12331}, {4, 13199}, {8, 6224}, {9, 5528}, {11, 6154}, {20, 153}, {40, 6326}, {104, 38665}, {119, 10993}, {149, 20095}, {191, 13146}, {901, 14513}, {1145, 10609}, {1155, 3689}, {1317, 13996}, {1490, 2950}, {1657, 38756}, {1768, 5531}, {2932, 5687}, {3218, 3935}, {3245, 4867}, {3913, 22560}, {5537, 44425}, {7991, 13253}, {11500, 12332}, {12119, 12751}, {12515, 12738}, {13269, 13270}, {18524, 35000}, {24466, 37725}, {32845, 32927}
X(100) = reflection of X(i) in X(j) for these {i,j}: {1, 214}, {2, 6174}, {3, 33814}, {4, 119}, {7, 10427}, {8, 1145}, {9, 6594}, {11, 3035}, {20, 24466}, {21, 35204}, {80, 10}, {104, 3}, {105, 46409}, {144, 6068}, {145, 1317}, {149, 11}, {153, 37725}, {382, 22799}, {765, 46973}, {908, 6745}, {962, 1537}, {1156, 9}, {1290, 36167}, {1320, 1}, {1482, 19907}, {1484, 140}, {1768, 46684}, {2611, 16598}, {2687, 46635}, {2975, 4996}, {3035, 35023}, {3065, 3647}, {3218, 1155}, {3244, 33812}, {3254, 142}, {3583, 3814}, {3868, 11570}, {3870, 41553}, {3935, 3689}, {4511, 5440}, {5057, 908}, {5080, 17757}, {5176, 6735}, {5375, 38310}, {5905, 12831}, {6075, 22102}, {6163, 765}, {6224, 10609}, {6264, 11715}, {6265, 22935}, {6595, 13089}, {6599, 12639}, {6909, 2077}, {7972, 33337}, {7982, 25485}, {7984, 31525}, {9318, 24685}, {9809, 13257}, {9897, 15863}, {10090, 25440}, {10265, 6684}, {10609, 9945}, {10698, 6265}, {10707, 2}, {10724, 4}, {10728, 10742}, {10738, 5}, {10742, 11698}, {10755, 6}, {10767, 113}, {10768, 114}, {10769, 115}, {10770, 116}, {10771, 117}, {10772, 118}, {10773, 120}, {10774, 121}, {10775, 122}, {10776, 123}, {10777, 124}, {10778, 125}, {10779, 126}, {10780, 127}, {10781, 1313}, {10782, 1312}, {11219, 10164}, {11256, 11260}, {11604, 442}, {11609, 2092}, {12248, 38761}, {12515, 3579}, {12528, 12665}, {12531, 8}, {12532, 72}, {12641, 12640}, {12649, 12832}, {12690, 12019}, {12699, 12611}, {12737, 1385}, {12761, 18242}, {12764, 1329}, {12773, 38602}, {12848, 25606}, {12868, 12631}, {13199, 10993}, {13243, 1768}, {13266, 659}, {13268, 402}, {13277, 9508}, {13278, 10087}, {13279, 10090}, {14217, 946}, {14923, 39776}, {14947, 6184}, {17636, 5836}, {17638, 960}, {17652, 9957}, {17654, 31788}, {17763, 4434}, {19914, 5690}, {19916, 44805}, {20067, 15326}, {20095, 6154}, {20119, 2550}, {21630, 1125}, {24297, 40587}, {24712, 24318}, {24852, 24850}, {25416, 12735}, {25438, 8715}, {26015, 3911}, {26726, 3244}, {31512, 3259}, {32198, 32157}, {32454, 39}, {34151, 15632}, {34195, 39778}, {34772, 41541}, {34789, 21635}, {34894, 6600}, {36002, 44425}, {36175, 5520}, {36237, 190}, {36845, 41556}, {36846, 41554}, {37726, 6713}, {38460, 1319}, {38521, 38643}, {38665, 12331}, {38669, 104}, {38693, 34474}, {38753, 550}, {41575, 41558}, {42322, 4394}, {43735, 18642}, {45393, 11517}, {46435, 6260}, {46685, 14740}, {47320, 3678}
X(100) = isogonal conjugate of X(513)
X(100) = isotomic conjugate of X(693)
X(100) = anticomplement of X(11)
X(100) = complement of X(149)
X(100) = Stevanovic circle inverse of X(919)
X(100) = Conway circle inverse of X(38478)
X(100) = polar circle inverse of X(5521)
X(100) = orthoptic circle of the Steiner inellipe inverse of X(120)
X(100) = orthoptic circle of the Steiner circumellipe inverse of X(20344)
X(100) = de Longchamps circle inverse of X(34188)
X(100) = Schoutte circle inverse of X(35107)
X(100) = second Brocard Circle inverse of X(38521)
X(100) = polar conjugate of X(17924)
X(100) = antitomic image of X(14947)
X(100) = anticomplement of the anticomplement of X(3035)
X(100) = anticomplement of the anticomplement of the anticomplement of X(6667)
X(100) = complement of the complement of X(20095)
X(100) = anticomplement of the isogonal conjugate of X(59)
X(100) = complement of the isogonal conjugate of X(3446)
X(100) = anticomplement of the isotomic conjugate of X(4998)
X(100) = complement of the isotomic conjugate of X(8047)
X(100) = isogonal conjugate of the anticomplement of X(513)
X(100) = isogonal conjugate of the complement of X(513)
X(100) = isotomic conjugate of the anticomplement of X(650)
X(100) = isotomic conjugate of the complement of X(17494)
X(100) = isotomic conjugate of the isogonal conjugate of X(692)
X(100) = isogonal conjugate of the isotomic conjugate of X(668)
X(100) = isotomic conjugate of the polar conjugate of X(1783)
X(100) = isogonal conjugate of the polar conjugate of X(6335)
X(100) = polar conjugate of the isotomic conjugate of X(1332)
X(100) = polar conjugate of the isogonal conjugate of X(906)
X(100) = Thomson isogonal conjugate of X(517)
X(100) = Lucas-isogonal conjugate of X(517)
X(100) = excentral isogonal conjugate of X(2957)
X(100) = tangential isogonal conjugate of X(38863)
X(100) = psi-transform of X(1083)
X(100) = circumcircle-antipode of X(104)
X(100) = Ψ(X(i),X(j)) for these (i,j): (1,2), 2,37), (3,63), (4,8), (6,1), (7,8), (48,3), (56,55), (68,72)
X(100) = X(1)-line conjugate of X(244)
X(100) = X(113)-of-the-hexyl-triangle
X(100) = concurrence of reflections in sides of ABC of line X(4)X(8)
X(100) = perspector of Hutson-Moses hyperbola
X(100) = trilinear pole of line X(1)X(6) (and PU(28)) (van Aubel line of excentral triangle)
X(100) = trilinear product of PU(33)
X(100) = trilinear product of intercepts of circumcircle and Nagel line
X(100) = polar conjugate of X(17924)
X(100) = pole wrt polar circle of trilinear polar of X(17924) (line X(11)X(2969))
X(100) = the point of intersection, other than A, B, and C, of the circumcircle and the circumellipse centered at X(1) (viz., {A,B,C,X(100),X(664),X(1120),X(1320)}})
X(100) = the point of intersection, other than A, B, and C, of the circumcircle and the circumellipse centered at X(9) (viz., {A,B,C,X(100),X(658),X(662),X(799),X(1821),X(2580),X(2581),PU(34)}})
X(100) = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {A,B,C,PU(8)}}
X(100) = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {A,B,C,PU(32)}}
X(100) = center of hyperbola passing through X(1), X(9), and the excenters
X(100) = X(125)-of-excentral-triangle
X(100) = trilinear pole wrt 1st circumperp triangle of line X(3)X(142)
X(100) = X(110)-of-1st-circumperp-triangle
X(100) = reflection of X(1290) in the Euler line
X(100) = reflection of X(2703) in the Brocard axis
X(100) = reflection of X(901) in line X(1)X(3)
X(100) = cevapoint of X(59) and inverse-in-circumcircle-of-X(59)
X(100) = inverse-in-{circumcircle, nine-point circle}-inverter of X(120)
X(100) = exsimilicenter of circumcircle and AC-incircle
X(100) = X(i)-aleph conjugate of X(j) for these (i,j) (1,1052), (100,1), (190,63), (643,411), (666,673), (765,100), (1016,190)
X(100) = X(i)-beth conjugate of X(j) for these (i,j): (8,80), (21,106), (100,109), (333,673), (643,100), (765,100)
X(100) = the point of intersection, other than A, B, and C, of the circumcircle and ellipse {A,B,C,PU(75)}}
X(100) = crossdifference of PU(27)
X(100) = homothetic center of 2nd Schiffler triangle and excenters-midpoints triangle
X(100) = Feuerbach image of X(2)
X(100) = Cundy-Parry Phi transform of X(14266)
X(100) = perspector of anti-Mandart-incircle and anticomplementary triangles
X(100) = intersection of antipedal lines of X(1381) and X(1382)
X(100) = eigencenter of Gemini triangle 2
X(100) = barycentric product of vertices of Gemini triangle 5
X(100) = barycentric product of vertices of Gemini triangle 6
X(100) = perspector of ABC and side-triangle of Gemini triangles 29 and 30
X(100) = homothetic center of 2nd Schiffler triangle and excenters-midpoints triangle
X(100) = barycentric product of vertices of Gemini triangle 29
X(100) = barycentric product of vertices of Gemini triangle 30
X(100) = intersection, other than A, B, C, of {ABC, Gemini 29}-circumconic and {ABC, Gemini 30}-circumconic
X(100) = barycentric product of circumcircle intercepts of line X(2)X(37)
X(100) = excentral-to-ABC barycentric image of X(1768)
X(100) = intouch-to-ABC barycentric image of X(11)
X(100) = ABC-to-excentral barycentric image of X(11)
X(100) = trilinear pole, wrt circumtangential triangle, of line X(1)X(3)
X(100) = BSS(a^2→a) of X(110)
X(100) = Collings transform of X(i) for these i: {1, 9, 10, 119, 142, 214, 442, 600, 1145, 2092, 3126, 3307, 3308, 3647, 5507, 6184, 6260, 6594, 6600, 10427, 10472, 11517, 11530, 12631, 12639, 12640, 12864, 13089, 15346, 15347, 15348, 17057, 17060, 18258, 18642, 19557, 19584, 22754, 34261, 35204, 39041, 39048, 40587, 40600, 40653, 41540, 41862, 41886, 43182, 45036}
X(100) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {31, 17036}, {59, 8}, {100, 33650}, {101, 37781}, {109, 149}, {249, 21273}, {651, 150}, {664, 21293}, {692, 39351}, {765, 3436}, {1016, 21286}, {1101, 2975}, {1110, 144}, {1252, 329}, {1262, 7}, {1275, 21285}, {1331, 34188}, {1415, 4440}, {2149, 2}, {4551, 3448}, {4552, 21294}, {4559, 21221}, {4564, 69}, {4567, 20245}, {4570, 3869}, {4619, 693}, {4620, 17137}, {4998, 6327}, {7012, 4}, {7045, 3434}, {7115, 5905}, {7339, 36845}, {9268, 5176}, {23357, 18662}, {23979, 3210}, {23990, 3177}, {24027, 145}, {24041, 35614}, {31615, 20295}, {39293, 20556}, {44717, 4329}, {46102, 21270}
X(100) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 5375}, {3446, 10}, {8047, 2887}, {42552, 124}
X(100) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 6163}, {2, 5375}, {6, 9266}, {59, 2975}, {99, 190}, {190, 644}, {249, 38871}, {643, 1331}, {662, 101}, {664, 651}, {666, 2284}, {668, 1332}, {765, 1}, {835, 4756}, {898, 23343}, {1016, 6}, {1252, 1621}, {1262, 38869}, {1275, 220}, {3257, 1023}, {4076, 145}, {4555, 4585}, {4564, 9}, {4567, 37}, {4570, 21}, {4596, 662}, {4600, 81}, {4601, 213}, {4603, 46148}, {4618, 3257}, {4998, 2}, {5376, 44}, {5377, 518}, {5378, 238}, {5379, 72}, {5381, 3230}, {5382, 1743}, {5383, 2176}, {5384, 984}, {5385, 45}, {5387, 16784}, {5388, 40728}, {6012, 1633}, {6079, 17780}, {6335, 1783}, {6606, 664}, {6648, 4559}, {7012, 3869}, {7035, 32911}, {7045, 63}, {8269, 934}, {8706, 3699}, {8707, 3952}, {8708, 4557}, {8709, 3570}, {9059, 4767}, {13136, 2427}, {15742, 8}, {23586, 38876}, {23984, 38875}, {24041, 33761}, {31615, 1252}, {31628, 650}, {34071, 932}, {34537, 38853}, {35574, 42720}, {36086, 3573}, {36147, 4579}, {36797, 1897}, {36802, 2398}, {37212, 1018}, {39272, 36086}, {39444, 36237}, {46102, 219}, {46649, 12531}
X(100) = X(i)-cross conjugate of X(j) for these (i,j): {1, 765}, {3, 59}, {6, 1016}, {9, 4564}, {35, 4570}, {36, 9268}, {37, 4567}, {40, 7012}, {43, 7035}, {44, 5376}, {45, 5385}, {55, 1252}, {72, 5379}, {101, 651}, {109, 13138}, {165, 7045}, {170, 24011}, {171, 4600}, {190, 932}, {197, 7115}, {198, 1262}, {213, 4601}, {219, 46102}, {220, 1275}, {238, 5378}, {512, 2298}, {513, 1}, {514, 2346}, {518, 5377}, {521, 8}, {522, 21}, {523, 943}, {610, 7128}, {644, 27834}, {647, 40406}, {649, 81}, {650, 2}, {652, 40399}, {654, 2990}, {656, 1257}, {659, 105}, {661, 1255}, {663, 23617}, {665, 2991}, {667, 6}, {692, 1783}, {798, 1258}, {846, 24041}, {890, 739}, {900, 104}, {906, 1332}, {926, 294}, {928, 8759}, {984, 5384}, {1018, 190}, {1019, 40433}, {1021, 40435}, {1023, 3257}, {1026, 660}, {1030, 249}, {1376, 4998}, {1415, 46640}, {1491, 1390}, {1615, 23586}, {1631, 15378}, {1633, 934}, {1635, 88}, {1734, 75}, {1743, 5382}, {1960, 40400}, {2176, 5383}, {2254, 1280}, {2283, 677}, {2284, 666}, {2427, 13136}, {2509, 30701}, {2516, 8056}, {2804, 45393}, {2820, 43736}, {2821, 9372}, {2915, 250}, {2933, 15386}, {2939, 24000}, {2947, 24032}, {2977, 15344}, {3063, 17743}, {3126, 518}, {3185, 2149}, {3197, 23984}, {3230, 5381}, {3251, 44}, {3257, 9271}, {3309, 7}, {3573, 37135}, {3659, 6733}, {3667, 1476}, {3733, 1126}, {3737, 1220}, {3738, 1320}, {3882, 3903}, {3887, 1156}, {3900, 9}, {3913, 4076}, {3939, 644}, {4040, 86}, {4057, 58}, {4063, 82}, {4083, 983}, {4105, 6605}, {4394, 57}, {4401, 1014}, {4427, 34594}, {4436, 99}, {4455, 292}, {4491, 106}, {4551, 1897}, {4553, 668}, {4557, 101}, {4559, 44765}, {4705, 37}, {4730, 2161}, {4775, 40401}, {4777, 15175}, {4782, 985}, {4790, 25417}, {4825, 45}, {4834, 2214}, {4893, 40434}, {4926, 15446}, {5277, 4590}, {5687, 15742}, {6003, 17097}, {6050, 39956}, {6161, 513}, {6182, 40779}, {6366, 34894}, {6586, 40403}, {6600, 6065}, {7234, 42}, {7239, 1978}, {7252, 40394}, {8043, 1224}, {8632, 20332}, {8640, 31}, {8641, 7123}, {8659, 1462}, {8662, 2221}, {8674, 80}, {8676, 1172}, {8678, 941}, {8683, 1293}, {8702, 7161}, {9001, 1000}, {9441, 39293}, {9508, 291}, {11124, 650}, {12331, 46649}, {13589, 1290}, {14392, 41798}, {14419, 34893}, {14589, 31615}, {15313, 4}, {15624, 1110}, {16553, 35049}, {16784, 5387}, {17494, 1621}, {18004, 15168}, {21003, 1438}, {21005, 251}, {21007, 83}, {21173, 1222}, {21189, 40436}, {21779, 34537}, {21786, 46638}, {21789, 2983}, {21791, 38810}, {21894, 39698}, {21901, 1221}, {23067, 1331}, {23224, 42019}, {23343, 898}, {23703, 36037}, {23832, 901}, {23845, 109}, {23865, 1174}, {23867, 38813}, {24052, 4632}, {24290, 335}, {28217, 15179}, {35057, 32635}, {35326, 43190}, {35338, 664}, {35341, 37206}, {35342, 662}, {38469, 43073}, {39199, 1167}, {39200, 36052}, {40728, 5388}, {43049, 3870}, {46611, 2687}
X(100) = X(i)-isoconjugate of X(j) for these (i,j): {1, 513}, {2, 649}, {3, 7649}, {4, 1459}, {6, 514}, {7, 663}, {8, 43924}, {9, 3669}, {10, 3733}, {11, 109}, {19, 905}, {21, 4017}, {25, 4025}, {27, 647}, {28, 656}, {31, 693}, {32, 3261}, {34, 521}, {37, 1019}, {38, 18108}, {39, 10566}, {41, 24002}, {42, 7192}, {43, 43931}, {44, 1022}, {48, 17924}, {54, 21102}, {55, 3676}, {56, 522}, {57, 650}, {58, 523}, {59, 21132}, {63, 6591}, {64, 21172}, {65, 3737}, {71, 17925}, {74, 11125}, {75, 667}, {76, 1919}, {77, 18344}, {78, 43923}, {79, 2605}, {81, 661}, {82, 2530}, {83, 21123}, {84, 6129}, {85, 3063}, {86, 512}, {87, 4083}, {88, 1635}, {89, 4893}, {91, 34948}, {92, 22383}, {99, 3122}, {100, 244}, {101, 1086}, {103, 676}, {104, 1769}, {105, 2254}, {106, 900}, {108, 7004}, {110, 3120}, {111, 4750}, {112, 4466}, {115, 4556}, {158, 23224}, {162, 18210}, {163, 16732}, {174, 6729}, {184, 46107}, {190, 1015}, {200, 43932}, {210, 7203}, {213, 7199}, {222, 3064}, {225, 23189}, {226, 7252}, {238, 876}, {239, 3572}, {241, 1024}, {249, 21131}, {250, 21134}, {251, 16892}, {256, 4367}, {257, 20981}, {266, 6728}, {267, 31947}, {269, 3900}, {273, 1946}, {274, 798}, {278, 652}, {279, 657}, {284, 7178}, {286, 810}, {291, 659}, {292, 812}, {306, 43925}, {310, 669}, {330, 20979}, {333, 7180}, {335, 8632}, {350, 875}, {386, 43927}, {393, 4091}, {479, 4105}, {516, 2424}, {518, 1027}, {519, 23345}, {520, 8747}, {525, 1474}, {536, 23892}, {560, 40495}, {561, 1980}, {593, 4024}, {595, 40086}, {596, 4057}, {603, 44426}, {604, 4391}, {608, 6332}, {651, 2170}, {653, 7117}, {654, 2006}, {658, 14936}, {660, 27846}, {662, 3125}, {664, 3271}, {665, 673}, {668, 3248}, {679, 3251}, {692, 1111}, {694, 4107}, {726, 23355}, {727, 3837}, {738, 4130}, {739, 4728}, {741, 4010}, {751, 4378}, {753, 4809}, {757, 4705}, {764, 765}, {788, 870}, {799, 3121}, {813, 27918}, {824, 40746}, {832, 977}, {834, 43531}, {849, 4036}, {850, 2206}, {871, 8630}, {884, 9436}, {885, 1458}, {890, 31002}, {891, 37129}, {893, 4369}, {897, 14419}, {898, 19945}, {899, 43928}, {901, 1647}, {902, 6548}, {903, 1960}, {904, 4374}, {908, 2423}, {909, 10015}, {918, 1438}, {932, 3123}, {934, 2310}, {959, 17418}, {961, 17420}, {963, 7661}, {967, 45745}, {983, 3777}, {985, 1491}, {996, 9002}, {998, 9001}, {1002, 4724}, {1014, 4041}, {1016, 21143}, {1018, 16726}, {1021, 1427}, {1026, 43921}, {1042, 7253}, {1043, 7250}, {1054, 6164}, {1088, 8641}, {1096, 4131}, {1106, 4397}, {1120, 6085}, {1126, 4977}, {1146, 1461}, {1149, 23836}, {1155, 35348}, {1156, 14413}, {1169, 21124}, {1170, 21127}, {1171, 4988}, {1173, 21103}, {1174, 21104}, {1175, 23752}, {1176, 21108}, {1177, 21109}, {1178, 2533}, {1193, 4581}, {1220, 6371}, {1222, 6363}, {1252, 6545}, {1255, 4979}, {1262, 42462}, {1279, 35355}, {1293, 3756}, {1318, 39771}, {1319, 23838}, {1323, 23351}, {1326, 18014}, {1331, 2969}, {1333, 1577}, {1334, 17096}, {1357, 3699}, {1358, 3939}, {1364, 36127}, {1365, 4636}, {1395, 35518}, {1396, 8611}, {1397, 35519}, {1400, 4560}, {1402, 18155}, {1407, 3239}, {1408, 4086}, {1411, 3738}, {1412, 3700}, {1413, 8058}, {1414, 4516}, {1415, 4858}, {1417, 4768}, {1421, 42552}, {1422, 14298}, {1431, 3907}, {1432, 3287}, {1434, 3709}, {1436, 14837}, {1437, 24006}, {1457, 43728}, {1472, 2517}, {1476, 6615}, {1486, 26721}, {1492, 4475}, {1509, 4079}, {1565, 8750}, {1576, 21207}, {1581, 4164}, {1638, 2291}, {1643, 37131}, {1646, 4607}, {1751, 43060}, {1783, 3942}, {1790, 2501}, {1795, 39534}, {1813, 8735}, {1826, 7254}, {1875, 37628}, {1876, 23696}, {1897, 3937}, {1911, 3766}, {1914, 4444}, {1924, 6385}, {1929, 9508}, {1967, 14296}, {1973, 15413}, {1977, 1978}, {2087, 3257}, {2109, 25381}, {2149, 40166}, {2156, 16757}, {2160, 14838}, {2161, 3960}, {2162, 3835}, {2163, 4777}, {2164, 21188}, {2183, 2401}, {2191, 3309}, {2194, 4077}, {2195, 43042}, {2203, 14208}, {2207, 30805}, {2208, 17896}, {2214, 14349}, {2215, 23882}, {2217, 21189}, {2218, 23800}, {2221, 6590}, {2226, 6544}, {2248, 21196}, {2258, 43067}, {2276, 4817}, {2279, 4762}, {2287, 7216}, {2296, 2978}, {2297, 8712}, {2299, 17094}, {2308, 4608}, {2311, 7212}, {2316, 30725}, {2319, 43051}, {2333, 15419}, {2334, 4778}, {2340, 43930}, {2348, 37626}, {2349, 14399}, {2350, 17494}, {2353, 21178}, {2354, 15420}, {2364, 43052}, {2382, 36848}, {2384, 14475}, {2395, 17209}, {2426, 15634}, {2432, 34050}, {2433, 18653}, {2486, 43076}, {2488, 21453}, {2489, 17206}, {2504, 9085}, {2509, 40188}, {2516, 36603}, {2526, 39958}, {2611, 13486}, {2616, 18180}, {2623, 17167}, {2718, 24457}, {2720, 35015}, {2786, 17962}, {2787, 17954}, {2832, 34893}, {2973, 32656}, {2983, 29162}, {2985, 23751}, {2998, 23572}, {3011, 35365}, {3022, 4626}, {3049, 44129}, {3119, 4617}, {3124, 4610}, {3224, 21191}, {3226, 6373}, {3227, 3768}, {3249, 31625}, {3250, 14621}, {3270, 36118}, {3285, 4049}, {3310, 34234}, {3423, 47123}, {3433, 21185}, {3435, 21186}, {3437, 21187}, {3444, 21192}, {3445, 3667}, {3446, 21201}, {3447, 21203}, {3449, 21118}, {3450, 21119}, {3451, 21120}, {3453, 21121}, {3455, 21205}, {3500, 21348}, {3668, 21789}, {3675, 36086}, {3762, 9456}, {3778, 7255}, {3798, 8770}, {3801, 38813}, {3803, 23051}, {3805, 40763}, {3862, 23597}, {3880, 37627}, {3912, 43929}, {3954, 39179}, {4040, 13476}, {4062, 43926}, {4063, 39798}, {4081, 6614}, {4128, 4594}, {4132, 39949}, {4142, 34250}, {4160, 34916}, {4162, 19604}, {4163, 7023}, {4303, 14775}, {4306, 23289}, {4373, 8643}, {4379, 30650}, {4382, 30651}, {4394, 8056}, {4401, 7241}, {4440, 9262}, {4449, 9309}, {4453, 6187}, {4455, 18827}, {4458, 8852}, {4459, 29055}, {4462, 38266}, {4467, 6186}, {4481, 40747}, {4491, 39697}, {4498, 39956}, {4521, 40151}, {4534, 38828}, {4551, 18191}, {4557, 17205}, {4559, 17197}, {4561, 42067}, {4565, 21044}, {4584, 39786}, {4598, 6377}, {4600, 8034}, {4603, 16592}, {4618, 42084}, {4637, 36197}, {4638, 35092}, {4707, 34079}, {4775, 39704}, {4784, 30571}, {4786, 21448}, {4790, 25430}, {4791, 28607}, {4813, 25417}, {4823, 34819}, {4834, 30598}, {4885, 9315}, {4932, 39967}, {4943, 16079}, {4957, 34073}, {4960, 28625}, {4978, 28615}, {4983, 40438}, {5009, 35352}, {5029, 6650}, {5209, 18002}, {5317, 24018}, {5331, 8672}, {5620, 42741}, {6005, 10013}, {6006, 41436}, {6149, 43082}, {6161, 46972}, {6336, 22086}, {6372, 40433}, {6381, 23349}, {6384, 8640}, {6549, 23344}, {6550, 9268}, {6551, 24188}, {6586, 14377}, {6588, 42467}, {6589, 13478}, {6610, 23893}, {6629, 9178}, {6730, 7370}, {7035, 8027}, {7077, 43041}, {7087, 20517}, {7121, 20906}, {7169, 21174}, {7234, 32010}, {7260, 21755}, {7316, 14432}, {7339, 23615}, {7658, 11051}, {8050, 8054}, {8059, 38357}, {8578, 44184}, {8648, 18815}, {8656, 36588}, {8659, 36807}, {8677, 36123}, {8690, 21963}, {8713, 10579}, {8714, 34444}, {8917, 17427}, {9217, 21200}, {9259, 42555}, {9265, 21211}, {9267, 9359}, {9269, 9325}, {9292, 17215}, {9299, 18149}, {9311, 20980}, {9361, 38238}, {9505, 38348}, {9506, 27929}, {10428, 23757}, {10490, 10495}, {10492, 18888}, {10509, 10581}, {14370, 21194}, {14554, 21786}, {15378, 21133}, {15382, 20504}, {16099, 42662}, {16606, 18197}, {16695, 42027}, {16702, 23894}, {16737, 40729}, {16887, 18105}, {17216, 32713}, {17217, 23493}, {17222, 45677}, {17435, 36146}, {17731, 18001}, {17758, 21007}, {17780, 43922}, {18018, 21122}, {18101, 46153}, {18359, 21758}, {18771, 21105}, {18772, 21106}, {18830, 38986}, {20295, 40148}, {20516, 34183}, {20518, 41528}, {20908, 34077}, {20974, 43190}, {21003, 39714}, {21110, 38826}, {21113, 42346}, {21138, 34071}, {21173, 34434}, {21175, 34436}, {21176, 34437}, {21179, 34441}, {21180, 34442}, {21183, 34446}, {21190, 34427}, {21202, 34179}, {21206, 36615}, {21208, 40519}, {21385, 39982}, {21763, 42328}, {21828, 24624}, {21832, 37128}, {22084, 26705}, {22108, 34578}, {22350, 43933}, {23100, 23990}, {23707, 30691}, {23723, 34429}, {23729, 38825}, {23807, 34248}, {23845, 40451}, {23989, 32739}, {24027, 42455}, {25426, 28840}, {26932, 32674}, {26933, 32691}, {28209, 41434}, {29198, 39972}, {29226, 36598}, {30723, 34820}, {30724, 33635}, {32039, 40610}, {32641, 42754}, {32714, 34591}, {34018, 46388}, {34051, 46393}, {34858, 36038}, {35014, 36110}, {36037, 42753}, {40076, 47234}, {40397, 40628}, {40409, 40627}, {40738, 45882}, {41799, 45877}, {42290, 45755}
X(100) = cevapoint of X(i) and X(j) for these (i,j): {1, 513}, {2, 17494}, {3, 521}, {6, 667}, {9, 3900}, {10, 522}, {11, 15914}, {37, 4705}, {42, 649}, {43, 8640}, {44, 3251}, {45, 4825}, {55, 650}, {57, 43049}, {101, 3939}, {119, 2804}, {142, 514}, {171, 7234}, {190, 4595}, {214, 3738}, {442, 523}, {512, 2092}, {518, 3126}, {525, 18642}, {656, 18673}, {659, 8299}, {661, 1962}, {663, 2347}, {665, 20455}, {669, 21753}, {678, 1635}, {692, 906}, {899, 38349}, {900, 1145}, {905, 7289}, {918, 17060}, {926, 6184}, {1018, 4557}, {1019, 18166}, {1021, 8021}, {1734, 15624}, {1960, 20972}, {2530, 18183}, {3158, 4394}, {3257, 9272}, {3293, 4057}, {3307, 3308}, {3309, 6600}, {3647, 35057}, {3667, 12640}, {3737, 4267}, {3795, 4782}, {3887, 6594}, {4041, 21811}, {4083, 41886}, {4097, 4401}, {4105, 8012}, {4477, 18235}, {4551, 23067}, {4730, 40988}, {4775, 20973}, {4777, 17057}, {4802, 41862}, {6161, 46973}, {6260, 8058}, {6366, 10427}, {6608, 42438}, {8043, 27787}, {8298, 9508}, {8632, 20663}, {8641, 30706}, {8674, 35204}, {8678, 34261}, {8702, 13089}, {10472, 23880}, {11124, 14589}, {11517, 15313}, {14077, 15346}, {15347, 30198}, {15348, 30199}
X(100) = crosspoint of X(i) and X(j) for these (i,j): {1, 9282}, {2, 8047}, {6, 9265}, {99, 662}, {101, 34071}, {190, 664}, {643, 36797}, {668, 6335}, {3257, 4618}, {4596, 37212}, {4600, 6632}, {4998, 31615}
X(100) = crosssum of X(i) and X(j) for these (i,j): {1, 1054}, {2, 9263}, {6, 16686}, {10, 22045}, {37, 22323}, {244, 764}, {512, 661}, {514, 3835}, {523, 31946}, {649, 663}, {650, 4162}, {659, 38348}, {667, 22383}, {693, 23807}, {812, 27854}, {891, 14434}, {1015, 8027}, {1635, 3251}, {1646, 14441}, {3120, 21132}, {3122, 21143}, {3259, 6550}, {4107, 4375}, {4979, 4983}, {6363, 6615}, {8677, 42769}, {33917, 39011}
X(100) = crossdifference of every pair of points on line {244, 665}
X(100) = X(i)-line conjugate of X(j) for these (i,j): {1, 244}, {101, 1635}, {292, 3121}, {294, 14936}, {4618, 14421}, {8649, 9283}, {10699, 3675}, {39443, 891}
X(100) = barycentric product X(i)*X(j) for these {i,j}: {1, 190}, {3, 6335}, {4, 1332}, {6, 668}, {7, 644}, {8, 651}, {9, 664}, {10, 662}, {11, 31615}, {12, 4612}, {19, 4561}, {21, 4552}, {31, 1978}, {32, 6386}, {35, 15455}, {36, 36804}, {37, 99}, {40, 44327}, {41, 4572}, {42, 799}, {43, 4598}, {44, 4555}, {45, 4597}, {55, 4554}, {56, 646}, {57, 3699}, {58, 4033}, {59, 4391}, {63, 1897}, {65, 645}, {69, 1783}, {71, 811}, {72, 648}, {74, 42716}, {75, 101}, {76, 692}, {78, 653}, {80, 4585}, {81, 3952}, {82, 4568}, {83, 4553}, {85, 3939}, {86, 1018}, {87, 4595}, {88, 17780}, {89, 4767}, {92, 1331}, {98, 42717}, {102, 42718}, {103, 42719}, {104, 2397}, {105, 42720}, {106, 24004}, {107, 3998}, {108, 345}, {109, 312}, {110, 321}, {111, 42721}, {112, 20336}, {145, 27834}, {162, 306}, {163, 313}, {171, 27805}, {181, 4631}, {192, 932}, {200, 658}, {210, 4573}, {212, 46404}, {213, 670}, {219, 18026}, {220, 4569}, {226, 643}, {228, 6331}, {238, 4562}, {239, 660}, {241, 36802}, {244, 6632}, {249, 4036}, {256, 18047}, {257, 4579}, {261, 21859}, {264, 906}, {269, 6558}, {273, 4587}, {274, 4557}, {278, 4571}, {279, 4578}, {281, 6516}, {286, 4574}, {291, 3570}, {292, 874}, {294, 883}, {304, 8750}, {314, 4559}, {318, 1813}, {322, 36049}, {329, 13138}, {333, 4551}, {335, 3573}, {341, 1461}, {344, 1292}, {346, 934}, {350, 813}, {386, 37218}, {476, 42701}, {480, 36838}, {512, 4601}, {513, 1016}, {514, 765}, {517, 13136}, {518, 666}, {519, 3257}, {521, 46102}, {522, 4564}, {523, 4567}, {524, 5380}, {525, 5379}, {536, 898}, {556, 6733}, {561, 32739}, {598, 3908}, {612, 37215}, {649, 7035}, {650, 4998}, {655, 4511}, {661, 4600}, {667, 31625}, {673, 1026}, {675, 42723}, {677, 30807}, {689, 21814}, {691, 42713}, {693, 1252}, {728, 4626}, {739, 41314}, {740, 4584}, {751, 4482}, {752, 5386}, {754, 5389}, {756, 4610}, {757, 4103}, {758, 47318}, {788, 5388}, {789, 2276}, {812, 5378}, {823, 3682}, {824, 5384}, {825, 33931}, {831, 17289}, {833, 32777}, {835, 28606}, {839, 4261}, {840, 42722}, {869, 37133}, {889, 3230}, {891, 5381}, {892, 21839}, {894, 3903}, {899, 4607}, {900, 5376}, {901, 4358}, {903, 1023}, {905, 15742}, {908, 36037}, {914, 36106}, {918, 5377}, {919, 3263}, {925, 42700}, {927, 3693}, {931, 31993}, {933, 42698}, {958, 32038}, {960, 6648}, {982, 4621}, {983, 33946}, {984, 4586}, {985, 3807}, {1001, 32041}, {1014, 30730}, {1020, 1043}, {1025, 14942}, {1042, 7258}, {1089, 4556}, {1098, 4605}, {1100, 6540}, {1110, 3261}, {1125, 37212}, {1212, 6606}, {1213, 4596}, {1214, 36797}, {1215, 4603}, {1220, 3882}, {1222, 21362}, {1253, 46406}, {1255, 4427}, {1257, 14543}, {1260, 13149}, {1262, 4397}, {1265, 32714}, {1267, 6135}, {1268, 35342}, {1275, 3900}, {1278, 29227}, {1290, 32849}, {1293, 18743}, {1296, 42724}, {1301, 42699}, {1302, 42704}, {1305, 27396}, {1308, 17264}, {1310, 2345}, {1319, 4582}, {1333, 27808}, {1334, 4625}, {1376, 30610}, {1400, 7257}, {1414, 2321}, {1415, 3596}, {1427, 7256}, {1429, 36801}, {1434, 4069}, {1441, 5546}, {1473, 42384}, {1476, 25268}, {1492, 3661}, {1500, 4623}, {1509, 40521}, {1575, 8709}, {1576, 27801}, {1577, 4570}, {1633, 30701}, {1698, 37211}, {1757, 35148}, {1824, 4563}, {1826, 4592}, {1911, 27853}, {1914, 4583}, {1918, 4602}, {1921, 34067}, {1930, 4628}, {1962, 4632}, {1969, 32656}, {1983, 20566}, {1997, 30236}, {2087, 6635}, {2149, 35519}, {2162, 36863}, {2176, 18830}, {2205, 4609}, {2214, 33948}, {2222, 32851}, {2223, 36803}, {2238, 4589}, {2283, 36796}, {2284, 2481}, {2287, 4566}, {2295, 4594}, {2323, 35174}, {2339, 14594}, {2340, 34085}, {2361, 46405}, {2398, 36101}, {2427, 18816}, {2702, 20947}, {2703, 17790}, {2715, 42703}, {2742, 37788}, {2743, 37758}, {2748, 37756}, {2753, 37857}, {2832, 5387}, {3006, 36087}, {3035, 31628}, {3112, 46148}, {3175, 8690}, {3198, 44326}, {3219, 6742}, {3222, 21877}, {3227, 23343}, {3239, 7045}, {3240, 37209}, {3262, 32641}, {3264, 32665}, {3290, 35574}, {3293, 37205}, {3436, 46640}, {3616, 4606}, {3666, 8707}, {3667, 5382}, {3668, 7259}, {3669, 4076}, {3672, 6574}, {3679, 4604}, {3681, 43190}, {3687, 36098}, {3692, 36118}, {3701, 4565}, {3717, 36146}, {3718, 32674}, {3719, 36127}, {3739, 8708}, {3747, 4639}, {3752, 8706}, {3762, 9268}, {3783, 37207}, {3797, 30664}, {3799, 14621}, {3869, 44765}, {3870, 37206}, {3888, 17743}, {3909, 40394}, {3912, 36086}, {3935, 37143}, {3943, 4622}, {3954, 4577}, {3969, 13486}, {3990, 6528}, {3992, 4591}, {3995, 34594}, {4024, 24041}, {4037, 36066}, {4039, 37134}, {4041, 4620}, {4043, 43076}, {4062, 36085}, {4079, 24037}, {4082, 4637}, {4083, 5383}, {4115, 40438}, {4357, 36147}, {4359, 8701}, {4370, 4618}, {4384, 37138}, {4417, 36050}, {4420, 38340}, {4421, 42343}, {4436, 32009}, {4441, 8693}, {4463, 44766}, {4505, 40746}, {4512, 4624}, {4515, 4616}, {4558, 41013}, {4576, 18098}, {4588, 4671}, {4590, 4705}, {4593, 21035}, {4599, 15523}, {4613, 40773}, {4614, 5257}, {4615, 21805}, {4617, 5423}, {4619, 24026}, {4629, 4647}, {4633, 37593}, {4636, 6358}, {4638, 4738}, {4663, 35177}, {4664, 29351}, {4687, 6013}, {4699, 29199}, {4752, 39704}, {4756, 25417}, {4777, 5385}, {4781, 40434}, {4812, 29026}, {4850, 9059}, {4980, 28176}, {4997, 23703}, {5222, 37223}, {5223, 32040}, {5291, 35147}, {5293, 8052}, {5297, 37210}, {5360, 43187}, {5375, 8047}, {5391, 6136}, {5435, 31343}, {5526, 35171}, {5545, 42712}, {6011, 33116}, {6012, 17279}, {6014, 30829}, {6065, 24002}, {6079, 16610}, {6163, 6630}, {6164, 6634}, {6332, 7012}, {6376, 34071}, {6381, 34075}, {6542, 37135}, {6554, 8269}, {6559, 41353}, {6577, 18137}, {6603, 35157}, {6605, 35312}, {6614, 30693}, {6631, 9282}, {6735, 37136}, {6745, 37139}, {6790, 46119}, {7017, 36059}, {7080, 37141}, {7081, 37137}, {7115, 35518}, {7239, 40415}, {7260, 20964}, {8050, 32911}, {8056, 43290}, {8652, 28605}, {8684, 33891}, {8694, 19804}, {8699, 20942}, {9058, 17740}, {9067, 17756}, {9070, 32779}, {9265, 9296}, {9266, 9295}, {9271, 17487}, {9278, 17934}, {9361, 9362}, {11124, 31619}, {11495, 42303}, {11611, 17944}, {13396, 17281}, {13397, 17776}, {14947, 40865}, {15322, 28653}, {15624, 31624}, {16514, 41072}, {16593, 39272}, {16777, 32042}, {16785, 35181}, {17459, 35572}, {17787, 29055}, {17796, 35156}, {17863, 29163}, {17930, 20693}, {18140, 40519}, {18147, 29014}, {19604, 30720}, {19799, 32691}, {20332, 23354}, {20440, 20640}, {20453, 20696}, {20568, 23344}, {20901, 31616}, {20911, 32736}, {20940, 40150}, {21272, 23617}, {21453, 35341}, {21802, 35137}, {21833, 31614}, {21874, 35136}, {21899, 37880}, {22003, 40430}, {22456, 42702}, {23067, 31623}, {23493, 36860}, {23704, 35160}, {23832, 36805}, {23845, 32017}, {23891, 37129}, {23981, 36795}, {23990, 40495}, {24589, 28210}, {25001, 43344}, {25660, 29151}, {26700, 42033}, {26706, 28420}, {26711, 33168}, {26714, 42711}, {28148, 42029}, {28162, 42034}, {28474, 41316}, {28583, 41315}, {28847, 30758}, {30555, 31130}, {30625, 42301}, {30963, 43077}, {31633, 42552}, {32008, 35338}, {32018, 35327}, {32094, 46972}, {32676, 40071}, {32718, 35543}, {32931, 35009}, {33113, 33637}, {33157, 43348}, {35280, 39749}, {35517, 36039}, {36077, 42706}, {36080, 44140}, {37204, 41267}, {38828, 44720}, {40728, 46132}, {41839, 43350}, {44426, 44717}
X(100) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 514}, {2, 693}, {3, 905}, {4, 17924}, {6, 513}, {7, 24002}, {8, 4391}, {9, 522}, {10, 1577}, {11, 40166}, {19, 7649}, {21, 4560}, {22, 16757}, {25, 6591}, {28, 17925}, {31, 649}, {32, 667}, {33, 3064}, {35, 14838}, {36, 3960}, {37, 523}, {38, 16892}, {39, 2530}, {40, 14837}, {41, 663}, {42, 661}, {43, 3835}, {44, 900}, {45, 4777}, {46, 21188}, {48, 1459}, {55, 650}, {56, 3669}, {57, 3676}, {58, 1019}, {59, 651}, {63, 4025}, {65, 7178}, {69, 15413}, {71, 656}, {72, 525}, {75, 3261}, {76, 40495}, {78, 6332}, {81, 7192}, {82, 10566}, {86, 7199}, {88, 6548}, {92, 46107}, {99, 274}, {101, 1}, {104, 2401}, {106, 1022}, {108, 278}, {109, 57}, {110, 81}, {112, 28}, {145, 4462}, {162, 27}, {163, 58}, {165, 7658}, {169, 21185}, {171, 4369}, {172, 4367}, {184, 22383}, {187, 14419}, {190, 75}, {191, 21192}, {192, 20906}, {194, 23807}, {197, 6588}, {198, 6129}, {200, 3239}, {210, 3700}, {212, 652}, {213, 512}, {218, 3309}, {219, 521}, {220, 3900}, {226, 4077}, {228, 647}, {238, 812}, {239, 3766}, {241, 43042}, {244, 6545}, {251, 18108}, {255, 4091}, {259, 6728}, {260, 10492}, {281, 44426}, {284, 3737}, {291, 4444}, {292, 876}, {294, 885}, {306, 14208}, {312, 35519}, {313, 20948}, {318, 46110}, {319, 18160}, {321, 850}, {326, 30805}, {329, 17896}, {333, 18155}, {345, 35518}, {346, 4397}, {354, 21104}, {385, 14296}, {386, 14349}, {391, 4811}, {394, 4131}, {405, 23882}, and many more
X(100) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 244, 3315}, {1, 404, 5253}, {1, 750, 37633}, {1, 1054, 244}, {1, 8715, 3871}, {1, 15015, 214}, {1, 25438, 13278}, {1, 25440, 404}, {2, 11, 31272}, {2, 55, 1621}, {2, 149, 11}, {2, 1621, 5284}, {2, 2550, 33108}, {2, 3434, 11680}, {2, 5274, 10584}, {2, 17784, 3434}, {2, 20075, 497}, {2, 20095, 149}, {2, 26007, 31226}, {2, 26073, 24988}, {2, 26795, 27134}, {2, 26846, 27190}, {2, 27134, 28743}, {2, 33110, 2886}, {2, 34607, 34611}, {3, 8, 2975}, {3, 104, 38693}, {3, 2932, 17100}, {3, 2975, 5303}, {3, 5687, 8}, {3, 12645, 32153}, {3, 12773, 38602}, {3, 32141, 11491}, {3, 33814, 34474}, {3, 38665, 38669}, {3, 45145, 47045}, {4, 5552, 11681}, {6, 1979, 1977}, {6, 37540, 17126}, {8, 17100, 104}, {8, 17740, 33089}, {9, 35445, 35258}, {10, 21, 5260}, {10, 35, 21}, {10, 21098, 21054}, {10, 32917, 5235}, {11, 149, 10707}, {11, 3035, 2}, {11, 6174, 3035}, {11, 31235, 6667}, {20, 7080, 3436}, {31, 43, 32911}, {35, 80, 10058}, {36, 7972, 10074}, {37, 21899, 21833}, {40, 78, 3869}, {40, 6796, 411}, {42, 171, 81}, {43, 3550, 31}, {46, 3811, 3868}, {55, 1376, 2}, {55, 4413, 1001}, {55, 4423, 4428}, {55, 11502, 497}, {55, 36497, 4972}, {56, 3913, 145}, {57, 3158, 3870}, {57, 3870, 3873}, {57, 37736, 5083}, {57, 41553, 14151}, {63, 200, 3681}, {65, 34772, 34195}, {65, 41541, 12739}, {75, 20940, 20901}, {88, 3315, 244}, {101, 1018, 644}, {101, 4752, 1023}, {104, 34474, 3}, {109, 3939, 1331}, {109, 4551, 651}, {119, 11248, 12775}, {119, 13199, 10724}, {145, 4188, 56}, {149, 3035, 31272}, {165, 200, 63}, {165, 1768, 46684}, {165, 5531, 1768}, {190, 3699, 3952}, {190, 3952, 4756}, {190, 17780, 4767}, {190, 43290, 3699}, {197, 37577, 22}, {198, 346, 38869}, {200, 1768, 46685}, {210, 4640, 3219}, {214, 5541, 1320}, {214, 8715, 10087}, {214, 9324, 14193}, {228, 32932, 11688}, {230, 21956, 17737}, {238, 899, 37680}, and many more


X(101) = Ψ(INCENTER, SYMMEDIAN POINT)

Trilinears    a/(b - c) : b/(c - a) : c/(a - b)
Trilinears    a(a - b)(a - c) : b(b - c)(b - a) : c(c - a)(c - b)
Barycentrics    a2/(b - c) : b2/(c - a) : c2/(a - b)
Barycentrics    a^2 (sin^2 A + sin B sin C - sin C sin A - sin A sin B) : :
Tripolars    |b c (b - c)| : :
X(101) = (r 2 + 6rR + 8R2 + s2)*X(1) - 6R(r + 4R)*X(2) - 2(r2 + 4rR - s2)*X(3)    (Peter Moses, April 2, 2013)

Let LA be the reflection of the line X(1)X(7) in line BC, and define LB and LC cyclically. Let A' = LB∩LC, B' = LC∩LA, C' = LA∩LB. The lines AA', BB', CC' concur in X(101). (Randy Hutson, 9/23/2011)

Let IaIbIc be the excentral triangle. The Brocard axes of BCIa, CAIb, ABIc concur in X(101). (Randy Hutson, February 10, 2016)

Let P be a point on line X(4)X(9) other than X(4). Let A' be the reflection of P in BC, and define B' and C' cyclically. The circumcircles of AA'B', BC'A', CA'B' concur in X(101). (Randy Hutson, February 10, 2016)

Let Q be a point on the Nagel line other than X(2). Let A'B'C' be the cevian triangle of Q. Let A″ be the {B,C}-harmonic conjugate of A' (or equivalently, A″ = BC∩B'C'), and define B″ and C″ cyclically. The circumcircles of AB″C″, BC″A″, CA″B″ concur in X(101). (Randy Hutson, February 10, 2016)

Let A', B', C' be the intersections of the antiorthic axis and lines BC, CA, AB, resp. The circumcircles of AB'C', BC'A', CA'B' concur in X(101). (Randy Hutson, February 10, 2016)

X(101) is the perspector of the anticevian triangle of X(109) and the unary cofactor triangle of the intangents triangle. (Randy Hutson, January 29, 2018)

X(101) lies on these lines: {1, 41}, {2, 116}, {3, 103}, {4, 118}, {5, 10739}, {6, 106}, {7, 2369}, {8, 1311}, {9, 48}, {10, 98}, {11, 10770}, {19, 913}, {20, 152}, {21, 3294}, {25, 3190}, {27, 22000}, {28, 3191}, {29, 22018}, {30, 2688}, {31, 609}, {32, 595}, {33, 20624}, {34, 2376}, {35, 1334}, {36, 672}, {37, 284}, {39, 21008}, {40, 972}, {42, 111}, {43, 9082}, {44, 2718}, {45, 2278}, {55, 2291}, {56, 218}, {57, 15728}, {58, 172}, {59, 657}, {63, 36016}, {71, 74}, {72, 2360}, {75, 767}, {78, 205}, {80, 19628}, {86, 2368}, {99, 190}, {100, 644}, {102, 198}, {107, 1897}, {108, 1783}, {109, 654}, {110, 163}, {112, 4249}, {119, 6506}, {140, 38774}, {142, 18162}, {145, 9083}, {154, 1260}, {162, 36077}, {165, 6602}, {171, 3997}, {182, 23095}, {184, 3046}, {187, 2712}, {199, 3690}, {200, 2187}, {212, 32726}, {226, 36019}, {228, 2249}, {230, 17734}, {238, 9454}, {239, 9073}, {242, 29016}, {269, 2377}, {281, 32706}, {306, 2373}, {312, 30901}, {313, 2367}, {321, 30905}, {329, 36023}, {346, 2370}, {354, 1174}, {376, 38773}, {382, 38767}, {386, 3033}, {404, 16549}, {476, 4024}, {477, 36026}, {480, 2371}, {501, 21879}, {511, 2700}, {512, 2702}, {513, 1308}, {514, 664}, {515, 2723}, {516, 2724}, {517, 910}, {518, 2725}, {519, 2726}, {520, 2727}, {521, 2728}, {522, 929}, {523, 2690}, {524, 2729}, {548, 38766}, {550, 38765}, {551, 16503}, {560, 713}, {574, 28563}, {579, 2178}, {580, 29015}, {583, 21773}, {584, 16777}, {594, 2372}, {604, 1743}, {607, 11399}, {610, 1295}, {631, 6712}, {643, 931}, {649, 901}, {650, 2222}, {651, 934}, {652, 2149}, {653, 4605}, {656, 2722}, {660, 1492}, {661, 1290}, {663, 919}, {667, 813}, {668, 789}, {673, 17761}, {677, 36136}, {689, 1978}, {691, 4079}, {692, 926}, {693, 2860}, {697, 1501}, {699, 30634}, {715, 2205}, {717, 18900}, {729, 1918}, {731, 4279}, {733, 904}, {743, 869}, {745, 21814}, {747, 14620}, {753, 2276}, {755, 21035}, {758, 3509}, {761, 984}, {765, 898}, {798, 2703}, {822, 2719}, {825, 34069}, {827, 4628}, {831, 4553}, {833, 7239}, {835, 3952}, {839, 4033}, {859, 14964}, {872, 2375}, {896, 2721}, {899, 9081}, {902, 2384}, {908, 2861}, {912, 8558}, {922, 35107}, {925, 36145}, {930, 36148}, {932, 4579}, {935, 4064}, {944, 6554}, {947, 14872}, {952, 1146}, {953, 2183}, {956, 37658}, {958, 3041}, {976, 9078}, {991, 24320}, {1012, 5781}, {1016, 4595}, {1025, 6183}, {1043, 21070}, {1054, 23622}, {1083, 8299}, {1100, 5049}, {1111, 9317}, {1113, 2576}, {1114, 2577}, {1125, 9108}, {1126, 17962}, {1141, 7110}, {1149, 9097}, {1191, 30435}, {1193, 5280}, {1201, 5299}, {1212, 1385}, {1250, 36738}, {1259, 2365}, {1262, 4091}, {1292, 1633}, {1293, 2429}, {1294, 2822}, {1296, 2824}, {1297, 2825}, {1300, 1826}, {1302, 36149}, {1304, 36131}, {1305, 4552}, {1309, 3239}, {1310, 1332}, {1317, 4534}, {1319, 2348}, {1324, 2708}, {1326, 12031}, {1376, 24264}, {1381, 2590}, {1382, 2591}, {1384, 3052}, {1407, 23089}, {1412, 4641}, {1420, 16572}, {1429, 3008}, {1473, 7123}, {1474, 22021}, {1475, 5563}, {1499, 2740}, {1500, 18755}, {1503, 2741}, {1565, 5845}, {1577, 2864}, {1580, 2664}, {1604, 3197}, {1615, 6244}, {1618, 2742}, {1634, 23161}, {1657, 38768}, {1691, 21830}, {1698, 9103}, {1729, 3868}, {1731, 8609}, {1753, 2910}, {1755, 2699}, {1757, 19554}, {1759, 3869}, {1762, 16577}, {1780, 2198}, {1781, 2171}, {1785, 2202}, {1790, 3219}, {1818, 3220}, {1819, 22005}, {1914, 2251}, {1924, 9266}, {1939, 20691}, {1944, 29069}, {1951, 20752}, {1953, 16547}, {1958, 3729}, {1959, 2856}, {1973, 3811}, {1981, 18026}, {2077, 2272}, {2092, 38453}, {2140, 2141}, {2160, 21863}, {2173, 2687}, {2175, 34247}, {2177, 9331}, {2182, 2716}, {2195, 9319}, {2210, 3009}, {2220, 16685}, {2238, 5291}, {2241, 9351}, {2245, 17796}, {2256, 4254}, {2259, 2294}, {2266, 11529}, {2268, 3731}, {2273, 2277}, {2275, 28574}, {2287, 21061}, {2295, 5277}, {2304, 5248}, {2308, 8700}, {2312, 2747}, {2317, 28219}, {2318, 5285}, {2325, 2757}, {2327, 5279}, {2332, 6198}, {2333, 3563}, {2344, 36480}, {2347, 38452}, {2359, 21033}, {2374, 4028}, {2388, 23398}, {2398, 9057}, {2425, 8059}, {2426, 26716}, {2481, 24455}, {2503, 20982}, {2646, 16601}, {2689, 3700}, {2692, 14321}, {2695, 7359}, {2697, 21017}, {2701, 3709}, {2731, 4521}, {2736, 3309}, {2737, 3667}, {2738, 6000}, {2739, 6001}, {2743, 4394}, {2751, 3693}, {2752, 3930}, {2758, 3943}, {2766, 8611}, {2768, 3712}, {2770, 4062}, {2802, 4919}, {2859, 14208}, {2862, 3912}, {2863, 4358}, {2870, 22310}, {2908, 27396}, {2948, 16562}, {2975, 16552}, {3002, 17966}, {3053, 14974}, {3061, 30144}, {3083, 9098}, {3084, 9099}, {3090, 38775}, {3091, 20401}, {3119, 5531}, {3146, 38769}, {3185, 29068}, {3208, 8715}, {3241, 9095}, {3257, 4604}, {3261, 31624}, {3290, 30117}, {3295, 4258}, {3496, 3878}, {3501, 15323}, {3508, 9417}, {3522, 33521}, {3579, 21872}, {3616, 9105}, {3635, 9106}, {3661, 9075}, {3678, 15168}, {3679, 4390}, {3681, 32664}, {3688, 23868}, {3691, 5258}, {3699, 4103}, {3720, 9110}, {3734, 4713}, {3747, 18266}, {3754, 23621}, {3781, 28844}, {3832, 38770}, {3870, 9061}, {3903, 30670}, {3915, 7031}, {3920, 9077}, {3923, 24480}, {3970, 34772}, {3990, 26701}, {4006, 4420}, {4040, 9323}, {4041, 9090}, {4051, 22837}, {4053, 12030}, {4055, 26717}, {4093, 8625}, {4109, 36974}, {4120, 15343}, {4128, 23648}, {4153, 7270}, {4169, 6558}, {4209, 14377}, {4268, 16885}, {4289, 16672}, {4362, 9074}, {4427, 30727}, {4436, 6013}, {4513, 5687}, {4517, 37586}, {4530, 7972}, {4554, 34083}, {4556, 4629}, {4565, 4627}, {4567, 4584}, {4578, 6574}, {4585, 13396}, {4588, 5549}, {4600, 9150}, {4775, 28875}, {4794, 36086}, {4859, 7225}, {4872, 5074}, {4893, 14513}, {4904, 26007}, {4998, 31286}, {5010, 28535}, {5060, 5127}, {5088, 10025}, {5120, 37519}, {5189, 24055}, {5199, 28236}, {5206, 28551}, {5226, 34929}, {5228, 37272}, {5239, 36941}, {5240, 36940}, {5247, 35108}, {5249, 34934}, {5267, 29308}, {5275, 30116}, {5282, 5692}, {5293, 7281}, {5313, 28539}, {5315, 21764}, {5378, 38367}, {5514, 37725}, {5603, 5819}, {5660, 33573}, {5752, 29218}, {5840, 10772}, {5881, 23058}, {5949, 14219}, {5951, 16553}, {6014, 8658}, {6017, 8656}, {6037, 36132}, {6065, 6078}, {6079, 30720}, {6163, 20981}, {6205, 9352}, {6386, 9065}, {6510, 34371}, {6545, 38019}, {6586, 23990}, {6633, 9089}, {6645, 17499}, {6647, 35102}, {6733, 13444}, {7012, 36140}, {7077, 19561}, {7079, 17857}, {7084, 9439}, {7115, 10397}, {7117, 13006}, {7119, 21077}, {7122, 35105}, {7193, 9321}, {7287, 28899}, {7296, 28505}, {7297, 17444}, {7368, 10310}, {7391, 24054}, {7760, 34063}, {8012, 15931}, {8301, 14839}, {8624, 16514}, {8666, 21384}, {8676, 35182}, {8687, 32736}, {8694, 34074}, {9059, 17780}, {9067, 23891}, {9076, 15523}, {9086, 21272}, {9094, 17018}, {9104, 23705}, {9264, 21788}, {9306, 14827}, {9404, 34921}, {10246, 34522}, {10267, 32561}, {10454, 27410}, {10571, 22131}, {10638, 36737}, {10703, 38345}, {10799, 18759}, {11349, 20367}, {11720, 17468}, {12437, 21096}, {13384, 34930}, {13597, 21012}, {14422, 32630}, {14439, 15015}, {15378, 35184}, {15586, 21864}, {15654, 20471}, {15792, 21816}, {15817, 15830}, {16283, 20995}, {16560, 16578}, {16680, 28847}, {16787, 30148}, {16822, 33945}, {16946, 21769}, {16970, 37817}, {16997, 30114}, {17062, 31284}, {17137, 29473}, {17170, 26658}, {17205, 18723}, {17277, 18042}, {17284, 25940}, {17349, 27348}, {17454, 19302}, {17743, 27091}, {17744, 33299}, {17793, 24294}, {17798, 20683}, {20269, 30617}, {20336, 30920}, {20696, 21791}, {20780, 28914}, {20834, 21795}, {20999, 23988}, {21049, 37730}, {21138, 24281}, {21208, 26273}, {21232, 24685}, {21253, 23674}, {21285, 28734}, {21290, 27546}, {21383, 34076}, {21747, 28310}, {22020, 27398}, {22054, 28173}, {22088, 37561}, {22126, 23361}, {22145, 23585}, {22147, 28235}, {22329, 37854}, {22357, 28211}, {23073, 28233}, {23165, 34986}, {23201, 26890}, {23343, 29351}, {23472, 25575}, {23646, 24488}, {23843, 29306}, {24170, 33828}, {24190, 33825}, {24502, 24815}, {24549, 30110}, {24625, 32911}, {24713, 25379}, {24779, 28081}, {24995, 30103}, {25066, 30618}, {26036, 26363}, {26265, 28916}, {27097, 33953}, {28163, 37499}, {28509, 31451}, {29092, 31737}, {29219, 31897}, {29479, 30997}, {30379, 34926}, {31561, 32592}, {31562, 32590}, {31563, 32556}, {31564, 32555}, {32094, 35574}, {32642, 32684}, {32653, 32661}, {32670, 36068}, {32671, 36069}, {32672, 36070}, {32673, 36071}, {32686, 32719}, {33946, 33951}, {34526, 37611}, {35128, 38617}, {36037, 36137}, {37211, 37212}, {37736, 38375}

X(101) = midpoint of X(20) and X(152)
X(101) = reflection of X(i) in X(j) for these (i,j): (4,118), (103,3), (150,116)
X(101) = isogonal conjugate of X(514)
X(101) = isotomic conjugate of X(3261)
X(101) = complement of X(150)
X(101) = anticomplement of X(116)
X(101) = X(59)-Ceva conjugate of X(55)
X(101) = cevapoint of X(354) and X(513)
X(101) = X(i)-cross conjugate of X(j) for these (i,j): (55,59), (199,250)
X(101) = crosssum of X(i) and X(j) for these (i,j): (513,650), (523,661), (649,1459)
X(101) = crossdifference of every pair of points on line X(11)X(244)
X(101) = X(i)-aleph conjugate of X(j) for these (i,j): (100,165), (509,1052), (662,572), (664,169)
X(101) = X(i)-beth conjugate of X(j) for these (i,j): (21,105), (644,644)
X(101) = circumcircle-antipode of X(103)
X(101) = Ψ(X(i),X(j)) for these (i,j): (1,6), (2,1), (3,48), (4,9), (6,31), (7,2), (8,9), (9,55), (57,56), (63,3), (69,63), (76,10)
X(101) = X(114)-of-the-hexyl-triangle
X(101) = trilinear product of PU(i) for these i: 26, 49
X(101) = barycentric product of PU(33)
X(101) = the point of intersection, other than A, B, C, of the circumcircle and hyperbola {A,B,C,PU(9)}}
X(101) = the point of intersection, other than A, B, C, of conic {A,B,C,X(1),PU(93)}}
X(101) = trilinear pole of line X(6)X(31) (the isogonal conjugate of the isotomic conjugate of the Nagel line)
X(101) = trilinear pole wrt 1st circumperp triangle of line X(9)X(165)
X(101) = X(99)-of -1st-circumperp-triangle
X(101) = crossdifference of PU(i) for these i: 121, 123
X(101) = concurrence of reflections of line X(4)X(9) in sides of ABC
X(101) = isogonal conjugate of isotomic conjugate of trilinear pole of Nagel line
X(101) = center of Kiepert hyperbola of excentral triangle (i.e. X(115) of excentral triangle)
X(101) = reflection of X(2690) in the Euler line
X(101) = reflection of X(2702) in the Brocard axis
X(101) = reflection of X(1308) in line X(1)X(3)
X(101) = reflection of X(5011) in antiorthic axis
X(101) = inverse-in-polar-circle of X(5190)
X(101) = inverse-in-{circumcircle, nine-point circle}-inverter of X(5513)
X(101) = X(6)-isoconjugate of X(693)
X(101) = X(92)-isoconjugate of X(1459)
X(101) = X(1577)-isoconjugate of X(58)
X(101) = eigencenter of 2nd circumperp triangle
X(101) = perspector of 3rd mixtilinear triangle and unary cofactor triangle of 5th mixtilinear triangle
X(101) = trilinear product of vertices of 1st circumperp triangle
X(101) = Thomson-isogonal conjugate of X(516)
X(101) = Lucas-isogonal conjugate of X(516)
X(101) = intersection of Lemoine axes of 1st & 2nd Montesdeoca bisector triangles
X(101) = focus of Yff parabola
X(101) = polar conjugate of isogonal conjugate of X(32656)
X(101) = trilinear product X(6)*X(100)
X(101) = trilinear product of circumcircle intercepts of line X(1)X(6)
X(101) = barycentric product of circumcircle intercepts of the Nagel line
X(101) = inverse-in-Stevanovic-circle of X(2222)
X(101) = polar conjugate of isotomic conjugate of X(1331)
X(101) = X(2)-Ceva conjugate of X(39026)
X(101) = perspector of hyperbola {A,B,C,X(59),X(677)}}
X(101) = X(63)-isoconjugate of X(7649)


X(102) = Λ(INCENTER, ORTHOCENTER)

Trilinears    1/[sin B (sec A - sec B) + sin C (sec A - sec C)] : :
Trilinears    a/[2a^4 - (b + c) a^3 - (b - c)^2 a^2 + (b - c)^2 (b + c) a - (b^2 - c^2)^2] : :
Barycentrics    a^2*(a^4 - 2*a^2*b^2 + b^4 - a^3*c + a^2*b*c + a*b^2*c - b^3*c + a^2*c^2 - 2*a*b*c^2 + b^2*c^2 + a*c^3 + b*c^3 - 2*c^4)*(a^4 - a^3*b + a^2*b^2 + a*b^3 - 2*b^4 + a^2*b*c - 2*a*b^2*c + b^3*c - 2*a^2*c^2 + a*b*c^2 + b^2*c^2 - b*c^3 + c^4) : :
Barycentrics    (sin A)/[sin B (sec A - sec B) + sin C (sec A - sec C)] : :
Tripolars    |sin B (sec A - sec B) + sin C (sec A - sec C)| : :
X(102) = X[10696] - 4 X[11713], 3 X[2] - 4 X[6711], 4 X[117] - 3 X[10709], X[151] - 4 X[6711], 2 X[151] - 3 X[10709], 8 X[6711] - 3 X[10709], 3 X[3] - X[38579], 3 X[3] - 2 X[38607], 2 X[3] + X[38667], 4 X[3] - X[38674], 2 X[3] - 3 X[38691], 4 X[3] - 3 X[38697], X[109] + 2 X[38573], 3 X[109] - 2 X[38579], X[109] - 4 X[38600], 3 X[109] - 4 X[38607], X[109] - 3 X[38691], 2 X[109] - 3 X[38697], 3 X[38573] + X[38579], X[38573] + 2 X[38600], 3 X[38573] + 2 X[38607], 4 X[38573] + X[38674], 2 X[38573] + 3 X[38691], 4 X[38573] + 3 X[38697], X[38579] - 6 X[38600], 2 X[38579] + 3 X[38667], 4 X[38579] - 3 X[38674], 2 X[38579] - 9 X[38691], 4 X[38579] - 9 X[38697], 3 X[38600] - X[38607], 4 X[38600] + X[38667], 8 X[38600] - X[38674], 4 X[38600] - 3 X[38691], 8 X[38600] - 3 X[38697], 4 X[38607] + 3 X[38667], 8 X[38607] - 3 X[38674], 4 X[38607] - 9 X[38691], 8 X[38607] - 9 X[38697], 2 X[38667] + X[38674], X[38667] + 3 X[38691], 2 X[38667] + 3 X[38697], X[38674] - 6 X[38691], X[38674] - 3 X[38697], 4 X[124] - X[10726], 2 X[5] - 3 X[38776], X[10740] - 3 X[38776], 3 X[10716] - X[10732], 3 X[10716] - 2 X[10747], 3 X[3576] - 2 X[11700], 4 X[140] - 5 X[38786], 3 X[165] - 2 X[14690], 3 X[376] - 2 X[38785], X[382] - 3 X[38779], 4 X[548] - 3 X[38778], 5 X[631] - 4 X[6718], 5 X[631] - 6 X[38784], 2 X[6718] - 3 X[38784], 7 X[3090] - 10 X[38787], X[3146] - 4 X[38781], 5 X[3522] - 4 X[38783], 5 X[3616] - 4 X[11727], 7 X[3832] - 12 X[38782], 3 X[5603] - 4 X[11734]

X(102) lies on the circumcircle, the conic {A,B,C,X(1), X(3)}}, the cubics K269, K685, and these lines: {1, 108}, {2, 117}, {3, 109}, {4, 124}, {5, 10740}, {6, 10757}, {11, 10771}, {19, 282}, {20, 33650}, {24, 41401}, {29, 107}, {30, 2689}, {36, 1795}, {40, 78}, {55, 1361}, {56, 1364}, {57, 12016}, {63, 43347}, {64, 18237}, {73, 947}, {74, 2773}, {77, 934}, {98, 2785}, {99, 332}, {101, 198}, {103, 928}, {104, 3738}, {105, 2814}, {106, 2815}, {110, 283}, {111, 2819}, {112, 284}, {140, 38786}, {165, 14690}, {186, 40081}, {226, 1065}, {376, 38785}, {382, 38779}, {386, 34455}, {476, 7424}, {511, 2701}, {512, 2708}, {513, 2716}, {514, 2723}, {515, 1309}, {516, 929}, {517, 1807}, {518, 2730}, {519, 2731}, {520, 2732}, {521, 2733}, {522, 2734}, {523, 2695}, {524, 2735}, {548, 38778}, {550, 38777}, {572, 29044}, {580, 15440}, {631, 6718}, {901, 2077}, {927, 31637}, {930, 16113}, {933, 35196}, {944, 32704}, {949, 2301}, {958, 3042}, {959, 37530}, {1036, 32691}, {1067, 12053}, {1069, 11249}, {1183, 37732}, {1292, 2835}, {1293, 2841}, {1294, 2846}, {1295, 2849}, {1296, 2852}, {1297, 2853}, {1304, 2075}, {1305, 4329}, {1311, 2399}, {1350, 28291}, {1376, 3040}, {1385, 7100}, {1420, 30239}, {1457, 36067}, {1499, 2768}, {1503, 2769}, {1633, 33810}, {1657, 38780}, {1794, 15439}, {1897, 31866}, {2291, 2432}, {2338, 40116}, {2359, 8687}, {2715, 5060}, {2728, 14203}, {2743, 13528}, {2751, 3309}, {2757, 3667}, {2762, 6000}, {2765, 6001}, {2968, 18339}, {3090, 38787}, {3146, 38781}, {3422, 36076}, {3430, 6011}, {3478, 9088}, {3522, 38783}, {3616, 11727}, {3832, 38782}, {4262, 26716}, {5053, 32685}, {5450, 21228}, {5603, 11734}, {5759, 44876}, {5840, 10777}, {6014, 6244}, {6614, 7215}, {7015, 29055}, {7152, 8064}, {8607, 32683}, {8686, 41343}, {9057, 36007}, {9058, 35996}, {9059, 10327}, {9107, 35973}, {10680, 37489}, {10902, 40442}, {11014, 26711}, {13329, 32682}, {13397, 36986}, {14127, 14987}, {15379, 35187}, {15501, 39763}, {16132, 44063}, {18446, 26706}, {21740, 30250}, {22765, 34921}, {24466, 39444}, {26712, 30272}, {32643, 32689}, {32667, 32688}, {32674, 47411}, {36984, 46964}

X(102) = midpoint of X(i) and X(j) for these {i,j}: {3, 38573}, {20, 33650}, {109, 38667}, {1657, 38780}
X(102) = reflection of X(i) in X(j) for these {i,j}: {1, 11713}, {3, 38600}, {4, 124}, {109, 3}, {117, 6711}, {151, 117}, {1897, 31866}, {10696, 1}, {10709, 2}, {10726, 4}, {10732, 10747}, {10740, 5}, {10757, 6}, {10771, 11}, {18339, 2968}, {38579, 38607}, {38667, 38573}, {38674, 109}, {38697, 38691}, {38777, 550}
X(102) = isogonal conjugate of X(515)
X(102) = isotomic conjugate of X(35516)
X(102) = anticomplement of X(117)
X(102) = complement of X(151)
X(102) = anticomplement of the anticomplement of X(6711)
X(102) = anticomplement of the isogonal conjugate of X(15379)
X(102) = complement of the isogonal conjugate of X(34180)
X(102) = isogonal conjugate of the anticomplement of X(515)
X(102) = isogonal conjugate of the complement of X(515)
X(102) = isotomic conjugate of the anticomplement of X(8607)
X(102) = isogonal conjugate of the isotomic conjugate of X(34393)
X(102) = Thomson isogonal conjugate of X(522)
X(102) = Collings transform of X(i) for these i: {124, 3137, 38983}
X(102) = X(15379)-anticomplementary conjugate of X(8)
X(102) = X(34180)-complementary conjugate of X(10)
X(102) = X(21)-beth conjugate of X(108)
X(102) = circumcircle-antipode of X(109)
X(102) = Λ(X(1), X(4))
X(102) = trilinear pole of line X(6)X(652)
X(102) = Ψ(X(6), X(652))
X(102) = Ψ(X(1), X(521))
X(102) = Ψ(X(108), X(1))
X(102) = Ψ(X(651), X(63))
X(102) = Ψ(X(653), X(2))
X(102) = trilinear product of circumcircle intercepts of line X(1)X(521)
X(102) = trilinear pole wrt 2nd circumperp triangle of line X(971)X(1001)
X(102) = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {A,B,C,X(4),X(58)}}
X(102) = reflection of X(2695) in the Euler line
X(102) = reflection of X(2708) in the Brocard axis
X(102) = reflection of X(2716) in line X(1)X(3)
X(102) = X(131)-of-excentral-triangle
X(102) = X(136)-of-hexyl-triangle
X(102) = X(925)-of-2nd-circumperp-triangle
X(102) = Thomson-isogonal conjugate of X(522)
X(102) = Lucas-isogonal conjugate of X(522)
X(102) = Cundy-Parry Phi transform of X(10571)
X(102) = Cundy-Parry Psi transform of X(10570)
X(102) = Ψ(X(19), X(650))
X(102) = polar-circle-inverse of nine-point circle antipode of X(38977)
X(102) = X(36100)-Ceva conjugate of X(15629)
X(102) = X(i)-cross conjugate of X(j) for these (i,j): {3, 15379}, {1457, 1}, {2342, 2316}, {8607, 2}, {8677, 109}, {46359, 34393} X(102) = X(i)-isoconjugate of X(j) for these (i,j): {1, 515}, {2, 2182}, {4, 46974}, {8, 1455}, {9, 34050}, {31, 35516}, {63, 8755}, {80, 11700}, {102, 24034}, {108, 39471}, {109, 14304}, {521, 23987}, {649, 42718}, {650, 2406}, {652, 24035}, {653, 46391}, {656, 7452}, {2425, 4391}, {6087, 13138}, {23986, 36100}, {34393, 42076}, {36037, 42755}, {36121, 38554}
X(102) = cevapoint of X(i) and X(j) for these (i,j): {48, 2361}, {55, 2183}, {523, 3137}
X(102) = crosssum of X(40) and X(6326)
X(102) = crossdifference of every pair of points on line {23986, 46391}
X(102) = barycentric product X(i)*X(j) for these {i,j}: {1, 36100}, {6, 34393}, {7, 15629}, {63, 36121}, {75, 32677}, {83, 46359}, {92, 36055}, {109, 2399}, {664, 2432}, {1262, 15633}, {4391, 36040}, {6081, 14837}, {6332, 36067}, {32643, 35519}, {32667, 35518}
X(102) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 35516}, {6, 515}, {25, 8755}, {31, 2182}, {48, 46974}, {56, 34050}, {100, 42718}, {108, 24035}, {109, 2406}, {112, 7452}, {604, 1455}, {650, 14304}, {652, 39471}, {1946, 46391}, {2182, 24034}, {2399, 35519}, {2432, 522}, {3310, 42755}, {6081, 44327}, {7113, 11700}, {8607, 117}, {15379, 2988}, {15629, 8}, {15633, 23978}, {32643, 109}, {32667, 108}, {32674, 23987}, {32677, 1}, {32683, 9056}, {32700, 26704}, {32720, 26715}, {34393, 76}, {35183, 44765}, {36040, 651}, {36055, 63}, {36067, 653}, {36100, 75}, {36121, 92}, {46359, 141}
X(102) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 151, 117}, {3, 109, 38697}, {3, 38501, 38559}, {3, 38579, 38607}, {3, 38600, 38691}, {3, 38667, 38674}, {109, 38691, 3}, {117, 151, 10709}, {117, 6711, 2}, {6718, 38784, 631}, {10716, 10732, 10747}, {10740, 38776, 5}, {38573, 38600, 109}, {38573, 38691, 38674}, {38579, 38607, 109}, {38600, 38667, 38697}, {38667, 38691, 109}, {38674, 38697, 109}


X(103) = ANTIPODE OF X(101)

Trilinears    a/[(a - b) cot C + (a - c) cot B] : b/[(b - c) cot A + (b - a) cot C] : c/[(c - a) cot B + (c - b) cot A]
Barycentrics    a2/[(a - b) cot C + (a - c) cot B] : :
Barycentrics    a^2 / (2 a^3 - a^2 (b + c) - (b - c)^2 (b + c)) : :
Tripolars    |b c ((a - b) cot C + (a - c) cot B)| : :

Let A'B'C' be the excentral triangle. The Lemoine axes of triangles A'BC, B'CA, C'AB bound a triangle perspective to ABC at X(103). (Randy Hutson, June 27, 2018)

X(103) lies on these lines: {1, 934}, {2, 118}, {3, 101}, {4, 116}, {5, 10741}, {6, 10758}, {11, 10772}, {20, 150}, {27, 107}, {30, 2690}, {33, 57}, {35, 1803}, {36, 4845}, {40, 1292}, {46, 18413}, {55, 109}, {56, 3022}, {58, 112}, {63, 100}, {64, 4306}, {74, 2774}, {84, 7719}, {98, 2786}, {99, 1043}, {102, 928}, {104, 3887}, {105, 2254}, {106, 2424}, {110, 1790}, {111, 2824}, {140, 38764}, {182, 30554}, {295, 813}, {354, 7073}, {376, 544}, {381, 38768}, {386, 34457}, {476, 5196}, {477, 37166}, {511, 2702}, {512, 2700}, {513, 2717}, {514, 2724}, {515, 929}, {516, 927}, {517, 1308}, {518, 2736}, {519, 2737}, {520, 2738}, {521, 2739}, {522, 2723}, {523, 2688}, {524, 2740}, {525, 2741}, {549, 38774}, {550, 38766}, {572, 825}, {573, 3033}, {580, 29041}, {631, 6710}, {654, 2291}, {672, 919}, {675, 2400}, {677, 901}, {789, 24728}, {831, 9943}, {835, 34255}, {910, 971}, {916, 15380}, {1146, 18328}, {1147, 3046}, {1155, 2222}, {1290, 5536}, {1293, 1350}, {1294, 2811}, {1295, 2812}, {1296, 2813}, {1297, 9518}, {1302, 7474}, {1304, 2073}, {1309, 34234}, {1310, 12520}, {1326, 2715}, {1376, 3041}, {1458, 24016}, {1461, 3270}, {1499, 2729}, {1593, 5185}, {1617, 2192}, {1656, 38767}, {1742, 7220}, {1764, 38479}, {1796, 8701}, {1810, 6078}, {2077, 2742}, {2078, 2342}, {2093, 34930}, {2221, 32691}, {2272, 3220}, {2280, 28899}, {2705, 18860}, {2716, 7634}, {2725, 3309}, {2726, 3667}, {2727, 6000}, {2728, 6001}, {3090, 38769}, {3190, 35987}, {3333, 14760}, {3428, 28291}, {3430, 6010}, {3522, 20096}, {3523, 38772}, {3525, 20401}, {3533, 38770}, {3576, 11712}, {3587, 34925}, {3616, 11728}, {3732, 31852}, {4256, 32722}, {4257, 26715}, {4300, 29279}, {5010, 34931}, {5188, 28486}, {5542, 10136}, {5584, 6575}, {5603, 11726}, {5840, 10770}, {6011, 30271}, {6135, 6213}, {6136, 6212}, {6282, 30237}, {6577, 15622}, {7072, 36082}, {7465, 9058}, {7466, 9107}, {7688, 20219}, {8608, 32684}, {8722, 28564}, {8750, 22084}, {9085, 35365}, {9320, 12032}, {9441, 9503}, {10164, 28346}, {10246, 32630}, {10299, 35024}, {10303, 38775}, {10884, 13395}, {11825, 34112}, {13397, 20243}, {13444, 16012}, {13478, 26704}, {15730, 30282}, {17729, 36028}, {21153, 28345}, {28162, 36942}, {28469, 30270}, {28474, 30269}, {28841, 37508}, {29044, 37469}, {29055, 37575}, {29091, 31732}, {29217, 37482}, {32642, 32682}

X(103) = midpoint of X(20) and X(150)
X(103) = reflection of X(i) in X(j) for these (i,j): (4,116), (101,3), (152,118)
X(103) = isogonal conjugate of X(516)
X(103) = isotomic conjugate of X(35517)
X(103) = complement of X(152)
X(103) = anticomplement of X(118)
X(103) = X(21)-beth conjugate of X(934)

X(103) = circumcircle-antipode of X(101)
X(103) = X(115)-of-the-hexyl-triangle
X(103) = perspector of ABC and the triangle formed by reflecting line PU(10) in the sidelines of ABC
X(103) = X(114)-of-excentral-triangle
X(103) = trilinear pole of line X(6)X(657)
X(103) = Ψ(X(i),X(j)) for these (i,j): (6,657), (101,3), (190,69)
X(103) = Λ(X(1), X(7))
X(103) = trilinear pole wrt 2nd circumperp triangle of line X(1001)X(1012)
X(103) = X(99)-of-2nd-circumperp-triangle
X(103) = reflection of X(2688) in the Euler line
X(103) = reflection of X(2700) in the Brocard axis
X(103) = reflection of X(2717) in line X(1)X(3)
X(103) = Thomson-isogonal conjugate of X(514)
X(103) = Lucas-isogonal conjugate of X(514)
X(103) = Cundy-Parry Phi transform of X(3730)
X(103) = Cundy-Parry Psi transform of X(14377)
X(103) = polar conjugate of isogonal conjugate of X(32657)
X(103) = areal center of pedal triangles of PU(10)
X(103) = X(19)-isoconjugate of X(26006)
X(103) = SR(P,U), where P and U are the circumcircle intercepts of the Soddy line

X(104) = ANTIPODE OF X(100)

Trilinears    1/(-1 + cos B + cos C) : 1/(-1 + cos C + cos A) : 1/(-1 + cos C + cos B)
Trilinears    1/[b^3 + c^3 - (a^2 + bc)(b + c) + 2abc] : :
Barycentrics   a/(-1 + cos B + cos C) : b/(-1 + cos C + cos A) : c/(-1 + cos C + cos B)
Tripolars    |-1 + cos B + cos C| : :
X(104) = 2R*X(1) - 3R*X(2) + (2R - 2r)*X(3)    (Peter Moses, April 2, 2013)

Let LA be the reflection of the line X(1)X(513) in line BC, and define LB and LC cyclically. Let A' = LB∩LC, B' = LC∩LA, C' = LA∩LB. The lines AA', BB', CC' concur in X(104). (Randy Hutson, 9/23/2011)

Let A', B', C' be the intersections of the antiorthic axis and lines BC, CA, AB, resp. Let Oa, Ob, Oc be the circumcenters of AB'C', BC'A', CA'B', resp. The lines AOa, BOb, COc conucr in X(104). (Randy Hutson, December 2, 2017)

Let A'B'C' be the excentral triangle (i.e. the antiorthic triangle). Let A″B″C″ be the triangle bounded by the orthic axes of A'BC, B'CA, C'AB. Then A″B″C″ is perspective to ABC at X(104); c.f. X(8068). (Randy Hutson, December 2, 2017)

Let A'B'C' be the excentral triangle. The de Longchamps lines of triangles A'BC, B'CA, C'AB bound a triangle perspective to ABC at X(104). (Randy Hutson, June 27, 2018)

Let A'B'C' be the excentral triangle. The Hatzipolakis axes of triangles A'BC, B'CA, C'AB bound a triangle perspective to ABC at X(104). (Randy Hutson, June 27, 2018)

In the plane of a triangle ABC, let
D = the point on line AC such that angle CAB = angle DAB and |AD| = |AB|
E = the point on line AB such that angle CAB = angle CAE and |AE| = |AC|
F = the point on line BC such that angle CBA = angle FBA and |BF| = |AB|
G = the point on line AB such that angle CBA = angle CBG and |BG| = |BC|
H = the point on line BC such that angle ACB = angle ACH and |CH| = |AC|
I = the point on line AC such that angle ACB = angle ICB and |CI| = |BC|
M = circumcenter of AGI
N = circumcenter of CDF
O = circumcenter of BEH
The lines AM, BO, CN concur in X(104). See the point X(104), labeled P,.
(Benjamin Warren, October 7, 2024)

X(104) lies on these lines: {1, 109}, {2, 119}, {3, 8}, {4, 11}, {5, 5253}, {6, 10759}, {7, 934}, {9, 48}, {10, 6940}, {12, 6952}, {20, 149}, {21, 110}, {24, 3435}, {25, 3420}, {28, 107}, {30, 1290}, {32, 12199}, {35, 5559}, {36, 80}, {40, 1293}, {55, 1000}, {57, 3577}, {63, 37611}, {65, 1389}, {74, 7429}, {79, 946}, {84, 1420}, {90, 6261}, {98, 2787}, {99, 314}, {102, 3738}, {103, 3887}, {105, 885}, {106, 1769}, {111, 2830}, {112, 1108}, {140, 5260}, {145, 11248}, {150, 6516}, {165, 4900}, {177, 13444}, {182, 13194}, {186, 2766}, {238, 9365}, {256, 1064}, {294, 919}, {355, 404}, {371, 19113}, {372, 19112}, {376, 528}, {378, 3433}, {381, 22799}, {382, 22938}, {386, 34458}, {388, 6833}, {402, 12752}, {405, 5811}, {411, 18481}, {474, 5818}, {476, 1325}, {484, 13143}, {493, 12765}, {494, 12766}, {497, 6938}, {499, 6256}, {511, 2703}, {512, 2699}, {513, 953}, {514, 2717}, {516, 1308}, {517, 901}, {518, 2742}, {519, 2077}, {520, 2744}, {521, 2745}, {522, 2716}, {523, 2687}, {524, 2746}, {525, 2747}, {545, 38531}, {549, 38762}, {550, 38754}, {551, 3255}, {573, 3032}, {614, 9088}, {631, 958}, {651, 34586}, {667, 2726}, {675, 7445}, {691, 37960}, {693, 2861}, {739, 2423}, {758, 6596}, {813, 2196}, {825, 2344}, {835, 37399}, {840, 3309}, {898, 13136}, {900, 14127}, {912, 4511}, {925, 16049}, {927, 2481}, {929, 14198}, {932, 7155}, {935, 37961}, {941, 32693}, {942, 17097}, {943, 2646}, {962, 10680}, {963, 18283}, {971, 1156}, {972, 6366}, {983, 8685}, {991, 29067}, {1001, 5851}, {1039, 1472}, {1056, 6935}, {1061, 36076}, {1066, 9363}, {1125, 6920}, {1147, 3045}, {1155, 17636}, {1181, 34046}, {1201, 3073}, {1251, 18471}, {1294, 2803}, {1295, 2804}, {1296, 2805}, {1297, 2806}, {1301, 30733}, {1302, 4228}, {1304, 2074}, {1305, 2997}, {1309, 37305}, {1310, 19262}, {1311, 7443}, {1319, 2720}, {1350, 9024}, {1376, 3036}, {1381, 3308}, {1382, 3307}, {1392, 1482}, {1415, 11998}, {1436, 7003}, {1455, 1870}, {1458, 1937}, {1468, 28624}, {1470, 12832}, {1476, 12672}, {1477, 35355}, {1478, 6830}, {1479, 5533}, {1483, 11849}, {1490, 38271}, {1499, 2721}, {1503, 2722}, {1519, 5193}, {1532, 15325}, {1593, 1862}, {1602, 9912}, {1603, 9798}, {1604, 37310}, {1610, 15654}, {1617, 3427}, {1621, 2320}, {1656, 34126}, {1698, 38133}, {1699, 5561}, {1709, 7284}, {1764, 35636}, {1783, 7117}, {1791, 8707}, {1811, 6079}, {1837, 20118}, {2078, 17010}, {2099, 14497}, {2217, 26704}, {2256, 2335}, {2291, 23893}, {2298, 8687}, {2346, 12755}, {2475, 26470}, {2550, 6955}, {2551, 6967}, {2648, 2701}, {2691, 7464}, {2715, 5006}, {2718, 3667}, {2719, 6000}, {2728, 3220}, {2734, 23224}, {2777, 10767}, {2794, 10768}, {2808, 14947}, {2818, 3937}, {2841, 38513}, {2865, 3263}, {2886, 6951}, {3057, 20586}, {3062, 13462}, {3065, 34921}, {3068, 13913}, {3069, 13977}, {3085, 6977}, {3090, 6667}, {3091, 18761}, {3098, 13235}, {3100, 8759}, {3149, 5704}, {3241, 10679}, {3295, 7320}, {3296, 3304}, {3333, 5665}, {3337, 31870}, {3338, 17098}, {3359, 3872}, {3413, 36736}, {3414, 36735}, {3417, 7412}, {3418, 4219}, {3434, 6948}, {3436, 6891}, {3448, 19478}, {3476, 8069}, {3486, 8071}, {3520, 34441}, {3522, 10993}, {3523, 38760}, {3524, 6174}, {3525, 20400}, {3528, 5584}, {3533, 38758}, {3560, 3616}, {3579, 28218}, {3600, 6847}, {3601, 7160}, {3612, 7162}, {3617, 38128}, {3618, 38119}, {3623, 37622}, {3624, 15017}, {3651, 4297}, {3653, 16858}, {3655, 10031}, {3658, 6740}, {3746, 13606}, {3813, 11826}, {3816, 6965}, {3817, 33709}, {3843, 38141}, {3869, 24467}, {3871, 26285}, {3873, 37533}, {3880, 11256}, {3916, 31786}, {4180, 20779}, {4183, 23204}, {4188, 11499}, {4189, 10267}, {4193, 26492}, {4220, 9070}, {4222, 15617}, {4223, 9057}, {4224, 9056}, {4231, 5322}, {4233, 9064}, {4294, 10806}, {4295, 26437}, {4305, 26357}, {4311, 6245}, {4313, 37287}, {4317, 6845}, {4341, 8809}, {4571, 6790}, {4853, 10270}, {4855, 5534}, {4861, 37562}, {4866, 5531}, {4881, 18857}, {4973, 5535}, {4999, 6853}, {5056, 38319}, {5080, 6882}, {5151, 37391}, {5204, 6942}, {5217, 7317}, {5248, 28162}, {5251, 5316}, {5258, 6684}, {5265, 6848}, {5267, 10902}, {5284, 7489}, {5288, 11362}, {5429, 29038}, {5433, 6949}, {5434, 7680}, {5537, 26726}, {5542, 34917}, {5552, 6961}, {5555, 10531}, {5556, 38038}, {5557, 11551}, {5558, 7373}, {5560, 5691}, {5587, 6702}, {5597, 12462}, {5598, 12463}, {5606, 5901}, {5663, 13868}, {5693, 30144}, {5720, 35262}, {5732, 28291}, {5734, 12001}, {5759, 5856}, {5776, 37519}, {5777, 17614}, {5779, 35272}, {5841, 6840}, {5842, 15326}, {5844, 35000}, {5848, 6776}, {5854, 8668}, {5881, 15863}, {5886, 6912}, {5903, 21398}, {6068, 21168}, {6073, 22102}, {6135, 13454}, {6136, 13426}, {6200, 35882}, {6284, 13274}, {6361, 22770}, {6396, 35883}, {6560, 35784}, {6561, 35785}, {6584, 6595}, {6594, 21153}, {6597, 16132}, {6705, 10106}, {6763, 31806}, {6770, 22773}, {6773, 22774}, {6796, 7280}, {6797, 37582}, {6831, 18990}, {6834, 7288}, {6850, 10527}, {6852, 25466}, {6879, 10590}, {6889, 30478}, {6890, 20076}, {6897, 19843}, {6903, 11827}, {6911, 18519}, {6915, 18480}, {6923, 11680}, {6924, 18525}, {6937, 26363}, {6943, 10526}, {6945, 18516}, {6956, 10599}, {6958, 11681}, {6968, 10589}, {6972, 20060}, {6973, 10584}, {6975, 10200}, {6976, 26105}, {6986, 8701}, {7091, 12705}, {7133, 12768}, {7161, 37616}, {7330, 19861}, {7411, 9963}, {7414, 30250}, {7458, 9084}, {7459, 9083}, {7469, 9060}, {7487, 22479}, {7580, 12690}, {7581, 19014}, {7582, 19013}, {7587, 13267}, {7588, 8104}, {7589, 12748}, {7676, 12730}, {7686, 16615}, {7707, 12760}, {7709, 22680}, {7966, 35445}, {7978, 31523}, {7982, 8699}, {7991, 12653}, {8074, 32625}, {8075, 8097}, {8076, 8098}, {8077, 8103}, {8107, 12733}, {8108, 12734}, {8109, 13260}, {8110, 13261}, {8224, 12744}, {8225, 13262}, {8227, 32557}, {8583, 13227}, {8609, 32685}, {8686, 23836}, {8698, 31663}, {8982, 26325}, {9059, 19649}, {9081, 28475}, {9104, 16434}, {9342, 38042}, {9540, 13922}, {9623, 21164}, {9655, 38163}, {9657, 10894}, {9778, 9802}, {9799, 37302}, {9840, 38470}, {9845, 28226}, {9862, 22504}, {9946, 18443}, {9951, 10860}, {9956, 17531}, {10039, 12749}, {10057, 12616}, {10072, 26333}, {10073, 10572}, {10101, 10295}, {10222, 28222}, {10299, 35023}, {10303, 38763}, {10306, 25416}, {10308, 12688}, {10309, 12246}, {10427, 21151}, {10429, 37252}, {10434, 12550}, {10476, 38484}, {10702, 35014}, {10769, 23698}, {10778, 17702}, {10779, 23699}, {10783, 22756}, {10784, 22757}, {10788, 22520}, {10864, 33576}, {10882, 13244}, {10942, 27529}, {11041, 37541}, {11240, 34629}, {11411, 22659}, {11414, 13222}, {11415, 30240}, {11492, 11844}, {11493, 11843}, {11822, 13228}, {11823, 13230}, {11824, 13269}, {11825, 13270}, {11828, 13275}, {11829, 13276}, {11845, 22755}, {11846, 22761}, {11847, 22762}, {12000, 20057}, {12032, 37998}, {12243, 22565}, {12244, 22583}, {12249, 22777}, {12250, 22778}, {12251, 22779}, {12252, 22780}, {12253, 19159}, {12254, 22781}, {12255, 22782}, {12256, 22595}, {12257, 22624}, {12383, 22586}, {12519, 30238}, {12650, 15803}, {12680, 37605}, {12771, 15997}, {12772, 18456}, {13172, 22514}, {13200, 19162}, {13396, 31394}, {13587, 23961}, {13674, 22783}, {13750, 18260}, {13794, 22784}, {13886, 22763}, {13935, 13991}, {13939, 22764}, {14513, 38707}, {14800, 37710}, {14882, 37734}, {15016, 30147}, {15071, 15446}, {15173, 18398}, {15175, 37525}, {15178, 28184}, {15179, 20323}, {15910, 34600}, {15931, 17009}, {15952, 34594}, {16371, 34627}, {16417, 38074}, {17501, 31673}, {17502, 28210}, {17579, 37820}, {18230, 38131}, {18469, 33653}, {18473, 28899}, {18492, 38161}, {18493, 38044}, {18524, 28224}, {19709, 38084}, {19860, 37534}, {20050, 35251}, {21495, 26594}, {22531, 22771}, {22532, 22772}, {22533, 22776}, {23361, 37115}, {23850, 37116}, {24016, 38459}, {24390, 31775}, {24475, 34195}, {24817, 24826}, {24852, 29057}, {25919, 37017}, {26287, 37733}, {26319, 26381}, {26320, 26405}, {26322, 26439}, {26323, 26440}, {26324, 26441}, {26705, 36009}, {26712, 35921}, {28148, 30389}, {28160, 36002}, {28170, 30392}, {28180, 31662}, {28206, 37624}, {28214, 31666}, {28444, 38314}, {28469, 30269}, {28486, 30272}, {30576, 36069}, {31509, 35242}, {31659, 37291}, {32198, 38455}, {32247, 32270}, {32337, 32363}, {32558, 37234}, {32669, 32689}, {32688, 32702}, {33667, 33857}, {33812, 34486}, {33994, 38307}, {36167, 38711}, {36476, 36484}, {36529, 36543}, {36560, 36575}

X(104) = midpoint of X(20) and X(149)
X(104) = reflection of X(i) in X(j) for these (i,j): (4,11), (100,3), (153,119), (1537,1387)
X(104) = isogonal conjugate of X(517)
X(104) = isotomic conjugate of X(3262)
X(104) = complement of X(153)
X(104) = anticomplement of X(119)
X(104) = cevapoint of X(i) and X(j) for these (i,j): (1,36), (44,55)
X(104) = X(21)-beth conjugate of X(109)
X(104) = circumcircle-antipode of X(100)
X(104) = point of intersection, other than A, B, and C, of the circumcircle and Feuerbach hyperbola
X(104) = Λ(X(1), X(3))
X(104) = Ψ(X(101), X(9))
X(104) = X(125)-of-the-hexyl-triangle
X(104) = X(113)-of-excentral-triangle
X(104) = X(110)-of-2nd-circumperp-triangle
X(104) = trilinear pole of line X(6)X(650)
X(104) = Ψ(X(6), X(650))
X(104) = Ψ(X(190), X(63))
X(104) = Ψ(X(1), X(522))
X(104) = Ψ(X(19), X(649))
X(104) = Ψ(X(109), X(1))
X(104) = Ψ(X(651), X(2))
X(104) = Feuerbach hyperbola antipode of X(4)
X(104) = trilinear pole wrt 2nd circumperp triangle of line X(3)X(142)
X(104) = reflection of X(2687) in the Euler line
X(104) = reflection of X(2699) in the Brocard axis
X(104) = reflection of X(953) in line X(1)X(3)
X(104) = crossdifference of every pair of points on line X(1769)X(3310)
X(104) = Thomson-isogonal conjugate of X(513)
X(104) = Lucas-isogonal conjugate of X(513)
X(104) = Cundy-Parry Phi transform of X(8)
X(104) = Cundy-Parry Psi transform of X(56)
X(104) = trilinear product of circumcircle intercepts of line X(1)X(522)
X(104) = trilinear pole, wrt Thomson triangle, of line X(40)X(392)
X(104) = areal center of pedal triangles of PU(3)
X(104) = areal center of pedal triangles of PU(6)


X(105) = Λ(INCENTER, SYMMEDIAN POINT)

Trilinears       1/[b2 + c2 - a(b + c)] : 1/[c2 + a2 - b(c + a)] : 1/[a2 + b2 - c(a + b)]
Barycentrics   a/[b2 + c2 - a(b + c)] : b/[c2 + a2 - b(c + a)] : c/[a2 + b2 - c(a + b)]
Tripolars    |b^2 + c^2 - a (b + c)| : :

X(105) is the perspector of ABC and the (degenerate) side-triangle of the circumcevian triangles of X(3513) and X(3514). (Randy Hutson, June 7, 2019)

X(105) lies on these lines: {1, 41}, {2, 11}, {3, 277}, {4, 5511}, {5, 10743}, {6, 1002}, {7, 3423}, {8, 16048}, {9, 4712}, {10, 20656}, {20, 34547}, {21, 99}, {22, 13397}, {23, 1290}, {25, 108}, {28, 112}, {30, 2691}, {31, 57}, {36, 1308}, {37, 1390}, {44, 6017}, {56, 279}, {65, 1170}, {74, 2775}, {75, 20628}, {81, 110}, {88, 901}, {89, 4588}, {98, 2788}, {102, 2814}, {103, 2254}, {104, 885}, {106, 1022}, {107, 2833}, {111, 2837}, {115, 37014}, {165, 1054}, {175, 30385}, {176, 30386}, {183, 9067}, {186, 10101}, {230, 9090}, {238, 291}, {243, 1309}, {330, 932}, {335, 6652}, {385, 33674}, {386, 3034}, {392, 29127}, {404, 6012}, {405, 26035}, {468, 2766}, {474, 34124}, {476, 7469}, {511, 2704}, {512, 2711}, {513, 840}, {514, 2725}, {515, 2730}, {516, 2736}, {517, 2742}, {518, 1280}, {519, 2748}, {520, 2749}, {521, 2750}, {522, 2751}, {523, 2752}, {524, 2753}, {525, 2754}, {551, 8691}, {595, 6577}, {612, 3722}, {644, 1083}, {651, 1362}, {659, 676}, {663, 12032}, {666, 898}, {667, 14665}, {691, 1325}, {693, 2862}, {739, 16501}, {743, 29956}, {789, 18031}, {805, 2106}, {825, 985}, {831, 1125}, {835, 37325}, {904, 1201}, {905, 28838}, {907, 17560}, {910, 919}, {927, 1447}, {931, 37870}, {935, 2074}, {957, 16483}, {958, 1219}, {959, 1191}, {960, 1257}, {961, 1104}, {984, 28883}, {993, 29351}, {995, 29067}, {999, 14074}, {1024, 1635}, {1036, 1310}, {1064, 36516}, {1086, 1633}, {1123, 6135}, {1156, 14191}, {1193, 27667}, {1200, 9445}, {1224, 5259}, {1255, 3748}, {1261, 7081}, {1283, 4220}, {1284, 8852}, {1289, 30733}, {1294, 9520}, {1295, 9521}, {1296, 4221}, {1297, 9523}, {1304, 37963}, {1305, 2218}, {1319, 14733}, {1320, 5376}, {1332, 25048}, {1336, 6136}, {1422, 2208}, {1428, 9455}, {1456, 24016}, {1458, 23694}, {1477, 37626}, {1486, 4000}, {1724, 29303}, {1929, 2702}, {1946, 2724}, {1995, 9058}, {2006, 2078}, {2108, 15485}, {2113, 9500}, {2144, 20332}, {2191, 7289}, {2223, 6185}, {2224, 30117}, {2264, 5572}, {2282, 28624}, {2292, 29119}, {2306, 36072}, {2347, 9440}, {2402, 9061}, {2651, 6083}, {2696, 37960}, {2703, 17946}, {2717, 37815}, {2720, 3660}, {2737, 6909}, {2743, 5121}, {2864, 16087}, {2865, 3262}, {2866, 35519}, {2982, 5173}, {2990, 6099}, {3056, 37659}, {3263, 3685}, {3309, 28914}, {3322, 7336}, {3428, 15747}, {3485, 13395}, {3565, 16049}, {3576, 15746}, {3601, 6575}, {3651, 30257}, {3659, 7589}, {3757, 8707}, {3827, 35185}, {3871, 32019}, {3924, 7132}, {4227, 30247}, {4231, 30250}, {4232, 9107}, {4236, 24617}, {4239, 9070}, {4310, 24320}, {4458, 26702}, {4511, 29241}, {4578, 24820}, {4585, 10755}, {4859, 24309}, {4904, 18343}, {5020, 15252}, {5089, 36122}, {5091, 35280}, {5205, 6079}, {5211, 25495}, {5222, 37580}, {5248, 6013}, {5262, 29143}, {5268, 28226}, {5273, 25494}, {5297, 28210}, {5310, 37261}, {5322, 6186}, {5323, 5545}, {5338, 32691}, {5695, 31130}, {5992, 31129}, {6014, 35445}, {6016, 20331}, {6081, 14203}, {6169, 11051}, {6180, 9309}, {6244, 16434}, {6553, 12513}, {7077, 23612}, {7175, 20978}, {7179, 8543}, {7465, 34879}, {7766, 29227}, {7965, 37456}, {8297, 8298}, {8647, 9441}, {8652, 17024}, {8684, 33676}, {8697, 26745}, {8699, 36603}, {8708, 16830}, {8709, 8851}, {8848, 16363}, {9057, 26246}, {9059, 26227}, {9068, 26281}, {9072, 9465}, {9073, 26277}, {9081, 26249}, {9086, 26229}, {9089, 26239}, {9096, 26279}, {9104, 26245}, {9317, 28850}, {9343, 9344}, {9456, 16507}, {9508, 28471}, {9511, 15731}, {10098, 37961}, {10246, 28536}, {10387, 25878}, {10789, 30554}, {10980, 28162}, {11284, 23858}, {12329, 37650}, {12589, 26540}, {13245, 29310}, {13444, 14596}, {13595, 26711}, {14625, 19309}, {15382, 34381}, {15571, 20045}, {16466, 36080}, {16485, 26715}, {16487, 26716}, {16684, 37033}, {16693, 20470}, {16752, 16876}, {20219, 24929}, {21214, 28469}, {21511, 23407}, {21793, 28899}, {23536, 28029}, {23834, 28583}, {24174, 28575}, {24178, 37328}, {24325, 24339}, {24540, 24669}, {24808, 38665}, {25304, 26657}, {26232, 26238}, {26237, 26243}, {27789, 28196}, {28156, 30350}, {28477, 37399}, {28480, 37431}, {28879, 38316}, {28895, 38315}, {29048, 30148}, {29363, 32911}, {29681, 33637}, {32630, 37587}, {33654, 36073}, {34084, 34085}, {34446, 37541}, {36740, 38053}

X(105) = reflection of X(i) in X(j) for these (i,j): (644,1083), (1292,3)
X(105) = isogonal conjugate of X(518)
X(105) = anticomplement of X(120)
X(105) = cevapoint of X(1) and X(238)
X(105) = X(1)-Hirst inverse of X(294)
X(105) = X(i)-beth conjugate of X(j) for these (i,j): (21,101), (927,105)
X(105) = Λ(X(1), X(6))
X(105) = isotomic conjugate of X(3263)
X(105) = crossdifference of every pair of points on line X(665)X(1642)
X(105) = Ψ(X(i), X(j)) for these (i,j): (1,514), (2,650), (6,513), (57,649), (59,651), (100,2), (101,1), (190,9), (650,11), (651,6)
X(105) = reflection of X(2752) in the Euler line
X(105) = reflection of X(2711) in the Brocard axis
X(105) = reflection of X(840) in line X(1)X(3)
X(105) = X(132)-of-excentral-triangle
X(105) = X(127)-of-hexyl-triangle
X(105) = X(6)-isoconjugate of X(3912)
X(105) = inverse-in-{circumcircle, nine-point circle}-inverter of X(11)
X(105) = trilinear pole of PU(i) for these i: 46, 54
X(105) = trilinear product of PU(96)
X(105) = bicentric sum of PU(142)
X(105) = Thomson-isogonal conjugate of X(3309)
X(105) = Lucas-isogonal conjugate of X(3309)
X(105) = polar conjugate of isogonal conjugate of X(32658)
X(105) = trilinear product of circumcircle intercepts of line X(1)X(514)
X(105) = trilinear pole, wrt Thomson triangle, of line X(354)X(612)
X(105) = Conway-circle-inverse of X(38479)
X(105) = X(19)-isoconjugate of X(25083)


X(106) = Λ(INCENTER, CENTROID)

Trilinears    a/(2a - b - c) : b/(2b - c - a) : c/(2c - a -b)
Barycentrics    a2/(2a - b - c) : b2/(2b - c - a) : c2/(2c - a - b)
Tripolars    |b c (2 a - b - c)| : :

X(106) lies on these lines: {1, 88}, {2, 121}, {3, 1293}, {4, 5510}, {5, 10744}, {6, 101}, {10, 1222}, {11, 10774}, {20, 34548}, {21, 34594}, {25, 9088}, {30, 2692}, {31, 2163}, {32, 30554}, {34, 108}, {35, 28218}, {36, 901}, {39, 9327}, {40, 14664}, {41, 9336}, {42, 16057}, {55, 5577}, {56, 109}, {58, 110}, {74, 2776}, {80, 1647}, {86, 99}, {87, 932}, {98, 2789}, {102, 2815}, {103, 2424}, {104, 1769}, {105, 1022}, {107, 2839}, {111, 2843}, {112, 1474}, {172, 28864}, {182, 28564}, {187, 2702}, {190, 34587}, {238, 898}, {269, 934}, {291, 5376}, {292, 813}, {347, 1305}, {386, 2334}, {476, 7478}, {511, 2705}, {512, 2712}, {513, 2718}, {514, 2726}, {515, 2731}, {516, 2737}, {517, 2743}, {518, 2748}, {519, 1120}, {520, 2755}, {521, 2756}, {522, 2757}, {523, 2758}, {524, 2759}, {525, 2760}, {535, 19634}, {572, 28467}, {614, 998}, {649, 2384}, {663, 840}, {664, 21208}, {665, 2291}, {667, 2382}, {672, 6017}, {675, 6548}, {691, 1326}, {693, 2863}, {726, 4582}, {727, 23355}, {739, 8632}, {789, 870}, {833, 977}, {835, 3616}, {905, 2751}, {919, 1055}, {927, 1323}, {931, 4658}, {952, 3756}, {953, 1459}, {958, 3038}, {979, 8666}, {991, 28291}, {997, 16496}, {1001, 29351}, {1068, 26704}, {1086, 1387}, {1125, 1220}, {1126, 1193}, {1130, 3659}, {1145, 24864}, {1168, 1319}, {1191, 28162}, {1203, 28176}, {1279, 1308}, {1290, 37919}, {1292, 2191}, {1294, 9524}, {1295, 9525}, {1296, 9526}, {1297, 9527}, {1309, 1785}, {1310, 16491}, {1350, 28295}, {1384, 26716}, {1385, 6011}, {1386, 8691}, {1413, 4306}, {1431, 29055}, {1457, 2720}, {1458, 14733}, {1462, 23890}, {1464, 34921}, {1468, 8652}, {1480, 10269}, {1616, 8699}, {1724, 28370}, {1739, 6095}, {1795, 35011}, {1914, 28875}, {1960, 4491}, {1964, 29189}, {2112, 28895}, {2215, 36080}, {2241, 28907}, {2275, 28883}, {2279, 8693}, {2280, 28911}, {2297, 3731}, {2308, 28180}, {2342, 19619}, {2390, 35186}, {2403, 9083}, {2441, 8659}, {2703, 17954}, {2716, 6129}, {2734, 21172}, {2742, 13329}, {2983, 29163}, {3098, 28469}, {3120, 16173}, {3226, 4555}, {3227, 3570}, {3242, 35272}, {3295, 8572}, {3333, 15839}, {3582, 36590}, {3699, 6789}, {3746, 8698}, {3898, 17596}, {3915, 8697}, {3939, 22560}, {3973, 33589}, {3976, 30144}, {4251, 28852}, {4252, 28192}, {4262, 28847}, {4432, 24416}, {4511, 4694}, {4615, 9150}, {4622, 16481}, {4677, 9350}, {4738, 9457}, {4752, 20331}, {4803, 31136}, {4867, 17449}, {4945, 25055}, {5024, 6184}, {5053, 8610}, {5092, 28486}, {5127, 6083}, {5258, 28352}, {5272, 9104}, {5288, 27627}, {5297, 31514}, {5400, 38669}, {5526, 5548}, {6010, 37508}, {6013, 10013}, {6099, 36052}, {6551, 9268}, {6575, 10579}, {6577, 34444}, {6765, 7963}, {7191, 9070}, {7208, 9318}, {7290, 14074}, {7312, 17960}, {7373, 28226}, {8028, 24858}, {8643, 9097}, {8658, 28302}, {8686, 37627}, {8688, 12513}, {8692, 11194}, {9067, 37670}, {9086, 36887}, {9093, 23598}, {9624, 36939}, {10179, 37599}, {10199, 37716}, {10571, 36082}, {10703, 12740}, {10987, 28868}, {11249, 34430}, {11636, 34476}, {13397, 28011}, {14028, 21630}, {14422, 23352}, {14438, 26278}, {14996, 27950}, {15306, 37541}, {15325, 17734}, {15663, 22770}, {15955, 20323}, {16466, 28148}, {16796, 32682}, {16801, 30664}, {16969, 24047}, {16971, 25426}, {17461, 17595}, {17724, 34123}, {18047, 27195}, {18976, 34355}, {21008, 28856}, {21842, 28082}, {22744, 29038}, {22837, 24174}, {23675, 24160}, {24516, 32454}, {24625, 37680}, {25920, 26698}, {26711, 38458}, {28096, 37710}, {28528, 33878}, {28552, 37517}, {28584, 30269}, {29227, 36598}, {34977, 37592}

X(106) = reflection of X(1293) in X(3)
X(106) = isogonal conjugate of X(519)
X(106) = isotomic conjugate of X(3264)
X(106) = complement of X(21290)
X(106) = anticomplement of X(121)
X(106) = X(36)-cross conjugate of X(58)
X(106) = X(i)-beth conjugate of X(j) for these (i,j): (21,100), (901,106)
X(106) = Λ(X(1), X(2))
X(106) = X(122)-of-hexyl-triangle
X(106) = trilinear pole of line X(6)X(649)
X(106) = Ψ(X(i), X(j)) for these (i,j): (1,513), (2,514), (6,649), (9,650), (100,1), (101,6), (190,2), (651,57)
X(106) = trilinear pole wrt 2nd circumperp triangle of line X(1)X(6)
X(106) = X(107) of 2nd circumperp triangle
X(106) = trilinear pole wrt circumsymmedial triangle of line X(6)X(31)
X(106) = reflection of X(2758) in the Euler line
X(106) = reflection of X(2712) in the Brocard axis
X(106) = reflection of X(2718) in line X(1)X(3)
X(106) = X(6)-isoconjugate of X(4358)
X(106) = X(133)-of-excentral triangle
X(106) = barycentric product of PU(50)
X(106) = trilinear product of PU(98)
X(106) = eigencenter of 1st circumperp triangle
X(106) = Thomson-isogonal conjugate of X(3667)
X(106) = Lucas-isogonal conjugate of X(3667)
X(106) = polar conjugate of isogonal conjugate of X(32659)
X(106) = trilinear product of circumcircle intercepts of line X(1)X(513)
X(106) = trilinear pole, wrt Thomson triangle, of line X(1201)X(2177)
X(106) = Conway-circle-inverse of X(35636)
X(106) = polar conjugate of isotomic conjugate of X(1797)
X(106) = X(63)-isoconjugate of X(8756)
X(106) = X(1)-Ceva conjugate of X(39148)


X(107) = Ψ(SYMMEDIAN POINT, ORTHOCENTER)

Trilinears    1/[cos A (sin 2B - sin 2C)] : :
Trilinears    (sec A)/(tan B - tan C) : :
Trilinears    bc/[(b2 - c2)(b2 + c2 - a2)2] : :
Barycentrics   1/[(b2 - c2)(b2 + c2 - a2)2] : 1/[(c2 - a2)(c2 + a2 - b2)2] : 1/[(a2 - b2)(a2 + b2 - c2)2]
Tripolars    |(cos A)(tan B - tan C)| : :

X(107) = center of the bianticevian conic of X(1) and X(4), the rectangular hyperbola passing through X(1), X(4), X(19), and the vertices of their anticevian triangles. This hyperbola is the excentral isogonal conjugate of line X(40)X(2939), the anticomplementary conjugate of line X(20)X(1330), and the anticomplementary isotomic conjugate of line X(1654)X(3164). (Randy Hutson, April 9, 2016)

X(107) lies on these lines: {1, 10701}, {2, 122}, {3, 1294}, {4, 74}, {5, 10745}, {6, 10762}, {11, 10775}, {19, 2249}, {20, 3184}, {21, 1295}, {22, 15466}, {23, 2697}, {24, 1093}, {25, 98}, {27, 103}, {28, 104}, {29, 102}, {30, 2693}, {51, 275}, {55, 7158}, {56, 3324}, {76, 2366}, {83, 27373}, {92, 26702}, {99, 2797}, {100, 823}, {101, 1897}, {105, 2833}, {106, 2839}, {108, 2845}, {109, 162}, {110, 648}, {111, 393}, {112, 1637}, {154, 15576}, {158, 759}, {184, 3168}, {186, 477}, {243, 14192}, {250, 687}, {264, 1995}, {284, 8764}, {297, 2710}, {324, 13595}, {333, 2365}, {373, 37124}, {376, 36876}, {382, 23241}, {403, 11657}, {415, 2708}, {419, 2698}, {422, 2699}, {423, 2700}, {425, 2707}, {427, 29011}, {428, 29316}, {450, 511}, {458, 3066}, {459, 32064}, {460, 23700}, {468, 842}, {470, 7684}, {471, 7685}, {472, 5479}, {473, 5478}, {476, 7480}, {512, 2713}, {513, 2719}, {514, 2727}, {515, 2732}, {516, 2738}, {517, 2744}, {518, 2749}, {519, 2755}, {520, 6080}, {521, 2761}, {522, 2762}, {523, 1304}, {524, 2763}, {525, 2764}, {631, 34842}, {685, 2501}, {691, 7473}, {699, 36417}, {729, 2207}, {739, 5317}, {741, 1096}, {755, 27376}, {811, 1310}, {841, 10295}, {850, 2867}, {859, 2734}, {877, 4563}, {915, 30733}, {925, 15329}, {930, 38342}, {933, 1576}, {934, 13149}, {935, 14618}, {953, 37168}, {972, 4183}, {1075, 6759}, {1105, 22467}, {1112, 9161}, {1141, 3518}, {1172, 32726}, {1249, 6793}, {1290, 37966}, {1292, 4238}, {1293, 9524}, {1296, 4235}, {1299, 3542}, {1305, 4243}, {1325, 2694}, {1559, 15311}, {1624, 1632}, {1625, 6570}, {1651, 34601}, {1783, 36080}, {1857, 28471}, {1981, 36516}, {1990, 15448}, {2073, 2688}, {2074, 2687}, {2075, 2695}, {2291, 8748}, {2367, 18027}, {2370, 7419}, {2374, 21447}, {2382, 34856}, {2404, 9064}, {2489, 9091}, {2691, 7476}, {2696, 7482}, {2714, 18344}, {2716, 17515}, {2722, 17924}, {2724, 37908}, {2752, 37963}, {2770, 37778}, {2868, 17984}, {3079, 11206}, {3090, 36520}, {3124, 6531}, {3146, 5896}, {3183, 6225}, {3563, 6353}, {3565, 4226}, {3616, 11732}, {3658, 13397}, {3952, 29163}, {4228, 26703}, {4242, 6011}, {4244, 26706}, {5064, 29322}, {5094, 14388}, {5379, 6099}, {5502, 33640}, {5640, 36794}, {5966, 14569}, {5994, 36309}, {5995, 36306}, {6015, 6059}, {6081, 7253}, {6587, 23590}, {6618, 11433}, {6761, 18400}, {6995, 29180}, {7009, 15168}, {7192, 24016}, {7337, 35108}, {7426, 37765}, {7488, 18284}, {7493, 17907}, {7722, 16169}, {7728, 11251}, {8057, 15384}, {8749, 13479}, {8750, 28624}, {8887, 19169}, {9076, 32085}, {9308, 35259}, {10101, 37965}, {10282, 14363}, {10301, 16264}, {10425, 14221}, {11005, 12828}, {11634, 20187}, {12032, 31905}, {12121, 32418}, {13195, 17409}, {13395, 18026}, {13398, 30512}, {13597, 34484}, {14006, 29056}, {14013, 28848}, {14185, 30252}, {14187, 30253}, {14615, 15259}, {14978, 18369}, {14979, 16337}, {15149, 28838}, {15440, 32676}, {16868, 22751}, {18809, 32417}, {20189, 36829}, {21396, 37814}, {23591, 23976}, {23701, 37855}, {23999, 36066}, {24000, 36069}, {24021, 36068}, {26714, 35325}, {28145, 31902}, {28173, 31900}, {28193, 31903}, {28197, 31901}, {28842, 31904}, {28844, 31909}, {28857, 31914}, {28861, 31907}, {28884, 31908}, {28892, 31921}, {32681, 32695}, {34473, 36176}, {36079, 36118}, {37760, 37766}

X(107) = reflection of X(i) in X(j) for these (i,j): (4,133), (1294,3)
X(107) = isogonal conjugate of X(520)
X(107) = isotomic conjugate of X(3265)
X(107) = complement of X(34186)
X(107) = anticomplement of X(122)
X(107) = cevapoint of X(4) and X(523)
X(107) = cevapoint of X(24007) and X(24008) (the Kiepert hyperbola intercepts of the orthic axis)
X(107) = X(i)-cross conjugate of X(j) for these (i,j): (24,250), (108,162), (523,4)
X(107) = trilinear pole of line X(4)X(6)
X(107) = Ψ(X(i),X(j)) for these (i,j): (1,29), (3,2), (6,4), (4,51), (54,4), (64,4), (65,4), (67,4), (69,4)
X(107) = intersection of reflections in sides of ABC of line X(4)X(51)
X(107) = reflection of X(1304) in the Euler line
X(107) = reflection of X(2713) in the Brocard axis
X(107) = reflection of X(2719) in line X(1)X(3)
X(107) = inverse-in-polar-circle of X(125)
X(107) = inverse-in-{circumcircle, nine-point circle}-inverter of X(132)
X(107) = pole wrt polar circle of trilinear polar of X(525) (line X(122)X(125))
X(107) = X(48)-isoconjugate (polar conjugate) of X(525)
X(107) = X(1577)-isoconjugate of X(577)
X(107) = crossdifference of every pair of points on line X(1636)X(2972)
X(107) = X(134)-of-excentral-triangle
X(107) = circumcircle intercept, other than A, B, C, of conic {A,B,C,PU(157)}}
X(107) = Thomson-isogonal conjugate of X(6000)
X(107) = Lucas-isogonal conjugate of X(6000)
X(107) = barycentric product of Steiner circumellipse intercepts of van Aubel line
X(107) = trilinear product of circumcircle intercepts of line X(1)X(29)


X(108) = Ψ(CIRCUMCENTER, INCENTER)

Trilinears       a/(sec B - sec C) : b/(sec C - sec A): c/(sec A - sec B)
                        = g(a,b,c) : g(b,c,a) : g(c,a,b) where g(a,b,c) = 1/[(b - c)(b + c - a)(b2 + c2 - a2)]
Barycentrics  a2/(sec B - sec C) : b2/(sec C - sec A): c2/(sec A - sec B)
Tripolars    |b c (sec B - sec C)| : :

X(108) lies on these lines: {1, 102}, {2, 123}, {3, 1295}, {4, 11}, {5, 10746}, {6, 10763}, {7, 1013}, {9, 3213}, {12, 451}, {19, 2291}, {20, 34550}, {23, 37798}, {24, 915}, {25, 105}, {28, 225}, {30, 2694}, {33, 57}, {34, 106}, {35, 37289}, {36, 1785}, {40, 207}, {53, 21773}, {55, 196}, {59, 6099}, {63, 2365}, {65, 74}, {73, 3194}, {81, 20122}, {92, 1311}, {98, 1402}, {99, 811}, {100, 653}, {101, 1783}, {107, 2845}, {109, 1020}, {110, 162}, {111, 1880}, {112, 1415}, {154, 34032}, {158, 32706}, {165, 1767}, {171, 29056}, {186, 2687}, {197, 17903}, {198, 1249}, {204, 223}, {222, 3195}, {226, 4183}, {240, 1758}, {241, 28838}, {242, 2726}, {243, 2723}, {273, 675}, {281, 3209}, {318, 404}, {331, 767}, {342, 1005}, {347, 35988}, {378, 1470}, {386, 34456}, {388, 406}, {393, 2178}, {411, 1895}, {429, 961}, {468, 2752}, {475, 7288}, {476, 34922}, {477, 36130}, {511, 2707}, {512, 2714}, {513, 2720}, {514, 2728}, {515, 2733}, {516, 2739}, {517, 2745}, {518, 2750}, {519, 2756}, {520, 2761}, {521, 6081}, {522, 2765}, {523, 2766}, {524, 2767}, {608, 739}, {644, 29163}, {648, 931}, {656, 2762}, {658, 6183}, {664, 1310}, {676, 6087}, {691, 7476}, {692, 15439}, {727, 1395}, {741, 1396}, {840, 1876}, {901, 7012}, {917, 2352}, {919, 6591}, {925, 3658}, {927, 13149}, {929, 17924}, {934, 36118}, {935, 37965}, {944, 8283}, {953, 1319}, {999, 34231}, {1014, 7282}, {1033, 1604}, {1035, 37818}, {1041, 28017}, {1055, 2202}, {1113, 2586}, {1114, 2587}, {1119, 15728}, {1148, 11491}, {1172, 1400}, {1214, 1297}, {1231, 2366}, {1284, 17985}, {1290, 37964}, {1292, 2283}, {1293, 9525}, {1294, 3651}, {1296, 9531}, {1299, 31385}, {1300, 31384}, {1305, 7437}, {1331, 24029}, {1361, 38513}, {1376, 7046}, {1388, 28203}, {1398, 4186}, {1403, 15323}, {1409, 26717}, {1423, 2212}, {1426, 11363}, {1429, 2356}, {1430, 1458}, {1435, 1477}, {1441, 2373}, {1447, 2862}, {1461, 8059}, {1465, 33849}, {1490, 1712}, {1577, 2769}, {1598, 15251}, {1621, 37295}, {1745, 7114}, {1750, 7008}, {1753, 15803}, {1763, 15278}, {1784, 2695}, {1824, 28471}, {1827, 38451}, {1828, 38452}, {1844, 15932}, {1847, 2369}, {1861, 2751}, {1872, 37582}, {1877, 2718}, {1887, 28173}, {1946, 23984}, {1947, 13588}, {1990, 19297}, {1995, 37800}, {2074, 12030}, {2078, 2717}, {2099, 28159}, {2149, 35182}, {2222, 7649}, {2223, 2724}, {2270, 7129}, {2333, 35106}, {2371, 7079}, {2405, 9107}, {2501, 9090}, {2689, 24006}, {2693, 36001}, {2697, 37959}, {2699, 5061}, {2715, 36104}, {2722, 7178}, {2725, 5236}, {2732, 3465}, {2734, 6905}, {2757, 38462}, {2818, 15501}, {2975, 17555}, {3176, 11500}, {3220, 34050}, {3339, 28149}, {3340, 28163}, {3361, 28193}, {3428, 37410}, {3518, 26707}, {3565, 4236}, {3600, 4194}, {3616, 11733}, {3666, 29206}, {3937, 34051}, {4200, 5265}, {4213, 15168}, {4223, 37695}, {4232, 9061}, {4559, 36080}, {4564, 29241}, {4566, 13395}, {4617, 24016}, {4998, 35574}, {5125, 19850}, {5142, 11392}, {5221, 28145}, {5253, 11109}, {5307, 16878}, {5379, 6083}, {5435, 35994}, {5757, 37538}, {5897, 30267}, {5930, 8885}, {5951, 14882}, {6079, 15742}, {6129, 24033}, {6223, 7338}, {6335, 8707}, {6353, 7337}, {6588, 23985}, {6851, 9645}, {7011, 7580}, {7071, 15731}, {7128, 14733}, {7146, 28844}, {7477, 10420}, {7677, 17923}, {8064, 36049}, {8807, 15324}, {9056, 24035}, {10058, 11798}, {10425, 36105}, {11011, 28185}, {11347, 18678}, {11510, 38295}, {12031, 15148}, {12832, 18341}, {13397, 13589}, {13462, 28233}, {14987, 37117}, {16757, 36093}, {17074, 26910}, {17080, 35996}, {17916, 20613}, {18421, 28171}, {21147, 36103}, {21189, 35183}, {23832, 30236}, {23890, 28291}, {24027, 35187}, {26377, 37384}, {32667, 32683}, {32685, 32702}, {32735, 35185}, {34588, 36100}, {35184, 36109}, {35186, 36112}, {35188, 36115}, {35189, 36116}, {36002, 37769}, {36098, 36099}, {36508, 37055}, {37371, 37797}

X(108) = reflection of X(1295) in X(3)
X(108) = isogonal conjugate of X(521)
X(108) = isotomic conjugate of X(35518)
X(108) = complement of X(34188)
X(108) = anticomplement of X(123)
X(108) = X(162)-Ceva conjugate of X(109)
X(108) = cevapoint of X(i) and X(j) for these (i,j): (56,513), (429,523)
X(108) = X(513)-cross conjugate of X(4)
X(108) = crosspoint of X(107) and X(162)
X(108) = crosssum of X(520) and X(656)
X(108) = X(i)-beth conjugate of X(j) for these (i,j): (21,102), (162,108)
X(108) = point of intersection, other than A, B, and C, of the circumcircle and hyperbola {A,B,C,PU(18)}}
X(108) = trilinear pole of line X(6)X(19) (the polar of X(4391) wrt polar circle)
X(108) = pole wrt polar circle of trilinear polar of X(4391) (line X(11)X(123))
X(108) = X(48)-isoconjugate (polar conjugate) of X(4391)
X(108) = X(1577)-isoconjugate of X(2193)
X(108) = concurrence of the reflections of line X(4)X(65) in the sidelines of ABC
X(108) = Ψ(X(i),Xj)) for these (i,j): (1,4), (3,1), (4,65), (6,19), (7,4), (8,4), (9,4), (29,1), (69,7), (80,4)
X(108) = reflection of X(2766) in the Euler line
X(108) = reflection of X(2714) in the Brocard axis
X(108) = reflection of X(2720) in line X(1)X(3)
X(108) = inverse-in-polar-circle of X(11)
X(108) = X(135)-of-excentral-triangle
X(108) = barycentric product of PU(76)
X(108) = trilinear product of PU(100)
X(108) = Thomson-isogonal conjugate of X(6001)
X(108) = Lucas-isogonal conjugate of X(6001)
X(108) = trilinear product of circumcircle intercepts of line X(1)X(4)
X(108) = barycentric product of circumcircle intercepts of line X(2)X(92)


X(109) = Ψ(INCENTER, CIRCUMCENTER)

Trilinears    a/(cos B - cos C) : b/(cos C - cos A) : c/(cos A - cos B)
Trilinears    a/[(b - c)(b + c - a)] : :
Barycentrics  a2/(cos B - cos C) : b2/(cos C - cos A): c2/(cos A - cos B)
Barycentrics    a^2 (cos^2 A + cos B cos C - cos C cos A - cos A cos B) : :
Tripolars    |b c (cos B - cos C)| : :

If the line X(1)X(4) is reflected in every side of triangle ABC, then the reflections concur in X(109). (Randy Hutson, 9/23/2011)

Let P be a point on line X(1)X(4) other than X(4). Let A' be the reflection of P in BC, and define B' and C' cyclically. The circumcircles of CA'B', BC'A', and CA'B' concur in X(109). (Randy Hutson, December 26, 2011)

Let Q be a point on line X(2)X(7) other than X(2). Let A'B'C' be the cevian triangle of Q. Let A″ be the {B,C}-harmonic conjugate of A' (or equivalently, A″ = BC∩B'C'), and define B″ and C″ cyclically. The circumcircles of AB″C″, BC″A″, CA″B″ concur in X(109). (Randy Hutson, December 26, 2011)

Let A', B', C' be the Fuhrmann triangle. The circumcircles of AB'C', BC'A', CA'B' concur in X(109). (Randy Hutson, December 26, 2011)

Let A', B', C' be the intersections of the Gergonne line and lines BC, CA, AB, respectively. The circumcircles of AB'C', BC'A', CA'B' concur in X(109). (Randy Hutson, December 26, 2011)

Let P and Q be the points where the line tangent to the incircle at X(11) intersects the circumcircle. Let L(P) be the line through P, other than PQ, that is tangent to the incircle; let L(Q) be the line through Q, other than PQ, that is tangent to the incircle. Then X(109) = L(P)∩L(Q). (Piotr Ambroszczyk, December 21, 2016)

X(109) lies on these lines: {1, 104}, {2, 124}, {3, 102}, {4, 117}, {5, 10747}, {6, 2291}, {7, 675}, {8, 2370}, {9, 2199}, {10, 1935}, {11, 10777}, {12, 2372}, {19, 8775}, {20, 151}, {21, 37558}, {30, 2695}, {31, 57}, {33, 1709}, {34, 46}, {35, 73}, {36, 953}, {37, 24863}, {40, 255}, {42, 2003}, {48, 32726}, {55, 103}, {56, 106}, {58, 65}, {59, 901}, {63, 8270}, {77, 35258}, {79, 1141}, {85, 767}, {98, 171}, {99, 643}, {100, 651}, {101, 654}, {107, 162}, {108, 1020}, {110, 1813}, {111, 1400}, {112, 163}, {140, 38776}, {154, 7011}, {165, 212}, {181, 20670}, {187, 17966}, {190, 6648}, {191, 201}, {204, 1767}, {213, 35106}, {220, 2371}, {225, 1300}, {227, 3579}, {238, 2726}, {241, 2725}, {260, 7597}, {269, 15728}, {278, 917}, {279, 2369}, {283, 4296}, {284, 296}, {307, 2373}, {349, 2367}, {376, 38783}, {381, 38780}, {386, 11509}, {388, 5264}, {394, 2365}, {476, 13486}, {477, 15228}, {478, 573}, {484, 2687}, {511, 2708}, {512, 2701}, {513, 2222}, {514, 929}, {515, 2734}, {516, 1936}, {517, 1455}, {518, 2751}, {519, 2757}, {520, 2762}, {521, 2765}, {522, 1309}, {523, 2689}, {524, 2768}, {525, 2769}, {549, 38786}, {550, 38778}, {553, 9108}, {572, 29068}, {577, 1630}, {579, 608}, {581, 11507}, {602, 15803}, {604, 739}, {631, 6711}, {644, 6574}, {649, 919}, {652, 7115}, {656, 2766}, {658, 927}, {660, 8684}, {661, 9090}, {662, 931}, {663, 1262}, {677, 4105}, {689, 4572}, {692, 1461}, {727, 1397}, {738, 2377}, {741, 1402}, {748, 31231}, {750, 5219}, {753, 1469}, {761, 7146}, {765, 6079}, {789, 4554}, {810, 2714}, {835, 4427}, {840, 902}, {846, 15168}, {896, 2752}, {898, 4564}, {905, 2728}, {932, 3573}, {933, 36134}, {934, 4617}, {940, 29310}, {946, 3075}, {952, 18339}, {958, 3040}, {959, 34281}, {993, 24806}, {995, 1470}, {999, 15306}, {1026, 4571}, {1035, 2122}, {1038, 12514}, {1042, 37583}, {1046, 15556}, {1086, 15253}, {1106, 1420}, {1110, 2742}, {1113, 1822}, {1114, 1823}, {1149, 5193}, {1155, 1456}, {1174, 20229}, {1201, 38452}, {1210, 3073}, {1214, 2328}, {1252, 6078}, {1253, 1419}, {1279, 3660}, {1284, 5061}, {1290, 4017}, {1292, 2814}, {1293, 2815}, {1294, 2816}, {1296, 2819}, {1297, 5285}, {1304, 36034}, {1305, 4566}, {1308, 3669}, {1310, 6516}, {1319, 2718}, {1365, 24836}, {1376, 3042}, {1393, 2964}, {1395, 1707}, {1401, 28485}, {1404, 2384}, {1405, 28317}, {1406, 4306}, {1407, 1477}, {1413, 10306}, {1423, 9082}, {1425, 3145}, {1428, 2382}, {1429, 14665}, {1434, 2368}, {1435, 2376}, {1436, 22124}, {1445, 9061}, {1447, 9073}, {1448, 37550}, {1450, 5315}, {1451, 3339}, {1459, 1618}, {1460, 20678}, {1464, 5172}, {1466, 16466}, {1468, 3340}, {1490, 2956}, {1492, 30670}, {1496, 1697}, {1497, 3333}, {1498, 12330}, {1499, 2735}, {1621, 17074}, {1656, 38779}, {1702, 3076}, {1703, 3077}, {1710, 1825}, {1724, 1788}, {1725, 1727}, {1735, 1870}, {1736, 1776}, {1737, 1877}, {1745, 6796}, {1758, 5018}, {1764, 34045}, {1780, 7098}, {1783, 35349}, {1794, 3466}, {1807, 2771}, {1818, 2750}, {1836, 5348}, {1918, 6015}, {1923, 3503}, {1943, 32932}, {1955, 2655}, {1995, 34141}, {2006, 3120}, {2077, 2745}, {2177, 28535}, {2187, 7125}, {2192, 38288}, {2223, 12032}, {2285, 4264}, {2293, 38451}, {2352, 29015}, {2360, 22341}, {2406, 9056}, {2425, 26715}, {2594, 8614}, {2605, 34921}, {2617, 3658}, {2688, 6357}, {2690, 7178}, {2699, 5143}, {2700, 17798}, {2702, 7180}, {2724, 9441}, {2729, 7181}, {2730, 3309}, {2731, 3667}, {2732, 6000}, {2733, 6001}, {2737, 30719}, {2739, 9371}, {2756, 5440}, {2760, 3712}, {2810, 22148}, {2829, 18340}, {2860, 24002}, {2861, 22464}, {2862, 9436}, {2864, 4077}, {2866, 3263}, {3028, 13868}, {3033, 20804}, {3072, 4292}, {3074, 6684}, {3090, 38781}, {3157, 11248}, {3173, 3190}, {3197, 15905}, {3218, 4318}, {3241, 14648}, {3295, 28193}, {3303, 28227}, {3451, 20228}, {3476, 37610}, {3485, 37522}, {3523, 38784}, {3533, 38782}, {3550, 15323}, {3563, 36051}, {3576, 11713}, {3585, 18426}, {3616, 11734}, {3652, 35194}, {3659, 6733}, {3699, 8706}, {3722, 37736}, {3744, 17625}, {3746, 28173}, {3750, 29308}, {3937, 20999}, {3952, 9059}, {3955, 29056}, {4220, 34027}, {4295, 37530}, {4303, 10902}, {4347, 37591}, {4418, 6358}, {4557, 8694}, {4569, 34083}, {4570, 6083}, {4587, 29163}, {4620, 9150}, {4654, 9103}, {4848, 5247}, {4998, 8709}, {5010, 28159}, {5089, 8558}, {5127, 12030}, {5204, 28235}, {5217, 28163}, {5248, 37523}, {5255, 10106}, {5297, 29007}, {5398, 36279}, {5399, 11849}, {5435, 9083}, {5537, 34143}, {5563, 28219}, {5603, 11727}, {5687, 9370}, {5840, 10771}, {5880, 37695}, {5897, 8803}, {5903, 14987}, {6014, 8683}, {6180, 29352}, {6244, 7074}, {6327, 28774}, {6589, 23979}, {6610, 19624}, {6745, 23693}, {6759, 20764}, {6788, 12832}, {7070, 10860}, {7179, 9075}, {7186, 29096}, {7248, 28574}, {7280, 28203}, {7292, 37789}, {7299, 24914}, {7339, 24016}, {7411, 34035}, {7580, 34032}, {7589, 34026}, {7676, 34028}, {7991, 34039}, {8059, 32652}, {8075, 34025}, {8076, 34034}, {8107, 34037}, {8108, 34038}, {8224, 34031}, {8543, 37633}, {8679, 34142}, {8757, 11499}, {9058, 24029}, {9095, 30653}, {9105, 21454}, {9363, 37588}, {9778, 18623}, {10303, 38787}, {10434, 34044}, {10700, 20586}, {11246, 26708}, {12709, 35672}, {13138, 31511}, {13589, 33637}, {13597, 37731}, {14074, 23890}, {14107, 37527}, {14115, 37815}, {14628, 24402}, {14647, 34231}, {14827, 20995}, {15252, 38357}, {15386, 35183}, {15439, 32651}, {15931, 22053}, {16173, 29008}, {18838, 30117}, {19369, 28559}, {22097, 29206}, {22329, 37856}, {22456, 36036}, {23071, 35000}, {23850, 30493}, {23987, 26704}, {24026, 24410}, {24033, 36044}, {25440, 37694}, {25577, 28847}, {25882, 36949}, {25938, 30827}, {25968, 26932}, {26227, 28968}, {26264, 29497}, {26740, 29821}, {28148, 35327}, {28777, 29473}, {28780, 33086}, {28844, 37586}, {28848, 37580}, {32643, 32683}, {32669, 32685}, {32675, 34073}, {32687, 36046}, {34234, 34589}, {34789, 35015}, {35448, 37483}, {36040, 36067}, {36048, 36118}

X(109) = midpoint of X(20) and X(151)
X(109) = reflection of X(i) in X(j) for these (i,j): (4,117), (102,3)
X(109) = isogonal conjugate of X(522)
X(109) = complement of X(33650)
X(109) = anticomplement of X(124)
X(109) = X(i)-Ceva conjugate of X(j) for these (i,j): (59,56), (162,108)
X(109) = cevapoint of X(65) and X(513)
X(109) = X(i)-cross conjugate of X(j) for these (i,j): (56,59), (513,58)
X(109) = crosspoint of X(110) and X(162)
X(109) = crosssum of X(i) and X(j) for these (i,j): (523,656), (652,663)
X(109) = crossdifference of every pair of points on line X(11)X(1146)
X(109) = X(i)-aleph conjugate of X(j) for these (i,j): (100,1079), (162,580), (651,223)
X(109) = X(i)-beth conjugate of X(j) for these (i,j): (21,104), (59,109), (100,100), (110,109), (765,109), (901,109)
X(109) = circumcircle-antipode of X(102)
X(109) = trilinear product X(1381)*X(1382)
X(109) = isogonal conjugate of isotomic conjugate of trilinear pole of line X(2)X(7)
X(109) = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {A,B,C,PU(19)}
X(109) = trilinear pole of line X(6)X(41)
X(109) = trilinear pole wrt 1st circumperp triangle of line X(971)X(1158)
X(109) = trilinear pole, wrt circumtangential triangle, of line X(3)X(10)
X(109) = X(925)-of-1st-circumperp-triangle
X(109) = X(136)-of-excentral-triangle
X(109) = X(131)-of-hexyl-triangle
X(109) = Ψ(X(i),X(j)) for these (i,j): (1,3), (2,7), (3,73), (4,1), (6,41), (7,20), (21,2), (69,73), (77,3)
X(109) = reflection of X(2689) in the Euler line
X(109) = reflection of X(2701) in the Brocard axis
X(109) = reflection of X(2222) in line X(1)X(3)
X(109) = X(6)-isoconjugate of X(4391)
X(109) = X(19)-isoconjugate of X(6332)
X(109) = X(92)-isoconjugate of X(652)
X(109) = X(1577)-isoconjugate of X(284)
X(109) = barycentric product of PU(57)
X(109) = trilinear product of PU(102)
X(109) = Thomson isogonal conjugate of X(515)
X(109) = Lucas isogonal conjugate of X(515)
X(109) = isotomic conjugate of polar conjugate of X(32674)
X(109) = polar conjugate of isogonal conjugate of X(32660)
X(109) = intersection of antipedal lines of circumcircle intercepts of line X(3)X(10)
X(109) = barycentric product of circumcircle intercepts of line X(2)X(7)
X(109) = Conway-circle-inverse of X(35649)


X(110) = FOCUS OF KIEPERT PARABOLA

Trilinears     csc(B - C) : csc(C - A) : csc(A -B)
Trilinears     cos(B - C) - cos(C - A) cos(A - B) : :
Trilinears     a/(b2 - c2) : :
Trilinears    sec(B - C) - sec(C - A) sec(A - B) : :
Barycentrics    a2/(b2 - c2) : :
Tripolars    |sin(B - C)| : :
Tripolars    |b c (b^2 - c^2)| : :

X(110) is the center of the Stammler hyperbola, SH, which is the rectangular hyperbola that passes through X(1), X(3), X(6), X(155), X(159), X(195), X(399), X(1498), X(2916), X(2917), X(2918), X(2929), X(2930), X(2931), X(2935), X(2948), X(3511), the excenters, and the vertices of the tangential triangle. SH is the bianticevian conic of X(1) and X(6) and the antipedal-anticevian conic of X(3). SH is the tangential isogonal conjugate of the Euler line, the tangential isotomic conjugate of the van Aubel line, the excentral isogonal conjugate of line X(30)X(40), and the excentral isotomic conjugate of line X(191)X(2938). SH is tangent to the Euler line at X(3) and meets the line at infinity (and the Jerabek hyperbola, other than at X(3) and X(6)) at X(2574) and X(2575). SH is the locus of a point P for which the P-Brocard triangle is perspective to ABC. (Randy Hutson, 9/23/2011, 1/29/2015)

J. W. Clawson, "Points on the circumcircle," American Mathematical Monthly 32 (1925) 169-174.

Roland H. Eddy and R. Fritsch, "The conics of Ludwig Kiepert: a comprehensive lesson in the geometry of the triangle," Mathematics Magazine 67 (1994) 188-205.

Benedetto Scimemi, "Paper-folding and Euler's Theorem Revisited," Forum Geometricorum.

Scimemi proves that if the Euler line is reflected in every side of triangle ABC, then the three reflections concur in X(110).

Seven constructions from Randy Hutson, January 29, 2015:

(1) Let P be a point, other than X(4), on Euler line. Let A' be the reflection of P in BC, and define B' and C' cyclically. The circumcircles of AA'B', BC'A', and CA'B' concur in X(110).

(2) Let Q be a point on line X(2)X(6) other than X(2). Let A'B'C' be the cevian triangle of Q. Let A″ be the {B,C}-harmonic conjugate of A' (or equivalently, A″ = BC∩B'C'), and define B″, C″ cyclically. The circumcircles of AB″C″, BC″A″, CA″B″ concur in X(110).

(3) Let La, Lb, Lc be the lines through A, B, C, resp. parallel to the Euler line. Let Ma, Mb, Mc be the reflections of BC, CA, AB in La, Lb, Lc, resp. Let A' = Mb∩Mc, and define B', C' cyclically. Triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. Let A″B″C″ be the reflection of A'B'C' in the Euler line. The triangle A″B″C″ is homothetic to ABC, with center of homothety X(125) and centroid X(110). (See Hyacinthos #16741/16782, Sep 2008.)

(4) Let A', B', C' be the intersections of the Lemoine axis and lines BC, CA, AB, resp. The circumcircles of AB'C', BC'A', CA'B' concur in X(110).

(5) Let A', B', C' be the intersections of line X(36)X(238) and lines BC, CA, AB, resp. The circumcircles of AB'C', BC'A', CA'B' concur in X(110).

(6) Let A', B', C' be the intersections of the Euler line and lines BC, CA, AB, resp. Let Oa, Ob, Oc be the circumcenters of AB'C', BC'A', CA'B', resp. The lines AOa, BOb, COc concur in X(110).

(7) Let Na be the reflection of X(5) in the perpendicular bisector of BC, and define Nb, Nc cyclically. X(110) = X(2070) of NaNbNc.

Let A2B2C2 and A3B3C3 be the 2nd and 3rd Parry triangles. Let A' be the barycentric product A2*A3, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(110). (Randy Hutson, February 10, 2016)

Let A'B'C' be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung). A'B'C' is homothetic to the anticomplementary triangle at X(110). The nine-point circle of A'B'C' touches the circumcircle of ABC at X(110). Also, X(110) = X(125)-of-A'B'C'. (Randy Hutson, Novermber 2, 2016)

Let A' be the trilinear pole of the perpendicular bisector of BC, and define B', C' cyclically. A'B'C' is homothetic to the circumanticevian triangle of X(4) at X(110). X(110) is also the barycentric product A'*B'*C'. (Randy Hutson, January 29, 2018)

Let OA be the circle centered at the A-vertex of the AAOA triangle and passing through A; define OB and OC cyclically. Circles OA, OB, OC concur in X(110). (Randy Hutson, August 30, 2020)

X(110) lies on the circumcircle, Walsmith rectangular hyperbola, Parry circle, and these lines: {1, 60}, {2, 98}, {3, 74}, {4, 113}, {5, 49}, {6, 111}, {10, 2372}, {11, 215}, {12, 2477}, {13, 3200}, {14, 3201}, {15, 2378}, {16, 2379}, {17, 3205}, {18, 3206}, {20, 146}, {21, 104}, {22, 154}, {23, 323}, {24, 155}, {25, 1112}, {26, 7731}, {27, 917}, {28, 915}, {29, 2988}, {30, 477}, {31, 593}, {32, 729}, {33, 9637}, {35, 501}, {36, 5127}, {39, 755}, {40, 37405}, {41, 35106}, {42, 28482}, {48, 2249}, {50, 32730}, {51, 1994}, {52, 2383}, {55, 3024}, {56, 3028}, {58, 106}, {59, 2222}, {61, 2380}, {62, 2381}, {63, 6061}, {64, 5896}, {65, 229}, {66, 15116}, {67, 141}, {68, 7505}, {69, 206}, {75, 21430}, {76, 2367}, {81, 105}, {83, 3203}, {86, 675}, {97, 418}, {99, 690}, {100, 643}, {101, 163}, {102, 283}, {103, 1790}, {107, 648}, {108, 162}, {109, 1813}, {112, 1625}, {115, 3044}, {116, 3046}, {126, 3048}, {140, 10264}, {143, 195}, {148, 16278}, {149, 19642}, {159, 12220}, {165, 9904}, {183, 9769}, {185, 17701}, {186, 7722}, {187, 352}, {190, 835}, {193, 1974}, {199, 22139}, {213, 2375}, {235, 22750}, {237, 2080}, {248, 9513}, {249, 512}, {250, 520}, {251, 694}, {255, 26701}, {270, 3562}, {274, 767}, {275, 30506}, {284, 2291}, {290, 36901}, {324, 436}, {325, 2857}, {333, 1311}, {343, 10192}, {351, 526}, {353, 574}, {355, 12889}, {371, 7598}, {372, 7599}, {373, 575}, {376, 541}, {378, 11472}, {381, 7699}, {382, 1539}, {384, 35399}, {385, 9418}, {386, 3031}, {389, 12227}, {395, 11134}, {396, 11137}, {402, 13212}, {403, 32123}, {467, 23233}, {468, 3564}, {476, 523}, {489, 30427}, {490, 30428}, {493, 13215}, {494, 13216}, {513, 1290}, {514, 2690}, {515, 2695}, {516, 2688}, {517, 1325}, {518, 2752}, {519, 2758}, {521, 2766}, {522, 2689}, {524, 2770}, {525, 935}, {539, 37943}, {543, 9144}, {546, 37472}, {548, 14677}, {549, 11694}, {550, 8718}, {560, 715}, {568, 12106}, {569, 3090}, {573, 34453}, {576, 9716}, {577, 26717}, {578, 3091}, {588, 8956}, {595, 849}, {599, 6325}, {620, 15357}, {631, 6699}, {632, 37471}, {645, 3952}, {647, 2715}, {649, 2702}, {650, 9090}, {660, 4599}, {663, 2701}, {664, 1305}, {667, 2703}, {668, 839}, {669, 805}, {670, 689}, {671, 5465}, {677, 23090}, {681, 823}, {685, 850}, {693, 2864}, {699, 1501}, {707, 18899}, {719, 1923}, {727, 2206}, {737, 3117}, {739, 1333}, {745, 1964}, {753, 3736}, {758, 12030}, {789, 799}, {813, 1252}, {825, 28860}, {826, 1287}, {827, 4611}, {833, 3888}, {840, 3286}, {841, 7464}, {846, 19655}, {858, 1503}, {859, 953}, {868, 6033}, {880, 4609}, {892, 8599}, {898, 4567}, {901, 1618}, {902, 1326}, {905, 2722}, {906, 36080}, {912, 2074}, {916, 2073}, {919, 5098}, {924, 10420}, {925, 1632}, {927, 4573}, {929, 4560}, {930, 13290}, {931, 4612}, {933, 13318}, {934, 1414}, {942, 1175}, {946, 37369}, {952, 3109}, {960, 1798}, {972, 1817}, {974, 1181}, {1010, 24809}, {1014, 15728}, {1019, 1308}, {1043, 2370}, {1055, 5060}, {1062, 9638}, {1098, 2975}, {1101, 2605}, {1109, 2606}, {1113, 2574}, {1114, 2575}, {1125, 13605}, {1151, 7601}, {1152, 7602}, {1154, 2070}, {1155, 11670}, {1157, 15907}, {1169, 2300}, {1171, 2054}, {1172, 8759}, {1173, 1493}, {1193, 38453}, {1194, 34945}, {1199, 5462}, {1205, 11574}, {1209, 9379}, {1216, 7512}, {1289, 2409}, {1291, 1510}, {1292, 2775}, {1293, 2776}, {1295, 2778}, {1296, 2434}, {1301, 2445}, {1302, 2407}, {1309, 5379}, {1310, 4592}, {1316, 2782}, {1320, 31523}, {1329, 9701}, {1340, 6142}, {1341, 6141}, {1370, 1660}, {1379, 5638}, {1380, 5639}, {1384, 9486}, {1396, 2376}, {1397, 17127}, {1408, 8686}, {1412, 1477}, {1415, 32693}, {1428, 7292}, {1434, 2369}, {1469, 32289}, {1473, 26910}, {1474, 9085}, {1478, 18968}, {1479, 12896}, {1498, 2935}, {1499, 2696}, {1506, 9697}, {1509, 2368}, {1531, 10296}, {1550, 36170}, {1553, 36172}, {1568, 3153}, {1593, 11439}, {1594, 9820}, {1599, 10132}, {1600, 10133}, {1619, 2063}, {1621, 2185}, {1633, 13397}, {1641, 10717}, {1649, 20404}, {1656, 9704}, {1657, 34584}, {1658, 18436}, {1691, 3231}, {1692, 3291}, {1698, 9587}, {1699, 9586}, {1793, 6326}, {1812, 26703}, {1853, 30744}, {1870, 19469}, {1897, 7452}, {1936, 23692}, {1946, 2714}, {1963, 19133}, {1971, 3289}, {1977, 18268}, {1992, 9084}, {2004, 21461}, {2005, 21462}, {2030, 9136}, {2051, 9562}, {2071, 2693}, {2072, 25739}, {2076, 8627}, {2106, 9455}, {2112, 2311}, {2175, 6015}, {2176, 35105}, {2177, 28559}, {2187, 29056}, {2193, 32726}, {2203, 2991}, {2223, 2711}, {2330, 5297}, {2341, 16554}, {2353, 28710}, {2365, 6514}, {2366, 3926}, {2382, 5009}, {2384, 3285}, {2393, 11416}, {2482, 11006}, {2644, 37135}, {2651, 3218}, {2682, 36174}, {2691, 3309}, {2692, 3667}, {2694, 6001}, {2700, 17209}, {2704, 8642}, {2705, 8643}, {2708, 37793}, {2709, 8644}, {2710, 18860}, {2717, 5536}, {2719, 23224}, {2720, 23189}, {2721, 16702}, {2723, 14956}, {2724, 14953}, {2725, 18206}, {2726, 5211}, {2727, 4091}, {2729, 6629}, {2740, 4786}, {2741, 26006}, {2746, 30234}, {2748, 6065}, {2754, 25083}, {2760, 3977}, {2769, 6332}, {2790, 36192}, {2794, 36163}, {2858, 14588}, {2859, 15413}, {2860, 7199}, {2861, 17139}, {2862, 30941}, {2863, 30939}, {2867, 3265}, {2868, 3266}, {2883, 11744}, {2886, 9702}, {2888, 10274}, {2892, 5596}, {2909, 7763}, {2915, 22136}, {2916, 35218}, {2917, 32338}, {2929, 22534}, {2937, 6101}, {2966, 6037}, {3016, 6787}, {3049, 9091}, {3052, 28531}, {3053, 15504}, {3056, 32290}, {3068, 8998}, {3069, 13990}, {3098, 7492}, {3100, 10118}, {3101, 10119}, {3108, 11205}, {3111, 9153}, {3129, 5611}, {3130, 5615}, {3133, 8883}, {3146, 13202}, {3147, 11411}, {3154, 31945}, {3219, 3955}, {3220, 22128}, {3229, 14602}, {3230, 5006}, {3258, 17511}, {3288, 14509}, {3357, 25564}, {3398, 37338}, {3428, 22583}, {3431, 4550}, {3511, 23180}, {3515, 12164}, {3517, 12160}, {3519, 18282}, {3520, 12038}, {3522, 10990}, {3523, 10984}, {3524, 11693}, {3525, 13336}, {3526, 20379}, {3529, 25712}, {3532, 15748}, {3533, 38725}, {3541, 15115}, {3542, 6193}, {3548, 11457}, {3565, 16680}, {3567, 7506}, {3574, 18428}, {3576, 11709}, {3579, 5951}, {3581, 7575}, {3589, 25328}, {3616, 11735}, {3618, 15118}, {3619, 5157}, {3620, 19126}, {3627, 37495}, {3628, 13353}, {3699, 9059}, {3732, 4241}, {3746, 7343}, {3747, 12031}, {3763, 6698}, {3796, 5621}, {3815, 9604}, {3818, 5169}, {3819, 15246}, {3832, 11424}, {3845, 13482}, {3851, 15046}, {3868, 13739}, {3869, 14529}, {3877, 17512}, {3909, 4585}, {3917, 6030}, {3939, 4627}, {4074, 10328}, {4215, 29009}, {4235, 30247}, {4238, 26706}, {4239, 15988}, {4242, 30250}, {4273, 28317}, {4296, 19505}, {4414, 5197}, {4557, 4629}, {4559, 8687}, {4584, 5384}, {4590, 9150}, {4600, 8709}, {4602, 9065}, {4603, 30670}, {4625, 34083}, {5008, 31609}, {5020, 5422}, {5026, 5108}, {5033, 8617}, {5050, 5544}, {5055, 15088}, {5070, 20396}, {5085, 5646}, {5091, 24617}, {5092, 5650}, {5094, 7703}, {5102, 31860}, {5104, 9831}, {5107, 13192}, {5111, 20977}, {5133, 23292}, {5135, 37633}, {5137, 33129}, {5138, 14996}, {5170, 8649}, {5189, 29012}, {5235, 9093}, {5250, 37032}, {5254, 9603}, {5285, 38822}, {5320, 37685}, {5333, 9103}, {5398, 19245}, {5418, 9677}, {5446, 34484}, {5449, 14940}, {5463, 11612}, {5464, 11613}, {5477, 6388}, {5505, 8542}, {5548, 32686}, {5562, 7488}, {5587, 9621}, {5597, 13208}, {5598, 13209}, {5603, 11723}, {5606, 17404}, {5612, 6105}, {5616, 6104}, {5618, 9200}, {5619, 9201}, {5641, 9141}, {5649, 35909}, {5661, 38352}, {5840, 10767}, {5846, 32298}, {5886, 12261}, {5890, 6644}, {5891, 18475}, {5899, 13391}, {5907, 13367}, {5943, 12834}, {5946, 13358}, {5965, 32223}, {5994, 6137}, {5995, 6138}, {6003, 34961}, {6011, 9810}, {6032, 11187}, {6043, 32911}, {6070, 22104}, {6093, 14653}, {6102, 11561}, {6150, 15770}, {6153, 11536}, {6197, 12661}, {6198, 12888}, {6200, 35299}, {6214, 12804}, {6215, 12803}, {6225, 30552}, {6233, 13242}, {6236, 32228}, {6240, 22660}, {6243, 37440}, {6284, 12374}, {6353, 6515}, {6368, 30716}, {6396, 35300}, {6413, 10962}, {6414, 10960}, {6467, 34470}, {6516, 13395}, {6528, 18831}, {6561, 9676}, {6564, 35834}, {6565, 35835}, {6574, 7259}, {6583, 18180}, {6617, 15324}, {6638, 23606}, {6640, 23294}, {6642, 7592}, {6676, 37636}, {6677, 11245}, {6739, 36154}, {6777, 30465}, {6778, 30468}, {6795, 9159}, {6816, 18925}, {6997, 11427}, {7058, 17135}, {7085, 26911}, {7256, 8706}, {7341, 36277}, {7354, 12373}, {7391, 31383}, {7417, 10753}, {7465, 37659}, {7476, 10101}, {7482, 10098}, {7487, 15473}, {7502, 23039}, {7503, 15056}, {7517, 16266}, {7519, 31670}, {7526, 15058}, {7527, 11430}, {7533, 18427}, {7542, 31831}, {7550, 10170}, {7556, 37478}, {7574, 19381}, {7577, 18474}, {7583, 19052}, {7584, 19051}, {7605, 25555}, {7666, 33541}, {7669, 14060}, {7689, 21844}, {7708, 8585}, {7715, 11566}, {7724, 10902}, {7725, 11824}, {7726, 11825}, {7779, 19558}, {7796, 18796}, {7799, 35568}, {8024, 19571}, {8025, 9108}, {8029, 12064}, {8041, 10329}, {8057, 22239}, {8151, 14695}, {8154, 14889}, {8200, 12466}, {8207, 12467}, {8220, 12894}, {8221, 12895}, {8227, 9622}, {8286, 24916}, {8287, 24955}, {8371, 30220}, {8537, 12596}, {8550, 37648}, {8584, 20192}, {8651, 10425}, {8673, 10423}, {8675, 9060}, {8681, 32127}, {8683, 8698}, {8690, 23831}, {8691, 23889}, {8705, 10510}, {8722, 34095}, {8758, 18609}, {8787, 9169}, {8911, 26912}, {8939, 19412}, {8943, 19413}, {8976, 13915}, {8994, 9540}, {9019, 19596}, {9027, 10102}, {9044, 9186}, {9064, 23347}, {9066, 23342}, {9073, 33295}, {9075, 30966}, {9080, 9182}, {9082, 27644}, {9101, 16285}, {9110, 18166}, {9126, 19902}, {9130, 9184}, {9160, 14270}, {9161, 23217}, {9162, 9202}, {9163, 9203}, {9171, 34539}, {9177, 15566}, {9208, 11636}, {9214, 10555}, {9301, 37914}, {9308, 37070}, {9514, 11794}, {9591, 31737}, {9626, 31738}, {9653, 10895}, {9666, 10896}, {9730, 11806}, {9781, 13861}, {9811, 13486}, {9818, 38396}, {9830, 14833}, {9833, 37444}, {9862, 35922}, {9919, 11414}, {9927, 16868}, {9937, 12271}, {9973, 15140}, {9996, 12501}, {10018, 12359}, {10111, 18912}, {10112, 14049}, {10203, 10227}, {10225, 15767}, {10263, 18378}, {10276, 34989}, {10294, 36852}, {10295, 15136}, {10298, 11202}, {10301, 21850}, {10303, 37515}, {10310, 12327}, {10516, 32274}, {10519, 32247}, {10533, 11417}, {10534, 11418}, {10537, 20243}, {10594, 11387}, {10601, 17809}, {10605, 15078}, {10610, 14128}, {10625, 12088}, {10627, 13564}, {10632, 10662}, {10633, 10661}, {10639, 32586}, {10640, 32585}, {10665, 10881}, {10666, 10880}, {10681, 11420}, {10682, 11421}, {10754, 25047}, {10766, 35901}, {10796, 12201}, {10942, 12905}, {10943, 12906}, {10989, 11645}, {11050, 15354}, {11078, 32461}, {11092, 32460}, {11107, 12528}, {11145, 13349}, {11146, 13350}, {11159, 11162}, {11176, 36255}, {11186, 13170}, {11199, 32437}, {11204, 35493}, {11248, 12381}, {11249, 12382}, {11250, 18439}, {11270, 33556}, {11328, 11842}, {11381, 12086}, {11423, 15024}, {11426, 15465}, {11499, 12334}, {11550, 31074}, {11559, 18364}, {11585, 12419}, {11656, 12243}, {11704, 34114}, {11793, 37126}, {11805, 15800}, {11820, 21312}, {11822, 12365}, {11823, 12366}, {11826, 12371}, {11827, 12372}, {11828, 12377}, {11829, 12378}, {11849, 35195}, {11898, 19154}, {12039, 29959}, {12068, 12079}, {12082, 37483}, {12084, 12290}, {12100, 22250}, {12113, 23240}, {12163, 32534}, {12225, 32330}, {12274, 12978}, {12275, 12979}, {12289, 18404}, {12293, 18504}, {12380, 15091}, {12588, 32243}, {12589, 32297}, {13015, 13055}, {13016, 13056}, {13163, 22051}, {13323, 16865}, {13348, 16661}, {13363, 15037}, {13364, 15038}, {13504, 15959}, {13509, 14961}, {13518, 19120}, {13558, 14687}, {13567, 19142}, {13582, 34306}, {13588, 15323}, {13596, 16194}, {13637, 13643}, {13757, 13762}, {13863, 30210}, {13935, 13969}, {13951, 13979}, {14008, 17188}, {14061, 15359}, {14133, 33021}, {14153, 20965}, {14247, 32027}, {14357, 36833}, {14360, 14928}, {14499, 14807}, {14500, 14808}, {14508, 36164}, {14528, 33537}, {14538, 34008}, {14539, 34009}, {14586, 32692}, {14590, 14696}, {14605, 35297}, {14606, 38366}, {14616, 24346}, {14651, 33511}, {14652, 34218}, {14658, 35006}, {14733, 21789}, {14781, 23105}, {14913, 21637}, {14933, 16169}, {14966, 26714}, {14981, 18337}, {14987, 37227}, {14998, 23969}, {15026, 32136}, {15028, 36752}, {15047, 22462}, {15069, 37638}, {15082, 20190}, {15138, 19374}, {15311, 16386}, {15322, 35342}, {15345, 32749}, {15431, 19124}, {15438, 37201}, {15439, 23067}, {15453, 35189}, {15535, 38224}, {15545, 15561}, {15731, 35997}, {15766, 36254}, {16039, 20626}, {16105, 37498}, {16168, 36193}, {16176, 25331}, {16238, 26879}, {16261, 31861}, {16270, 19347}, {16659, 23335}, {16806, 35337}, {16807, 35336}, {16876, 28838}, {16981, 37517}, {17167, 26708}, {17187, 28485}, {17222, 33628}, {17414, 32694}, {17455, 34079}, {17524, 28173}, {17714, 37484}, {17821, 17835}, {17977, 32849}, {18331, 33512}, {18358, 37454}, {18381, 32743}, {18403, 30522}, {18435, 18570}, {18488, 35482}, {18829, 33514}, {18916, 18932}, {19118, 19588}, {19119, 32241}, {19123, 32249}, {19125, 32251}, {19129, 32272}, {19131, 32275}, {19132, 32276}, {19134, 32293}, {19135, 32294}, {19137, 32300}, {19167, 19189}, {19171, 32258}, {19185, 19195}, {19406, 19507}, {19407, 19508}, {19424, 19482}, {19425, 19483}, {19440, 19484}, {19441, 19485}, {19459, 26206}, {19478, 22758}, {19576, 36214}, {19924, 37901}, {20063, 29317}, {20301, 38317}, {20766, 20877}, {20987, 26284}, {20999, 36942}, {21008, 25435}, {21294, 23674}, {21444, 23216}, {21663, 37941}, {22135, 33652}, {22333, 22334}, {22528, 36982}, {22648, 33499}, {22649, 33501}, {23208, 28724}, {23235, 31854}, {23344, 28210}, {23390, 23861}, {23582, 32725}, {23698, 36181}, {23700, 35296}, {23832, 28218}, {23878, 36886}, {23964, 32687}, {23967, 36904}, {23997, 29055}, {24000, 36068}, {24041, 36066}, {24281, 30927}, {24542, 25536}, {24550, 29310}, {24987, 37152}, {25577, 28868}, {25738, 26917}, {26156, 26926}, {26227, 27958}, {26257, 35301}, {26733, 36075}, {26874, 26880}, {26875, 26886}, {26889, 27003}, {26890, 27065}, {26898, 37068}, {28624, 32656}, {28708, 36851}, {29181, 37900}, {29300, 37619}, {29330, 36015}, {29352, 35983}, {30212, 37964}, {30528, 32733}, {30715, 30717}, {31133, 36990}, {31378, 31379}, {31626, 32078}, {31874, 34209}, {32072, 32563}, {32073, 32570}, {32074, 32434}, {32124, 37969}, {32225, 37907}, {32262, 37485}, {32264, 34774}, {32267, 37909}, {32315, 32599}, {32428, 32439}, {32445, 32547}, {32515, 37906}, {32608, 37922}, {32640, 32681}, {32676, 32691}, {33543, 35446}, {33586, 37672}, {33852, 33883}, {33873, 33876}, {34146, 37929}, {34150, 36169}, {34380, 37897}, {34381, 37963}, {34382, 37777}, {34394, 36759}, {34395, 36760}, {34783, 37814}, {34830, 38535}, {35002, 37916}, {35056, 35057}, {35447, 38623}, {36034, 36064}, {36065, 36084}, {36070, 36142}, {36078, 36134}, {37496, 37924}, {37649, 37990}

X(110) is the {X(5),X(49)}-harmonic conjugate of X(54). For a list of other harmonic conjugates of X(110), click Tables at the top of this page.

X(110) = midpoint of X(i) and X(j) for these (i,j): (3,399), (20,146), (23,323), (1495,3292)
X(110) = reflection of X(i) in X(j) for these (i,j): (3,1511), (4,113), (23,1495), (67,141), (74,3), (265,5), (382,1539), (895,6), (1177,206), (3580,468)
X(110) = circumcircle-antipode of X(74)
X(110) = isogonal conjugate of X(523)
X(110) = isotomic conjugate of X(850)
X(110) = isogonal conjugate of the isotomic conjugate of X(99)
X(110) = inverse of X(2) in the Brocard circle
X(110) = complement of X(3448)
X(110) = anticomplement of X(125)
X(110) = X(i)-Ceva conjugate of X(j) for these (i,j): (249,6), (250,3)
X(110) = cevapoint of X(i) and X(j) for these (i,j): (3,520), (5,523), (6,512), (141,525)
X(110) = X(i)-cross conjugate of X(j) for these (i,j): (1,59), (3,250), (6,249), (109,162), (351,111), (512,6), (520,3), (523,54), (526,74)
X(110) = crosssum of X(i) and X(j) for these (i,j): (2,148), (512,647), (520,647)
X(110) = crossdifference of every pair of points on line X(115)X(125)
X(110) = X(i)-Hirst inverse of X(j) for these (i,j): (1,245), (2,125), (3,246), (4,247)
X(110) = X(i)-beth conjugate of X(j) for these (i,j): (21,759), (643,643)
X(110) = X(23)-of-1st-Brocard triangle
X(110) = X(111)-of-circumsymmedial-triangle
X(110) = X(323)-of-orthocentroidal-triangle
X(110) = X(137)-of-excentral-triangle
X(110) = X(128)-of-hexyl-triangle
X(110) = trilinear pole of the Brocard axis
X(110) = trilinear pole of PU(29) (see ETC→Tables→Bicentric Pairs)
X(110) = crosssum of polar circle intercepts of Euler line
X(110) = perspector of ABC and vertex-triangle of anticevian triangles of X(3) and X(6)
X(110) = Johnson-circumconic antipode of X(265)
X(110) = MacBeath-circumconic antipode of X(895)
X(110) = perspector of conic {A,B,C,PU(2)}
X(110) = intersection of trilinear polars of P(2) and U(2)
X(110) = intersection of tangents to Steiner circumellipse at X(99) and X(648)
X(110) = crosspoint of X(99) and X(648)
X(110) = reflection of X(476) in the Euler line
X(110) = reflection of X(691) in the Brocard axis
X(110) = reflection of X(23) in the Lemoine axis
X(110) = reflection of X(1290) in line X(1)X(3)
X(110) = reflection of X(111) in line X(3)X(351)
X(110) = inverse-in-polar-circle of X(136)
X(110) = inverse-in-{circumcircle, nine-point circle}-inverter of X(114)
X(110) = inverse-in-Moses-radical-circle of X(2715)
X(110) = inverse-in-O(15,16) of X(843), where O(15,16) is the circle having segment X(15)X(16) as diameter
X(110) = X(i)-isoconjugate of X(j) for these (i,j): (6,1577), (92,647), (1577,6)
X(110) = perspector of circumorthic triangle and Johnson triangle (the reflection triangles of X(4) and X(3), resp.)
X(110) = trilinear product of vertices of circumtangential triangle
X(110) = {X(3),X(156)}-harmonic conjugate of X(1614)
X(110) = orthocentroidal-to-ABC similarity image of X(2)
X(110) = 4th-Brocard-to-circumsymmedial similarity image of X(2)
X(110) = isogonal conjugate of isotomic conjugate of trilinear pole of line X(2)X(6)
X(110) = the point of intersection, other than A, B, C, of the circumcircle and Johnson circumconic
X(110) = the point of intersection, other than A, B, C, of the circumcircle and MacBeath circumconic
X(110) = the point of intersection, other than A, B, C, of the circumcircle and circumconic {A,B,C,PU(5)}}
X(110) = Collings transform of X(5)
X(110) = Collings transform of X(6)
X(110) = intersection of tangents at X(61) and X(62) to the Napoleon-Feuerbach cubic K005
X(110) = SR(PU(4))
X(110) = insimilicenter of nine-point circle and sine-triple-angle circle
X(110) = insimilicenter of circumcircle and nine-point circle of tangential triangle; the exsimilicenter is X(1614)
X(110) = X(7972)-of-Trinh-triangle
X(110) = Ψ(X(i), X(j)) for these (i,j): (1,21), (2,6), (3,49), (4,2), (5,51), (6,3), (7,21), (8,21), (9,21), (19,1), (53,5), (54,3), (64,3), (65,1), (66,3), (67,3), (68,3), (69,3), (73,3), (74,3), (75,1), (76,2), (93,4), (95,54), (115,125), (190,99), (250,186)
X(110) = X(110)-of-1st-Parry-triangle
X(110) = X(74)-of-2nd-Parry-triangle
X(110) = center of similitude of ABC and 1st Parry triangle
X(110) = inverse-in-Parry-isodynamic-circle of X(111); see X(2)
X(110) = barycentric product of PU(i) for these i: 78, 145
X(110) = perspector of unary cofactor triangles of outer and inner Napoleon triangles
X(110) = X(6792)-of-4th-anti-Brocard-triangle
X(110) = X(111)-of-anti-Artzt-triangle
X(110) = perspector of circumcevian triangle of X(36) and cross-triangle of ABC and 2nd circumperp triangle
X(110) = perspector of circumcevian triangle of X(187) and cross-triangle of ABC and circumsymmedial triangle
X(110) = Kiepert image of X(3)
X(110) = Jerabek image of X(2)
X(110) = Cundy-Parry Phi transform of X(14264)
X(110) = endo-homothetic center of X(2)-altimedial and X(2)-anti-altimedial triangles
X(110) = endo-homothetic center of X(20)-altimedial and X(3)-anti-altimedial triangles
X(110) = Thomson isogonal conjugate of X(30)
X(110) = Lucas isogonal conjugate of X(30)
X(110) = center of the perspeconic of these triangles: inner and outer Napoleon
X(110) = intersection of antipedal lines of X(1113) and X(1114)
X(110) = X(104)-of-circumorthic-triangle if ABC is acute
X(110) = perspector, wrt circumorthic triangle, of polar circle
X(110) = trilinear product of circumcircle intercepts of line X(1)X(21)
X(110) = barycentric product of circumcircle intercepts of line X(2)X(6)
X(110) = barycentric product X(6)*X(99)
X(110) = barycentric product of Steiner circumellipse intercepts of Brocard axis
X(110) = Feuerbach point of the tangential triangle if ABC is acute; otherwise, a vertex of the Feuerbach triangle of the tangential triangle
X(110) = barycentric product of MacBeath circumconic intercepts of Euler line
X(110) = excentral-to-ABC functional image of X(1768)
X(110) = intouch-to-ABC functional image of X(11)
X(110) = trilinear pole, wrt circumtangential triangle, of Euler line
X(110) = trilinear pole, wrt 1st Parry triangle, of Lemoine axis
X(110) = trilinear product of circumcircle intercepts of line X(1)X(21)
X(110) = de-Longchamps-circle-inverse of X(34186)
X(110) = antipode of X(3580) in Walsmith rectangular hyperbola
X(110) = orthocenter of X(6)X(74)X(3569)
X(110) = Conway-circle-inverse of X(38480)


X(111) = PARRY POINT

Trilinears    a/(2a2 - b2 - c2) : b/(2b2 - c2 - a2) : c/(2c2 - a2 - b2)
Barycentrics    a2/(2a2 - b2 - c2) : b2/(2b2 - c2 - a2) : c2/(2c2 - a2 - b2)
Tripolars    |b c (2 a^2 - b^2 - c^2)| : :

Let L be a line tangent to the Brocard circle. Let P be the trilinear pole of L, and let P' be the isogonal conjugate of P. As L varies, P' traces a parabola with focus at X(111). The parabola meets the line at infinity at X(524). Also, X(111) is the QA-P4 center (Isogonal Center) of quadrangle X(13)X(14)X(15)X(16) (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/25-qa-p4.html) Also, let A' = BC∩X(115)X(125), and define B' and C' cyclically. The circumcircles of AB'C', BC'A', CA'B' concur in X(111). (Randy Hutson, October 13, 2015)

Let A″B″C″ be the 2nd Ehrmann triangle. Let Pa be the pole of line B″C″ wrt the A-Ehrmann circle, and define Pb and Pc cyclically. Let Pa' be the pole of line BC wrt the A-Ehrmann circle, and define Pb' and Pc' cyclically. The lines APaPa', BPbPb', CPcPc' concur in X(111). Also, let A* be the trilinear pole of line B″C″, and define B*, C* cyclically. The lines AA*, BB*, CC* concur in X(111). (Randy Hutson, November 18, 2015)

Let A1B1C1 and A3B3C3 be the 1st and 3rd Parry triangles. Let A' be the barycentric product A1*A3, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(111). (Randy Hutson, February 10, 2016)

Let A'B'C' and A″B″C″ be the 4th Brocard and 4th anti-Brocard triangles, resp. Let A* be the barycentric product A'*A'', and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(111). (Randy Hutson, December 2, 2017)

X(111) lies on the Parry circle and these lines: {1, 8691}, {2, 99}, {3, 1296}, {4, 1560}, {5, 10748}, {6, 110}, {11, 10779}, {15, 9202}, {16, 9203}, {20, 20187}, {22, 2079}, {23, 187}, {24, 8428}, {25, 112}, {30, 2696}, {32, 1383}, {37, 100}, {39, 7711}, {42, 101}, {51, 10558}, {55, 6019}, {56, 3325}, {74, 2433}, {76, 16055}, {98, 1637}, {102, 2819}, {103, 2824}, {104, 2830}, {105, 2837}, {106, 2843}, {107, 393}, {108, 1880}, {109, 1400}, {125, 35902}, {140, 38806}, {141, 36883}, {182, 353}, {183, 9066}, {184, 3048}, {186, 10098}, {230, 476}, {232, 1304}, {251, 827}, {263, 9463}, {265, 14846}, {308, 689}, {316, 35569}, {323, 2987}, {325, 2858}, {351, 2492}, {352, 511}, {376, 30256}, {381, 6032}, {382, 38799}, {385, 892}, {468, 935}, {477, 5915}, {493, 1306}, {494, 1307}, {512, 843}, {513, 2721}, {514, 2729}, {515, 2735}, {516, 2740}, {517, 2746}, {518, 2753}, {519, 2759}, {520, 2763}, {521, 2767}, {522, 2768}, {523, 2770}, {524, 6082}, {526, 9184}, {538, 5971}, {542, 6791}, {548, 38798}, {550, 38797}, {576, 8600}, {612, 15322}, {631, 38804}, {647, 842}, {649, 2712}, {650, 2752}, {667, 35107}, {669, 5970}, {675, 26273}, {694, 805}, {729, 5027}, {739, 5040}, {741, 3572}, {755, 5113}, {759, 23894}, {831, 1961}, {840, 5098}, {850, 2868}, {858, 24855}, {880, 34087}, {898, 5291}, {901, 17735}, {902, 2054}, {907, 1180}, {919, 20989}, {925, 2165}, {927, 37798}, {930, 2963}, {931, 941}, {932, 16606}, {933, 8882}, {934, 1427}, {1055, 2701}, {1084, 14948}, {1113, 8106}, {1114, 8105}, {1171, 6578}, {1194, 3108}, {1239, 6572}, {1241, 35567}, {1287, 10416}, {1289, 6353}, {1290, 3290}, {1291, 14579}, {1292, 4220}, {1293, 9526}, {1294, 9529}, {1295, 9531}, {1297, 13114}, {1302, 3018}, {1305, 26252}, {1310, 17019}, {1340, 6141}, {1341, 6142}, {1344, 8427}, {1345, 8426}, {1350, 33979}, {1379, 5639}, {1380, 5638}, {1495, 1976}, {1499, 6093}, {1611, 36616}, {1640, 14932}, {1645, 9468}, {1648, 9140}, {1657, 38800}, {1691, 32694}, {1692, 35265}, {1799, 6573}, {1992, 10552}, {2080, 13225}, {2177, 5147}, {2248, 33774}, {2291, 5075}, {2373, 9979}, {2374, 9131}, {2378, 6138}, {2379, 6137}, {2393, 5166}, {2408, 9084}, {2444, 9135}, {2469, 14899}, {2470, 35607}, {2690, 3011}, {2691, 37959}, {2697, 6587}, {2698, 3288}, {2703, 3230}, {2713, 3331}, {2748, 5524}, {2760, 3239}, {2766, 5089}, {2782, 9775}, {2981, 10409}, {2998, 3222}, {3003, 9060}, {3016, 9160}, {3055, 10276}, {3060, 10559}, {3090, 38807}, {3098, 8617}, {3143, 34171}, {3146, 38801}, {3163, 5304}, {3430, 28295}, {3448, 6388}, {3457, 5995}, {3458, 5994}, {3522, 38803}, {3542, 30251}, {3815, 30537}, {3832, 38802}, {3920, 9281}, {4231, 26706}, {4240, 6531}, {4563, 25047}, {4588, 28658}, {5012, 10485}, {5024, 9155}, {5033, 7712}, {5038, 5643}, {5085, 14688}, {5092, 14660}, {5099, 30718}, {5108, 5969}, {5169, 18424}, {5182, 35356}, {5251, 29351}, {5468, 10754}, {5475, 11640}, {5477, 9143}, {5622, 35901}, {5916, 9200}, {5917, 9201}, {6034, 7698}, {6037, 9154}, {6083, 20461}, {6094, 9080}, {6128, 7736}, {6151, 10410}, {6200, 7598}, {6221, 7601}, {6323, 9208}, {6325, 9769}, {6339, 20080}, {6396, 7599}, {6398, 7602}, {6781, 37901}, {7467, 14678}, {7485, 34866}, {7492, 8588}, {7496, 8589}, {7575, 32229}, {7606, 10166}, {7610, 9829}, {7792, 9069}, {7954, 14075}, {8030, 38020}, {8429, 37930}, {8586, 9225}, {8644, 9136}, {8651, 14659}, {8652, 28625}, {8681, 15387}, {8707, 14624}, {8709, 27809}, {8744, 10422}, {8859, 18818}, {8869, 19121}, {9059, 26244}, {9067, 26243}, {9070, 26242}, {9076, 22105}, {9091, 16098}, {9096, 26241}, {9126, 19901}, {9137, 10102}, {9159, 9828}, {9169, 9830}, {9176, 13233}, {9189, 18007}, {9206, 11081}, {9207, 11086}, {9605, 30734}, {9810, 14126}, {9831, 12434}, {10304, 37749}, {10313, 10420}, {10560, 11002}, {10645, 14704}, {10646, 14705}, {11175, 36213}, {11215, 11622}, {11332, 21444}, {11632, 14694}, {11635, 38463}, {11639, 31489}, {11645, 11647}, {12041, 35447}, {12149, 25424}, {12212, 31609}, {12834, 14153}, {13397, 26253}, {13398, 26283}, {14733, 17966}, {15018, 30535}, {15066, 35575}, {15271, 34227}, {15360, 15993}, {16081, 22456}, {16167, 16310}, {16318, 22239}, {16806, 21461}, {16807, 21462}, {19561, 28895}, {21906, 34574}, {23297, 33013}, {23701, 30209}, {30249, 33630}, {31884, 37751}, {32237, 38010}, {32526, 35002}, {32849, 35574}, {34079, 36069}, {34482, 34572}, {36066, 36085}, {36201, 36203}, {36894, 37643}

X(111) = reflection of X(1296) in X(3)
X(111) = isogonal conjugate of X(524)
X(111) = isotomic conjugate of X(3266)
X(111) = inverse-in-Brocard-circle of X(353)
X(111) = anticomplement of X(126)
X(111) = cevapoint of X(6) and X(187)
X(111) = X(i)-cross conjugate of X(j) for these (i,j): (23,251), (187,6), (351,110)
X(111) = crossdifference of every pair of points on line X(351)X(690)
X(111) = perspector of ABC and the triangle formed by the reflections of line PU(7) in the sides of ABC
X(111) = point of intersection, other than A, B, C, of circumcircle and hyperbola {A,B,C,X(2),X(6)}}
X(111) = trilinear pole of line X(6)X(512)
X(111) = Λ(X(2), X(6))
X(111) = Ψ(X(i),X(j)) for these (i,j): (6,512), (21,650), (190,10)
X(111) = trilinear pole wrt circumsymmedial triangle of Brocard axis
X(111) = trilinear pole wrt circummedial triangle of line X(2)X(6)
X(111) = X(110)-of-circumsymmedial-triangle
X(111) = X(23)-of-4th-Brocard-triangle
X(111) = X(352)-of-orthocentroidal-triangle
X(111) = X(138)-of-excentral-triangle
X(111) = reflection of X(2770) in the Euler line
X(111) = reflection of X(843) in the Brocard axis
X(111) = reflection of X(2721) in line X(1)X(3)
X(111) = reflection of X(110) in line X(3)X(351)
X(111) = inverse-in-polar-circle of X(1560)
X(111) = inverse-in-{circumcircle, nine-point circle}-inverter of X(115)
X(111) = inverse-in-Moses-radical-circle of X(842)
X(111) = inverse-in-circle-O(15,16) of X(691)
X(111) = X(1577)-isoconjugate of X(5467)
X(111) = SR(P,U), where P and U are the circumcircle intercepts of line X(2)X(6)
X(111) = one of two harmonic traces of the McCay circles; X(2) is the other
X(111) = X(1296)-of-1st-Parry-triangle
X(111) = X(111)-of-2nd-Parry-triangle
X(111) = X(691)-of-3rd-Parry-triangle
X(111) = center of similitude of ABC and 2nd Parry triangle
X(111) = inverse-in-Parry-isodynamic-circle of X(110); see X(2)
X(111) = 3rd-Parry-to-circumsymmedial similarity image of X(352)
X(111) = center of similitude of ABC and circumsymmedial triangle of Artzt triangle
X(111) = eigencenter of circumtangential triangle
X(111) = perspector of ABC and unary cofactor triangle of 2nd Brocard triangle
X(111) = X(2)-of-4th-anti-Brocard-triangle
X(111) = anti-Artzt-to-4th-anti-Brocard similarity image of X(2)
X(111) = Thomson-isogonal conjugate of X(1499)
X(111) = Lucas-isogonal conjugate of X(1499)
X(111) = Cundy-Parry Phi transform of X(14262)
X(111) = Cundy-Parry Psi transform of X(13608)
X(111) = X(6)-isoconjugate of X(14210)
X(111) = inverse-in-circle-{X(1687),X(1688),PU(1),PU(2)}} of X(32694)
X(111) = barycentric product of circumcircle intercepts of line X(2)X(523)
X(111) = inverse-in-Stevanovic-circle of X(2758)
X(111) = trilinear pole, wrt Thomson triangle, of line X(6)X(5646)
X(111) = trilinear pole, wrt 2nd Parry triangle, of line X(511)X(3569)
X(111) = areal center of pedal triangles of PU(7)
X(111) = X(2)-Ceva conjugate of X(15899)
X(111) = perspector of circumconic centered at X(15899)
X(111) = McCay-bisector-circle-inverse of X(2)


X(112) = Ψ(ORTHOCENTER, SYMMEDIAN POINT)

Trilinears    a/(sin 2B - sin 2C) : b/(sin 2C - sin 2A) : c/(sin 2A - sin 2B)
Trilinears    a/[(b2 - c2)(b2 + c2 - a2)]
Trilinears    tan A csc(B - C) : :
Barycentrics    a2/(sin 2B - sin 2C) : b2/(sin 2C - sin 2A) : c2/(sin 2A - sin 2B)
Tripolars    |cot A sin(B - C)| : :

If the line X(4)X(6) is reflected in every side of triangle ABC, then the reflections concur in X(112). (Randy Hutson, 9/23/2011)

Let P be a point on the van Aubel line other than X(4). Let A' be the reflection of P in BC, and define B' and C' cyclically. The circumcircles of AA'B', BC'A', and CA'B' concur at X(112). (Randy Hutson, December 26, 2015)

Let Q be a point on the Euler line other than X(2). Let A'B'C' be the cevian triangle of Q. Let A″ be the {B,C}-harmonic conjugate of A' (or equivalently, A″ = BC∩B'C'), and define B″ and C″ cyclically. The circumcircles of AB″C″, BC″A″, CA″B″ concur in X(112). (Randy Hutson, December 26, 2015)

Let A', B', C' be the intersections of the orthic axis and lines BC, CA, AB, respectively. The circumcircles of AB'C', BC'A', CA'B' concur in X(112). (Randy Hutson, December 26, 2015)

Let A' be the reflection of X(6) in BC, and define B' and C' cyclically. The circumcircles of AB'C', BC'A', CA'B' concur in X(112). (Randy Hutson, December 26, 2015)

Let A'B'C' be the circummedial triangle. Let A″ be the pole, wrt the polar circle, of line B'C', and define B″ and C″ cyclically. The lines A'A″, B'B″, C'C″ concur in X(112). (Randy Hutson, December 26, 2015)

Let L1 be the line that is the barycentric product of the Euler line and P(4). Let L2 be the line that is the barycentric product of the Euler line and U(4). Then X(112) = L1∩L2. (Randy Hutson, July 11, 2019)

X(112) lies on these lines: {1, 10705}, {2, 127}, {3, 1297}, {4, 32}, {5, 10749}, {6, 74}, {10, 13280}, {11, 10780}, {12, 13295}, {19, 759}, {20, 10316}, {21, 26703}, {22, 3162}, {23, 14580}, {24, 2079}, {25, 111}, {27, 675}, {28, 105}, {29, 1311}, {30, 2697}, {31, 2249}, {33, 609}, {34, 7031}, {35, 13116}, {36, 13117}, {39, 3520}, {48, 26701}, {50, 477}, {53, 1141}, {54, 217}, {55, 6020}, {56, 3320}, {58, 103}, {69, 2366}, {83, 37125}, {97, 34579}, {99, 648}, {100, 162}, {101, 4249}, {102, 284}, {104, 1108}, {106, 1474}, {107, 1637}, {108, 1415}, {109, 163}, {110, 1625}, {165, 12408}, {172, 6198}, {182, 12207}, {184, 26717}, {186, 187}, {193, 35940}, {216, 11587}, {230, 403}, {246, 19504}, {249, 10425}, {250, 691}, {251, 427}, {264, 2367}, {283, 2365}, {286, 767}, {297, 2857}, {340, 35568}, {371, 19094}, {372, 19093}, {376, 577}, {381, 19163}, {382, 19160}, {384, 1235}, {385, 2868}, {389, 16225}, {393, 571}, {402, 13281}, {420, 8623}, {451, 5277}, {462, 34376}, {463, 34374}, {468, 2770}, {476, 2501}, {493, 13298}, {494, 13299}, {511, 2710}, {512, 2715}, {513, 2722}, {515, 12784}, {516, 2741}, {517, 2747}, {518, 2754}, {519, 2760}, {520, 2764}, {522, 2769}, {523, 935}, {525, 2867}, {574, 14388}, {601, 29056}, {607, 28471}, {631, 34841}, {647, 1304}, {650, 2766}, {651, 13395}, {652, 2762}, {653, 1305}, {662, 1310}, {669, 9091}, {670, 35567}, {681, 36126}, {685, 6037}, {689, 6331}, {692, 36080}, {693, 2859}, {699, 11325}, {729, 1974}, {733, 11380}, {739, 2203}, {741, 1973}, {755, 1843}, {789, 811}, {805, 9218}, {831, 35338}, {835, 1897}, {839, 6335}, {843, 2030}, {898, 5379}, {907, 1634}, {915, 5317}, {917, 8747}, {925, 4226}, {927, 17925}, {930, 35318}, {933, 14586}, {934, 4565}, {953, 3285}, {972, 3194}, {1003, 9308}, {1021, 2765}, {1033, 5897}, {1113, 8105}, {1114, 8106}, {1287, 30716}, {1289, 1632}, {1290, 6591}, {1292, 9523}, {1293, 9527}, {1295, 2193}, {1296, 5467}, {1299, 1609}, {1301, 1624}, {1302, 4240}, {1309, 17926}, {1326, 2700}, {1344, 8426}, {1345, 8427}, {1370, 8879}, {1396, 15728}, {1414, 6183}, {1459, 2727}, {1461, 36079}, {1498, 15324}, {1506, 6143}, {1529, 19158}, {1562, 2777}, {1576, 2492}, {1593, 12145}, {1594, 7745}, {1597, 15433}, {1614, 14585}, {1691, 2211}, {1692, 23700}, {1826, 2372}, {1870, 1914}, {1885, 5305}, {1886, 2688}, {1971, 3331}, {1983, 6011}, {1995, 34108}, {2071, 14961}, {2074, 2752}, {2138, 11413}, {2194, 32726}, {2212, 35106}, {2222, 7115}, {2291, 2299}, {2322, 2370}, {2331, 35192}, {2333, 28482}, {2354, 38453}, {2374, 6353}, {2378, 8739}, {2379, 8740}, {2380, 10642}, {2381, 10641}, {2383, 14576}, {2393, 34107}, {2421, 35575}, {2485, 10423}, {2548, 37119}, {2549, 19220}, {2687, 14571}, {2689, 3064}, {2690, 7649}, {2693, 3284}, {2694, 37960}, {2695, 8755}, {2699, 5006}, {2706, 3289}, {2708, 5060}, {2709, 9181}, {2713, 3049}, {2714, 3063}, {2716, 4282}, {2719, 22383}, {2720, 7252}, {2726, 37168}, {2728, 3737}, {2749, 20752}, {2755, 22356}, {2758, 8756}, {2761, 36054}, {2763, 3292}, {2862, 15149}, {2864, 17924}, {2966, 6528}, {3003, 32710}, {3043, 9696}, {3068, 13923}, {3069, 13992}, {3087, 6128}, {3098, 12503}, {3151, 18686}, {3164, 35952}, {3192, 4262}, {3199, 3518}, {3222, 17941}, {3286, 28838}, {3291, 37777}, {3428, 19159}, {3516, 9605}, {3545, 10314}, {3565, 11634}, {3567, 16224}, {3575, 19157}, {3576, 12265}, {3736, 28844}, {3815, 37118}, {4232, 9084}, {4233, 9061}, {4236, 13397}, {4241, 9057}, {4242, 9070}, {4246, 9058}, {4248, 9083}, {4267, 29206}, {4273, 28159}, {4276, 29045}, {4278, 29042}, {4559, 15439}, {4567, 35574}, {4570, 29241}, {4574, 29163}, {5007, 14865}, {5008, 13596}, {5009, 12032}, {5013, 35477}, {5017, 6403}, {5023, 32534}, {5024, 11410}, {5039, 19124}, {5041, 35478}, {5063, 35485}, {5094, 6032}, {5140, 14659}, {5206, 21844}, {5210, 35472}, {5254, 18560}, {5301, 36420}, {5412, 32420}, {5413, 32422}, {5475, 7577}, {5477, 18331}, {5597, 13229}, {5598, 13231}, {5649, 35911}, {5663, 22146}, {5702, 14482}, {5896, 34570}, {6000, 8779}, {6080, 32320}, {6081, 23090}, {6093, 15471}, {6099, 32698}, {6200, 35828}, {6239, 10881}, {6240, 27376}, {6284, 12955}, {6323, 8541}, {6396, 35829}, {6400, 10880}, {6587, 22239}, {6623, 37689}, {6644, 15355}, {6748, 13597}, {6749, 13338}, {6753, 7468}, {6781, 13619}, {6792, 12828}, {7054, 17512}, {7254, 24016}, {7354, 12945}, {7391, 13854}, {7435, 9107}, {7452, 9056}, {7471, 16167}, {7480, 9060}, {7502, 18472}, {7526, 26216}, {7746, 16868}, {7749, 14940}, {7750, 28724}, {7754, 37199}, {7772, 29322}, {7787, 37337}, {8362, 26224}, {8429, 36176}, {8537, 13330}, {8553, 8746}, {8673, 15388}, {8707, 36797}, {8719, 15576}, {8735, 19628}, {8748, 32706}, {8883, 23233}, {8889, 15437}, {8962, 15218}, {9073, 31905}, {9075, 31909}, {9078, 17520}, {9090, 18344}, {9103, 31902}, {9105, 31903}, {9108, 31900}, {9112, 11612}, {9113, 11613}, {9150, 18020}, {9161, 15463}, {9209, 31510}, {9540, 13918}, {10102, 11580}, {10310, 12340}, {10594, 15745}, {10632, 19781}, {10633, 19780}, {10985, 33842}, {11062, 11063}, {11107, 17916}, {11248, 13118}, {11249, 13119}, {11414, 12413}, {11822, 12478}, {11823, 12479}, {11824, 12805}, {11825, 12806}, {11826, 12925}, {11827, 12935}, {11828, 12996}, {11829, 12997}, {12084, 22120}, {12111, 23128}, {12150, 36794}, {12593, 23701}, {13238, 19127}, {13881, 35488}, {13935, 13985}, {14039, 32000}, {14577, 33643}, {14579, 33631}, {14907, 17907}, {15013, 30737}, {15311, 15341}, {15340, 18400}, {15344, 30733}, {15352, 16813}, {15513, 17506}, {15515, 23040}, {17943, 29329}, {18859, 22121}, {19121, 35924}, {21789, 32642}, {22329, 37855}, {23096, 38463}, {23590, 32646}, {23616, 38233}, {26704, 32653}, {26912, 35949}, {27377, 35920}, {28624, 32739}, {32643, 36067}, {32649, 32687}, {32675, 36078}, {32681, 32715}, {32692, 32734}, {32699, 35182}, {32700, 35183}, {32701, 35184}, {32703, 35185}, {32705, 35186}, {32707, 35187}, {32709, 32729}, {32711, 35189}, {32730, 34397}, {33513, 35178}, {34137, 34146}, {34205, 35569}, {34870, 35476}, {35902, 36201}, {36064, 36131}, {36065, 36104}

X(112) = reflection of X(i) in X(j) for these (i,j): (4,132), (1297,3)
X(112) = isogonal conjugate of X(525)
X(112) = isotomic conjugate of X(3267)
X(112) = anticomplement of X(127)
X(112) = X(i)-Ceva conjugate of X(j) for these (i,j): (249,24), (250,25)
X(112) = cevapoint of X(i) and X(j) for these (i,j): (32,512), (427,523)
X(112) = X(i)-cross conjugate of X(j) for these (i,j): (25,250), (512,4), (523,251)
X(112) = crossdifference of every pair of points on line X(122)X(125)
X(112) = barycentric product of X(1113) and X(1114)
X(112) = isogonal conjugate of isotomic conjugate of trilinear pole of Euler line
X(112) = point of intersection, other than A, B, C, of circumcircle and hyperbola {A,B,C,X(4),PU(39)}}
X(112) = trilinear pole of line X(6)X(25)
X(112) = X(647)-cross conjugate of X(6)
X(112) = pole wrt polar circle of trilinear polar of X(850) (line X(115)X(127))
X(112) = X(48)-isoconjugate (polar conjugate) of X(850)
X(112) = X(92)-isoconjugate of X(520)
X(112) = X(1577)-isoconjugate of X(3)
X(112) = trilinear pole wrt circumsymmedial triangle of line X(6)X(647)
X(112) = reflection of X(935) in the Euler line
X(112) = reflection of X(2715) in the Brocard axis
X(112) = reflection of X(2722) in line X(1)X(3)
X(112) = inverse-in-polar-circle of X(115)
X(112) = inverse-in-{circumcircle, nine-point circle}-inverter of X(1560)
X(112) = inverse-in-Moses-radical-circle of X(1304)
X(112) = inverse-in-[circle with diameter X(15)X(16) and center X(187)] of X(842)
X(112) = inverse-in-circle-{X(1687),X(1688),PU(1),PU(2)} of X(2698)
X(112) = inverse-in-Stevanovic-circle of X(2766)
X(112) = X(139)-of-excentral-triangle
X(112) = barycentric product of PU(74)
X(112) = trilinear product of PU(108)
X(112) = eigencenter of circumnormal triangle
X(112) = Thomson-isogonal conjugate of X(1503)
X(112) = Lucas-isogonal conjugate of X(1503)
X(112) = perspector of circumcevian triangle of X(468) and cross-triangle of ABC and circumcevian triangle of X(25)
X(112) = trilinear product of circumcircle intercepts of line X(1)X(19)
X(112) = barycentric product X(3)*X(107)
X(112) = barycentric product X(4)*X(110)
X(112) = X(6)-of-1st-anti-orthosymmedial-triangle
X(112) = cevapoint of Jerabek hyperbola intercepts of Lemoine axis
X(112) = Ψ(X(i),X(j)) for these (i,j): (1,19), (2,3), (3,6), (4,6), (5,53), (6,25), (69,2), (76,4), (125,115)
X(112) = Vu circlecevian point of PU(39)

leftri

Centers X(113)-X(139)

rightri
Centers X(113)-X(139) lie on the nine-point circle.

Suppose that X is a point on the nine-point circle, and let X' be the reflection of X in the orthocenter, H. Then X is the anticenter of the cyclic quadrilateral ABCX'. Let HA be the orthocenter of triangle BCX, Let HB be the orthocenter of CAX, and let HC be the orthocenter of triangle ABX. Then the quadrilateral HHAHBHC is homothetic to and congruent to the cyclic quadrilateral ABCX', and X is the center of homothety. (Randy Hutson, 9/23/2011)


X(113) = JERABEK ANTIPODE

Trilinears    sin B sin C [(sin C)/(cos C - 2 cos A cos B) + (sin B)/(cos B - 2 cos A cos C)] : :
Barycentrics    (sin C)/(cos C - 2 cos A cos B) + (sin B)/(cos B - 2 cos A cos C) : :
Barycentrics    b2/(b2SB - 2SASC) + c2/(c2SC - 2SASB) : : (Peter J. C. Moses, 3/2003)
Barycentrics    (2a^4 - b^4 - c^4 - a^2b^2 - a^2c^2 + 2b^2c^2)[a^4(b^2 + c^2) - 2a^2(b^4 - b^2c^2 + c^4) + (b^2 - c^2)^2(b^2 + c^2)] : :

Let A'B'C' be the orthic triangle. Let L be the line through A' parallel to the Euler line, and define M and N cyclically. Let L' be the reflection of L in sideline BC, and define M' and N' cyclically. The lines L', M', N' concur in X(113). (Randy Hutson, August 26, 2014)

Let A'B'C' be the orthic triangle. Let MA be the reflection of the orthic axis in line B'C', and define Let MB and Let MC cyclically. Let A'' = Let MB∩MC, and define B'' and C'' cyclically. The lines A'A'', B'B'' C'C'' concur in X(113). (Randy Hutson, August 26, 2014)

Let A'B'C' be the orthic triangle. Let NA be the orthic axis of AB'C', and define NB and NC cyclically. Let A'' = NB∩NC, B'' = NC∩AC, C'' = NA∩BC. Then triangle A''B''C'' is inversely similar to ABC, with similitude center X(6), and the lines A'A'', B'B'', C'C'' concur in X(113). Also, X(113) = X(3)-of-A''B''C''. (Randy Hutson, August 26, 2014)

X(113) lies on the bicevian conic of X(2) and X(110), the Walsmith rectangular hyperbola, and these lines: 2,74   3,122   4,110   5,125   6,13   11,942   52,135   114,690   123,960   127,141   137,546

X(113) = midpoint of X(i) and X(j) for these (i,j): (4,110), (74,146), (265,399), (1553,3258)
X(113) = reflection of X(i) in X(j) for these (i,j): (52,1112), (125,5), (32110, 468)
X(113) = complementary conjugate of X(30)
X(113) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,30), (2,3003)
X(113) = crosspoint of X(4) and X(403)
X(113) = crossdifference of every pair of points on line X(526)X(686)
X(113) = nine-point-circle-antipode of X(125)
X(113) = X(74)-of-medial-triangle
X(113) = X(104)-of-orthic-triangle if ABC is acute
X(113) = X(186)-of-X(4)-Brocard-triangle
X(113) = center of rectangular circumhyperbola that passes through X(110)
X(113) = center of rectangular hyperbola {X(3),X(4),X(110),X(155),X(1351),X(1352),X(2574,X(2575)}}
X(113) = perpsector of circumconic centered at X(3003)
X(113) = inverse-in-polar-circle-of X(1300)
X(113) = inverse-in-{circumcircle, nine-point circle}-inverter of X(1302)
X(113) = anticenter of cyclic quadrilateral ABCX(110)
X(113) = Λ(X(2),X(3))-with-respect-to-orthic-triangle
X(113) = barycentric product X(30)*X(3580)
X(113) = complement of X(74)
X(113) = reflection of X(3) in X(5972)
X(113) = antipode of X(3) in the bicevian conic of X(2) and X(110)
X(113) = antipode of X(52) in the Hatzipolakis-Lozada hyperbola
X(113) = orthopole of line X(3)X(523)
X(113) = perspector of Ehrmann mid-triangle and orthic triangle
X(113) = excentral-to-ABC functional image of X(104)
X(113) = antipode of X(32110) in Walsmith rectangular hyperbola
X(113) = orthocenter of X(6)X(125)X(3569)
X(113) = crosssum of circumcircle intercepts of line PU(161) (line X(3)X(523))
X(113) = homothetic center of ABC and X(30)-Fuhrmann triangle


X(114) = KIEPERT ANTIPODE

Trilinears   bc[b sec(B + ω) + c sec(C + ω)] : :
Trilinears   cos(B - C) cos 2ω - sin ω sin(A + ω) (Peter J. C. Moses, 9/12/03)
Barycentrics   b sec(B + ω) + c sec(C + ω) : :
Barycentrics   (b^4 + c^4 - a^2b^2 - a^2c^2)(2a^4 + b^4 + c^4 - a^2b^2 - a^2c^2 - 2b^2c^2) : :

X(114) is the QA-P30 center (Reflection of QA-P2 in QA-P11) of quadrangle ABCX(2) (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/58-qa-p30.html)

Let A'B'C' be the orthic triangle. Let La be the Lemoine axis of triangle AB'C', and define Lb and Lc cyclically. Let A″ = Lb∩Lc, and define B″ and C″ cyclically. Triangle A″B″C″ is inversely similar to ABC, with similicenter X(2). The lines A'A″, B'B″, C'C″ concur in X(114), which is X(3)-of-A″B″C″.

Let OA be the circle centered at the A-vertex of the obverse triangle of X(69) and passing through A; define OB and OC cyclically. X(114) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

X(114) lies on the bicevian conic of X(2) and X(99), and on these lines: 2,98   3,127   4,99   5,39   25,135   52,211   113,690   132,684   136,427   325,511   381,543

X(114) = midpoint of X(i) and X(j) for these (i,j): (4,99), (98,147)
X(114) = reflection of X(i) in X(j) for these (i,j): (3,620), (115,5)
X(114) = isogonal conjugate of X(2065)
X(114) = complement of X(98)
X(114) = complementary conjugate of X(511)
X(114) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,230), (4,511)
X(114) = crosspoint of X(2) and X(325)
X(114) = orthojoin of X(230)
X(114) = Moses-radical-circle-inverse of X(38975)
X(114) = nine-point-circle-antipode of X(115)
X(114) = X(98)-of-medial triangle
X(114) = X(103)-of-orthic triangle if ABC is acute
X(114) = perspector of circumconic centered at X(230)
X(114) = center of circumconic that is locus of trilinear poles of lines passing through X(230)
X(114) = center of rectangular hyperbola {X(4),X(76),X(99),X(376),X(487),X(488)}}
X(114) = X(1513)-of-1st-Brocard-triangle
X(114) = inverse-in-polar-circle of X(3563)
X(114) = inverse-in-{circumcircle, nine-point circle}-inverter of X(110)
X(114) = center of inverse-in-{circumcircle, nine-point circle}-inverter of Brocard circle
X(114) = X(5)-of-1st-anti-Brocard-triangle
X(114) = anticenter of cyclic quadrilateral ABCX(99)
X(114) = Λ(X(3), X(6)), wrt orthic triangle
X(114) = orthocenter of X(98)X(115)X(31953)
X(114) = centroid of mid-triangle of X(15)- and X(16)-Fuhrmann triangles


X(115) = CENTER OF KIEPERT HYPERBOLA

Trilinears    bc(b2 - c2)2 : ca(c2 - a2)2 : ab(a2 - b2)2
Trilinears    cos A - 2 cos(B - C) + cot ω sin A : : (Peter J. C. Moses, 9/12/03)
Trilinears    sin A sin2(B - C) : :
Barycentrics    (b2 - c2)2 : (c2 - a2)2 : (a2 - b2)2
Barycentrics    (SB - SC)^2 : :
Barycentrics    (SA - SW) (SB + SC) + 4 SB SC : :
X(115) = 2(tan ω sin 2ω)*X(5) - X(39)
X(115) = X(13) + X(14) (Randy Hutson, July 23, 2015) X(115) = 3 X[2] + X[148], 3 X[2] - 4 X[6722], 5 X[2] - X[8591], 7 X[2] + X[8596], X[2] - 3 X[9166], 4 X[2] - 3 X[9167], 3 X[2] - 5 X[14061], 2 X[2] - 3 X[14971], 4 X[2] - X[15300], 9 X[2] - X[20094], 5 X[2] - 4 X[22247], 6 X[2] - 5 X[31274], 9 X[2] - 4 X[35022], X[3] - 4 X[20398], 3 X[4] + X[9862], 3 X[4] - X[10722], 2 X[4] + X[10991], X[4] + 2 X[11623], X[4] - 3 X[14639], X[4] + 3 X[14651], 4 X[5] - X[14981], 3 X[5] - 4 X[15092], 2 X[5] - 3 X[23514], X[6] - 3 X[6034], X[13] + 3 X[5469], X[13] - 3 X[5470], 3 X[13] + X[6777], 3 X[13] - X[6778], X[14] - 3 X[5469], X[14] + 3 X[5470], 3 X[14] - X[6777], 3 X[14] + X[6778], X[15] - 3 X[22510], X[16] - 3 X[22511], X[20] - 3 X[34473], 3 X[98] - X[9862], 3 X[98] + X[10722], X[98] + 3 X[14639], X[98] - 3 X[14651], X[99] + 3 X[671], 2 X[99] - 3 X[2482], X[99] - 6 X[5461], X[99] - 4 X[6722], 5 X[99] - 3 X[8591], 7 X[99] + 3 X[8596], X[99] - 9 X[9166], 4 X[99] - 9 X[9167], X[99] - 5 X[14061], 2 X[99] - 9 X[14971], 4 X[99] - 3 X[15300], 3 X[99] - X[20094], 5 X[99] - 12 X[22247], 2 X[99] - 5 X[31274]

The circumcircle of the incentral triangle intersects the nine-point circle at 2 points, X(11) and X(115), and X(115) lies on the incentral circle and the cevian circle of every point on the Kiepert hyperbola. Let A'B'C' be the orthic triangle. The Brocard axes of AB'C', BC'A', CA'B' concur in X(115). Let P be a point on the Brocard circle, and let L be the line tangent to the Brocard circle at P. Let P' be the trilinear pole of L, and let P" be the isotomic conjugate of P'. As P varies, P" traces an ellipse with center at X(115). (Randy Hutson, July 23, 2015)

Let La, Lb, Lc be the lines through A, B, C, resp. parallel to the Brocard axis. Let Ma, Mb, Mc be the reflections of BC, CA, AB in La, Lb, Lc, resp. Let A' = Mb∩Mc, and define B', C' cyclically. Triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. Let A″B″C″ be the reflection of A'B'C' in the Brocard axis. The triangle A″B″C″ is homothetic to ABC, with center of homothety X(115); see Hyacinthos #16741/16782, Sep 2008.

X(115) is the QA-P2 center (Euler-Poncelet Point) of quadrangle ABCX(2) (see http://www.chrisvantienhoven.nl/quadrangle-objects/10-mathematics/quadrangle-objects/12-qa-p2.html)

Let F be the Feuerbach point, X(11), and FaFbFc be the Feuerbach triangle (the extraversion triangle of X(11)). Let A' be the barycentric product F*Fa, and define B', C' cyclically. The lines AA', BB', CC' concur in X(115). (Randy Hutson, January 29, 2018)

If you have The Geometer's Sketchpad, you can view Kiepert Hyperbola, showing X(115).

Roland H. Eddy and R. Fritsch, "The conics of Ludwig Kiepert: a comprehensive lesson in the geometry of the triangle," Mathematics Magazine 67 (1994) 188-205.

For a construction of X(115) as intersection point of four conics see Peter Moses, euclid 5328.

For an artistic design generated by X(115), see X(244).

X(115) lies on these curves: nine-point circle, Steiner inellipse, bicevian conic of X(2) and X(98), 3rd Lemoine circle, Moses circle, Moses-Parry circle, the cubics K203, K237, K239, K301, K392, K483, K599, K629, K886, K942, and on these lines: {1, 11725}, {2, 99}, {3, 2079}, {4, 32}, {5, 39}, {6, 13}, {8, 7983}, {10, 3029}, {11, 1015}, {12, 1500}, {15, 6771}, {16, 6774}, {17, 11602}, {18, 11603}, {20, 5206}, {25, 3455}, {30, 187}, {33, 9636}, {37, 3822}, {50, 231}, {53, 133}, {54, 9697}, {55, 9664}, {56, 9651}, {61, 16002}, {62, 16001}, {69, 7818}, {75, 21431}, {76, 626}, {83, 4027}, {100, 10769}, {110, 3044}, {116, 1086}, {117, 1901}, {118, 1834}, {119, 2092}, {120, 442}, {121, 1213}, {122, 2797}, {123, 2798}, {124, 1146}, {125, 245}, {127, 338}, {128, 233}, {129, 389}, {130, 24862}, {131, 216}, {135, 2970}, {136, 8901}, {140, 10992}, {141, 5969}, {142, 35074}, {147, 2548}, {149, 21341}, {172, 3585}, {182, 13517}, {183, 7761}, {193, 32827}, {194, 7752}, {213, 21935}, {226, 11608}, {232, 403}, {235, 3199}, {244, 23063}, {274, 17669}, {302, 14904}, {303, 14905}, {312, 34542}, {315, 7751}, {316, 385}, {321, 11611}, {325, 538}, {371, 8980}, {372, 13967}, {373, 27779}, {376, 8588}, {378, 21397}, {382, 3053}, {384, 5152}, {393, 6529}, {395, 530}, {396, 531}, {397, 22832}, {398, 22831}, {427, 1196}, {429, 20621}, {485, 1504}, {486, 1505}, {498, 9598}, {499, 9597}, {511, 15980}, {512, 2679}, {513, 35079}, {514, 35080}, {515, 11710}, {516, 20666}, {517, 5164}, {518, 35084}, {519, 10026}, {522, 35086}, {523, 5099}, {524, 5107}, {525, 34943}, {536, 35089}, {546, 5007}, {547, 3055}, {549, 3054}, {550, 15513}, {567, 9604}, {571, 31723}, {577, 2165}, {590, 6306}, {591, 13927}, {593, 1029}, {594, 4013}, {597, 3363}, {599, 19662}, {609, 18513}, {615, 6302}, {616, 11489}, {617, 11488}, {618, 6670}, {619, 6669}, {623, 31711}, {624, 31712}, {631, 13172}, {645, 26081}, {647, 3258}, {650, 5520}, {656, 35091}, {661, 2170}, {662, 24957}, {694, 34359}, {698, 5031}, {726, 20546}, {732, 5103}, {759, 21004}, {799, 25472}, {804, 1084}, {805, 31513}, {857, 35093}, {858, 3291}, {958, 13181}, {1007, 32984}, {1062, 9635}, {1078, 6655}, {1089, 16886}, {1107, 25639}, {1109, 21833}, {1111, 1577}, {1125, 11711}, {1131, 6568}, {1132, 6569}, {1141, 14586}, {1184, 5064}, {1194, 5133}, {1210, 24472}, {1211, 13466}, {1250, 13076}, {1281, 5051}, {1312, 8106}, {1313, 8105}, {1316, 8429}, {1329, 1574}, {1346, 8427}, {1347, 8426}, {1348, 2033}, {1349, 2034}, {1352, 5028}, {1365, 3708}, {1376, 13173}, {1384, 3830}, {1415, 13273}, {1478, 2242}, {1479, 2241}, {1503, 1692}, {1509, 6625}, {1513, 2021}, {1565, 4403}, {1568, 3289}, {1570, 3564}, {1571, 1698}, {1572, 1699}, {1573, 2886}, {1575, 3814}, {1587, 19055}, {1588, 19056}, {1597, 34809}, {1598, 9861}, {1609, 18534}, {1611, 34609}, {1650, 13179}, {1656, 5013}, {1657, 5023}, {1676, 2035}, {1677, 2036}, {1691, 29012}, {1737, 20461}, {1865, 2791}, {1914, 3583}, {1970, 13403}, {1971, 18400}, {1975, 3788}, {1990, 18809}, {1991, 13850}, {1992, 11161}, {1995, 13233}, {2028, 2039}, {2029, 2040}, {2030, 11645}, {2051, 9561}, {2072, 14961}, {2076, 29317}, {2086, 9427}, {2176, 24045}, {2238, 5526}, {2240, 5990}, {2245, 31841}, {2275, 7741}, {2276, 7951}, {2310, 15607}, {2321, 34528}, {2378, 5618}, {2379, 5619}, {2386, 5140}, {2387, 5167}, {2394, 14223}, {2395, 14998}, {2420, 12295}, {2475, 5277}, {2476, 5283}, {2489, 34978}, {2501, 12079}, {2502, 5642}, {2503, 5540}, {2566, 32481}, {2567, 32482}, {2643, 4092}, {2653, 12047}, {2770, 10415}, {2788, 5511}, {2789, 5510}, {2793, 5512}, {2896, 7911}, {2936, 5020}, {2963, 14367}, {2971, 4516}, {2980, 9233}, {2996, 3926}, {3003, 11799}, {3008, 35115}, {3011, 33329}, {3068, 19109}, {3069, 19108}, {3070, 5062}, {3071, 5058}, {3087, 20774}, {3090, 7738}, {3094, 7697}, {3096, 7933}, {3104, 16630}, {3105, 16631}, {3117, 22735}, {3136, 5513}, {3141, 16614}, {3153, 10313}, {3154, 9209}, {3229, 21531}, {3239, 35122}, {3271, 5509}, {3284, 10297}, {3290, 30447}, {3329, 7827}, {3438, 8742}, {3439, 8741}, {3448, 15342}, {3452, 35130}, {3454, 20532}, {3524, 12117}, {3526, 15815}, {3534, 5210}, {3545, 6054}, {3548, 15075}, {3552, 7857}, {3566, 34953}, {3589, 5026}, {3614, 31460}, {3618, 5182}, {3619, 33223}, {3627, 35007}, {3628, 31652}, {3695, 7230}, {3700, 24040}, {3721, 34829}, {3729, 27688}, {3735, 3944}, {3741, 23917}, {3785, 32982}, {3817, 21636}, {3825, 16604}, {3832, 5319}, {3836, 35118}, {3839, 5304}, {3840, 23918}, {3843, 5346}, {3845, 5008}, {3849, 8352}, {3850, 5041}, {3851, 9605}, {3856, 34571}, {3875, 27556}, {3912, 20337}, {3933, 7821}, {3934, 5976}, {3936, 17310}, {3948, 35126}, {3954, 21029}, {3972, 7806}, {3981, 21243}, {3992, 20483}, {4016, 21018}, {4037, 21057}, {4052, 34899}, {4129, 6547}, {4136, 22036}, {4428, 12326}, {4721, 24995}, {4995, 12354}, {5024, 5055}, {5034, 12177}, {5038, 25555}, {5046, 5985}, {5052, 5480}, {5054, 12355}, {5056, 31400}, {5065, 18537}, {5066, 9300}, {5068, 31404}, {5070, 31457}, {5071, 23234}, {5072, 14692}, {5076, 22331}, {5077, 7610}, {5079, 22332}, {5080, 5291}, {5097, 14160}, {5104, 19924}, {5111, 5965}, {5134, 17734}, {5149, 5989}, {5162, 5999}, {5169, 6032}, {5215, 27088}, {5275, 17532}, {5276, 17577}, {5298, 18969}, {5318, 5478}, {5321, 5479}, {5334, 6770}, {5335, 6773}, {5418, 9674}, {5432, 15452}, {5449, 8571}, {5463, 16645}, {5464, 16644}, {5466, 9180}, {5473, 11481}, {5474, 11480}, {5485, 5503}, {5515, 6377}, {5521, 8735}, {5552, 13189}, {5569, 8860}, {5585, 15688}, {5587, 9620}, {5590, 6320}, {5591, 6319}, {5599, 13176}, {5600, 13177}, {5603, 7970}, {5613, 18582}, {5617, 18581}, {5663, 15535}, {5705, 31442}, {5886, 11724}, {5912, 16092}, {5939, 7792}, {5972, 20998}, {5977, 26601}, {5980, 11304}, {5981, 11303}, {5986, 7394}, {5987, 7533}, {5992, 27040}, {6179, 7823}, {6201, 6226}, {6202, 6227}, {6328, 8574}, {6337, 32969}, {6381, 20541}, {6390, 31275}, {6392, 7758}, {6422, 31481}, {6423, 23251}, {6424, 23261}, {6528, 16081}, {6560, 13790}, {6561, 9675}, {6587, 16177}, {6683, 32992}, {6697, 33324}, {6704, 7859}, {6739, 8649}, {6748, 18402}, {6753, 16178}, {6776, 7694}, {6779, 16963}, {6780, 16962}, {6782, 10612}, {6783, 10611}, {6792, 9140}, {6794, 11005}, {7031, 18514}, {7051, 18975}, {7200, 35131}, {7257, 25685}, {7336, 8061}, {7486, 31450}, {7514, 9609}, {7576, 10985}, {7585, 13640}, {7586, 13760}, {7669, 11641}, {7684, 31701}, {7685, 31702}, {7728, 14849}, {7750, 7780}, {7754, 7759}, {7757, 7777}, {7760, 7785}, {7762, 7805}, {7763, 7781}, {7766, 7812}, {7767, 7873}, {7768, 7885}, {7769, 7783}, {7771, 7833}, {7774, 7775}, {7776, 7855}, {7778, 7801}, {7779, 7809}, {7782, 7907}, {7784, 7854}, {7786, 7864}, {7787, 7856}, {7789, 7874}, {7791, 7815}, {7793, 7802}, {7795, 7867}, {7796, 7912}, {7799, 7925}, {7800, 7935}, {7803, 7808}, {7807, 7816}, {7811, 7898}, {7814, 7906}, {7819, 7852}, {7822, 7866}, {7824, 7847}, {7831, 7924}, {7832, 7901}, {7836, 7899}, {7837, 7926}, {7839, 7858}, {7840, 11054}, {7846, 7932}, {7860, 7893}, {7865, 16990}, {7868, 33219}, {7869, 33283}, {7875, 7884}, {7876, 7918}, {7877, 7900}, {7878, 7920}, {7891, 7940}, {7892, 7942}, {7894, 7921}, {7896, 33290}, {7904, 7910}, {7905, 7941}, {7908, 32833}, {7915, 8363}, {7930, 14065}, {7931, 14046}, {7937, 16986}, {7939, 33289}, {7943, 16895}, {7988, 9592}, {7989, 9593}, {8029, 12076}, {8068, 13006}, {8145, 20387}, {8176, 11163}, {8182, 23055}, {8196, 12179}, {8203, 12180}, {8212, 12186}, {8213, 12187}, {8222, 13184}, {8223, 13185}, {8227, 9619}, {8253, 9600}, {8258, 25607}, {8259, 20394}, {8260, 20395}, {8265, 34845}, {8355, 22110}, {8362, 31239}, {8553, 12083}, {8573, 18535}, {8587, 17503}, {8597, 8859}, {8598, 32479}, {8623, 14957}, {8770, 30771}, {8779, 13851}, {8799, 12233}, {8882, 9378}, {8902, 20625}, {8960, 12962}, {8962, 15234}, {9114, 22578}, {9116, 22577}, {9151, 23301}, {9293, 10278}, {9302, 14492}, {9346, 11269}, {9479, 15449}, {9574, 31441}, {9603, 18350}, {9608, 13861}, {9650, 10895}, {9665, 10896}, {9669, 16781}, {9766, 22253}, {9855, 26613}, {9875, 25055}, {9881, 19875}, {9927, 23128}, {9971, 13249}, {10054, 10072}, {10056, 10070}, {10061, 10078}, {10062, 10077}, {10104, 32152}, {10151, 14581}, {10153, 32532}, {10175, 31398}, {10314, 18420}, {10315, 18406}, {10316, 18404}, {10317, 18403}, {10527, 13190}, {10531, 12189}, {10532, 12190}, {10575, 15575}, {10590, 31409}, {10638, 13075}, {10641, 12141}, {10642, 12142}, {10653, 16635}, {10654, 16634}, {10753, 14853}, {10893, 12182}, {10894, 12183}, {10986, 18559}, {10989, 11580}, {11056, 26257}, {11081, 15929}, {11082, 16807}, {11086, 15930}, {11087, 16806}, {11121, 11129}, {11122, 11128}, {11168, 15810}, {11231, 31443}, {11284, 34013}, {11287, 15271}, {11295, 22575}, {11296, 22576}, {11485, 13102}, {11486, 13103}, {11496, 12178}, {11539, 11614}, {11585, 22401}, {11625, 27550}, {11627, 27551}, {11680, 16975}, {11742, 15689}, {11897, 12181}, {12077, 30460}, {12100, 26614}, {12191, 14712}, {12900, 33512}, {13192, 15360}, {13479, 17983}, {13509, 25739}, {13540, 35111}, {13703, 13705}, {13823, 13825}, {13893, 31437}, {14023, 32006}, {14157, 15340}, {14269, 21309}, {14389, 30685}, {14585, 21659}, {14873, 23537}, {14907, 17008}, {14989, 32640}, {15081, 18331}, {15118, 32740}, {15538, 15545}, {15655, 15681}, {15903, 17056}, {16267, 22997}, {16268, 22998}, {16303, 18487}, {16317, 24855}, {16529, 16960}, {16530, 16961}, {16989, 33016}, {17006, 33273}, {17116, 27707}, {17423, 34952}, {17448, 24387}, {17677, 26244}, {17702, 32661}, {17757, 21956}, {18140, 33841}, {18472, 18564}, {18584, 19709}, {18591, 30445}, {18974, 19373}, {19053, 19058}, {19054, 19057}, {19102, 22502}, {19105, 22501}, {19106, 19780}, {19107, 19781}, {19905, 20423}, {20065, 32996}, {20595, 23912}, {20976, 24981}, {21138, 21253}, {21448, 32216}, {21674, 21816}, {21675, 21810}, {22425, 30449}, {22491, 22579}, {22492, 22580}, {22495, 22571}, {22496, 22572}, {22504, 22753}, {23288, 34206}, {23536, 27555}, {24345, 24711}, {24443, 30436}, {25561, 25562}, {26235, 31107}, {26363, 31456}, {26582, 27076}, {27318, 33061}, {27374, 27375}, {27570, 33932}, {27706, 33933}, {27966, 33940}, {28473, 34959}, {30466, 30469}, {30714, 35324}, {31411, 31412}, {31416, 31418}, {31422, 31423}, {31433, 31434}, {31448, 31501}, {31471, 31472}, {31477, 31479}, {31482, 31484}, {31490, 31493}, {31685, 31686}, {31689, 31691}, {31690, 31692}, {32190, 32476}, {32456, 35297}, {32490, 32494}, {32491, 32497}, {32822, 32955}, {32826, 32973}, {32829, 32988}, {32834, 33200}, {32838, 32990}, {32986, 34229}, {34989, 34990}

p> X(115) = midpoint of X(i) and X(j) for these {i,j}: {1, 13178}, {2, 671}, {3, 6321}, {4, 98}, {6, 11646}, {8, 7983}, {10, 11599}, {13, 14}, {15, 23004}, {16, 23005}, {17, 11602}, {18, 11603}, {20, 10723}, {69, 10754}, {76, 1916}, {83, 11606}, {99, 148}, {100, 10769}, {125, 16278}, {226, 11608}, {265, 18332}, {316, 385}, {321, 11611}, {381, 11632}, {625, 32457}, {805, 31513}, {1131, 6568}, {1132, 6569}, {1650, 13179}, {1992, 11161}, {2009, 2010}, {2394, 14223}, {2679, 6071}, {2996, 8781}, {3448, 15342}, {3830, 14830}, {4052, 34899}, {5466, 9180}, {5469, 5470}, {5485, 5503}, {5912, 34169}, {6033, 12188}, {6054, 12243}, {6055, 9880}, {6108, 31710}, {6109, 31709}, {6772, 6775}, {6777, 6778}, {6779, 25156}, {6780, 25166}, {6782, 33517}, {6783, 33518}, {7809, 19570}, {7840, 11054}, {8029, 14443}, {8352, 22329}, {8587, 17503}, {9114, 22578}, {9116, 22577}, {9140, 9144}, {9293, 13187}, {9302, 14492}, {9862, 10722}, {10153, 32532}, {10418, 14832}, {11005, 22265}, {12042, 22515}, {13173, 13180}, {13181, 22514}, {14041, 14568}, {14639, 14651}, {16001, 25560}, {16002, 25559}, {19905, 20423}, {22512, 22513}, {24345, 24711}
X(115) = midpoint of PU(40)
X(115) = reflection of X(i) in X(j) for these {i,j}: {1, 11725}, {2, 5461}, {3, 6036}, {39, 2023}, {98, 11623}, {99, 620}, {114, 5}, {125, 15359}, {187, 230}, {325, 625}, {599, 19662}, {618, 6670}, {619, 6669}, {620, 6722}, {1569, 39}, {2482, 2}, {4027, 7829}, {5026, 3589}, {5099, 14120}, {5477, 6}, {5976, 3934}, {6036, 20398}, {6292, 9478}, {6781, 187}, {6782, 11543}, {6783, 11542}, {7813, 325}, {8290, 6704}, {9115, 395}, {9117, 396}, {9167, 14971}, {10991, 98}, {10992, 33813}, {11711, 1125}, {12829, 5305}, {14501, 2040}, {14502, 2039}, {14928, 5026}, {14971, 9166}, {14981, 114}, {15300, 2482}, {15357, 125}, {18800, 597}, {20387, 8145}, {22110, 8355}, {22505, 546}, {22566, 5066}, {22848, 22847}, {22892, 22893}, {25486, 8176}, {25559, 20415}, {25560, 20416}, {25562, 25561}, {25607, 8258}, {25608, 8259}, {25609, 8260}, {31274, 14061}, {32135, 25555}, {32458, 626}, {33512, 12900}, {33813, 140}, {34602, 22846}
X(115) = isogonal conjugate of X(249)
X(115) = isotomic conjugate of X(4590)
X(115) = complement of X(99)
X(115) = anticomplement of X(620)
X(115) = circumcircle-inverse of X(2079)
X(115) = orthocentroidal-circle-inverse of X(6)
X(115) = Stevanovic-circle-inverse of X(5520)
X(115) = Lester-circle-inverse of X(10413)
X(115) = Spieker-radical-circle-inverse of X(5213)
X(115) = polar-circle-inverse of X(112)
X(115) = Dao-Moses-Telv-circle-inverse of X(125)
X(115) = orthoptic-circle-of-Steiner-inellipse-inverse of X(111)
X(115) = orthoptic-circle-of-Steiner-circumellipe-inverse of X(20099)
X(115) = Moses-radical-circle-inverse of X(3258)
X(115) = Hutson-Parry-circle-inverse of X(1648)
X(115) = inverse-in-circle-{X(3102),X(3103),PU(1)}} of X(32448)
X(115) = polar conjugate of X(18020)
X(115) = antigonal image of X(6328)
X(115) = complement of the isogonal conjugate of X(512)
X(115) = complement of the isotomic conjugate of X(523)
X(115) = isotomic conjugate of the anticomplement of X(23991)
X(115) = isotomic conjugate of the isogonal conjugate of X(3124)
X(115) = isogonal conjugate of the isotomic conjugate of X(338)
X(115) = isotomic conjugate of the polar conjugate of X(8754)
X(115) = isogonal conjugate of the polar conjugate of X(2970)
X(115) = polar conjugate of the isotomic conjugate of X(125)
X(115) = polar conjugate of the isogonal conjugate of X(20975)
X(115) = medial-isogonal conjugate of X(512)
X(115) = orthic-isogonal conjugate of X(512)
X(115) = psi-transform of X(6792)
X(115) = tripolar centroid of X(5466)
X(115) = X(2)-Hirst inverse of X(148)
X(115) = X(99)-of-medial triangle
X(115) = X(101)-of-orthic triangle if ABC is acute
X(115) = X(325)-of-1st-Brocard-triangle
X(115) = X(187)-of-4th-Brocard-triangle
X(115) = X(187)-of-orthocentroidal-triangle
X(115) = X(141)-of-1st-antiBrocard-triangle
X(115) = barycentric product X(11)*X(12)
X(115) = {X(5),X(39)}-harmonic conjugate of X(1506)
X(115) = projection from Steiner circumellipse to Steiner inellipse of X(671)
X(115) = center of similitude of incentral and Feuerbach triangles
X(115) = center of circumconic that is locus of trilinear poles of lines parallel to the orthic axis (i.e. lines that pass through X(523))
X(115) = perspector of circumconic centered at X(523) (parabola {A,B,C,X(476),X(523),X(685)})
X(115) = trilinear pole wrt medial triangle of line X(2)X(6)
X(115) = inverse-in-circumcircle of X(2079)
X(115) = inverse-in-polar-circle of X(112)
X(115) = inverse-in-{circumcircle, nine-point circle}-inverter of X(111)
X(115) = inverse-in-Moses-radical-circle of X(3258)
X(115) = inverse-in-Steiner-circumellipse of X(148)
X(115) = inverse-in-excircles-radical-circle of X(5213)
X(115) = {X(99),X(671)}-harmonic conjugate of X(148)
X(115) = {X(6108),X(6109)}-harmonic conjugate of X(6055)
X(115) = inverse-in-circle-{X(2),X(13),X(14),X(111),X(476)}} of X(1648)
X(115) = orthopole of Brocard axis
X(115) = orthic isogonal conjugate of X(512)
X(115) = incentral isogonal conjugate of X(512)
X(115) = perspector of orthic triangle and tangential triangle of hyperbola {A,B,C,X(2),X(6)}}
X(115) = similitude center of (equilateral) pedal triangles of X(15) and X(16)
X(115) = exsimilicenter of Moses circle and the nine-point circle
X(115) = anticenter of cyclic quadrilateral ABCX(98)
X(115) = Λ(X(187), X(237))-wrt-orthic-triangle
X(115) = X(1101)-isoconjugate of X(2)
X(115) = harmonic center of nine-point circle and Gallatly circle
X(115) = perspector of medial triangle and Schroeter triangle
X(115) = trilinear pole of line X(1648)X(8029)
X(115) = barycentric square of X(523)
X(115) = inverse-in-Hutson-Parry-circle of X(1648)
X(115) = {X(13636),X(13722)}-harmonic conjugate of X(1648)
X(115) = homothetic center of medial triangle and mid-triangle of antipedal triangles of X(13) and X(14)
X(115) = homothetic center of Ehrmann vertex-triangle and mid-triangle of 1st and 2nd Kenmotu diagonals triangles
X(115) = centroid of reflection triangle of X(187)
X(115) = centroid of mid-triangle of 1st and 2nd isodynamic-Dao triangles
X(115) = centroid of mid-triangle of 3rd and 4th isodynamic-Dao triangles
X(115) = point of concurrence of cevian circles of the excenters
X(115) = intersection of Fermat axes of outer and inner Vecten triangles
X(115) = Schroeter-isogonal conjugate of X(35605)
X(115) = orthocenter of X(98)X(114)X(31953)
X(115) = point of concurrence of cevian circles of vertices of anticevian triangle of X(4)
X(115) = cevapoint of X(i) and X(j) for these (i,j): {512, 8574}, {523, 10278}, {1648, 23992}, {1649, 33906}, {3124, 20975}, {21043, 21833}
X(115) = crosspoint of X(i) and X(j) for these (i,j): {2, 523}, {4, 14618}, {6, 2623}, {10, 31010}, {25, 18105}, {68, 525}, {252, 15412}, {338, 2970}, {393, 2501}, {512, 27375}, {594, 4024}, {661, 2171}, {850, 1502}, {1089, 1577}, {2394, 5627}, {3120, 16732}, {3267, 6340}, {3413, 3414}, {4036, 28654}, {8024, 31067}, {8105, 8106}, {11080, 20578}, {11085, 20579}, {12028, 14592}
X(115) = crosssum of X(i) and X(j) for these (i,j): {2, 14570}, {3, 32661}, {6, 110}, {24, 112}, {69, 4576}, {99, 1078}, {101, 33771}, {143, 1625}, {163, 849}, {394, 4558}, {593, 4556}, {662, 2185}, {1379, 1380}, {1501, 1576}, {1511, 2420}, {1634, 8041}, {1986, 14591}, {4636, 7054}, {5546, 35193}, {8115, 8116}, {11130, 17403}, {11131, 17402}
X(115) = trilinear pole of line {1648, 8029}
X(115) = crossdifference of every pair of points on line {110, 351}
X(115) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 512}, {4, 21259}, {6, 4369}, {10, 21260}, {19, 30476}, {25, 8062}, {31, 523}, {32, 14838}, {37, 3835}, {42, 513}, {57, 17066}, {65, 17072}, {75, 23301}, {76, 21263}, {110, 21254}, {115, 21253}, {163, 620}, {213, 514}, {228, 20315}, {251, 8060}, {321, 21262}, {351, 16597}, {512, 10}, {513, 3741}, {514, 21240}, {523, 2887}, {560, 647}, {604, 17069}, {647, 18589}, {649, 3739}, {650, 21246}, {656, 1368}, {661, 141}, {663, 960}, {667, 1125}, {669, 37}, {688, 16587}, {741, 4155}, {756, 31946}, {798, 2}, {810, 3}, {822, 6389}, {850, 21235}, {872, 661}, {875, 740}, {881, 18904}, {922, 1649}, {923, 690}, {1015, 17761}, {1018, 27076}, {1042, 3900}, {1084, 16592}, {1096, 520}, {1101, 10190}, {1245, 8678}, {1333, 21196}, {1334, 20317}, {1356, 16613}, {1400, 4885}, {1402, 522}, {1438, 24285}, {1500, 4129}, {1576, 16598}, {1577, 626}, {1824, 20316}, {1910, 24284}, {1911, 9508}, {1918, 650}, {1919, 3666}, {1924, 39}, {1927, 2491}, {1946, 34851}, {1960, 34587}, {1964, 3005}, {1967, 804}, {1973, 525}, {1974, 16612}, {2084, 6292}, {2179, 18314}, {2203, 21187}, {2205, 6586}, {2206, 31947}, {2238, 27854}, {2258, 8672}, {2422, 16609}, {2485, 21247}, {2489, 226}, {2491, 16591}, {2501, 20305}, {2616, 3819}, {2623, 21231}, {2642, 126}, {2643, 125}, {2786, 20548}, {3005, 21249}, {3049, 1214}, {3063, 5745}, {3112, 688}, {3120, 21252}, {3121, 1086}, {3122, 11}, {3124, 8287}, {3125, 116}, {3223, 3221}, {3248, 244}, {3271, 34589}, {3402, 23878}, {3669, 17050}, {3700, 21244}, {3708, 127}, {3709, 3452}, {4010, 20542}, {4017, 2886}, {4024, 21245}, {4041, 1329}, {4069, 3038}, {4077, 17047}, {4079, 1211}, {4117, 1084}, {4455, 17793}, {4516, 124}, {4557, 24003}, {4559, 21232}, {4705, 3454}, {4729, 2885}, {4730, 121}, {4770, 21251}, {5027, 19563}, {5029, 20529}, {5466, 21256}, {6591, 34830}, {7178, 17046}, {7180, 142}, {7216, 21258}, {7250, 11019}, {7252, 21233}, {7257, 3037}, {8061, 21248}, {9178, 4892}, {9406, 5664}, {9426, 16584}, {9508, 20339}, {14407, 16594}, {14574, 23993}, {16606, 21191}, {18105, 1215}, {18344, 34831}, {18793, 6373}, {20975, 34846}, {21759, 31286}, {21832, 20333}, {21834, 21250}, {22260, 24040}, {23493, 4083}, {23503, 6374}, {23894, 625}, {24006, 21243}, {24290, 20540}, {28625, 4932}, {32676, 5972}, {32713, 23998}
X(115) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 12071}, {2, 523}, {4, 512}, {6, 12077}, {7, 12072}, {8, 12069}, {10, 6367}, {12, 4705}, {76, 826}, {83, 7927}, {99, 12076}, {148, 13187}, {190, 12078}, {264, 12075}, {338, 125}, {393, 2501}, {427, 2514}, {523, 8029}, {594, 4024}, {598, 12073}, {671, 690}, {850, 22260}, {1029, 513}, {1109, 4092}, {1119, 12070}, {1141, 14270}, {1502, 850}, {1916, 2799}, {1989, 1637}, {2165, 647}, {2970, 8754}, {2980, 669}, {2986, 9033}, {2996, 525}, {3120, 2643}, {3613, 3005}, {4024, 21131}, {4080, 6370}, {5392, 6368}, {5395, 3800}, {5466, 33919}, {5485, 3906}, {5627, 15475}, {6504, 520}, {6625, 514}, {7332, 2611}, {8754, 6388}, {8791, 14273}, {8818, 661}, {8901, 20975}, {10415, 9178}, {11080, 20578}, {11082, 6137}, {11085, 20579}, {11087, 6138}, {11121, 23870}, {11122, 23871}, {11538, 20188}, {11606, 804}, {12066, 30}, {12067, 12064}, {13576, 4155}, {13579, 924}, {13580, 2575}, {13581, 2574}, {13582, 526}, {13583, 3900}, {13584, 4083}, {13585, 1510}, {13854, 2489}, {14618, 23105}, {16081, 16230}, {16732, 1109}, {18023, 9134}, {18840, 7950}, {21043, 6627}, {21044, 3708}, {24624, 6089}, {28654, 4036}, {34294, 3124}, {34449, 34952}, {34816, 3806}
X(115) = X(i)-cross conjugate of X(j) for these (i,j): {512, 6328}, {2643, 1365}, {3124, 8754}, {8029, 523}, {14443, 690}, {20975, 125}, {21043, 1109}, {21131, 4024}, {21833, 2643}, {22260, 850}, {23991, 2}, {23992, 1648}, {33919, 5466}
X(115) = X(i)-isoconjugate of X(j) for these (i,j): {1, 249}, {2, 1101}, {6, 24041}, {31, 4590}, {32, 24037}, {41, 7340}, {48, 18020}, {58, 4567}, {59, 2185}, {60, 4564}, {63, 250}, {75, 23357}, {76, 23995}, {81, 4570}, {99, 163}, {100, 4556}, {109, 4612}, {110, 662}, {112, 4592}, {162, 4558}, {255, 23582}, {261, 2149}, {326, 23964}, {394, 24000}, {560, 34537}, {561, 23963}, {577, 23999}, {593, 765}, {604, 6064}, {643, 4565}, {648, 4575}, {651, 4636}, {691, 23889}, {692, 4610}, {757, 1252}, {758, 9273}, {798, 31614}, {799, 1576}, {811, 32661}, {849, 1016}, {873, 23990}, {1098, 1262}, {1110, 1509}, {1333, 4600}, {1414, 5546}, {1634, 4599}, {1725, 18879}, {1790, 5379}, {2150, 4998}, {2194, 4620}, {2206, 4601}, {2617, 18315}, {2966, 23997}, {3936, 9274}, {4563, 32676}, {4576, 34072}, {4602, 14574}, {4623, 32739}, {6507, 32230}, {6578, 35342}, {7045, 7054}, {7058, 24027}, {9268, 30576}, {10411, 32678}, {14213, 14587}, {18042, 27867}, {24039, 32729}, {35049, 35193}
X(115) = barycentric product X(i)*X(j) for these {i,j}: {1, 1109}, {3, 2970}, {4, 125}, {5, 8901}, {6, 338}, {7, 4092}, {8, 1365}, {10, 3120}, {11, 12}, {13, 30465}, {14, 30468}, {19, 20902}, {25, 339}, {27, 21046}, {30, 12079}, {31, 23994}, {32, 23962}, {37, 16732}, {42, 21207}, {68, 136}, {69, 8754}, {75, 2643}, {76, 3124}, {79, 21054}, {86, 21043}, {92, 3708}, {94, 2088}, {98, 868}, {99, 8029}, {107, 5489}, {110, 23105}, {122, 6526}, {127, 13854}, {135, 32132}, {137, 252}, {141, 34294}, {158, 2632}, {181, 34387}, {190, 21131}, {226, 21044}, {244, 1089}, {264, 20975}, {265, 35235}, {274, 21833}, {298, 30452}, {299, 30453}, {305, 2971}, {313, 3122}, {321, 3125}, {327, 6784}, {393, 15526}, {459, 1562}, {477, 6070}, {512, 850}, {513, 4036}, {514, 4024}, {523, 523}, {525, 2501}, {526, 10412}, {594, 1086}, {598, 8288}, {647, 14618}, {656, 24006}, {661, 1577}, {670, 22260}, {671, 1648}, {690, 5466}, {693, 4705}, {756, 1111}, {762, 16727}, {798, 20948}, {879, 16230}, {882, 14295}, {892, 33919}, {1015, 28654}, {1084, 1502}, {1093, 2972}, {1096, 17879}, {1118, 7068}, {1146, 6354}, {1254, 24026}, {1312, 1313}, {1358, 6057}, {1367, 1857}, {1425, 21666}, {1441, 4516}, {1500, 23989}, {1565, 7140}, {1637, 2394}, {1640, 14223}, {1647, 4013}, {1826, 4466}, {1897, 21134}, {1928, 4117}, {2052, 3269}, {2086, 18896}, {2170, 6358}, {2171, 4858}, {2395, 2799}, {2489, 3267}, {2611, 6757}, {2616, 2618}, {2623, 18314}, {2793, 34246}, {2969, 3695}, {2973, 3690}, {2996, 6388}, {3121, 27801}, {3258, 5627}, {3261, 4079}, {3268, 15475}, {3271, 34388}, {3318, 7157}, {3413, 13722}, {3414, 13636}, {3448, 6328}, {3613, 7668}, {3700, 7178}, {3906, 8599}, {3937, 7141}, {4010, 35352}, {4017, 4086}, {4041, 4077}, {4049, 4120}, {4052, 21950}, {4064, 7649}, {4081, 6046}, {4103, 6545}, {4608, 6367}, {4609, 23099}, {4988, 31010}, {5099, 10415}, {5139, 6340}, {5485, 6791}, {5512, 32133}, {5514, 13853}, {5949, 7332}, {6344, 16186}, {6506, 7363}, {6529, 23616}, {6535, 17205}, {6625, 6627}, {7018, 21725}, {7372, 12071}, {7927, 31065}, {8034, 27808}, {8061, 18070}, {8287, 8818}, {8371, 9180}, {8735, 26942}, {8736, 26932}, {8769, 17876}, {9033, 18808}, {9293, 10278}, {10413, 13582}, {10555, 14357}, {11792, 34110}, {12028, 16221}, {12077, 15412}, {14052, 34953}, {14086, 14086}, {14273, 14977}, {14431, 35353}, {14998, 18312}, {15320, 21045}, {18003, 18015}, {18004, 18014}, {18006, 18013}, {18007, 34763}, {18023, 21906}, {18105, 23285}, {20578, 23870}, {20579, 23871}, {21824, 30690}, {23283, 23284}, {23286, 23290}, {23287, 23288}
X(115) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 24041}, {2, 4590}, {4, 18020}, {6, 249}, {7, 7340}, {8, 6064}, {10, 4600}, {11, 261}, {12, 4998}, {25, 250}, {31, 1101}, {32, 23357}, {37, 4567}, {42, 4570}, {75, 24037}, {76, 34537}, {99, 31614}, {125, 69}, {127, 34254}, {136, 317}, {148, 31632}, {158, 23999}, {181, 59}, {226, 4620}, {244, 757}, {321, 4601}, {338, 76}, {339, 305}, {351, 5467}, {393, 23582}, {512, 110}, {514, 4610}, {523, 99}, {525, 4563}, {526, 10411}, {560, 23995}, {594, 1016}, {647, 4558}, {649, 4556}, {650, 4612}, {656, 4592}, {661, 662}, {663, 4636}, {669, 1576}, {690, 5468}, {693, 4623}, {756, 765}, {798, 163}, {804, 17941}, {810, 4575}, {826, 4576}, {850, 670}, {868, 325}, {872, 1110}, {879, 17932}, {881, 17938}, {882, 805}, {888, 5118}, {1015, 593}, {1084, 32}, {1086, 1509}, {1089, 7035}, {1096, 24000}, {1109, 75}, {1111, 873}, {1146, 7058}, {1254, 7045}, {1356, 1397}, {1357, 7341}, {1358, 552}, {1365, 7}, {1367, 7055}, {1500, 1252}, {1501, 23963}, {1577, 799}, {1637, 2407}, {1640, 14999}, {1645, 33875}, {1648, 524}, {1824, 5379}, {2086, 1691}, {2087, 30576}, {2088, 323}, {2170, 2185}, {2171, 4564}, {2207, 23964}, {2310, 1098}, {2395, 2966}, {2422, 2715}, {2485, 4611}, {2489, 112}, {2491, 14966}, {2501, 648}, {2610, 4585}, {2623, 18315}, {2632, 326}, {2642, 23889}, {2643, 1}, {2682, 5642}, {2793, 34245}, {2799, 2396}, {2970, 264}, {2971, 25}, {2972, 3964}, {3005, 1634}, {3022, 6061}, {3049, 32661}, {3120, 86}, {3121, 1333}, {3122, 58}, {3124, 6}, {3125, 81}, {3248, 849}, {3258, 6148}, {3269, 394}, {3271, 60}, {3569, 2421}, {3700, 645}, {3708, 63}, {3709, 5546}, {3906, 9146}, {4017, 1414}, {4024, 190}, {4036, 668}, {4041, 643}, {4049, 4615}, {4064, 4561}, {4077, 4625}, {4079, 101}, {4086, 7257}, {4092, 8}, {4103, 6632}, {4117, 560}, {4155, 3573}, {4171, 7259}, {4391, 4631}, {4466, 17206}, {4516, 21}, {4530, 30606}, {4705, 100}, {5099, 7664}, {5139, 6353}, {5466, 892}, {5489, 3265}, {6057, 4076}, {6137, 17402}, {6138, 17403}, {6328, 13485}, {6354, 1275}, {6367, 4427}, {6388, 193}, {6524, 32230}, {6627, 1654}, {6784, 182}, {6791, 1992}, {7063, 2175}, {7064, 6065}, {7068, 1264}, {7109, 23990}, {7140, 15742}, {7143, 7339}, {7178, 4573}, {7180, 4565}, {7216, 4637}, {7668, 1078}, {7669, 14366}, {7927, 10330}, {8029, 523}, {8034, 3733}, {8287, 34016}, {8288, 599}, {8371, 9182}, {8599, 35138}, {8663, 35327}, {8754, 4}, {8901, 95}, {9148, 23342}, {9171, 9181}, {9178, 691}, {9180, 9170}, {9426, 14574}, {9427, 1501}, {10278, 31998}, {10412, 35139}, {10630, 34539}, {11123, 14588}, {11182, 14607}, {12071, 6758}, {12073, 35356}, {12077, 14570}, {12079, 1494}, {13636, 6189}, {13722, 6190}, {14086, 14089}, {14223, 6035}, {14273, 4235}, {14295, 880}, {14398, 2420}, {14423, 33921}, {14443, 1649}, {14444, 8030}, {14618, 6331}, {14910, 18879}, {14936, 7054}, {14998, 5649}, {15328, 18878}, {15422, 16813}, {15449, 7794}, {15451, 23181}, {15475, 476}, {15525, 439}, {15526, 3926}, {15630, 1976}, {16230, 877}, {16726, 763}, {16732, 274}, {17205, 6628}, {17414, 9145}, {17876, 18156}, {17989, 17944}, {17990, 17943}, {17992, 17942}, {17993, 23348}, {17994, 4230}, {18001, 17940}, {18002, 17939}, {18003, 17935}, {18004, 17934}, {18006, 17933}, {18007, 34760}, {18012, 17937}, {18013, 17931}, {18014, 17930}, {18015, 17929}, {18070, 4593}, {18105, 827}, {18210, 1444}, {18808, 16077}, {19610, 9217}, {20382, 20380}, {20578, 23895}, {20579, 23896}, {20902, 304}, {20948, 4602}, {20975, 3}, {21043, 10}, {21044, 333}, {21045, 33297}, {21046, 306}, {21054, 319}, {21131, 514}, {21134, 4025}, {21138, 7304}, {21141, 16755}, {21207, 310}, {21709, 21085}, {21723, 21081}, {21725, 171}, {21731, 15329}, {21823, 172}, {21824, 3219}, {21833, 37}, {21859, 31615}, {21906, 187}, {21944, 3879}, {21961, 33116}, {22260, 512}, {23099, 669}, {23105, 850}, {23216, 14575}, {23282, 33948}, {23610, 9426}, {23616, 4143}, {23943, 32004}, {23962, 1502}, {23991, 620}, {23992, 2482}, {23994, 561}, {24006, 811}, {27375, 27867}, {28654, 31625}, {30452, 13}, {30453, 14}, {30465, 298}, {30468, 299}, {31010, 4632}, {31065, 35137}, {31644, 14061}, {33919, 690}, {34079, 9273}, {34294, 83}, {34387, 18021}, {34980, 1092}, {34981, 7752}, {34982, 13198}, {35078, 4027}, {35088, 32458}, {35235, 340}, {35352, 4589}, {35364, 10425}
X(115) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 99, 620}, {2, 111, 10418}, {2, 148, 99}, {2, 620, 31274}, {2, 2482, 9167}, {2, 2549, 574}, {2, 3734, 7820}, {2, 5461, 14971}, {2, 7790, 4045}, {2, 9166, 5461}, {2, 11185, 3734}, {2, 14061, 6722}, {2, 26138, 25530}, {2, 26794, 27071}, {2, 26845, 27008}, {3, 7746, 7749}, {3, 7748, 7756}, {3, 13881, 7746}, {4, 32, 7747}, {4, 3767, 32}, {4, 7735, 7737}, {4, 9862, 10722}, {4, 11623, 10991}, {4, 14651, 98}, {5, 39, 1506}, {5, 3815, 7603}, {5, 5254, 39}, {5, 7765, 9698}, {5, 15048, 3815}, {6, 13, 5472}, {6, 14, 5471}, {6, 381, 5475}, {6, 1989, 3018}, {6, 3018, 3163}, {6, 5309, 5355}, {6, 5475, 7753}, {6, 18353, 1879}, {10, 4115, 21711}, {10, 24070, 4115}, {13, 5469, 14}, {13, 6777, 6778}, {14, 5470, 13}, {14, 6778, 6777}, {32, 3767, 7755}, {39, 1506, 9698}, {39, 5254, 7765}, {39, 7603, 3815}, {53, 1596, 33842}, {75, 21604, 21431}, {76, 626, 7794}, {76, 5025, 626}, {76, 7934, 3314}, {83, 7797, 7829}, {83, 15031, 16044}, {98, 10722, 9862}, {98, 14639, 4}, {98, 14651, 11623}, {99, 620, 2482}, {99, 671, 148}, {99, 6722, 31274}, {99, 9166, 14061}, {99, 14061, 2}, {114, 23514, 5}, {125, 1562, 3269}, {125, 3124, 6388}, {125, 6791, 1648}, {127, 339, 15526}, {141, 33184, 7853}, {148, 5461, 31274}, {148, 6722, 2482}, {148, 9166, 6722}, {148, 14061, 620}, {148, 31274, 15300}, {183, 7761, 7810}, {183, 7841, 7761}, {194, 7752, 7764}, {194, 32966, 7752}, {235, 27376, 3199}, {315, 7751, 7826}, {315, 14063, 7825}, {316, 14568, 385}, {325, 33228, 625}, {376, 21843, 8588}, {381, 5309, 7753}, {381, 12188, 6033}, {384, 7828, 6680}, {385, 14041, 316}, {403, 5523, 232}, {498, 9598, 31451}, {546, 5305, 7745}, {574, 11648, 2549}, {597, 20112, 3363}, {620, 5461, 6722}, {620, 6722, 2}, {620, 31274, 9167}, {631, 13172, 21166}, {661, 2170, 20982}, {671, 5461, 2482}, {671, 7919, 14931}, {671, 9166, 2}, {671, 14061, 99}, {671, 14971, 15300}, {1078, 6655, 7830}, {1086, 8287, 17058}, {1107, 25639, 31488}, {1506, 7765, 39}, {1648, 3124, 6791}, {1648, 6791, 6388}, {1648, 8288, 125}, {1656, 5013, 31455}, {1656, 13188, 15561}, {1656, 15561, 6721}, {1975, 3788, 7863}, {1975, 7887, 3788}, {1989, 6128, 3163}, {2079, 34866, 3}, {2276, 7951, 31476}, {2482, 14971, 2}, {2482, 31274, 620}, {2548, 5286, 7772}, {2643, 21043, 4092}, {2996, 32972, 3926}, {3018, 6128, 6}, {3090, 7738, 31401}, {3090, 23235, 20399}, {3091, 5286, 2548}, {3120, 21044, 3125}, {3124, 8288, 1648}, {3314, 5025, 7934}, {3314, 7934, 626}, {3545, 7736, 31415}, {3545, 12243, 6054}, {3734, 7844, 2}, {3734, 8178, 99}, {3734, 18546, 11185}, {3767, 7737, 7735}, {3815, 5254, 15048}, {3815, 7603, 1506}, {3815, 15048, 39}, {3845, 5306, 14537}, {3934, 6656, 6292}, {3934, 7861, 6656}, {5007, 5305, 5368}, {5008, 14537, 18907}, {5024, 5055, 31489}, {5080, 17737, 5291}, {5134, 17734, 17735}, {5139, 8754, 2971}, {5305, 7745, 5007}, {5306, 18907, 5008}, {5309, 5475, 6}, {5309, 18424, 5475}, {5355, 7753, 6}, {5461, 6722, 14061}, {5471, 5472, 5477}, {5475, 18424, 381}, {5989, 7770, 5149}, {6033, 11632, 12188}, {6034, 11632, 5309}, {6034, 11646, 6}, {6108, 6109, 6055}, {6114, 6115, 114}, {6392, 32816, 7758}, {6392, 32980, 32816}, {6564, 6565, 3818}, {6722, 14061, 14971}, {7668, 34294, 1084}, {7735, 7737, 32}, {7739, 31415, 7736}, {7746, 7748, 3}, {7747, 7755, 32}, {7748, 13881, 7749}, {7749, 7756, 3}, {7750, 33229, 7842}, {7751, 7825, 315}, {7754, 7759, 7890}, {7754, 7773, 7759}, {7758, 32816, 7903}, {7760, 7785, 7838}, {7763, 32961, 7862}, {7770, 7834, 7889}, {7770, 7851, 7834}, {7771, 17004, 34506}, {7775, 7798, 7774}, {7780, 7842, 7750}, {7781, 7862, 7763}, {7783, 32967, 7769}, {7789, 8361, 7874}, {7791, 32832, 7815}, {7792, 8370, 7804}, {7793, 33019, 7802}, {7795, 14064, 7867}, {7797, 16044, 83}, {7797, 32528, 4027}, {7800, 32974, 7935}, {7803, 16924, 7808}, {7804, 7817, 7792}, {7805, 7843, 7762}, {7806, 11361, 3972}, {7807, 32819, 7816}, {7808, 7902, 7803}, {7815, 7872, 7791}, {7816, 7886, 7807}, {7818, 17131, 69}, {7833, 17004, 7771}, {7844, 11185, 7820}, {7844, 18546, 3734}, {7853, 9466, 141}, {7864, 16921, 7786}, {7867, 17130, 7795}, {7885, 17129, 7768}, {7901, 17128, 7832}, {7912, 20081, 7796}, {7933, 31276, 3096}, {8591, 22247, 2482}, {9167, 15300, 2482}, {9293, 10278, 13187}, {9607, 31406, 39}, {10723, 34473, 20}, {10896, 16502, 9665}, {11318, 34505, 7801}, {11485, 13102, 23013}, {11486, 13103, 23006}, {11606, 16044, 32528}, {11648, 18362, 2}, {13636, 13722, 1648}, {13653, 13773, 2}, {13831, 13832, 1992}, {13873, 13926, 6036}, {14045, 17129, 7885}, {14136, 20252, 10611}, {14137, 20253, 10612}, {14644, 22265, 11005}, {15359, 16278, 15357}, {16092, 34169, 17964}, {16592, 16613, 1015}, {16960, 25236, 16529}, {16961, 25235, 16530}, {16962, 25166, 6780}, {16963, 25156, 6779}, {17008, 33017, 14907}, {20337, 23947, 3912}, {21043, 23938, 21709}, {22510, 23004, 15}, {22511, 23005, 16}, {25468, 25683, 2}, {27133, 27189, 2}, {30465, 30468, 125}, {31709, 31710, 9880}, {31862, 31863, 6}, {32457, 33228, 7813}, {32828, 32974, 7800}, {33813, 34127, 140}


X(116) = MIDPOINT OF X(4) AND X(103)

Trilinears    bc[(b - c)2(b2 + bc + c2 - ab - ac)] : :
Barycentrics   (b - c)2(b2 + bc + c2 - ab - ac) : :

Let A'B'C' be the orthic triangle. Let La be the Soddy line of triangle AB'C', and define Lb and Lc cyclically. Let A″ = Lb∩Lc, and define B″ and C″ cyclically. The lines A'A″, B'B″, C'C″ concur in X(116), which is X(3)-of-A″B″C″. (Randy Hutson, July 31 2018)

X(116) lies on the nine-point circle and these lines: 2,101   4,103   5,118   10,120   115,1086   119,142   121,141   124,928

X(116) = midpoint of X(i) and X(j) for these (i,j): (4,103), (101,150)
X(116) = reflection of X(118) in X(5)
X(116) = isotomic conjugate of isogonal conjugate of X(20974)
X(116) = complement of X(101)
X(116) = complementary conjugate of X(514)
X(116) = X(4)-Ceva conjugate of X(514)
X(116) = polar conjugate of isogonal conjugate of X(22084)
X(116) = inverse-in-excircles-radical-circle of X(3034)
X(116) = inverse-in-orthoptic-circle-of-Steiner-inellipse of X(675)
X(116) = anticenter of cyclic quadrilateral ABCX(103)
X(116) = Λ(Gergonne line), wrt orthic triangle
X(116) = X(2)-Ceva conjugate of X(6586)
X(116) = X(101)-of-medial triangle.


X(117) = MIDPOINT OF X(4) AND X(109)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = g(b,c,a) + g(c,b,a), and
                        g(b,c,a) = b2c/[c(sec B - sec C) + a(sec B - sec A)]

Barycentrics  h(a,b,c) : h(b,c,a) : h(c,a,b) where h(a,b,c) = af(a,b,c)

X(117) lies on the nine-point circle
X(117) = X(102)-of-medial triangle.

X(117) lies on these lines: 2,102   4,109   5,124   10,123   11,65   118,928   136,407

X(117) = midpoint of X(i) and X(j) for these (i,j): (4,109), (102,151)
X(117) = reflection of X(124) in X(5)
X(117) = complement of X(102)
X(117) = complementary conjugate of X(515)
X(117) = X(4)-Ceva conjugate of X(515)
X(117) = inverse-in-polar-circle of X(32706)
X(117) = anticenter of cyclic quadrilateral ABCX(109)
X(117) = Λ(X(1), X(4)), wrt orthic triangle
X(117) = Spieker-radical-circle inverse of X(34456)


X(118) = MIDPOINT OF X(4) AND X(101)

Trilinears    g(b,c,a) + g(c,b,a) : : ,where g(b,c,a) = b3c/[(b - c) cot A + (b - a) cot C]

Let A'B'C' be the orthic triangle. Let La be the Gergonne line of triangle AB'C', and define Lb and Lc cyclically. Let A″ = Lb∩Lc, and define B″ and C″ cyclically. The lines A'A″, B'B″, C'C″ concur in X(118), which is X(3)-of-A″B″C″. (Randy Hutson, July 31 2018)

X(118) lies on the nine-point circle and these lines: 2,103   4,101   5,116   11,226   117,928   122,440   136,430   381,544   516,910

X(118) = midpoint of X(i) and X(j) for these (i,j): (4,101), (103,152)
X(118) = reflection of X(116) in X(5)
X(118) = complementary conjugate of X(516)
X(118) = X(4)-Ceva conjugate of X(516)
X(118) = complement of X(103)
X(118) = inverse-in-polar-circle of X(917)
X(118) = anticenter of cyclic quadrilateral ABCX(101)
X(118) = X(103)-of-medial triangle.
X(118) = Λ(X(1), X(7)), wrt orthic triangle
X(118) = Λ(X(4), X(9)), wrt orthic triangle


X(119) = FEUERBACH ANTIPODE

Trilinears    (csc A)(-1 + cos B + cos C)[sin 2B + sin 2C + 2(-1 + cos A)(sin B + sin C)] : :
Barycentrics    (-1 + cos B + cos C)[sin 2B + sin 2C + 2(-1 + cos A)(sin B + sin C)] : :

Let Na = X(5)-BCX(1), Nb = X(5)-of-CAX(1), Nc = X(5)-of-ABX(1). Then X(119) = X(2071)-of-NaNbNc. (Randy Hutson, January 29, 2018)

Let A'B'C' be the orthic triangle. Let La be the antiorthic axis of AB'C', and define Lb and Lc cyclically. Let A″ = Lb∩Lc, B″ = Lc∩La, C″ = La∩Lb. Then triangle A″B″C″ is inversely similar to ABC, with similicenter X(9). The lines A'A″, B'B″, C'C″ concur in X(119). Also, X(119) = X(3) of A″B″C″. (Randy Hutson, January 29, 2018)

X(119) lies on these lines: 1,5   2,104   3,123   4,100   10,124   116,142   125,442   135,431   136,429   214,515   381,528   517,908

X(119) = midpoint of X(i) and X(j) for these (i,j): (4,100), (104,153)
X(119) = reflection of X(i) in X(j) for these (i,j): (11,5), (3,3035)
X(119) = isogonal conjugate of X(15381)
X(119) = complement of X(104)
X(119) = complementary conjugate of X(517)
X(119) = X(4)-Ceva conjugate of X(517)
X(119) = nine-point-circle-antipode of X(11)
X(119) = X(104)-of-medial triangle
X(119) = X(2072)-of-Fuhrmann-triangle
X(119) = inverse-in-polar-circle of X(915)
X(119) = anticenter of cyclic quadrilateral ABCX(100)
X(119) = Λ(X(1), X(3)), wrt orthic triangle
X(119) = Λ(X(4), X(8)), wrt orthic triangle
X(119) = Spieker-radical-circle-inverse of X(34459)
X(119) = center of rectangular circumhyperbola passing through isogonal and isotomic conjugates of X(3657)


X(120) = X(105)-OF-MEDIAL-TRIANGLE

Trilinears    bc[2abc - (b + c)(a2 + (b - c)2)](b2 + c2 - ab -ac) : :
Barycentrics    [2abc - (b + c)(a2 + (b - c)2)](b2 + c2 - ab -ac) : :

X(120) lies on the nine-point circle and these lines: 2,11   10,116   12,85   115,442

X(120) = complementary conjugate of X(518)
X(120) = X(4)-Ceva conjugate of X(518)
X(120) = X(105)-of-medial triangle. X(120) = complement of X(105)
X(120) = perspector of circumconic centered at X(3290)
X(120) = center of circumconic that is locus of trilinear poles of lines passing through X(3290)
X(120) = X(2)-Ceva conjugate of X(3290)
X(120) = polar conjugate of isogonal conjugate of X(20728)
X(120) = inverse-in-excircles-radical-circle of X(3033)
X(120) = inverse-in-{circumcircle, nine-point circle}-inverter of X(100)
X(120) = X(1292)-of-Euler-triangle
X(120) = midpoint of X(4) and X(1292)
X(120) = Λ(X(1), X(6)), wrt orthic triangle
X(120) = orthopole of PU(44) (line X(3)X(667))
X(120) = crosssum of circumcircle intercepts of line PU(44) (line X(3)X(667))


X(121) = X(106)-OF-MEDIAL-TRIANGLE

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where
                        f(a,b,c) = bc(b + c - 2a)[b3 + c3 + a(b2 + c2) - 2bc(b + c)]

Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b) where g(a,b,c) = (b + c - 2a)[b3 + c3 + a(b2 + c2) - 2bc(b + c)]

X(121) lies on the nine-point circle
X(121) = X(106)-of-medial triangle.

X(121) lies on these lines: 2,106   10,11   116,141

X(121) = complementary conjugate of X(519)
X(121) = X(4)-Ceva conjugate of X(519)
X(121) = complement of X(106)
X(121) = polar conjugate of isogonal conjugate of X(22428)
X(121) = inverse-in-excircles-radical-circle of X(3032)
X(121) = Λ(X(1), X(2)), wrt orthic triangle
X(121) = isotomic conjugate of isogonal conjugate of X(23644)


X(122) = X(107)-OF-MEDIAL-TRIANGLE

Trilinears    (b2 - c2)2(cos A - cos B cos C) cot2A : :
Barycentrics    a(b2 - c2)2(cos A - cos B cos C) cot2A : :

Barycentrics    (b^2 - c^2)^2 (a^2 - b^2 - c^2)^2 (3 a^4 - 2 a^2 (b^2 + c^2) - (b^2 - c^2)^2) : :

X(122) lies on the nine-point circle, the cevian circle of X(20), and these lines: 2,107   3,113   5,133   118,440   125,684   138,233

X(122) = reflection of X(133) in X(5)
X(122) = isogonal conjugate of X(15384)
X(122) = complementary conjugate of X(520)
X(122) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,520), (253,525)
X(122) = crosssum of X(i) and X(j) for these (i,j): (64,1301), (112,154)
X(122) = crosspoint of X(253) and X(525)
X(122) = crossdifference of every pair of points on line X(112)X(1301)
X(122) = X(107)-of-medial triangle
X(122) = center of the rectangular hyperbola that passes through A, B, C, and X(20)
X(122) = X(1293)-of-orthic-triangle if ABC is acute
X(122) = complement of X(107)
X(122) = perspector of orthic triangle and tangential triangle of hyperbola {A,B,C,X(2),X(3)}}
X(122) = inverse-in-polar-circle of X(1301)
X(122) = inverse-in-orthoptic-circle-of-Steiner-inellipse of X(1297)
X(122) = inverse-in-Moses-radical-circle of X(33504)
X(122) = X(2)-Ceva conjugate of X(6587)
X(122) = barycentric product X(20)*X(15526) = X(20)*X(525)^2
X(122) = crosssum of circumcircle intercepts of line X(3)X(64)
X(122) = orthopole of line X(3)X(64)
X(122) = Kirikami-six-circles image of X(20)


X(123) =  X(108)-OF-MEDIAL-TRIANGLE

Trilinears    (csc A)(sec B - sec C)[(sec A)(sin2B - sin2C) + sin C tan C - sin B tan B] : :
Barycentrics    (sec B - sec C)[(sec A)(sin2B - sin2C) + sin C tan C - sin B tan B]
Barycentrics    (a - b - c)(b - c)^2(a^2 - b^2 - c^2)(a^4 - b^4 - c^4 + 2a^2bc - 2ab^2c - 2abc^2 + 2b^2c^2) : :

X(123) lies on the nine-point circle and hese lines: 2,108   3,119   10,117   113,960

X(123) = isogonal conjugate of X(15385)
X(123) = complement of X(108)
X(123) = complementary conjugate of X(521)
X(123) = X(2)-Ceva conjugate of X(6588)
X(123) = X(4)-Ceva conjugate of X(521)
X(123) = Spieker-radical-circle-inverse of X(34455)
X(123) = X(108)-of-medial triangle
X(123) = Stevanovic-circle-inverse of X(38972)
X(123) = crossdifference of every pair of points on line X(1415)X(2443)
X(123) = trilinear product X(i)*X(j) for these {i,j}: {3436, 7004}, {6332, 6588}


X(124) = X(109)-OF-MEDIAL-TRIANGLE

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where
                        f(a,b,c) = bc(b + c - a)(b - c)2[(b + c)(b2 + c2 - a2 - bc) + abc]

Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where
                        g(a,b,c) = (b + c - a)(b - c)2 [(b + c)(b2 + c2 - a2 - bc) + abc]

X(124) lies on the nine-point circle
X(124) = X(109)-of-medial triangle
X(124) = center of the rectangular hyperbola that passes through A, B, C, and X(58)

X(124) lies on these lines: 2,109   4,102   5,117   10,119   116,928

X(124) = midpoint of X(4) and X(102)
X(124) = reflection of X(117) in X(5)
X(124) = complementary conjugate of X(522)
X(124) = X(4)-Ceva conjugate of X(522)
X(124) = complement of X(109)
X(124) = crosssum of circumcircle intercepts of line X(3)X(10)
X(124) = orthopole of line X(3)X(10)
X(124) = focus of Mandart parabola
X(124) = anticenter of cyclic quadrilateral ABCX(102)
X(124) = inverse-in-orthoptic-circle-of-Steiner-inellipse of X(1311)
X(124) = Spieker-radical-circle-inverse of X(34458)
X(124) = X(2)-Ceva conjugate of X(6589)
X(124) = Kirikami-six-circles image of X(58)
X(124) = center of rectangular circumhyperbola that is isogonal conjugate of line X(3)X(10)


X(125) = CENTER OF JERABEK HYPERBOLA

Trilinears    cos A sin2(B - C) : cos B sin2(C - A) : cos C sin2(A - B)
Trilinears    (sec A)(c cos C - b cos B)2 : (sec B)(a cos A - c cos C)2 : (sec C)(b cos B - a cos A)2
Trilinears    bc(b2 + c2 - a2)(b2 - c2)2 : :
Barycentrics    (sin 2A)[sin(B - C)]2 : (sin 2B)[sin(C - A)]2 : (sin 2C)[sin(A - B)]2

Roland H. Eddy and R. Fritsch, "The conics of Ludwig Kiepert: a comprehensive lesson in the geometry of the triangle," Mathematics Magazine 67 (1994) 188-205.

X(125) is the pole of the Fermat axis with respect to the Dao-Moses-Telv circle. (Randy Hutson, December 14, 2014)

Let La, Lb, Lc be the lines through A, B, C, resp. parallel to the Euler line. Let Ma, Mb, Mc be the reflections of BC, CA, AB in La, Lb, Lc, resp. Let A' = Mb∩Mc, and define B', C' cyclically. Triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. Let A″B″C″ be the reflection of A'B'C' in the Euler line. The triangle A″B″C″ is homothetic to ABC, with center of homothety X(125); see Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, March 25, 2016)

Let A'B'C' be the orthic triangle. The Euler lines of AB'C', BC'A', CA'B' concur in X(125). (Randy Hutson, March 25, 2016)

X(125) lies on these curves: nine-point circle, Walsmith rectangular hyperbola, orthic inconic, symmedial circle, Johnson circumconic of the medial triangle, cevian circle of every point on the Jerabek hyperbola, and bicevian conic of X(2) and X(72). X(125) also lies on these lines: 2,98   3,131   4,74   5,113   6,67   51,132   68,1092   69,895   115,245   119,442   122,684   126,141   128,140   136,338   381,541   511,858

X(125) = midpoint of X(i) and X(j) for these (i,j): (3,265), (4,74), (6,67), (110,3448)
X(125) = reflection of X(i) in X(j) for these (i,j): (113,5), (185,974), (1495,468), (1511,140), (1539,546)
X(125) = isogonal conjugate of X(250)
X(125) = isotomic conjugate of X(18020)
X(125) = inverse-in-Brocard-circle of X(184)
X(125) = complement of X(110)
X(125) = complementary conjugate of X(523)
X(125) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,523), (66,512), (68,520), (69,525), (338,115)
X(125) = crosspoint of X(i) and X(j) for these (i,j): (4,523), (69,525), (338,339)
X(125) = crosssum of X(i) and X(j) for these (i,j): (3,110), (25,112), (162,270), (1113,1114)
X(125) = crossdifference of every pair of points on the line X(110)X(112)
X(125) = X(115)-Hirst inverse of X(868)
X(125) = X(2)-line conjugate of X(110)
X(125) = orthopole of the Euler line
X(125) = perspector of orthic triangle and Schroeter triangle
X(125) = X(110)-of-medial triangle
X(125) = X(100)-of-orthic triangle, if ABC is acute
X(125) = X(858)-of-1st-Brocard triangle
X(125) = anticenter of cyclic quadrilateral ABCX(74)
X(125) = Λ(X(230), X(231)), wrt orthic triangle
X(125) = anticomplement of X(5972)
X(125) = pole of Fermat axis wrt Dao-Moses-Telv circle
X(125) = orthic-isogonal conjugate of X(523)
X(125) = perspector of circumconic centered at X(647)
X(125) = center of circumconic that is locus of trilinear poles of lines passing through X(647)
X(125) = X(2)-Ceva conjugate of X(647)
X(125) = trilinear pole wrt orthic triangle of van Aubel line
X(125) = inverse-in-polar-circle of X(107)
X(125) = inverse-in-{circumcircle, nine-point circle}-inverter of X(98)
X(125) = inverse-in-orthosymmedial-circle of X(51)
X(125) = centroid of (degenerate) pedal triangle of X(74)
X(125) = X(i)-isoconjugate of X(j) for these {i,j}: {4,1101}, {92,249}
X(125) = inverse-in-Hutson-Parry-circle of X(868)
X(125) = {X(13636),X(13722)}-harmonic conjugate of X(868)
X(125) = crosssum of MacBeath circumconic intercepts of Brocard axis
X(125) = excentral-to-ABC functional image of X(100)
X(125) = 1st-Brocard-isotomic conjugate of X(35901)
X(125) = antipode of X(1495) in Walsmith rectangular hyperbola
X(125) = orthocenter of X(6)X(113)X(3569)
X(125) = orthocenter of X(110)X(125)X(3569)
X(125) = orthocenter of X(1495)X(3569)X(3580)
X(125) = polar conjugate of X(23582)
X(125) = Moses-radical-circle-inverse of X(38974)


X(126) = X(111)-OF-MEDIAL-TRIANGLE

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where
                        f(a,b,c) = bc(2a2 - b2 - c2)[b4 + c4 + a2(b2 + c2) - 4b2c2]

Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b) where g(a,b,c) = (2a2 - b2 - c2)[b4 + c4 + a2(b2 + c2) - 4b2c2]

X(126) lies on the nine-point circle
X(126) = X(111)-of-medial triangle.

X(126) lies on these lines: 2,99   125,141   625,858

X(126) = complement of X(111)
X(126) = complementary conjugate of X(524)
X(126) = X(4)-Ceva conjugate of X(524)
X(126) = perspector of circumconic centered at X(3291)
X(126) = center of circumconic that is locus of trilinear poles of lines passing through X(3291)
X(126) = X(2)-Ceva conjugate of X(3291)
X(126) = one of two intersections (X(3258) is the other) of the nine-point circle of ABC and the Parry circle of the X(2)-Brocard triangle
X(126) = inverse-in-polar-circle of X(2374)
X(126) = inverse-in-{circumcircle, nine-point circle}-inverter of X(99)
X(126) = Λ(X(2), X(6)), wrt orthic triangle


X(127) = X(112)-OF-MEDIAL-TRIANGLE

Trilinears    bc(sin 2B - sin 2C)[(b2 - c2)sin 2A - b2sin 2B + c2sin 2C] : :
Barycentrics   (sin 2B - sin 2C)[(b2 - c2)sin 2A - b2sin 2B + c2sin 2C] : :
Barycentrics   (b^2 - c^2)^2(b^2 + c^2 - a^2)(b^4 + c^4 - a^4) : :

Let A'B'C' be the orthic triangle. Let La be the van Aubel line of triangle AB'C', and define Lb and Lc cyclically. Let A″ = Lb∩Lc, and define B″ and C″ cyclically. The lines A'A″, B'B″, C'C″ concur in X(127), which is X(3)-of-A″B″C″. (Randy Hutson, July 31 2018)

X(127) lies on the nine-point circle, the cevian circle of X(22), and on these lines: 2,112   3,114   5,132   113,141   115,338   133,381   125,140

X(127) = reflection of X(132) in X(5)
X(127) = isogonal conjugate of X(15388)
X(127) = isotomic conjugate of isogonal conjugate of X(38356)
X(127) = complement of X(112)
X(127) = anticomplementary conjugate of X(525)
X(127) = X(4)-Ceva conjugate of X(525)
X(127) = X(1292)-of-orthic-triangle if ABC is acute
X(127) = perspector of circumconic centered at X(2485)
X(127) = center of circumconic that is locus of trilinear poles of lines passing through X(2485)
X(127) = X(2)-Ceva conjugate of X(2485)
X(127) = inverse-in-polar-circle of X(1289)
X(127) = inverse-in-orthoptic-circle-of-Steiner-inellipse of X(2373)
X(127) = X(112)-of-medial triangle
X(127) = center of the rectangular hyperbola that passes through A, B, C, and X(22)
X(127) = crosssum of circumcircle intercepts of line X(3)X(66)
X(127) = orthopole of line X(3)X(66)
X(127) = Kirikami-six-circles image of X(22)


X(128) = X(74)-OF-ORTHIC-TRIANGLE

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where
                        f(A,B,C) = (sec A)(cos 2B + cos 2C)(1 + 2 cos 2A)(cos 2A + 2 cos 2B cos 2C)

Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B) where g(A,B,C) = (sin A) f(A,B,C)

X(128) lies on the nine-point circle
X(128) = X(74)-of-orthic triangle.

X(128) lies on these lines: 5,137   52,134   53,139   115,233   125,140

X(128) = reflection of X(137) in X(5)
X(128) = isogonal conjugate of X(15401)
X(128) = complement of X(1141)
X(128) = anticomplement of X(34837)
X(128) = perspector of circumconic centered at X(231)
X(128) = center of circumconic that is locus of trilinear poles of lines passing through X(231)
X(128) = inverse-in-polar-circle of X(2383)
X(128) = X(2)-Ceva conjugate of X(231)
X(128) = orthojoin of X(231)
X(128) = excentral-to-ABC functional image of X(74)


X(129) = X(98)-OF-ORTHIC-TRIANGLE

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where
                        f(A,B,C) = (sec A)(sin 2A)(sin 2B + sin 2C) s(A,B,C) t(A,B,C),
                        s(A,B,C) = sin4(2B) + sin4(2C) - sin2(2A) sin2(2B) - sin2(2A) sin2(2C),
                        t(A,B,C) = sin4(2A) + sin2(2A) u(A,B,C) + v(A,B,C),
                        u(A,B,C) = sin 2B sin 2C - sin2(2B) - sin2(2C),
                        v(A,B,C) = (sin 2B sin 2C)(sin 2B - sin 2C)2

Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B) where g(A,B,C) = (sin A) f(A,B,C)

X(129) lies on the nine-point circle
X(129) = X(98)-of-orthic triangle.

X(129) lies on these lines: 5,130   51,137   52,139   115,389

X(129) = reflection of X(130) in X(5)
X(129) = complement of X(1298)
X(129) = anticomplement of X(34838)
X(129) = complementary conjugate of X(32428)
X(129) = Λ(X(5), X(53)), wrt orthic triangle
X(129) = excentral-to-ABC functional image of X(98)


X(130) = X(99)-OF-ORTHIC-TRIANGLE

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where
                        f(A,B,C) = (sin A)(sin 2B + sin 2C)[(sin 2B - sin 2C)2][sin2(2A) + sin 2B sin 2C]

Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(130 lies on the nine-point circle
X(130) = X(99)-of-orthic triangle
X(130) = center of the rectangular hyperbola that passes through A, B, C, and X(51)

X(130) lies on these lines: 5,129   51,138

X(130) = reflection of X(129) in X(5)
X(130) = complement of X(1303)
X(130) = anticomplement of X(34839)
X(130) = crosssum of circumcircle intercepts of line X(3)X(95)
X(130) = excentral-to-ABC functional image of X(99)
X(130) = trilinear pole wrt orthic triangle of line X(51)X(53)
X(130) = orthopole of line X(3)X(95)
X(130) = Kirikami-six-circles image of X(51)


X(131) = INTERSECTION OF LINES X(3)X(125) AND X(4)X(135)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where
                        f(A,B,C) = (sec A)[2T - S(sec 2B + sec 2C)](T - S sec 2A),
                        S = sin 2A + sin 2B + sin 2C, T = tan 2A + tan 2B + tan 2C

Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(131) lies on the nine-point circle
X(131) = X(102)-of-orthic triangle if ABC is acute.

X(131) lies on these lines: 3,125   4,135   5,136   115,216

X(131) = reflection of X(136) in X(5)
X(131) = complement of X(1300)
X(131) = anticomplement of X(34840)
X(131) = complementary conjugate of X(13754)
X(131) = inverse-in-polar-circle of X(1299)
X(131) = Λ(X(4), X(52)), wrt orthic triangle


X(132) = X(2)X(107)∩X(4)X(32)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where
                        f(A,B,C) = (sec A) u(A,B,C) v(A,B,C),
                        u(A,B,C) = [sin2(2A) + (sin 2B - sin 2C)2 + (sin 2A)(sin 2A - sin 2B - sin 2C)],
                        v(A,B,C) = [sin2(2B) + sin2(2C) - (sin 2A sin 2B) - (sin 2A sin 2C)]

Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(132) lies on the nine-point circle and these lines: 2,107   4,32   5,127   25,136   51,125   114,684   137,428   147,648

X(132) = midpoint of X(4) and X(112)
X(132) = reflection of X(127) in X(5)
X(132) = isogonal conjugate of X(15407)
X(132) = anticomplement of X(34841)
X(132) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,232), (4,1503)
X(132) = X(4)-line conjugate of X(248)
X(132) = crossdifference of every pair of points on line X(248)X(684)
X(132) = X(105)-of-orthic triangle if ABC is acute
X(132) = complement of X(1297)
X(132) = perspector of circumconic centered at X(232)
X(132) = center of rectangular hyperbola {A,B,C,X(4),X(112),PU(39)}} that is locus of trilinear poles of lines passing through X(232)
X(132) = center of rectangular hyperbola {X(4),X(112),X(371),X(372),X(378),X(1064)}
X(132) = inverse-in-polar-circle of X(98)
X(132) = circumcircle-inverse of X(34131)
X(132) = inverse-in-{circumcircle, nine-point circle}-inverter of X(107)
X(132) = anticenter of cyclic quadrilateral ABCX(112)
X(132) = Λ(X(4), X(6)), wrt orthic triangle
X(132) = crosssum of circumcircle intercepts of line PU(37) (line X(3)X(525))
X(132) = orthopole of PU(37)


X(133) = INTERSECTION OF LINES X(4)X(74) AND X(5)X(122)

Trilinears       (sec A)[(sin 2B - sin 2C)2 + (sin 2A)(sin 2B) + (sin 2A)(sin 2C) - 2(sin 2B)(sin 2C)](2 sin 2A - sin 2B - sin 2C) : :
Barycentrics  (tan A)[(sin 2B - sin 2C)2 + (sin 2A)(sin 2B) + (sin 2A)(sin 2C) - 2(sin 2B)(sin 2C)](2 sin 2A - sin 2B - sin 2C) : :

X(133) lies on the nine-point circle
X(133) = X(106)-of-orthic triangle is ABC is acute.

X(133) lies on these lines: 2,1294   4,74   5,122   53,115   127,381   136,235

X(133) = midpoint of X(4) and X(107)
X(133) = reflection of X(122) in X(5)
X(133) = isogonal conjugate of X(15404)
X(133) = complement of X(1294)
X(133) = anticomplement of X(34842)
X(133) = perspector of circumconic centered at X(1990)
X(133) = center of circumconic that is locus of trilinear poles of lines passing through X(1990)
X(133) = X(2)-Ceva conjugate of X(1990)
X(133) = trilinear pole wrt Euler triangle of van Aubel line
X(133) = inverse-in-polar-circle of X(74)
X(133) = anticenter of cyclic quadrilateral ABCX(107)
X(133) = Λ(X(4), X(51)), wrt orthic triangle


X(134) = X(107)-OF-ORTHIC-TRIANGLE

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where
                        f(A,B,C) = (sec A) u(A,B,C) [v(B,C,A) - v(C,B,A)],
                        u(A,B,C) = (sin 2A)[sin2(2B) - sin2(2C)][sin2(2B) + sin2(2C) - sin2(2A)]2,
                        v(B,C,A) = sin 2C [sin2(2A) - sin2(2B)][sin2(2A) + sin2(2B) - sin2(2C)]2

Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(134) lies on the nine-point circle
X(134) = X(107)-of-orthic triangle
X(134) = center of the rectangular hyperbola that passes through A, B, C, and X(52)

X(134) lies on this line: 52,128

X(134) = trilinear pole wrt orthic triangle of line X(52)X(53)
X(134) = crosssum of circumcircle intercepts of line X(3)X(96)
X(134) = orthopole of line X(3)X(96)
X(134) = Kirikami-six-circles-image of X(52)


X(135) = INTERSECTION OF LINE X(4)X(131) AND X(25)X(114)

Trilinears    (sec A)[(sin 2B)/(sec 2C - sec 2A) + (sin 2C)/(sec 2A - sec 2B)] : :
Barycentrics    (tan A)[(sin 2B)/(sec 2C - sec 2A) + (sin 2C)/(sec 2A - sec 2B)] : :

X(135) lies on the nine-point circle
X(135) = X(108)-of-orthic-triangle if ABC is acute
X(135) = center of the rectangular hyperbola that passes through A, B, C, and X(24)

X(135) lies on these lines: 4,131   25,114   52,113   119,431

X(135) = anticomplement of X(34843)
X(135) = inverse-in-polar-circle of X(925)
X(135) = crosssum of circumcircle intercepts of line X(3)X(68)
X(135) = orthopole of line X(3)X(68)
X(135) = Kirikami-six-circles-image of X(24)


X(136) = INTERSECTION OF LINE X(4)X(110) AND X(25)X(132)

Barycentrics    (b^2 - c^2)^2 (a^4 + b^4 + c^4 - 2 a^2 b^2 - 2 a^2 c^2)/(a^2 - b^2 - c^2) : :
Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where
                        f(A,B,C) = (sec A)[(sin 2B - sin 2C)2](sin 2B + sin 2C - sin 2A) u(A,B,C),
                        u(A,B,C) = [sin2(2B) + sin2(2C) - sin2(2A)]

Barycentrics  af(A,B,C) : bf(B,C,A) : cf(C,A,B)

X(136) lies on the nine-point circle
X(136) =X(109)-of-orthic triangle if ABC is acute
X(136) = center of the rectangular hyperbola that passes through A, B, C, and X(93)

X(136) lies on these lines: 2,925   4,110   5,131   25,132   68,254   114,427   117,407   118,430   119,429   125,338   127,868   133,235

X(136) = reflection of X(131) in X(5)
X(136) = complement of X(925)
X(136) = anticomplement of X(34844)
X(136) = crosssum of circumcircle intercepts of line X(3)X(49)
X(136) = complementary conjugate of X(924)
X(136) = X(254)-Ceva conjugate of X(523)
X(136) = perspector of circumconic centered at X(2501)
X(136) = center of circumconic that is locus of trilinear poles of lines passing through X(2501) (hyperbola {A,B,C,X(4),X(93)}})
X(136) = X(2)-Ceva conjugate of X(2501)
X(136) = inverse-in-polar-circle of X(110)
X(136) = inverse-in-orthoptic-circle-of-Steiner-inellipse of X(3563)
X(136) = orthopole of line X(3)X(49)
X(136) = Kirikami-six-circles image of X(93)
X(136) = Dou-circles-radical-circle-inverse of X(36472)
X(136) = Moses-radical-circle-inverse of X(38970)
X(136) = {X(39240),X(39241)}-harmonic conjugate of polar conjugate of X(18879)


X(137) = X(110)-OF-ORTHIC-TRIANGLE

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where
                        f(A,B,C) = (sec A)(sin 2B + sin 2C)[(sin 2B - sin 2C)2] u(A,B,C),
                        u(A,B,C) = [sin2(2A) - sin2(2B) - sin2(2C) - sin 2B sin 2C]

Barycentrics  af(A,B,C) : bf(B,C,A) : cf(C,A,B)

X(137) lies on the nine-point circle
X(137) = X(110)-of-orthic triangle
X(137) = center of the rectangular hyperbola that passes through A, B, C, X(5), and X(53)

X(137) lies on the cevian circle of X(5) and these lines: 5,128   51,129   53,138   113,546   132,428

X(137) = reflection of X(128) in X(5)
X(137) = complement of X(930)
X(137) = X(4)-Ceva conjugate of X(1510)
X(137) = crosssum of X(252) and X(930)
X(137) = trilinear pole wrt orthic triangle of line X(5)X(53)
X(137) = inverse-in-polar-circle of X(933)
X(137) = excentral-to-ABC functional image of X(110)
X(137) = crosssum of circumcircle intercepts of line X(3)X(54)
X(137) = orthopole of line X(3)X(54)
X(137) = Kirikami-six-circles image of X(5)


X(138) = X(111)-OF-ORTHIC-TRIANGLE

Trilinears       (v + w) sec A : (w + u) sec B : (u + v) sec C, where
                        u = u(A,B,C) = (sin 2A)/(2 sin22A - sin22B - sin22C), v = u(B,C,A), w = u(C,A,B)

Barycentrics  (v + w) tan A : (w + u) tan B : (u + v) tan C

X(138) lies on the nine-point circle
X(138) = X(111)-of-orthic triangle

X(138) lies on these lines: 51,130   53,137   122,233

X(138) = Λ(X(51), X(53)), wrt orthic triangle


X(139) = X(112)-OF-ORTHIC-TRIANGLE

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where
                        f(A,B,C) = (sec A)(sin 2B + sin 2C)[(sin 2B - sin 2C)2][sin2(2B) + sin2(2C) - sin2(2A)] u(A,B,C),
                        u(A,B,C) = (sin 2B)4 + (sin 2C)4 - (sin 2A)4 + (sin 2B sin 2C)[sin2(2B) + sin2(2C) - sin2(2A)]

X(139) lies on the nine-point circle and these lines: 52,129   53,128

X(139) = inverse-in-polar-circle of X(32692)
X(139) = X(112)-of-orthic-triangle

leftri

Centers X(140)-X(170)

rightri

Centers in this section, by index:

113- 127, 140- 143: centers of the medial triangle
128- 139: centers of the orthic triangle
144- 153: centers of the anticomplementary triangle
154- 157, 159- 163: centers of the tangential triangle
164- 170: centers of the excentral triangle


X(140) = MIDPOINT OF X(3) AND X(5)

Trilinears    2 cos A + cos(B - C) : 2 cos B + cos(C - A) : 2 cos C + cos(A - B)
Trilinears    cos A + 2 sin B sin C : cos B + 2 sin C sin A : cos C + 2 sin A sin B
Trilinears    3 cos A + 2 cos B cos C : 3 cos B + 2 cos C cos A : 3 cos C + 2 cos A cos B
Trilinears    2 sec A + 3 sec B sec C : 2 sec B + 3 sec C sec A : 2 sec C + 3 sec A sec B
Trilinears    bc[b cos(C - A) + c cos(B - A)] : :
Barycentrics   b cos(C - A) + c cos(B - A) : :
Barycentrics   3 - cot B cot C : :
Barycentrics   2a^4 - 3a^2(b^2 + c^2) + (b^2 - c^2)^2 : :
Barycentrics    3 S^2 - SB SC : :
X(140) = 3*X(2) + X(3) = 3*X(2) - X(5) = 3*X(3) + X(4) = X(4) - 3*X(5)

As a point on the Euler line, X(140) has Shinagawa coefficients (3, -1).

Let A' be the midpoint between A and X(3), and define B' and C' cyclically; the triangle A'B'C' is homothetic to the medial triangle, and the center of homothety is X(140). Let A'' be the centroid of the triangle BCX(3), and define B'' and C'' cyclically; then A''B''C'' is homothetic to ABC, and the center of homothety is X(140). Also, X(140) is the center of the conic consisting of the centers of all the conics which pass through A, B, C, and X(3). (Randy Hutson, 9/23/2011) This conic is also the locus of crosssums of the intersections of the circumcircle and lines through X(4). Furthmore, this conic is the bicevian conic of X(2) and X(3). (Randy Hutson, 9/14/2016)

Let Oa be the circle centered at A and passing through the A-vertex of the Euler triangle; define Ob and Oc cyclically. The radical center of Oa, Ob, Oc is X(140). (Randy Hutson, September 14, 2016)

Let Oa be the circle centered at A with radius 1/2*sqrt(b^2 + c^2), and define Ob and Oc cyclically. The radical center of Oa, Ob, Oc is X(140). (Randy Hutson, September 14, 2016)

Let P be a point on the circumcircle. The bicevian conic of X(2) and P is a rectangular hyperbola, H. Let X be the center of H. As P varies, X traces a circle centered at X(140). (Randy Hutson, November 2, 2017)

X(140) is the centroid of the six circumcenters in the construction of the van Lamoen circle. (Randy Hutson, October 15, 2018)

X(140) lies on these lines: {1,5432}, {2,3}, {6,5418}, {8,1483}, {9,5843}, {10,214}, {11,35}, {12,36}, {13,5237}, {14,5238}, {15,18}, {16,17}, {32,3815}, {39,230}, {40,3624}, {46,11375}, {49,5012}, {51,10263}, {52,3917}, {53,10979}, {54,252}, {55,496}, {56,495}, {57,6147}, {61,395}, {62,396}, {69,1353}, {72,10202}, {76,6390}, {79,5131}, {83,2080}, {95,340}, {98,7832}, {100,1484}, {104,5260}, {110,10264}, {113,10990}, {114,6292}, {115,10992}, {119,5251}, {125,128}, {141,182}, {142,5762}, {143,511}, {156,9306}, {165,8227}, {183,3933}, {184,13336}, {185,5876}, {187,1506}, {195,323}, {216,1990}, {231,570}, {233,6748}, {236,8129}, {262,7846}, {298,628}, {299,627}, {302,633}, {303,634}, {325,1078}, {343,569}, {355,1698}, {371,615}, {372,590}, {385,13571}, {389,1154}, {392,11729}, {394,12161}, {484,5443}, {485,1152}, {486,1151}, {497,10386}, {515,3634}, {516,9955}, {517,1125}, {518,13373}, {523,1116}, {524,575}, {539,11264}, {542,6698}, {551,10222}, {567,3580}, {568,7998}, {572,1213}, {574,3054}, {576,597}, {577,6749}, {578,13567}, {601,748}, {602,750}, {618,630}, {619,629}, {620,2782}, {623,13350}, {624,13349}, {625,7830}, {626,13335}, {671,10185}, {758,5885}, {908,3916}, {912,5044}, {930,1263}, {936,5791}, {942,3911}, {944,5790}, {946,3579}, {956,5552}, {958,3820}, {970,6703}, {971,6666}, {993,1329}, {999,3085}, {1001,10200}, {1007,3785}, {1040,8144}, {1056,5265}, {1058,5281}, {1071,12691}, {1141,11016}, {1145,4861}, {1155,12047}, {1173,12834}, {1210,12433}, {1319,10039}, {1351,3618}, {1352,3763}, {1376,10267}, {1387,3057}, {1388,12647}, {1478,5204}, {1479,5217}, {1482,3616}, {1493,13366}, {1503,5092}, {1587,6398}, {1588,6221}, {1621,11849}, {1697,11373}, {1737,2646}, {1768,3652}, {1834,4256}, {1837,3612}, {1853,9833}, {2077,5259}, {2095,5761}, {2548,3053}, {2777,5893}, {2794,6721}, {2800,13145}, {2808,6710}, {2818,6711}, {2831,11259}, {2883,3357}, {2888,7666}, {2896,7925}, {2979,3567}, {3019,13329}, {3068,3312}, {3069,3311}, {3070,6396}, {3071,6200}, {3086,3295}, {3095,7786}, {3096,7940}, {3098,5480}, {3167,11411}, {3216,5396}, {3303,10072}, {3304,10056}, {3316,3590}, {3317,3591}, {3336,3649}, {3337,5557}, {3419,4855}, {3487,5435}, {3488,5704}, {3532,4846}, {3581,5888}, {3582,3746}, {3583,7173}, {3584,5298}, {3585,3614}, {3600,8164}, {3601,5722}, {3617,7967}, {3619,6776}, {3620,11898}, {3622,10247}, {3626,13607}, {3630,7916}, {3631,5965}, {3653,3679}, {3654,7982}, {3655,5881}, {3656,7991}, {3678,12005}, {3740,12675}, {3767,5013}, {3793,7762}, {3813,8715}, {3814,5267}, {3816,5248}, {3822,5841}, {3824,12436}, {3825,5840}, {3826,6796}, {3841,5842}, {3898,10284}, {3925,10902}, {3927,5744}, {4045,7886}, {4255,5292}, {4292,5122}, {4293,9654}, {4294,9669}, {4297,10175}, {4299,10895}, {4302,10896}, {4309,11238}, {4317,11237}, {4413,11499}, {4423,10310}, {5007,9300}, {5010,6284}, {5023,7737}, {5024,5286}, {5045,13405}, {5080,5303}, {5086,10609}, {5097,6329}, {5119,11376}, {5126,10106}, {5157,13562}, {5171,7808}, {5188,7889}, {5206,5475}, {5306,7772}, {5309,9607}, {5318,10646}, {5321,10645}, {5339,11480}, {5340,11481}, {5398,5718}, {5403,8160}, {5404,8161}, {5414,9661}, {5437,5709}, {5438,5705}, {5440,6734}, {5446,5943}, {5449,12038}, {5486,8548}, {5489,5664}, {5534,8580}, {5550,5603}, {5562,5650}, {5569,7775}, {5587,7987}, {5590,5874}, {5591,5875}, {5609,5642}, {5640,11465}, {5646,5654}, {5651,10539}, {5656,13093}, {5658,12684}, {5663,5907}, {5687,10527}, {5694,5884}, {5720,8726}, {5731,5818}, {5743,13323}, {5777,11227}, {5878,10606}, {5883,11281}, {5889,7999}, {5890,11444}, {5894,11204}, {5944,12134}, {6000,6696}, {6033,7944}, {6055,9167}, {6130,8552}, {6153,13433}, {6194,7875}, {6199,7582}, {6247,6759}, {6248,7820}, {6287,9751}, {6321,7847}, {6368,10213}, {6395,7581}, {6409,6561}, {6410,6560}, {6417,7586}, {6418,7585}, {6425,9680}, {6449,6459}, {6450,6460}, {6455,9541}, {6502,9646}, {6592,8902}, {6669,6673}, {6670,6674}, {6688,10110}, {6722,7861}, {6723,11801}, {6746,12363}, {7028,8130}, {7280,7354}, {7308,7330}, {7603,7747}, {7610,12040}, {7616,8859}, {7622,7781}, {7697,7835}, {7709,7891}, {7735,9605}, {7743,10624}, {7750,7752}, {7751,13468}, {7756,8589}, {7758,8667}, {7761,7862}, {7778,7800}, {7801,11168}, {7806,12251}, {7810,7821}, {7811,7814}, {7831,7899}, {7834,9737}, {7844,9734}, {7854,7888}, {7863,9466}, {7868,9744}, {7881,9755}, {7904,7912}, {7914,9996}, {7956,8167}, {8071,10320}, {8125,8128}, {8126,8127}, {8141,10319}, {8148,10595}, {8151,10190}, {8666,12607}, {8721,9756}, {8722,10358}, {9301,10357}, {9655,10590}, {9668,10591}, {9704,11003}, {9707,11457}, {9729,9820}, {9781,11451}, {9826,13416}, {10006,11247}, {10163,12506}, {10171,12512}, {10189,10280}, {10198,11249}, {10225,11813}, {10277,13582}, {10517,11916}, {10518,11917}, {10541,11179}, {10572,12019}, {10574,11459}, {10584,11928}, {10585,11929}, {10586,12000}, {10587,12001}, {10628,11561}, {10915,11260}, {10950,11545}, {11015,12690}, {11017,13474}, {11176,11615}, {11219,12738}, {11246,11544}, {11255,11511}, {11265,11513}, {11266,11514}, {11267,11515}, {11268,11516}, {11426,11433}, {11427,11432}, {11430,12241}, {11438,12233}, {11485,11489}, {11486,11488}, {11703,11792}, {11808,13365}, {12162,13491}, {12325,13432}, {12358,13148}, {13142,13352}, {13470,13565}

X(140) is the {X(2),X(3)}-harmonic conjugate of X(5). For a list of other harmonic conjugates of X(140), click Tables at the top of this page.

X(140) = midpoint of X(i) and X(j) for these (i,j): (3,5), (141,182), (389,1216), (2883, 3357)
X(140) = reflection of X(i) in X(j) for these (i,j): (546,5), (547,2), (548,3)
X(140) = isogonal conjugate of X(1173)
X(140) = complement of X(5)
X(140) = anticomplement of X(3628)
X(140) = complementary conjugate of X(1209)
X(140) = polar-circle-inverse of X(37943)
X(140) = X(2)-Ceva conjugate of X(233)
X(140) = crosspoint of X(i) and X(j) for these (i,j): (2,95), (17,18)
X(140) = crosssum of X(i) and X(j) for these (i,j): (6,51), (61,62)
X(140) = crosspoint of the two Napoleon points, X(17) and X(18)
X(140) = inverse-in-orthocentroidal-circle of X(1656)
X(140) = X(5)-of-medial triangle
X(140) = centroid of the quadrangle ABCX(3)
X(140) = perspector of circumconic centered at X(233)
X(140) = center of circumconic that is locus of trilinear poles of lines passing through X(233)
X(140) = intersection of tangents to Evans conic at X(3070) and X(3071)
X(140) = centroid of X(2)X(3)X(115)X(2482)
X(140) = pole of Brocard axis wrt conic {X(5),X(13),X(14),X(15),X(16)}}
X(140) = X(3) of polar triangle of complement of polar circle
X(140) = inverse-in-complement-of-polar-circle of X(2072)
X(140) = inverse-in-{circumcircle, nine-point circle}-inverter of X(5189)
X(140) = center of inverse-in-{circumcircle, nine-point circle}-inverter of anticomplementary circle
X(140) = centroid of the six circumcenters in the construction of the van Lamoen circle.
X(140) = centroid of ABCX(3)
X(140) = Kosnita(X(3),X(2)) point
X(140) = center of circle that is locus of crosssums of antipodes on the 2nd Lemoine circle
X(140) = {X(2),X(5)}-harmonic conjugate of X(3628)
X(140) = {X(3),X(4)}-harmonic conjugate of X(550)
X(140) = {X(4),X(5)}-harmonic conjugate of X(3850)
X(140) = homothetic center of X(4)-altimedial and X(140)-anti-altimedial triangles
X(140) = X(3579)-of-orthic-triangle if ABC is acute
X(140) = pole of van Aubel line wrt conic {X(2),X(15),X(16),X(17),X(18)}}
X(140) = homothetic center of McCay and Moses-Steiner osculatory triangles


X(141) = COMPLEMENT OF SYMMEDIAN POINT

Trilinears    bc(b2 + c2) : :
Trilinears    csc2A sin(A + ω) : :
Barycentrics    b2 + c2 : :
Barycentrics    cot A + cot ω : :
Barycentrics    SA + SW : :
X(141) = 3*X(2) + X(69) = 3*X(2) - X(6)

Let P be a point on the circumcircle, and let L be the line tangent to the circumcircle at P. Let P' be the trilinear pole of L, and let P" be the isotomic conjugate of P'. As P traces the circumcircle, P" traces an ellipse inscribed in ABC with center at X(141). (Randy Hutson, December 26, 2015)

Let A'B'C' be the 2nd Brocard triangle. Let A″ be the cevapoint of B' and C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(141). (Randy Hutson, December 26, 2015)

For an artistic design generated by X(141), see X(244).

X(141) lies on the bicevian conic of X(2) and X(110) and on these lines: 2,6   3,66   5,211   10,142   37,742   39,732   45,344   53,264   67,110   75,334   76,698   95,287   99,755   113,127   116,121   125,126   140,182   239,319   241,307   308,670   311,338   317,458   320,894   384,1031   441,577   498,611   499,613   523,882   542,549   575,629   997,1060

X(141) is the {X(2),X(69)}-harmonic conjugate of X(6). For a list of other harmonic conjugates of X(141), click Tables at the top of this page.

X(141) = midpoint of X(i) and X(j) for these (i,j): (1,3416), (6,69), (8,3242), (66,159), (67,110), (69,3313), (1843,3313) (2930, 3448)
X(141) = reflection of X(i) in X(j) for these (i,j): (182,140), (597,2), (1353,575), (1386,1125)
X(141) = isogonal conjugate of X(251)
X(141) = isotomic conjugate of X(83)
X(141) = inverse-in-nine-point-circle of X(625)
X(141) = complement of X(6)
X(141) = complementary conjugate of X(2)
X(141) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,39), (67,524), (110,525)
X(141) = X(39)-cross conjugate of X(427)
X(141) = crosspoint of X(2) and X(76)
X(141) = crosssum of X(6) and X(32)
X(141) = X(39)-Hirst inverse of X(732)
X(141) = X(645)-beth conjugate of X(141)
X(141) = X(6)-of-medial triangle
X(141) = anticomplement of X(3589)
X(141) = centroid of ABCX(69)
X(141) = Kosnita(X(69),X(2)) point
X(141) = perspector of circumconic centered at X(39)
X(141) = center of circumconic that is locus of trilinear poles of lines passing through X(39)
X(141) = bicentric sum of PU(11)
X(141) = midpoint of PU(11)
X(141) = polar conjugate of X(32085)
X(141) = X(6)-of-X(2)-Brocard-triangle
X(141) = X(115)-of-1st-Brocard-triangle
X(141) = crosspoint of X(2) and X(2896) wrt excentral triangle
X(141) = crosspoint of X(2) and X(2896) wrt anticomplementary triangle
X(141) = crosspoint of X(6) and X(2916) wrt excentral triangle
X(141) = crosspoint of X(6) and X(2916) wrt tangential triangle
X(141) = {X(2),X(6)}-harmonic conjugate of X(3589)
X(141) = {X(395),X(396)}-harmonic conjugate of X(5306)
X(141) = trilinear cube root of X(14125)
X(141) = perspector of 2nd Brocard triangle and cross-triangle of ABC and 2nd Brocard triangle
X(141) = intersection, other than X(3), of the orthosymmedial circles of the 1st and 2nd Ehrmann inscribed triangles


X(142) = COMPLEMENT OF X(9)

Trilinears    b + c - [(b - c)2]/a : c + a - [(c - a)2]/b : a + b - [(a - b)2]/c
Barycentrics    ab + ac - (b - c)2 : bc + ba - (c - a)2 : ca + cb - (a - b)2
X(142) = 3*X(2) + X(7) = 3*X(2) - X(9)

Let A' be the midpoint between A and X(7), and define B' and C' cyclically; the triangle A'B'C' is homothetic to the medial triangle, and the center of homothety is X(142). (Randy Hutson, 9/23/2011)

Let A' be the intersection of these three lines:

(1) through midpoint of CA perpendicular to BX(1)
(2) through midpoint of AB perpendicular to CX(1)
(3) through midpoint of AX(1) perpendicular to BC.

Define B' and C' cyclically. Then X(142) = X(6)-of-A'B'C'. The triangle A'B'C' is the complement of the excentral triangle, and also the extraversion triangle of X(10). (Randy Hutson, September 14, 2016)

X(142) lies on these lines: 1,277   2,7   3,516   5,971   10,141   37,1086   86,284   116,119   214,528   269,948   354,3059   377,950   474,954

X(142) is the {X(2),X(7)}-harmonic conjugate of X(9). For a list of other harmonic conjugates, click Tables at the top of this page.

X(142) = midpoint of X(i) and X(j) for these (i,j): (7,9), (8,3243), (100,3254)
X(142) = reflection of X(1001) in X(1125)
X(142) = isogonal conjugate of X(1174)
X(142) = isotomic conjugate of X(32008)
X(142) = complement of X(9)
X(142) = X(100)-Ceva conjugate of X(514)
X(142) = crosspoint of X(2) and X(85)
X(142) = crosssum of X(6) and X(41)
X(142) = X(190)-beth conjugate of X(142)
X(142) = X(9)-of- medial triangle
X(142) = polar conjugate of isogonal conjugate of X(22053)
X(142) = centroid of the set {X(1), X(4), X(7), X(40)}
X(142) = perspector of circumconic centered at X(1212)
X(142) = X(2)-Ceva conjugate of X(1212)
X(142) = centroid of ABCX(7)
X(142) = center of circumconic that is locus of trilinear poles of lines passing through X(1212)
X(142) = X(9969)-of-excentral-triangle


X(143) = NINE-POINT CENTER OF ORTHIC TRIANGLE

Trilinears    (sec A)[cos(2C - 2A) + cos(2A - 2B)] : :
Trilinears    (1 - 2 cos 2A)cos(B - C)]: :
Trilinears    sec A cos(3A) cos(B - C) : :
Barycentrics    (tan A)[cos(2C - 2A) + cos(2A - 2B)] : :
Barycentrics    a^2 (a^2 b^2 + a^2 c^2 + 2 b^2 c^2 - b^4 - c^4) (a^4 + b^4 + c^4 - 2 a^2 b^2 - 2 a^2 c^2 - b^2 c^2) : :
X(143) = X[5] - 3 X[51] = 3 X[51] + X[52] = X[4] + 3 X[568] = X[3] + 3 X[3060] = 3 X[2979] - 7 X[3526] = X[3] - 5 X[3567] = 3 X[3060] + 5 X[3567] = 5 X[632] - 3 X[3917] = 3 X[140] - 2 X[5447] = X[5447] - 3 X[5462] = 3 X[5] - X[5562] = 9 X[51] - X[5562] = 3 X[52] + X[5562] = 5 X[1656] - 9 X[5640] = 3 X[381] - X[5876] = 3 X[381] + X[5889] = X[382] + 3 X[5890] = 5 X[5562] - 9 X[5891] = 5 X[5] - 3 X[5891] = 5 X[51] - X[5891] = 5 X[52] + 3 X[5891] = 2 X[3530] - 3 X[5892] = X[1216] - 3 X[5943] = 2 X[3628] - 3 X[5943] = X[3] - 3 X[5946] = 5 X[3567] - 3 X[5946] = 3 X[568] - X[6102] = 3 X[3830] + X[6241] = 3 X[2] + X[6243] = 3 X[5093] + X[6403] = X[195] + 3 X[7730] = 11 X[5070] - 7 X[7999]

X(143) is the third of three Spanish Points developed by Antreas P. Hatzipolakis and Javier Garcia Capitan in 2009; see X(3567).

Let A'B'C' be the cevian triangle of X(5). Let A″, B″, C″ be the inverse-in-circumcircle of A', B', C'. The lines AA″, BB″, CC″ concur in X(143). Also, X(143) = intersection of the tangent to hyperbola {A,B,C,X(4),X(15)}} at X(61) and the tangent to the hyperbola {A,B,C,X(4),X(16)}} at X(62). (Randy Hutson, July 23, 2015)

X(143) is the QA-P13 center (Nine-point Center of the QA-Diagonal Triangle) of quadrangle ABCX(4) (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/41-qa-p13.html). Also, X(143) is the QA-P22 center (Midpoint QA-P1 and QA-P20) of quadrangle ABCX(4) (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/50-qa-p22.html)

X(143) lies on the curves K054, K416, K464, Q106, and these lines: {2,6101}, {3,1173}, {4,94}, {5,51}, {6,26}, {23,1199}, {25,156}, {30,389}, {49,1493}, {54,2070}, {61,2912}, {62,2913}, {110,195}, {140,511}, {182,7525}, {185,3627}, {324,565}, {381,5876}, {382,5890}, {567,7488}, {569,7502}, {575,7555}, {576,1147}, {578,1658}, {632,3917}, {970,7508}, {1181,7530}, {1216,3628}, {1351,6642}, {1353,1843}, {1656,5640}, {1993,7506}, {2392,5885}, {2937,5012}, {2979,3526}, {3517,5093}, {3530,5892}, {3580,5576}, {3830,6241}, {3850,5907}, {3853,6000}, {5070,7999}, {5609,7545}, {6515,7528}, {7517,7592}

X(143) = midpoint of X(i) and X(j) for these {i,j}: {4, 6102}, {5, 52}, {185, 3627}, {389, 5446}, {1353, 1843}, {1493, 6152}, {3060, 5946}, {5876, 5889}, {6101, 6243}
X(143) = reflection of X(i) in X(j) for these {i,j}: {140, 5462}, {1216, 3628}, {5907,3850}
X(143) = isogonal conjugate of X(252)
X(143) = anticomplement of X(32142)
X(143) = X(137)-cross conjugate of X(1510)
X(143) = X(5)-of-orthic triangle
X(143) = X(249)-Ceva conjugate of X(1625)
X(143) = X(137)-cross conjugate of X(1510)
X(143) = excentral-to-ABC functional image of X(5)
X(143) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,6243,6101), (3,3567,5946), (4,568,6102), (49,1994,1493), (51,52,5), (54,2070,5944), (381,5889,5876), (1112,6746,4), (1216,5943,3628), (1994,3518,49), (3060,3567,3)
X(143) = X(i)-isoconjugate of X(j) for these {i,j}: {1,252}, {54,2962}, {93,2169}, {930,2616}, {2167,2963}, {2190,3519}


X(144) = ANTICOMPLEMENT OF X(7)

Trilinears    (csc A)(tan B/2 + tan C/2 - tan A/2) : :
Barycentrics    tan B/2 + tan C/2 - tan A/2 : :
Barycentrics    1/(a - b - c) + 1/(a - b + c) + 1/(a + b - c) : :
Barycentrics    3a^2 - 2a(b + c) - (b - c)^2 : :
Barycentrics    3*a^2 - 2*a*b - b^2 - 2*a*c + 2*b*c - c^2 : :
X(144) = 3 X[2] - 4 X[9],9 X[2] - 8 X[142],5 X[2] - 4 X[6173],15 X[2] - 16 X[6666],9 X[2] - 10 X[18230],21 X[2] - 20 X[20195],2 X[3] - 3 X[21168],3 X[4] - 2 X[31671],3 X[7] - 4 X[142],X[7] - 3 X[6172],5 X[7] - 6 X[6173],5 X[7] - 8 X[6666],3 X[7] - 5 X[18230],7 X[7] - 10 X[20195],3 X[9] - 2 X[142],2 X[9] - 3 X[6172],5 X[9] - 3 X[6173],5 X[9] - 4 X[6666],6 X[9] - 5 X[18230],4 X[9] - X[20059],7 X[9] - 5 X[20195],2 X[75] - 3 X[27484],4 X[142] - 9 X[6172],10 X[142] - 9 X[6173],5 X[142] - 6 X[6666],4 X[142] - 5 X[18230],8 X[142] - 3 X[20059],14 X[142] - 15 X[20195],X[145] - 4 X[5698],3 X[210] - 2 X[15587],3 X[210] - X[31391]

X(144) lies on the Feuerbach circumhyperbola of the anticomplementary triangle, the Mandart hyperbola, the cubics K200, K202, K308, K710, K1044, K1084, and on these lines:: {2, 7}, {3, 5843}, {4, 2894}, {6, 3672}, {8, 516}, {10, 4312}, {12, 18231}, {20, 72}, {21, 954}, {37, 3945}, {40, 5815}, {42, 4335}, {44, 4000}, {45, 4648}, {65, 27288}, {69, 190}, {71, 27544}, {75, 391}, {77, 2324}, {78, 3522}, {85, 25001}, {86, 4747}, {92, 6994}, {100, 480}, {145, 192}, {149, 1156}, {153, 1145}, {165, 21060}, {175, 30557}, {176, 30556}, {191, 3085}, {194, 20036}, {200, 2951}, {210, 3474}, {213, 4352}, {218, 17691}, {219, 347}, {220, 279}, {238, 4310}, {239, 4452}, {241, 26669}, {269, 25930}, {281, 7282}, {306, 25734}, {320, 344}, {321, 14552}, {345, 33066}, {348, 10004}, {376, 3940}, {405, 11036}, {443, 15650}, {452, 3868}, {524, 17262}, {528, 12531}, {536, 5839}, {545, 4361}, {573, 21362}, {597, 17323}, {599, 17340}, {631, 31657}, {658, 17113}, {666, 30228}, {673, 4373}, {758, 18412}, {912, 6987}, {920, 15518}, {938, 10398}, {942, 5129}, {950, 20008}, {956, 6912}, {960, 3600}, {962, 6766}, {966, 4363}, {984, 4307}, {1001, 2975}, {1086, 16885}, {1119, 26003}, {1212, 24554}, {1213, 4470}, {1260, 7411}, {1266, 4402}, {1278, 20248}, {1418, 25067}, {1419, 3160}, {1434, 24557}, {1441, 4047}, {1633, 12329}, {1654, 2475}, {1721, 28043}, {1742, 2340}, {1743, 3663}, {1757, 24248}, {1761, 14543}, {1766, 7291}, {1788, 8165}, {1901, 31043}, {1959, 4704}, {1992, 4360}, {1999, 10889}, {2095, 6939}, {2245, 27039}, {2267, 18162}, {2287, 5781}, {2293, 24708}, {2321, 32099}, {2325, 17296}, {2345, 4643}, {2478, 5729}, {2664, 25570}, {2801, 6224}, {2895, 2897}, {3008, 3973}, {3059, 3681}, {3086, 6763}, {3091, 5805}, {3161, 3912}, {3174, 3935}, {3241, 30331}, {3243, 3623}, {3247, 4667}, {3339, 18250}, {3419, 3543}, {3434, 16112}, {3475, 3683}, {3476, 31165}, {3487, 17558}, {3523, 3916}, {3589, 17255}, {3616, 5542}, {3618, 4389}, {3619, 17273}, {3620, 4741}, {3621, 5853}, {3629, 17318}, {3630, 17309}, {3631, 17269}, {3640, 30334}, {3641, 30333}, {3648, 5696}, {3650, 5687}, {3664, 3731}, {3671, 5234}, {3686, 4659}, {3687, 10443}, {3715, 11246}, {3739, 7222}, {3758, 17258}, {3812, 28646}, {3826, 11681}, {3832, 5735}, {3839, 18482}, {3870, 4326}, {3873, 5572}, {3876, 6904}, {3946, 16670}, {3949, 24683}, {3952, 4019}, {3958, 8680}, {4001, 34255}, {4021, 16667}, {4073, 4712}, {4098, 29602}, {4110, 25278}, {4182, 20534}, {4190, 5784}, {4192, 22149}, {4292, 5785}, {4293, 5692}, {4294, 5904}, {4301, 24644}, {4308, 15829}, {4313, 11523}, {4321, 19861}, {4329, 5227}, {4343, 17018}, {4344, 7174}, {4353, 16469}, {4359, 20921}, {4370, 17267}, {4371, 4686}, {4384, 31995}, {4422, 7232}, {4430, 7671}, {4462, 6008}, {4468, 6006}, {4473, 16593}, {4512, 10578}, {4552, 20082}, {4640, 5281}, {4645, 27549}, {4652, 15717}, {4661, 7674}, {4670, 28640}, {4675, 16814}, {4678, 24393}, {4683, 33163}, {4699, 24633}, {4715, 4851}, {4748, 17303}, {4758, 28626}, {4788, 20016}, {4795, 28639}, {4847, 9812}, {4855, 21734}, {4859, 4887}, {4882, 5493}, {4888, 29571}, {4896, 25072}, {4902, 31183}, {4912, 17348}, {4915, 28228}, {5044, 17580}, {5175, 17578}, {5176, 17488}, {5177, 11662}, {5231, 9779}, {5274, 24477}, {5290, 18249}, {5302, 28629}, {5440, 10304}, {5587, 5775}, {5703, 31424}, {5708, 17559}, {5709, 5811}, {5714, 5791}, {5738, 31049}, {5739, 32933}, {5758, 7330}, {5766, 7675}, {5809, 12649}, {5832, 6871}, {5833, 9612}, {5857, 20060}, {5880, 15481}, {6007, 20012}, {6067, 11680}, {6144, 17388}, {6147, 16845}, {6244, 10307}, {6350, 32849}, {6360, 20211}, {6542, 20080}, {6600, 7676}, {6604, 32024}, {6605, 10509}, {6762, 9785}, {6764, 10624}, {6885, 31835}, {6908, 26921}, {6926, 24467}, {6995, 7717}, {7155, 17794}, {7172, 32937}, {7175, 9310}, {7201, 17451}, {7227, 17251}, {7238, 17265}, {7262, 33144}, {7277, 16777}, {7321, 17335}, {7613, 32857}, {7670, 16019}, {7672, 13601}, {7677, 24558}, {8055, 30567}, {8163, 12513}, {9533, 31627}, {9780, 30424}, {9797, 12575}, {10005, 32850}, {10177, 11025}, {10392, 24391}, {10442, 11679}, {10446, 21061}, {10580, 30330}, {10590, 17057}, {11008, 17377}, {11037, 31435}, {11160, 17373}, {12125, 12632}, {12514, 15298}, {12560, 19860}, {12630, 20014}, {12670, 14872}, {14555, 32939}, {14986, 15299}, {15254, 30340}, {15374, 28071}, {15492, 17278}, {15680, 20013}, {15913, 31527}, {16020, 24231}, {16284, 21872}, {16435, 23089}, {16552, 17753}, {16566, 28795}, {16669, 17301}, {16675, 17392}, {16713, 17139}, {17002, 26245}, {17007, 26032}, {17045, 24441}, {17051, 26105}, {17116, 17331}, {17118, 17330}, {17120, 17247}, {17132, 17151}, {17137, 27523}, {17147, 20043}, {17170, 17742}, {17183, 18206}, {17234, 31333}, {17243, 28333}, {17253, 17369}, {17261, 17316}, {17264, 17361}, {17272, 17355}, {17279, 17345}, {17281, 17344}, {17285, 21356}, {17288, 17339}, {17289, 17329}, {17298, 25101}, {17300, 29621}, {17327, 26039}, {17343, 21286}, {17375, 29583}, {17496, 20296}, {17582, 24470}, {17615, 17668}, {17776, 32859}, {18161, 21801}, {18600, 27644}, {18661, 24435}, {19742, 19789}, {19993, 20068}, {19998, 22312}, {20009, 20077}, {20018, 25264}, {20020, 20064}, {20101, 31087}, {20905, 30854}, {20930, 28974}, {20992, 21320}, {21039, 24341}, {21075, 27525}, {21084, 28124}, {21219, 30662}, {22003, 24048}, {24597, 33151}, {25000, 27541}, {26034, 32938}, {26768, 27136}, {26871, 32863}, {27268, 27475}, {27543, 28420}, {28628, 28645}, {30225, 32028}, {32007, 32100}, {33099, 33137}

X(144) = reflection of X(i) in X(j) for these {i,j}: {2, 6172}, {4, 5779}, {7, 9}, {8, 5223}, {20, 5759}, {100, 6068}, {145, 390}, {149, 1156}, {390, 5698}, {962, 11372}, {2550, 5220}, {3868, 5728}, {4312, 10}, {4430, 7671}, {4440, 673}, {4452, 5838}, {5880, 15481}, {5905, 8545}, {8581, 960}, {9965, 12848}, {17314, 17262}, {20014, 12630}, {20059, 7}, {20533, 190}, {25722, 3059}, {30628, 14100}, {31391, 15587}
X(144) = isogonal conjugate of X(11051)
X(144) = isotomic conjugate of X(10405)
X(144) = complement of X(20059)
X(144) = anticomplement of X(7)
X(144) = anticomplementary conjugate of X(3434)
X(144) = anticomplementary-isogonal conjugate of X(3434)
X(144) = polar conjugate of the isogonal of X(22117)
X(144) = perspector of anticomplementary triangle and its intouch triangle (inner Conway triangle)
X(144) = perspector of anticomplementary triangle and the extouch triangle of ABC
X(144) = perspector of extouch triangle and inner Conway triangle (Gemini triangle 30)
X(144) = X(7)-of-anticomplementary triangle
X(144) = polar conjugate of isogonal conjugate of X(22117)
X(144) = X(i)-beth conjugate of X(j) for these (i,j): (190,144), (645,346)
X(144) = Conway-triangle-to-inner-Conway-triangle similarity image of X(7)
X(144) = X(i)-Ceva conjugate of X(j) for these (i,j): {8, 2}, {516, 20533}, {3729, 192}, {4416, 1654}, {4480, 17487}, {5223, 27484}, {31627, 3160}
X(144) = X(i)-cross conjugate of X(j) for these (i,j): {165, 3160}, {3160, 2}, {21060, 16284}, {21872, 165}
X(144) = X(i)-isoconjugate of X(j) for these (i,j): {1, 11051}, {6, 3062}, {31, 10405}, {56, 19605}
X(144) = cevapoint of X(i) and X(j) for these (i,j): {7, 15913}, {9, 2951}, {57, 7955}, {220, 6244}, {3207, 22117}, {21060, 21872}
X(144) = crosspoint of X(i) and X(j) for these (i,j): {190, 1275}, {16284, 31627}
X(144) = crosssum of X(649) and X(14936)
X(144) = crossdifference of every pair of points on line {663, 20980}
X(144) = barycentric product X(i)*X(j) for these {i,j}: {1, 16284}, {8, 3160}, {9, 31627}, {75, 165}, {76, 3207}, {86, 21060}, {190, 7658}, {264, 22117}, {274, 21872}, {312, 1419}, {341, 17106}, {346, 9533}, {1275, 13609}
X(144) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3062}, {2, 10405}, {6, 11051}, {9, 19605}, {165, 1}, {1419, 57}, {3160, 7}, {3207, 6}, {7658, 514}, {9533, 279}, {13609, 1146}, {15856, 30330}, {16284, 75}, {17106, 269}, {21060, 10}, {21872, 37}, {22117, 3}, {23058, 24856}, {31627, 85}
X(144) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 3434}, {2, 21285}, {6, 7}, {8, 6327}, {9, 69}, {21, 17135}, {25, 12649}, {29, 20242}, {31, 145}, {32, 3210}, {33, 4}, {37, 2893}, {41, 2}, {42, 2475}, {43, 20350}, {48, 347}, {55, 8}, {57, 6604}, {58, 3873}, {60, 17140}, {71, 2897}, {75, 21280}, {78, 1370}, {81, 20244}, {100, 21302}, {101, 693}, {109, 3900}, {163, 4467}, {192, 20559}, {198, 5932}, {200, 3436}, {210, 1330}, {212, 20}, {213, 17778}, {219, 4329}, {220, 329}, {228, 3152}, {251, 20247}, {281, 21270}, {282, 21279}, {283, 20243}, {284, 75}, {294, 20347}, {312, 315}, {314, 17138}, {318, 11442}, {333, 17137}, {346, 21286}, {522, 21293}, {604, 4452}, {607, 5905}, {643, 512}, {644, 20295}, {645, 17217}, {646, 21304}, {650, 150}, {662, 4374}, {663, 149}, {692, 522}, {765, 3888}, {904, 29840}, {911, 9436}, {923, 4442}, {983, 25304}, {1110, 100}, {1126, 20292}, {1172, 17220}, {1174, 85}, {1212, 2890}, {1252, 21272}, {1253, 144}, {1320, 21282}, {1333, 3875}, {1334, 2895}, {1395, 11851}, {1397, 17480}, {1412, 17158}, {1415, 4025}, {1812, 18659}, {1857, 5906}, {1973, 30699}, {2053, 10453}, {2149, 664}, {2150, 4360}, {2175, 192}, {2176, 20537}, {2185, 17143}, {2188, 280}, {2191, 6601}, {2192, 962}, {2193, 17134}, {2194, 1}, {2195, 518}, {2200, 18667}, {2204, 3187}, {2212, 193}, {2251, 30577}, {2258, 388}, {2259, 1441}, {2287, 20245}, {2289, 6527}, {2299, 3868}, {2311, 30941}, {2316, 320}, {2319, 21281}, {2320, 21283}, {2321, 21287}, {2328, 3869}, {2329, 30660}, {2332, 92}, {2338, 4872}, {2340, 20344}, {2341, 17139}, {2342, 517}, {2344, 4441}, {2360, 20221}, {2361, 6224}, {3063, 4440}, {3207, 31527}, {3596, 21275}, {3683, 2891}, {3684, 20345}, {3685, 20554}, {3688, 21289}, {3689, 21290}, {3693, 20552}, {3699, 21301}, {3700, 21294}, {3709, 21221}, {3711, 21291}, {3939, 513}, {4041, 3448}, {4166, 20346}, {4182, 20555}, {4548, 21215}, {4612, 17159}, {4636, 17166}, {4845, 5057}, {4876, 20553}, {5546, 7192}, {5547, 17491}, {5548, 21297}, {6065, 3952}, {6602, 30695}, {7037, 9799}, {7054, 21273}, {7069, 2888}, {7070, 6225}, {7071, 5942}, {7072, 11415}, {7074, 6223}, {7075, 32548}, {7077, 4645}, {7084, 17784}, {7110, 21276}, {7115, 4566}, {7118, 9965}, {7156, 14361}, {7367, 189}, {8611, 13219}, {8750, 521}, {8851, 20352}, {9439, 497}, {9447, 194}, {9448, 17486}, {9456, 1266}, {10482, 3681}, {11051, 32003}, {13455, 637}, {14547, 2894}, {14827, 3177}, {14942, 20556}, {15374, 9801}, {18265, 17759}, {18889, 527}, {20967, 5484}, {21059, 7674}, {23990, 4552}, {24019, 23683}, {28615, 3879}, {32635, 20290}, {32652, 8058}, {32665, 4453}, {32674, 17896}, {32677, 22464}, {32739, 17496}, {33299, 1369}
X(144) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 9965, 21454}, {2, 20059, 7}, {2, 20078, 9965}, {2, 20214, 5905}, {6, 3672, 17014}, {6, 4419, 3672}, {6, 17334, 4419}, {7, 9, 2}, {7, 6172, 9}, {7, 18230, 142}, {7, 29007, 8232}, {8, 3729, 4461}, {8, 4488, 3729}, {8, 30625, 30695}, {8, 30695, 10405}, {9, 142, 18230}, {9, 6173, 6666}, {20, 72, 20007}, {37, 3945, 29624}, {37, 4644, 3945}, {44, 17276, 4000}, {45, 17365, 4648}, {57, 18228, 2}, {63, 329, 2}, {63, 908, 5744}, {63, 17781, 329}, {69, 190, 346}, {69, 346, 29616}, {142, 18230, 2}, {190, 17347, 69}, {192, 193, 145}, {192, 20072, 193}, {193, 20073, 192}, {193, 20111, 20110}, {210, 31391, 15587}, {226, 3929, 5273}, {226, 5273, 2}, {307, 27382, 2}, {320, 344, 4869}, {320, 17336, 344}, {329, 5744, 908}, {329, 20348, 20245}, {391, 4454, 75}, {480, 11495, 100}, {672, 1423, 27624}, {672, 30946, 2}, {894, 17257, 2}, {894, 17333, 17257}, {908, 5744, 2}, {984, 24695, 4307}, {1001, 11038, 3622}, {1743, 3663, 5222}, {2287, 8822, 14953}, {2345, 4643, 5232}, {2550, 5220, 5686}, {2550, 5686, 3617}, {3161, 21296, 3912}, {3177, 3869, 20535}, {3177, 20111, 145}, {3218, 31018, 2}, {3219, 5905, 2}, {3243, 8236, 3623}, {3305, 9776, 2}, {3452, 3928, 5435}, {3452, 5435, 2}, {3487, 31445, 17558}, {3662, 26685, 2}, {3664, 3731, 5308}, {3681, 25722, 3059}, {3686, 4659, 32087}, {3715, 11246, 26040}, {3729, 4416, 8}, {3729, 4480, 4488}, {3758, 17258, 17321}, {3869, 30616, 20111}, {3911, 5328, 2}, {3911, 31142, 5328}, {3912, 25728, 3161}, {3973, 4862, 3008}, {4000, 17276, 4346}, {4357, 5749, 2}, {4363, 17332, 966}, {4416, 4480, 3729}, {4416, 4488, 4461}, {4640, 25568, 5281}, {4643, 17351, 2345}, {4652, 27383, 15717}, {4661, 20075, 20015}, {4704, 20090, 29585}, {4741, 17280, 3620}, {5226, 5745, 2}, {5296, 10436, 2}, {5698, 10394, 6872}, {5745, 28609, 5226}, {5805, 5817, 3091}, {6646, 17350, 2}, {12246, 31793, 20}, {12526, 12527, 8}, {16552, 17753, 27304}, {17120, 17247, 26626}, {17183, 18206, 26818}, {17261, 17364, 17316}, {17272, 17355, 29611}, {17273, 17354, 3619}, {17288, 17339, 29579}, {17298, 25101, 29627}, {18228, 28610, 57}, {20072, 20073, 145}, {21151, 31658, 3523}, {24477, 24703, 5274}, {24909, 25679, 2}, {24952, 25461, 2}, {26059, 26125, 2}, {26065, 27184, 2}, {27058, 27170, 2}, {27282, 27334, 2}, {27509, 28739, 2}, {29621, 32093, 17300}, {31547, 31548, 8}


X(145) = ANTICOMPLEMENT OF NAGEL POINT

Trilinears    bc(3a - b - c) : :
Trilinears    -1 + csc A/2 sin B/2 sin C/2 : :
Trilinears    2(r/R) - sin B sin C : :
Barycentrics   3a - b - c : :
X(145) = 4 X[1] - 3 X[2], 3 X[1] - 2 X[10], 7 X[1] - 6 X[551], 5 X[1] - 4 X[1125], 7 X[1] - 5 X[1698], 2 X[1] - 3 X[3241], 6 X[1] - 5 X[3616], 8 X[1] - 5 X[3617], 4 X[1] - X[3621], 8 X[1] - 7 X[3622], 4 X[1] - 5 X[3623], 9 X[1] - 7 X[3624], 5 X[1] - 2 X[3625], 7 X[1] - 4 X[3626], 3 X[1] - X[3632], 11 X[1] - 8 X[3634], 3 X[1] - 4 X[3635], 9 X[1] - 8 X[3636], 5 X[1] - 3 X[3679], 17 X[1] - 12 X[3828], 9 X[1] - 5 X[4668], 11 X[1] - 6 X[4669], 7 X[1] - 3 X[4677], 12 X[1] - 7 X[4678], 13 X[1] - 8 X[4691], 9 X[1] - 4 X[4701], 19 X[1] - 12 X[4745], 15 X[1] - 8 X[4746], 11 X[1] - 5 X[4816], 14 X[1] - 11 X[5550], 10 X[1] - 7 X[9780], 17 X[1] - 14 X[15808], 13 X[1] - 10 X[19862], 23 X[1] - 17 X[19872], 13 X[1] - 9 X[19875], 29 X[1] - 21 X[19876], 18 X[1] - 13 X[19877], 21 X[1] - 16 X[19878], 23 X[1] - 18 X[19883], 4 X[1] + X[20014], 4 X[1] + 3 X[20049], 2 X[1] + X[20050], 12 X[1] - 5 X[20052], 6 X[1] - X[20053], 8 X[1] - X[20054], 6 X[1] - 7 X[20057], 53 X[1] - 38 X[22266], 11 X[1] - 9 X[25055], 8 X[1] - 3 X[31145], 27 X[1] - 20 X[31253], 17 X[1] - 13 X[34595], 13 X[1] - 6 X[34641], and many others

Let A' be the reflection of the midpoint of segment BC in X(1), and define B' and C' cyclically. The lines AA', BB', CC' concur in X(145). Let A'' be the reflection of the A in X(1), and define B'' and C'' cyclically. Let A'''B'''C''' be the intouch triangle. The lines A''A''', B''B''', C''C''' concur in X(145). (Randy Hutson, 9/23/2011)

Let OA be the circle tangent to side BC at its midpoint and to the circumcircle on the same side of BC as A. Define OB, OC cyclically. Then X(145) is the trilinear pole of the line of the exsimilicenters (the Monge line) of OA, OB, OC. See the reference at X(1001).

Let Ha be the hyperbola passing through A, with foci at B and C. This is called the A-Soddy hyperbola in (Paul Yiu, Introduction to the Geometry of the Triangle, 2002; 12.4 The Soddy hyperbolas, p. 143). Let La be the polar of X(2) with respect to Ha. Define Lb and Lc cyclically. The lines La, Lb, Lc concur in X(145). (Randy Hutson, September 5, 2015)

Let A'B'C' and A″B″C″ be the intouch and extouch triangles. X(145) is the radical center of the circumcircles of AA'A″, BB'B″, CC'C″. (Randy Hutson, September 14, 2016)

X(145) lies on on the Jerabek circumhyperbola of the intouch triangle, the Feuerbach circumhyperbola of the anticomplementary triangle, the cubics K201, K295, K360, K365, K372, K571, K830, K1011, K1078, K1083, K1086, and on these lines: {1, 2}, {3, 1483}, {4, 149}, {5, 10247}, {6, 346}, {7, 1266}, {9, 4029}, {11, 5154}, {12, 3813}, {20, 517}, {21, 956}, {22, 8192}, {23, 9798}, {30, 8148}, {31, 37588}, {35, 8666}, {36, 8715}, {37, 391}, {38, 37598}, {40, 3218}, {44, 49714}, {45, 4969}, {46, 23958}, {55, 2975}, {56, 100}, {57, 2136}, {58, 17539}, {63, 1697}, {65, 3189}, {69, 3242}, {72, 452}, {75, 3945}, {80, 1392}, {81, 1043}, {85, 17158}, {86, 17589}, {92, 7518}, {99, 28531}, {101, 9083}, {104, 11248}, {125, 27715}, {140, 37624}, {141, 26104}, {142, 4402}, {144, 192}, {146, 7978}, {147, 7970}, {148, 7983}, {150, 10695}, {151, 10696}, {152, 10697}, {165, 21734}, {174, 7057}, {186, 47476}, {188, 1488}, {190, 1992}, {194, 7976}, {213, 27523}, {218, 644}, {226, 4323}, {238, 27549}, {244, 24440}, {263, 27809}, {278, 5174}, {279, 664}, {280, 1433}, {291, 38247}, {304, 31130}, {312, 4696}, {318, 34231}, {319, 5232}, {320, 4346}, {321, 4673}, {322, 17863}, {329, 950}, {330, 1002}, {333, 17588}, {335, 27480}, {341, 4358}, {344, 1279}, {345, 3744}, {347, 2897}, {350, 20943}, {354, 3893}, {355, 3091}, {376, 12702}, {377, 1056}, {381, 37705}, {382, 28224}, {385, 50776}, {388, 2099}, {392, 3876}, {402, 16211}, {404, 999}, {405, 6767}, {411, 22770}, {443, 15934}, {474, 7373}, {485, 35810}, {486, 35811}, {487, 7980}, {488, 7981}, {491, 26515}, {492, 26514}, {495, 2476}, {496, 4193}, {497, 2098}, {511, 50419}, {514, 38371}, {515, 962}, {516, 5059}, {522, 14812}, {523, 36171}, {524, 4419}, {527, 30332}, {528, 4440}, {529, 6284}, {535, 34719}, {536, 4454}, {537, 24695}, {542, 50883}, {543, 50888}, {594, 16884}, {595, 1331}, {597, 17269}, {599, 17395}, {616, 7975}, {617, 7974}, {621, 51689}, {622, 51691}, {627, 22912}, {628, 22867}, {631, 5690}, {651, 34040}, {659, 25574}, {668, 18135}, {672, 3208}, {704, 40907}, {718, 40906}, {726, 4788}, {730, 20081}, {740, 1278}, {752, 49708}, {754, 34685}, {758, 3648}, {760, 20065}, {856, 38284}, {858, 47536}, {860, 38295}, {872, 27291}, {894, 3886}, {900, 24097}, {901, 44873}, {902, 35284}, {908, 4345}, {912, 23340}, {940, 3996}, {942, 3889}, {946, 3832}, {957, 39698}, {958, 1621}, {959, 39694}, {960, 3681}, {961, 15375}, {964, 19717}, {966, 16777}, {982, 4642}, {984, 4704}, {986, 4392}, {993, 3746}, {996, 1126}, {1001, 5260}, {1006, 16202}, {1010, 4720}, {1018, 4253}, {1046, 4427}, {1058, 2478}, {1066, 26050}, {1086, 50120}, {1100, 2345}, {1104, 17776}, {1108, 27396}, {1145, 6921}, {1146, 27541}, {1150, 16347}, {1155, 34711}, {1158, 12703}, {1159, 11112}, {1191, 1265}, {1220, 2334}, {1270, 5604}, {1271, 5605}, {1284, 12642}, {1319, 1788}, {1329, 36972}, {1334, 21384}, {1376, 3304}, {1385, 3523}, {1386, 17280}, {1387, 6931}, {1388, 7288}, {1389, 6839}, {1398, 35974}, {1400, 3169}, {1420, 3158}, {1434, 7268}, {1449, 2321}, {1453, 3710}, {1468, 5255}, {1469, 25304}, {1470, 38901}, {1475, 3501}, {1478, 11009}, {1479, 5080}, {1484, 6971}, {1500, 16975}, {1587, 35641}, {1588, 35642}, {1616, 4383}, {1617, 37301}, {1656, 10283}, {1657, 28212}, {1699, 16189}, {1706, 3306}, {1724, 40091}, {1738, 23675}, {1743, 3161}, {1757, 49696}, {1829, 6995}, {1834, 3936}, {1837, 5048}, {1854, 37781}, {1870, 4200}, {1909, 4441}, {1938, 25271}, {1997, 6556}, {2049, 19740}, {2089, 12643}, {2093, 4311}, {2094, 34701}, {2102, 14807}, {2103, 14808}, {2176, 3780}, {2177, 32919}, {2238, 16969}, {2241, 5291}, {2256, 2287}, {2257, 3692}, {2275, 17756}, {2276, 17448}, {2280, 2329}, {2292, 7226}, {2320, 5559}, {2325, 16670}, {2403, 3667}, {2550, 11038}, {2551, 5289}, {2646, 3769}, {2771, 17652}, {2784, 5992}, {2802, 3874}, {2809, 17489}, {2810, 42448}, {2886, 15888}, {2888, 7979}, {2895, 26117}, {2896, 7977}, {2899, 26791}, {2979, 50586}, {3036, 31272}, {3052, 4884}, {3060, 16980}, {3061, 3930}, {3068, 44635}, {3069, 44636}, {3090, 5790}, {3160, 9436}, {3194, 14954}, {3212, 3598}, {3219, 5250}, {3246, 49702}, {3247, 3686}, {3263, 18156}, {3305, 51779}, {3307, 24648}, {3308, 24649}, {3315, 17054}, {3332, 29016}, {3333, 27003}, {3336, 5541}, {3339, 4315}, {3416, 3620}, {3419, 3487}, {3448, 7984}, {3452, 37723}, {3475, 4863}, {3485, 5086}, {3524, 34718}, {3525, 38028}, {3526, 38112}, {3529, 28174}, {3543, 11278}, {3545, 18357}, {3560, 12000}, {3576, 11362}, {3578, 49728}, {3579, 3655}, {3586, 51423}, {3589, 17309}, {3601, 5744}, {3614, 3829}, {3618, 17233}, {3619, 17295}, {3629, 17262}, {3630, 17255}, {3631, 17323}, {3640, 30333}, {3641, 30334}, {3644, 28582}, {3650, 15678}, {3653, 15721}, {3654, 13624}, {3656, 3839}, {3662, 3755}, {3663, 21296}, {3664, 17151}, {3666, 37655}, {3671, 50725}, {3673, 30806}, {3678, 3898}, {3684, 9310}, {3685, 3751}, {3689, 20323}, {3693, 26690}, {3695, 17526}, {3696, 4699}, {3698, 3742}, {3699, 6552}, {3701, 4737}, {3702, 4385}, {3704, 33089}, {3707, 16676}, {3714, 46897}, {3716, 23057}, {3717, 7290}, {3723, 17275}, {3726, 3959}, {3727, 49509}, {3730, 45751}, {3736, 17178}, {3739, 4371}, {3750, 10448}, {3753, 5045}, {3754, 3892}, {3756, 31227}, {3758, 49484}, {3790, 16475}, {3791, 17715}, {3812, 17609}, {3814, 37720}, {3816, 21031}, {3817, 37714}, {3823, 17241}, {3834, 4000}, {3836, 49677}, {3851, 38138}, {3854, 9779}, {3855, 38034}, {3878, 4127}, {3881, 5902}, {3883, 7174}, {3884, 3988}, {3887, 21222}, {3894, 4084}, {3897, 24929}, {3899, 4067}, {3900, 17496}, {3901, 4302}, {3902, 4968}, {3905, 21285}, {3907, 4804}, {3915, 5247}, {3927, 11111}, {3940, 5084}, {3946, 17296}, {3952, 17460}, {3974, 32942}, {3976, 24443}, {3977, 35261}, {3983, 4711}, {3984, 5795}, {3987, 4694}, {3991, 25082}, {3993, 49448}, {3994, 25253}, {4007, 5750}, {4021, 17272}, {4025, 45290}, {4026, 17238}, {4030, 17599}, {4033, 41316}, {4034, 5257}, {4050, 17474}, {4051, 17451}, {4065, 17461}, {4083, 31291}, {4085, 33087}, {4090, 4903}, {4101, 26580}, {4115, 24049}, {4160, 31290}, {4162, 4462}, {4194, 6198}, {4195, 37685}, {4197, 31419}, {4201, 32863}, {4202, 48847}, {4205, 27081}, {4225, 23853}, {4231, 11401}, {4232, 11363}, {4240, 11910}, {4248, 20818}, {4295, 10052}, {4297, 7990}, {4298, 5586}, {4301, 5691}, {4305, 5119}, {4310, 4645}, {4314, 9819}, {4318, 8271}, {4339, 32929}, {4349, 17116}, {4352, 17137}, {4353, 17288}, {4356, 17247}, {4357, 32099}, {4361, 4648}, {4363, 4747}, {4364, 28337}, {4366, 16920}, {4388, 32928}, {4390, 41239}, {4395, 17313}, {4399, 15668}, {4405, 49738}, {4421, 5204}, {4423, 45085}, {4429, 4966}, {4431, 7229}, {4433, 21010}, {4439, 16477}, {4445, 17045}, {4449, 21302}, {4470, 4665}, {4474, 48172}, {4478, 17327}, {4487, 4849}, {4504, 4729}, {4514, 5016}, {4527, 4649}, {4535, 49473}, {4547, 10176}, {4552, 12848}, {4555, 34342}, {4643, 4725}, {4646, 4850}, {4647, 41819}, {4657, 17372}, {4658, 26860}, {4659, 4667}, {4660, 49464}, {4662, 10179}, {4663, 4676}, {4664, 49515}, {4670, 28329}, {4675, 50125}, {4686, 7222}, {4690, 4748}, {4693, 32935}, {4695, 9335}, {4716, 50288}, {4723, 46937}, {4727, 16666}, {4732, 40328}, {4740, 49483}, {4741, 11160}, {4754, 50257}, {4764, 49525}, {4771, 21921}, {4772, 24325}, {4774, 47834}, {4780, 24231}, {4806, 4879}, {4814, 48242}, {4821, 49474}, {4831, 10385}, {4862, 33800}, {4865, 4892}, {4873, 4982}, {4883, 19804}, {4887, 50108}, {4894, 36974}, {4900, 5558}, {4901, 17353}, {4902, 36606}, {4922, 50343}, {4923, 4967}, {4930, 51409}, {4943, 30719}, {4973, 37572}, {4981, 30711}, {4986, 33942}, {4996, 10087}, {5011, 17736}, {5014, 7270}, {5015, 30991}, {5047, 9708}, {5049, 5439}, {5051, 31037}, {5054, 50823}, {5056, 5818}, {5057, 12701}, {5067, 38042}, {5068, 5587}, {5081, 7952}, {5083, 39776}, {5090, 7378}, {5091, 16380}, {5100, 5300}, {5189, 47537}, {5208, 41723}, {5217, 5303}, {5218, 34471}, {5220, 47357}, {5221, 39777}, {5223, 17261}, {5226, 9578}, {5229, 13463}, {5245, 30414}, {5246, 30415}, {5248, 5258}, {5259, 17544}, {5263, 17379}, {5266, 33168}, {5273, 5837}, {5284, 8162}, {5295, 31025}, {5372, 19278}, {5376, 11607}, {5434, 34502}, {5440, 24928}, {5442, 21842}, {5542, 48627}, {5543, 25719}, {5563, 25440}, {5564, 17394}, {5597, 5602}, {5598, 5601}, {5640, 23841}, {5695, 28503}, {5711, 14996}, {5719, 6856}, {5720, 5804}, {5722, 6919}, {5728, 25243}, {5737, 19333}, {5748, 9581}, {5761, 6844}, {5768, 37531}, {5770, 33596}, {5772, 17368}, {5774, 19270}, {5802, 22021}, {5815, 31018}, {5828, 37704}, {5847, 6646}, {5856, 12730}, {5880, 30340}, {5883, 50190}, {5920, 12533}, {5921, 39898}, {5933, 39773}, {6018, 33551}, {6144, 17334}, {6168, 34497}, {6172, 50110}, {6175, 16137}, {6193, 9933}, {6194, 22713}, {6225, 7973}, {6264, 9803}, {6265, 6979}, {6327, 49454}, {6366, 47695}, {6376, 25278}, {6462, 8210}, {6463, 8211}, {6548, 44314}, {6554, 6603}, {6600, 7677}, {6601, 17097}, {6630, 35313}, {6636, 8193}, {6645, 16919}, {6653, 33823}, {6701, 47033}, {6702, 32558}, {6763, 37563}, {6796, 32905}, {6826, 10597}, {6827, 10806}, {6830, 10943}, {6838, 21740}, {6840, 12116}, {6842, 32213}, {6847, 37533}, {6848, 37700}, {6850, 10805}, {6867, 14497}, {6875, 37621}, {6882, 32214}, {6893, 10596}, {6903, 35457}, {6905, 10680}, {6906, 10679}, {6909, 10306}, {6911, 12001}, {6924, 12331}, {6933, 8164}, {6938, 14988}, {6940, 16203}, {6941, 10942}, {6942, 22765}, {6948, 24475}, {6950, 11849}, {6953, 45770}, {6958, 19914}, {6959, 19907}, {6960, 10786}, {6972, 10785}, {6981, 11729}, {6987, 14054}, {7046, 11109}, {7052, 36930}, {7184, 25573}, {7408, 49542}, {7426, 47493}, {7486, 9956}, {7487, 41722}, {7492, 37546}, {7500, 44662}, {7504, 31479}, {7520, 37547}, {7580, 8158}, {7585, 7969}, {7586, 7968}, {7672, 7674}, {7681, 37725}, {7760, 30225}, {7787, 10800}, {7925, 50772}, {7951, 24387}, {7987, 43174}, {7989, 38155}, {7995, 18452}, {8000, 9874}, {8055, 8834}, {8056, 35577}, {8071, 37293}, {8094, 8422}, {8113, 12633}, {8114, 12634}, {8125, 8242}, {8126, 11924}, {8168, 25524}, {8186, 49556}, {8187, 49555}, {8197, 11367}, {8204, 11366}, {8227, 15022}, {8239, 11687}, {8240, 11688}, {8241, 11690}, {8243, 12638}, {8275, 35258}, {8390, 11685}, {8392, 11686}, {8581, 12448}, {8591, 9884}, {8668, 11509}, {8726, 12658}, {8732, 34489}, {8972, 13902}, {9316, 9363}, {9331, 25092}, {9352, 32636}, {9451, 17136}, {9535, 44039}, {9538, 27505}, {9540, 35763}, {9541, 35610}, {9543, 9616}, {9575, 14930}, {9588, 30392}, {9589, 28164}, {9624, 10175}, {9654, 17577}, {9669, 37375}, {9709, 17531}, {9776, 11518}, {9782, 11524}, {9783, 11527}, {9787, 11528}, {9789, 11532}, {9791, 11533}, {9793, 11534}, {9795, 11899}, {9799, 12650}, {9800, 12651}, {9801, 12652}, {9802, 11280}, {9804, 12654}, {9807, 12656}, {9809, 13253}, {9845, 10860}, {9848, 12125}, {9859, 17616}, {9897, 21630}, {9911, 12087}, {9945, 37545}, {9961, 12680}, {10025, 30695}, {10026, 23903}, {10031, 10609}, {10044, 17647}, {10074, 17100}, {10164, 30389}, {10167, 31798}, {10267, 37106}, {10303, 15178}, {10371, 33075}, {10404, 20292}, {10405, 14942}, {10427, 14151}, {10436, 32087}, {10441, 50702}, {10446, 11521}, {10465, 10890}, {10473, 35634}, {10480, 35614}, {10481, 32098}, {10483, 34690}, {10531, 13729}, {10569, 12128}, {10570, 44765}, {10572, 11415}, {10588, 15950}, {10590, 37710}, {10593, 43734}, {10624, 20214}, {10699, 20344}, {10700, 21290}, {10701, 34186}, {10702, 34188}, {10703, 33650}, {10704, 14360}, {10705, 13219}, {10707, 10896}, {10711, 45631}, {10866, 11678}, {10895, 11235}, {10965, 22760}, {11008, 17347}, {11024, 11525}, {11061, 32298}, {11108, 15935}, {11113, 15172}, {11114, 15171}, {11200, 28870}, {11230, 46936}, {11234, 18258}, {11249, 11491}, {11252, 11844}, {11253, 11843}, {11281, 37703}, {11365, 13595}, {11376, 32537}, {11499, 13279}, {11501, 18967}, {11517, 37313}, {11535, 11891}, {11541, 28190}, {11544, 15679}, {11604, 21398}, {11875, 32147}, {11876, 32146}, {11900, 45289}, {12019, 50890}, {12047, 37708}, {12114, 38669}, {12115, 37437}, {12383, 12898}, {12384, 13099}, {12389, 12400}, {12391, 12395}, {12438, 16212}, {12527, 12575}, {12528, 12672}, {12529, 12709}, {12530, 12721}, {12532, 12758}, {12534, 12876}, {12535, 12877}, {12542, 12655}, {12543, 12657}, {12641, 41554}, {12667, 18243}, {12751, 25485}, {12763, 13271}, {12782, 23473}, {12849, 13100}, {12854, 37544}, {13161, 33134}, {13541, 17777}, {13601, 17625}, {13678, 13702}, {13745, 49718}, {13798, 13822}, {13869, 36154}, {13935, 35762}, {13941, 13959}, {14077, 17494}, {14210, 33937}, {14552, 28606}, {14555, 34064}, {14584, 51562}, {14759, 25272}, {14774, 24068}, {14872, 45776}, {14997, 16483}, {15015, 33812}, {15325, 17566}, {15558, 46685}, {15569, 27268}, {15570, 38053}, {15600, 17312}, {15640, 28208}, {15672, 18253}, {15674, 21677}, {15676, 35016}, {15683, 28194}, {15688, 50809}, {15702, 38066}, {15703, 38081}, {15705, 50817}, {15707, 50832}, {15708, 50821}, {15709, 50830}, {15863, 16173}, {15971, 48909}, {16062, 19823}, {16210, 51712}, {16284, 26563}, {16394, 50042}, {16397, 37540}, {16466, 17697}, {16468, 49685}, {16469, 17339}, {16478, 33166}, {16486, 37680}, {16597, 26147}, {16610, 21896}, {16667, 17355}, {16672, 17330}, {16705, 33297}, {16706, 17386}, {16781, 33854}, {16930, 20132}, {16971, 17750}, {17037, 20212}, {17056, 46875}, {17063, 46190}, {17079, 32007}, {17103, 32004}, {17117, 17391}, {17119, 17392}, {17141, 24282}, {17143, 34284}, {17160, 17378}, {17163, 49598}, {17166, 29298}, {17236, 49511}, {17237, 50076}, {17243, 37650}, {17246, 40341}, {17256, 50077}, {17276, 28566}, {17287, 17396}, {17290, 50112}, {17301, 17374}, {17305, 21356}, {17311, 17366}, {17320, 17360}, {17338, 35227}, {17343, 17772}, {17358, 38047}, {17369, 50087}, {17387, 37756}, {17441, 50698}, {17469, 33163}, {17491, 24851}, {17579, 18990}, {17597, 18141}, {17614, 51788}, {17615, 17622}, {17619, 51362}, {17690, 43993}, {17721, 28808}, {17724, 25529}, {17728, 37828}, {17740, 37539}, {17742, 33950}, {17766, 24248}, {17774, 17783}, {17781, 41864}, {17794, 21219}, {17800, 28216}, {18230, 24393}, {18231, 30478}, {18283, 37420}, {18444, 37108}, {18446, 37421}, {18467, 37736}, {18519, 21669}, {18600, 21281}, {18623, 34039}, {19692, 51710}, {19738, 48862}, {19741, 27797}, {19743, 48863}, {19796, 26729}, {19824, 23537}, {19945, 24418}, {19954, 27918}, {20061, 20074}, {20089, 28850}, {20173, 30807}, {20257, 30949}, {20293, 48303}, {20330, 38149}, {20347, 36854}, {20530, 24761}, {20533, 32029}, {20669, 23559}, {20760, 28376}, {20895, 44735}, {20911, 39731}, {20970, 27040}, {21075, 27131}, {21077, 30384}, {21283, 33112}, {21301, 48333}, {21343, 46403}, {21616, 37721}, {21620, 31019}, {21746, 50577}, {21870, 30861}, {21933, 27395}, {21935, 33141}, {22370, 27624}, {22647, 22969}, {22758, 37622}, {22793, 50688}, {23528, 48380}, {23536, 33131}, {24004, 24485}, {24416, 33910}, {24473, 50193}, {24474, 50701}, {24914, 36920}, {25048, 34434}, {25241, 43161}, {25244, 27340}, {25261, 27288}, {25269, 51196}, {25270, 33888}, {25280, 30963}, {25466, 33108}, {25592, 33298}, {25639, 37719}, {25723, 31721}, {25875, 42884}, {25962, 51416}, {26051, 37635}, {26062, 35262}, {26064, 43990}, {26098, 32920}, {26137, 30543}, {26242, 41015}, {26394, 26395}, {26418, 26419}, {26487, 33281}, {26494, 26495}, {26503, 26504}, {26777, 48285}, {26818, 27334}, {26824, 29066}, {26853, 29350}, {27065, 31435}, {27086, 37579}, {27484, 31342}, {27506, 38955}, {28029, 42461}, {28146, 49140}, {28160, 49135}, {28178, 49138}, {28186, 33703}, {28292, 47676}, {28522, 49532}, {28600, 41836}, {28610, 34716}, {28629, 44840}, {28633, 28640}, {28634, 28639}, {28635, 28641}, {28646, 44447}, {28849, 40868}, {29366, 48143}, {30283, 37022}, {30312, 41555}, {30424, 51101}, {30588, 43533}, {30720, 40621}, {30852, 50443}, {30962, 34063}, {31162, 31673}, {31246, 33559}, {31313, 50293}, {31423, 38127}, {31447, 31662}, {31547, 31567}, {31548, 31568}, {31551, 31559}, {31552, 31560}, {31730, 34632}, {31786, 37423}, {31992, 32212}, {32095, 41838}, {32354, 32394}, {32760, 45392}, {32784, 50315}, {32846, 49472}, {32859, 50065}, {33100, 38456}, {33118, 42378}, {33655, 36931}, {33930, 49779}, {34607, 36004}, {34610, 37299}, {34628, 51082}, {34629, 50910}, {34648, 50871}, {34689, 49736}, {34707, 36005}, {34744, 37568}, {34753, 50843}, {34930, 34932}, {35058, 51223}, {35104, 39780}, {35239, 37105}, {35242, 51705}, {35448, 37403}, {35659, 35669}, {35661, 35665}, {35979, 39783}, {36037, 36944}, {36404, 50026}, {36855, 37191}, {36991, 43166}, {37339, 37592}, {37407, 37615}, {37462, 40587}, {37532, 48363}, {37553, 38000}, {37631, 49734}, {37760, 47321}, {37901, 47538}, {37907, 47472}, {37909, 47540}, {37913, 49553}, {38057, 42819}, {38071, 50797}, {38074, 51709}, {38087, 51006}, {38089, 50953}, {38176, 46935}, {38514, 47274}, {39349, 39363}, {40420, 44794}, {41135, 50885}, {41792, 41794}, {41821, 41921}, {42045, 49745}, {43511, 49227}, {43512, 49226}, {43816, 43824}, {44367, 50250}, {45508, 45572}, {45509, 45573}, {46716, 50626}, {47359, 51146}, {47490, 51725}, {47721, 47869}, {47724, 50761}, {47793, 48294}, {47796, 48287}, {47805, 48327}, {47824, 48344}, {47825, 48289}, {47836, 48328}, {47840, 48347}, {48037, 48338}, {48164, 48332}, {48208, 48290}, {48798, 48819}, {48800, 48820}, {48806, 48824}, {48877, 48903}, {48936, 50422}, {49456, 49503}, {49457, 49689}, {49508, 51035}, {49700, 49712}, {49716, 49735}, {49743, 50171}, {49746, 50074}, {50041, 51591}, {50043, 50054}, {50044, 50061}, {50050, 50068}, {50055, 50067}, {50154, 50164}, {50155, 50259}, {50175, 50183}, {50179, 50279}, {50247, 50248}, {50398, 51715}, {50690, 51118}, {50794, 51706}, {50949, 51149}, {50950, 51193}, {51156, 51169}
midpoint of X(i) and X(j) for these {i,j}: {1, 3633}, {2, 20049}, {7, 12630}, {8, 20050}, {3242, 49679}, {3621, 20014}, {3868, 3885}, {7972, 26726}, {8148, 18526}, {12536, 12541}, {34631, 50818}, {34747, 51093}, {34748, 50805}, {45719, 45720}, {49469, 49498}, {49490, 49678}
reflection of X(i) in X(j) for these {i,j}: {1, 3244}, {2, 3241}, {3, 1483}, {4, 1482}, {6, 51147}, {7, 3243}, {8, 1}, {10, 3635}, {20, 944}, {40, 5882}, {65, 34791}, {69, 3242}, {72, 9957}, {75, 49478}, {100, 1317}, {144, 390}, {146, 7978}, {147, 7970}, {148, 7983}, {149, 1320}, {150, 10695}, {151, 10696}, {152, 10697}, {153, 10698}, {192, 49470}, {193, 51192}, {194, 7976}, {238, 49691}, {239, 49771}, {329, 7962}, {355, 10222}, {487, 7980}, {488, 7981}, {551, 51091}, {597, 51145}, {599, 50998}, {616, 7975}, {617, 7974}, {621, 51689}, {622, 51691}, {627, 22912}, {628, 22867}, {944, 37727}, {956, 37728}, {962, 7982}, {984, 49471}, {1145, 12735}, {1278, 24349}, {1320, 25416}, {1757, 49696}, {1992, 51000}, {2136, 12437}, {2475, 34195}, {2550, 42871}, {2888, 7979}, {2896, 7977}, {2975, 37734}, {3146, 962}, {3241, 51093}, {3416, 49465}, {3419, 50194}, {3434, 2099}, {3436, 2098}, {3448, 7984}, {3578, 49739}, {3579, 32900}, {3617, 3623}, {3621, 8}, {3625, 1125}, {3632, 10}, {3644, 49461}, {3648, 5441}, {3655, 51087}, {3679, 51071}, {3681, 5919}, {3751, 49684}, {3828, 51095}, {3868, 3555}, {3869, 3057}, {3893, 5836}, {3952, 17460}, {4240, 12626}, {4295, 12559}, {4307, 50284}, {4419, 17318}, {4440, 24841}, {4454, 4644}, {4461, 4344}, {4462, 4162}, {4645, 49675}, {4659, 4667}, {4660, 49464}, {4661, 3877}, {4664, 50778}, {4677, 551}, {4678, 20057}, {4701, 3636}, {4729, 4504}, {4740, 51055}, {4764, 49525}, {4814, 48325}, {4924, 4856}, {4929, 3950}, {5086, 11011}, {5176, 5048}, {5223, 30331}, {5541, 33337}, {5691, 4301}, {5881, 946}, {5903, 3874}, {5904, 3878}, {5921, 39898}, {6193, 9933}, {6194, 22713}, {6223, 7971}, {6224, 7972}, {6225, 7973}, {6327, 49454}, {6361, 18481}, {6764, 12629}, {6796, 32905}, {7057, 12646}, {7426, 47493}, {7672, 15185}, {7991, 4297}, {8591, 9884}, {9797, 12127}, {9799, 12650}, {9800, 12651}, {9801, 12652}, {9802, 12653}, {9803, 6264}, {9804, 12654}, {9807, 12656}, {9809, 13253}, {9812, 11224}, {9859, 17644}, {9874, 8000}, {9897, 21630}, {9961, 12680}, {10449, 50637}, {10914, 942}, {11041, 36867}, {11061, 32298}, {11160, 50999}, {11362, 13607}, {11415, 30323}, {11525, 14563}, {11684, 10543}, {11691, 8422}, {12125, 9848}, {12245, 3}, {12247, 12737}, {12383, 12898}, {12384, 13099}, {12389, 12400}, {12391, 12395}, {12526, 4314}, {12527, 12575}, {12528, 12672}, {12529, 12709}, {12530, 12721}, {12531, 11}, {12532, 12758}, {12533, 5920}, {12534, 12876}, {12535, 12877}, {12541, 3680}, {12542, 12655}, {12543, 12657}, {12625, 21627}, {12632, 3189}, {12645, 5}, {12648, 3870}, {12649, 36846}, {12672, 13600}, {12699, 11278}, {12702, 34773}, {12751, 25485}, {12849, 13100}, {13219, 10705}, {13678, 13702}, {13798, 13822}, {14360, 10704}, {14450, 16126}, {14807, 2102}, {14808, 2103}, {14872, 45776}, {14923, 65}, {15971, 48909}, {16017, 8241}, {17164, 2650}, {17299, 49467}, {17363, 3883}, {17494, 47729}, {17777, 13541}, {17784, 3476}, {18525, 22791}, {20008, 9797}, {20012, 20037}, {20014, 20050}, {20016, 50015}, {20017, 49687}, {20050, 3633}, {20051, 20040}, {20052, 3616}, {20053, 3632}, {20054, 3621}, {20055, 36534}, {20070, 20}, {20072, 49709}, {20076, 36977}, {20085, 149}, {20095, 6224}, {20293, 48303}, {20344, 10699}, {21290, 10700}, {21301, 48333}, {21302, 4449}, {22647, 22969}, {24248, 49455}, {24349, 49490}, {25304, 1469}, {25413, 24475}, {25722, 8581}, {26117, 41813}, {26824, 48304}, {28610, 34716}, {31145, 2}, {31162, 51077}, {31298, 17794}, {31302, 192}, {31547, 31567}, {31548, 31568}, {31551, 31559}, {31552, 31560}, {31888, 15680}, {32354, 32394}, {32850, 4864}, {33090, 17015}, {33650, 10703}, {33703, 48661}, {34186, 10701}, {34188, 10702}, {34605, 34749}, {34611, 34699}, {34627, 3656}, {34628, 51082}, {34631, 50805}, {34632, 50811}, {34641, 51103}, {34689, 49736}, {34718, 50824}, {34747, 51096}, {34748, 50831}, {34790, 31792}, {34932, 34930}, {35659, 35669}, {35661, 35665}, {36154, 13869}, {36972, 1329}, {36991, 43166}, {38514, 47274}, {39351, 14942}, {39776, 5083}, {42020, 38496}, {46403, 21343}, {46685, 15558}, {47321, 47491}, {47490, 51725}, {47724, 50761}, {47745, 13464}, {48304, 50767}, {48798, 48819}, {48800, 48820}, {48806, 48824}, {48877, 48903}, {49060, 49329}, {49061, 49330}, {49168, 22837}, {49169, 22836}, {49447, 49462}, {49448, 3993}, {49450, 37}, {49459, 24325}, {49470, 49475}, {49474, 49479}, {49493, 49491}, {49501, 49523}, {49503, 49456}, {49532, 49535}, {49677, 3836}, {49688, 1386}, {49689, 49457}, {49690, 49524}, {49698, 1279}, {49702, 3246}, {49704, 49695}, {49707, 238}, {49709, 49699}, {49712, 49700}, {49714, 44}, {49719, 5434}, {50043, 50070}, {50055, 50072}, {50074, 49746}, {50107, 50130}, {50154, 50182}, {50155, 50259}, {50172, 42045}, {50248, 50247}, {50277, 49735}, {50279, 50179}, {50289, 3879}, {50295, 50281}, {50343, 4922}, {50810, 3655}, {50818, 34748}, {50835, 47357}, {50864, 31162}, {50871, 34648}, {50872, 34631}, {51192, 49681}, {51515, 10283}

X(145) = midpoint of X(i) and X(j) for these {i,j}: {1, 3633}, {2, 20049}, {7, 12630}, {8, 20050}, {3242, 49679}, {3621, 20014}, {3868, 3885}, {7972, 26726}, {8148, 18526}, {12536, 12541}, {34631, 50818}, {34747, 51093}, {34748, 50805}, {45719, 45720}, {49469, 49498}, {49490, 49678}
X(145) = reflection of X(i) in X(j) for these {i,j}: {1, 3244}, {2, 3241}, {3, 1483}, {4, 1482}, {6, 51147}, {7, 3243}, {8, 1}, {10, 3635}, {20, 944}, {40, 5882}, {65, 34791}, {69, 3242}, {72, 9957}, {75, 49478}, {100, 1317}, {144, 390}, {146, 7978}, {147, 7970}, {148, 7983}, {149, 1320}, {150, 10695}, {151, 10696}, {152, 10697}, {153, 10698}, {192, 49470}, {193, 51192}, {194, 7976}, {238, 49691}, {239, 49771}, {329, 7962}, {355, 10222}, {487, 7980}, {488, 7981}, {551, 51091}, {597, 51145}, {599, 50998}, {616, 7975}, {617, 7974}, {621, 51689}, {622, 51691}, {627, 22912}, {628, 22867}, {944, 37727}, {956, 37728}, {962, 7982}, {984, 49471}, {1145, 12735}, {1278, 24349}, {1320, 25416}, {1757, 49696}, {1992, 51000}, {2136, 12437}, {2475, 34195}, {2550, 42871}, {2888, 7979}, {2896, 7977}, {2975, 37734}, {3146, 962}, {3241, 51093}, {3416, 49465}, {3419, 50194}, {3434, 2099}, {3436, 2098}, {3448, 7984}, {3578, 49739}, {3579, 32900}, {3617, 3623}, {3621, 8}, {3625, 1125}, {3632, 10}, {3644, 49461}, {3648, 5441}, {3655, 51087}, {3679, 51071}, {3681, 5919}, {3751, 49684}, {3828, 51095}, {3868, 3555}, {3869, 3057}, {3893, 5836}, {3952, 17460}, {4240, 12626}, {4295, 12559}, {4307, 50284}, {4419, 17318}, {4440, 24841}, {4454, 4644}, {4461, 4344}, {4462, 4162}, {4645, 49675}, {4659, 4667}, {4660, 49464}, {4661, 3877}, {4664, 50778}, {4677, 551}, {4678, 20057}, {4701, 3636}, {4729, 4504}, {4740, 51055}, {4764, 49525}, {4814, 48325}, {4924, 4856}, {4929, 3950}, {5086, 11011}, {5176, 5048}, {5223, 30331}, {5541, 33337}, {5691, 4301}, {5881, 946}, {5903, 3874}, {5904, 3878}, {5921, 39898}, {6193, 9933}, {6194, 22713}, {6223, 7971}, {6224, 7972}, {6225, 7973}, {6327, 49454}, {6361, 18481}, {6764, 12629}, {6796, 32905}, {7057, 12646}, {7426, 47493}, {7672, 15185}, {7991, 4297}, {8591, 9884}, {9797, 12127}, {9799, 12650}, {9800, 12651}, {9801, 12652}, {9802, 12653}, {9803, 6264}, {9804, 12654}, {9807, 12656}, {9809, 13253}, {9812, 11224}, {9859, 17644}, {9874, 8000}, {9897, 21630}, {9961, 12680}, {10449, 50637}, {10914, 942}, {11041, 36867}, {11061, 32298}, {11160, 50999}, {11362, 13607}, {11415, 30323}, {11525, 14563}, {11684, 10543}, {11691, 8422}, {12125, 9848}, {12245, 3}, {12247, 12737}, {12383, 12898}, {12384, 13099}, {12389, 12400}, {12391, 12395}, {12526, 4314}, {12527, 12575}, {12528, 12672}, {12529, 12709}, {12530, 12721}, {12531, 11}, {12532, 12758}, {12533, 5920}, {12534, 12876}, {12535, 12877}, {12541, 3680}, {12542, 12655}, {12543, 12657}, {12625, 21627}, {12632, 3189}, {12645, 5}, {12648, 3870}, {12649, 36846}, {12672, 13600}, {12699, 11278}, {12702, 34773}, {12751, 25485}, {12849, 13100}, {13219, 10705}, {13678, 13702}, {13798, 13822}, {14360, 10704}, {14450, 16126}, {14807, 2102}, {14808, 2103}, {14872, 45776}, {14923, 65}, {15971, 48909}, {16017, 8241}, {17164, 2650}, {17299, 49467}, {17363, 3883}, {17494, 47729}, {17777, 13541}, {17784, 3476}, {18525, 22791}, {20008, 9797}, {20012, 20037}, {20014, 20050}, {20016, 50015}, {20017, 49687}, {20050, 3633}, {20051, 20040}, {20052, 3616}, {20053, 3632}, {20054, 3621}, {20055, 36534}, {20070, 20}, {20072, 49709}, {20076, 36977}, {20085, 149}, {20095, 6224}, {20293, 48303}, {20344, 10699}, {21290, 10700}, {21301, 48333}, {21302, 4449}, {22647, 22969}, {24248, 49455}, {24349, 49490}, {25304, 1469}, {25413, 24475}, {25722, 8581}, {26117, 41813}, {26824, 48304}, {28610, 34716}, {31145, 2}, {31162, 51077}, {31298, 17794}, {31302, 192}, {31547, 31567}, {31548, 31568}, {31551, 31559}, {31552, 31560}, {31888, 15680}, {32354, 32394}, {32850, 4864}, {33090, 17015}, {33650, 10703}, {33703, 48661}, {34186, 10701}, {34188, 10702}, {34605, 34749}, {34611, 34699}, {34627, 3656}, {34628, 51082}, {34631, 50805}, {34632, 50811}, {34641, 51103}, {34689, 49736}, {34718, 50824}, {34747, 51096}, {34748, 50831}, {34790, 31792}, {34932, 34930}, {35659, 35669}, {35661, 35665}, {36154, 13869}, {36972, 1329}, {36991, 43166}, {38514, 47274}, {39351, 14942}, {39776, 5083}, {42020, 38496}, {46403, 21343}, {46685, 15558}, {47321, 47491}, {47490, 51725}, {47724, 50761}, {47745, 13464}, {48304, 50767}, {48798, 48819}, {48800, 48820}, {48806, 48824}, {48877, 48903}, {49060, 49329}, {49061, 49330}, {49168, 22837}, {49169, 22836}, {49447, 49462}, {49448, 3993}, {49450, 37}, {49459, 24325}, {49470, 49475}, {49474, 49479}, {49493, 49491}, {49501, 49523}, {49503, 49456}, {49532, 49535}, {49677, 3836}, {49688, 1386}, {49689, 49457}, {49690, 49524}, {49698, 1279}, {49702, 3246}, {49704, 49695}, {49707, 238}, {49709, 49699}, {49712, 49700}, {49714, 44}, {49719, 5434}, {50043, 50070}, {50055, 50072}, {50074, 49746}, {50107, 50130}, {50154, 50182}, {50155, 50259}, {50172, 42045}, {50248, 50247}, {50277, 49735}, {50279, 50179}, {50289, 3879}, {50295, 50281}, {50343, 4922}, {50810, 3655}, {50818, 34748}, {50835, 47357}, {50864, 31162}, {50871, 34648}, {50872, 34631}, {51192, 49681}, {51515, 10283}
X(145) = isogonal conjugate of X(3445)
X(145) = isotomic conjugate of X(4373)
X(145) = complement of X(3621)
X(145) = anticomplement of X(8)
X(145) = incircle-inverse of X(37743)
X(145) = Steiner-circumellipse-inverse of X(3008)
X(145) = orthoptic-circle-of-Steiner-circumellipse-inverse of X(5121)
X(145) = anticomplement of the isogonal conjugate of X(56)
X(145) = anticomplement of the isotomic conjugate of X(7)
X(145) = complement of the isotomic conjugate of X(36606)
X(145) = isogonal conjugate of the anticomplement of X(2885)
X(145) = isogonal conjugate of the complement of X(42020)
X(145) = isotomic conjugate of the anticomplement of X(3161)
X(145) = isotomic conjugate of the isogonal conjugate of X(3052)
X(145) = polar conjugate of the isogonal conjugate of X(20818)
X(145) = anticomplementary isogonal conjugate of X(3436)
X(145) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 3436}, {2, 21286}, {6, 329}, {7, 6327}, {25, 5942}, {31, 144}, {32, 3177}, {34, 4}, {41, 30695}, {56, 8}, {57, 69}, {58, 3869}, {59, 3952}, {65, 1330}, {77, 1370}, {81, 20245}, {85, 315}, {87, 20557}, {101, 4462}, {106, 5176}, {108, 20293}, {109, 513}, {221, 6223}, {222, 4329}, {226, 21287}, {241, 20552}, {251, 20248}, {269, 3434}, {273, 11442}, {278, 21270}, {279, 21285}, {284, 18750}, {513, 33650}, {560, 21218}, {593, 21273}, {603, 20}, {604, 2}, {608, 5905}, {649, 37781}, {651, 20295}, {664, 21301}, {667, 39351}, {692, 4468}, {738, 6604}, {757, 35614}, {849, 2975}, {934, 21302}, {951, 7270}, {961, 17751}, {1014, 17135}, {1037, 10327}, {1042, 2475}, {1088, 21280}, {1106, 145}, {1118, 5906}, {1262, 21272}, {1319, 21290}, {1333, 63}, {1393, 2888}, {1394, 6225}, {1395, 193}, {1396, 17220}, {1397, 192}, {1398, 12649}, {1399, 3648}, {1400, 2895}, {1401, 21289}, {1402, 1654}, {1404, 30578}, {1407, 7}, {1408, 1}, {1409, 3151}, {1410, 3152}, {1411, 5080}, {1412, 75}, {1413, 962}, {1414, 512}, {1415, 514}, {1416, 518}, {1417, 519}, {1418, 2890}, {1420, 42020}, {1422, 21279}, {1424, 32548}, {1427, 2893}, {1428, 17794}, {1429, 20345}, {1431, 4388}, {1434, 17137}, {1436, 189}, {1438, 30807}, {1447, 20554}, {1455, 151}, {1456, 152}, {1457, 153}, {1458, 20344}, {1459, 34188}, {1461, 693}, {1462, 20347}, {1474, 92}, {1477, 32850}, {1973, 30694}, {2006, 21277}, {2099, 21291}, {2149, 190}, {2162, 20348}, {2194, 45738}, {2199, 20211}, {2215, 26872}, {3213, 14361}, {3248, 17036}, {3450, 25253}, {3451, 312}, {3669, 150}, {3676, 21293}, {4017, 3448}, {4554, 21304}, {4564, 668}, {4565, 7192}, {4573, 17217}, {4617, 46402}, {4625, 44445}, {4637, 4374}, {6063, 21275}, {6611, 5932}, {6614, 3900}, {7023, 36845}, {7045, 3888}, {7099, 347}, {7121, 20535}, {7125, 6527}, {7153, 21281}, {7175, 30660}, {7178, 21294}, {7180, 21221}, {7251, 21215}, {7316, 17491}, {7341, 17140}, {7366, 4452}, {8059, 4397}, {8809, 32064}, {9447, 46706}, {9456, 908}, {16945, 3621}, {16947, 17147}, {18268, 1959}, {20567, 33796}, {23971, 35312}, {23979, 4552}, {24027, 100}, {28607, 5744}, {28615, 17781}, {32636, 2891}, {32669, 3904}, {32674, 4391}, {32675, 3762}, {32714, 46400}, {34079, 14206}, {36049, 20296}, {36059, 20294}, {36098, 6371}, {36110, 8677}, {36141, 30565}, {36146, 3766}, {38828, 4106}, {40151, 21296}, {41280, 17486}, {41526, 21219}, {42467, 8048}, {43924, 149}, {51640, 13219}, {51641, 34186}, {51642, 148}, {51649, 12384}, {51653, 147}, {51654, 34547}, {51655, 14360}, {51656, 146}, {51658, 34548}, {51662, 34550}
X(145) = X(i)-complementary conjugate of X(j) for these (i,j): {8699, 513}, {36603, 141}, {36606, 2887}, {36621, 17046}, {38255, X(145) = 21244}, {40026, 626}
X(145) = X(i)-Ceva conjugate of X(j) for these (i,j): {7, 2}, {190, 31182}, {664, 30719}, {1016, 30720}, {1266, 30577}, {3875, 3210}, {3879, 17778}, {4076, 100}, {4998, 25737}, {9312, 3177}, {10106, 5484}, {18743, 3161}, {35577, 3622}, {39126, 5435}, {40420, 9}, {41629, 1743}, {43290, 3667}, {44724, 43290}
X(145) = X(i)-cross conjugate of X(j) for these (i,j): {1, 44301}, {1743, 5435}, {3158, 3161}, {3161, 2}, {3667, 43290}, {3756, 3667}, {3950, 18743}, {4534, 4462}, {4849, 1743}, {4856, 41629}, {4943, 30720}, {4953, 4521}, {6555, 24150}, {12640, 44720}, {31182, 190}, {40621, 30719}, {44720, 39701}, {45219, 1420}, {51658, 100}
X(145) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3445}, {2, 38266}, {6, 8056}, {8, 16945}, {9, 40151}, {31, 4373}, {32, 40014}, {41, 27818}, {55, 19604}, {56, 3680}, {513, 1293}, {514, 34080}, {604, 6557}, {607, 27832}, {649, 27834}, {650, 38828}, {900, 36042}, {1015, 5382}, {1022, 2429}, {1106, 6556}, {1261, 46367}, {1333, 4052}, {1420, 33963}, {3158, 16079}, {3762, 32645}, {31343, 43924}
X(145) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 3680}, {1, 45036}, {2, 4373}, {3, 3445}, {9, 8056}, {37, 4052}, {145, 3621}, {223, 19604}, {478, 40151}, {514, 40621}, {522, 3756}, {900, 5516}, {1086, 4521}, {1293, 39026}, {1358, 3669}, {1420, 34039}, {1743, 3928}, {3052, 36641}, {3160, 27818}, {3161, 6557}, {3667, 3756}, {3950, 22034}, {4162, 36639}, {4849, 22312}, {4859, 24392}, {4862, 30827}, {5375, 27834}, {5435, 36640}, {6376, 40014}, {6552, 6556}, {10029, 36905}, {10563, 11530}, {14321, 17058}, {21255, 21342}, {32664, 38266}, {39048, 51839}
X(145) = cevapoint of X(i) and X(j) for these (i,j): {1, 2136}, {8, 8834}, {200, 45193}, {1743, 3158}, {3052, 20818}, {3161, 15519}, {3667, 3756}, {3950, 4849}, {4162, 4534}, {4404, 4939}, {4521, 4953}, {4943, 40621}, {12640, 45219}, {12643, 12646}
X(145) = crosspoint of X(i) and X(j) for these (i,j): {1, 39969}, {2, 36606}, {7, 5435}, {664, 1016}, {18743, 39126}, {43290, 44724}
X(145) = crosssum of X(i) and X(j) for these (i,j): {1, 21214}, {6, 21000}, {244, 6615}, {663, 1015}, {3271, 17424}
X(145) = trilinear pole of line {2976, 3667}
X(145) = crossdifference of every pair of points on line {649, 6363}
X(145) = X(643)-beth conjugate of X(56)
X(145) = exsimilicenter of incircle and AC-incircle
X(145) = X(64)-of-intouch-triangle
X(145) = trilinear pole of line X(2976)X(3667) (radical axis of incircle and AC-incircle, and the pole of X(2) wrt the Spieker circle)
X(145) = X(8)-of-anticomplementary-triangle
X(145) = X(11381)-of-excentral-triangle
X(145) = perspector of intouch triangle and Gemini triangle 29
X(145) = {X(37794),X(37795)}-harmonic conjugate of X(2)
barycentric product X(i)*X(j) for these {i,j}: {1, 18743}, {7, 3161}, {8, 5435}, {9, 39126}, {10, 41629}, {43, 27496}, {56, 44723}, {57, 44720}, {75, 1743}, {76, 3052}, {83, 4884}, {85, 3158}, {86, 3950}, {88, 4487}, {92, 4855}, {99, 14321}, {100, 4462}, {190, 3667}, {222, 44721}, {264, 20818}, {274, 4849}, {278, 44722}, {279, 6555}, {306, 4248}, {312, 1420}, {313, 33628}, {321, 16948}, {333, 4848}, {514, 43290}, {519, 31227}, {646, 51658}, {658, 4546}, {662, 4404}, {664, 4521}, {666, 4925}, {668, 4394}, {673, 4899}, {799, 4729}, {1016, 3756}, {1086, 44724}, {1088, 4936}, {1222, 45204}, {1268, 4856}, {1275, 4953}, {1978, 8643}, {2403, 17780}, {3210, 39701}, {3676, 30720}, {3699, 30719}, {4076, 40617}, {4162, 4554}, {4452, 24150}, {4504, 27805}, {4534, 4998}, {4555, 14425}, {4564, 4939}, {4573, 44729}, {4600, 21950}, {4881, 18359}, {4891, 32009}, {4898, 30598}, {4918, 14534}, {4935, 39962}, {4949, 32042}, {6049, 6557}, {6357, 44727}, {6632, 23764}, {8055, 44301}, {12640, 40420}, {15519, 27818}, {32017, 45219}, {34234, 51433}
barycentric quotient X(i)/X(j) for these {i,j}: {1, 8056}, {2, 4373}, {6, 3445}, {7, 27818}, {8, 6557}, {9, 3680}, {10, 4052}, {31, 38266}, {56, 40151}, {57, 19604}, {75, 40014}, {77, 27832}, {100, 27834}, {101, 1293}, {109, 38828}, {346, 6556}, {604, 16945}, {644, 31343}, {692, 34080}, {765, 5382}, {1122, 45205}, {1200, 45229}, {1279, 51839}, {1420, 57}, {1743, 1}, {2136, 24151}, {2403, 6548}, {2441, 23345}, {2976, 6084}, {3052, 6}, {3158, 9}, {3161, 8}, {3210, 27835}, {3434, 27826}, {3667, 514}, {3731, 10563}, {3756, 1086}, {3870, 27819}, {3873, 27827}, {3875, 27813}, {3879, 27820}, {3891, 27817}, {3896, 27823}, {3905, 27821}, {3907, 27831}, {3950, 10}, {4162, 650}, {4248, 27}, {4394, 513}, {4404, 1577}, {4449, 27837}, {4452, 27828}, {4453, 27836}, {4462, 693}, {4487, 4358}, {4504, 4369}, {4521, 522}, {4534, 11}, {4546, 3239}, {4729, 661}, {4848, 226}, {4849, 37}, {4855, 63}, {4856, 1125}, {4881, 3218}, {4884, 141}, {4891, 3739}, {4898, 1698}, {4899, 3912}, {4917, 3305}, {4918, 1211}, {4924, 29571}, {4925, 918}, {4929, 17284}, {4936, 200}, {4939, 4858}, {4943, 4521}, {4949, 4802}, {4952, 17279}, {4953, 1146}, {4964, 25666}, {5435, 7}, {6049, 5435}, {6555, 346}, {8643, 649}, {9312, 27829}, {9436, 10029}, {12640, 3452}, {14100, 45202}, {14284, 21120}, {14321, 523}, {14350, 28161}, {14351, 28225}, {14425, 900}, {15519, 3161}, {16948, 81}, {17780, 2415}, {18211, 16726}, {18743, 75}, {20818, 3}, {21950, 3120}, {23344, 2429}, {23511, 47636}, {23764, 6545}, {24150, 6553}, {27496, 6384}, {27818, 16078}, {30719, 3676}, {30720, 3699}, {31182, 3667}, {31227, 903}, {32665, 36042}, {32719, 32645}, {33628, 58}, {37764, 17951}, {39126, 85}, {39701, 39694}, {40151, 16079}, {40617, 1358}, {40621, 3756}, {41629, 86}, {43290, 190}, {44301, 8051}, {44720, 312}, {44721, 7017}, {44722, 345}, {44723, 3596}, {44724, 1016}, {44729, 3700}, {45036, 3928}, {45204, 3663}, {45219, 3752}, {51433, 908}, {51658, 3669}
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 2, 3622}, {1, 8, 2}, {1, 10, 3616}, {1, 43, 1201}, {1, 200, 19861}, {1, 978, 1149}, {1, 1125, 38314}, {1, 1698, 551}, {1, 1722, 28011}, {1, 3214, 28370}, {1, 3241, 3623}, {1, 3244, 3241}, {1, 3293, 995}, {1, 3617, 46934}, {1, 3621, 3617}, {1, 3624, 3636}, {1, 3625, 9780}, {1, 3626, 5550}, {1, 3632, 10}, {1, 3635, 20057}, {1, 3679, 1125}, {1, 3811, 4511}, {1, 3870, 34772}, {1, 3938, 36565}, {1, 4668, 3624}, {1, 4677, 1698}, {1, 4701, 19877}, {1, 4816, 3634}, {1, 4853, 19860}, {1, 4882, 8583}, {1, 5262, 17024}, {1, 5312, 50604}, {1, 5554, 10586}, {1, 6738, 10580}, {1, 6765, 78}, {1, 7080, 24558}, {1, 10573, 3086}, {1, 11519, 4853}, {1, 12629, 3872}, {1, 12647, 3085}, {1, 12648, 10528}, {1, 12649, 10529}, {1, 16830, 29570}, {1, 18391, 14986}, {1, 20014, 31145}, {1, 20018, 20036}, {1, 20049, 20054}, {1, 20050, 3621}, {1, 20052, 46932}, {1, 20053, 4678}, {1, 31145, 46933}, {1, 31397, 5703}, {1, 34595, 51105}, {1, 34747, 3633}, {1, 36846, 38460}, {1, 39584, 38475}, {1, 39587, 29624}, {1, 41575, 12649}, {1, 41684, 499}, {1, 49168, 10527}, {1, 49169, 5552}, {1, 49458, 36534}, {1, 49476, 17316}, {1, 49495, 239}, {1, 49762, 29579}, {1, 49772, 16020}, {1, 50016, 16825}, {1, 50286, 29569}, {1, 50581, 1193}, {1, 50582, 29840}, {1, 51093, 3244}, {2, 8, 3617}, {2, 10, 46932}, {2, 239, 24599}, {2, 3006, 30762}, {2, 3617, 46933}, {2, 3621, 8}, {2, 3622, 46934}, {2, 3623, 1}, {2, 3912, 30833}, {2, 4393, 17014}, {2, 4678, 10}, {2, 6542, 29616}, {2, 9780, 46930}, {2, 17316, 29621}, {2, 20008, 12649}, {2, 20011, 20012}, {2, 20013, 20007}, {2, 20014, 3621}, {2, 20040, 20036}, {2, 20045, 26245}, {2, 20048, 19998}, {2, 20050, 20054}, {2, 20052, 4678}, {2, 20056, 7172}, {2, 26773, 27112}, {2, 26820, 27165}, {2, 29585, 29624}, {2, 29588, 29585}, {2, 29824, 30948}, {2, 46933, 46931}, {2, 51093, 51092}, {3, 1483, 7967}, {5, 10247, 10595}, {6, 17314, 346}, {6, 17388, 17314}, {7, 3875, 4452}, {7, 4452, 4373}, {7, 4460, 3875}, {7, 9312, 43983}, {7, 25718, 9312}, {8, 10, 4678}, {8, 938, 5554}, {8, 3241, 1}, {8, 3244, 3623}, {8, 3616, 10}, {8, 3621, 31145}, {8, 3622, 46933}, {8, 3623, 3622}, {8, 3632, 20052}, {8, 3633, 20014}, {8, 3957, 10587}, {8, 4511, 7080}, {8, 9780, 3679}, {8, 9797, 36845}, {8, 10453, 17751}, {8, 10578, 24987}, {8, 20011, 20051}, {8, 20014, 20054}, {8, 20018, 20012}, {8, 20037, 20036}, {8, 20053, 3632}, {8, 20057, 3616}, {8, 27383, 6735}, {8, 34772, 10528}, {8, 36845, 12649}, {8, 38314, 9780}, {8, 38460, 10529}, {8, 39567, 24599}, {8, 49771, 39567}, {10, 551, 19878}, {10, 3244, 3635}, {10, 3616, 2}, {10, 3624, 19877}, {10, 3625, 4746}, {10, 3632, 8}, {10, 3635, 1}, {10, 3636, 3624}, {10, 4062, 27558}, {10, 4678, 3617}, {10, 4701, 4668}, {10, 4746, 3679}, {10, 19878, 1698}, {10, 20053, 20052}, {10, 46932, 46933}, {11, 11681, 5154}, {11, 12607, 11681}, {12, 3813, 11680}, {12, 11680, 5141}, {20, 3868, 9965}, {20, 20075, 20066}, {20, 20076, 20067}, {37, 5839, 391}, {40, 5731, 3522}, {40, 5882, 5731}, {42, 10453, 2}, {42, 50001, 30942}, {45, 4969, 37654}, {55, 2975, 4189}, {55, 12513, 2975}, {56, 100, 4188}, {56, 3913, 100}, {63, 4313, 17576}, {65, 3189, 17784}, {65, 3476, 3600}, {65, 3600, 21454}, {65, 34791, 3873}, {65, 37738, 3476}, {69, 4360, 3672}, {72, 3488, 452}, {72, 9957, 3877}, {75, 21605, 21432}, {78, 938, 2}, {78, 6765, 3935}, {81, 1043, 11115}, {149, 20060, 4}, {192, 193, 144}, {192, 3177, 25237}, {192, 20072, 20073}, {192, 49704, 390}, {193, 20073, 20072}, {200, 10580, 2}, {226, 12625, 5175}, {239, 17316, 2}, {239, 17389, 17316}, {239, 29575, 29628}, {239, 29619, 17244}, {279, 6604, 51351}, {319, 17321, 5232}, {319, 17393, 17321}, {320, 50101, 4346}, {344, 3759, 37681}, {344, 49698, 10005}, {354, 3893, 5836}, {355, 5603, 3091}, {355, 10222, 5603}, {377, 5082, 33110}, {388, 3434, 2475}, {392, 34790, 3876}, {495, 24390, 2476}, {496, 17757, 4193}, {497, 3436, 5046}, {499, 27529, 2}, {499, 45701, 27529}, {551, 1698, 5550}, {551, 3626, 1698}, {664, 6604, 279}, {899, 21214, 27625}, {899, 30947, 2}, {944, 3868, 20076}, {944, 3885, 20075}, {944, 6361, 18481}, {946, 16200, 5734}, {950, 7962, 9785}, {950, 11523, 329}, {956, 3295, 21}, {958, 1621, 16865}, {958, 3303, 1621}, {960, 5919, 3890}, {978, 1149, 28370}, {995, 50575, 3293}, {999, 5687, 404}, {1001, 5260, 16859}, {1056, 5082, 377}, {1058, 3421, 2478}, {1100, 17299, 2345}, {1125, 3625, 3679}, {1125, 3626, 51069}, {1125, 3679, 9780}, {1125, 4746, 10}, {1125, 9780, 2}, {1125, 51071, 1}, {1149, 3214, 978}, {1150, 19765, 16347}, {1193, 35633, 29824}, {1193, 50581, 3240}, {1210, 6735, 25005}, {1210, 27383, 2}, {1319, 1788, 5265}, {1319, 41687, 1788}, {1376, 3304, 5253}, {1376, 5253, 17572}, {1385, 5657, 3523}, {1388, 40663, 7288}, {1420, 3158, 4855}, {1420, 4848, 5435}, {1420, 4855, 4881}, {1449, 2321, 5749}, {1468, 5255, 17126}, {1482, 18525, 22791}, {1697, 6762, 63}, {1698, 4677, 3626}, {1698, 5550, 2}, {1698, 51069, 9780}, {1722, 28011, 7292}, {1743, 3950, 3161}, {1743, 4898, 3950}, {1743, 4929, 4899}, {1829, 7718, 6995}, {1837, 32049, 5176}, {1909, 17144, 4441}, {1998, 3870, 3935}, {1999, 50582, 12649}, {2098, 10950, 497}, {2098, 41711, 12635}, {2099, 10944, 388}, {2176, 3780, 37657}, {2275, 20691, 17756}, {2340, 28124, 28057}, {2550, 42871, 11038}, {3006, 26228, 2}, {3008, 29573, 29627}, {3008, 29627, 2}, {3011, 30741, 2}, {3057, 3486, 390}, {3057, 37740, 3486}, {3085, 3086, 10320}, {3085, 10527, 2}, {3085, 34625, 10527}, {3086, 5552, 2}, {3086, 34619, 5552}, {3160, 32003, 9436}, {3177, 20111, 144}, {3189, 3873, 4190}, {3189, 34791, 3600}, {3240, 29824, 2}, {3241, 3616, 20057}, {3241, 3621, 3622}, {3241, 3633, 3621}, {3241, 9797, 36846}, {3241, 15519, 37743}, {3241, 20014, 3617}, {3241, 20049, 31145}, {3241, 20050, 8}, {3241, 20053, 3616}, {3241, 20054, 46934}, {3241, 20057, 3635}, {3241, 34747, 20049}, {3241, 36845, 38460}, {3241, 38314, 51071}, {3241, 49495, 39567}, {3243, 3340, 11520}, {3243, 3680, 3340}, {3244, 3625, 51071}, {3244, 3632, 20057}, {3244, 3633, 8}, {3244, 3634, 51097}, {3244, 15808, 51095}, {3244, 20014, 3622}, {3244, 20049, 3617}, {3244, 20050, 2}, {3244, 34747, 20050}, {3244, 41575, 38460}, {3247, 3686, 5296}, {3340, 10106, 7}, {3419, 3487, 5177}, {3434, 3891, 30699}, {3476, 14923, 4190}, {3485, 5086, 6871}, {3485, 5252, 5261}, {3486, 3869, 6872}, {3555, 3868, 4430}, {3555, 37727, 36977}, {3579, 32900, 3655}, {3579, 51087, 32900}, {3600, 12632, 17784}, {3600, 17784, 4190}, {3601, 24391, 5744}, {3616, 3632, 4678}, {3616, 4678, 46932}, {3616, 19877, 3624}, {3616, 20050, 20053}, {3616, 20052, 3617}, {3616, 20053, 8}, {3616, 20057, 1}, {3616, 27558, 24946}, {3617, 3622, 2}, {3617, 20054, 31145}, {3617, 31145, 8}, {3617, 46932, 10}, {3617, 46934, 46931}, {3617, 51092, 3623}, {3621, 3623, 2}, {3621, 4678, 20052}, {3621, 20013, 20015}, {3621, 20018, 20051}, {3621, 20040, 20012}, {3621, 20041, 20036}, {3621, 20049, 20014}, {3621, 20052, 3632}, {3621, 20057, 46932}, {3621, 29588, 39587}, {3621, 51071, 46930}, {3622, 20054, 8}, {3622, 31145, 3617}, {3622, 46930, 1125}, {3623, 3633, 20054}, {3623, 20011, 20036}, {3623, 20014, 8}, {3623, 20049, 3621}, {3623, 20050, 31145}, {3623, 20052, 3616}, {3623, 20053, 46932}, {3623, 20054, 46933}, {3623, 31145, 46934}, {3624, 3632, 4668}, {3624, 3636, 3616}, {3624, 4668, 10}, {3624, 19877, 2}, {3625, 3679, 8}, {3625, 51071, 1125}, {3626, 4677, 8}, {3626, 19878, 10}, {3632, 3635, 3616}, {3632, 4668, 4701}, {3632, 20052, 31145}, {3632, 20053, 3621}, {3632, 20057, 2}, {3633, 3635, 20053}, {3633, 20039, 20051}, {3633, 20050, 20049}, {3633, 51093, 1}, {3634, 41150, 1125}, {3635, 4701, 3636}, {3635, 20050, 20052}, {3635, 20052, 3622}, {3635, 20053, 2}, {3635, 20057, 3623}, {3636, 4668, 19877}, {3636, 4701, 10}, {3648, 5441, 15680}, {3655, 50810, 10304}, {3656, 34627, 3839}, {3661, 26626, 2}, {3661, 29584, 26626}, {3663, 21296, 45789}, {3664, 17151, 31995}, {3679, 38314, 2}, {3679, 51071, 38314}, {3681, 3890, 960}, {3685, 3751, 17350}, {3693, 40133, 26690}, {3699, 42020, 6552}, {3702, 4385, 4671}, {3717, 7290, 26685}, {3746, 5288, 993}, {3754, 3892, 18398}, {3755, 4684, 3662}, {3759, 17315, 344}, {3811, 18391, 7080}, {3828, 15808, 34595}, {3870, 12649, 10528}, {3870, 20008, 3617}, {3870, 36845, 2}, {3870, 36846, 1}, {3870, 41575, 8}, {3873, 3896, 3210}, {3873, 14923, 65}, {3873, 17784, 21454}, {3875, 3879, 7}, {3875, 4452, 32105}, {3875, 4464, 4460}, {3879, 4452, 32093}, {3879, 4460, 4452}, {3879, 4464, 3875}, {3883, 7174, 17257}, {3885, 36977, 20}, {3912, 5222, 2}, {3912, 16834, 5222}, {3912, 30036, 29986}, {3915, 5247, 17127}, {3940, 12433, 5084}, {3950, 4856, 1743}, {3950, 4924, 4899}, {3987, 4694, 24046}, {3991, 43065, 25082}, {4000, 4851, 4869}, {4000, 4916, 4851}, {4029, 4700, 9}, {4051, 51058, 17451}, {4297, 7991, 9778}, {4297, 9778, 50693}, {4301, 5691, 9812}, {4360, 17377, 69}, {4361, 17390, 4648}, {4373, 32093, 7}, {4373, 32105, 4452}, {4384, 5308, 2}, {4384, 29574, 5308}, {4393, 6542, 2}, {4429, 4966, 17232}, {4430, 20075, 9965}, {4511, 14986, 24558}, {4511, 18391, 2}, {4651, 29814, 2}, {4659, 4667, 35578}, {4662, 10179, 25917}, {4663, 4702, 4676}, {4668, 4701, 8}, {4669, 4816, 8}, {4669, 51071, 41150}, {4677, 51093, 51091}, {4678, 20014, 20053}, {4678, 20052, 8}, {4678, 20053, 31145}, {4678, 20057, 3622}, {4685, 26102, 26038}, {4690, 41312, 4748}, {4691, 19862, 19875}, {4691, 51103, 19862}, {4701, 19877, 4678}, {4727, 16666, 17281}, {4814, 48325, 48242}, {4847, 10578, 2}, {4848, 12640, 51433}, {4849, 4891, 18743}, {4851, 4852, 4000}, {4851, 4889, 4916}, {4851, 4910, 4852}, {4852, 4889, 4851}, {4852, 4916, 4869}, {4853, 6737, 8}, {4855, 4917, 3158}, {4856, 4898, 3161}, {4863, 5794, 5178}, {4871, 4946, 43}, {4889, 4910, 4000}, {4915, 6743, 8}, {4946, 42057, 4871}, {4952, 45219, 44722}, {4969, 50113, 45}, {5051, 41014, 31037}, {5217, 11194, 5303}, {5252, 11011, 3485}, {5256, 34255, 2}, {5256, 50292, 34255}, {5274, 25568, 46873}, {5435, 6049, 1420}, {5543, 31994, 40719}, {5550, 51091, 3623}, {5552, 11240, 3086}, {5563, 48696, 25440}, {5597, 12455, 5602}, {5598, 12454, 5601}, {5604, 12628, 1270}, {5605, 12627, 1271}, {5690, 10246, 631}, {5691, 9812, 17578}, {5691, 11224, 4301}, {5698, 49447, 20073}, {5703, 6734, 2}, {5704, 27385, 2}, {5730, 37730, 2478}, {5734, 5881, 3832}, {5790, 5901, 3090}, {5795, 15829, 18228}, {5818, 5886, 5056}, {5880, 51099, 30340}, {5881, 16200, 946}, {5903, 6224, 37256}, {6361, 18481, 20}, {6735, 27383, 27525}, {6736, 11019, 24982}, {6911, 12001, 45977}, {7080, 14986, 2}, {7191, 10327, 2}, {7320, 8236, 37556}, {7962, 11523, 11682}, {7967, 12245, 3}, {7968, 19065, 7586}, {7969, 19066, 7585}, {8148, 34748, 18526}, {8192, 12410, 22}, {8210, 12636, 6462}, {8211, 12637, 6463}, {8666, 25439, 35}, {9436, 25716, 3160}, {9779, 19925, 3854}, {9780, 38314, 1125}, {9797, 20050, 41575}, {9957, 37739, 3488}, {9997, 12495, 2896}, {10074, 25438, 17100}, {10247, 12645, 5}, {10449, 19767, 2}, {10527, 11239, 3085}, {10528, 10529, 2}, {10543, 11684, 15677}, {10572, 30323, 30305}, {10573, 22836, 5552}, {10573, 49169, 8}, {10800, 12195, 7787}, {10912, 10944, 3434}, {10915, 49627, 1737}, {10916, 49626, 10039}, {10950, 12635, 3436}, {11009, 37707, 1478}, {11239, 34625, 2}, {11240, 34619, 2}, {11362, 13607, 3576}, {11396, 12135, 4}, {11520, 12536, 37435}, {11522, 19925, 9779}, {11522, 37712, 19925}, {11849, 32153, 6950}, {12247, 46920, 6972}, {12632, 34791, 21454}, {12644, 12646, 174}, {12647, 22837, 10527}, {12647, 49168, 8}, {12648, 12649, 8}, {12648, 36846, 10529}, {12648, 38460, 2}, {12649, 34772, 2}, {12653, 16126, 11280}, {12702, 34773, 376}, {13464, 47745, 5587}, {13902, 13911, 8972}, {13959, 13973, 13941}, {14923, 30614, 11851}, {15680, 20086, 20077}, {16569, 26103, 2}, {16777, 17362, 966}, {16816, 29569, 2}, {16830, 39581, 2}, {16830, 50310, 39581}, {16833, 29602, 29571}, {16834, 29605, 3912}, {17014, 29616, 2}, {17018, 17135, 2}, {17023, 17294, 29611}, {17023, 29611, 2}, {17023, 49761, 17294}, {17024, 33091, 2}, {17121, 17242, 26685}, {17150, 33093, 2}, {17160, 17378, 42697}, {17244, 17389, 29619}, {17244, 29619, 17316}, {17295, 17380, 3619}, {17302, 17373, 3620}, {17310, 17367, 29579}, {17316, 39567, 3622}, {17316, 50129, 239}, {17319, 17363, 17257}, {17367, 29579, 2}, {17389, 50129, 2}, {18357, 18493, 3545}, {18493, 50798, 18357}, {18525, 22791, 4}, {18526, 50805, 8148}, {19862, 34641, 4691}, {19875, 34641, 51072}, {19877, 20053, 4701}, {19993, 20009, 3622}, {19993, 20011, 20043}, {19993, 20013, 20036}, {19993, 20020, 2}, {19993, 20035, 20007}, {19994, 20016, 20012}, {19998, 20011, 20048}, {19998, 20048, 20012}, {20009, 20018, 20007}, {20011, 20020, 20015}, {20011, 20039, 20037}, {20011, 20040, 20018}, {20011, 20041, 20040}, {20011, 20044, 19994}, {20014, 20037, 20051}, {20014, 20039, 20036}, {20014, 20041, 20012}, {20014, 20049, 20050}, {20014, 51092, 46934}, {20015, 20043, 20012}, {20016, 29588, 2}, {20018, 20037, 20040}, {20020, 20040, 20007}, {20035, 20040, 20013}, {20039, 20040, 20041}, {20039, 20049, 20012}, {20040, 20041, 20037}, {20041, 20050, 20051}, {20045, 29832, 2}, {20048, 20052, 20047}, {20050, 20057, 3632}, {20050, 51092, 46933}, {20050, 51093, 3623}, {20053, 20057, 10}, {20053, 38314, 4746}, {20054, 31145, 3621}, {20054, 46932, 20052}, {20054, 51092, 1}, {20064, 20068, 20078}, {20066, 20067, 20}, {20072, 20073, 144}, {20072, 49704, 49709}, {20075, 20076, 20}, {20247, 21272, 3212}, {21075, 41012, 27131}, {21627, 37709, 5175}, {22765, 32141, 6942}, {22836, 49169, 34619}, {22837, 49168, 34625}, {23891, 29400, 29715}, {23891, 29438, 29400}, {24349, 50284, 20090}, {24393, 38316, 18230}, {24599, 29621, 2}, {24883, 25650, 2}, {24936, 25446, 2}, {24968, 25479, 2}, {25415, 45287, 4295}, {25459, 25663, 2}, {25719, 40719, 31994}, {25746, 25805, 2}, {25773, 25843, 2}, {25879, 25965, 2}, {26029, 26093, 2}, {26038, 26102, 2}, {26395, 48493, 26394}, {26419, 48494, 26418}, {26495, 49402, 26494}, {26504, 49401, 26503}, {26514, 49060, 492}, {26515, 49061, 491}, {26531, 26658, 2}, {26626, 50079, 3661}, {26757, 26805, 2}, {26964, 27096, 2}, {26965, 27248, 2}, {26980, 27043, 2}, {27025, 27146, 2}, {27097, 27299, 2}, {27253, 27304, 2}, {27264, 27313, 2}, {27271, 29986, 30833}, {27627, 49984, 6048}, {28740, 41785, 2}, {29569, 40891, 16816}, {29571, 50019, 16833}, {29572, 29590, 2}, {29574, 49770, 4384}, {29584, 50079, 2}, {29586, 29593, 2}, {29815, 33090, 2}, {29829, 33175, 2}, {29830, 33139, 2}, {29831, 31079, 2}, {29839, 33137, 2}, {29966, 49774, 30057}, {30619, 30628, 193}, {30694, 39351, 10405}, {30941, 33296, 18600}, {30942, 50001, 10453}, {31162, 50864, 50687}, {31792, 34790, 392}, {32093, 32105, 4373}, {34595, 51105, 15808}, {34641, 51103, 19875}, {34718, 50824, 3524}, {34729, 34748, 34667}, {34772, 36845, 10529}, {34772, 38460, 1}, {35810, 35842, 485}, {35811, 35843, 486}, {36479, 36534, 2}, {36500, 36565, 2}, {36845, 41575, 20008}, {36846, 41575, 36845}, {37571, 51111, 2320}, {37794, 37795, 2}, {38098, 51108, 19876}, {38112, 51700, 3526}, {38314, 46930, 46934}, {38315, 49524, 3618}, {38315, 49690, 49524}, {39567, 49476, 29621}, {39581, 48856, 16830}, {42871, 49486, 32922}, {44635, 49232, 3068}, {44636, 49233, 3069}, {44720, 44722, 6555}, {45476, 49329, 492}, {45477, 49330, 491}, {45572, 48746, 45508}, {45573, 48747, 45509}, {46933, 46934, 2}, {48856, 50310, 2}, {49447, 49462, 192}, {49447, 49470, 49462}, {49447, 49709, 5698}, {49451, 49466, 8}, {49470, 49681, 49704}, {49470, 51192, 390}, {49476, 49495, 8}, {49476, 49771, 1}, {49490, 49493, 49491}, {49491, 49493, 24349}, {49497, 49534, 49707}, {49695, 49709, 49699}, {49699, 49709, 49704}, {49765, 50114, 17284}, {50017, 50286, 16816}, {50043, 50070, 51592}, {50805, 50818, 50872}, {50805, 50831, 50818}, {51068, 51105, 2}, {51069, 51091, 51071}, {51072, 51103, 2}


X(146) = REFLECTION OF X(20) IN X(110)

Trilinears       bc(-avw + bwu + cuv) : : , where u = u(A,B,C) = cos A - 2 cos B cos C, v = u(B,C,A), w = u(C,A,B)
Barycentrics  -avw + bwu + cuv : -bwu + cuv + avw : -cuv + avw + bwu
Barycentrics    a^10 + a^8 (b^2 + c^2) - a^6 (8 b^4 - 9 b^2 c^2 + 8 c^4) + 2 a^4 (b^2 + c^2) (4 b^4 - 7 b^2 c^2 + 4 c^4) - a^2 (b^2 - c^2)^2 (b^4 + 9 b^2 c^2 + c^4) - (b^2 - c^2)^4 (b^2 + c^2) : :

X(146) lies on the anticomplementary circle and these lines: 2,74   4,94   20,110   30,323   147,690   148,193

Let T be the P-Brocard triangle for any P on the Euler line. Let OA be the circle centered at the A-vertex of T and passing through A; define OB and OC cyclically. X(146) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

X(146) is the {X(74),X(113)}-harmonic conjugate of X(2). For a list of harmonic conjugates, click Tables at the top of this page.

X(146) = reflection of X(i) in X(j) for these (i,j): (20,110), (74,113), (265,1539)
X(146) = isogonal conjugate of X(34178)
X(146) = isotomic conjugate of anticomplement of X(36896)
X(146) = anticomplementary conjugate of X(30)
X(146) = X(74)-of-anticomplementary triangle
X(146) = crosspoint of X(399) and X(2935) wrt both the excentral and tangential triangles


X(147) = TARRY POINT OF ANTICOMPLEMENTARY TRIANGLE

Trilinears    bc[a8 + (b2 + c2)a6 - (2b4 + 3b2c2 + 2c4)a4 + (b6 + b4c2 + b2c4 + c6)a2 - b8 + b6c2 + b2c6 - c8] : :
Barycentrics    a sec(A + ω) - b sec(B + ω) - c sec(C + ω) : :

X(147) lies on the anticomplementary circle and these lines: 1,150   2,98   4,148   20,99   132,648   146,690   684,804

Let T be the P-Brocard triangle for any P on the Brocard axis (including the 1st Brocard triangle, for P = X(6)). Let OA be the circle centered at the A-vertex of T and passing through A; define OB and OC cyclically. X(147) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

X(147) is the {X(98),X(114)}-harmonic conjugate of X(2). For a list of harmonic conjugates, click Tables at the top of this page.

X(147) = reflection of X(i) in X(j) for these (i,j): (20,99), (98,114), (148,4), (385,1513)
X(147) = complement of X(5984)
X(147) = anticomplement of X(98)
X(147) = anticomplementary conjugate of X(511)
X(147) = X(325)-Ceva conjugate of X(2)
X(147) = anticomplementary isotomic conjugate of X(385)
X(147) = X(4) of 1st anti-Brocard triangle
X(147) = perspector of anticomplementary and 2nd Neuberg triangles
X(147) = perspector of 1st anti-Brocard and 2nd Neuberg triangles
X(147) = perspector of 2nd Neuberg triangle and cross-triangle of ABC and 1st Neuberg triangle
X(147) = X(98)-of-anticomplementary triangle
X(147) = isogonal conjugate of X(34130)
X(147) = isotomic conjugate of X(9473)


X(148) = STEINER POINT OF ANTICOMPLEMENTARY TRIANGLE

Trilinears    bc[a4 - (b2 - c2)2 + b2c2 - a2b2 - a2c2]
Barycentrics    a4 - (b2 - c2)2 + b2c2 - a2b2 - a2c2
X(148) = 3 X[2] - 4 X[115], 9 X[2] - 8 X[620], 5 X[2] - 4 X[2482], 7 X[2] - 8 X[5461], 15 X[2] - 16 X[6722], 5 X[2] - 6 X[9166], 13 X[2] - 12 X[9167], 9 X[2] - 10 X[14061], 11 X[2] - 12 X[14971], 7 X[2] - 4 X[15300], 17 X[2] - 16 X[22247], 21 X[2] - 20 X[31274], 21 X[2] - 16 X[35022], 2 X[3] - 3 X[14651], 3 X[4] - 2 X[6033], 7 X[4] - 2 X[14692], 5 X[4] - 4 X[22505], 3 X[4] - 4 X[22515], 3 X[13] - 2 X[32552], 3 X[14] - 2 X[32553], 3 X[99] - 4 X[620], X[99] - 3 X[671], 5 X[99] - 6 X[2482], 7 X[99] - 12 X[5461], 5 X[99] - 8 X[6722], 4 X[99] - 3 X[8591], 2 X[99] + 3 X[8596], 5 X[99] - 9 X[9166], 13 X[99] - 18 X[9167], 3 X[99] - 5 X[14061], 11 X[99] - 18 X[14971], 7 X[99] - 6 X[15300], 17 X[99] - 24 X[22247], 7 X[99] - 10 X[31274], 7 X[99] - 8 X[35022], 4 X[114] - 5 X[3091], 2 X[114] - 3 X[14639], 3 X[115] - 2 X[620], 2 X[115] - 3 X[671], 5 X[115] - 3 X[2482], 7 X[115] - 6 X[5461], 5 X[115] - 4 X[6722], 8 X[115] - 3 X[8591], 4 X[115] + 3 X[8596], 10 X[115] - 9 X[9166], 13 X[115] - 9 X[9167], 6 X[115] - 5 X[14061], 11 X[115] - 9 X[14971], 7 X[115] - 3 X[15300]

X(148) lies on the anticomplementary circle, the permutation ellipse E(X(4440)), the cubics K484, K895, K966, and these lines: {1, 6625}, {2, 99}, {3, 13172}, {4, 147}, {5, 7783}, {6, 11361}, {8, 13178}, {10, 13174}, {13, 617}, {14, 616}, {20, 98}, {22, 13175}, {23, 19577}, {30, 385}, {32, 6658}, {39, 16044}, {63, 11608}, {69, 5969}, {76, 2896}, {83, 7765}, {100, 13173}, {110, 16278}, {114, 3091}, {141, 7924}, {145, 7983}, {146, 193}, {149, 2787}, {150, 2786}, {151, 2792}, {152, 2784}, {153, 2783}, {183, 7833}, {187, 14568}, {190, 21711}, {192, 1478}, {230, 13586}, {257, 24851}, {265, 18331}, {274, 17685}, {287, 1562}, {315, 20081}, {316, 538}, {325, 14041}, {330, 1479}, {339, 35923}, {376, 11632}, {377, 27269}, {381, 7777}, {382, 7754}, {384, 5254}, {388, 3027}, {485, 35878}, {486, 35879}, {497, 3023}, {512, 31513}, {516, 9860}, {519, 9875}, {523, 9293}, {524, 8597}, {530, 3181}, {531, 3180}, {532, 25156}, {533, 25166}, {546, 38743}, {549, 17006}, {598, 33694}, {618, 5469}, {619, 5470}, {621, 23004}, {622, 23005}, {625, 7799}, {627, 11602}, {628, 11603}, {631, 33813}, {668, 6653}, {690, 3448}, {698, 5207}, {726, 20558}, {804, 25051}, {805, 6071}, {938, 24472}, {1003, 7806}, {1007, 33006}, {1029, 11611}, {1031, 37888}, {1078, 7756}, {1086, 26147}, {1131, 6462}, {1132, 6463}, {1270, 6320}, {1271, 6319}, {1281, 26117}, {1330, 9902}, {1506, 15031}, {1569, 2548}, {1654, 2796}, {1655, 2475}, {1656, 38229}, {1699, 21636}, {1975, 5025}, {1992, 9830}, {2023, 7738}, {2086, 9431}, {2170, 24504}, {2478, 27318}, {2640, 21089}, {2679, 14509}, {2785, 33650}, {2788, 34547}, {2789, 34548}, {2790, 34549}, {2791, 34550}, {2794, 3146}, {2797, 34186}, {2798, 34188}, {2799, 13219}, {2895, 35103}, {2936, 13595}, {2975, 22514}, {2998, 16098}, {3029, 9534}, {3044, 9544}, {3053, 33257}, {3068, 13657}, {3069, 13777}, {3085, 10086}, {3086, 10089}, {3090, 15561}, {3096, 7872}, {3125, 6650}, {3269, 30227}, {3314, 7841}, {3329, 8370}, {3407, 10336}, {3434, 13180}, {3436, 13181}, {3455, 37913}, {3522, 11623}, {3523, 6036}, {3524, 38739}, {3525, 34127}, {3528, 38731}, {3529, 36864}, {3545, 8724}, {3552, 3767}, {3571, 21725}, {3585, 25264}, {3616, 11711}, {3618, 5026}, {3620, 33210}, {3622, 11725}, {3627, 7762}, {3785, 32997}, {3815, 33013}, {3830, 7837}, {3832, 14981}, {3839, 6054}, {3849, 11054}, {3926, 7912}, {3933, 7885}, {3934, 7847}, {3972, 5309}, {4027, 5286}, {4240, 13179}, {4293, 10069}, {4294, 10053}, {5013, 16921}, {5023, 33268}, {5032, 8593}, {5046, 38499}, {5056, 23514}, {5059, 10991}, {5068, 36519}, {5071, 15092}, {5080, 17759}, {5139, 36898}, {5149, 7803}, {5182, 14928}, {5218, 15452}, {5225, 12185}, {5229, 12184}, {5283, 33030}, {5304, 12191}, {5305, 19687}, {5355, 12150}, {5459, 9114}, {5460, 9116}, {5475, 7757}, {5478, 36776}, {5503, 11148}, {5523, 15014}, {5601, 13176}, {5602, 13177}, {5731, 11710}, {5870, 12297}, {5871, 12296}, {5889, 21661}, {5938, 37902}, {5939, 7735}, {5976, 7791}, {5978, 31709}, {5979, 31710}, {5985, 6872}, {5986, 7500}, {5987, 7519}, {5988, 26051}, {5999, 38654}, {6055, 10304}, {6248, 37336}, {6298, 16809}, {6299, 16808}, {6337, 32961}, {6390, 7925}, {6542, 20349}, {6543, 9509}, {6560, 35825}, {6561, 35824}, {6626, 23897}, {6656, 17128}, {6721, 7486}, {6792, 34344}, {6960, 38556}, {7492, 13233}, {7533, 31088}, {7585, 19109}, {7586, 19108}, {7709, 37348}, {7714, 12132}, {7736, 11152}, {7737, 7766}, {7745, 7839}, {7746, 7782}, {7747, 7760}, {7750, 17129}, {7751, 7802}, {7752, 7781}, {7758, 7900}, {7763, 32966}, {7767, 19695}, {7768, 7842}, {7770, 7864}, {7773, 7906}, {7786, 33020}, {7789, 7901}, {7794, 7911}, {7795, 7933}, {7796, 7825}, {7798, 7812}, {7801, 7934}, {7804, 7827}, {7809, 7813}, {7811, 17131}, {7814, 35005}, {7816, 7828}, {7819, 7923}, {7822, 7918}, {7831, 9466}, {7832, 7861}, {7834, 19689}, {7840, 8352}, {7843, 7905}, {7846, 7902}, {7848, 14711}, {7851, 7892}, {7855, 7860}, {7856, 19693}, {7858, 32450}, {7863, 7899}, {7875, 11286}, {7876, 9478}, {7887, 7891}, {7897, 32833}, {7904, 33234}, {7907, 13881}, {7920, 14034}, {7921, 12830}, {7926, 11055}, {7928, 8357}, {7931, 33184}, {7932, 14001}, {7938, 32974}, {7941, 14044}, {7945, 14064}, {7946, 32006}, {7947, 14045}, {8029, 13187}, {8267, 8878}, {8289, 14033}, {8369, 16984}, {8371, 10556}, {8584, 10488}, {8592, 14482}, {8594, 22573}, {8595, 22574}, {8598, 8859}, {8781, 32831}, {8972, 8997}, {9143, 9144}, {9149, 36182}, {9180, 36955}, {9510, 35173}, {9535, 34454}, {9855, 22329}, {9865, 39266}, {9870, 10989}, {9903, 17499}, {9939, 33192}, {10163, 11056}, {10303, 20398}, {10352, 32971}, {10385, 12354}, {10449, 38481}, {10528, 13189}, {10529, 13190}, {11001, 14830}, {11053, 20998}, {11112, 16999}, {11113, 17000}, {11114, 16998}, {11160, 11161}, {11287, 16986}, {11317, 15484}, {11541, 38627}, {11641, 37915}, {12041, 14849}, {12066, 14223}, {12177, 14853}, {12258, 38314}, {12383, 18332}, {12829, 33280}, {13582, 18301}, {13584, 36857}, {13941, 13989}, {14588, 31644}, {14645, 20080}, {14683, 15342}, {14731, 31296}, {14850, 20304}, {14904, 34540}, {14905, 34541}, {14907, 33264}, {15022, 20399}, {15035, 33511}, {15398, 31655}, {15589, 33272}, {15680, 38557}, {15682, 19569}, {15717, 38737}, {15719, 26614}, {15720, 38635}, {15815, 33015}, {15903, 26109}, {15928, 37077}, {16001, 22113}, {16002, 22114}, {16041, 32817}, {16084, 35524}, {16613, 37128}, {16711, 26145}, {16990, 32986}, {16997, 17579}, {17035, 39120}, {17103, 23903}, {17389, 17778}, {17538, 38742}, {17565, 18140}, {17677, 31090}, {17702, 22265}, {18135, 33823}, {20533, 27295}, {20939, 21899}, {22601, 22603}, {22630, 22632}, {22735, 37190}, {23991, 33799}, {24505, 24715}, {24711, 36223}, {25332, 35965}, {26058, 27262}, {26124, 27312}, {26978, 33820}, {27040, 33831}, {31372, 31373}, {31400, 32962}, {31401, 33002}, {31404, 32995}, {31683, 31695}, {31684, 31696}, {32458, 32830}, {32488, 33340}, {32489, 33341}, {32816, 32996}, {32824, 33290}, {32828, 32965}, {32829, 32963}, {32832, 33004}, {32834, 33023}, {32836, 33278}, {32838, 33012}, {32839, 33270}, {32867, 33188}, {33008, 34229}, {33273, 37688}, {33274, 37637}, {33623, 35752}, {33625, 35749}, {33703, 36859}, {33824, 34284}, {35927, 37689}, {37182, 38642}, {37437, 38498}

X(148) = midpoint of X(i) and X(j) for these {i,j}: {2, 8596}, {3146, 5984}
X(148) = reflection of X(i) in X(j) for these {i,j}: {1, 11599}, {2, 671}, {4, 6321}, {8, 13178}, {20, 98}, {63, 11608}, {69, 11646}, {99, 115}, {110, 16278}, {145, 7983}, {147, 4}, {149, 10769}, {187, 32457}, {193, 10754}, {194, 1916}, {376, 11632}, {616, 14}, {617, 13}, {621, 23004}, {622, 23005}, {627, 11602}, {628, 11603}, {805, 6071}, {2896, 11606}, {3146, 10723}, {4240, 13179}, {5978, 31709}, {5979, 31710}, {5989, 5254}, {6033, 22515}, {6054, 9880}, {7779, 316}, {7840, 8352}, {8591, 2}, {8594, 22573}, {8595, 22574}, {8782, 76}, {9114, 5459}, {9116, 5460}, {9143, 9144}, {9855, 22329}, {9862, 12188}, {9890, 18546}, {10488, 8584}, {10992, 6036}, {11001, 14830}, {11148, 5503}, {11160, 11161}, {11177, 12243}, {12117, 6055}, {12383, 18332}, {13172, 3}, {13174, 10}, {13188, 5}, {14509, 2679}, {14588, 31644}, {14683, 15342}, {14712, 385}, {15300, 5461}, {17147, 11611}, {18331, 265}, {20094, 99}, {22507, 16002}, {22509, 16001}, {23235, 114}, {33265, 14568}, {33799, 23991}
X(148) = isogonal conjugate of X(9217)
X(148) = isotomic conjugate of X(35511)
X(148) = complement of X(20094)
X(148) = anticomplement of X(99)
X(148) = anticomplementary conjugate of X(512)
X(148) = X(523)-Ceva conjugate of X(2)
X(148) = X(2)-Hirst inverse of X(115)
X(148) = crosssum of PU(2)
X(148) = crosspoint of PU(40)
X(148) = polar conjugate of isogonal conjugate of X(22143)
X(148) = polar-circle-inverse of X(5186)
X(148) = orthoptic-circle-of-Steiner-inellipse-inverse of X(6719)
X(148) = orthoptic-circle-of-Steiner-circumellipe-inverse of X(111)
X(148) = de-Longchamps-circle-inverse of X(3565)
X(148) = inverse-in-Steiner-circumellipse of X(115)
X(148) = antitomic image of X(31632)
X(148) = anticomplement of the isogonal conjugate of X(512)
X(148) = anticomplement of the isotomic conjugate of X(523)
X(148) = isotomic conjugate of the anticomplement of X(31998)
X(148) = isotomic conjugate of the complement of X(31372)
X(148) = isotomic conjugate of the isogonal conjugate of X(20998)
X(148) = polar conjugate of the isogonal conjugate of X(22143)
X(148) = anticomplementary isogonal conjugate of X(512)
X(148) = psi-transform of X(32525)
X(148) = X(99)-of-anticomplementary triangle
X(148) = intersection of tangents at PU(40) to conic {A,B,C,PU(40)}} (i.e., the Steiner circumellipse)
X(148) = trilinear pole wrt anticomplementary triangle of line X(2)X(6)
X(148) = X(69)-of-1st-anti-Brocard-triangle
X(148) = center of conic through X(2), X(8), and the extraversions of X(8)
X(148) = pole of line X(115)X(125) wrt Steiner circumellipse
X(148) = X(i)-Ceva conjugate of X(j) for these (i,j): {523, 2}, {31632, 31998}, {31998, 10278}
X(148) = X(i)-cross conjugate of X(j) for these (i,j): {2640, 17085}, {10278, 31998}, {21089, 20939}, {21899, 2640}, {31998, 2}
X(148) = X(i)-isoconjugate of X(j) for these (i,j): {1, 9217}, {6, 9395}, {110, 9396}, {163, 9293}, {19610, 24041}
X(148) = cevapoint of X(i) and X(j) for these (i,j): {2, 31372}, {115, 13187}, {20998, 22143}, {21089, 21899}
X(148) = crosspoint of X(i) and X(j) for these (i,j): {523, 10278}, {31632, 31998}
X(148) = trilinear pole of line {10278, 11053}
X(148) = crossdifference of every pair of points on line {351, 10567}
X(148) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 512}, {2, 17217}, {4, 21300}, {6, 7192}, {10, 21301}, {19, 850}, {25, 7253}, {31, 523}, {32, 4560}, {37, 20295}, {42, 513}, {48, 6563}, {57, 4374}, {58, 17166}, {65, 21302}, {76, 21305}, {81, 17159}, {110, 21295}, {115, 21294}, {163, 99}, {213, 514}, {228, 20294}, {321, 21304}, {512, 8}, {513, 17135}, {514, 17137}, {523, 6327}, {560, 31296}, {604, 4467}, {647, 4329}, {649, 75}, {650, 20245}, {656, 1370}, {657, 18750}, {661, 69}, {662, 4576}, {663, 3869}, {667, 1}, {669, 192}, {688, 21217}, {692, 4427}, {693, 17138}, {741, 4155}, {798, 2}, {799, 670}, {810, 20}, {813, 874}, {822, 6527}, {850, 21275}, {872, 31290}, {875, 740}, {881, 17493}, {905, 18659}, {923, 690}, {1018, 668}, {1019, 17143}, {1042, 3900}, {1084, 21220}, {1096, 520}, {1245, 8678}, {1333, 17161}, {1334, 4462}, {1400, 693}, {1402, 522}, {1415, 17136}, {1459, 20243}, {1576, 6758}, {1577, 315}, {1821, 14295}, {1824, 20293}, {1918, 17494}, {1919, 17147}, {1924, 194}, {1967, 804}, {1973, 525}, {1974, 17498}, {1980, 17148}, {2054, 4010}, {2084, 2896}, {2155, 3265}, {2156, 3267}, {2159, 3268}, {2205, 21225}, {2258, 8672}, {2333, 4391}, {2350, 7199}, {2357, 4397}, {2485, 21288}, {2489, 5905}, {2501, 21270}, {2578, 22340}, {2579, 22339}, {2616, 2979}, {2623, 21271}, {2624, 1272}, {2642, 14360}, {2643, 3448}, {2786, 20560}, {3005, 21289}, {3049, 6360}, {3063, 63}, {3112, 688}, {3120, 21293}, {3121, 4440}, {3122, 149}, {3124, 21221}, {3125, 150}, {3223, 3221}, {3248, 17154}, {3402, 23878}, {3572, 30941}, {3669, 20244}, {3700, 21286}, {3708, 13219}, {3709, 329}, {3733, 17140}, {4010, 20554}, {4017, 3434}, {4024, 21287}, {4041, 3436}, {4077, 21280}, {4079, 2895}, {4117, 25054}, {4455, 17794}, {4516, 33650}, {4551, 3888}, {4557, 3952}, {4559, 21272}, {4705, 1330}, {4730, 21290}, {4770, 21291}, {4983, 2891}, {5029, 20538}, {5466, 21298}, {6010, 7257}, {6591, 17220}, {7121, 4367}, {7178, 21285}, {7180, 7}, {7216, 6604}, {7252, 21273}, {7649, 20242}, {8061, 1369}, {9178, 17491}, {9288, 3005}, {9426, 17486}, {9508, 20351}, {14407, 30578}, {18002, 32842}, {18105, 17165}, {18108, 17142}, {18197, 34086}, {18785, 3766}, {18793, 6373}, {20948, 33796}, {20979, 17149}, {21759, 649}, {21832, 20345}, {21837, 17732}, {22383, 17134}, {23345, 17145}, {23493, 4083}, {23503, 32747}, {23894, 316}, {24006, 11442}, {24290, 20552}, {28615, 4608}, {29055, 3903}, {32666, 3573}, {32676, 110}, {34072, 10330}, {34248, 669}
X(148) = barycentric product X(i)*X(j) for these {i,j}: {1, 20939}, {8, 17085}, {75, 2640}, {76, 20998}, {86, 21089}, {99, 10278}, {115, 31632}, {190, 21200}, {264, 22143}, {274, 21899}, {523, 31998}, {671, 11053}, {850, 9218}, {1577, 2644}
X(148) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 9395}, {6, 9217}, {523, 9293}, {661, 9396}, {2640, 1}, {2644, 662}, {3124, 19610}, {9218, 110}, {10278, 523}, {11053, 524}, {17085, 7}, {20939, 75}, {20998, 6}, {21089, 10}, {21200, 514}, {21899, 37}, {22143, 3}, {31632, 4590}, {31998, 99}
X(148) ={X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 20094, 99}, {4, 194, 7785}, {69, 33017, 7898}, {76, 6655, 2896}, {76, 7748, 6655}, {76, 7910, 7854}, {99, 115, 2}, {99, 671, 115}, {99, 14061, 620}, {99, 20094, 8591}, {114, 14639, 3091}, {115, 620, 14061}, {115, 2482, 6722}, {115, 6722, 9166}, {115, 15300, 31274}, {115, 31274, 5461}, {187, 32457, 14568}, {194, 7785, 13571}, {194, 32528, 147}, {381, 31859, 7777}, {382, 7754, 7823}, {384, 5254, 7797}, {384, 7797, 10583}, {620, 14061, 2}, {671, 8596, 8591}, {671, 16093, 111}, {1078, 7756, 33260}, {1506, 15031, 33024}, {1916, 32528, 7785}, {1975, 5025, 7836}, {2482, 9166, 2}, {2549, 11185, 2}, {3023, 13183, 497}, {3027, 13182, 388}, {3096, 7872, 19690}, {3146, 6392, 20065}, {3734, 7790, 2}, {3734, 11648, 7790}, {3926, 14063, 7912}, {3933, 33229, 7885}, {3934, 7847, 33021}, {5026, 6034, 3618}, {5254, 32819, 384}, {5286, 14035, 7787}, {5286, 32826, 14035}, {5461, 35022, 31274}, {6033, 6321, 22515}, {6033, 22515, 4}, {6036, 10992, 21166}, {6036, 21166, 3523}, {6054, 9880, 3839}, {6055, 12117, 10304}, {6390, 33228, 7925}, {7737, 7766, 34604}, {7746, 7782, 33259}, {7747, 7760, 20088}, {7816, 7828, 33225}, {7820, 7919, 2}, {7835, 7844, 2}, {7839, 14042, 7745}, {7872, 17130, 3096}, {7900, 20105, 7758}, {7906, 14062, 7773}, {8370, 15048, 3329}, {9862, 12188, 11177}, {9862, 12243, 12188}, {9878, 19570, 11177}, {10418, 30786, 2}, {13172, 14651, 3}, {14639, 23235, 114}, {14712, 19570, 385}, {14832, 16093, 2}, {14931, 32815, 8591}, {15300, 31274, 35022}, {15300, 35022, 99}, {17129, 33256, 7750}, {20081, 33019, 315}, {20099, 31125, 7665}, {26072, 26138, 2}, {26564, 26691, 2}, {26794, 26845, 2}, {27008, 27133, 2}, {27071, 27189, 2}


X(149) = REFLECTION OF X(20) IN X(104)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where
                        f(a,b,c) = bc[b3 + c3 - a3 + (a2 - bc)(b + c) + a(bc - b2 - c2)]

Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where
                        g(a,b,c) = b3 + c3 - a3 + (a2 - bc)(b + c) + a(bc - b2 - c2)

Let A' be the reflection of the midpoint of segment BC in X(11), and define B' and C' cyclically. The lines AA', BB', CC' concur in X(149). (Randy Hutson, 9/23/2011)

X(149) lies on the anticomplementary circle and these lines: 2,11   4,145   8,80   20,104   151,962   377,1058   404,496

X(149) is the {X(11),X(100)}-harmonic conjugate of X(2). For a list of harmonic conjugates, click Tables at the top of this page.

X(149) = reflection of X(i) in X(j) for these (i,j): (3,1484), (8,80), (20,104), (100,11), (144,1156), (145,1320), (153,4)
X(149) = isogonal conjugate of X(3446)
X(149) = isotomic conjugate of X(8047)
X(149) = anticomplementary conjugate of X(513)
X(149) = trilinear pole wrt anticomplementary triangle of line X(7)X(8)
X(149) = center of conic through X(7), X(8), and the extraversions of X(8)
X(149) = X(100)-of-anticomplementary-triangle
X(149) = homothetic center of 2nd Schiffler triangle and polar triangle of AC-incircle


X(150) = REFLECTION OF X(20) IN X(103)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where
                        f(a,b,c) = bc[b4 + c4 - a4 + a(bc2 +cb2 - b3 - c3) - bc(a2 + b2 + c2) + (b + c)a3]

Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where
                        g(a,b,c) = b4 + c4 - a4 + a(bc2 +cb2 - b3 - c3) - bc(a2 + b2 + c2) + (b + c)a3

X(150) = X(101)-of-anticomplementary triangle

X(150) lies on the anticomplementary circle and these lines: 1,147   2,101   4,152   7,80   20,103   69,668   85,355   295,334   348,944   664,952

X(150) = reflection of X(i) in X(j) for these (i,j): (20,103), (101,116), (152,4), (664,1565)
X(150) = isogonal conjugate of X(34179)
X(150) = isotomic conjugate of isogonal conjugate of X(20999)
X(150) = isotomic conjugate of anticomplement of X(39026)
X(150) = polar conjugate of isogonal conjugate of X(22145)
X(150) = anticomplementary conjugate of X(514)


X(151) = REFLECTION OF X(20) IN X(109)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where
                        f(a,b,c) = (by + cz - ax)/a, where x : y : z = X(102)
Barycentrics    a^12 + a^11 (b + c) + a^10 b c + a^9 (3 b^3 - b^2 c - b c^2 + 3 c^3) - a^8 (3 b^4 + b^3 c + b c^3 + 3 c^4) - 2 a^7 (7 b^5 + b^4 c - 4 b^3 c^2 - 4 b^2 c^3 + b c^4 + 7 c^5) - 2 a^6 b c (b^4 - 2 b^3 c - 6 b^2 c^2 - 2 b c^3 + c^4) + 2 a^5 (7 b^7 + b^6 c - 5 b^5 c^2 - 3 b^4 c^3 - 3 b^3 c^4 - 5 b^2 c^5 + b c^6 + 7 c^7) + a^4 (3 b^8 + 2 b^7 c - 4 b^6 c^2 - 2 b^5 c^3 + 2 b^4 c^4 - 2 b^3 c^5 - 4 b^2 c^6 + 2 b c^7 + 3 c^8) - a^3 (3 b^9 - b^8 c + 8 b^6 c^3 - 10 b^5 c^4 - 10 b^4 c^5 + 8 b^3 c^6 - b c^8 + 3 c^9) + a^2 b c (b^8 - 4 b^7 c - 12 b^6 c^2 + 4 b^5 c^3 + 22 b^4 c^4 + 4 b^3 c^5 - 12 b^2 c^6 - 4 b c^7 + c^8) - a (b^11 + b^10 c - 3 b^9 c^2 - 3 b^8 c^3 + 2 b^7 c^4 + 2 b^6 c^5 + 2 b^5 c^6 + 2 b^4 c^7 - 3 b^3 c^8 - 3 b^2 c^9 + b c^10 + c^11) - b^12 - b^11 c + 4 b^10 c^2 + 3 b^9 c^3 - 7 b^8 c^4 - 2 b^7 c^5 + 8 b^6 c^6 - 2 b^5 c^7 - 7 b^4 c^8 + 3 b^3 c^9 + 4 b^2 c^10 - b c^11 - c^12 : :

X(151) lies on the anticomplementary circle and these lines: 2,102   20,109   49,962   152,928

X(151) = reflection of X(i) in X(j) for these (i,j): (20,109), (102,117)
X(151) = isogonal conjugate of X(34180)
X(151) = anticomplementary conjugate of X(515)


X(152) = REFLECTION OF X(20) IN X(101)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where
                        f(a,b,c) = (by + cz - ax)/a, where x : y : z = X(103)
Barycentrics    a^8 - a^7 (b + c) + a^6 (2 b^2 + b c + 2 c^2) - a^5 (5 b^3 - b^2 c - b c^2 + 5 c^3) - a^4 b c (b^2 - 6 b c + c^2) + a^3 (5 b^5 + b^4 c - 6 b^3 c^2 - 6 b^2 c^3 + b c^4 + 5 c^5) - a^2 (2 b^6 + b^5 c + 2 b^4 c^2 - 10 b^3 c^3 + 2 b^2 c^4 + b c^5 + 2 c^6) + a (b^7 - b^6 c - 3 b^5 c^2 + 3 b^4 c^3 + 3 b^3 c^4 - 3 b^2 c^5 - b c^6 + c^7) - b^8 + b^7 c + 2 b^6 c^2 - b^5 c^3 - 2 b^4 c^4 - b^3 c^5 + 2 b^2 c^6 + b c^7 - c^8 : :

X(152) lies on the anticomplementary circle and these lines: 2,103   4,150   20,101   151,928

X(152) = reflection of X(i) in X(j) for these (i,j): (20,101), (103,118), (150,4)
X(152) = isogonal conjugate of X(34181)
X(152) = anticomplementary conjugate of X(516)
X(152) = X(103)-of-anticomplementary triangle


X(153) = REFLECTION OF X(20) IN X(100)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where
                        f(a,b,c) = (by + cz - ax)/a, where x : y : z = X(104)

Barycentrics    a^7 - a^6 (b + c) - a^5 (b^2 - 7 b c + c^2) + a^4 (b + c) (b^2 - 6 b c + c^2) - a^3 (b^4 + 2 b^3 c - 10 b^2 c^2 + 2 b c^3 + c^4) + a^2 (b - c)^2 (b + c) (b^2 + 6 b c + c^2) + a (b^2 - c^2)^2 (b^2 - 5 b c + c^2) - (b - c)^4 (b + c)^3 : :

Let T be the P-Brocard triangle for any P on line X(1)X(3). Let OA be the circle centered at the A-vertex of T and passing through A; define OB and OC cyclically. X(153) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

X(153) lies on the anticomplementary circle and these lines: 2,104   4,145   7,80   11,388   20,100   515,908

X(153) = reflection of X(i) in X(j) for these (i,j): (20,100), (104,119), (149,4), (1320,1537)
X(153) = isogonal conjugate of X(34182)
X(153) = anticomplementary conjugate of X(517)
X(153) = X(104)-of-anticomplementary triangle


X(154) = X(3)-CEVA CONJUGATE OF X(6)

Trilinears    (cos A - cos B cos C)a2 : (cos B - cos C cos A)b2 : (cos C - cos A cos B)c2
Trilinears    a(tan B + tan C - tan A) : b(tan C + tan A - tan B) : c(tan A + tan B - tan C)
Trilinears    (sec A - sec B sec C)a^2 : :
Barycentrics    a^2(3a^4 - b^4 - c^4 - 2a^2b^2 - 2a^2c^2 + 2b^2c^2) : :
Barycentrics    (sin2 A)(tan B + tan C - tan A) : :

Let A'B'C' be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung). A'B'C' is homothetic to the Ara triangle at X(154).

X(154) lies on these lines: 3,64   6,25   22,110   26,155   31,56   48,55   160,418   197,692   198,212   205,220   237,682

X(154) is the {X(26),X(156)}-harmonic conjugate of X(155). For a list of harmonic conjugates of X(154), click Tables at the top of this page.

X(154) = isogonal conjugate of X(253)
X(154) = complement of X(32064)
X(154) = X(3)-Ceva conjugate of X(6)
X(154) = crosssum of X(i) and X(j) for these (i,j): (64,1073), (122,525)
X(154) = X(109)-beth conjugate of X(154)
X(154) = X(2)-of-tangential triangle
X(154) = centroid of cevian triangle of X(20)
X(154) = pole wrt circumcircle of the trilinear polar of X(3)
X(154) = Thomson isogonal conjugate of X(4)
X(154) = intouch-to-ABC functional image of X(2)


X(155) = EIGENCENTER OF ORTHIC TRIANGLE

Trilinears    (cos A)[cos2B + cos2C - cos2A] : :
Barycentrics   a^2 SA [a^2 (SA^2 - SB SC) - SA (SB^2 + SC^2)] : :
Barycentrics    a^2 (a^2 - b^2 - c^2) (a^6 - 3 a^4 (b^2 + c^2) + a^2 (3 b^4 - 2 b^2 c^2 + 3 c^4) - (b^2 - c^2)^2 (b^2 + c^2)) : :
X(155) = 4 R^2 X[5] - SW X[6], (J^2 - 3) X[3] - 2(J^2 - 2) X[49], (J^2 + 1) X[24] - 2 J^2 X[110], (J^2 + 3) X[25] - 4 X[52]
3 X[2] - 4 X[9820],X[3] - 3 X[3167],3 X[3] - 2 X[7689],3 X[3] - 4 X[12038],X[3] + 2 X[15083],2 X[5] - 3 X[5654],2 X[5] + X[9936],4 X[5] - 3 X[14852],3 X[6] - 2 X[8548],2 X[26] - 3 X[154],3 X[51] - 2 X[12235],3 X[51] - X[21651],X[68] - 3 X[5654],2 X[68] - 3 X[14852],4 X[110] - X[15085],3 X[154] - 4 X[156],3 X[154] - X[17834],4 X[156] - X[17834],3 X[381] - 4 X[5448],3 X[381] - 2 X[9927],3 X[381] - X[12429],2 X[1147] - 3 X[3167],3 X[1147] - X[7689],3 X[1147] - 2 X[12038],4 X[1147] - X[12163],2 X[1147] + X[12164],X[1498] + 2 X[16266],5 X[1656] - 4 X[5449],4 X[1658] - 5 X[17821],3 X[1853] - 4 X[13371],3 X[1853] - 2 X[32140],3 X[3060] + X[12271],9 X[3167] - 2 X[7689],9 X[3167] - 4 X[12038],6 X[3167] - X[12163],3 X[3167] + X[12164],3 X[3167] + 2 X[15083],5 X[3567] - X[12282],9 X[5054] - 8 X[20191],3 X[5093] - X[6391],4 X[5448] - X[12429],3 X[5587] - X[9896],3 X[5654] + X[9936],3 X[5886] - 2 X[12259],2 X[6193] + X[12293],X[6193] + 2 X[22660],4 X[7689] - 3 X[12163],2 X[7689] + 3 X[12164],X[7689] + 3 X[15083],X[8548] - 3 X[19139],5 X[8567] - 4 X[32138],4 X[9820] - X[11411],3 X[9909] - 5 X[14530],2 X[9925] + X[11477],2 X[9936] + 3 X[14852],4 X[10282] - 3 X[14070],3 X[10606] - 4 X[11250],3 X[11206] - X[31305],4 X[11255] - 3 X[17813],8 X[12038] - 3 X[12163],4 X[12038] + 3 X[12164],2 X[12038] + 3 X[15083],X[12163] + 2 X[12164],X[12163] + 4 X[15083],X[12293] - 4 X[22660],2 X[12893] - 3 X[32609],4 X[13561] - 5 X[31283],3 X[18324] - 4 X[32171],4 X[18377] - 3 X[18405],5 X[19132] - 4 X[19154]

Let (A) be the pedal circle of A wrt the tangential triangle, and define (B), (C) cyclically. The radical center of (A), (B), (C) = X(155). (Randy Hutson, December 10, 2016)

X(155) = X(4)-of-tangential-triangle. This point is also the center of the circle which cuts (extended) lines BC, CA, AB in pairs of points A' and A″, B' and B″, C' and C″, respectively, such that angles A'AA″, B'BB″, C'CC″ are all right angles. This is the Dou circle, described in Jordi Dou, Problem 1140, Crux Mathematicorum, 28 (2002) 461-462.

Let A' be the isogonal conjugate of A with respect to the triangle BCX(4), and define B' and C' cyclically. Let A''B''C'' be the orthic triangle. Then the lines A'A'', B'B'', C'C'' concur in X(155). (Randy Hutson, 9/23/2011)

Let OA be the circle centered at the A-vertex of the 2nd anti-extouch triangle and passing through A; define OB and OC cyclically. X(155) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

X(155) lies on the following curves: Feuerbach hyperbola of T, Jerabek hyperbola of T, Kiepert circumhyperbola of T, where T = tangential triangle; the Feuerbach circumhyperbola of the orthic triangle; the cubics K006, K044, K176, K339, K364, K389, K500, K742; and on these lines: {1, 90}, {2, 7592}, {3, 49}, {4, 254}, {5, 6}, {11, 10071}, {12, 10055}, {19, 12417}, {20, 323}, {22, 1614}, {24, 110}, {25, 52}, {26, 154}, {30, 1498}, {33, 9931}, {34, 19471}, {51, 7529}, {54, 7503}, {55, 6237}, {56, 7352}, {64, 2935}, {69, 3547}, {81, 6824}, {113, 19504}, {125, 19360}, {136, 20422}, {140, 17811}, {143, 13861}, {159, 511}, {182, 7393}, {193, 3089}, {195, 381}, {215, 9672}, {219, 26921}, {221, 14988}, {222, 24467}, {225, 8757}, {265, 15317}, {275, 19196}, {287, 28695}, {343, 3549}, {355, 12422}, {371, 8909}, {372, 8943}, {378, 9938}, {382, 399}, {389, 6642}, {403, 2904}, {427, 23307}, {450, 1075}, {454, 6503}, {517, 9928}, {524, 16252}, {525, 8151}, {542, 8549}, {550, 16936}, {568, 7506}, {569, 5891}, {576, 8681}, {578, 5907}, {610, 2323}, {631, 15032}, {648, 1093}, {651, 1068}, {858, 11457}, {916, 8053}, {940, 6862}, {952, 9933}, {1060, 19349}, {1062, 19354}, {1160, 8904}, {1161, 8903}, {1199, 3090}, {1209, 32341}, {1350, 2916}, {1351, 1598}, {1368, 18914}, {1478, 18970}, {1479, 12428}, {1493, 15060}, {1495, 9714}, {1503, 14790}, {1593, 11472}, {1594, 11442}, {1596, 13142}, {1611, 10011}, {1616, 19907}, {1620, 15646}, {1625, 2207}, {1656, 5449}, {1658, 17821}, {1740, 3072}, {1829, 7497}, {1839, 7534}, {1853, 13371}, {1899, 11585}, {1974, 19141}, {1994, 3091}, {1995, 3567}, {2063, 18913}, {2072, 25738}, {2192, 8144}, {2477, 9659}, {2914, 11271}, {2929, 6102}, {2930, 5609}, {2979, 10323}, {2990, 23697}, {3047, 12219}, {3053, 32661}, {3060, 6152}, {3098, 15606}, {3197, 8141}, {3216, 6911}, {3289, 23115}, {3435, 10692}, {3448, 23306}, {3511, 19597}, {3517, 8780}, {3518, 16880}, {3527, 5093}, {3542, 6515}, {3546, 18909}, {3548, 11064}, {3580, 7505}, {3627, 15811}, {3628, 17825}, {3629, 15873}, {3832, 11004}, {3851, 14627}, {4383, 6959}, {4846, 31829}, {5012, 7509}, {5020, 5462}, {5050, 10170}, {5054, 20191}, {5070, 15037}, {5072, 15038}, {5085, 7516}, {5095, 16534}, {5374, 31386}, {5398, 16471}, {5412, 12424}, {5413, 12425}, {5453, 6914}, {5587, 9896}, {5706, 6917}, {5876, 7526}, {5886, 12259}, {5890, 17928}, {5898, 12310}, {5972, 12227}, {6000, 12085}, {6090, 9730}, {6146, 12420}, {6212, 19216}, {6213, 19215}, {6241, 11413}, {6243, 7517}, {6623, 32605}, {6643, 6776}, {6756, 31802}, {6800, 7512}, {6804, 14912}, {6944, 32911}, {7074, 32141}, {7391, 16659}, {7401, 14826}, {7404, 11427}, {7484, 13336}, {7485, 7999}, {7486, 15018}, {7487, 15741}, {7488, 9544}, {7507, 18474}, {7514, 11591}, {7527, 9716}, {7552, 12325}, {7553, 31383}, {7723, 19457}, {8200, 12415}, {8207, 12416}, {8220, 12426}, {8221, 12427}, {8227, 16472}, {8251, 19350}, {8276, 12239}, {8277, 12240}, {8538, 10602}, {8541, 9926}, {8567, 32138}, {8718, 33524}, {8954, 10962}, {8976, 13909}, {9019, 15581}, {9545, 14118}, {9645, 10535}, {9705, 11464}, {9715, 26864}, {9815, 10127}, {9909, 14530}, {9923, 9996}, {9967, 19459}, {10112, 18390}, {10224, 18356}, {10255, 32539}, {10257, 26937}, {10282, 14070}, {10575, 12174}, {10606, 11250}, {10620, 12901}, {10625, 11414}, {10628, 12412}, {10634, 19363}, {10635, 19364}, {10641, 10659}, {10642, 10660}, {10796, 12193}, {10897, 19355}, {10898, 19356}, {10942, 12430}, {10943, 12431}, {11206, 31305}, {11249, 23361}, {11255, 17813}, {11265, 17819}, {11266, 17820}, {11267, 17826}, {11268, 17827}, {11403, 16194}, {11422, 13434}, {11424, 15030}, {11426, 11479}, {11433, 18934}, {11449, 32534}, {11499, 12328}, {11557, 22550}, {11560, 12168}, {11562, 12165}, {11819, 31815}, {11898, 19362}, {12082, 23061}, {12112, 33703}, {12233, 18420}, {12290, 14094}, {12315, 14915}, {12319, 14683}, {12358, 13198}, {12362, 31804}, {12601, 19461}, {12602, 19462}, {12603, 19463}, {12604, 19464}, {12605, 19467}, {12606, 19468}, {12825, 15463}, {12893, 32609}, {12902, 19479}, {13039, 19465}, {13040, 19466}, {13321, 18369}, {13561, 31283}, {13567, 18951}, {13951, 13970}, {14128, 32136}, {14216, 23335}, {14480, 15112}, {15033, 15058}, {15106, 15115}, {15750, 32110}, {15800, 32332}, {15818, 31807}, {16618, 16789}, {17849, 32428}, {17975, 20764}, {18324, 32171}, {18377, 18405}, {18396, 18404}, {18534, 26883}, {19125, 19131}, {19132, 19154}, {19170, 19179}, {19180, 19210}, {19189, 19194}, {19358, 19428}, {19359, 19429}, {19446, 19486}, {19447, 19487}, {19460, 22834}, {22120, 22146}, {22808, 22953}, {23070, 23144}, {23294, 30744}, {26913, 31282}, {26920, 26922}, {26944, 30771}, {32153, 34046}, {32251, 32275}

X(155) = midpoint of X(i) and X(j) for these {i,j}: {3, 12164}, {4, 6193}, {68, 9936}, {1147, 15083}, {1351, 19588}, {6237, 6238}, {12160, 12166}, {12319, 14683}, {12422, 12423}, {16266, 32139}, {17838, 17847}
X(155) = reflection of X(i) in X(j) for these {i,j}: {3, 1147}, {4, 22660}, {6, 19139}, {26, 156}, {64, 12084}, {68, 5}, {1498, 32139}, {2931, 110}, {3448, 23306}, {7387, 6759}, {7689, 12038}, {9927, 5448}, {10620, 12901}, {11411, 12359}, {12085, 13346}, {12163, 3}, {12164, 15083}, {12293, 4}, {12302, 5504}, {12359, 9820}, {12421, 13292}, {12429, 9927}, {12902, 19479}, {14216, 23335}, {14852, 5654}, {15085, 2931}, {15136, 3292}, {16003, 15115}, {17834, 26}, {18356, 10224}, {19458, 12161}, {21651, 12235}, {32140, 13371}
X(155) = isogonal conjugate of X(254)
X(155) = isotomic conjugate of the polar conjugate of X(1609)
X(155) = X(4)-of-tangential-triangle
X(155) = complement of X(11411)
X(155) = anticomplement of X(12359)
X(155) = circumcircle-inverse of X(12095)
X(155) = polar-circle inverse of X(16172)
X(155) = isogonal conjugate of the polar conjugate of X(6515)
X(155) = polar conjugate of the isotomic conjugate of X(6503)
X(155) = excentral-isogonal conjugate of X(2960)
X(155) = tangential-isogonal conjugate of X(26)
X(155) = orthic-isogonal conjugate of X(3)
X(155) = X(155) = X(i)-Ceva conjugate of X(j) for these (i,j): {4, 3}, {1993, 6}, {3193, 1}, {3542, 454}, {6193, 9937}, {6515, 1609}, {11441, 1498}, {12134, 2918}, {14516, 2917}, {15242, 18126}
X(155) = X(i)-isoconjugate of X(j) for these (i,j): {1, 254}, {4, 921}, {19, 6504}, {158, 15316}, {2190, 8800}, {13398, 24006}
X(155) = crosspoint of X(4) and X(3542)
X(155) = crosssum of X(i) and X(j) for these (i,j): {3, 15316}, {4, 3147}, {136, 523}, {647, 8754}
X(155) = crossdifference of every pair of points on line {924, 2501}
X(155) = eigencenter of cevian triangle of X(4)
X(155) = eigencenter of anticevian triangle of X(3)
X(155) = X(84)-of-orthic triangle if ABC is acute
X(155) = perspector of orthic triangle and tangential triangle of the MacBeath circumconic, which is also the anticevian triangle of X(3)
X(155) = perspector of orthic triangle and cross-triangle of ABC and 2nd Hyacinth triangle
X(155) = orthic-to-ABC barycentric image of X(3)
X(155) = intouch-to-ABC functional image of X(4)
X(155) = barycentric product X(i)*X(j) for these {i,j}: {3, 6515}, {4, 6503}, {48, 33808}, {63, 920}, {69, 1609}, {343, 8883}, {394, 3542}, {454, 6504}, {3580, 15478}
X(155) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 6504}, {6, 254}, {48, 921}, {216, 8800}, {454, 6515}, {577, 15316}, {920, 92}, {1609, 4}, {3003, 16172}, {3542, 2052}, {6503, 69}, {6515, 264}, {8883, 275}, {15478, 2986}, {18126, 13579}, {32661, 13398}, {33808, 1969}
X(155) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 11411, 12359}, {3, 49, 19357}, {3, 3167, 1147}, {3, 18445, 1181}, {4, 11441, 18451}, {5, 68, 14852}, {5, 12161, 6}, {5, 15068, 17814}, {5, 31831, 1352}, {6, 17814, 5}, {6, 17836, 19458}, {24, 9932, 2931}, {25, 12160, 52}, {25, 12166, 9937}, {26, 156, 154}, {49, 18436, 3}, {51, 21651, 12235}, {52, 10539, 25}, {54, 11459, 7503}, {68, 5654, 5}, {110, 5889, 24}, {143, 13861, 17810}, {154, 17834, 26}, {182, 11793, 7393}, {184, 5562, 3}, {185, 1092, 3}, {185, 3292, 1092}, {381, 12429, 9927}, {389, 9306, 6642}, {394, 1181, 3}, {485, 486, 9722}, {568, 18350, 7506}, {569, 5891, 7395}, {578, 5907, 9818}, {1069, 3157, 1}, {1147, 7689, 12038}, {1147, 12164, 12163}, {1147, 19908, 19357}, {1199, 3090, 5422}, {1351, 1598, 5446}, {1593, 12162, 11472}, {1598, 19588, 12309}, {1614, 11412, 22}, {1993, 6193, 15316}, {1993, 11441, 4}, {3167, 12164, 3}, {3167, 15083, 12163}, {3917, 10984, 3}, {5012, 11444, 7509}, {5020, 11432, 5462}, {5448, 9927, 381}, {5562, 9908, 12163}, {5654, 9936, 68}, {6102, 6644, 9786}, {6193, 22660, 12293}, {6243, 7517, 33586}, {6243, 10540, 7517}, {7395, 11402, 569}, {7488, 9544, 9707}, {7689, 12038, 3}, {9820, 12359, 2}, {10661, 10662, 6}, {10665, 10666, 6}, {11422, 15056, 13434}, {11591, 32046, 7514}, {12161, 15068, 5}, {12162, 13352, 1593}, {12174, 21312, 10575}, {13371, 32140, 1853}, {14516, 22661, 12293}, {15068, 19139, 5654}, {15316, 18451, 12293}, {17814, 17836, 68}, {17814, 19458, 14852}, {18436, 19908, 12163}


X(156) = X(5)-OF-TANGENTIAL-TRIANGLE

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[b2y/v + c2z/w - a2x/u],
                        u = u(A,B,C) = sin 2A, v = u(B,C,A), w = u(C,A,B);
                        x = x(A,B,C) = u2(v2 + w2) - (v2 - w2)2, y = x(B,C,A), z = x(C,A,B)
Barycentrics    a^2 (a^8 - 3 a^6 (b^2 + c^2) + a^4 (3 b^4 + 2 b^2 c^2 + 3 c^4) - a^2 (b^6 + c^6) + b^2 c^2 (b^2 - c^2)^2) : :

Let OA be the reflection of X(3) in BC, and define OB and OC cyclically. Let O'A be the circumcenter of AOBOC, and define O'B and O'C cyclically. X(156) is the circumcenter of O'AO'BO'C. (Randy Hutson, July 11, 2019)

Let A'B'C' be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung). Then X(156) = X(5)-of-A'B'C'. (Randy Hutson, July 11, 2019)

X(156) lies on these lines: 3,74   4,49   5,184   25,143   26,154   54,381   546,578   550,1092

X(156) is the {X(154),X(155)}-harmonic conjugate of X(26). For a list of harmonic conjugates, click Tables at the top of this page.

X(156) = midpoint of X(26) and X(155)
X(156) = X(5)-of-tangential-triangle
X(156) = {X(110),X(1614)}-harmonic conjugate of X(3)
X(156) = intouch-to-ABC functional image of X(5)


X(157) = X(6)-OF-TANGENTIAL-TRIANGLE

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[b3cos B + c3cos C - a3cos A]
                        = g(a,b,c) : g(b,c,a) : g(c,a,b), where
                        g(a,b,c) = a[a6 - b6 - c6 - a4(b2 + c2) + b4(c2 + a2) + c4(a2 + b2)]

Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(157) lies on these lines: 3,66   6,248   22,183   25,53   161,418   206,216

X(157) = isogonal conjugate of anticomplement of X(22391)
X(157) = isotomic conjugate of isogonal conjugate of X(2909)
X(157) = X(264)-Ceva conjugate of X(6)
X(157) = crosssum of X(127) and X(520)
X(157) = X(6)-of-tangential-triangle
X(157) = perspector of circumcircle wrt Schroeter triangle
X(157) = perspector of polar circle wrt tangential triangle
X(157) = intouch-to-ABC functional image of X(6)


X(158) = X(19)-CROSS CONJUGATE OF X(92)

Trilinears       sec2A : sec2B : sec2C
                        = 1/(1 + cos 2A) : 1/(1 + cos 2B) : 1/(1 + cos 2C)

Barycentrics  sec A tan A : sec B tan B : sec C tan C

Let A'B'C' be the hexyl triangle. Let A″ be the pole, wrt the polar circle, of line B'C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(158). (Randy Hutson, October 15, 2018)

Let A'B'C' and A″B″C″ be the Euler and anti-Euler triangles, resp. Let A* be the trilinear product A'*A″, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(158). (Randy Hutson, October 15, 2018)

X(158) lies on these lines: 1,29   3,243   4,65   10,318   37,281   46,412   47,162   75,240   107,759   225,1093   255,775   286,969   823,897   920,921   1068,3542

X(158) = isogonal conjugate of X(255)
X(158) = isotomic conjugate of X(326)
X(158) = X(i)-cross conjugate of X(j) for these (i,j): (19,92), (225,4)
X(158) = crosssum of X(520) and X(1364)
X(158) = crossdifference of every pair of points on line X(680)X(822)
X(158) = X(i)-aleph conjugate of X(j) for these (i,j): (821,158), (1105,255)
X(158) = X(107)-beth conjugate of X(34)
X(158) = trilinear pole of polar of X(63) wrt polar circle (line X(661)X(3064))
X(158) = pole wrt polar circle of trilinear polar of X(63) (line X(521)X(656))
X(158) = polar conjugate of X(63)
X(158) = trilinear product of X(1123) and X(1336)
X(158) = trilinear square of X(4)


X(159) = X(9)-OF-TANGENTIAL-TRIANGLE

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where
                        f(a,b,c) = a[(a2 + b2 + c2)sin 2A + (c2 - b2 - a2)sin 2B + (b2 - c2 - a2)sin 2C]

Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(159) = X(9)-of-tangential triangle if ABC is acute

X(159) lies on these lines: 3,66   6,25   22,69   23,193   155,511   197,200

X(159) = reflection of X(i) in X(j) for these (i,j): (6,206), (66,141)
X(159) = isogonal conjugate of X(13575)
X(159) = crosspoint of X(110) and X(15388)
X(159) = crossdifference of every pair of points on line X(525)X(2485)
X(159) = trilinear product X(i)*X(j) for these {i,j}: {31, 1370}, {38, 8793}, {63, 3162}
X(159) = complement of X(36851)
X(159) = X(i)-Ceva conjugate of X(j) for these (i,j): (22,3), (69,6)
X(159) = crosssum of X(i) and X(j) for these {i,j}: {2, 7500}, {127, 523}
X(159) = tangential-isogonal conjugate of X(25)
X(159) = tangential-isotomic conjugate of X(32445)
X(159) = intouch-to-ABC functional image of X(9)


X(160) = X(37)-OF-TANGENTIAL-TRIANGLE

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where
                        f(a,b,c) = a[(b2 + c2)sin 2A + (c2 - a2)sin 2B + (b2 - a2)sin 2C]

Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(160) lies on these lines: 3,66   6,237   22,325   95,327   154,418   206,57

X(160) = isogonal conjugate of anticomplement of complementary conjugate of X(34845)
X(160) = anticomplement of X(34845)
X(160) = X(95)-Ceva conjugate of X(6)
X(160) = crosssum of X(338) and X(512)
X(160) = X(37)-of-tangential triangle if ABC is acute


X(161) = X(63)-OF-TANGENTIAL-TRIANGLE

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where
                        f(a,b,c) = a[(a2 + b2 + c2)sin2(2A) + (c2 - b2 - a2)sin2(2B) + (b2 - c2 - a2)sin2(2C)]

Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(161) = X(63)-of-tangential triangle if ABC is acute

X(161) lies on these lines: 6,25   22,343   26,68   157,418

X(161) = X(68)-Ceva conjugate of X(6)


X(162) = CEVAPOINT OF X(108) AND X(109)

Trilinears    1/(sin 2B - sin 2C) : 1/(sin 2C - sin 2A) : 1/(sin 2A - sin 2B)
Trilinears    1/[(b2 - c2)(b2 + c2 - a2)]
Trilinears    sec A csc(B - C) : :
Trilinears    1/(tan B - tan C) : :
Barycentrics  a/(sin 2B - sin 2C) : b/(sin 2C - sin 2A) : c/(sin 2A - sin 2B)

Let La be the A-extraversion of line X(1)X(19), and define Lb and Lc cyclically. Let A' = Lb ∩ Lc, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(162). (Randy Hutson, January 29, 2018)

Let La be the A-extraversion of line X(8)X(29), and define Lb and Lc cyclically. Let A' = Lb ∩ Lc, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(162). (Randy Hutson, January 29, 2018)

Let La be the A-extraversion of line X(9)X(21), and define Lb and Lc cyclically. Let A' = Lb ∩ Lc, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(162). (Randy Hutson, January 29, 2018)

See (and hear) Dan Reznik's Dance of the Swans: X(88) and X(162) and their Never-Touching Motion Over the Elliptic Billiard (March 4, 2020)

X(162) lies on these lines: 4,270   6,1013   19,897   27,673   28,88   29,58   31,92   47,158   63,204   100,112   107,109   108,110   190,643   238,415   240,896   242,422   255,1099   412,580   799,811

X(162) = isogonal conjugate of X(656)
X(162) = isotomic conjugate of X(14208)
X(162) = complement of X(34846)
X(162) = X(2)-Ceva conjugate of X(39052)
X(162) = X(250)-Ceva conjugate of X(270)
X(162) = cevapoint of X(i) and X(j) for this (i,j): (108,109)
X(162) = X(i)-cross conjugate of X(j) for these (i,j): (108,107), (109,110)
X(162) = crosssum of X(810) and X(822)
X(162) = X(i)-aleph conjugate of X(j) for these (i,j): (28,1052), (107,920), (162,1), (648,63)
X(162) = trilinear pole of line X(1)X(19)
X(162) = trilinear product of X(1113) and X(1114)
X(162) = crossdifference of PU(75)
X(162) = pole wrt polar circle of trilinear polar of X(1577) (line X(1109)X(2632))
X(162) = X(48)-isoconjugate (polar conjugate) of X(1577)
X(162) = X(92)-isoconjugate of X(822)
X(162) = X(6)-isoconjugate of X(525)
X(162) = crosspoint of X(811) and X(823)
X(162) = trilinear product of PU(74)
X(162) = perspector of conic {A,B,C,PU(74)}}
X(162) = barycentric product of circumcircle intercepts of line X(19)X(27)


X(163) = TRILINEAR PRODUCT X(6)*X(110)

Trilinears    (sin 2A)/(sin 2B - sin 2C) : (sin 2B)/(sin 2C - sin 2A) : (sin 2C)/(sin 2A - sin 2B)
Trilinears    a2/(b2 - c2) : :
Barycentrics  a3/(b2 - c2) : b3/(c2 - a2) : c3/(a2 - b2)

Let A'B'C' be the circumcevian triangle of X(512). Let A″ be the trilinear product B'*C', and define B″ and C″ cyclically. A″, B″, C″ are collinear on line X(667)X(788). The lines AA″, BB″, CC″ concur in X(163). (Randy Hutson, August 19, 2019)

X(163) lies on these lines: 1,293   19,563   31,923   32,849   48,1094   99,825   101,110   109,112   284,909   643,1018   692,906   798,1101   813,827]

X(163) = isogonal conjugate of X(1577)
X(163) = isotomic conjugate of X(20948)
X(163) = complement of X(21294)
X(163) = anticomplement of X(21253)
X(163) = barycentric product of circumcircle intercepts of line X(1)X(21)
X(163) = crosssum of X(656) and X(661)
X(163) = X(i)-aleph conjugate of X(j) for these (i,j): (648,19), (662,610)
X(163) = trilinear product of PU(2)
X(163) = barycentric product of PU(70)
X(163) = trilinear product of X(58)X(101)
X(163) = trilinear product of the 6 vertices of the 1st and 2nd circumperp triangles
X(163) = trilinear pole of line X(31)X(48)
X(163) = X(92)-isoconjugate of X(656)
X(163) = crossdifference of every pair of points on line X(1109)X(2632)
X(163) = trilinear product X(6)*X(110)


X(164 = INCENTER OF EXCENTRAL TRIANGLE

Trilinears       sin B/2 + sin C/2 - sin A/2 : sin C/2 + sin A/2 - sin B/2 : sin A/2 + sin B/2 - sin C/2
Barycentrics  a(sin B/2 + sin C/2 - sin A/2) : b(sin C/2 + sin A/2 - sin B/2) : c(sin A/2 + sin B/2 - sin C/2)
X(164) = X[1] - 3 X[55168], X[1] + 2 X[55170], X[1] - 4 X[55171], 3 X[1] - 4 X[55172], 3 X[1] - 2 X[55173], 2 X[1] - 3 X[55175], 5 X[1] - 6 X[55176], 4 X[12523] - X[12656], 2 X[12523] - 3 X[55168], 2 X[12523] + X[55169], 3 X[12523] - 2 X[55172], 3 X[12523] - X[55173], 4 X[12523] - 3 X[55175], 5 X[12523] - 3 X[55176], X[12656] - 6 X[55168], X[12656] + 2 X[55169], X[12656] + 4 X[55170], X[12656] - 8 X[55171], 3 X[12656] - 8 X[55172], and many others

Let OA be the circle centered at the A-vertex of the 1st-circumperp-of-1st-circumperp triangle (which is also the 2nd-circumperp-of-2nd-circumperp triangle) and passing through A; define OB and OC cyclically. X(164) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

Let OA be the circle centered at the A-vertex of the 1st-circumperp-of-2nd-circumperp triangle (which is also the 2nd-circumperp-of-1st-circumperp triangle) and passing through A; define OB and OC cyclically. X(164) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

X(164) lies on the the Feuerbach circumhyperbola of the excentral trianjgle, the curves KK414, K654, Q068, and these lines: {1, 258}, {2, 9807}, {3, 3659}, {8, 55331}, {9, 168}, {40, 188}, {55, 17641}, {57, 177}, {63, 11691}, {84, 42017}, {165, 167}, {166, 7991}, {173, 504}, {354, 58614}, {361, 503}, {362, 845}, {517, 8112}, {846, 13091}, {1376, 17657}, {1445, 7670}, {1488, 11924}, {1697, 8422}, {1698, 12622}, {1699, 12614}, {1764, 12554}, {3333, 12908}, {3339, 31768}, {3645, 6213}, {5437, 58444}, {5708, 12813}, {6212, 7001}, {6726, 8108}, {6767, 31796}, {8075, 10967}, {8076, 8084}, {8081, 60554}, {8091, 58777}, {8231, 13090}, {8351, 16015}, {8580, 12450}, {9819, 31767}, {9837, 60598}, {10215, 20114}, {10490, 11923}, {10980, 58616}, {11234, 37556}, {11527, 55363}, {13443, 43192}, {31393, 32183}, {31790, 37560}

X(164) = midpoint of X(i) and X(j) for these {i,j}: {1, 55169}, {9807, 58706}, {11691, 12539}, {12523, 55170}, {21633, 58705}
X(164) = reflection of X(i) in X(j) for these {i,j}: {1, 12523}, {2, 58709}, {167, 12518}, {177, 12443}, {188, 52797}, {9807, 21633}, {12450, 58689}, {12523, 55171}, {12656, 1}, {12694, 18258}, {12844, 3}, {12879, 188}, {21633, 58440}, {55169, 55170}, {55173, 55172}, {55175, 55168}, {58706, 58705}, {58710, 9807}, {58711, 2}, {58712, 58708}, {58714, 58715}, {58715, 58718}, {58716, 58719}, {58720, 58712}, {58721, 58713}
X(164) = isogonal conjugate of X(505)
X(164) = complement of X(9807)
X(164) = anticomplement of X(21633)
X(164) = Thomson-isogonal conjugate of X(55175)
X(164) = excentral -sogonal conjugate of X(164)
X(164) = X(i)-Ceva conjugate of X(j) for these (i,j): {188, 1}, {20183, 168}
X(164) = X(i)-isoconjugate of X(j) for these (i,j): {1, 505}, {2, 60555}, {259, 16664}
X(164) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 505}, {164, 9807}, {174, 4146}, {32664, 60555}
X(164) = X(1)-of-excentral triangle
X(164) = X(8)-of-1st-circumperp-triangle
X(164) = X(944)-of-2nd-circumperp-triangle
X(164) = excentral-isogonal conjugate of X(164)
X(164) = excentral-isotomic conjugate of X(844)
X(164) = X(188)-Ceva conjugate of X(1)
X(164) = X(i)-aleph conjugate of X(j) for these (i,j): (1,361), (2,362), (9,844), (188,164), (366,173)
X(164) = centroid of curvatures of extraversions of Conway circle
X(164) = ABC-to-excentral trilinear image of X(1)
X(164) = barycentric product X(i)*X(j) for these {i,j}: {1, 16017}, {174, 60598}, {188, 15495}
X(164) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 505}, {31, 60555}, {266, 16664}, {15495, 4146}, {16017, 75}, {60598, 556}
X(164) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 12523, 55175}, {1, 18291, 10231}, {1, 55168, 12523}, {2, 9807, 21633}, {2, 21633, 58712}, {2, 58440, 58713}, {2, 58705, 58710}, {2, 58706, 9807}, {2, 58708, 58440}, {2, 58710, 58721}, {2, 58711, 58722}, {2, 58716, 58717}, {2, 58717, 58718}, {40, 188, 505}, {40, 52797, 363}, {165, 167, 12518}, {258, 8078, 1}, {266, 15997, 1}, {7588, 8093, 1}, {7589, 11032, 1}, {8077, 8094, 1}, {9807, 21633, 58711}, {9807, 58440, 58712}, {9807, 58708, 2}, {9807, 58709, 58713}, {9807, 58711, 58720}, {9807, 58712, 58721}, {9807, 58713, 58722}, {9807, 58716, 58715}, {9807, 58717, 58716}, {12523, 55169, 12656}, {12523, 55171, 55168}, {12523, 55173, 55172}, {12656, 55175, 1}, {20114, 52800, 24242}, {20183, 52797, 505}, {21633, 58440, 2}, {21633, 58706, 58710}, {21633, 58708, 58713}, {21633, 58709, 58440}, {21633, 58710, 58720}, {21633, 58711, 58721}, {21633, 58712, 58722}, {21633, 58714, 58715}, {21633, 58715, 58716}, {21633, 58718, 58717}, {21633, 58719, 58718}, {55168, 55169, 1}, {55168, 55170, 12656}, {55169, 55171, 55175}, {55170, 55171, 1}, {55172, 55173, 1}, {58440, 58705, 9807}, {58440, 58706, 58711}, {58440, 58709, 58708}, {58440, 58710, 58722}, {58440, 58714, 58717}, {58440, 58715, 58718}, {58440, 58718, 58719}, {58705, 58708, 58711}, {58705, 58709, 21633}, {58705, 58719, 58714}, {58706, 58708, 21633}, {58706, 58709, 58712}, {58706, 58713, 58720}, {58706, 58717, 58714}, {58708, 58716, 58719}, {58709, 58714, 58719}, {58710, 58711, 9807}, {58710, 58712, 58711}, {58710, 58713, 21633}, {58711, 58712, 21633}, {58711, 58713, 58712}, {58712, 58713, 2}, {58714, 58715, 9807}, {58714, 58717, 58711}, {58714, 58718, 58716}, {58714, 58719, 21633}, {58715, 58716, 58711}, {58715, 58718, 21633}, {58715, 58719, 58717}, {58716, 58717, 21633}, {58716, 58719, 58712}, {58717, 58718, 58712}, {58718, 58719, 2}, {58720, 58721, 58711}, {58720, 58722, 58721}, {58721, 58722, 21633}


X(165) = CENTROID OF THE EXCENTRAL TRIANGLE

Trilinears    tan(B/2) + tan(C/2) - tan(A/2) : tan(C/2) + tan(A/2) - tan(B/2) : tan(A/2) + tan(B/2) - tan(C/2)
Trilinears    3a2 - 2a(b + c) - (b - c)2 : :
Trilinears    (r/R) - 4 cos A : :
Barycentrics    a[tan(B/2) + tan(C/2) - tan(A/2)] : :
X(165) = X(1) - 4 X(3)

Problem 916, proposed by H. Demir, M.E.T.U., Ankara, Turkey, in Mathematics Magazine 47, no. 5, November 1974, p. 286: "Let XYZ be the pedal triangle of a point P with regard to the triangle ABC. Then find the trilinear coordinates x, y, z of P such that YA + AZ = ZB + BX = XC + CY." Solution: X(165).

If DEF is the pedal triangle of X(165), then |AE| + |AF| = |BF| + |BD| = |CD| + |CE|. (Seiichi Kirikami, October 8, 2010.)

Let A'B'C' be the anticevian triangle, wrt intouch triangle, of X(1). Let A″ be the reflection of A' in A, and define B' and C' cyclically. The centroid of A″B″C″ is X(165). (Randy Hutson, December 2, 2017)

Let A'B'C' be the excentral triangle. Let A″ be the symmedian point of triangle A'BC, and define B″ and C″ cyclically. The lines A'A″, B'B″, C'C″ concur in X(165). (Randy Hutson, July 31 2018)

If you have The Geometer's Sketchpad, you can view X(165).

X(165) lies on these lines: 1,3   2,516   4,1698   9,910   10,20   32,1571   42,991   43,573   63,100   71,610   105,1054   108,1767   109,212   164,167   166,168   191,1079   210,971   218,1190   220,1615   227,1394   255,1103   269,1253   355,550   371,1703   372,1702   376,515   380,579   386,1695   411,936   479,1323   498,1770   572,1051   574,1572   580,601   612,990   614,902   631,946   750,968   846,1719   950,1788   958,1706   962,1125   1011,1730   1342,1701   1343,1700

X(165) is the {X(3),X(40)}-harmonic conjugate of X(1). For a list of harmonic conjugates of X(165), click Tables at the top of this page.

X(165) = isogonal conjugate of X(3062)
X(165) = X(9)-Ceva conjugate of X(1)
X(165) = anticomplement of X(3817)
X(165) = X(2)-of-1st-circumperp-triangle
X(165) = homothetic center of ABC and orthic triangle of 1st circumperp triangle
X(165) = homothetic center of excentral triangle and medial triangle of 1st circumperp triangle
X(165) = excentral isogonal conjugate of X(9)
X(165) = excentral isotomic conjugate of X(165)
X(165) = excentral polar conjugate of X(1)
X(165) = Thomson-isogonal conjugate of X(1)
X(165) = reflection of X(1699) in X(2)
X(165) = cyclocevian conjugate of X(1) wrt anticevian triangle of X(1)
X(165) = X(i)-beth conjugate of X(j) for these (i,j): (100,165), (643,200)
X(165) = X(i)-aleph conjugate of X(j) for these (i,j):
(2,169), (9,165), (21,572), (100,101), (188,9), (259,43), (365,978), (366,57), (650,1053)
X(165) = centroid of the triangle with vertices X(1), X(8), X(20)
X(165) = centroid of the triangle with vertices X(4), X(20), X(40)
X(165) = ABC-to-excentral barycentric image of X(2)


X(166) = GERGONNE POINT OF EXCENTRAL TRIANGLE

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where
                        f(A,B,C) = (tan A/2)/(cos B/2 + cos C/2 - cos A/2) - (tan B/2)/(cos C/2 + cos A/2 - cos B/2) - (tan C/2)/(cos A/2 + cos B/2 - cos C/2)

Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(166) = X(7)-of-excentral triangle

X(166) lies on these lines: 1,1488   165,168   167,188

X(166) = X(266)-cross conjugate of X(57)
X(166) = cevapoint of X(266) and X(289)


X(167) = NAGEL POINT OF EXCENTRAL TRIANGLE

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where
                        f(A,B,C) = s(B,C,A)t(B,C,A) + s(C,A,B)t(C,A,B) - s(A,B,C)t(A,B,C),
                        where s(A,B,C) = sin(A/2) and t(A,B,C) = (cos B/2 + cos C/2 - cos A/2) sec A/2

Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(167) = X(8)-of-excentral triangle

X(167) lies on these lines: 1,174   164,165   166,188

X(167) = X(9)-aleph conjugate of X(166)


X(168) = MITTENPUNKT OF EXCENTRAL TRIANGLE

Trilinears    sin A - sin B - sin C + 2[cos A/2 + sin(B/2 - A/2) + sin(C/2 - A/2)] : :
Trilinears    b/(1 - sin B/2) + c/(1 - sin C/2) - a/(1 - sin A/2) : :

X(168) lies on these lines: 1,173   9,164   165,166

X(168) = X(188)-aleph conjugate of X(363)

X(168) = X(9)-of-excentral triangle
X(168) = homothetic center of the excentral and outer Hutson triangles; see X(363).
X(168) = X(7)-of-1st-circumperp-triangle
X(168) = homothetic center of ABC and orthic triangle of outer Hutson triangle

X(169) = X(85)-CEVA CONJUGATE OF X(1)

Trilinears    - (sin A)cos2(A/2) + (sin B)cos2(B/2) + (sin C)cos2(C/2) : :
Trilinears    a^3 - a^2 (b + c) + a (b^2 + c^2) - (b - c)^2 (b + c) : :

X(169) lies on these lines: 1,41   3,910   4,9   6,942   46,672   57,277   63,379   65,218   220,517   572,610

X(169) = complement of X(17170)
X(169) = anticomplement of X(34847)
X(169) = X(85)-Ceva conjugate of X(1)
X(169) = crosssum of X(6) and X(1473)
X(169) = X(i)-aleph conjugate of X(j) for these (i,j):
(2,165), (85,169), (86,572), (174,43), (188,170), (508,1), (514,1053), (664,101)


X(170) = X(9)-ALEPH CONJUGATE OF X(9)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where
                        f(A,B,C) = - (tan A/2)sec2(A/2) + (tan B/2)sec2(B/2) + (tan C/2)sec2(C/2)

Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(170) = X(76)-of-excentral triangle

X(170) lies on these lines: 1,7   43,218

X(170) = anticomplement of X(34848)
X(170) = X(220)-Ceva conjugate of X(1)
X(170) = X(i)-aleph conjugate of X(j) for these (i,j): (9,9), (55,43), (188,169), (220,170), (644,1018)
X(170) = X(664)-beth conjugate of X(170)


X(171) = {X(2), X(31)}-HARMONIC CONJUGATE OF X(238)

Trilinears       a2 + bc : b2 + ca : c2 + ab
Barycentrics  a3 + abc : b3 + abc : c3 + abc

X(171) lies on these lines: 1,3   2,31   4,601   6,43   7,983   10,58   37,846   42,81   47,498   63,612   72,1046   84,989   98,109   181,511   222,611   292,893   319,757   385,894   388,603   474,978   595,1125   602,631   756,896

X(171) = isogonal conjugate of X(256)
X(171) = isotomic conjugate of X(7018)
X(171) = complement of X(4388)
X(171) = anticomplement of X(3846)
X(171) = perspector of Gemini triangle 33 and cross-triangle of ABC and Gemini triangle 33
X(171) = trilinear pole line X(3287)X(3805) (the perspectrix of ABC and Gemini triangle 34)
X(171) = X(292)-Ceva conjugate of X(238)
X(171) = X(i)-beth conjugate of X(j) for these (i,j): (100,171), (643,42)
X(171) = crosssum of PU(6)
X(171) = crosspoint of PU(8)
X(171) = intersection of tangents at PU(8) to hyperbola {A,B,C,X(100),PU(8)}}
X(171) = {X(1),X(40)}-harmonic conjugate of X(37598)
X(171) = {X(55),X(56)}-harmonic conjugate of X(23853)


X(172) = TRILINEAR PRODUCT X(6)*X(171)

Trilinears       a3 + abc : b3 + abc : c3 + abc
Barycentrics  a4 + bca2 : b4 + cab2 : c4 + abc2

X(172) lies on these lines: 1,32   6,41   12,230   21,37   35,187   36,39   42,199   58,101   65,248   350,384   577,1038   694,904   699,932

X(172) = isogonal conjugate of X(257)
X(172) = crossdifference of every pair of points on line X(522)X(1491)
X(172) = X(101)-beth conjugate of X(172)
X(172) = {X(1),X(32)}-harmonic conjugate of X(1914)
X(172) = intersection of tangents at PU(9) to hyperbola {A,B,C,X(101),PU(9)}
X(172) = crosspoint of PU(9)
X(172) = crosssum of PU(10)
X(172) = homothetic center of anti-tangential midarc triangle and mid-triangle of 1st and 2nd Kenmotu diagonals triangles


X(173) = CONGRUENT ISOSCELIZERS POINT

Trilinears    cos B/2 + cos C/2 - cos A/2 : :
Trilinears    tan A/2 + sec A/2 : tan B/2 + sec B/2 : tan C/2 + sec C/2     (M. Iliev, 4/12/07)
Trilinears    b(csc B/2) + c(csc C/2) - a(csc A/2) : :
Trilinears    cot A'/2 : :, where A'B'C' = excentral triangle
Trilinears    b' + c' - a' : :, where a', b', c' are sidelengths of excentral triangle
Trilinears    cot A' + csc A' : :, where A'B'C' = excentral triangle
Trilinears    [distance from A to far side of A-excircle] : :

Let PB on sideline AC and QC be equidistant from A, so that APBQC is an isosceles triangle. The line PBQC is called an isoscelizer. The lines PBQC, PCQA, PAQB concur in X(173). (P. Yff, unpublished notes, 1989)

The intouch triangle of the intouch triangle of triangle ABC is perspective to triangle ABC, and X(173) is the perspector. (Eric Danneels, Hyacinthos 7892, 9/13/03)

Also, X(173) = X(1486)-of-the-intouch-triangle. (Darij Grinberg; see notes at X(1485) and X(1486).)

If you have The Geometer's Sketchpad, you can view Congruent Isoscelizers Point.

X(173) lies on the cubics K220, K748, K1079 and these lines: {1, 168}, {2, 11891}, {3, 7590}, {6, 61072}, {7, 21623}, {9, 177}, {10, 12582}, {40, 8351}, {46, 30408}, {55, 10502}, {57, 174}, {63, 8126}, {65, 11899}, {164, 504}, {165, 7589}, {167, 11032}, {180, 483}, {191, 16151}, {223, 34026}, {266, 1743}, {363, 3973}, {503, 844}, {505, 1130}, {846, 8425}, {942, 8082}, {1376, 17631}, {1445, 8389}, {1490, 12685}, {1652, 30410}, {1653, 30409}, {1697, 11924}, {1698, 8382}, {1699, 8379}, {1721, 12728}, {1764, 11896}, {1768, 13267}, {2136, 12646}, {3306, 8125}, {3333, 8092}, {3338, 30420}, {3339, 8094}, {3361, 7588}, {3576, 18454}, {3729, 40893}, {4860, 10501}, {4882, 12130}, {5119, 30411}, {5285, 8132}, {5290, 8088}, {5437, 7028}, {5541, 12748}, {5708, 8100}, {5709, 8130}, {6203, 30406}, {6204, 30407}, {6212, 31592}, {6213, 31593}, {6326, 12774}, {8056, 41799}, {8090, 10980}, {8128, 55104}, {8129, 37534}, {8231, 11996}, {8388, 60938}, {8545, 30404}, {8580, 11860}, {8915, 35681}, {9795, 21454}, {10215, 58868}, {11529, 18456}, {12396, 12406}, {12514, 12570}, {12565, 12716}, {12658, 12873}, {12659, 13074}, {12660, 13127}, {13443, 18885}, {30423, 51816}, {34924, 34925}, {45707, 60955}, {46370, 52799}, {52999, 53118}

X(173) = isogonal conjugate of X(258)
X(173) = anticomplement of X(34849)
X(173) = X(174)-Ceva conjugate of X(1)
X(173) = excentral-isogonal conjugate of X(845)
X(173) = X(31)-complementary conjugate of X(13443)
X(173) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 13443}, {57, 8078}, {174, 1}, {2089, 52999}
X(173) = X(i)-isoconjugate of X(j) for these (i,j): {1, 258}, {2, 60554}, {6, 7048}, {55, 21456}, {174, 53119}, {188, 289}, {259, 1488}, {260, 16015}, {266, 7028}, {3659, 10492}, {10495, 45875}, {18887, 59467}
X(173) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 258}, {9, 7048}, {188, 556}, {223, 21456}, {236, 53123}, {5430, 312}, {10493, 2090}, {13443, 2}, {32664, 60554}
X(173) = cevapoint of X(7707) and X(18888)
X(173) = SS(A→A')-of-X(9), where A'B'C' is the excentral triangle
X(173) = X(19)-of-excentral triangle
X(173) = X(19)-of-Yff central triangle
X(173) = homothetic center of excentral triangle and Yff central triangle
X(173) = homothetic center of ABC and orthic triangle of Yff central triangle
X(173) = homothetic center of ABC and extangents triangle of excentral triangle
X(173) = excentral isogonal conjugate of X(845)
X(173) = X(i)-aleph conjugate of X(j) for these (i,j): (1,503), (2,504), (174,173), (188,845), (366,164), (507,1), (508,362), (509,361) X(173) = barycentric product X(i)*X(j) for these {i,j}: {1, 7057}, {9, 18886}, {75, 42622}, {174, 236}, {188, 2089}, {266, 53122}, {4146, 53118}, {7001, 53076}, {7010, 53077}, {7048, 52999}, {45877, 55341}
X(173) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 7048}, {6, 258}, {31, 60554}, {57, 21456}, {188, 53123}, {236, 556}, {259, 7028}, {266, 1488}, {2089, 4146}, {7057, 75}, {7707, 2090}, {16012, 42017}, {18885, 21624}, {18886, 85}, {18888, 16015}, {42622, 1}, {43192, 45876}, {45878, 10495}, {52999, 7057}, {53118, 188}, {58968, 45875}, {60539, 53119}, {61072, 21623}
X(173) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 12445, 11535}, {2, 11891, 21624}, {3, 12491, 7590}, {9, 18888, 8078}, {57, 174, 258}, {165, 8423, 7589}, {165, 30394, 8423}, {236, 45708, 7593}, {259, 7707, 1}, {7001, 7010, 8078}, {7587, 12445, 1}, {7589, 11195, 8423}, {7593, 45708, 41855}, {8076, 8083, 1}, {8090, 10980, 11033}, {8423, 30394, 11195}, {8729, 13098, 174}, {16015, 20183, 258}


X(174) = YFF CENTER OF CONGRUENCE

Trilinears    sec A/2 : sec B/2 : sec C/2
Trilinears    [bc/(b + c - a)]1/2 : [ca(c + a - b)]1/2 : [ab(a + b - c)]1/2
Trilinears    csc A' : csc B' : csc C', where A'B'C' = excentral triangle
Trilinears    |AA'| : |BB'| : |CC'|, where A'B'C' = excentral triangle
Trilinears    (csc A')(cos B' + cos C') : :, where A'B'C' = excentral triangle
Barycentrics  sin A/2 : sin B/2 : sin C/2

Let Ea be the ellipse with B and C as foci and passing through X(1), and define Eb and Ec cyclically. Let La be the line tangent to Ea at X(1), and define Lb and Lc cyclically. Let A' = La∩BC, B' = Lb∩CA, C' = Lc∩AB. Then A', B', C' are collinear, and X(174) = trilinear pole of line A'B'C'. The line A'B'C' meets the line at infinity at the isogonal conjugate of X(3659). Alternately, let A″ be the trilinear pole of line La, and define B″ and C″ cyclically. The lines AA″, BB″ and CC″ concur at X(174); see also X(188). The points A″, B″, C″ lie on the circumconic centered at X(9). (Randy Hutson, December 10, 2016)

Let A'B'C' be the excentral triangle. X(174) is the trilinear pole of the Monge line of the incircles of BCA', CAB', ABC'. (Randy Hutson, December 10, 2016)

In notes dated 1987, Yff raises this question concerning certain triangles lying within ABC: can isoscelizers (defined at X(173)), PBQC, PCQA, PAQB, be constructed so that, on putting

RA = PAQB∩PBQC,       RB = PBQC∩PCQA,      RC = PCQA∩PAQB,

the following four triangles are congruent:

PAQARA,       PBQBRB,       PCQCRC,       RARBRC ?

After proving that the answer is yes, Yff moves the three isoscelizers in such a way that the three outer triangles, stay congruent and the inner triangle (called the Yff central triangle), RARBRC, shrinks to X(174).

Let D be the point on side BC such that (angle BID) = (angle DIC), and likewise for point E on side CA and point F on side AB. The lines AD, BE, CF concur in X(174). (Seiichi Kirikami, Jan. 29, 2010)

Generalization: if I is replaced by an arbitrary point P = p : q : r (trilinears), then the lines AD, BE, CF concur in the point K(P) = f(p,q,r,A) : f(q,r,p,B) : f(r,p,q,C), where f(p,q,r,A) = (q2 + r2 + 2qr cos A)-1/2. Moreover, if P* is the inverse of P in the circumcircle, then K(P*) = K(P). (Peter Moses, Feb. 1, 2010, based on Seiichi Kirikami's construction of X(174))

X(174) is the homothetic center of ABC and the extangents triangle of the intouch triangle. (Randy Hutson, 9/23/2011)

If you have The Geometer's Sketchpad, you can view Yff Center of Congruence (1) and Yff Center of Congruence (2) and Yff Center of Congruence (3). For access to a sketch of the Yff central triangle, see X(177).

X(174) lies on the curves K134, K365, K747, K748, K965, Q027, and these lines: {1, 167}, {2, 236}, {3, 8129}, {7, 234}, {11, 8086}, {12, 8088}, {55, 7589}, {56, 7587}, {57, 173}, {65, 8094}, {145, 7057}, {175, 483}, {176, 1274}, {178, 60598}, {188, 266}, {222, 34026}, {223, 16664}, {226, 7593}, {259, 7370}, {260, 12518}, {289, 45875}, {312, 40893}, {354, 8083}, {481, 1127}, {482, 10230}, {515, 9837}, {556, 4146}, {557, 13389}, {558, 1489}, {631, 8127}, {658, 59463}, {942, 8100}, {1071, 8096}, {1284, 8250}, {1317, 8098}, {3210, 16018}, {3340, 11535}, {3645, 34495}, {3649, 16147}, {3671, 12569}, {3752, 61072}, {4298, 12581}, {4654, 41855}, {5435, 7002}, {5542, 59442}, {5902, 18408}, {6732, 7670}, {6733, 60539}, {8084, 10500}, {8134, 8136}, {8137, 8139}, {8243, 8248}, {8581, 11859}, {9850, 9854}, {10164, 59464}, {10473, 35625}, {10497, 13444}, {10503, 10967}, {10980, 30394}, {11570, 12772}, {12402, 12406}, {12711, 12715}, {12723, 12727}, {12854, 12871}, {12912, 13073}, {12913, 13125}, {13390, 39122}, {15730, 34924}, {17625, 17630}, {35671, 35681}, {39121, 55331} 481,1127   558,1489

X(174) = reflection of X(i) in X(j) for these {i,j}: {8241, 1}, {16017, 2090}
X(174) = isotomic conjugate of X(556)
X(174) = complement of X(16017)
X(174) = isogonal conjugate of X(259)
X(174) = anticomplement of X(2090)
X(174) = complement of the isogonal conjugate of X(60555)
X(174) = isotomic conjugate of the anticomplement of X(16015)
X(174) = isotomic conjugate of the complement of X(16018)
X(174) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {45877, 33650}, {45878, 37781}
X(174) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 15495}, {505, 141}, {60555, 10}
X(174) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 15495}, {7, 2089}, {555, 7371}, {4146, 188}, {21456, 1488}, {46891, 558}, {46892, 557}, {55328, 10492}, {55329, 10495}
X(174) = X(i)-isoconjugate of X(j) for these (i,j): {1, 259}, {2, 60539}, {6, 188}, {9, 266}, {21, 60533}, {31, 556}, {37, 6727}, {41, 4146}, {55, 174}, {56, 6731}, {57, 6726}, {58, 6725}, {100, 6729}, {101, 6728}, {109, 6730}, {173, 53119}, {200, 7370}, {220, 7371}, {236, 60554}, {258, 53118}, {260, 7707}, {284, 6724}, {365, 60534}, {366, 60530}, {509, 4166}, {555, 1253}, {604, 7027}, {650, 6733}, {1172, 7591}, {3659, 45877}, {4182, 60538}, {7001, 7014}, {7028, 42622}, {18753, 55336}, {45878, 55331}, {60555, 60598}
X(174) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 6731}, {2, 556}, {3, 259}, {9, 188}, {10, 6725}, {11, 6730}, {174, 16017}, {223, 174}, {236, 8}, {266, 504}, {478, 266}, {1015, 6728}, {3160, 4146}, {3161, 7027}, {5431, 53120}, {5452, 6726}, {6609, 7370}, {8054, 6729}, {10493, 16016}, {13443, 236}, {15495, 2}, {17113, 555}, {32664, 60539}, {39122, 1143}, {40374, 55336}, {40589, 6727}, {40590, 6724}, {40611, 60533}
X(174) = cevapoint of X(i) and X(j) for these (i,j): {1, 173}, {2, 16018}, {177, 7707}, {236, 12646}, {259, 266}, {513, 61072}, {514, 21623}, {650, 10501}, {6724, 60533}, {6732, 45877}, {16015, 41799}
X(174) = trilinear pole of line {6728, 10492}
X(174) = crossdifference of every pair of points on line {6729, 45878}
X(174) = X(i)-cross conjugate of X(j) for these (i,j): (1,1488), (177,7), (259,188)
X(174) = crosssum of X(1) and X(503)
X(174) = X(556)-beth conjugate of X(556)
X(174) = SS(A→A')-of-X(2), where A'B'C' is the excentral triangle
X(174) = SS(A→A')-of-X(226), where A'B'C' is the excentral triangle
X(174) = isotomic conjugate of X(556)
X(174) = X(55)-of-intouch triangle
X(174) = X(55)-of-Yff central triangle
X(174) = homothetic center of intouch triangle and Yff central triangle
X(174) = homothetic center of ABC and the intangents triangle of the intouch triangle.
X(174) = {X(8134),X(8136)}-harmonic conjugate of X(8123)
X(174) = {X(8137),X(8139)}-harmonic conjugate of X(8124)
X(174) = X(1824)-of-excentral-triangle
X(174) = excentral-to-ABC trilinear image of X(165)
X(174) = barycentric product X(i)*X(j) for these {i,j}: {1, 4146}, {7, 188}, {8, 7371}, {9, 555}, {57, 556}, {75, 266}, {85, 259}, {86, 6724}, {176, 5451}, {236, 21456}, {269, 7027}, {274, 60533}, {279, 6731}, {286, 7591}, {312, 7370}, {366, 508}, {509, 18297}, {557, 1143}, {558, 1274}, {658, 6730}, {664, 6728}, {693, 6733}, {1088, 6726}, {1434, 6725}, {1441, 6727}, {1488, 7057}, {1489, 46892}, {2089, 7048}, {4554, 6729}, {6063, 60539}, {7001, 53121}, {7010, 53120}, {7028, 18886}, {10492, 55341}, {16017, 16664}, {41885, 46891}
X(174) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 188}, {2, 556}, {6, 259}, {7, 4146}, {8, 7027}, {9, 6731}, {31, 60539}, {37, 6725}, {55, 6726}, {56, 266}, {58, 6727}, {65, 6724}, {73, 7591}, {109, 6733}, {164, 60598}, {173, 236}, {177, 178}, {188, 8}, {258, 7028}, {259, 9}, {266, 1}, {269, 7371}, {279, 555}, {289, 258}, {365, 60534}, {366, 55336}, {508, 18297}, {509, 366}, {513, 6728}, {555, 85}, {556, 312}, {557, 1274}, {558, 1143}, {649, 6729}, {650, 6730}, {1400, 60533}, {1407, 7370}, {1488, 7048}, {2089, 7057}, {3659, 55363}, {4146, 75}, {5451, 40700}, {6724, 10}, {6725, 2321}, {6726, 200}, {6727, 21}, {6728, 522}, {6729, 650}, {6730, 3239}, {6731, 346}, {6733, 100}, {7001, 3082}, {7010, 483}, {7027, 341}, {7048, 53123}, {7057, 53122}, {7370, 57}, {7371, 7}, {7591, 72}, {7707, 16016}, {8078, 12646}, {10490, 177}, {12646, 5430}, {13444, 43192}, {14088, 60532}, {14596, 234}, {15495, 16017}, {15997, 42017}, {16011, 15997}, {16015, 2090}, {18753, 60530}, {18888, 7707}, {24242, 12644}, {41799, 16015}, {42622, 53118}, {43192, 55342}, {45874, 3659}, {45875, 55331}, {53116, 7001}, {53117, 7010}, {55328, 55341}, {55331, 55332}, {60530, 4166}, {60531, 60544}, {60532, 60545}, {60533, 37}, {60534, 4182}, {60537, 4179}, {60538, 365}, {60539, 55}, {60540, 60546}, {60541, 60547}, {60542, 60548}, {60543, 39131}, {60554, 53119}
X(174) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 177, 2089}, {1, 503, 16012}, {1, 8092, 8242}, {1, 8351, 11924}, {1, 13092, 177}, {1, 30408, 8351}, {1, 30420, 8092}, {1, 58868, 13385}, {2, 8125, 7028}, {2, 8126, 236}, {2, 16017, 2090}, {7, 8388, 45707}, {7, 8389, 45708}, {7, 18886, 234}, {57, 16015, 15495}, {173, 258, 57}, {177, 13385, 58868}, {236, 7028, 2}, {258, 16015, 1488}, {266, 6724, 188}, {354, 10501, 11033}, {354, 10502, 8083}, {6732, 11923, 8113}, {7587, 7588, 56}, {7589, 8076, 55}, {7590, 8082, 1}, {7593, 8080, 226}, {7707, 8138, 8114}, {8083, 11033, 354}, {8083, 11195, 10502}, {8083, 11217, 11033}, {8084, 11032, 17641}, {8086, 8379, 11}, {8088, 8382, 12}, {8090, 8423, 1}, {8092, 8351, 1}, {8092, 30408, 11924}, {8094, 12445, 65}, {8096, 12685, 1071}, {8098, 12748, 1317}, {8100, 12491, 942}, {8102, 8734, 258}, {8102, 13098, 57}, {8104, 13267, 11}, {8125, 8126, 2}, {8127, 8128, 631}, {8129, 8130, 3}, {8131, 8132, 3}, {8242, 11924, 1}, {8248, 11996, 8243}, {8250, 8425, 1284}, {8351, 30420, 8242}, {8388, 8389, 7}, {8388, 18886, 8114}, {8388, 30404, 30405}, {8389, 30405, 30404}, {8729, 8734, 57}, {8729, 13098, 173}, {9795, 11889, 45707}, {9795, 11891, 7}, {9854, 12130, 9850}, {10235, 10236, 1}, {10500, 17641, 11032}, {10501, 10502, 354}, {11033, 11195, 8083}, {11033, 11217, 10501}, {11195, 11217, 354}, {11535, 11899, 3340}, {11859, 11860, 8581}, {11889, 11890, 7}, {11890, 11891, 45708}, {11895, 11896, 1}, {11923, 53118, 43192}, {12406, 13475, 12402}, {12569, 12570, 3671}, {12581, 12582, 4298}, {12644, 12646, 145}, {12715, 12716, 12711}, {12727, 12728, 12723}, {12772, 12774, 11570}, {12871, 12873, 12854}, {13073, 13074, 12912}, {13125, 13127, 12913}, {13385, 58868, 2089}, {16147, 16151, 3649}, {17630, 17631, 17625}, {18408, 18409, 5902}, {18454, 18456, 1}, {21623, 21624, 226}, {30394, 30395, 10980}, {30404, 30405, 7}, {30406, 30407, 45708}, {30406, 30418, 7}, {30407, 30419, 7}, {30408, 30420, 1}, {30408, 30423, 30411}, {30409, 30410, 45708}, {30409, 30421, 7}, {30410, 30422, 7}, {30411, 30420, 30423}, {30411, 30423, 1}, {30418, 30419, 45707}, {30421, 30422, 45707}, {31580, 31592, 481}, {31581, 31593, 482}, {34026, 34034, 222}, {35625, 35627, 10473}, {35681, 35900, 35671}, {45707, 45708, 7}, {46891, 53977, 1143}, {46892, 53979, 1274}


X(175) = ISOPERIMETRIC POINT

Trilinears    -1 + sec A/2 cos B/2 cos C/2 : :
Barycentrics    (sin A)(-1 + sec A/2 cos B/2 cos C/2) : :
Barycentrics    -2a + (a + b + c) tan(A/2) : :
Barycentrics    a - ra : b - rb : c - rc, where ra, rb, rc are the exradii
Barycentrics    (a + b - c)*(a - b + c)*(a*(a - b - c) + S) : :
X(175) = 2s*X(1) - (r + 4R)*X(7) = 3X[2]-4X[31534]
X(175) = 3 X[2] - 4 X[31534]

The points X(175) and X(176) are discussed in an 1890 article by Emile Lemoine, accessible at Gallica. The article begins on page 111, and the two points are considered beginning on page 128.

A point X is defined as an isoperimetric point of triangle ABC if |XB| + |XC| + |BC| = |XC| + |XA| + |CA| = |XA| + |XB| + |AB|. Veldkamp established that X = X(175), uniquely, for some triangles ABC, but the conditions he gives are not correct. Hajja and Yff proved that the condition tan(A/2) + tan(B/2) + tan(C/2) < 2 is necessary and sufficient. See also X(176) and the 1st and 2nd Eppstein points, X(481), X(482).

In unpublished notes, Yff proved that X(175) is the center of the outer Soddy circle. His proof later appeared in the paper by Hajja and Yff cited below.

Every point on the Soddy line has barycentric coordinates of the form a + k/sa : b + k/sb : c + k/sc, where k is a symmetric function in a,b,c, and sa=(b+c-a)/2, sb=(c+a-b)/2, sc=(a+b-c)/2. Writing S for 4*area(ABC):

X(175) = 2a - S/sa : 2b - S/sb : 2c - S/sc
X(176) = 2a + S/sa : 2b + S/sb : 2c + S/sc
X(481) = a - S/sa : b - S/sb : c - S/sc
X(482) = a + S/sa : b + S/sb : c + S/sc
X(1371) = a + 2S/(3 sa) : b + 2S/(3 sb) : c + 2S/(3 sc)
X(1372) = a - 2S/(3 sa) : b - 2S/(3 sb) : c - 2S/(3 sc)
X(1373) = a + 2S/sa : b + 2S/sb : c + 2S/sc
X(1374) = a - 2S/sa : b - 2S/sb : c - 2S/sc

Clark Kimberling and R. W. Wagner, Problem E 3020 and Solution, American Mathematical Monthly 93 (1986) 650-652 [proposed 1983].

G. R. Veldkamp, "The isoperimetric point and the point(s) of equal detour," American Mathematical Monthly 92 (1985) 546-558.

Muwaffaq Hajja and Peter Yff, "The isoperimetric point and the point(s) of equal detour in a triangle," Journal of Geometry 87 (2007) 76-82.

Randy Hutson (August 23, 2011) noted that "There are exactly two points P such that the incircles of the triangles PBC, PCA, PAB are pairwise tangent to one another; the two points are X(175) and X(176). There are exactly two points P such that the radical center of the incircles of PBC, PCA, PAB is P; the two points are X(175) and X(176).'' However, an anonymous contributor noted that there are cases in which the second statement does not hold. His counterexamples and conditions for such points P to exist are quoted here:

For the isoperimetric point, the incircles become excircles, so really we need two separate claims, as follows:

(1) P is an isoperimetric point, in the sense that triangles PBC, PCA, and PAB have the same perimeter, if and only if P is the radical center of the excircles of triangles PBC, PCA, and PAB opposite P.

(2) P is an equal detour point, in the sense that |XB| + |XC| - |BC| = |XC| + |XA| - |CA| = |XA| + |XB| - |AB|, if and only if P is the radical center of the incircles of triangles PBC, PCA, and PAB.'

The proofs are clear: P is these circles' radical center if and only if the tangents from P to them have the same length, and we can express these tangents explicitly in terms of the triangles' sides.

The claim about pairwise tangency deserves further examination. Once again we must change the incircles to excircles for the isoperimetric point, but this time around that doesn't really fix everything. It is indeed true that if P is an isoperimetric point, then the excircles are pairwise tangent and if P is an equal detour point, then so are the incircles. However, the converses do not hold. Following is a counterexample. Start with an isosceles triangle ABC where AB = AC and angle A is sufficiently small. Let M be the midpoint of side BC and let Q be some point on ray AM beyond M. Observe that the excircles omega_B and omega_C of triangles QCA and QAB are always tangent on line QA, by symmetry. When Q is close to M, the excircle omega_A of triangle QBC intersects both omega_B and omega_C. Conversely, when Q tend to infinity away from M, however, both of omega_B and omega_C are disjoint from omega_A and lie outside of it. (This step is why we need angle A to be sufficiently small.) Therefore, there exists some intermediate position of point Q such that omega_A, omega_B, and omega_C are pairwise externally tangent. But then Q cannot be an isoperimetric point because the tangency points of omega_A with omega_B and omega_C do not lie on lines QC and QB, respectively.

A counterexample for incircles is similar. Start with an isosceles triangle ABC where AB = AC and angle A is sufficiently large. (For example, at least 60 degrees.) Let M be the midpoint of side BC and let Q be some point on ray MA beyond A. The rest of the construction is fully analogous, but here we have disjoint incircles when Q is close to A and intersecting incircles when Q tends to infinity away from A. We can still salvage some of this if we explicitly require P to be an interior point, as follows:

(1') Let point P lie in the interior of triangle ABC. Then P is an isoperimetric point, in the sense that triangles PBC, PCA, and PAB have the same perimeter, if and only if the excircles of triangles PBC, PCA, and PAB opposite P are pairwise tangent.'

(2') Let point P lie in the interior of triangle ABC. Then P is an equal detour point, in the sense that |XB| + |XC| - |BC| = |XC| + |XA| - |CA| = |XA| + |XB| - |AB|, if and only if the incircles of triangles PBC, PCA, PAB are pairwise tangent.'

Now lines PA, PB, PC separate the incircles or excircles from one another, so that their tangency points must necessarily lie on these lines, and from here on we can finish the proofs in the exact same way as for the pair of radical center claims.

Note that X(175) and X(176) do not correspond exactly to the isoperimetric and equal detour points! For the outer Soddy circle, whether we get equal perimeters or equal detours depends on whether this circle is tangent to the three small circles with centers A, B, and C and radii (-a + b + c)/2, etc., internally or externally. When the tangencies are internal, we get an isoperimetric point; when they are external we get an equal detour point; and when the outer Soddy circle degenerates into a straight line, X(175) is undefined.

Therefore, if we define the isoperimetric and equal detour points in terms of lengths, then some triangles will have one of each, others will have no isoperimetric point and two equal detour points, and still others will have no isoperimetric point and just a single equal detour point -- all depending on what happens with the outer Soddy circle.

Something like this is also in the entry for X(176), attributed to Hajja and Yff.

If you have GeoGebra, you can view X(175), X(176).ggb

Let Ha be the hyperbola through A having foci B and C, and define Hb and Hc cyclically. The three hyperbolas meet in in two points: X(175) and X(176). The 6 vertices of the hyperbolas lie on the Privalov conic (an ellipse). Specifically, if the vertices are labeled as A1, A2; B1, B2; C1,C2; then A1, B1, C1 are the vertices of the extouch triangle, and A2, B2, C2 are the vertices of the intouch triangle; see X(5452) and this Figure. (Liliana Gheorghe, Dan Reznik, Peter Moses, December, 2021)

In the plane of a triangle ABC, let

Oa = circle with diameter BC
C' = the point, other than B, where Oa meets AB
B' = the point, other than C, where Oa meets AC
Ub = B-mixtilinear incircle of BCC'
Uc = C-mixtilinear incircle of CBB'
Ab = touchpoint of Ub and BC; define Bc and Ca cyclically
Ac = touchpoint of Uc and BC; define Ba and Cb cyclically
The points Ab, Ac, Ba, Ba, Cb, Cb line on a conic, of which the center is X(176). A barycentric equation for this conic follows:

2 (a-b-c) (a^2-b^2-c^2) (a^2 (b+c)-(b-c)^2 (b+c)+2 a S) x^2 - ((a+b-c) (a-b+c) (3 a^4-2 a^3 (b+c)-3 (b^2-c^2)^2+2 a (b+c) (b^2+c^2))+2 (a^4-3 (b^2-c^2)^2+2 a^2 (b^2+c^2)) S) y z + (cyclic) = 0.

See X(175) and X(176). Continuing, let

Vb = B-mixtilinear excircle of BCC'
Vc = C-mixtilinear excircle of CBB'
A'b = touchpoint of Vb and BC; define B'c and C'a cyclically
A'c = touchpoint of Vc and BC; define B'a and C'b cyclically
The points A'b, A'c, B'a, B'c, C'a, C'b line on a conic, of which the center is X(175). A barycentric equation for this conic follows:

(a+b-c) (a-b+c) (2 (a-b-c) (2 a^3 b c+a^4 (b+c)-2 a b c (b+c)^2+(b-c)^2 (b+c)^3-2 a^2 (b^3+b^2 c+b c^2+c^3)+2 (a^3-2 b c (b+c)-a (b+c)^2) S) x^2 + (3 a^6-4 a^5 (b+c)-4 a (b-c)^2 (b+c)^3-3 (b-c)^2 (b+c)^4+a^2 (b+c)^2 (9 b^2+2 b c+9 c^2)-a^4 (9 b^2+14 b c+9 c^2)+8 a^3 (b^3+b^2 c+b c^2+c^3)-2 (a^4+6 a^3 (b+c)-4 a^2 (b+c)^2+3 (b^2-c^2)^2-2 a (3 b^3+7 b^2 c+7 b c^2+3 c^3)) S) y z) + (cyclic) = 0. (Angel Montesdeoca, September 9, 2022)

X(175) lies on the curves K032, K199, K200, K,1175, Q074, Q092, Q104 and on these lines: {1, 7}, {2, 13386}, {4, 10905}, {8, 1270}, {40, 34495}, {65, 6252}, {69, 10908}, {105, 30385}, {144, 30557}, {174, 483}, {226, 1131}, {280, 40700}, {329, 3084}, {388, 10911}, {490, 664}, {517, 39795}, {651, 1335}, {1086, 44636}, {1336, 10253}, {1463, 7353}, {2082, 6203}, {2550, 45713}, {3062, 10973}, {3083, 9776}, {3296, 34216}, {3297, 5228}, {3298, 6180}, {4000, 7968}, {4644, 7969}, {5045, 39794}, {5222, 18992}, {5226, 5393}, {5261, 9907}, {5405, 5435}, {5932, 31528}, {5933, 31530}, {6283, 49537}, {6360, 46422}, {6405, 20358}, {7090, 10405}, {7585, 30325}, {9778, 10135}, {9789, 21147}, {10580, 16663}, {11293, 17086}, {13387, 20211}, {13389, 21454}, {13459, 16441}, {15913, 31536}, {16214, 32081}, {17365, 44635}, {17950, 43134}, {21453, 30335}, {30347, 31588}, {30413, 31547}, {31544, 32201}

X(175) = reflection of X(i) in X(j) for these {i,j}: {176, 3160}, {10405, 7090}, {14121, 31534}, {30334, 1}
X(175) = isogonal conjugate of X(30336)
X(175) = isotomic conjugate of X(40699)
X(175) = anticomplement of X(14121)
X(175) = anticomplement of the isogonal conjugate of X(2067) X(175) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {6, 13386}, {603, 176}, {1659, 21270}, {1805, 3869}, {2067, 8}, {2362, 4}, {5414, 329}, {6502, 31552}, {13388, 69}, {30557, 3436}, {34121, 13387}
X(175) = X(8)-Ceva conjugate of X(176)
X(175) = X(10135)-cross conjugate of X(7)
X(175) = X(i)-isoconjugate of X(j) for these (i,j): {1, 30336}, {6, 15891}, {31, 40699}, {56, 34911}
X(175) = cevapoint of X(i) and X(j) for these (i,j): {1, 34495}, {10253, 32058}
X(175) = crosssum of X(55) and X(19037)
X(175) = reflection of X(175) in the Soddy line
X(175) = X(6406)-of-excentral-triangle
X(175) = X(1152)-of-intouch-triangle
X(175) = X(7353)-of-(inner)-tangential-mid-arc-triangle (TCCT 6.15)
X(175) = X(12224)-of-first-circumperp-triangle (TCCT 6.21)
X(175) = X(6400)-of-second-circumperp-triangle (TCCT 6.22)
X(175) = barycentric product X(i)*X(j) for these {i,j}: {7, 30413}, {8, 16662}, {1659, 31547}
X(175) = barycentric quotient X(i)/X(j) for these {i,j}: 1, 15891}, {2, 40699}, {6, 30336}, {9, 34911}, {9778, 30412}, {16662, 7}, {30413, 8}
X(175) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 7, 176}, {1, 15891}, {2, 40699}, {6, 30336}, {9, 34911}, {9778, 30412}, {16662, 7}, {30413, 8}
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 7, 176}, {1, 481, 7}, {1, 482, 17805}, {1, 1372, 481}, {1, 1373, 31538}, {1, 1374, 482}, {1, 17803, 31539}, {1, 30342, 11038}, {1, 31539, 17802}, {1, 31568, 8236}, {7, 176, 21169}, {7, 482, 21170}, {7, 1372, 17801}, {7, 8236, 31566}, {7, 17802, 1}, {7, 17805, 482}, {7, 31601, 1373}, {7, 31602, 481}, {20, 347, 176}, {77, 962, 176}, {176, 21169, 17804}, {176, 21170, 482}, {269, 9785, 176}, {279, 390, 176}, {481, 482, 1374}, {481, 1372, 31602}, {481, 17802, 176}, {481, 17803, 17802}, {481, 31539, 1}, {481, 31602, 17801}, {482, 1374, 7}, {482, 17805, 176}, {482, 21170, 21169}, {1323, 30332, 176}, {1372, 17803, 1}, {1372, 31539, 7}, {1373, 31538, 31601}, {1374, 17805, 21170}, {1442, 4295, 176}, {1443, 30305, 176}, {3600, 3672, 176}, {3638, 3639, 1372}, {3663, 4308, 176}, {3664, 4323, 176}, {3668, 4313, 176}, {3674, 4344, 176}, {4296, 4329, 176}, {4297, 36640, 176}, {4318, 17170, 176}, {4862, 6049, 176}, {5542, 5543, 176}, {5731, 22464, 176}, {7190, 11037, 176}, {8236, 10481, 176}, {10135, 32083, 16662}, {10481, 31568, 31566}, {13388, 13390, 2}, {14121, 31534, 2}, {17801, 17802, 21169}, {17802, 31602, 7}, {30424, 31721, 176}, {31538, 31601, 176}, {31539, 31602, 176}


X(176) = EQUAL DETOUR POINT

Trilinears    1 + sec A/2 cos B/2 cos C/2 : :
Barycentrics    (sin A)(1 + sec A/2 cos B/2 cos C/2)

Barycentrics    2a + (a + b + c) tan(A/2) : :
Barycentrics    a + ra : b + rb : c + rc, where ra, rb, rc are the exradii
Barycentrics    (a + b - c)*(a - b + c)*(a*(a - b - c) - S) : :
X(176) = 2s*X(1) + (r + 4R)*X(7) = 3X[2]-4X[31535]

The points X(175) and X(176) are discussed in an 1890 article by Emile Lemoine, accessible at Gallica. The article begins on page 111, and the two points are considered beginning on page 128.

The following construction was found by Elkies: call two circles within ABC companion circles if they are the incircles of two triangles formed by dividing ABC into two smaller triangles by passing a line through one of the vertices and some point on the opposite side; chain of circles O(1), O(2), ... such that O(n),O(n+1) are companion incircles for every n consists of at most six distinct circles; there is a unique chain consisting of only three distinct circles; and for this chain, the three subdividing lines concur in X(176).

A point X is defined as a point of equal detour of triangle ABC if |XB| + |XC| - |BC| = |XC| + |XA| - |CA| = |XA| + |XB| - |AB|. Veldkamp established that X = X(176) for some triangles ABC, but the conditions he gives are not correct. Hajja and Yff proved that the condition tan(A/2) + tan(B/2) + tan(C/2) < 2 is necessary and sufficient for the existence of exactly two points of equal detour and that the condition tan(A/2) + tan(B/2) + tan(C/2) = 2 is necessary and sufficient for the existence of exactly one point of equal detour. Yff found that X(176) is also is the center of the inner Soddy circle. See also X(175) and the 1st and 2nd Eppstein points, X(481), X(482).

G. R. Veldkamp, "The isoperimetric point and the point(s) of equal detour," American Mathematical Monthly 92 (1985) 546-558.

Noam D. Elkies and Jiro Fukuta, Problem E 3236 and Solution, American Mathematical Monthly 97 (1990) 529-531 [proposed 1987].

Mowaffaq Hajja and Peter Yff, "The isoperimetric point and the point(s) of equal detour in a triangle," Journal of Geometry 87 (2007) 76-82.

If you have GeoGebra, you can view X(175), X(176).ggb

Let Ia, Ib, Ic be the centers of the Elkies companion incircles. Let A' be the trilinear product Ib*Ic, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(482). The lines IaA', IbB', IcC' concur in X(176). (Randy Hutson, December 2, 2017)

Let Q be a point. Let A' be the incenter of triangle BCQ, and define B' and C' cyclically. If Q is an equal detour point, in the sense that |QB| + |QC| - |BC| = |QC| + |QA| - |CA| = |QA| + |QB| - |AB|, then Q is either the incenter or an excenter of triangle A'B'C'. (Do there exist other points with this property?) (Randy Hutson, 9/23/2011)''

For the Moses-Gheorghe-Reznick conic, see X(175).

In the plane of a triangle ABC, let

Oa = circle with diameter BC
C' = the point, other than B, where Oa meets AB
B' = the point, other than C, where Oa meets AC
Ub = B-mixtilinear incircle of BCC'
Uc = C-mixtilinear incircle of CBC'
Ab = touchpoint of Ub and BC; define Bc and Ca cyclically
Ac = touchpoint of Uc and BC; define Ba and Cb cyclically
The points Ab, Ac, Ba, Ba, Cb, Cb line on a conic, of which the center is X(176). A barycentric equation for this conic follows:

2 (a-b-c) (a^2-b^2-c^2) (a^2 (b+c)-(b-c)^2 (b+c)+2 a S) x^2 - ((a+b-c) (a-b+c) (3 a^4-2 a^3 (b+c)-3 (b^2-c^2)^2+2 a (b+c) (b^2+c^2))+2 (a^4-3 (b^2-c^2)^2+2 a^2 (b^2+c^2)) S) y z + (cyclic) = 0.

See X(175) and X(176). Continuing, let

Vb = B-mixtilinear excircle of BCC'
Vc = C-mixtilinear excircle of CBC'
A'b = touchpoint of Vb and BC; define B'c and C'a cyclically
A'c = touchpoint of Vc and BC; define B'a and C'b cyclically
The points A'b, A'c, B'a, B'c, C'a, C'b line on a conic, of which the center is X(175). A barycentric equation for this conic follows:

(a+b-c) (a-b+c) (2 (a-b-c) (2 a^3 b c+a^4 (b+c)-2 a b c (b+c)^2+(b-c)^2 (b+c)^3-2 a^2 (b^3+b^2 c+b c^2+c^3)+2 (a^3-2 b c (b+c)-a (b+c)^2) S) x^2 + (3 a^6-4 a^5 (b+c)-4 a (b-c)^2 (b+c)^3-3 (b-c)^2 (b+c)^4+a^2 (b+c)^2 (9 b^2+2 b c+9 c^2)-a^4 (9 b^2+14 b c+9 c^2)+8 a^3 (b^3+b^2 c+b c^2+c^3)-2 (a^4+6 a^3 (b+c)-4 a^2 (b+c)^2+3 (b^2-c^2)^2-2 a (3 b^3+7 b^2 c+7 b c^2+3 c^3)) S) y z) + (cyclic) = 0. (Angel Montesdeoca, September 9, 2022)

X(176) lies on K032, K199, K200, K1175, Q074, Q092, Q104 and these lines: {1, 7}, {2, 1659}, {4, 10904}, {8, 1271}, {40, 34494}, {65, 6404}, {69, 10907}, {105, 30386}, {144, 30556}, {174, 1274}, {226, 1132}, {241, 38487}, {280, 40699}, {329, 3083}, {388, 10910}, {489, 664}, {517, 39794}, {651, 1124}, {1086, 44635}, {1123, 10252}, {1463, 7362}, {1587, 8953}, {1588, 8978}, {2082, 6204}, {2550, 45714}, {3062, 10972}, {3084, 9776}, {3177, 31408}, {3296, 34215}, {3297, 6180}, {3298, 5228}, {3622, 8243}, {4000, 7969}, {4644, 7968}, {5045, 39795}, {5222, 18991}, {5226, 5405}, {5261, 9906}, {5393, 5435}, {5932, 31529}, {5933, 31531}, {6283, 20358}, {6360, 37881}, {6405, 49537}, {7586, 30324}, {8973, 23259}, {9778, 10134}, {10405, 14121}, {10580, 16662}, {11294, 17086}, {13386, 20211}, {13388, 21454}, {13437, 16440}, {15913, 31537}, {16213, 32080}, {17365, 44636}, {17950, 43133}, {21453, 30336}, {30346, 31589}, {30412, 31548}, {31545, 32200}

X(176) = reflection of X(i) in X(j) for these {i,j}: {20, 8984}, {175, 3160}, {7090, 31535}, {10405, 14121}, {30333, 1}
X(176) = isogonal conjugate of X(30335)
X(176) = isotomic conjugate of X(40700)
X(176) = anticomplement of X(7090)
X(176) = anticomplement of the isogonal conjugate of X(6502)
X(176) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {6, 13387}, {603, 175}, {1806, 3869}, {2066, 329}, {2067, 31551}, {6502, 8}, {13389, 69}, {13390, 21270}, {16232, 4}, {30556, 3436}, {34125, 13386}
X(176) = X(8)-Ceva conjugate of X(175)
X(176) = X(10134)-cross conjugate of X(7)
X(176) = X(i)-isoconjugate of X(j) for these (i,j): {1, 30335}, {6, 15892}, {31, 40700}, {56, 34912}
X(176) = cevapoint of X(i) and X(j) for these (i,j): {1, 34494}, {10252, 32057}
X(176) = crosssum of X(55) and X(19038)
X(176) = reflection of X(176) in the Soddy line
X(176) = X(6291)-of-excentral-triangle
X(176) = X(1151)-of-intouch-triangle
X(176) = X(7362)-of-(inner)-tangential-mid-arc-triangle (TCCT 6.15)
X(176) = X(12223)-of-first-circumperp-triangle (TCCT 6.21)
X(176) = X(6239)-of-second-circumperp-triangle (TCCT 6.22)
X(176) = barycentric product X(i)*X(j) for these {i,j}: {7, 30412}, {8, 16663}, {13390, 31548}
X(176) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 15891}, {2, 40699}, {6, 30336}, {9, 34911}, {9778, 30412}, {16662, 7}, {30413, 8}
X(176) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 7, 175}, {1, 481, 17802}, {1, 482, 7}, {1, 1371, 482}, {1, 1373, 481}, {1, 1374, 31539}, {1, 17804, 21170}, {1, 17806, 31538}, {1, 21170, 17801}, {1, 21171, 31602}, {1, 30341, 11038}, {1, 31538, 17805}, {1, 31567, 8236}, {1, 31601, 21169}, {7, 482, 21169}, {7, 1371, 17804}, {7, 8236, 31565}, {7, 17802, 481}, {7, 17805, 1}, {7, 21169, 21170}, {7, 31601, 482}, {7, 31602, 1374}, {20, 347, 175}, {77, 962, 175}, {175, 482, 21170}, {175, 17804, 21169}, {175, 21169, 7}, {269, 9785, 175}, {279, 390, 175}, {481, 482, 1373}, {481, 1373, 7}, {481, 17802, 175}, {482, 1371, 31601}, {482, 17805, 175}, {482, 17806, 17805}, {482, 31538, 1}, {482, 31539, 21171}, {482, 31601, 17804}, {1323, 30332, 175}, {1371, 17805, 21169}, {1371, 17806, 1}, {1371, 31538, 7}, {1374, 21171, 7}, {1374, 31539, 31602}, {1442, 4295, 175}, {1443, 30305, 175}, {1659, 13389, 2}, {3600, 3672, 175}, {3638, 3639, 1371}, {3663, 4308, 175}, {3664, 4323, 175}, {3668, 4313, 175}, {3674, 4344, 175}, {4296, 4329, 175}, {4297, 36640, 175}, {4318, 17170, 175}, {4862, 6049, 175}, {5542, 5543, 175}, {5731, 22464, 175}, {7090, 31535, 2}, {7190, 11037, 175}, {8236, 10481, 175}, {10134, 32082, 16663}, {10481, 31567, 31565}, {17804, 21169, 482}, {17805, 31601, 7}, {21171, 31539, 1374}, {30424, 31721, 175}, {31538, 31601, 175}, {31539, 31602, 175}


X(177) = 1st MID-ARC POINT

Trilinears    (cos B/2 + cos C/2) sec A/2 : (cos C/2 + cos A/2) sec B/2 : (cos A/2 + cos B/2) sec C/2
Trilinears    (b' + c')/a' : :, where a', b', c' are sidelengths of excentral triangle
Barycentrics    (Cos[B/2] + Cos[C/2])*Sin[A/2] : :

Let A',B',C' be the first points of intersection of the angle bisectors of ABC with its incircle. The tangents at A', B', C' form a triangle A″B″C″, and the lines AA″,BB″,CC″ concur in X(177). Also, X(177) = X(1) of the intouch triangle.

Clark Kimberling and G. R. Veldkamp, Problem 1160 and Solution, Crux Mathematicorum 13 (1987) 298-299 [proposed 1986].

X(177) is the perspector of ABC and the Yff central triangle, and X(177) is X(65)-of-the-Yff-central-triangle . (Darij Grinberg, Hyacinthos #7689, 8/25/2003)

If you have The Geometer's Sketchpad, you can view X(177) and Yff Central Triangle.

X(177) lies on on the Feuerbach circumhyperbola, the curves K746, K747, K748, and these lines: {1, 167}, {2, 11691}, {4, 8095}, {7, 555}, {8, 556}, {9, 173}, {11, 12614}, {12, 12622}, {21, 7587}, {40, 31790}, {55, 12518}, {56, 12523}, {57, 164}, {65, 31768}, {80, 12771}, {84, 9836}, {104, 13444}, {165, 31801}, {178, 2090}, {210, 58689}, {226, 12694}, {234, 10489}, {260, 7589}, {266, 8133}, {314, 35624}, {354, 5571}, {481, 12492}, {553, 58707}, {942, 12813}, {1130, 10215}, {1156, 30367}, {1251, 30373}, {1284, 13091}, {1319, 55172}, {1320, 8097}, {1388, 55176}, {1420, 55175}, {2099, 55173}, {2346, 8076}, {3057, 31766}, {3058, 31770}, {3339, 55169}, {3340, 12656}, {3361, 55168}, {3576, 31791}, {3577, 9837}, {3660, 58614}, {3680, 11534}, {3911, 58440}, {3982, 58715}, {4031, 58705}, {4114, 58714}, {4654, 58711}, {5049, 31796}, {5219, 58712}, {5221, 55170}, {5226, 58717}, {5434, 31734}, {5435, 58708}, {5558, 9795}, {5919, 31767}, {6284, 31769}, {6597, 13124}, {6724, 10493}, {7133, 8247}, {7354, 31735}, {7371, 45086}, {7707, 8135}, {7962, 8392}, {7991, 31800}, {8084, 11033}, {8085, 8379}, {8087, 8382}, {8099, 12491}, {8101, 13098}, {8103, 13267}, {8113, 12879}, {8114, 12884}, {8120, 34025}, {8126, 11690}, {8243, 13090}, {8372, 21619}, {8581, 12450}, {8729, 8733}, {9853, 12130}, {10390, 45707}, {10439, 31784}, {10473, 35644}, {10490, 15997}, {10498, 55329}, {10501, 46695}, {10504, 10506}, {10505, 12809}, {11192, 11195}, {12406, 12916}, {12435, 31783}, {12568, 12570}, {12580, 12582}, {12714, 12716}, {12726, 12728}, {12870, 12873}, {13072, 13074}, {17625, 17657}, {17629, 17631}, {21454, 58706}, {21465, 30369}, {30371, 30394}, {30372, 30409}, {31231, 58713}, {31578, 31592}, {31579, 31593}, {32636, 55171}, {35681, 35899}, {46876, 53076}, {53006, 53007}

X(177) = midpoint of X(i) and X(j) for these {i,j}: {7, 7670}, {9807, 12539}
X(177) = reflection of X(i) in X(j) for these {i,j}: {1, 12908}, {40, 31790}, {65, 31768}, {164, 12443}, {942, 12813}, {3057, 31766}, {5571, 58616}, {6284, 31769}, {7354, 31735}, {7991, 31800}, {8422, 1}, {10501, 46695}, {11234, 11191}, {11691, 18258}, {12435, 31783}, {12694, 21633}, {17641, 5571}, {18258, 58444}, {42017, 178}
X(177) = isogonal conjugate of X(260)
X(177) = complement of X(11691)
X(177) = anticomplement of X(18258)
X(177) = crosspoint of X(7) and X(174)
X(177) = crosssum of X(55) and X(259)
X(177) = incircle-inverse of X(13385)
X(177) = X(i)-Ceva conjugate of X(j) for these (i,j): {7, 234}, {174, 7707}, {2089, 10490}, {7057, 178}, {10215, 13385}, {55329, 10492}
X(177) = X(i)-isoconjugate of X(j) for these (i,j): {1, 260}, {101, 10492}, {6733, 10495}, {16012, 59467}
X(177) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 260}, {177, 11691}, {178, 8}, {1015, 10492}, {2090, 7048}, {10493, 188}, {16015, 53123}, {16016, 556}
X(177) = cevapoint of X(16012) and X(18888)
X(177) = trilinear pole of line {650, 6728}
X(177) = SS(A→A')-of-X(10), where A'B'C' is the excentral triangle
X(177) = X(4)-of-mid-arc triangle
X(177) = X(1829)-of-excentral triangle
X(177) = perspector of ABC and mid-triangle of 1st tangential mid-arc triangle and Yff central triangle
barycentric product X(i)*X(j) for these {i,j}: {7, 16016}, {8, 14596}, {75, 18888}, {85, 16012}, {174, 178}, {188, 234}, {514, 55342}, {522, 55328}, {556, 10490}, {2089, 2090}, {4146, 7707}, {4391, 13444}, {6728, 55341}, {6730, 55329}, {7057, 16015}, {18886, 42017}, {41799, 53122}
X(177) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 260}, {178, 556}, {234, 4146}, {513, 10492}, {2090, 53123}, {2091, 21456}, {6729, 10495}, {7707, 188}, {10490, 174}, {10502, 16016}, {13444, 651}, {14596, 7}, {15997, 7028}, {16011, 258}, {16012, 9}, {16015, 7048}, {16016, 8}, {18887, 42017}, {18888, 1}, {41799, 1488}, {55328, 664}, {55342, 190}, {58968, 6733}
X(177) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 2089, 13385}, {1, 8422, 11234}, {1, 12908, 11191}, {1, 13092, 174}, {1, 46370, 6585}, {1, 52999, 13443}, {1, 58868, 2089}, {2, 11691, 18258}, {174, 2089, 1}, {174, 58868, 13385}, {266, 8133, 58777}, {354, 17641, 5571}, {1143, 1274, 11891}, {2089, 11924, 13443}, {5571, 58616, 354}, {7589, 8075, 260}, {7590, 8081, 1}, {8083, 10967, 11032}, {8089, 8423, 1}, {8091, 8351, 1}, {8092, 11044, 1}, {8241, 11924, 1}, {8241, 52999, 13385}, {8422, 11191, 1}, {10489, 10499, 234}, {10502, 10503, 10500}, {11894, 11896, 1}, {13092, 58868, 1}, {18258, 58444, 2}, {18448, 18454, 1}, {30370, 30408, 1}, {30374, 30411, 1}


X(178) = 2nd MID-ARC POINT

Trilinears       (cos B/2 + cos C/2) csc A : (cos C/2 + cos A/2) csc B : (cos A/2 + cos B/2) csc C
Barycentrics  cos B/2 + cos C/2 : cos C/2 + cos A/2 : cos A/2 + cos B/2
X(178) = 3 X[2] + X[7057], X[12879] - 5 X[58712]

Let A',B',C' be the first points of intersection of the angle bisectors of ABC with its incircle. Let A″,B″,C″ be the midpoints of egments BC,CA,AB, respectively. The lines A'A″,B'B″,C'C″ concur in X(178).

Clark Kimberling, Problem 804, Nieuw Archikef voor Wiskunde 6 (1988) 170.

X(178) lies on these lines: {2, 188}, {8, 236}, {10, 12908}, {85, 4146}, {142, 18258}, {174, 60598}, {177, 2090}, {189, 39122}, {312, 7027}, {946, 12489}, {1121, 55341}, {1311, 58968}, {5430, 6557}, {5431, 5451}, {6245, 12443}, {8422, 21623}, {10489, 16016}, {12622, 45304}, {12879, 58712}, {34234, 43192}, {52156, 55329}, {52797, 58440}

X(178) = midpoint of X(i) and X(j) for these {i,j}: {8, 12646}, {177, 42017}, {188, 7057}
X(178) = reflection of X(52797) in X(58440)
X(178) = complement of X(188)
X(178) = crosspoint of X(2) and X(508)
X(178) = complement of the isogonal conjugate of X(266)
X(178) = complement of the isotomic conjugate of X(4146)
X(178) = X(i)-complementary conjugate of X(j) for these (i,j): {6, 2090}, {31, 16016}, {56, 178}, {174, 141}, {188, 1329}, {259, 3452}, {266, 10}, {289, 34849}, {508, 20543}, {509, 20334}, {555, 17046}, {556, 21244}, {604, 16015}, {4146, 2887}, {6724, 3454}, {6727, 960}, {6728, 124}, {6729, 26932}, {6733, 513}, {7370, 142}, {7371, 2886}, {7591, 21530}, {18753, 14218}, {60533, 1211}, {60538, 20527}, {60539, 9}
X(178) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 16016}, {7057, 177}
X(178) = X(i)-isoconjugate of X(j) for these (i,j): {109, 10495}, {260, 266}
X(178) = X(i)-Dao conjugate of X(j) for these (i,j): {11, 10495}, {178, 188}, {2090, 1488}, {10493, 1}, {16015, 7048}, {16016, 2}
X(178) = trilinear pole of line {522, 6730}
X(178) = barycentric product X(i)*X(j) for these {i,j}: {8, 234}, {75, 7707}, {177, 556}, {312, 10490}, {522, 55341}, {2090, 7057}, {3239, 55329}, {4146, 16016}, {4391, 43192}, {7027, 14596}, {16015, 53122}, {35519, 58968}
X(178) = barycentric quotient X(i)/X(j) for these {i,j}: {177, 174}, {234, 7}, {259, 260}, {650, 10495}, {2090, 7048}, {6728, 10492}, {7707, 1}, {10489, 234}, {10490, 57}, {10502, 7707}, {14596, 7371}, {15997, 258}, {16011, 289}, {16012, 259}, {16015, 1488}, {16016, 188}, {18885, 10490}, {18887, 15997}, {18888, 266}, {42017, 7028}, {43192, 651}, {55329, 658}, {55341, 664}, {58968, 109}
{X(2),X(7057)}-harmonic conjugate of X(188)


X(179) = 1st AJIMA-MALFATTI POINT

Trilinears    sec4(A/4) : sec4(B/4) : sec4(C/4)
Barycentrics    sin A sec4(A/4) : sin B sec4(B/4) : sin C sec4(C/4)

The famous Malfatti Problem is to construct three circles O(A), O(B), O(C) inside ABC such that each is externally tangent to the other two, O(A) is tangent to lines AB and AC, O(B) is tangent to BC and BA, and O(C) is tangent to CA and CB. Let A' = O(B)∩O(C), B' = O(C)∩O(A), C' = O(A)∩O(B). The lines AA',BB',CC' concur in X(179). Trilinears are found in Yff's unpublished notes. See also the Yff-Malfatti Point, X(400), having trilinears csc4(A/4) : csc4(B/4) : csc4(C/4), and the references for historical notes.

H. Fukagawa and D. Pedoe, Japanese Temple Geometry Problems (San Gaku), The Charles Babbage Research Centre, Winnipeg, Canada, 1989.

Michael Goldberg, "On the original Malfatti problem," Mathematics Magazine, 40 (1967) 241-247.

Clark Kimberling and I. G. MacDonald, Problem E 3251 and Solution, American Mathematical Monthly 97 (1990) 612-613.

Let A', B', C' be the centers of the Malfatti circles. Let A″ be the trilinear product B'*C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(1142). The lines A'A″, B'B″, C'C″ concur in X(179). (Randy Hutson, July 11, 2019)

If you have The Geometer's Sketchpad, you can view X(179). For an artistic design generated by X(179), see X(244).

X(179) lies on these lines: {1, 1142}, {483, 8242}, {1143, 9795}, {31957, 53078}

X(179) = X(400)-isoconjugate of X(1106)
X(179) = X(6552)-Dao conjugate of X(400)
X(179) = barycentric quotient X(i)/X(j) for these {i,j}: {346, 400}, {483, 557}, {7014, 53116}
X(179) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1142, 21455}, {1142, 31495, 1}


X(180) = 2nd AJIMA-MALFATTI POINT

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/t(B,C,A) + 1/t(C,B,A) - 1/t(A,C,B),
                        t(A,B,C) = 1 + 2(sec A/4 cos B/4 cos C/4)2
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)
Barycentrics   (1 + Cos[B/2])*(1 + Cos[C/2])*((2 + Cos[A/2] + Cos[B/2] + Cos[C/2])^2 + Cos[A/2]*(-(2 + Cos[A/2])^2 + Cos[B/2]^2 + Cos[C/2]^2))*Sin[A] : :

Let A″,B″,C″ be the excenters of ABC, and let A',B',C' be as in the construction of X(179). The lines A'A″,B'B″,B'B″ concur in X(180). Trilinears are found in Yff's unpublished notes. See X(179).

If you have The Geometer's Sketchpad, you can view a class="bold" href="X(180).gsp">X(180) and X(180) External.

X(180) lies on this line: 173,483


X(181) = APOLLONIUS POINT

Trilinears       a(b + c)2/(b + c - a) : b(c + a)2/(c + a - b) : c(a + b)2/(a + b - c)
                        = a2cos2(B/2 - C/2) : b2cos2(C/2 - A/2) : c2cos2(A/2 - B/2)

Trilinears        h(a,b,c) : h(b,c,a) : h(c,a,b), where h(a,b,c) = [r cos(A/2) + s sin(A/2)]2, s = semiperimeter, r = inradius

Barycentrics  a3cos2(B/2 - C/2) : b3cos2(C/2 - A/2) : c3cos2(A/2 - B/2)

Let O(A),O(B),O(C) be the excircles. Apollonius's Problem includes the construction of the circle O tangent to the three excircles and encompassing them. (The circle is called the Apollonius circle.) Let A' = O∩O(A), B'=O∩O(B), C'=O∩O(C). The lines AA',BB',CC' concur in X(181). Yff derived trilinears in 1992.

X(181) is the external center of similitude (or exsimilicenter) of the incircle and Apollonius circle. The internal center is X(1682). (Peter J. C. Moses, 8/22/2003)

X(181) is the isogonal conjugate of the isotomic conjugate of X(12); also, X(181) is the {X(i) ,X(j) }-harmonic conjugate of X(k) for these (i,j,k)): (31,51,3271), (42,1400,1402), (57,1401,1357), (57,1469,1401). (Peter J. C. Moses, 6/20/2014)

A proof of the the concurrence of lines AA',BB',CC' follows.
        A = exsimilicenter(incircle, A-excircle)
        A' = exsimilicenter(A-excircle, Apollonius circle)
        Let J = exsimilicenter(incircle, Apollonius circle).
By Monge's theorem, the points A, A', J are collinear. In particular, J lies on line AA', and cyclically, J lies on lines BB' and CC'. Therefore, J = X(181). (Darij Grinberg, Hyacinthos, 7461, 8/10/03)

Let A' be the inverse-in-excircles-radical-circle of the midpoint of BC, and define B', C' cyclically. Triangle A'B'C' is inscribed in the Apollonius circle and homothetic to the intouch triangle at X(181). (Randy Hutson, March 21, 2019)

See also Clark Kimberling, Shiko Iwata, and Hidetosi Fukagawa, Problem 1091 and Solution, Crux Mathematicorum 13 (1987) 128-129; 217-218. [proposed 1985].

X(181) lies on these lines: {1,970}, {6,197}, {8,959}, {10,12}, {11,2051}, {25,2175}, {31,51}, {33,3022}, {42,228}, {43,57}, {44,375}, {55,573}, {56,386}, {58,1324}, {81,5061}, {171,511}, {182,5329}, {200,3779}, {213,2333}, {373,748}, {389,3072}, {518,3687}, {553,1463}, {575,5363}, {612,3688}, {750,3917}, {756,2171}, {942,5530}, {994,1361}, {1124,1685}, {1254,1425}, {1317,3032}, {1335,1686}, {1356,5213}, {1358,3034}, {1364,5348}, {1376,4259}, {1395,1843}, {1672,1683}, {1673,1684}, {1674,1693}, {1675,1694}, {1695,1697}, {2007,2019}, {2008,2020}, {2330,5285}, {2534,2538}, {2535,2539}, {3027,3029}, {3028,3031}, {3056,5269}, {3340,4517}, {3781,5268}, {3792,3819}, {4276,5172}

X(181) = isogonal conjugate of X(261)
X(181) = complement of X(35614)
X(181) = X(i)-Ceva conjugate of X(j) for these (i,j): (12,2197), (59,4559), (65,2171), (2171,1500)
X(181) = X(i)-cross conjugate of X(j) for these (i,j): (872,1500), (2643,512)
X(181) = crosspoint of X(i) and X(j) for these (i,j): (42,1824), (59,4559), (65,1400), (1354,2171)
X(181) = crosssum of X(i) and X(j) for these (i,j): (2,2975), (11,4560), (21,333), (81,4225), (86,1444), (1098,2185)
X(181) = crossdifference of X(3904) and X(3910)
X(181) = X(i)-beth conjugate of X(j) for these (i,j): (42,181), (660,181), (756,756)
X(181) = {X(1),X(970)}-harmonic conjugate of X(1682)
X(181) = X(60)-isoconjugate of X(75)
X(181) = trilinear product of vertices of extangents triangle

X(181) = trilinear product of X(i) and X(j) for these {I,J}:
{1,181}, {6,2171}, {7,872}, {10,1402}, {12,31}, {19,2197}, {25,201}, {33,1425}, {34,3690}, {37,1400}, {42,65}, {55,1254}, {56,756}, {57,1500}, {59,2643}, {71,1880}, {73,1824}, {109,4705}, {115,2149}, {210,1042}, {213,226}, {225,228}, {227,2357}, {349,2205}, {512,4551}, {594,604}, {608,3949}, {651,4079}, {661,4559}, {762,1412}, {798,4552}, {1020,3709}, {1089,1397}, {1110,1365}, {1214,2333}, {1334,1427}, {1395,3695}, {1409,1826}, {1415,4024}, {1426,2318}, {1441,1918}, {3063,4605}, {3124,4564}, {4017,4557}

X(181) = barycentric product of X(i) and X(j) for these {I,J}:
{1,2171}, {2,181}, {4,2197}, {6,12}, {7,1500}, {9,1254}, {10,1400}, {19,201}, {34,3949}, {37,65}, {42,226}, {56,594}, {57,756}, {59,115}, {71,225}, {72,1880}, {73,1826}, {85,872}, {109,4024}, {210,1427}, {213,1441}, {227,1903}, {278,3690}, {281,1425}, {307,2333}, {321,1402}, {349,1918}, {512,4552}, {523,4559}, {604,1089}, {608,3695}, {651,4705}, {661,4551}, {663,4605}, {664,4079}, {762,1014}, {1018,4017}, {1020,4041}, {1042,2321}, {1091,2150}, {1109,2149}, {1214,1824}, {1252,1365}, {1262,4092}, {1334,3668}, {1404,4013}, {1411,4053}, {1415,4036}, {1426,3694}, {2222,2610}, {2643,4564}, {3124,4998}, {3709,4566}

X(181) = X(i)-isoconjugate of X(j) for these (i,j):
(1,261), (2,2185), (7,1098), (8,757), (9,1509), (21,86), (27,1812), (28,332), (29,1444), (55,873), (58,314), (60,75), (69,270), (76,2150), (81,333), (99,3737), (200,552), (249,4858), (274,284), (283,286), (304,2189), (310,2194), (312,593), (348,2326), (514,4612), (645,1019), (649,4631), (650,4610), (662,4560), (663,4623), (693,4636), (763,2321), (849,3596), (1014,1043), (1021,4573), (1434,2287), (2170,4590), (4391,4556)


X(182) = MIDPOINT OF BROCARD DIAMETER

Trilinears       cos(A- ω) : cos(B - ω) : cos(C -ω)
Trilinears       cos A + sin A tan ω : cos B + sin B tan ω : cos C + sin C tan ω
Trilinears       sin A - sin(A - 2ω) : sin B - sin(B - 2ω) : sin C - sin(C - 2ω)
Trilinears       cos A + cos(A - 2ω) : cos B + cos(B - 2ω) : cos C + cos(C - 2ω) (cf., X(39))
Trilinears       a + 2R cot ω cos A : b + 2R cot ω cos B: c + 2R cot ω cos C (cf., X(1350), X(1351))
Trilinears       sin A + cos A cot ω : sin B + cos B cot ω : sin C + cos C cot ω (cf., X(575), X(576),,X(1350), X(1351))
Trilinears       cos A + (2 - 2 cot ω) sin A : cos B + (2 - 2 cot ω) sin B : cos C + (2 - 2 cot ω) sin C
Barycentrics  sin A cos(A - ω) : sin B cos(B - ω) : sin C cos(C -ω)
Barycentrics   a^2(a^4 - a^2b^2 - a^2c^2 - 2b^2c^2) : :
X(182) = X(3) + X(6) X(182) is the midpoint of the Brocard diameter (the segment X(3)-to-X(6)); also the center of the 1st Lemoine circle, and the center of the Brocard circle. If you have The Geometer's Sketchpad, you can view X(1316), which includes X(182).

X(182) = radical center of Lucas(2 tan ω) circles, where 2 tan ω is the value of t for which the Brocard circle is the radical circle of the Lucas(t) circles. (Randy Hutson, January 29, 2015)

X(182) lies on these lines: 1,983   2,98   3,6   4,83   5,206   10,1678   22,51   24,1843   25,3066  30,597   36,1469   40,1700   54,69   55,613   56,611   111,353   140,141   171,1397   373,1495   474,1437   517,1386   518,1385   524,549   691,2698   692,1001   727,1293   729,1296   952,996

X(182) is the {X(371),X(372)}-harmonic conjugate of X(39). For a list of other harmonic conjugates of X(182), click Tables at the top of this page.

X(182) = midpoint of X(3) and X(6)
X(182) = reflection of X(i) in X(j) for these (i,j): (6,575), (141,140), (576,6)
X(182) = isogonal conjugate of X(262)
X(182) = isotomic conjugate of X(327)
X(182) = complement of X(1352)
X(182) = X(3)-of-1st-Brocard triangle
X(182) = X(3)- of 2nd Brocard triangle
X(182) = X(182)-of-circumsymmedial triangle
X(182) = {X(3),X(6)}-harmonic conjugate of X(511)
X(182) = {X(6),X(1350)}-harmonic conjugate of X(1351)
X(182) = {X(1340),X(1341)}-harmonic conjugate of X(3)
X(182) = {X(1687),X(1688)}-harmonic conjugate of X(6)
X(182) = vertex conjugate of PU(191)
X(182) = X(5)-of-obverse-triangle-of-X(69)
X(182) = inverse-in-circumcircle of X(2080)
X(182) = inverse-in-2nd-Brocard-circle of X(3095)
X(182) = inverse-in-circle-{X(3102),X(3103),PU(1)}} of X(32452)
X(182) = exsimilicenter of circle centered at X(371) through X(1151) and circle centered at X(1152) through X(372)
X(182) = exsimilicenter of circle centered at X(372) through X(1152) and circle centered at X(1151) through X(371)
X(182) = radical trace of circles with diameters X(371)X(372) and X(1151)X(1152)
X(182) = harmonic center of 1st and 2nd Kenmotu circles
X(182) = {X(15),X(16)}-harmonic conjugate of X(574)
X(182) = harmonic center of Lucas radical circle and Lucas(-1) radical circle
X(182) = harmonic center of Lucas inner circle and Lucas(-1) inner circle
X(182) = harmonic center of 2nd Lemoine circle and circle {X(1687),X(1688),PU(1),PU(2)}}
X(182) = radical trace of circles O(15,16) and O(61,62)
X(182) = exsimilicenter of circles {X(15),X(62),PU(1)} and {X(16),X(61),PU(1)}; the insimilicenter is X(32)
X(182) = X(2)-of-1st-Ehrmann-triangle
X(182) = {X(9738),X(9739)}-harmonic conjugate of X(9737)
X(182) = Artzt-to-McCay similarity image of X(381)
X(182) = X(3)-of-6th-anti-Brocard-triangle
X(182) = X(5476)-of-4th-anti-Brocard-triangle
X(182) = homothetic center of 5th anti-Brocard triangle and cevian triangle of X(3)
X(182) = homothetic center of 6th anti-Brocard triangle and 1st Brocard triangle
X(182) = endo-homothetic center of 6th Brocard triangle and 1st anti-Brocard triangle
X(182) = perspector of 1st Neuberg triangle and cross-triangle of 1st and 2nd Neuberg triangles
X(182) = Cundy-Parry Phi transform of X(39)
X(182) = Cundy-Parry Psi transform of X(83)
X(182) = endo-similarity image of reflection triangles of PU(1); the similitude center of these triangles is X(6)
X(182) = 1st-Brocard-isogonal conjugate of X(1352)


X(183) = TRILINEAR PRODUCT X(75)X(182)

Trilinears    b2c2cos(A- ω) : c2a2cos(B - ω) : a2b2cos(C - ω)
Barycentrics   csc A cos(A - ω) : csc B cos(B - ω) : csc C cos(C - ω)
Barycentrics    cot A + tan ω : :
X(183) = 3*X(2) - 2(cos ω)2*X(6)

Let A'B'C' be the circummedial triangle. Let La be the line through A parallel to B'C', and define Lb and Lc cyclically. Let A″ = Lb∩Lc, B″ = Lc∩La, C″ = La∩Lb. The lines A'A″, B'B″, C'C″ concur in X(183). (Randy Hutson, December 26, 2015)

X(183) lies on these lines: 2,6   3,76   5,315   22,157   25,264   55,350   95,305   187,1003   274,474   316,381   317,427   383,621   538,574   622,1080   668,956

X(183) is the {X(2),X(69)}-harmonic conjugate of X(325). For a list of other harmonic conjugates of X(183), click Tables at the top of this page.

X(183) = isogonal conjugate of X(263)
X(183) = isotomic conjugate of X(262)
X(183) = X(645)-beth conjugate of X(183)
X(183) = {X(2),X(69)}-harmonic conjugate of X(325)
X(183) = X(6)-of-circummedial-triangle
X(183) = pole wrt circumcircle of trilinear polar of X(3114) (line X(669)X(804))
X(183) = insimilicenter of Artzt and anti-Artzt circles; the exsimilicenter is X(2)
X(183) = crossdifference of every pair of points on the line through [U(2) of pedal triangle of P(1)] and [P(2) of pedal triangle of U(1)]
X(183) = X(5034)-of-6th-Brocard-triangle


X(184) = INVERSE OF X(125) IN THE BROCARD CIRCLE

Trilinears       a2cos A : :
Trilinears        sin A sin 2A : :
Barycentrics  a3cos A : :

X(184) is the homothetic center of triangles ABC and A'B'C', the latter defined as follows: let B1 and C1 be the points where the perpendicular bisector BC meets sidelines CA and AB, and cyclically define C2, A2; A3, B3. Then A'B'C' is formed by the perpendicular bisectors of segments B1C1, C2A2, A3B3. (Fred Lang, Hyacinthos #1190)

X(184) is the subject of Hyacinthos messages 5423-5441 (May, 2002). In #5423, Alexei Myakishev notes that X(184) serves as a common vertex of three triangles inside ABC, mutually congruent and similar to ABC. (The triangles can be labeled XBCCB, XCAAC, XABBA, with BC and CB on side BC, CA and AC on side CA, and AB and BA on side AB.) See

Alexei Myakishev, On the Procircumcenter and Related Points, Forum Geometricorum 3 (2003) 29-34.

In #5435, Paul Yiu cites Fred Lang's construction of X(184) and notes that the three triangles are then easily constructed from X(184). The triangles determine three other triangles with common vertex X(184); in #5437, Nikos Dergiades notes that the vertex angles of these are 4A - π, 4B - π, 4C - π, and that

if ABC is acute, then X(184) = X(63)-of-the-orthic-triangle = X(226)-of-the-tangential-triangle
X(184) = homothetic center of the orthic triangle and the medial triangle of the tangential triangle.

Randy Hutson notes that X(184) is the exsimilicenter of the circumcircle and sine-triple-angle circle. (December 14, 2014)

Let A'B'C' be the intersections, other than X(3), of the X(3)-cevians and the Brocard circle. Let A″B″C″ be the intersections, other than X(6), of the X(6)-cevians and the Brocard circle. Then A'B'C' and A″B″C″ are perspective at X(184). Also, X(184) = U∩V, where U = isotomic conjugate of polar conjugate of Brocard axis (i.e., line X(3)X(49)), and V = polar conjugate of isotomic conjugate of Brocard axis (i.e., line X(6)X(25)). Let DEF be the orthic triangle. Let D' be the isotomic conjugate of X(4) wrt AEF, and define E' and F' cyclically; then the lines AD', BE', CF' concur in X(184). (Randy Hutson, June 1, 2015)

Let A'B'C' be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung). A'B'C' is homothetic to ABC at X(184).

In the plane of a triangle ABC, let
A'B'C' = tangential triangle;
Oa = circle with diameter BC, and define Ob and Oc cyclically;
Ab = A'BC'∩Oa, and define Bc and Ca cyclically;
Ac = B'CA'∩Oa, and define Ba and Cb cyclically;
A″= BcBa;∩CaCb, define B″ and C″ cyclically.
The triangle A″B″C″ is perspective to ABC, and the perspector is X(184).
(Dasari Naga Vijay Krishna, April 15, 2021)

X(184) lies on these lines: 2,98   3,49   4,54   5,156   6,25   23,576   24,389   26,52   22,511   31,604   32,211   48,212   55,215   157,570   160,571   199,573   205,213   251,263   351,686   381,567   397,463   398,462   418,577   572,1011   647,878

X(184) is the {X(6),X(25)}-harmonic conjugate of X(51). For a list of other harmonic conjugates of X(184), click Tables at the top of this page.

X(184) = isogonal conjugate of X(264)
X(184) = isotomic conjugate of X(18022)
X(184) = complement of X(11442)
X(184) = inverse-in-Brocard-circle of X(125)
X(184) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,22391), (6,32), (54,6), (74,50)
X(184) = X(217)-cross conjugate of X(6)
X(184) = crosspoint of X(3) and X(6)
X(184) = crosssum of X(i) and X(j) for these (i,j): (2,4), (5,324), (6, 157), (92,318), (273,342), (338,523), (339,850), (427,1235), (491,492)
X(184) = crossdifference of every pair of points on line X(297)X(525)
X(184) = X(32)-Hirst inverse of X(237)
X(184) = X(i)-beth conjugate of X(j) for these (i,j): (212,212), (692,184)
X(184) = X(22) of 1st Brocard triangle
X(184) = trilinear product of PU(19)
X(184) = trilinear pole of line X(3049)X(39201)
X(184) = {X(3),X(49)}-harmonic conjugate of X(1147)
X(184) = vertex conjugate of PU(157) (the polar conjugates of PU(38)
X(184) = X(75)-isoconjugate of X(4)
X(184) = {X(8880),X(8881)}-harmonic conjugate of X(25)
X(184) = homothetIc center of orthic triangle and X(3)-Ehrmann triangle; see X(25)
X(184) = perspector of ABC and unary cofactor triangle of tangential-of-tangential triangle
X(184) = perspector of ABC and unary cofactor triangle of MacBeath triangle
X(184) = intersection of tangents to Moses-Jerabek conic at X(6) and X(34469)
X(184) = 1st-Brocard-isogonal conjugate of X(24270)


X(185) = NAGEL POINT OF THE ORTHIC TRIANGLE

Trilinears    (cos A)[1 - cos A cos(B - C)] : :
Trilinears    (cos A)(cos^2 B + cos^2 C) : :
Trilinears    a(b^2 + c^2 - a^2)[2a^2(b^2 - c^2)^2 - a^4(b^2 + c^2) - (b^2 - c^2)^2(b^2 + c^2)] : :
Barycentrics  (sin 2A)[1 - cos A cos(B - C)] : (sin 2B)[1 - cos B cos(C - A)] : (sin 2C)[1 - cos C cos(A - B)]

Alexei Myakishev has noted that X(185) is the Nagel point of the orthic triangle only is ABC is an acute triangle.

Let Ha be the foot of the A-altitude. Let Ba and Ca be the feet of perpendiculars from Ha to CA and AB, respectively. Let Ga be the centroid of HaBaCa. Define Gb and Gc cyclically. The lines HaGa, HbGb, HcGc concur in X(185). (Randy Hutson, December 26, 2015)

Let Ha, Hb, Hc be the orthocenters of the A-, B-, and C-altimedial triangles. X(185) is the orthocenter of HaHbHc. (Randy Hutson, March 25, 2016)

Let P be a point on the circumcircle. Let Pa be the orthogonal projection of P on the A-altitude, and define Pb, Pc cyclically. The locus of the orthocenter of PaPbPc as P varies is an ellipse centered at X(185). See also X(9730). (Randy Hutson, March 25, 2016)

Let A'B'C' be the midheight triangle. Let AB, AC be the orthogonal projections of A' on CA, AB, resp. Define BC, BA, CA, CB cyclically. Let A″ = CAAC∩ABBA, and define B″ and C″ cyclically. Triangle A″B″C″ is homothetic to ABC at X(6) and perspective to the orthic triangle at X(185). (Randy Hutson, March 29, 2020)

X(185) lies on these lines: 1,296   3,49   4,51   5,113   6,64   20,193   25,1498   30,52   39,217   54,74   72,916   287,384   378,578   382,568   411,970   648,1105

X(185) = reflection of X(i) in X(j) for these (i,j): (4,389), (125,974)
X(185) = isogonal conjugate of X(1105)
X(185) = X(i)-Ceva conjugate of X(j) for these (i,j): (3,417), (4,235)
X(185) = crosspoint of X(3) and X(4)
X(185) = crosssum of X(i) and X(j) for these (i,j): (3,4), (25,1249)
X(185) = anticomplement of X(5907)
X(185) = bicentric sum of PU(17)
X(185) = PU(17)-harmonic conjugate of X(647)
X(185) = orthology center of orthic and half-altitude triangles
X(185) = half-altitude isogonal conjugate of X(4)
X(185) = orthic-isogonal conjugate of X(235)
X(185) = orthic-isotomic conjugate of X(1843)
X(185) = X(20)-of-X(4)-Brocard-triangle
X(185) = anticomplement of X(4) wrt orthic triangle
X(185) = X(4)-of-tangential-triangle-of-Jerabek-hyperbola
X(185) = eigencenter of cevian triangle of X(648)
X(185) = eigencenter of anticevian triangle of X(647)
X(185) = trilinear product of vertices of 2nd Hyacinth triangle
X(185) = X(10)-of-circumorthic-triangle if ABC is acute
X(185) = excentral-to-ABC functional image of X(8)
X(185) = polar-circle-inverse of X(34170)


X(186) = INVERSE-IN-CIRCUMCIRCLE OF X(4)

Trilinears    4 cos A - sec A : 4 cos B - sec B : 4 cos C - sec C
Trilinears    sin 3A csc 2A : sin 3B csc 2B : sin 3C csc 2C
Barycentrics    (sin A)(4 cos A - sec A) : (sin B)(4 cos B - sec B) : (sin C)(4 cos C - sec C)
Barycentrics    a^2 ((a^2 - b^2 - c^2)^2 - b^2 c^2) / (a^2 - b^2 - c^2) : :
Tripolars    |cos A| : :
X(186) = 3*X(2) + (J^2 - 3)*X(3) = 2 X(3) + X(23) = (a^2 - b^2 - c^2)(a^2 + b^2 - c^2)(a^2 - b^2 + c^2)*X(3) + (a^2b^2c^2)*X(4)

As a point on the Euler line, X(186) has Shinagawa coefficients (4F, -E - 4F).

Let AA1A2, BB1B2, CC1C2 be the circumcircle-inscribed equilateral triangles used in the construction of the Trinh triangle. Let A' be the cevapoint of A1 and A2, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(186). (Randy Hutson, October 8, 2019

Let P and Q be circumcircle antipodes. X(186) is the Euler line intercept, other than X(20), of circle {X(20),P,Q}} for all P, Q. (Randy Hutson, August 30, 2020)

X(186) lies on these lines: 2,3   54,389   93,252   98,935   107,477   112,187   249,250   2931,3580

X(186) is the {X(3),X(24)}-harmonic conjugate of X(4). For a list of other harmonic conjugates of X(100), click Tables at the top of this page.

X(186) = reflection of X(i) in X(j) for these (i,j): (4,403), (403,468)
X(186) = isogonal conjugate of X(265)
X(186) = isotomic conjugate of X(328)
X(186) = complement of X(3153)
X(186) = anticomplement of X(2072)
X(186) = circumcircle-inverse of X(4)
X(186) = polar-circle-inverse of X(5)
X(186) = Kosnita-circle-inverse of X(3)
X(186) = de-Longchamps-circle-inverse of X(37444)
X(186) = X(340)-Ceva conjugate of X(323)
X(186) = X(50)-cross conjugate of X(323)
X(186) = crosspoint of X(54) and X(74)
X(186) = crosssum of X(i) and X(j) for these (i,j): (5,30), (621,622)
X(186) = crossdifference of every pair of points on line X(216)X(647)
X(186) = anticomplementary conjugate of anticomplement of X(38534)
X(186) = pole wrt polar circle of trilinear polar of X(94) (line X(5)X(523))
X(186) = X(48)-isoconjugate (polar conjugate) of X(94)
X(186) = perspector of ABC and the reflection of the circumorthic triangle in the Euler line
X(186) = perspector of ABC and the reflection of the Kosnita triangle in the Euler line
X(186) = perspector of ABC and the reflection of the orthic triangle in the orthic axis
X(186) = reflection of X(403) in the orthic axis
X(186) = crosspoint of X(3) and X(2931) wrt both the excentral and tangential triangles
X(186) = homothetic center of circumorthic and Kosnita triangles
X(186) = perspector of circumconic through polar conjugates of PU(5)
X(186) = Hofstadter 3 point
X(186) = antigonal image of X(5962)
X(186) = X(484)-of-orthic-triangle if ABC is acute
X(186) = Thomson-isogonal conjugate of X(15131)
X(186) = Ehrmann-vertex-to-orthic similarity image of X(3153)
X(186) = {X(3),X(4)}-harmonic conjugate of X(3520)
X(186) = intersection of the tangent to hyperbola {A,B,C,X(3),X(15)}} at X(15) and the tangent to hyperbola {A,B,C,X(3),X(16)}} at X(16)


X(187) = INVERSE-IN-CIRCUMCIRCLE OF X(6) (SCHOUTE CENTER)

Trilinears    a(2a2 - b2 - c2) : :
Trilinears    sin A - 3 cos A tan ω : :
Trilinears    2 sin(A - 2ω) - sin(A + 2ω) + sin A : :
Trilinears    sin A + sin A cos 2ω - 3 cos A sin 2ω : :
Trilinears    cos(A + ω) sin 2ω - e^2 sin(A - ω) : :
Barycentrics    a2(2a2 - b2 - c2) : :
Tripolars    b c Sqrt[2(b^2 + c^2) - a^2] : :
X(187) = X(15) + X(16)

Let L denote the line having trilinears of X(187) as coefficients. Then L is the line passing through X(2) perpendicular to the Euler line.

Let A'B'C' be the 1st Brocard triangle. Let A″B″C″ be the 2nd Brocard triangle. Let A* = Λ((A',A″), and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(187). (Randy Hutson, December 26, 2015)

Let (OA) be the circumcircle of BCX(2). Let PA be the perspector of (OA). Let LA be the polar of PA wrt (OA). Define LB and LC cyclically. Let A' = LB∩LC, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(187). (Randy Hutson, June 7, 2019)

If you have The Geometer's Sketchpad, you can view X(1316), which includes X(187).

X(187) lies on the Darboux quintic and these lines: 2,316   3,6   23,111   30,115   35,172   36,1015   74,248   99,385   110,352   112,186   183,1003   237,351   249,323   325,620   353,3117   395,531   396,530   729,805

X(187) is the {X(3),X(6)}-harmonic conjugate of X(574). For a list of other harmonic conjugates of X(187), click Tables at the top of this page.

X(187) = midpoint of X(i) and X(j) for these (i,j): (15,16), (99,385)
X(187) = reflection of X(i) in X(j) for these (i,j): (115,230), (316,625), (325,620), (5107,6)
X(187) = isogonal conjugate of X(671)
X(187) = isotomic conjugate of X(18023)
X(187) = inverse-in-circumcircle of X(6)
X(187) = inverse-in-Brocard-circle of X(574)
X(187) = inverse-in-van-Lamoen-circle-of-X(2)
X(187) = radical trace of the circumcircle and Brocard circle
X(187) = complement of X(316)
X(187) = anticomplement of X(625)
X(187) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,6593), (111,6)
X(187) = crosspoint of X(i) and X(j) for these (i,j): (2,67), (6,111), (468,524)
X(187) = crosssum of X(i) and X(j) for these (i,j): (2,524), (6,23), (111,895), (115,690)
X(187) = crossdifference of every pair of points on line X(2)X(523)
X(187) = X(55)-beth conjugate of X(187)
X(187) = inverse-in-Moses-radical-circle of X(1495)
X(187) = radical trace of Moses radical circle and Parry circle
X(187) = radical trace of Lucas radical circle and Lucas(-1) radical circle
X(187) = radical trace of Lucas inner and Lucas(-1) inner circle
X(187) = radical trace of circles {P(1),U(2),U(39)}} and {U(1),P(2),P(39)}}
X(187) = intersection of Brocard axis and Lemoine axis
X(187) = intersection of Brocard axis (or Lemoine axis) and non-transverse axis of hyperbola {A,B,C,PU(2)}}
X(187) = intersection of Brocard axis (or Lemoine axis) and tangent at X(691) to hyperbola {A,B,C,PU(2)}}
X(187) = midpoint of PU(2)
X(187) = bicentric sum of PU(2)
X(187) = perspector of ABC and the reflection of the circumsymmedial triangle in the Brocard axis
X(187) = perspector of ABC and the reflection of the circumsymmedial triangle in the Lemoine axis
X(187) = intersection of tangents to hyperbola {A,B,C,X(2),X(6)} at X(6) and X(111)
X(187) = inverse-in-Parry-circle of X(2502)
X(187) = X(187)-of-2nd-Brocard-triangle
X(187) = X(187)-of-circumsymmedial-triangle
X(187) = polar conjugate of isogonal conjugate of X(23200)
X(187) = polar conjugate of isotomic conjugate of X(3292)
X(187) = X(92)-isoconjugate of X(895)
X(187) = X(1577)-isoconjugate of X(691)
X(187) = {X(1687),X(1688)}-harmonic conjugate of X(2080)
X(187) = trilinear pole of PU(107)
X(187) = inverse-in-Parry-isodynamic-circle of X(351); see X(2)
X(187) = radical trace of 3rd and 4th Lozada circles
X(187) = radical trace of 6th and 7th Lozada circles
X(187) = radical trace of 8th and 9th Lozada circles
X(187) = radical trace of 10th and 11th Lozada circles
X(187) = radical trace of circumcircles of outer and inner Grebe triangles
X(187) = X(115)-of-4th-anti-Brocard-triangle
X(187) = X(187)-of-X(3)PU(1)
X(187) = Thomson-isogonal conjugate of X(6054)
X(187) = Cundy-Parry Phi transform of X(576)
X(187) = Cundy-Parry Psi transform of X(7607)
X(187) = X(13)-antipedal-to-X(16)-pedal similarity image of X(13)
X(187) = X(14)-antipedal-to-X(15)-pedal similarity image of X(14)
X(187) = homothetic center of Trinh triangle and mid-triangle of 1st and 2nd Kenmotu diagonals triangles
X(187) = QA-P4 (Isogonal Center of quadrangle ABCX(2); see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/25-qa-p4.html)


X(188) = 2nd MID-ARC POINT OF ANTICOMPLEMENTARY TRIANGLE

Trilinears    csc A/2 : csc B/2 : csc C/2
Trilinears    [bc(b + c - a)]1/2 : :
Trilinears    csc A cos A/2 : :
Trilinears    |AX(1)| : |BX(1)| : |CX(1)|
Trilinears    sec A' : sec B', sec C', where A'B'C' = excentral triangle
Barycentrics   1/(sin(B/2) sin(C/2) + sin(A/2)) : :       c.f., X(5451)
Barycentrics   1/(csc(B/2) csc(C/2) + csc(A/2)) : :       c.f., X(5451)
Barycentrics   cos A/2 : cos B/2 : cos C/2

Let A'B'C' be the excentral triangle of ABC, so that A' = -1 : 1 : 1 (trilinears). Let A'' be the point where the bisector of angle BA'C meets the line BC. Define B'' and C'' cyclically. The lines AA'', BB'', CC'' concur in X(188). (Seiichi Kirikami, February 14, 2010)

Let Ea be the ellipse with B and C as foci and passing through the A-excenter, and define Eb and Ec cyclically. Let La be the line tangent to Ea at the A-excenter, and define Lb and Lc cyclically. Let A' = La∩BC, B' = Lb∩CA, C' = Lc∩AB. Then A', B', C' are collinear, and the trilinear pole of line A'B'C' = X(188). Note: The triangle formed by La, Lb, Lc is also the excentral triangle of the excentral triangle. Alternately, let A″ be the trilinear pole of line La, and define B″, C″ cyclically. The lines AA″, BB″ and CC″ concur at X(188); see also X(174). (Randy Hutson, December 2, 2017)

X(188) lies on these lines: 1,361   2,178   9,173   40,164   166,167   174,266

X(188) = isogonal conjugate of X(266)
X(188) = isotomic conjugate of X(4146)
X(188) = anticomplement of X(178)
X(188) = X(2)-Ceva conjugate of X(236)
X(188) = cevapoint of X(1) and X(164)
X(188) = X(259)-cross conjugate of X(174)
X(188) = crosssum of X(1) and X(361)
X(188) = X(188)-beth conjugate of X(266)
X(188) = SS(A→A') of X(4), where A'B'C' is the excentral triangle
X(188) = isotomic conjugate of X(4146)
X(188) = X(65)-of-excentral-triangle
X(188) = perspector of circumconic centered at X(236)
X(188) = center of circumconic that is locus of trilinear poles of lines passing through X(236)
X(188) = excentral-to-ABC trilinear image of X(1)
X(188) = excentral-to-ABC barycentric image of X(164)


X(189) = CYCLOCEVIAN CONJUGATE OF X(8)

Trilinears       bc/(cos B + cos C - cos A - 1) : ca/(cos C + cos A - cos B - 1) : ab/(cos A + cos B - cos C - 1)
Barycentrics  1/(cos B + cos C - cos A - 1) : 1/(cos C + cos A - cos B - 1) : 1/(cos A + cos B - cos C - 1)

X(189) is the perspector of triangle ABC and the pedal triangle of X(84).

X(189) lies on the Lucas cubic and these lines: 2,77   7,92   8,20   29,81   69,309   222,281

X(189) = isogonal conjugate of X(198)
X(189) = isotomic conjugate of X(329)
X(189) = cyclocevian conjugate of X(8)
X(189) = anticomplement of X(223)
X(189) = X(309)-Ceva conjugate of X(280)
X(189) = cevapoint of X(84) and X(282)
X(189) = X(i)-cross conjugate of X(j) for these (i,j): (4,7), (57,2), (282,280)
X(189) = trilinear pole of line X(522)X(905)
X(189) = perspector of ABC and the reflection in X(282) of the pedal triangle of X(282)


X(190) = YFF PARABOLIC POINT

Trilinears    bc/(b - c) : ca/(c - a) : ab/(a - b)
Barycentrics  1/(b - c) : 1/(c - a) : 1/(a - b)

In unpublished notes, Yff has studied the parabola tangent to sidelines BC, CA, AB and having focus X(101). If A',B',C' are the respective points of tangency, then the lines AA', BB', CC' concur in X(190).

The line X(100)X(190) is tangent to the Steiner circumellipse at X(190) and to the circumcircle at X(100). (Peter Moses, July 7, 2009)

Let Ha be the hyperbola passing through A, and with foci at B and C. This is called the A-Soddy hyperbola in (Paul Yiu, Introduction to the Geometry of the Triangle, 2002; 12.4 The Soddy hyperbolas, p. 143). Let La be the polar of X(8) with respect to Ha. Define Lb and Lc cyclically. Let A' = Lb∩Lc, B' = Lc∩La, C' = La∩Lb. The triangle A'B'C' is perspective to ABC, and the perspector is X(190). (Randy Hutson, December 26, 2015)

Let Ia be the reflection of X(1) in the perpendicular bisector of BC, and define Ib and Ic cyclically. X(190) = X(238) of IaIbIc. (Randy Hutson, December 26, 2015)

Let A5B5C5 and A6B6C6 be Gemini triangles 5 and 6, resp. Let LA be the tangent at A to conic {A,B5,C5,B6,C6}}, and define LB and LC cyclically. Let A' = LB∩LC, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(190). (Randy Hutson, January 15, 2019)

If you have The Geometer's Sketchpad, you can view X(190).

X(190) lies on the Steiner circumellipse and these lines: 1,537   2,45   6,192   7,344   8,528   9,75   10,671   37,86   40,341   44,239   63,312   69,144   71,290   72,1043   99,101   100,659   110,835   162,643   191,1089   238,726   320,527   321,333   329,345   350,672   513,660   514,1016   522,666   644,651   646,668   649,889   658,1020   670,799   789,813   872,1045   1222, 3057

X(190) = reflection of X(i) in X(j) for these (i,j): (239,44), (335,37), (673,9), (903,2)
X(190) = isogonal conjugate of X(649)
X(190) = isotomic conjugate of X(514)
X(190) = complement of X(4440)
X(190) = anticomplement of X(1086)
X(190) = X(i)-Ceva conjugate of X(j) for these (i,j): (99,100), (666,3570)
X(190) = cevapoint of X(i) and X(j) for these (i,j): (2,514), (9,522), (37,513), (440,525)
X(190) = X(i)-cross conjugate of X(j) for these (i,j): (513,86), (514,2), (522,75)
X(190) = crosssum of X(512) and X(798)
X(190) = crossdifference of every pair of points on line X(1015)X(1960)
X(190) = X(i)-aleph conjugate of X(j) for these (i,j): (2,1052), (190,1), (645,411), (668,63), (1016,100)
X(190) = X(i)-beth conjugate of X(j) for these (i,j): (9,292), (190,651), (333,88), (645,190), (646,646), (1016,190)
X(190) = trilinear pole of the line X(1)X(2)
X(190) = Steiner-circumellipse-antipode of X(903)
X(190) = barycentric product of PU(24)
X(190) = crossdifference of PU(25)
X(190) = trilinear product of PU(58)
X(190) = Steiner-circumellipse-X(1)-antipode of X(3227)
X(190) = Steiner-circumellipse-X(6)-antipode of X(3226)
X(190) = perspector of ABC and tangential triangle (wrt excentral triangle) of hyperbola passing through X(1), X(9) and the excenters (the Jerabek hyperbola of the excentral triangle)
X(190) = X(6)-isoconjugate of X(513)
X(190) = perspector of ABC and vertex-triangle of Gemini triangles 5 and 6
X(190) = ABC-to-Gemini-triangle-19 parallelogic center
X(190) = areal center of cevian triangles of PU(24)

leftri

Centers X(191)-X(236)

rightri
Centers X(191)-X(236) are Ceva conjugates. (The P-Ceva conjugate of Q is the perspector of the cevian triangle of P and the anticevian triangle of Q.)

X(191) = X(10)-CEVA CONJUGATE OF X(1)

Trilinears    (b + c - a)(bc + ca + ab) + b3 + c3 - a3 : :
Trilinears    SA + rR : SB + rR : SC + rR

X(191) = X(1) - 2 X(21)

Let Ia be the reflection of X(1) in the perpendicular bisector of BC, and define Ib and Ic cyclically. X(191) = X(21) of IaIbIc. (Randy Hutson, December 2, 2017)

Let IaIbIc be the excentral triangle. Let Na be the nine-point center of BCIa, and define Nb and Nc cyclically. The lines IaNa, IbNb, IcNc concur in X(191); c.f. X(5506). (Randy Hutson, December 2, 2017)

X(191) lies on these lines: 1,21   9,46   10,267   30,40   35,72   36,960   109,201   165,1079   190,1089   329,498

X(191) = reflection of X(i) in X(j) for these (i,j): (1,21), (79,442)
X(191) = isogonal conjugate of X(267)
X(191) = X(10)-Ceva conjugate of X(1)
X(191) = crosspoint of X(i) and X(j) for these (i,j): (10,502)
X(191) = crosssum of X(58) and X(501)
X(191) = excentral-isogonal conjugate of X(3)
X(191) = X(i)-aleph conjugate of X(j) for these (i,j): (2,2), (8,20), (10,191), (37,1045), (188,3), (366,6)
X(191) = X(643)-beth conjugate of X(191)
X(191) = crossdifference of every pair of points on line X(661)X(2605)
X(191) = X(54)-of-excentral-triangle
X(191) = perspector of excentral triangle and Fuhrmann triangle
X(191) = intersection of Euler lines of outer and inner Garcia triangles
X(191) = {X(1),X(21)}-harmonic conjugate of X(5426)
X(191) = complement of X(14450)
X(191) = anticomplement of X(11263)


X(192) = X(1)-CEVA CONJUGATE OF X(2)
(CONGRUENT PARALLELIANS POINT)

Trilinears       bc(ca + ab - bc) : ca(ab + bc - ca) : ab(bc + ca - ab)
Barycentrics  ca + ab - bc : ab + bc - ca : bc + ca - ab

The segments through X(192) parallel to the sidelines with endpoints on the sidelines have equal length. For references as early as 1881, see Hyacinthos message 2929 (Paul Yiu, May 29, 2001). See also

Sabrina Bier, "Equilateral Triangles Intercepted by Oriented Parallelians," Forum Geometricorum 1 (2001) 25-32.

X(192) lies on these lines: 1,87   2,37   6,190   7,335   8,256   9,239   55,385   69,742   144,145   315,746   869,1045

X(192) = reflection of X(i) in X(j) for these (i,j): (8,984), (75,37), (1278,75)
X(192) = isogonal conjugate of X(2162)
X(192) = isotomic conjugate of X(330)
X(192) = complement of X(1278)
X(192) = anticomplement of X(75)
X(192) = X(1)-Ceva conjugate of X(2)
X(192) = crosspoint of X(1) and X(43)
X(192) = crosssum of X(1) and X(87)
X(192) = X(9)-Hirst inverse of X(239)
X(192) = X(646)-beth conjugate of X(192)
X(192) = perspector of anticomplementary triangle and Gemini triangle 4
X(192) = perspector of Gemini triangles 4 and 6
X(192) = X(19)-isoconjugate of X(23086)
X(192) = perspector of incentral triangle and inverse of n(Incentral)*n(Medial)
X(192) = trilinear pole of line X(3835)X(4083) (the perspectrix of ABC and Gemini triangle 16)
X(192) = {X(37),X(75)}-harmonic conjugate of X(2)


X(193) = X(4)-CEVA CONJUGATE OF X(2)

Trilinears       (csc A)(cot B + cot C - cot A) : (csc B)(cot C + cot A - cot B) : (csc C)(cot A + cot B - cot C)
Trilinears        (SA - a2)/a : (SB - b2)/b : (SC - c2)/c
Barycentrics  cot B + cot C - cot A : cot C + cot A - cot B : cot A + cot B - cot C
                        = 3a2 - b2 - c2 : 3b2 - c2 - a2 : 3c2 - a2 - b2 (Milorad Stevanovic, 5/12/2003)

Let A' be the reflection of the midpoint of segment BC in X(6), and define B' and C' cyclically. The lines AA', BB', CC' concur in X(193). (Randy Hutson, 9/23/2011)

Let A' be the trilinear pole of the perpendicular bisector of BC, and define B', C' cyclically. A'B'C' is also the anticomplement of the anticomplement of the midheight triangle. A'B'C' is perspective to the orthic and anticomplementary triangles at X(193). (Randy Hutson, January 29, 2018)

X(193) lies on these lines: 2,6   4,1351   7,239   8,894   20,185   23,159   44,344   66,895   144,145   146,148   253,287   317,393   330,959   371,488   372,487   608,651   1839,3187

X(193) = reflection of X(i) in X(j) for these (i,j): (3,1353), (4,1351), (69,6), (1352,576)
X(193) = isogonal conjugate of X(8770)
X(193) = isotomic conjugate of X(2996)
X(193) = complement of X(20080)
X(193) = anticomplement of X(69)
X(193) = anticomplementary conjugate of X(1370)
X(193) = complementary conjugate of complement of X(36616)
X(193) = X(4)-Ceva conjugate of X(2)
X(193) = X(2)-Hirst inverse of X(230)
X(193) = polar conjugate of X(34208)
X(193) = X(i)-beth conjugate of X(j) for these (i,j): (645,193), (662,608)
X(193) = perspector of pedal and antipedal triangles of X(4) (orthic and anticomplementary triangles)
X(193) = perspector, wrt orthic triangle, of polar circle
X(193) = anticomplementary isotomic conjugate of X(20)
X(193) = orthic-isogonal conjugate of X(2)
X(193) = trilinear pole of polar, wrt complement of polar circle, of X(69)
X(193) = pole of orthic axis wrt Steiner circumellipse
X(193) = {X(385),X(7774)}-harmonic conjugate of X(2)
X(193) = endo-homothetic center of 3rd and 4th tri-squares central triangles
X(193) = perspector, wrt anticomplementary triangle, of polar circle


X(194) = X(6)-CEVA CONJUGATE OF X(2)

Trilinears    bc[a2b2 + a2c2 - b2c2] : :
Barycentrics    a2b2 + a2c2 - b2c2 : :
Barycentrics    cot2A - csc2A cos 2ω : :      (M. Iliev, 5/13/07)
X(194) = 2X(39) - X(76) - P(1) + U(1) - X(76)

Let Oa be the circle through A and tangent to BC at its midpoint. Define Ob and Oc cyclically. The radical center of Oa, Ob, Oc is X(194). (Randy Hutson, December 26, 2015)

Let AaBaCa, AbBbCb, AcBcCc be the A-, B- and C-anti-altimedial triangles. Let A' be the trilinear product Aa*Ab*Ac, and define B', C' cyclically. Triangle A'B'C' is the anticomplementary triangle of the 1st Brocard triangle, and is perspective to ABC at X(4), and to the anticomplementary triangle at X(194). (Randy Hutson, November 2, 2017)

Let OA be the circle centered at the A-vertex of the 1st Neuberg triangle and passing through A; define OB and OC cyclically. X(194) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

X(194) lies on these lines: {1,87}, {2,39}, {3,385}, {4,147}, {6,384}, {8,730}, {10,3097}, {20,185}, {32,99}, {63,239}, {69,695}, {75,1107}, {83,3734}, {183,5013}, {184,3492}, {190,2176}, {257,986}, {262,2996}, {263,3498}, {304,3797}, {315,736}, {325,5025}, {350,2275}, {401,1993}, {487,1587}, {488,1588}, {548,3793}, {574,1078}, {616,3104}, {617,3105}, {627,3106}, {628,3107}, {648,1968}, {712,4393}, {1007,2023}, {1593,1941}, {1654,4201}, {1670,2547}, {1671,2546}, {1909,2276}, {2128,2285}, {3096,4045}, {3212,3503}, {3314,3933}, {3413,3557}, {3414,3558}, {3522,5188}, {3770,4261}, {3906,5652}, {3972,5007}

X(194) is the {X(39),X(76)}-harmonic conjugate of X(2). For a list of other harmonic conjugates of X(194), click Tables at the top of this page.

X(194) = reflection of X(76) in X(39)
X(194) = isogonal conjugate of X(3224)
X(194) = isotomic conjugate of X(2998)
X(194) = isotomic conjugate of the complement of X(32747)
X(194) = complement of X(20081)
X(194) = anticomplement of X(76)
X(194) = circumcircle-inverse of X(32517)
X(194) = polar circle inverse of X(32527)
X(194) = anticomplementary conjugate of X(315)
X(194) = complementary conjugate of complement of X(36615)
X(194) = eigencenter of cevian triangle of X(6)
X(194) = eigencenter of anticevian triangle of X(2)
X(194) = radical center of the Neuberg circles.
X(194) = X(6)-Ceva conjugate of X(2)
X(194) = X(3)-Hirst inverse of X(385)
X(194) = anticomplementary-isotomic conjugate of X(69)
X(194) = X(6374)-cross conjugate of X(2)
X(194) = vertex conjugate of PU(140)
X(194) = 1st-Brocard-to-6th-Brocard similarity image of X(6)
X(194) = X(99)-of-6th-Brocard-triangle
X(194) = anticomplementary-circle-inverse of X(32528)
X(194) = orthoptic-circle-of-Steiner-inellipse-inverse of X(32530)
X(194) = orthoptic-circle-of-Steiner-circumellipe-inverse of X(32526)
X(194) = de Longchamps-circle-inverse of X(32529)
X(194) = 2nd-Brocard-circle-inverse of X(32531)
X(194) = perspector of 3rd Brocard triangle and 1st Brocard-reflected triangle


X(195) = X(5)-CEVA CONJUGATE OF X(3)

Trilinears    (cos A)(v + w - u) : : , where u = u(A,B,C) = cos A cos(B - A) cos(C - A)
Trilinears    a[a^8 + b^8 + c^8 - 4a^6(b^2 + c^2) + a^4(6b^4 + 6c^4 + 5b^2c^2) - a^2(4b^6 + 4c^6 - b^4c^2 - b^2c^4) - 2b^2c^2(b^4 + c^4 - b^2c^2)] : :
Barycentrics    4 cos 2A + cot2A - cot A cot ω : :(   (M. Iliev, 5/13/07)

Let A' be the isogonal conjugate of the A-vertex of the outer Napoleon triangle, and define B' and C' cyclically. Let A″ be the isogonal conjugate of the A-vertex of the inner Napoleon triangle, and define B″ and C″ cyclically. The lines A'A″, B'B″, C'C″ concur in X(195). (Randy Hutson, November 18, 2015)

The Napoleon axis and Napoleon-Feuerbach cubic K005 meet in three points: X(17), X(18), and X(195). (Randy Hutson, November 18, 2015)

A construction of X(195) is given by Antreas Hatipolakis and Angel Montesdeoca at 24180.

X(195) lies on the Napoleon cubic and these lines: 1,3467   3,54   4,399   5,3459   6,17   49,52   110,143   140,323   155,381   382,1498   2121,3462   3461,3468

X(195) = reflection of X(i) in X(j) for these (i,j): (3,54), (54,1493), (3519,1209)
X(195) = isogonal conjugate of X(3459)
X(195) = complement of X(12325)
X(195) = anticomplement of X(21230)
X(195) = X(5)-Ceva conjugate of X(3)
X(195) = crosssum of X(137) and X(523)
X(195) = X(3)-of-reflection-triangle
X(195) = X(79)-of-tangential-triangle if ABC is acute
X(195) = tangential isogonal conjugate of X(2937)
X(195) = 2nd isogonal perspector of X(5); see X(36)
X(195) = Yiu-isogonal conjugate of X(1157)
X(195) = perspector of [cross-triangle of ABC and outer Napoleon triangle] and [cross-triangle of ABC and inner-Napoleon triangle]


X(196) = X(7)-CEVA CONJUGATE OF X(4)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos B + cos C - cos A - 1) sec A tan A/2
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (cos B + cos C - cos A - 1) tan A tan A/2

X(196) lies on these lines: 1,207   2,653   4,65   7,92   19,57   34,937   40,208   55,108   226,281   329,342

X(196) = isogonal conjugate of X(268)
X(196) = X(i)-Ceva conjugate of X(j) for these (i,j): (7,4), (92,278)
X(196) = cevapoint of X(19) and X(207)
X(196) = X(221)-cross conjugate of X(347)
X(196) = X(i)-beth conjugate of X(j) for these (i,j): (648,2), (653,196)
X(196) = Danneels point of X(653)
X(196) = pole wrt polar circle of trilinear polar of X(280) (line X(521)X(3239))
X(196) = polar conjugate of X(280)


X(197) = X(8)-CEVA CONJUGATE OF X(6)

Trilinears    a[-a2tan A/2 + b2tan B/2 + c2tan C/2] : :

Let A'B'C' be the extouch triangle. Let A″ be the crosspoint of the circumcircle intercepts of line B'C', and define B″, C″ cyclically. The lines A'A″, B'B″, C'C″ concur in X(197). (Randy Hutson, July 31 2018)

For a construction of X(197), see Dasari Naga Vijay Krishna, "On A Simple Construction of Triangle Centers X(8), X(197), X(K) & X(594)", Scientific Inquiry and Review, Vol. 2, Issue 3, July 2018.

X(197) lies on these lines: 3,10   6,181   19,25   22,100   42,48   56,227   159,200

X(197) = X(8)-Ceva conjugate of X(6)
X(197) = crosssum of X(124) and X(514)
X(197) = isogonal conjugate of X(8048)
X(197) = crossdifference of every pair of points on line X(905)X(3910)
X(197) = crosspoint of circumcircle intercepts of excircles radical circle


X(198) = X(9)-CEVA CONJUGATE OF X(6)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = a(cos B + cos C - cos A - 1)
Trilinears    a (a^3 + a^2 (b + c) - a (b + c)^2 - (b - c)^2 (b + c)) : :
Trilinears    s cos A - r cot(A/2) : :

X(198) lies on these lines: 3,9   6,41   19,25   5,1030   64,71   100,346   101,102   154,212   208,227   218,579   284,859   478,577   958,966

X(198) = isogonal conjugate of X(189)
X(198) = complement of X(21279)
X(198) = anticomplement of X(21239)
X(198) = X(i)-Ceva conjugate of X(j) for these (i,j): (3,55), (9,6), (223,221)
X(198) = crosspoint of X(40) and X(223)
X(198) = crosssum of X(i) and X(j) for these (i,j): (57,1422), (84,282), (513,1146), (650,1364), (1433,1436)
X(198) = crossdifference of every pair of points on line X(522)X(905)
X(198) = X(i)-beth conjugate of X(j) for these (i,j): (9,19), (101,198)
X(198) = perspector of Apus and tangential triangles


X(199) = X(10)-CEVA CONJUGATE OF X(6)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[b4 + c4 - a4 + (b2 + c2 - a2)(bc + ca + ab)]
Barycentrics  af(A,B,C) : bf(B,C,A) : cf(C,A,B)

As a point on the Euler line, X(199) has Shinagawa coefficients (E + 2F + $bc$, -2E - 2F + $bc$).

Let IA, IB, IC be the excenters. Let (OA) be the circle tangent to the circumcircle at A and passing through IA. Let A' be the antipode of A in (OA). Let LA be the tangent to (OA) at A'. Define LB and LC cyclically. Let TA = LB∩LC, and define TB and TC cyclically. Triangle TATBTC is homothetic to the tangential triangle at X(199). (Randy Hutson, June 7, 2019)

X(199) lies on these lines: 2,3   42,172   51,572   55,1030   184,573

X(199) = isogonal conjugate of X(8044)
X(199) = anticomplement of X(34119)
X(199) = X(10)-Ceva conjugate of X(6)
X(199) = crosspoint of X(101) and X(250)
X(199) = crosssum of X(125) and X(514)
X(199) = tangential isogonal conjugate of X(8053)
X(199) = orthic-to-tangential similarity image of X(430)


X(200) = X(8)-CEVA CONJUGATE OF X(9)

Trilinears    cot2(A/2) : :
Trilinears    (b + c - a)2 : :
Trilinears    (1 + cos A)/(1 - cos A) : : (Randy Hutson, 9/23/2011)
Trilinears    1 - csc^2(A/2) : :
Trilinears    squared distance of A to Gergonne line : :
Barycentrics   a(b + c - a)2 : :

Let A'B'C' be the extouch triangle. Let A″ be the trilinear product of the circumcircle intercepts of line B'C', and define B″ and C″ cyclically. The lines A'A″, B'B″, C'C″ concur in X(200). (Randy Hutson, July 31 2018)

Let A' be the trilinear product of the circumcircle intercepts of the A-excircle. Define B' and C' cyclically. The lines AA', BB', CC' concur in X(200). (Randy Hutson, July 31 2018)

X(200) lies on these lines: 1,2   3,963   9,55   33,281   40,64   46,1004   57,518   63,100   69,269   159,197   219,282   220,728   255,271   318,1089   319,326   329,516   341,1043   756,968

X(200) is the {X(8),X(78)}-harmonic conjugate of X(1). For a list of harmonic conjugates of X(200), click Tables at the top of this page.

X(200) = reflection of X(i) in X(j) for these (i,j): (1,997), (57,1376)
X(200) = isogonal conjugate of X(269)
X(200) = isotomic conjugate of X(1088)
X(200) = complement of X(36845)
X(200) = X(8)-Ceva conjugate of X(9)
X(200) = cevapoint of X(220) and X(480)
X(200) = X(220)-cross conjugate of X(9)
X(200) = crosspoint of X(8) and X(346)
X(200) = crosssum of X(i) and X(j) for these (i,j): (56,1407), (57,1420), (1042,1427)
X(200) = X(i)-beth conjugate of X(j) for these (i,j): (100,223), (200,55), (643,165)
X(200) = {X(1),X(8)}-harmonic conjugate of X(4853)
X(200) = {X(2),X(8)}-harmonic conjugate of X(4847)
X(200) = homothetic center of anticomplementary triangle and 3rd antipedal triangle of X(1)
X(200) = homothetic center of ABC and medial triangle of 3rd antipedal triangle of X(1)
X(200) = Danneels point of X(8)
X(200) = polar conjugate of X(1847)
X(200) = trilinear square of X(9)
X(200) = trilinear product of the circumcircle intercepts with the excircles
X(200) = X(1899)-of-excentral-triangle
X(200) = complement of polar conjugate of isogonal conjugate of X(22153)
X(200) = excentral-to-ABC barycentric image of X(57)
X(200) = mixtilinear-incentral-to-ABC barycentric image of X(1)
X(200) = mixtilinear-excentral-to-ABC barycentric image of X(1)


X(201) = X(10)-CEVA CONJUGATE OF X(12)

Trilinears       (cos A)[1 + cos(B - C)] : (cos B)[1 + cos(C - A)] : (cos C)[1 + cos(A - B)]
Barycentrics  (sin 2A)[1 + cos(B - C)] : (sin 2B)[1 + cos(C - A)] : (sin 2C)[1 + cos(A - B)]

X(201) lies on these lines: 1,212   9,34   10,225   12,756   33,40   37,65   38,56   55,774   57,975   63,603   72,73   109,191   210,227   220,221   255,1060   337,348   388,984   601,920

X(201) = isogonal conjugate of X(270)
X(201) = X(10)-Ceva conjugate of X(12)
X(201) = crosspoint of X(10) and X(72)
X(201) = crosssum of X(i) and X(j) for these (i,j): (1,580), (28,58)
X(201) = X(i)-beth conjugate of X(j) for these (i,j): (72,201), (1018,201)


X(202) = X(1)-CEVA CONJUGATE OF X(15)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = u(v + w - u),
                         u = u(A,B,C) = sin(A + π/3), v = u(B,C,A), w = u(C,A,B)

Trilinears       1 - cos(A + π/3) : 1 - cos(B + π/3) : 1 - cos(C + π/3)   (Joe Goggins, Oct. 19, 2005)

Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(202) lies on these lines: 1,62   6,101   11,13   12,18   15,36   16,55   17,499   56,61   395,495   397,496

X(202) = X(1)-Ceva conjugate of X(15)


X(203) = X(1)-CEVA CONJUGATE OF X(16)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = u(v + w - u),
                         u = u(A,B,C) = sin(A - π/3), v = u(B,C,A), w = u(C,A,B)

Trilinears       1 + cos(A + 2π/3) : 1 + cos(B + 2π/3) : 1 + cos(C + 2π/3)   (Joe Goggins, Oct. 19, 2005)

Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(203) lies on these lines: 1,61   6,101   11,14   12,17   15,55   16,36   18,499   56,62   396,495   398,496

X(203) = X(1)-Ceva conjugate of X(16)


X(204) = X(1)-CEVA CONJUGATE OF X(19)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (tan A)(tan B + tan C - tan A)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(204) lies on these lines: 6,33   19,31   25,34   55,1033   63,162   108,223   207,221

X(204) = isogonal conjugate of X(19611)
X(204) = crosspoint of X(1) and X(610)
X(204) = crosssum of X(i) and X(j) for these {i,j}: {1, 2184}, {525, 7358}
X(204) = X(1)-Ceva conjugate of X(19)
X(204) = X(i)-beth conjugate of X(j) for these (i,j): (108,204), (162,223)


X(205) = X(9)-CEVA CONJUGATE OF X(31)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2[b2tan B/2 + c2tan C/2 - a2tan A/2]
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(205) lies on these lines: 25,41   37,48   78,101   154,220   184,213

X(205) = X(9)-Ceva conjugate of X(31)


X(206) = X(2)-CEVA CONJUGATE OF X(32)

Trilinears    a3(b4 + c4 - a4) : :
Trilinears    (sin A)(sin 2A - tan ω) : :

Let A'B'C' be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung); then X(206) = X(6)-of-A'B'C'. (Randy Hutson, July 31 2018)

For another construction see Antreas P. Hatzipolakis and Peter Moses, Euclid 1713 .

X(206) lies on these lines: 2,66   5,182   6,25   26,511   69,110   157,216   160,577   219,692   237,571

X(206) = midpoint of X(i) and X(j) for these (i,j): (6,159), (110,1177)
X(206) = isogonal conjugate of X(18018)
X(206) = isotomic conjugate of isogonal conjugate of X(20968)
X(206) = complement of X(66)
X(206) = complementary conjugate of X(427)
X(206) = X(2)-Ceva conjugate of X(32)
X(206) = crosspoint of X(2) and X(315)
X(206) = crosssum of X(i) and X(j) for these {i,j}: {2, 7391}, {6, 2353}, {66, 14376}, {339, 523}, {826, 15526}, {13854, 17407}
X(206) = X(66)-of-medial triangle
X(206) = perspector of circumconic centered at X(32)
X(206) = isogonal conjugate of the isotomic conjugate of X(22)
X(206) = center of conic that is the locus of centers of conics passing through X(6) and the vertices of the tangential triangle
X(206) = centroid of X(6) plus the vertices of the tangential triangle
X(206) = crosssum of circumcircle intercepts of de Longchamps line
X(206) = polar conjugate of isogonal conjugate of X(22075)
X(206) = center of circumconic that is locus of trilinear poles of lines passing through X(32); this conic is the isogonal conjugate of the de Longchamps line
X(206) = trilinear product X(i)*X(j) for these {i,j}: {2, 17453}, {6, 2172}, {9, 7251}, {19, 10316}, {22, 31}, {32, 1760}, {37, 17186}, {48, 8743}, {57, 4548}, {63, 17409}, {163, 2485}, {315, 560}, {798, 4611}, {1333, 4456}, {1397, 4123}, {1755, 11610}, {2175, 7210}, {2206, 4463}, {9247, 17907}, {9447, 17076}


X(207) = X(1)-CEVA CONJUGATE OF X(34)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (1 - sec A)(sec B + sec C - sec A - 1)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(207) lies on these lines: 1,196   19,56   33,64   34,1042   40,108   78,653   204,221

X(207) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,34), (196,19)
X(207) = X(1)-beth conjugate of X(64)
X(207) = trilinear product X(34)*X(1490)


X(208) = X(4)-CEVA CONJUGATE OF X(34)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (1 - sec A)(cos B + cos C - cos A - 1)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(208) lies on these lines: 1,102   4,57   19,225   25,34   33,64   40,196   198,227   226,406   318,653

X(208) = isogonal conjugate of X(271)
X(208) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,34), (57,19), (342,223)
X(208) = crosssum of X(3) and X(1433)
X(208) = polar conjugate of X(34404)
X(208) = X(i)-beth conjugate of X(j) for these (i,j): (108,208), (162,1)


X(209) = X(4)-CEVA CONJUGATE OF X(37)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sin B + sin C)[sin A + sin(A - B) + sin(A - C)]
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(209) lies on these lines: 6,31   10,12   44,51   306,518

X(209) = isogonal conjugate of X(272)
X(209) = X(4)-Ceva conjugate of X(37)


X(210) = X(10)-CEVA CONJUGATE OF X(37)

Trilinears       (b + c)(b + c - a) : (c + a)(c + a - b) : (a + b)(a + b - c)
Trilinears    (cos B - cos C)^2 + (cos A + 1)(cos B + cos C - 2) : :
Barycentrics  a(b + c)(b + c - a) : b(c + a)(c + a - b) : c(a + b)(a + b - c) : :

X(210) lies on these lines: 1,2334   2,354   6,612   8,312   9,55   10,12   31,44   33,220   37,42   38,899   43,984   45,968   51,374   56,936   63,1004   78,958   165,971   201,227   213,762   381,517   392,519   430,594   869,1107   956,997   976,1104

X(210) = X(2)-of-extouch triangle, so that X(210)X(1158) = Euler line of the extouch triangle

X(210) = reflection of X(i) in X(j) for these (i,j): (51,375), (354,2)
X(210) = isogonal conjugate of X(1014)
X(210) = complement of X(3873)
X(210) = anticomplement of X(3742)
X(210) = X(10)-Ceva conjugate of X(37)
X(210) = crosspoint of X(8) and X(9)
X(210) = crosssum of X(i) and X(j) for these (i,j): (56,57), (58,1412)
X(210) = crossdifference of every pair of points on line X(1019)X(1429)
X(210) = X(i)-beth conjugate of X(j) for these (i,j): (200,210), (210,42)
X(210) = centroid of Bevan circle intercepts with sidelines of ABC
X(210) = centroid of AbAcBcBaCaCb as defined at X(3588)
X(210) = centroid of AbAcBcBaCaCb as used in the construction of the inner-Conway triangle; see preamble before X(11677)
X(210) = trilinear pole of line X(3709)X(4041)
X(210) = excentral-to-ABC barycentric image of X(2)
X(210) = {X(37),X(42)}-harmonic conjugate of X(37593)


X(211) = X(4)-CEVA CONJUGATE OF X(39)

Trilinears    sin(A + ω)[cos B sin(B + ω) + cos C sin(C + ω) - cos A sin(A + ω)] : :
Barycentrics    a^4*(b^2 + c^2)*(a^2*b^2 - b^4 + a^2*c^2 + b^2*c^2 - c^4) : :

X(211) lies on the cubic K1088 and these lines: {5, 141}, {6, 11360}, {32, 184}, {51, 1506}, {52, 114}, {232, 15897}, {263, 2548}, {571, 2909}, {632, 15510}, {688, 2518}, {754, 3491}, {1078, 11673}, {2794, 13419}, {2979, 3096}, {3060, 7752}, {3202, 20960}, {3456, 17970}, {3917, 6292}, {4173, 5007}, {5017, 20987}, {5167, 7747}, {7807, 14962}, {11674, 12110}, {14575, 18796}

X(211) = X(4)-Ceva conjugate of X(39)
X(211) = crossdifference of every pair of points on line {850, 3050}
X(211) = barycentric product X(i)*X(j) for these {i,j}: {39, 3060}, {1964, 18041}, {3051, 7752}
X(211) = barycentric quotient X(i)/X(j) for these {i,j}: {3060, 308}, {18041, 18833}
X(211) = {X(3051),X(23208)}-harmonic conjugate of X(3203)


X(212) = X(9)-CEVA CONJUGATE OF X(41)

Trilinears       (cos A)(1 + cos A) : (cos B)(1 + cos B) : (cos C)(1 + cos C)
                        = (cos A)cos2(A/2) : (cos B)cos2(B/2) : (cos C)cos2(C/2)
                        = a2(b + c - a)(b2 + c2 - a2) : b2(c + a - b)(c2 + a2 - b2) : c2(a + b - c)(a2 + b2 - c2)

Barycentrics  (sin 2A)(1 + cos A) : (sin 2B)(1 + cos B) : (sin 2C)(1 + cos C)

The trilinear polar of X(212) passes through X(1946). (Randy Hutson, June 7, 2019)

X(212) lies on these lines: 1,201   3,73   6,31   9,33   11,748   34,40   35,47   48,184   56,939   63,1040   78,283   109,165   154,198   238,497   312,643   582,942

X(212) = isogonal conjugate of X(273)
X(212) = crossdifference of every pair of points on line X(514)X(3064)
X(212) = X(4)-isoconjugate of X(7)
X(212) = X(57)-isoconjugate of X(92)
X(212) = X(i)-Ceva conjugate of X(j) for these (i,j): (3,48), (9,41), (283,219)
X(212) = X(228)-cross conjugate of X(55)
X(212) = crosspoint of X(i) and X(j) for these (i,j): (3,219), (9,78)
X(212) = crosssum of X(i) and X(j) for these (i,j): (4,278), (34,57)
X(212) = X(212)-beth conjugate of X(184)


X(213) = X(6)-CEVA CONJUGATE OF X(42)

Trilinears       (b + c)a2 : (c + a)b2 : (a + b)c2
Trilinears       a2(ar - S) : b2(br - S) : c2(cr - S)
Barycentrics  (b + c)a3 : (c + a)b3 : (a + b)c3

X(213) lies on these lines: 1,6   8,981   31,32   39,672   58,101   63,980   83,239   100,729   184,205   274,894   607,1096   667,875   692,923

X(213) = isogonal conjugate of X(274)
X(213) = isotomic conjugate of X(6385)
X(213) = complement of X(17137)
X(213) = anticomplement of X(21240)
X(213) = X(i)-Ceva conjugate of X(j) for these (i,j): (6,42), (37,228)
X(213) = crosspoint of X(6) and X(31)
X(213) = crosssum of X(i) and X(j) for these (i,j): (2,75), (81,1444), (85,348)
X(213) = crossdifference of every pair of points on line X(320)X(350)
X(213) = X(i)-beth conjugate of X(j) for these (i,j): (41,213), (101,65), (644,213)
X(213) = bicentric sum of PU(9)
X(213) = PU(9)-harmonic conjugate of X(667)
X(213) = barycentric product of PU(85)
X(213) = trilinear pole of line X(669)X(798)
X(213) = X(92)-isoconjugate of X(1444)
X(213) = {X(1),X(9)}-harmonic conjugate of X(5283)


X(214) = X(2)-CEVA CONJUGATE OF X(44)

Trilinears     (b + c - 2a)(b2 + c2 - a2 - bc) : :

X(214) lies on the bicevian conic of X(1) and X(2), which is also QA-Co1 (Nine-point Conic) of quadrangle ABCX(1) (see http://www.chrisvantienhoven.nl/other-quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/76-qa-co1.html) (Randy Hutson, July 20, 2016)

X(214) lies on these lines: 1,88   2,80   9,48   10,140   11,442   36,758   44,1017   119,515   142,528   535,908   662,759   1015,1100

X(214) = midpoint of X(1) and X(100)
X(214) = reflection of X(11) in X(1125)
X(214) = isogonal conjugate of X(1168)
X(214) = complement of X(80)
X(214) = X(2)-Ceva conjugate of X(44)
X(214) = crosspoint of X(2) and X(320)
X(214) = X(21)-beth conjugate of X(244)
X(214) = perspector of circumconic centered at X(44)
X(214) = center of circumconic that is locus of trilinear poles of lines passing through X(44)
X(214) = X(36) of X(1)-Brocard triangle
X(214) = inner-Garcia-to-ABC similarity image of X(10)
X(214) = X(7687)-of-excentral-triangle
X(214) = crosssum of circumcircle intercepts of line PU(55) (line X(1)X(513))
X(214) = center of conic {A,B,C,X(1),X(100)}}
X(214) = complementary conjugate of X(3814)
X(214) = QA-P3 (Gergonne-Steiner Point) of quadrangle ABCX(1) (see http://www.chrisvantienhoven.nl/quadrangle-objects/10-mathematics/quadrangle-objects/18-qa-p3.html)


X(215) = X(1)-CEVA CONJUGATE OF X(50)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sin 3A)(sin 3B + sin 3C - sin 3A)
Trilinears       cos2(3A/2) : cos2(3B/2) : cos2(3C/2)     (M. Iliev, 4/12/07)
Trilinears       1 + cos 3A : 1 + cos 3B : 1+ cos 3C     (M. Iliev, 4/12/07)
Trilinears    a^3 (a^2 - b^2 - c^2 + b c)^2 (a - b - c) : :

X(215) is the insimilicenter of the incircle and the sine-triple-angle circle. (Randy Hutson, December 14, 2014)

Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(215) lies on these lines: 1,49   11,110   12,54   55,184

X(215) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,50)


X(216) = X(5)-CEVA CONJUGATE OF X(51)

Trilinears    sin 2A cos(B - C) : :
Trilinears    cos A (tan A + tan B + tan C) + sin A : :
Trilinears    cos A + sin A (cot A cot B cot C) : :
Trilinears    (sin 2B + sin 2C) cos A : :
Trilinears    sin A (cos 2B + cos 2C) : :
Trilinears    (sin A)(1 - sin^2 B - sin^2 C) : :
Trilinears    sec B sec C + csc B csc C : :
Trilinears    cos(A + T) : :, T as at X(389)
Barycentrics   csc 2B + csc 2C : :
Barycentrics   a^2(b^2 + c^2 - a^2)[a^2(b^2 + c^2) - (b^2 - c^2)^2] : :
Barycentrics   (sin A)(sin 2A)cos(B - C) : :

X(216) is the perspector of triangle ABCand the tangential triangle of the Johnson circumconic. (Randy Hutson, 9/23/2011)

Let Ea be the ellipse with B and C as foci and passing through X(5), and define Eb, Ec cyclically. Let La be the line tangent to Ea at X(5), and define Lb, Lc cyclically. Let A' be the trilinear pole of line La, and define B', C' cyclically. A', B', C' lie on the circumconic centered at X(216). (Randy Hutson, July 20, 2016)

X(216) = intersection of isogonal conjugate of polar conjugate of Euler line (i.e., line X(3)X(6)) and the polar conjugate of isogonal conjugate of Euler line (i.e., line X(2)X(216)) (Randy Hutson, July 20, 2016)

X(216) lies on hyperbola {X(2),X(6),X(216),X(233),X(1249),X(1560),X(3162)}}, which is a circumconic of the medial triangle, as well as the locus of the perspector of circumconics centered at a point on the Euler line. Also, this hyperbola is tangent to Euler line at X(2). (Randy Hutson, July 20, 2016)

X(216) lies on these lines: 2,232   3,6   5,53   51,418   95,648   97,288   115,131   157,206   373,852   395,465   396,466   631,1075   1015,1060   2493,3054

X(216) = isogonal conjugate of X(275)
X(216) = isotomic conjugate of X(276)
X(216) = inverse-in-Brocard-circle of X(577)
X(216) = complement of X(264)
X(216) = complementary conjugate of X(21243)
X(216) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,5), (3,418), (5,51), (324,52)
X(216) = cevapoint of X(217) and X(418)
X(216) = X(217)-cross conjugate of X(51)
X(216) = crosspoint of X(i) and X(j) for these (i,j): (2,3), (5,343)
X(216) = crosssum of X(4) and X(6)
X(216) = crossdifference of every pair of points on line X(186)X(523)
X(216) = inverse-in-Brocard-circle of X(577)
X(216) = center of circumconic that is locus of trilinear poles of lines passing through X(5)
X(216) = intersection of trilinear polars of any 2 points on the Johnson circumconic
X(216) = perspector of cevian triangle of X(3) and tangential triangle, wrt cevian triangle of X(3), of circumconic of cevian triangle of X(3) centered at X(3)
X(216) = pole wrt polar circle of trilinear polar of X(8795)
X(216) = X(48)-isoconjugate (polar conjugate) of X(8795)
X(216) = X(92)-isoconjugate of X(54)
X(216) = X(1577)-isoconjugate of X(933)
X(216) = X(573)-of-orthic-triangle if ABC is acute
X(216) = perspector of ABC and unary cofactor triangle of circumorthic triangle
X(216) = {X(61),X(62)}-harmonic conjugate of X(37505)


X(217) = X(6)-CEVA CONJUGATE OF X(51)

Trilinears    (sin3A) cos A cos(B - C) : :
Barycentrics    a^4*(a^2 - b^2 - c^2)*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4) : :

X(217) lies on the cubic K1088 and these lines: {3, 3289}, {4, 6}, {5, 1625}, {32, 184}, {39, 185}, {51, 3199}, {54, 112}, {83, 287}, {125, 1506}, {187, 13367}, {213, 20230}, {216, 5562}, {232, 389}, {263, 10790}, {394, 3785}, {574, 1204}, {577, 10984}, {578, 1968}, {648, 9291}, {1015, 1425}, {1404, 1409}, {1500, 3270}, {1562, 7765}, {1614, 1971}, {1691, 11674}, {1692, 5167}, {1899, 2548}, {1993, 20065}, {1994, 20088}, {2909, 20968}, {2965, 14533}, {3053, 19357}, {3172, 11402}, {3398, 17974}, {4173, 6752}, {4846, 15075}, {5007, 8779}, {5013, 10605}, {5038, 5622}, {5052, 6467}, {5058, 21640}, {5062, 21641}, {5889, 22240}, {6423, 19356}, {6424, 19355}, {6759, 10311}, {7736, 18909}, {7737, 19467}, {7747, 21659}, {7757, 9289}, {8571, 10254}, {10547, 14600}, {10986, 26882}, {12111, 26216}, {13330, 15073}, {13366, 14581}, {13754, 22416}, {15043, 15355}, {16502, 19349}, {18445, 23128}

X(217) = isogonal conjugate of X(276)
X(217) = anticomplement of X(34850)
X(217) = X(i)-Ceva conjugate of X(j) for these (i,j): {6, 51}, {216, 418}, {1625, 15451}, {1987, 237}
X(217) = X(i)-isoconjugate of X(j) for these (i,j): {1, 276}, {54, 1969}, {63, 8795}, {75, 275}, {76, 2190}, {92, 95}, {264, 2167}, {304, 8884}, {326, 8794}, {561, 8882}, {811, 15412}, {933, 20948}, {1577, 18831}, {2148, 18022}, {2169, 18027}, {2616, 6331}, {14208, 16813}
X(217) = crosspoint of X(i) and X(j) for these (i,j): {6, 184}, {51, 216}
X(217) = crossdifference of every pair of points on line {340, 520}
X(217) = crosssum of X(i) and X(j) for these (i,j): {2, 264}, {76, 7763}, {95, 275}
X(217) = X(92)-isoconjugate of X(95)
X(217) = barycentric product X(i)*X(j) for these {i,j}: {3, 51}, {4, 418}, {5, 184}, {6, 216}, {25, 5562}, {32, 343}, {48, 1953}, {52, 2351}, {53, 577}, {63, 2179}, {110, 15451}, {112, 17434}, {154, 8798}, {212, 1393}, {213, 16697}, {228, 18180}, {255, 2181}, {311, 14575}, {324, 14585}, {394, 3199}, {512, 23181}, {560, 18695}, {603, 7069}, {647, 1625}, {810, 2617}, {1092, 14569}, {1437, 21807}, {1576, 6368}, {1820, 2180}, {2200, 17167}, {3049, 14570}, {3078, 20574}, {3527, 26907}, {9247, 14213}, {13450, 23606}, {14587, 24862}, {17500, 20775}
X(217) = barycentric quotient X(i)/X(j) for these {i,j}: {5, 18022}, {6, 276}, {25, 8795}, {32, 275}, {51, 264}, {53, 18027}, {184, 95}, {216, 76}, {343, 1502}, {418, 69}, {560, 2190}, {1501, 8882}, {1576, 18831}, {1625, 6331}, {1953, 1969}, {1974, 8884}, {2179, 92}, {2207, 8794}, {3049, 15412}, {3199, 2052}, {5562, 305}, {9247, 2167}, {9418, 19189}, {14574, 933}, {14575, 54}, {14585, 97}, {15451, 850}, {16697, 6385}, {17434, 3267}, {18695, 1928}, {23181, 670}
X(217) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (32, 184, 14585), (39, 185, 3269), (54, 112, 1970), (1614, 10312, 1971), (7592, 8743, 6)


X(218) = X(7)-CEVA CONJUGATE OF X(55)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where
                        f(A,B,C) = cos2(A/2) [cos4(B/2) + cos4(C/2) - cos4(A/2)]
Barycentrics    a^2 (a^2 + b^2 + c^2 - 2 a b - 2 a c) : :

Let A' be the center of the conic through the contact points of the B- and C- excircles with the sidelines of ABC. Define B' and C' cyclically. The triangle A'B'C' is perspective to the intouch triangle at X(218). See also X(6), X(25), X(222), X(940), X(1743). (Randy Hutson, July 23, 2015)

A'B'C' is also the unary cofactor triangle of the intangents triangle. (Randy Hutson, June 7, 2019)

X(218) lies on these lines: 1,6   3,41   4,294   7,277   32,906   43,170   46,910   56,101   65,169   145,644   198,579   222,241   279,651

X(218) = isogonal conjugate of X(277)
X(218) = eigencenter of cevian triangle of X(7)
X(218) = eigencenter of anticevian triangle of X(55)
X(218) = X(7)-Ceva conjugate of X(55)
X(218) = crosssum of X(650) and X(1086)
X(218) = X(644)-beth conjugate of X(218)
X(218) = crossdifference of every pair of points on the de Longchamps line of the intouch triangle
X(218) = perspector of 2nd mixtilinear triangle and unary cofactor triangle of 5th mixtilinear triangle
X(218) = perspector of excentral triangle and unary cofactor triangle of inverse-in-incircle triangle


X(219) = X(8)-CEVA CONJUGATE OF X(55)

Trilinears    cos A cot A/2 : :
Trilinears    (sin A)/(1 - sec A) : :
Trilinears    1/(csc A - 2 csc 2A) : :
Trilinears    a(b + c - a)(b2 + c2 - a2) : :
Trilinears    (b + c - a) cos A : :
Barycentrics  sin 2A cot A/2 : :

Let A'B'C' be the extouch triangle. Let A″ be the barycentric product of the circumcircle intercepts of line B'C', and define B″ and C″ cyclically. The lines A'A″, B'B″, C'C″ concur in X(219). (Randy Hutson, July 31 2018)

X(219) lies on these lines: 1,6   3,48   8,29   10,965   19,517   40,610   41,1036   55,284   56,579   63,77   69,1332   101,102   144,347   200,282   206,692   255,268   278,329   332,345   346,644   572,947   577,906   604,672   1993,3219

X(219) = isogonal conjugate of X(278)
X(219) = isotomic conjugate of X(331)
X(219) = X(i)-Ceva conjugate of X(j) for these (i,j): (8,55), (63,3), (283,212)
X(219) = X(i)-cross conjugate of X(j) for these (i,j): (48,268), (71,9), (212,3)
X(219) = crosspoint of X(i) and X(j) for these (i,j): (8,345), (64,78)
X(219) = crosssum of X(i) and X(j) for these (i,j): (19,34), (56,608)
X(219) = X(i)-beth conjugate of X(j) for these (i,j): (101,478), (219,48), (644,219)
X(219) = trilinear pole of line X(652)X(1946)
X(219) = crossdifference of every pair of points on line X(513)X(1835)
X(219) = X(92)-isoconjugate of X(56)
X(219) = perspector of extouch triangle and unary cofactor triangle of intouch triangle
X(219) = perspector of ABC and unary cofactor triangle of Gemini triangle 37
X(219) = polar conjugate of isogonal conjugate of X(6056)
X(219) = polar conjugate of isotomic conjugate of X(1259)
X(219) = X(63)-isoconjugate of X(1118)


X(220) = X(9)-CEVA CONJUGATE OF X(55)

Trilinears    a(b + c - a)2 : :
Trilinears    (1 + cos A)2/sin A : :     (M. Iliev, 4/12/07)
Barycentrics    a2(b + c - a)2 : :

The trilinear polar of X(220) passes through X(657) (Randy Hutson, July 20, 2016)

Let A' be the barycentric product of the circumcircle intercepts of the A-excircle. Define B' and C' cyclically. The lines AA', BB', CC' concur in X(220). (Randy Hutson, July 11, 2019)

X(220) lies on these lines: 1,6   3,101   8,294   33,210   40,910   41,55   48,963   63,241   64,71   78,949   144,279   154,205   169,517   200,728   201,221   268,577   277,1086   281,594   329,948   346,1043

X(220) = isogonal conjugate of X(279)
X(220) = anticomplement of X(21258)
X(220) = X(i)-Ceva conjugate of X(j) for these (i,j): (9,55), (200,480)
X(220) = cevapoint of X(1) and X(170)
X(220) = crosspoint of X(9) and X(200)
X(220) = crosssum of X(57) and X(269)
X(220) = crossdifference of every pair of points on line X(513)X(676)
X(220) = X(i)-beth conjugate of X(j) for these (i,j): (101,221), (220,41), (644,220), (728,728)
X(220) = {X(1),X(9)}-harmonic conjugate of X(1212)
X(220) = perspector of ABC and unary cofactor triangle of inverse-in-incircle triangle
X(220) = barycentric square of X(9)


X(221) = X(1)-CEVA CONJUGATE OF X(56)

Trilinears    (sin2A/2)(cos B + cos C - cos A - 1) : :
Trilinears    (1 - cos A)(1 + cos A - cos B - cos C) : :

Let I = X(1) = incenter, and A'B'C' = medial triangle. Let AB = AC∩IA', and define BC and CA cyclically. Let AC = AB∩IA', and define BA and CB cyclically. Let OA be the circumcircle of AABAC, and define OB and OC cyclically. Then X(221) is the radical center of OA, OB, OC. (Angel Montesdeoca, April 27, 2021)

X(221) lies on these lines: 1,84   3,102   6,19   8,651   31,56   40,223   55,64   201,220   204,207   960,1038

X(221) = isogonal conjugate of X(280)
X(221) = anticomplement of X(20306)
X(221) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,56), (222,6), (223,198)
X(221) = crosspoint of X(i) and X(j) for these (i,j): (1,40), (196,347)
X(221) = crosssum of X(1) and X(84)
X(221) = X(i)-beth conjugate of X(j) for these (i,j): (1,34), (40,40), (101,220), (109,221), (110,3)
X(221) = trilinear product X(40)*X(56)
X(221) = X(92)-isoconjugate of X(268
X(221) = perspector of unary cofactor triangles of 1st and 3rd extouch triangles


X(222) = X(7)-CEVA CONJUGATE OF X(56)

Trilinears       cos A tan A/2 : cos B tan B/2 : cos C tan C/2
                        = 1/(csc A + 2 csc 2A) : 1/(csc B + 2 csc 2B) : 1/(csc A + 2 csc 2C)
                        = a(b2 + c2 - a2)/(b + c - a) : b(c2 + a2 - b2)/(c + a - b) : c(a2 + b2 - c2)/(a + b - c)

Barycentrics  a2/(1 + sec A) : b2/(1 + sec B) : c2/(1 + sec C)

Let A' be the center of the conic through the contact points of the incircle and the A-excircle with the sidelines of ABC. Define B' and C' cyclically. The triangle A'B'C' is perspective to the intouch triangle at X(222). See also X(6), X(25), X(218), X(940), X(1743). (Randy Hutson, July 23, 2015)

Let A'B'C' be the intouch triangle. Let A″ be the barycentric product of the circumcircle intercepts of line B'C', and define B″ and C″ cyclically. The lines A'A″, B'B″, C'C″ concur in X(222). (Randy Hutson, July 31 2018)

X(222) lies on these lines: 1,84   2,651   3,73   6,57   7,27   33,971   34,942   46,227   55,103   56,58   63,77   72,1038   171,611   189,281   218,241   226,478   268,1073   581,1035   601,1066   613,982   912,1060   1355,1363   1993,3218

X(222) = isogonal conjugate of X(281)
X(222) = isotomic conjugate of X(7017)
X(222) = X(i)-Ceva conjugate of X(j) for these (i,j): (7,56), (77,3), (81,57)
X(222) = cevapoint of X(6) and X(221)
X(222) = X(i)-cross conjugate of X(j) for these (i,j): (48,3), (73,77)
X(222) = crosspoint of X(7) and X(348)
X(222) = crosssum of X(i) and X(j) for these (i,j): (55,607), (650,1146)
X(222) = crossdifference of every pair of points on line X(3064)X(3700)
X(222) = trilinear pole of line X(1459)X(1946)
X(222) = Danneels point of X(651) (see notes at X(3078))
X(222) = X(4)-isoconjugate of X(9)
X(222) = intouch-isogonal conjugate of X(12723)
X(222) = perspector of intouch triangle and unary cofactor triangle of extouch triangle
X(222) = perspector of ABC and unary cofactor triangle of Gemini triangle 38
X(222) = vertex conjugate of foci of inconic that is the isotomic conjugate of the polar conjugate of the incircle (centered at X(17073))
X(222) = X(i)-beth conjugate of X(j) for these (i,j): (21,1012), (63,63), (110,222), (287,222), (648,222), (651,222), (662,2), (895,222)


X(223) = X(2)-CEVA CONJUGATE OF X(57)

Trilinears    (tan A/2)(cos B + cos C - cos A - 1)
Trilinears    sec B + sec C - sec A + 1 : :
Trilinears    (a^3 + a^2 b - a b^2 - b^3 + a^2 c - 2 a b c + b^2 c - a c^2 + b c^2 - c^3)/(b + c - a) : :

Let A' be the homothetic center of ABC and the orthic triangle of the A-extouch triangle and define B' and C' cyclically. Triangle A'B'C' is perspective to the 3rd extouch triangle at X(223). (Randy Hutson, September 14, 2016)

Let Va, Vb, Vc be the antipodes of V=X(40) in the circles (VBC), (VCA), (VAB), respectively. The triangle VaVbVc is here named the Bevan antipodal triangle (Hyacinthos #29638; however, see next note). The point X(223) is the unique finite fixed point of the affine transformation that maps the reference triangle ABC onto VaVbVc. (Angel Montesdeoca, October 15, 2019)

The construction of the triangle VaVbVc can be generalized for an arbitrary point P (instead of V=X(40)), and is, in fact, the antipedal triangle of P (Randy Hutson, message in Anopolis Group). Thus, "Bevan antipodal triangle" = antipedal triangle of X(40). (Angel Montesdeoca, October 16, 2019)

X(223) lies on the Thomson cubic and these lines: 4   2,77   3,1035   6,57   9,1073   40,221   55,1456   56,937   63,651   108,204   109,165   312,664   329,347   380,608   580,603   936,1038   1249,3352   3341,3349   3351,3356

X(223) = isogonal conjugate of X(282)
X(223) = isotomic conjugate of X(34404)
X(223) = complement of X(189)
X(223) = anticomplement of X(20205)
X(223) = complementary conjugate of X(21239)
X(223) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,57), (77,1), (342,208), (347,40)
X(223) = cevapoint of X(198) and X(221)
X(223) = X(i)-cross conjugate of X(j) for these (i,j): (198,40), (227,347)
X(223) = crosspoint of X(2) and X(329)
X(223) = crosssum of X(6) and X(1436)
X(223) = perspector of ABC and antipedal triangle of X(3345)
X(223) = perspector of pedal and anticevian triangles of X(3182)
X(223) = perspector of ABC and medial triangle of pedal triangle of X(1490)
X(223) = perspector of circumconic centered at X(57)
X(223) = center of circumconic that is locus of trilinear poles of lines passing through X(57)
X(223) = X(92)-isoconjugate of X(2188)
X(223) = {X(1),X(1745)}-harmonic conjugate of X(1490)
X(223) = pole wrt polar circle of trilinear polar of X(7020)
X(223) = X(48)-isoconjugate (polar conjugate) of X(7020)
X(223) = trilinear product X(40)*X(57)
X(223) also lies on line 55,1456. X(223) = X(i)-aleph conjugate of X(j) for these (i,j):
(63,1079), (77,223), (81,580), (174,46), (651,109)
X(223) = X(i)-beth conjugate of X(j) for these (i,j):
(2,278), (100,200), (162,204), (329,329), (651,223), (662,63)


X(224) = X(7)-CEVA CONJUGATE OF X(63)

Trilinears    [cot B cos2(B/2) + cot C cos2(C/2) - cot A cos2(A/2)]cot A
Trilinears    (b^2 + c^2 - a^2) (a^4 - b^4 - c^4 - 2 a^3 (b + c) - 2 a^2 b c + 2 a (b^3 + c^3) + 2 b^2 c^2) : :

X(224) lies on these lines: 1,377   3,63   8,914   21,90   46,100   65,1004   908,1079

X(224) = X(7)-Ceva conjugate of X(63)
X(224) = barycentric product of vertices of Gemini triangle 7
X(224) = barycentric product of vertices of Gemini triangle 8


X(225) = X(4)-CEVA CONJUGATE OF X(65)

Trilinears    (sec A)(cos B + cos C) : (sec B)(cos C + cos A) : (sec C)(cos A + cos B)
Barycentrics    (tan A)(cos B + cos C) : (tan B)(cos C + cos A) : (tan C)(cos A + cos B)
Barycentrics    (b + c)/((b^2 + c^2 - a^2) (b + c - a)) : :

X(225) lies on these lines: 1,4   3,1074   7,969   10,201   12,37   19,208   28,108   46,254   65,407   75,264   91,847   158,1093   377,1038   412,775   653,897

X(225) = isogonal conjugate of X(283)
X(225) = isotomic conjugate of X(332)
X(225) = anticomplement of X(34851)
X(225) = X(4)-Ceva conjugate of X(65)
X(225) = X(407)-cross conjugate of X(4)
X(225) = crosspoint of X(i) and X(j) for these (i,j): (4,158), (273,278)
X(225) = crosssum of X(i) and X(j) for these (i,j): (3,255), (212,219)
X(225) = X(i)-beth conjugate of X(j) for these (i,j): (4,225), (10,227), (108,1042), (318,10)
X(225) = pole wrt polar circle of trilinear polar of X(333) (line X(522)X(663))
X(225) = polar conjugate of X(333)


X(226) = X(7)-CEVA CONJUGATE OF X(65)

Trilinears    (csc A)(cos B + cos C) : (csc B)(cos C + cos A) : (csc C)(cos A + cos B)
Trilinears    bc(b + c)/(b + c - a) : ca(c + a)/(c + a - b) : ab(a + b)/(a + b - c)
Trilinears    cos(angle BIC) : cos(angle CIA) : cos(angle AIB)
Barycentrics    (b + c)/(b + c - a) : (c + a)/(c + a - b) : (a + b)/(a + b - c)
Barycentrics    area(A'BC) : : , where A'B'C' is the 2nd extouch triangle
Barycentrics    rA + r : rB + r : rC + r, where rA, rB, rC are the exradii

X(226) is the homothetic center of the intouch triangle and the triangle formed by the lines of the external pairs of extouch points of the excircles. (Randy Hutson, 9/23/2011)

Let A' be the radical center of the incircle and the B- and C-excircles; define B' and C' cyclically. A'B'C' is also the complement of the excentral triangle, and the triangle formed by the radical axes of the incircle and each excircle. Then X(226) is the homothetic center of A'B'C' and the intouch triangle. (Randy Hutson, December 26, 2015)

Let A' be the homothetic center of the orthic triangles of the intouch and A-extouch triangles, and define B' and C' cyclically. Let AaBaCa be the orthic triangle of the A-extouch triangle, and define AbBbCb, and AcBcCc cyclically. Let A″ be the centroid of AaAbAc, and define B″ and C″ cyclically. Then A'B'C' and A″B″C″ are homothetic to each other and to the medial triangle and the orthic triangle of the intouch triangle at X(226). (Randy Hutson, December 26, 2015)

Let (A') be the pedal circle of the A-vertex of the hexyl triangle, and define (B') and (C') cyclically. Then X(226) is the radical center of circles (A'), (B'), (C'). (Randy Hutson, December 26, 2015)

Let IaIbIc be the reflection triangle of X(1). Let A' be the trilinear pole of line IbIc, and define B', C' cyclically. The lines AA', BB', CC' concur in X(226). (Randy Hutson, July 20, 2016)

Let A13B13C13 be Gemini triangle 13. Let A' be the perspector of conic {A,B,C,B13,C13}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(226). (Randy Hutson, January 15, 2019)

Let A14B14C14 be Gemini triangle 14. Let A' be the center of conic {A,B,C,B14,C14}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(226). (Randy Hutson, January 15, 2019)

X(226) lies on these lines: 1,4   2,7   5,912   10,12   11,118   13,1082   14,554   27,284   29,951   35,79   36,1006   37,440   41,379   46,498   55,516   56,405   76,85   78,377   81,651   83,1429   86,1412   92,342   98,109   102,1065   175,1131   176,1132   196,281   208,406   222,478   228,851   262,982   273,469   306,321   429,1426   443,936   452,1420   474,1466   481,485   482,486   495,517   535,551   664,671   673,1174   748,1471   857,1446   975,1038   990,1040   1029,1442   1260,1376   1284,1402   1401,1463

X(226) = reflection of X(993) in X(1125)
X(226) = isogonal conjugate of X(284)
X(226) = isotomic conjugate of X(333)
X(226) = complement of X(63)
X(226) = X(i)-Ceva conjugate of X(j) for these (i,j): (7,65), (349,307)
X(226) = cevapoint of X(37) and X(65)
X(226) = X(i)-cross conjugate of X(j) for these (i,j): (37,10), (73,307)
X(226) = crosspoint of X(2) and X(92)
X(226) = crosssum of X(i) and X(j) for these (i,j): (6,48), (41,55)
X(226) = crossdifference of every pair of points on line X(652)X(663)
X(226) = X(63)-of-medial-triangle
X(226) = bicentric sum of PU(20)
X(226) = midpoint of PU(20)
X(226) = radical center of cevian circles of the extraversions of X(8)
X(226) = trilinear pole of line X(523)X(656) (the polar of X(29) wrt polar circle)
X(226) = pole wrt polar circle of trilinear polar of X(29) (the line X(243)X(522))
X(226) = X(48)-isoconjugate (polar conjugate) of X(29)
X(226) = X(6)-isoconjugate of X(21)
X(226) = X(184)-of-2nd-extouch-triangle
X(226) = {X(2),X(57)}-harmonic conjugate of X(3911)
X(226) = {X(9),X(57)}-harmonic conjugate of X(1708)
X(226) = homothetic center of intouch triangle and the complement of excentral triangle)
X(226) = homothetic center of 3rd Euler tringle and inverse-in-incircle triangle
X(226) = perspector of intouch triangle and Gergonne line extraversion triangle
X(226) = perspector of 2nd extouch triangle and Gergonne line extraversion triangle
X(226) = X(i)-beth conjugate of X(j) for these (i,j): (2,226), (21,1064), (100,42), (190,226), (312,306), (321,321), (335,226), (835,226)
X(226) = barycentric product X(109)*X(850)
X(226) = perspector of Gemini triangle 9 and cross-triangle of ABC and Gemini triangle 9
X(226) = trilinear pole of perspectrix of ABC and Gemini triangle 10
X(226) = perspector of Gemini triangle 40 and cross-triangle of ABC and Gemini triangle 40


X(227) = X(10)-CEVA CONJUGATE OF X(65)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(cos B + cos C - cos A - 1)tan A/2
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(227) lies on these lines: 12,37   34,55   40,221   42,65   46,222   56,197   198,208   201,210   322,347   607,910

X(227) = isogonal conjugate of X(285)
X(227) = complement of X(20220)
X(227) = X(10)-Ceva conjugate of X(65)
X(227) = crosspoint of X(223) and X(347)
X(227) = crosssum of X(84) and X(1433)
X(227) = X(i)-beth conjugate of X(j) for these (i,j): (10,225), (40,227), (100,72)


X(228) = X(3)-CEVA CONJUGATE OF X(71)

Trilinears       (sin 2A)(sin B + sin C) : (sin 2B)(sin C + sin A) : (sin 2C)(sin A + sin B)
Barycentrics  (sin A sin 2A)(sin B + sin C) : (sin B sin 2B)(sin C + sin A) : (sin C sin 2C)(sin A + sin B)

X(228) lies on these lines: 3,63   9,1011   12,407   19,25   28,943   31,32   35,846   42,181   48,184   73,408   98,100   226,851

X(228) = isogonal conjugate of X(286)
X(228) = complement of X(20242)
X(228) = X(i)-Ceva conjugate of X(j) for these (i,j): (3,71), (37,213), (55,42)
X(228) = crosspoint of X(i) and X(j) for these (i,j): (3,48), (37,72), (55,212), (71,73)
X(228) = crosssum of X(i) and X(j) for these (i,j): (4,92), (7,273), (27,29), (28,81)
X(228) = crossdifference of every pair of points on line X(693)X(905)
X(228) = X(212)-beth conjugate of X(228)
X(228) = X(28)-isoconjugate of X(75)
X(228) = X(81)-isoconjugate of X(92)
X(228) = trilinear pole of line X(810)X(3049)
X(228) = perspector of unary cofactor triangles of Gemini triangles 1 and 2
X(228) = anticomplement of complementary conjugate of X(18591)
X(228) = {X(3),X(63)}-harmonic conjugate of X(22060)


X(229) = X(7)-CEVA CONJUGATE OF X(81)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (v + w - u)/(b + c),
                         u = u(a,b,c) = a(b + c - a)/(b + c), v = u(b,c,a), w = u(c,a,b)

Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(229) lies on these lines: 1,267   21,36   28,60   58,244   65,110   593,1104

X(229) = midpoint of X(1) and X(267)
X(229) = X(7)-Ceva conjugate of X(81)


X(230) = X(2)-CEVA CONJUGATE OF X(114)

Trilinears    bc[a2(2a2 - b2 - c2) + (b2 - c2)2] : :
Barycentrics    2*a^4 - a^2*b^2 + b^4 - a^2*c^2 - 2*b^2*c^2 + c^4 : :
X(230) = X(13) + X(14) + X(15) + X(16)
X(230) = 3 X[2] + X[385], 9 X[2] - X[7779], 5 X[2] - X[7840], 9 X[2] - 5 X[7925], X[2] + 3 X[8859], 6 X[2] + X[15480], 5 X[2] - 3 X[41133], 11 X[2] - 3 X[41136], 2 X[2] - 3 X[41139], 7 X[2] + X[44367], 3 X[2] - 4 X[44381], X[4] + 3 X[21445], X[4] - 3 X[39663], X[15] + 3 X[22511], X[16] + 3 X[22510], X[98] + 3 X[38227], 3 X[98] + X[43460], X[99] + 3 X[14568], X[99] - 3 X[35297], 3 X[115] + X[6781], 5 X[115] - 3 X[39563], 4 X[140] - 3 X[10256], X[148] + 3 X[13586], 3 X[187] - X[6781], 5 X[187] + 3 X[39563], X[187] + 2 X[43291], X[316] - 5 X[14061], X[316] - 3 X[33228], 3 X[325] - X[7779], 5 X[325] - 3 X[7840], 3 X[325] - 5 X[7925], X[325] + 9 X[8859], 2 X[325] + X[15480], 2 X[325] - 3 X[22110], X[325] + 3 X[22329], 5 X[325] - 9 X[41133], 11 X[325] - 9 X[41136], 2 X[325] - 9 X[41139], 7 X[325] + 3 X[44367], X[325] - 4 X[44381], X[325] - 6 X[44401], 3 X[385] + X[7779], 5 X[385] + 3 X[7840], 3 X[385] + 5 X[7925], X[385] - 9 X[8859], 2 X[385] + 3 X[22110], X[385] - 3 X[22329], 5 X[385] + 9 X[41133], 11 X[385] + 9 X[41136], 2 X[385] + 9 X[41139], 7 X[385] - 3 X[44367], X[385] + 2 X[44377], X[385] + 4 X[44381], X[385] + 6 X[44401], 3 X[468] - X[16316]

X(230) is the midpoint of the centers of the (equilateral) pedal triangles of X(15) and X(16).

X(230) = QA-P6 (Parabola Axes Crosspoint) of quadrangle ABCX(2); see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/27-qa-p6.html

X(230) lies on the cubics K492, K518, K780, K1166, and these lines: {1, 21965}, {2, 6}, {3, 2549}, {4, 3053}, {5, 32}, {10, 4544}, {11, 1914}, {12, 172}, {13, 41406}, {14, 41407}, {15, 21156}, {16, 21157}, {20, 5023}, {21, 44517}, {22, 8553}, {23, 11063}, {24, 27376}, {25, 53}, {26, 44527}, {30, 115}, {37, 6690}, {39, 140}, {50, 858}, {67, 9769}, {76, 7789}, {83, 32992}, {98, 1503}, {99, 14568}, {100, 17737}, {101, 17734}, {105, 9090}, {111, 476}, {112, 403}, {114, 1692}, {125, 8779}, {132, 38970}, {147, 35006}, {148, 13586}, {160, 20885}, {182, 37451}, {185, 15575}, {186, 2079}, {194, 7907}, {216, 1196}, {231, 232}, {233, 11548}, {235, 1968}, {248, 2450}, {251, 2963}, {262, 7607}, {263, 13240}, {274, 17694}, {297, 2966}, {315, 7887}, {316, 14061}, {338, 30737}, {350, 26629}, {376, 5210}, {381, 1384}, {383, 5321}, {384, 44530}, {387, 7410}, {393, 6353}, {397, 37463}, {398, 37464}, {404, 44542}, {427, 571}, {428, 1879}, {439, 2996}, {442, 5277}, {444, 2178}, {460, 39072}, {485, 6423}, {486, 6424}, {495, 2242}, {496, 2241}, {499, 16502}, {511, 2023}, {538, 620}, {542, 2030}, {543, 27088}, {546, 7747}, {547, 5008}, {548, 7756}, {549, 574}, {550, 5206}, {566, 7495}, {570, 1194}, {577, 1368}, {594, 7081}, {609, 7951}, {625, 754}, {626, 7767}, {631, 5013}, {632, 5346}, {641, 45576}, {642, 45577}, {671, 8598}, {691, 15546}, {694, 36897}, {698, 5976}, {732, 2024}, {736, 3934}, {800, 6375}, {868, 35906}, {1003, 11185}, {1015, 15325}, {1030, 4220}, {1040, 9594}, {1078, 6656}, {1080, 5318}, {1086, 1447}, {1100, 24239}, {1107, 4999}, {1108, 36227}, {1146, 35081}, {1151, 26516}, {1152, 26521}, {1180, 13351}, {1249, 38282}, {1281, 28530}, {1285, 3545}, {1312, 46629}, {1313, 46628}, {1329, 4426}, {1333, 37360}, {1352, 37071}, {1353, 39764}, {1379, 31862}, {1380, 31863}, {1504, 8981}, {1505, 13966}, {1506, 3628}, {1560, 16221}, {1562, 21663}, {1570, 34380}, {1572, 5886}, {1575, 3035}, {1576, 44089}, {1594, 10312}, {1625, 45938}, {1627, 2965}, {1656, 2548}, {1799, 16890}, {1834, 6998}, {1865, 4231}, {1901, 2305}, {1909, 26686}, {1970, 12241}, {1975, 16925}, {2021, 2782}, {2071, 34866}, {2072, 10317}, {2076, 5999}, {2080, 15980}, {2207, 3542}, {2243, 37691}, {2271, 5292}, {2275, 5433}, {2276, 5432}, {2280, 29662}, {2459, 13873}, {2460, 13926}, {2482, 5215}, {2502, 35266}, {2886, 4386}, {2896, 7901}, {3016, 46817}, {3058, 10987}, {3070, 6811}, {3071, 6813}, {3086, 16781}, {3087, 8889}, {3091, 22331}, {3094, 22712}, {3096, 7942}, {3124, 32269}, {3147, 41361}, {3162, 8746}, {3163, 23992}, {3172, 11404}, {3199, 21841}, {3229, 11672}, {3266, 36953}, {3269, 15341}, {3284, 5159}, {3285, 8229}, {3363, 7617}, {3424, 43537}, {3509, 17719}, {3522, 44519}, {3523, 7738}, {3525, 31400}, {3526, 5319}, {3530, 7765}, {3534, 15655}, {3548, 23115}, {3552, 32819}, {3553, 5268}, {3554, 5272}, {3582, 16784}, {3584, 16785}, {3624, 9575}, {3684, 33140}, {3705, 17362}, {3712, 4037}, {3726, 17724}, {3734, 8369}, {3785, 7784}, {3788, 3933}, {3818, 41412}, {3830, 43618}, {3845, 18362}, {3849, 5461}, {3850, 39590}, {3925, 10315}, {3926, 32970}, {3943, 37764}, {3972, 8370}, {4045, 7817}, {4071, 4434}, {4136, 8669}, {4189, 44520}, {4251, 45939}, {4395, 33891}, {4478, 30179}, {4590, 37803}, {4892, 4987}, {5000, 41200}, {5001, 41201}, {5012, 9604}, {5017, 5480}, {5020, 8573}, {5024, 5054}, {5025, 7750}, {5034, 38110}, {5039, 38317}, {5041, 5368}, {5052, 18583}, {5055, 15484}, {5058, 7584}, {5062, 7583}, {5063, 30739}, {5066, 14537}, {5067, 31404}, {5070, 43136}, {5071, 18584}, {5077, 8182}, {5094, 6749}, {5104, 6034}, {5106, 44215}, {5111, 40336}, {5116, 37455}, {5124, 19649}, {5184, 38220}, {5204, 9597}, {5217, 9598}, {5218, 31477}, {5280, 31460}, {5283, 7483}, {5291, 17757}, {5418, 6422}, {5420, 6421}, {5471, 6114}, {5472, 6115}, {5503, 10153}, {5585, 10304}, {5641, 44576}, {5965, 6721}, {5977, 35101}, {5980, 10617}, {5981, 10616}, {5988, 17768}, {6128, 39602}, {6146, 14585}, {6179, 7752}, {6194, 44453}, {6292, 7852}, {6321, 38225}, {6337, 6392}, {6409, 26295}, {6410, 26294}, {6560, 13711}, {6561, 13834}, {6564, 26438}, {6565, 18539}, {6636, 44521}, {6640, 22120}, {6642, 9608}, {6675, 16589}, {6683, 7829}, {6691, 16604}, {6719, 40553}, {6720, 44340}, {6756, 27371}, {6772, 35304}, {6775, 35303}, {6791, 32225}, {7000, 23261}, {7031, 7741}, {7179, 17365}, {7181, 7200}, {7374, 23251}, {7386, 36748}, {7467, 8266}, {7488, 44523}, {7492, 44522}, {7494, 36751}, {7505, 8743}, {7512, 44525}, {7525, 9700}, {7576, 10986}, {7615, 11159}, {7616, 10336}, {7754, 7763}, {7759, 7862}, {7760, 7769}, {7761, 7844}, {7764, 7805}, {7768, 7899}, {7770, 32832}, {7771, 7790}, {7773, 20065}, {7776, 14023}, {7783, 33259}, {7785, 32967}, {7786, 7856}, {7787, 16921}, {7791, 7851}, {7794, 7874}, {7795, 32954}, {7796, 7940}, {7797, 7824}, {7800, 7866}, {7801, 17131}, {7802, 33229}, {7803, 11285}, {7810, 7853}, {7811, 7934}, {7813, 31274}, {7814, 7877}, {7815, 7834}, {7818, 14929}, {7820, 8368}, {7821, 7826}, {7822, 33185}, {7823, 32966}, {7830, 7861}, {7836, 17129}, {7839, 16923}, {7841, 14907}, {7845, 31275}, {7847, 43459}, {7854, 7867}, {7855, 7888}, {7864, 33004}, {7876, 7932}, {7879, 33218}, {7889, 31239}, {7891, 20081}, {7892, 31276}, {7893, 7912}, {7904, 7933}, {7920, 33015}, {7923, 33021}, {7938, 14065}, {7968, 26300}, {7969, 26301}, {8352, 9166}, {8354, 46893}, {8355, 14971}, {8550, 9744}, {8557, 24345}, {8587, 43535}, {8588, 8703}, {8589, 12100}, {8623, 21531}, {8716, 33216}, {8744, 37943}, {8749, 40347}, {8778, 37197}, {8791, 14910}, {8854, 10962}, {8855, 10960}, {8882, 23295}, {8901, 41270}, {8960, 44647}, {8976, 31411}, {9112, 16267}, {9113, 16268}, {9164, 17968}, {9167, 39785}, {9475, 44887}, {9593, 31423}, {9595, 37696}, {9620, 26446}, {9650, 10592}, {9665, 10593}, {9675, 42215}, {9696, 40111}, {9699, 12106}, {9759, 34319}, {9760, 41746}, {9762, 41745}, {9764, 41747}, {9765, 41749}, {9855, 41135}, {10018, 39575}, {10154, 34481}, {10164, 31443}, {10257, 14961}, {10264, 14901}, {10272, 46301}, {10298, 44538}, {10303, 22332}, {10316, 11585}, {10352, 13196}, {10691, 22052}, {10988, 34612}, {11054, 41134}, {11173, 20423}, {11179, 40248}, {11231, 31398}, {11272, 46305}, {11317, 20112}, {11360, 23208}, {11413, 44528}, {11485, 20426}, {11486, 20425}, {11632, 37461}, {11645, 38010}, {11668, 11669}, {11676, 14651}, {11799, 46633}, {11812, 39593}, {12007, 43461}, {12088, 44537}, {12108, 31652}, {12359, 23128}, {12815, 35018}, {12962, 31454}, {13178, 38221}, {13337, 15302}, {13345, 37439}, {13449, 23514}, {14001, 32828}, {14041, 14712}, {14043, 46226}, {14482, 15709}, {14581, 37942}, {14586, 40631}, {14588, 19577}, {14694, 14995}, {14881, 46321}, {15109, 15246}, {15122, 44468}, {15301, 35022}, {15355, 22353}, {15448, 20998}, {15515, 15712}, {15602, 41983}, {15603, 15689}, {15905, 30771}, {16081, 16089}, {16196, 22401}, {16252, 32445}, {16434, 36743}, {16529, 16961}, {16530, 16960}, {16808, 41408}, {16809, 41409}, {16968, 46835}, {16976, 40349}, {17128, 33225}, {17602, 41269}, {17735, 17747}, {18253, 21879}, {18353, 34603}, {18365, 30745}, {18806, 32189}, {18840, 32952}, {18860, 38737}, {19103, 43430}, {19104, 43431}, {19130, 41413}, {19312, 23905}, {19544, 36744}, {21158, 36772}, {21397, 44269}, {21448, 34288}, {21554, 33863}, {21840, 29683}, {22253, 34511}, {22512, 41017}, {22513, 41016}, {22682, 40927}, {23004, 39555}, {23005, 39554}, {23972, 35080}, {23976, 35088}, {23980, 35079}, {23986, 35086}, {24243, 45595}, {24244, 45596}, {25488, 30489}, {25555, 44500}, {25681, 39248}, {26242, 29665}, {26369, 44635}, {26370, 44636}, {26468, 42262}, {26469, 42265}, {26517, 44646}, {26518, 44644}, {26522, 44645}, {26523, 44643}, {28697, 41009}, {29012, 35021}, {30478, 31490}, {30516, 40130}, {30537, 39389}, {30677, 44095}, {31409, 31479}, {31416, 31493}, {31467, 46219}, {32006, 32972}, {32431, 33628}, {32452, 32521}, {32479, 36523}, {32494, 39388}, {32497, 39387}, {32518, 45900}, {32640, 34150}, {32661, 44665}, {32762, 39828}, {32815, 32985}, {32816, 32969}, {32818, 32959}, {32823, 32958}, {32826, 33239}, {32827, 32984}, {32829, 32977}, {32830, 33203}, {32834, 33181}, {32838, 32968}, {32867, 32975}, {32980, 39143}, {33758, 39087}, {33900, 35298}, {34254, 42406}, {34828, 41760}, {35002, 38739}, {35085, 40621}, {35087, 35133}, {35606, 45662}, {36759, 46054}, {36760, 46053}, {37182, 44882}, {37362, 41332}, {37453, 45141}, {37897, 40350}, {37910, 46216}, {37911, 40135}, {40107, 44499}, {40236, 44536}, {40323, 41489}, {40884, 41254}, {40938, 40939}, {42006, 43528}, {44214, 46634}, {44648, 45514}, {45524, 45868}, {45525, 45869}

X(230) = midpoint of X(i) and X(j) for these {i,j}: {2, 22329}, {6, 15993}, {98, 1513}, {115, 187}, {193, 44369}, {297, 2966}, {325, 385}, {395, 396}, {468, 16315}, {671, 8598}, {2080, 15980}, {3229, 35078}, {3580, 14999}, {5912, 5913}, {6108, 6109}, {6782, 6783}, {7426, 16092}, {9127, 16341}, {11537, 11549}, {11632, 37461}, {11799, 46633}, {14568, 35297}, {21445, 39663}, {32456, 32457}, {44376, 44388}, {44392, 44394}
X(230) = reflection of X(i) in X(j) for these {i,j}: {2, 44401}, {69, 44395}, {99, 32459}, {114, 10011}, {115, 43291}, {325, 44377}, {625, 6722}, {6390, 620}, {15301, 35022}, {15480, 385}, {16320, 468}, {22110, 2}, {31173, 8355}, {35088, 44334}, {37350, 5461}, {37459, 14693}, {44377, 44381}, {44380, 3589}
X(230) = complement of X(325)
X(230) = anticomplement of X(44377)
X(230) = circumcircle-inverse of X(5941)
X(230) = orthoptic-circle-of-Steiner-inellipse-inverse of X(6792)
X(230) = Moses-Parry-circle-inverse of X(2493)
X(230) = isogonal conjugate of X(2987)
X(230) = isotomic conjugate of X(8781)
X(230) = polar conjugate of X(35142)
X(230) = complement of the isogonal conjugate of X(1976)
X(230) = complement of the isotomic conjugate of X(98)
X(230) = isotomic conjugate of the isogonal conjugate of X(1692)
X(230) = isotomic conjugate of the polar conjugate of X(460)
X(230) = isogonal conjugate of the polar conjugate of X(44145)
X(230) = polar conjugate of the isotomic conjugate of X(3564)
X(230) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 114}, {32, 16591}, {98, 2887}, {248, 18589}, {290, 21235}, {293, 1368}, {560, 11672}, {661, 36471}, {685, 21259}, {798, 35088}, {878, 34846}, {1821, 626}, {1910, 141}, {1933, 46840}, {1973, 15595}, {1976, 10}, {2395, 21253}, {2422, 8287}, {2715, 4369}, {2966, 42327}, {3404, 21248}, {6531, 20305}, {9154, 21256}, {11610, 21247}, {14600, 1214}, {14601, 37}, {15628, 21244}, {15630, 24040}, {32676, 41167}, {32696, 8062}, {36036, 23301}, {36084, 512}, {36104, 30476}, {36120, 21243}, {43187, 21263}, {46273, 40379}, {46289, 36213}
X(230) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 114}, {297, 1503}, {648, 38359}, {2396, 804}, {2966, 523}, {6531, 6}, {31635, 157}, {39645, 5254}, {40120, 25}, {44145, 460}, {44767, 525}
X(230) = X(i)-cross conjugate of X(j) for these (i,j): {1692, 460}, {36472, 2501}
X(230) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2987}, {2, 36051}, {6, 8773}, {19, 43705}, {31, 8781}, {48, 35142}, {63, 3563}, {75, 32654}, {92, 42065}, {647, 36105}, {656, 32697}, {661, 10425}, {662, 35364}, {1755, 40428}, {1821, 34157}, {1959, 2065}, {2159, 36891}
X(230) = crosspoint of X(i) and X(j) for these (i,j): {2, 98}, {22456, 34537}
X(230) = crosssum of X(i) and X(j) for these (i,j): {6, 511}, {520, 41172}, {1084, 39469}, {32654, 42065}
X(230) = trilinear pole of line {5477, 42663}
X(230) = crossdifference of every pair of points on line {3, 512}
X(230) = X(2)-Hirst inverse of X(193)
X(230) = X(i)-beth conjugate of X(j) for these (i,j): (281,230), (645,230)
X(230) = centroid of quadrangle X(13)X(14)X(15)X(16)
X(230) = radical center of cirumcircle, nine-point circle and Lester circle
X(230) = perspector of circumconic centered at X(114)
X(230) = center of circumconic that is locus of trilinear poles of lines passing through X(114)
X(230) = inverse-in-Steiner-inellipse of X(6)
X(230) = X(910)-of-orthic-triangle if ABC is acute
X(230) = PU(4)-harmonic conjugate of X(2501)
X(230) = X(13)-antipedal-to-X(16)-pedal similarity image of X(14)
X(230) = X(14)-antipedal-to-X(15)-pedal similarity image of X(13)
X(230) = barycentric product X(i)*X(j) for these {i,j}: {1, 1733}, {3, 44145}, {4, 3564}, {13, 6782}, {14, 6783}, {30, 36875}, {69, 460}, {75, 8772}, {76, 1692}, {98, 114}, {305, 44099}, {511, 14265}, {523, 4226}, {542, 34174}, {670, 42663}, {671, 5477}, {1821, 17462}, {1916, 12829}, {2782, 46039}, {2974, 3563}, {7612, 10011}, {11606, 12830}, {31842, 40120}, {38359, 44768}
X(230) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 8773}, {2, 8781}, {3, 43705}, {4, 35142}, {6, 2987}, {25, 3563}, {30, 36891}, {31, 36051}, {32, 32654}, {98, 40428}, {110, 10425}, {112, 32697}, {114, 325}, {162, 36105}, {184, 42065}, {237, 34157}, {460, 4}, {512, 35364}, {1692, 6}, {1733, 75}, {1976, 2065}, {3564, 69}, {4226, 99}, {5477, 524}, {6782, 298}, {6783, 299}, {8772, 1}, {10011, 1007}, {12829, 385}, {12830, 7779}, {14265, 290}, {17462, 1959}, {34174, 5641}, {35296, 14253}, {36875, 1494}, {39072, 37183}, {42663, 512}, {44099, 25}, {44145, 264}, {46039, 46142}
X(230) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 6, 3815}, {2, 69, 7778}, {2, 183, 141}, {2, 193, 1007}, {2, 325, 44377}, {2, 385, 325}, {2, 1611, 40326}, {2, 1654, 30761}, {2, 1992, 11184}, {2, 2895, 30760}, {2, 3815, 3055}, {2, 5276, 37661}, {2, 5304, 7736}, {2, 5306, 9300}, {2, 5912, 44398}, {2, 5913, 24855}, {2, 7610, 11168}, {2, 7735, 6}, {2, 7736, 31489}, {2, 7766, 7777}, {2, 7777, 37647}, {2, 7779, 7925}, {2, 7792, 3589}, {2, 7806, 7792}, {2, 7840, 41133}, {2, 8859, 22329}, {2, 8860, 15597}, {2, 11163, 9771}, {2, 11174, 15491}, {2, 11580, 5913}, {2, 13638, 590}, {2, 13758, 615}, {2, 16989, 11174}, {2, 16990, 7868}, {2, 16997, 37664}, {2, 17004, 37688}, {2, 17008, 183}, {2, 23055, 7610}, {2, 26243, 1211}, {2, 26244, 1213}, {2, 26282, 5718}, {2, 34229, 15271}, {2, 37637, 3054}, {2, 37667, 69}, {2, 37668, 37690}, {2, 37689, 7735}, {2, 42850, 21358}, {2, 44401, 41139}, {3, 3767, 5254}, {5, 32, 7745}, {5, 18907, 5475}, {6, 3054, 3055}, {6, 3815, 9300}, {6, 7735, 5306}, {6, 8253, 31463}, {6, 21001, 40805}, {6, 31489, 7736}, {6, 37637, 2}, {32, 5475, 18907}, {32, 7746, 5}, {39, 7749, 140}, {39, 7755, 5305}, {50, 44529, 858}, {69, 37667, 8667}, {76, 7807, 7789}, {76, 7857, 7807}, {98, 36899, 41932}, {98, 38227, 1513}, {99, 35297, 32459}, {100, 17737, 21956}, {114, 12829, 12830}, {140, 5305, 39}, {141, 13468, 183}, {141, 15598, 16990}, {183, 7868, 16990}, {183, 16990, 15598}, {183, 17008, 13468}, {193, 1007, 9766}, {231, 3003, 16310}, {232, 6103, 16318}, {232, 16318, 1990}, {316, 14061, 33228}, {325, 22329, 385}, {325, 44377, 22110}, {376, 43448, 44526}, {381, 1384, 7737}, {385, 3329, 39097}, {385, 7925, 7779}, {385, 39095, 6}, {385, 39101, 39093}, {385, 44381, 22110}, {427, 10311, 6748}, {468, 6103, 1990}, {468, 16306, 16308}, {468, 16310, 2493}, {468, 16313, 16335}, {468, 16317, 10418}, {468, 16318, 232}, {468, 16325, 16321}, {549, 15048, 574}, {571, 9722, 6748}, {574, 5309, 15048}, {590, 615, 141}, {597, 15491, 11174}, {597, 15597, 2}, {626, 7780, 7767}, {626, 7886, 8361}, {631, 5286, 5013}, {632, 31406, 31455}, {671, 26613, 8598}, {1078, 7828, 6656}, {1078, 7919, 7831}, {1609, 2165, 53}, {1656, 30435, 2548}, {1691, 44534, 1513}, {2076, 44531, 5999}, {2080, 38224, 15980}, {2493, 3003, 16308}, {2493, 16306, 1990}, {2549, 21843, 3}, {3003, 3018, 16303}, {3003, 3291, 2493}, {3003, 6103, 16306}, {3003, 16310, 1990}, {3018, 10418, 2493}, {3018, 16303, 1990}, {3053, 13881, 4}, {3054, 3815, 2}, {3054, 5306, 3815}, {3054, 7735, 9300}, {3055, 9300, 3815}, {3068, 3069, 193}, {3068, 26456, 44594}, {3068, 44365, 44394}, {3068, 44595, 6}, {3069, 26463, 44597}, {3069, 44364, 44392}, {3069, 44596, 6}, {3096, 7942, 8363}, {3291, 6103, 16310}, {3291, 10418, 16317}, {3314, 37671, 3631}, {3329, 17006, 2}, {3523, 7738, 15815}, {3526, 5319, 9606}, {3526, 9605, 31401}, {3767, 21843, 2549}, {3785, 14064, 7784}, {3788, 7751, 3933}, {3815, 5306, 6}, {3934, 6680, 7819}, {5008, 7603, 7753}, {5013, 5286, 9607}, {5013, 44535, 631}, {5023, 44518, 20}, {5025, 7793, 7750}, {5055, 15484, 31415}, {5055, 21309, 15484}, {5206, 7748, 550}, {5210, 44526, 376}, {5304, 7736, 6}, {5306, 37637, 3055}, {5319, 31401, 9605}, {5346, 31455, 7772}, {5368, 9698, 5041}, {5475, 18907, 7745}, {5585, 44541, 10304}, {6103, 10418, 3018}, {6179, 7752, 7762}, {6189, 6190, 193}, {6292, 7852, 8364}, {6392, 32989, 6337}, {7610, 15271, 34229}, {7612, 9752, 9756}, {7615, 37809, 11159}, {7735, 7736, 5304}, {7735, 37637, 3815}, {7736, 31489, 3815}, {7737, 43620, 381}, {7747, 39565, 546}, {7749, 7755, 39}, {7754, 33233, 7763}, {7756, 15513, 548}, {7761, 7844, 33184}, {7762, 33249, 7752}, {7766, 7777, 41624}, {7766, 41624, 32455}, {7767, 8361, 626}, {7771, 7790, 8356}, {7772, 31455, 31406}, {7774, 14614, 3629}, {7778, 8667, 69}, {7779, 7925, 325}, {7780, 7886, 626}, {7792, 37688, 2}, {7806, 17004, 2}, {7806, 37688, 3589}, {7815, 7834, 8362}, {7817, 34506, 8359}, {7828, 7831, 7919}, {7830, 7861, 8357}, {7831, 7919, 6656}, {7868, 16990, 141}, {7891, 20081, 32820}, {8105, 8106, 2493}, {8553, 44524, 22}, {8584, 9771, 11163}, {8960, 45515, 44647}, {8974, 13950, 9740}, {9605, 31401, 9606}, {9744, 9755, 8550}, {9753, 13860, 5480}, {11063, 44533, 23}, {11174, 16989, 597}, {11174, 39093, 39101}, {11488, 11489, 3620}, {13638, 13758, 183}, {13846, 13847, 15533}, {13910, 13972, 8177}, {13910, 45871, 590}, {13972, 45872, 615}, {14537, 39601, 43457}, {14971, 31173, 8355}, {15271, 34229, 11168}, {15480, 41139, 44377}, {16303, 16310, 3018}, {16303, 16317, 2493}, {16644, 16645, 21358}, {17129, 33245, 7836}, {20065, 32961, 7773}, {22110, 41139, 2}, {22329, 44377, 15480}, {22329, 44401, 22110}, {23302, 23303, 34573}, {24855, 44398, 22110}, {26456, 44594, 6}, {26463, 44597, 6}, {32787, 32788, 8584}, {35007, 39565, 7747}, {36899, 41932, 34369}, {37637, 37689, 5306}, {37640, 37641, 5032}, {37647, 41624, 7777}, {39022, 39023, 6}, {39107, 39108, 1992}, {39601, 43457, 5066}, {43622, 43623, 3763}, {44192, 44193, 157}, {44364, 44365, 44369}, {44377, 44381, 2}, {44377, 44401, 44381}, {44390, 44391, 141}, {44594, 44595, 26456}, {44596, 44597, 26463}


X(231) = X(2)-CEVA CONJUGATE OF X(128)

Trilinears    u(-au + bv + cw), u : v : w = X(128)
Trilinears    (cos B) (4 sin^2 C - 1) (cos A sin B - sin A cos B) - (cos C) (4 sin^2 B - 1) (cos C sin A - sin C cos A) : :
Barycentrics    2*a^8 - 4*a^6*b^2 + 3*a^4*b^4 - 2*a^2*b^6 + b^8 - 4*a^6*c^2 + 2*a^2*b^4*c^2 - 4*b^6*c^2 + 3*a^4*c^4 + 2*a^2*b^2*c^4 + 6*b^4*c^4 - 2*a^2*c^6 - 4*b^2*c^6 + c^8 : :
X(231) = X[35727] + 3 X[38394]

X(231) lies on the cubic K752 and these lines: {2, 1273}, {3, 15551}, {4, 96}, {5, 15552}, {6, 17}, {50, 115}, {140, 570}, {216, 34833}, {230, 232}, {340, 44375}, {395, 33530}, {396, 33529}, {566, 7749}, {1609, 3517}, {1989, 2070}, {2072, 3284}, {2965, 36412}, {3001, 6036}, {3054, 33992}, {3523, 14806}, {3767, 5063}, {5158, 6639}, {5306, 37439}, {5355, 13337}, {6749, 9722}, {7735, 33872}, {7747, 9220}, {10018, 40939}, {10510, 38740}, {10619, 14533}, {11077, 16337}, {11082, 11136}, {11087, 11135}, {16534, 45938}, {32223, 46155}, {35727, 38394}, {37972, 44533}, {41770, 44134}

X(231) = complement of X(1273)
X(231) = circumcircle-inverse of X(15551)
X(231) = nine-point-circle-inverse of X(15552)
X(231) = complement of the isotomic conjugate of X(1141)
X(231) = polar conjugate of the isotomic conjugate of X(539)
X(231) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 128}, {1141, 2887}, {11077, 18589}, {46138, 21235}
X(231) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 128}, {14918, 18400}
X(231) = radical center of cirumcircle, nine-point circle and Hutson-Parry circle
X(231) = crosspoint of X(i) and X(j) for these (i,j): {2, 1141}, {11082, 11087}
X(231) = crosssum of X(i) and X(j) for these (i,j): {6, 1154}, {6368, 10413}, {11126, 11127}
X(231) = crossdifference of every pair of points on line {3, 1510}
X(231) = X(281)-beth conjugate of X(230)
X(231) = perspector of circumconic centered at X(128)
X(231) = center of circumconic that is locus of trilinear poles of lines passing through X(128)
X(231) = X(63)-isoconjugate of X(2383)
X(231) = barycentric product X(i)*X(j) for these {i,j}: {4, 539}, {5, 40631}, {93, 45083}, {128, 1141}, {1263, 27423}, {3459, 10615}, {10412, 43969}
X(231) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 2383}, {128, 1273}, {539, 69}, {10615, 45799}, {40631, 95}, {43969, 10411}, {45083, 44180}
X(231) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 2963, 233}, {17, 18, 1209}, {230, 16310, 3003}, {233, 40136, 6}, {571, 2165, 1879}, {3003, 16310, 3018}, {3767, 46262, 5063}


X(232) = X(2)-CEVA CONJUGATE OF X(132)

Trilinears    tan A cos(A + ω) : tan B cos(B + ω) : tan C cos(C + ω)
Barycentrics    sin A tan A cos(A + ω) : sin B tan B cos(B + ω) : sin C tan C cos(C + ω)
Barycentrics    a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^2*b^2 - b^4 + a^2*c^2 - c^4) : :

X(232) lies on the cubics K222, K493, K756, K781, K782, K783, K785, K787, K1097 and these lines: {2, 216}, {3, 1968}, {4, 39}, {5, 27376}, {6, 25}, {19, 444}, {20, 22401}, {22, 577}, {23, 250}, {24, 32}, {26, 10316}, {30, 14961}, {33, 2276}, {34, 2275}, {50, 3447}, {53, 427}, {69, 37187}, {98, 41204}, {99, 15014}, {111, 1304}, {112, 186}, {114, 38970}, {115, 403}, {132, 1513}, {136, 38975}, {183, 9308}, {185, 32445}, {211, 15897}, {217, 389}, {230, 231}, {233, 37990}, {235, 5254}, {237, 9475}, {240, 42702}, {248, 32649}, {251, 8882}, {297, 325}, {316, 40889}, {317, 7774}, {340, 7779}, {378, 574}, {385, 648}, {406, 5283}, {419, 6531}, {420, 3229}, {428, 5421}, {451, 16589}, {458, 11174}, {511, 2211}, {538, 41676}, {543, 37855}, {566, 5094}, {571, 1485}, {607, 26377}, {685, 41932}, {800, 1196}, {1015, 1870}, {1033, 1611}, {1107, 46878}, {1113, 15167}, {1114, 15166}, {1172, 2092}, {1180, 3087}, {1184, 8573}, {1216, 41480}, {1235, 3934}, {1297, 32687}, {1368, 42459}, {1384, 11181}, {1500, 6198}, {1506, 1594}, {1560, 3258}, {1575, 1861}, {1585, 8962}, {1593, 5013}, {1595, 31406}, {1596, 15048}, {1597, 5024}, {1598, 9605}, {1609, 3162}, {1625, 13754}, {1627, 41758}, {1691, 34130}, {1692, 2065}, {1783, 5291}, {1785, 13006}, {1865, 37362}, {1875, 43039}, {1906, 9607}, {1907, 9606}, {1970, 13367}, {1986, 46301}, {1989, 8791}, {1995, 5158}, {2023, 12131}, {2070, 10317}, {2071, 40349}, {2079, 37917}, {2165, 13854}, {2176, 41320}, {2197, 7120}, {2202, 7117}, {2204, 20832}, {2322, 26244}, {2332, 18755}, {2373, 46239}, {2374, 9091}, {2409, 35906}, {2971, 5140}, {3053, 3172}, {3055, 43458}, {3088, 31400}, {3089, 5286}, {3091, 26216}, {3092, 6422}, {3093, 6421}, {3094, 12294}, {3108, 33631}, {3163, 7426}, {3269, 3331}, {3329, 36794}, {3516, 15815}, {3517, 30435}, {3518, 5007}, {3520, 37512}, {3541, 31401}, {3542, 3767}, {3569, 32112}, {3575, 7745}, {3917, 40805}, {4213, 21838}, {4220, 5317}, {4230, 5968}, {4232, 5304}, {4233, 40129}, {4235, 32456}, {5000, 41196}, {5001, 41197}, {5008, 10986}, {5023, 8778}, {5034, 39588}, {5041, 34484}, {5052, 6403}, {5058, 10880}, {5062, 10881}, {5063, 9609}, {5064, 13351}, {5133, 36412}, {5206, 32534}, {5276, 35973}, {5305, 21841}, {5462, 41334}, {5661, 6795}, {5899, 22121}, {5907, 22416}, {5978, 11093}, {5979, 11094}, {6114, 6116}, {6115, 6117}, {6240, 7747}, {6241, 41367}, {6529, 41368}, {6623, 43448}, {6636, 22052}, {6683, 37125}, {6749, 10301}, {6759, 39643}, {6781, 10295}, {6998, 8747}, {7071, 31477}, {7387, 23115}, {7413, 8748}, {7418, 35908}, {7467, 11574}, {7484, 36751}, {7485, 10979}, {7505, 7746}, {7517, 22120}, {7576, 7753}, {7577, 7603}, {7713, 9575}, {7714, 13341}, {7737, 18533}, {7749, 10018}, {7755, 41366}, {7756, 18560}, {7772, 10594}, {7786, 37337}, {7868, 11331}, {8430, 17994}, {8585, 14685}, {8588, 35472}, {8589, 35473}, {8750, 17735}, {8752, 17969}, {8753, 15268}, {8770, 41489}, {9229, 40413}, {9596, 11392}, {9599, 11393}, {9698, 15559}, {9753, 41371}, {9909, 15905}, {10151, 35903}, {10184, 11548}, {10282, 14585}, {10539, 23128}, {10540, 22146}, {10605, 41376}, {10632, 16258}, {10633, 16257}, {11063, 37920}, {11325, 40325}, {11380, 34870}, {11381, 38297}, {11398, 16502}, {11403, 22332}, {11456, 39913}, {13357, 27369}, {13481, 37453}, {13509, 14157}, {13860, 33971}, {14002, 15860}, {14356, 34349}, {14537, 18559}, {14865, 31652}, {15262, 37777}, {15340, 25739}, {15484, 18494}, {15513, 21844}, {15515, 35477}, {16068, 17980}, {16868, 39565}, {16968, 36103}, {16990, 44134}, {17966, 32674}, {17984, 19566}, {18907, 37458}, {18993, 45502}, {18994, 45503}, {19189, 37123}, {19220, 44269}, {20850, 38292}, {20960, 27373}, {21525, 34859}, {21843, 35486}, {25985, 37661}, {27377, 41624}, {28710, 32816}, {31455, 37119}, {32697, 35296}, {33630, 38282}, {33828, 40411}, {34481, 46432}, {34818, 39951}, {34990, 44377}, {35007, 44879}, {35764, 45513}, {35765, 45512}, {35907, 36166}, {36414, 36416}, {36415, 36423}, {37121, 44732}, {37197, 44518}, {37942, 43291}, {37969, 39176}, {37984, 44468}, {44438, 44526}

X(232) = midpoint of X(i) and X(j) for these {i,j}: {237, 38368}, {3269, 3331}, {41676, 44146}
X(232) = isogonal conjugate of X(287)
X(232) = complement of X(30737)
X(232) = polar-circle-inverse of X(34235)
X(232) = Moses-radical-circle-inverse of X(468)
X(232) = Moses-Parry-circle inverse of X(6103)
X(232) = polar conjugate of X(290)
X(232) = complement of the isotomic conjugate of X(1297)
X(232) = isogonal conjugate of the anticomplement of X(15595)
X(232) = isogonal conjugate of the complement of X(40867)
X(232) = isotomic conjugate of the isogonal conjugate of X(2211)
X(232) = isogonal conjugate of the isotomic conjugate of X(297)
X(232) = isotomic conjugate of the polar conjugate of X(34854)
X(232) = isogonal conjugate of the polar conjugate of X(6530)
X(232) = polar conjugate of the isotomic conjugate of X(511)
X(232) = polar conjugate of the isogonal conjugate of X(237)
X(232) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 132}, {560, 23976}, {1297, 2887}, {8767, 21243}, {32649, 8062}, {34212, 21253}, {35140, 21235}, {36046, 30476}, {43717, 20305}, {44770, 21259}
X(232) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 132}, {297, 511}, {685, 512}, {1976, 1968}, {2409, 2881}, {3563, 25}, {4230, 17994}, {6530, 34854}, {16081, 4}, {19189, 237}, {32649, 2485}, {43717, 6}
X(232) = X(i)-cross conjugate of X(j) for these (i,j): {237, 511}, {446, 694}, {2211, 34854}, {9475, 6}, {17994, 4230}, {36425, 27375}, {38368, 523}, {38974, 647}, {41172, 3569}, {44114, 16230}
X(232) = cevapoint of X(i) and X(j) for these (i,j): {237, 2211}, {419, 41204}, {1692, 42671}, {2491, 44114}, {3569, 41172}
X(232) = crosspoint of X(i) and X(j) for these (i,j): {2, 1297}, {4, 16081}, {6, 1987}, {25, 17980}, {250, 32697}, {297, 6530}, {685, 23582}, {1976, 9292}, {23964, 32687}, {41200, 41201}
X(232) = crosssum of X(i) and X(j) for these (i,j): {2, 401}, {3, 3289}, {6, 1503}, {69, 12215}, {248, 17974}, {325, 1975}, {684, 3269}, {15526, 39473}, {41198, 41199}
X(232) = trilinear pole of line {3569, 17994}
X(232) = crossdifference of every pair of points on line {3, 525}
X(232) = orthojoin of X(132)
X(232) = X(6)-Hirst inverse of X(25)
X(232) = X(281)-beth conjugate of X(232)
X(232) = perspector of hyperbola {A,B,C,X(4),X(112),PU(39)}} (centered at X(132))
X(232) = center of circumconic that is locus of trilinear poles of lines passing through X(132)
X(232) = intersection of trilinear polars of X(112), P(39), and U(39)
X(232) = crossdifference of PU(37)
X(232) = PU(4)-harmonic conjugate of X(647)
X(232) = pole wrt polar circle of trilinear polar of X(290) (line X(2)X(647))
X(232) = X(48)-isoconjugate (polar conjugate) of X(290)
X(232) = X(19)-isoconjugate of X(6394)
X(232) = X(i)-isoconjugate of X(j) for these (i,j): {1, 287}, {2, 293}, {3, 1821}, {6, 336}, {19, 6394}, {48, 290}, {63, 98}, {69, 1910}, {75, 248}, {77, 15628}, {92, 17974}, {184, 46273}, {255, 16081}, {304, 1976}, {326, 6531}, {394, 36120}, {525, 36084}, {561, 14600}, {647, 36036}, {656, 2966}, {661, 17932}, {662, 879}, {685, 24018}, {799, 878}, {810, 43187}, {822, 22456}, {1577, 43754}, {1799, 3404}, {1820, 31635}, {1966, 15391}, {2349, 35912}, {2395, 4592}, {2715, 14208}, {3265, 36104}, {4575, 43665}, {8766, 9476}, {9247, 18024}, {14601, 40364}, {20021, 34055}
X(232) = barycentric product X(i)*X(j) for these {i,j}: {1, 240}, {3, 6530}, {4, 511}, {5, 19189}, {6, 297}, {19, 1959}, {25, 325}, {30, 35908}, {31, 40703}, {32, 44132}, {34, 44694}, {54, 39569}, {64, 44704}, {69, 34854}, {76, 2211}, {92, 1755}, {98, 2967}, {99, 17994}, {107, 684}, {110, 16230}, {112, 2799}, {114, 3563}, {132, 1297}, {186, 14356}, {237, 264}, {248, 36426}, {250, 868}, {281, 43034}, {286, 5360}, {324, 41270}, {393, 36212}, {419, 40810}, {468, 5968}, {512, 877}, {523, 4230}, {648, 3569}, {685, 41167}, {694, 39931}, {935, 33752}, {1309, 42751}, {1503, 39265}, {1513, 40801}, {1826, 17209}, {1843, 20022}, {1969, 9417}, {1973, 46238}, {1990, 35910}, {2052, 3289}, {2203, 42703}, {2207, 6393}, {2396, 2489}, {2421, 2501}, {2491, 6331}, {3267, 34859}, {3405, 17442}, {4235, 8430}, {4240, 32112}, {5000, 5001}, {5140, 36892}, {5523, 36823}, {5976, 17980}, {6103, 46787}, {6330, 9475}, {6333, 32713}, {6528, 39469}, {6531, 36790}, {6591, 42717}, {7473, 23350}, {8743, 34138}, {8884, 44716}, {9155, 17983}, {9307, 15143}, {9418, 18022}, {10311, 46807}, {11672, 16081}, {14251, 17984}, {14618, 14966}, {15595, 43717}, {18020, 44114}, {20410, 36884}, {23582, 41172}, {23996, 36120}, {23997, 24006}, {34129, 38652}, {34157, 44145}, {40804, 41204}, {44778, 44781}, {44779, 44780}
X(232) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 336}, {3, 6394}, {4, 290}, {6, 287}, {19, 1821}, {24, 31635}, {25, 98}, {31, 293}, {32, 248}, {92, 46273}, {107, 22456}, {110, 17932}, {112, 2966}, {132, 30737}, {162, 36036}, {184, 17974}, {237, 3}, {240, 75}, {264, 18024}, {297, 76}, {325, 305}, {393, 16081}, {419, 14382}, {460, 14265}, {511, 69}, {512, 879}, {607, 15628}, {648, 43187}, {669, 878}, {684, 3265}, {868, 339}, {877, 670}, {1096, 36120}, {1495, 35912}, {1501, 14600}, {1576, 43754}, {1755, 63}, {1843, 20021}, {1959, 304}, {1973, 1910}, {1974, 1976}, {2207, 6531}, {2211, 6}, {2421, 4563}, {2450, 41009}, {2489, 2395}, {2491, 647}, {2501, 43665}, {2799, 3267}, {2967, 325}, {3289, 394}, {3563, 40428}, {3569, 525}, {4230, 99}, {5000, 42812}, {5001, 42811}, {5140, 36874}, {5360, 72}, {5968, 30786}, {6000, 36893}, {6103, 46786}, {6530, 264}, {6531, 34536}, {8430, 14977}, {8743, 31636}, {8753, 9154}, {9155, 6390}, {9417, 48}, {9418, 184}, {9419, 3289}, {9468, 15391}, {9475, 441}, {10311, 46806}, {11672, 36212}, {14251, 36214}, {14356, 328}, {14581, 35906}, {14966, 4558}, {15143, 1975}, {16230, 850}, {17209, 17206}, {17409, 11610}, {17980, 36897}, {17994, 523}, {19189, 95}, {23582, 41174}, {23997, 4592}, {32112, 34767}, {32676, 36084}, {32696, 41173}, {32713, 685}, {34157, 43705}, {34397, 14355}, {34854, 4}, {34859, 112}, {35908, 1494}, {36212, 3926}, {36213, 12215}, {36426, 44132}, {36790, 6393}, {38368, 3150}, {39265, 35140}, {39469, 520}, {39569, 311}, {39931, 3978}, {40601, 14965}, {40703, 561}, {40810, 40708}, {41167, 6333}, {41172, 15526}, {41270, 97}, {42067, 43920}, {42068, 15630}, {42671, 34156}, {42702, 3998}, {43034, 348}, {43717, 9476}, {44080, 11653}, {44089, 40820}, {44102, 5967}, {44114, 125}, {44132, 1502}, {44162, 14601}, {44467, 37858}, {44694, 3718}, {44704, 14615}, {44778, 32619}, {44779, 32618}, {46238, 40364}, {46522, 36822}
X(232) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 22240, 216}, {3, 2207, 1968}, {4, 33885, 33842}, {4, 39575, 39}, {6, 25, 10311}, {6, 1971, 8779}, {24, 8743, 32}, {25, 10311, 10985}, {25, 40938, 1194}, {25, 44089, 44099}, {25, 44090, 44089}, {25, 45141, 6}, {39, 3199, 4}, {39, 33842, 33843}, {53, 3815, 427}, {112, 186, 187}, {112, 8744, 14581}, {186, 8744, 112}, {187, 14581, 112}, {230, 1990, 16318}, {230, 16318, 6103}, {403, 5523, 115}, {468, 1990, 6103}, {468, 14580, 3291}, {468, 16318, 230}, {800, 1196, 7735}, {1249, 6353, 7735}, {1495, 8779, 1971}, {1506, 27371, 1594}, {1513, 6530, 132}, {1990, 2493, 14580}, {1990, 6103, 41358}, {1990, 11062, 3003}, {2493, 3003, 3291}, {2493, 11062, 468}, {2493, 16308, 3003}, {2971, 46522, 5140}, {3003, 14580, 6103}, {3172, 3515, 3053}, {3199, 33842, 33885}, {3199, 33843, 33842}, {3542, 41361, 3767}, {5412, 5413, 1974}, {6995, 37665, 3087}, {8105, 8106, 6103}, {8739, 8740, 44102}, {8778, 15750, 5023}, {10641, 10642, 44091}, {14576, 40938, 10311}, {15355, 22240, 2}, {18533, 41370, 7737}, {19128, 41363, 1692}, {33842, 33843, 4}, {38867, 43717, 248}, {41676, 46511, 44146}, {44089, 44090, 44102}, {44099, 44102, 44089}

X(233) = X(2)-CEVA CONJUGATE OF X(140)

Trilinears    [b cos(C - A) + c cos(B - A)]cos(B - C) : :
Trilinears    b sec(A - B) + c sec(A - C) : :
Barycentrics    (a^2 (b^2 + c^2) - (b^2 - c^2)^2) (2 a^4 - 3 a^2 (b^2 + c^2) + (b^2 - c^2)^2) : :

X(233) lies on hyperbola {X(2),X(6),X(216),X(233),X(1249),X(1560),X(3162)}}. This hyperbola is a circumconic of the medial triangle, and the locus of perspectors of circumconics centered at a point on the Euler line. It is tangent to Euler line at X(2). (Randy Hutson, March 21, 2019)

X(233) lies on these lines: 2,95   5,53   6,17   115,128   122,138

X(233) = isogonal conjugate of X(288)
X(233) = isotomic conjugate of X(31617)
X(233) = perspector of circumconic centered at X(140)
X(233) = complement of X(95)
X(233) = X(2)-Ceva conjugate of X(140)
X(233) = crosspoint of X(2) and X(5)
X(233) = crosspoint of X(36300) and X(36301)
X(233) = crosssum of X(6) and X(54)
X(233) = crossdifference of every pair of points on line X(1157)X(1510)
X(233) = polar conjugate of the isogonal conjugate of X(32078)
X(233) = center of circumconic that is locus of trilinear poles of lines passing through X(140)


X(234) = X(7)-CEVA CONJUGATE OF X(177)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos B/2 + cos C/2)(cos B/2 cos C/2)2
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(234) lies on these lines: 2,178   7,174   57,362   75,556   555,1088

X(234) = X(7)-Ceva conjugate of X(177)
X(234) = X(31)-of-intouch-triangle


X(235) = X(4)-CEVA CONJUGATE OF X(185)

Trilinears     sec A - cos(B - C) : :
Trilinears     (sec A)(cos2B + cos2C) : :
Barycentrics    (tan A)(cos2B + cos2C) : :
Barycentrics    (a^4 (b^2 + c^2) - 2 a^2 (b^2 - c^2)^2 + (b^2 - c^2)^2 (b^2 + c^2))/(a^2 - b^2 - c^2) : :

As a point on the Euler line, X(235) has Shinagawa coefficients (F, F - E).

X(235) lies on these lines: 2,3   11,34   12,33   52,113   133,136

X(235) = midpoint of X(4) and X(24)
X(235) = complement of X(11413)
X(235) = anticomplement of X(16196)
X(235) = circumcircle-inverse of X(37917)
X(235) = excentral-to-ABC functional image of X(55)
X(235) = X(4)-Ceva conjugate of X(185)
X(235) = crosssum of X(3) and X(1092)
X(235) = orthic-isogonal conjugate of X(185)
X(235) = X(56) of orthic triangle if ABC is acute
X(235) = insimilicenter of nine-point circle and incircle of orthic triangle if ABC is acute; the exsimilicenter is X(427)
X(235) = pole wrt polar circle of trilinear polar of X(801) (line X(523)X(2071))
X(235) = X(48)-isoconjugate (polar conjugate) of X(801)
X(235) = perspector of ABC and cross-triangle of ABC and 2nd Hyacinth triangle
X(235) = radical center of the polar-circle-inverses of the power circles
X(235) = intersection of tangents to Hatzipolakis-Lozada hyperbola at X(4) and X(155)
X(235) = crosspoint, wrt orthic triangle, of X(4) and X(155)


X(236) = X(2)-CEVA CONJUGATE OF X(188)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (csc A/2)(cos B/2 + cos C/2 - cos A/2)
Trilinears       [1 + sin(A/2)]/sin A : [1 + sin(B/2)]/sin B : [1 + sin(C/2)]//sin C     (M. Iliev, 4/12/07)

X(236) lies on these lines: 2,174   8,178   9,173

X(236) = isogonal conjugate of X(289)
X(236) = complement of X(7048)
X(236) = X(2)-Ceva conjugate of X(188)
X(236) = perspector of circumconic centered at X(188)
X(236) = center of circumconic that is locus of trilinear poles of lines passing through X(188)

leftri

Centers X(237)-X(248)

rightri

Centers X(237)-X(248) are line conjugates. (The P-line conjugate of Q is the point where line PQ meets the trilinear polar of the isogonal conjugate of Q.)

See Clark Kimberling and Peter J. C. Moses, Line conjugates in the plane of a triangle, Aequationes Mathematicae, August 2022.


X(237) = X(3)-LINE CONJUGATE OF X(2)

Trilinears    a2cos(A + ω) : b2cos(B + ω) : c2cos(C + ω)
Trilinears    a3(b4 + c4 - a2b2 - a2c2) (Darij Grinberg, 3/29/03)
Barycentrics    a3cos(A + ω) : b3cos(B + ω) : c3cos(C + ω)
Barycentrics    a^4*(a^2*b^2 - b^4 + a^2*c^2 - c^4) : :
X(237) = X(237) = 3 X[2] + X[46518], 2 X[3001] - 3 X[22087], 3 X[9155] - 2 X[36212], X[14957] - 6 X[44215], X[21531] - 3 X[44215], 2 X[21531] + X[46518], 3 X[22087] - 4 X[34990], X[34175] - 3 X[38227], 6 X[44215] + X[46518]

As a point on the Euler line, X(237) has Shinagawa coefficients (EF + F2 + S2, -(E + F)2 - S2).

X(237) is the point of intersection of the Euler line and the Lemoine axis (defined as the radical axis of the circumcircle and the Brocard circle).

If you have The Geometer's Sketchpad, you can view X(1316), which includes X(237).

X(237) lies on the cubics K380, K532, K782, K783, K789, K1119, and these lines: {2, 3}, {6, 160}, {15, 14186}, {16, 14188}, {31, 904}, {32, 184}, {35, 40790}, {39, 51}, {42, 18758}, {50, 1576}, {55, 20471}, {95, 32085}, {98, 6037}, {99, 3978}, {110, 2080}, {111, 46316}, {141, 8266}, {154, 682}, {157, 8553}, {159, 1609}, {187, 351}, {193, 20794}, {206, 571}, {216, 1843}, {232, 9475}, {248, 1971}, {323, 9301}, {325, 20022}, {373, 21163}, {385, 3511}, {476, 43654}, {511, 9155}, {524, 1634}, {566, 9971}, {570, 9969}, {574, 34417}, {577, 1974}, {672, 20777}, {694, 1691}, {800, 6467}, {804, 46302}, {805, 39092}, {1030, 22369}, {1084, 33875}, {1194, 13357}, {1284, 43920}, {1384, 9463}, {1503, 20021}, {1624, 15448}, {1660, 40320}, {1661, 40321}, {1755, 5360}, {1899, 8721}, {1915, 34870}, {1975, 20023}, {2021, 3124}, {2076, 34214}, {2183, 23198}, {2187, 18756}, {2211, 36425}, {2269, 22389}, {2351, 2353}, {2352, 20284}, {2393, 3003}, {2482, 38998}, {2936, 9890}, {2979, 9821}, {2980, 45838}, {3001, 9019}, {3049, 43112}, {3060, 3095}, {3164, 3186}, {3284, 44102}, {3289, 9418}, {3313, 20819}, {3398, 5012}, {3564, 25046}, {3589, 35222}, {3785, 14826}, {3917, 5188}, {3926, 9917}, {3964, 37491}, {5007, 13366}, {5008, 44109}, {5013, 17810}, {5023, 21001}, {5024, 13192}, {5041, 34565}, {5063, 19136}, {5065, 33578}, {5092, 34236}, {5104, 33876}, {5158, 8541}, {5162, 37841}, {5171, 9306}, {5206, 44082}, {5210, 41424}, {5467, 32217}, {5640, 11171}, {5651, 8722}, {5938, 15340}, {5943, 13334}, {5976, 8840}, {6000, 44437}, {6324, 13531}, {6403, 30258}, {6530, 19189}, {6752, 44088}, {7113, 22096}, {7669, 11063}, {7716, 36751}, {7772, 15004}, {7782, 41259}, {7795, 43653}, {8265, 41331}, {8299, 20878}, {8570, 10329}, {8573, 19459}, {8617, 15655}, {8705, 46127}, {8724, 15360}, {8880, 12049}, {8881, 12048}, {9142, 12367}, {9149, 22329}, {9157, 32526}, {9468, 32748}, {9512, 44375}, {9605, 9777}, {9752, 22655}, {10132, 39648}, {10133, 39679}, {10313, 44089}, {10316, 23606}, {10317, 23357}, {11002, 32447}, {11003, 11842}, {11257, 39906}, {11402, 30435}, {11649, 18114}, {12143, 26166}, {12215, 24729}, {14251, 23611}, {14547, 23199}, {14601, 32654}, {14908, 32640}, {14961, 44084}, {14981, 41586}, {15066, 34095}, {15080, 26316}, {15107, 33873}, {15166, 44125}, {15167, 44126}, {15624, 17454}, {15905, 19118}, {16872, 23868}, {16985, 39652}, {17984, 18024}, {18265, 18266}, {18755, 23212}, {18860, 23217}, {19121, 37893}, {19599, 25332}, {20080, 22152}, {20968, 44078}, {21352, 37575}, {21639, 40135}, {22052, 44091}, {22143, 37784}, {22401, 44079}, {23098, 34157}, {23158, 41588}, {23181, 32269}, {23197, 40956}, {25054, 44371}, {32224, 40879}, {32428, 44145}, {32564, 39649}, {32571, 39658}, {32713, 44096}, {33581, 40319}, {33582, 44200}, {33871, 46327}, {33972, 34013}, {34175, 38227}, {34566, 34571}, {35007, 44110}, {35383, 38873}, {36822, 46777}, {37479, 43650}, {37512, 44106}, {38354, 41167}, {38383, 46807}, {39557, 40850}, {41005, 41584}, {41196, 44778}, {41197, 44779}, {42329, 43976}, {44192, 45429}, {44193, 45428}, {46286, 46306}

X(237) = midpoint of X(i) and X(j) for these {i,j}: {23, 7468}, {1634, 5201}, {14957, 46518}
X(237) = reflection of X(i) in X(j) for these {i,j}: {2, 44215}, {3, 44221}, {4, 44227}, {3001, 34990}, {14957, 21531}, {20975, 3003}, {36189, 468}, {38368, 232}, {46522, 21177}
X(237) = isogonal conjugate of X(290)
X(237) = isotomic conjugate of X(18024)
X(237) = complement of X(14957)
X(237) = anticomplement of X(21531)
X(237) = circumcircle-inverse of X(1316)
X(237) = orthocentroidal-circle-inverse of X(37988)
X(237) = orthoptic-circle-of-Steiner-inellipse-inverse of X(36183)
X(237) = isogonal conjugate of the anticomplement of X(11672)
X(237) = isogonal conjugate of the complement of X(39355)
X(237) = isotomic conjugate of the isogonal conjugate of X(9418)
X(237) = isogonal conjugate of the isotomic conjugate of X(511)
X(237) = isotomic conjugate of the polar conjugate of X(2211)
X(237) = isogonal conjugate of the polar conjugate of X(232)
X(237) = polar conjugate of the isotomic conjugate of X(3289)
X(237) = psi-transform of X(15920)
X(237) = X(31)-complementary conjugate of X(40601)
X(237) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 40601}, {3, 9475}, {6, 9419}, {98, 6}, {232, 2211}, {511, 3289}, {694, 3051}, {1691, 32748}, {1987, 217}, {2715, 3049}, {4230, 3569}, {6037, 3288}, {9468, 14251}, {14966, 2491}, {17938, 669}, {19189, 232}, {22456, 2451}, {32654, 32}, {34157, 11672}, {44770, 32320}, {46302, 3231}
X(237) = X(i)-cross conjugate of X(j) for these (i,j): {2491, 14966}, {9418, 2211}, {9419, 6}, {11672, 14251}
X(237) = cevapoint of X(i) and X(j) for these (i,j): {3, 3511}, {6, 46272}, {1691, 1971}
X(237) = crosspoint of X(i) and X(j) for these (i,j): {6, 98}, {32, 9468}, {56, 18268}, {232, 511}, {250, 2715}, {325, 34138}, {19189, 41270}, {32654, 34157}
X(237) = crosssum of X(i) and X(j) for these (i,j): {2, 511}, {8, 3948}, {76, 3978}, {98, 287}, {125, 2799}, {264, 16089}, {525, 3150}, {1976, 11610}, {2395, 15630}, {3766, 34387}, {18018, 34237}, {34239, 41194}, {34240, 41195}, {44780, 44781}
X(237) = trilinear pole of line {2491, 9419}
X(237) = crossdifference of every pair of points on line {2, 647}
X(237) = X(32)-Hirst inverse of X(184)
X(237) = X(3)-line conjugate of X(2)
X(237) = X(55)-beth conjugate of X(237)
X(237) = crosspoint of X(3) and X(3511) wrt excentral triangle
X(237) = crosspoint of X(3) and X(3511) wrt tangential triangle
X(237) = X(92)-isoconjugate of X(287)
X(237) = trilinear pole of PU(89)
X(237) = X(i)-isoconjugate of X(j) for these (i,j): {1, 290}, {2, 1821}, {4, 336}, {6, 46273}, {31, 18024}, {63, 16081}, {69, 36120}, {75, 98}, {76, 1910}, {85, 15628}, {91, 31635}, {92, 287}, {158, 6394}, {248, 1969}, {264, 293}, {304, 6531}, {308, 3404}, {523, 36036}, {561, 1976}, {656, 22456}, {661, 43187}, {662, 43665}, {685, 14208}, {799, 2395}, {811, 879}, {850, 36084}, {1577, 2966}, {1581, 14382}, {1733, 40428}, {1926, 34238}, {1928, 14601}, {1934, 40820}, {1959, 34536}, {1966, 36897}, {2422, 4602}, {2715, 20948}, {3112, 20021}, {3267, 36104}, {3708, 41174}, {5967, 46277}, {7035, 43920}, {8773, 14265}, {9154, 14210}, {11610, 46244}, {17932, 24006}, {33805, 35906}, {41932, 46238}
X(237) = barycentric product X(i)*X(j) for these {i,j}: {1, 1755}, {3, 232}, {4, 3289}, {5, 41270}, {6, 511}, {25, 36212}, {28, 42702}, {31, 1959}, {32, 325}, {42, 17209}, {48, 240}, {50, 14356}, {55, 43034}, {69, 2211}, {75, 9417}, {76, 9418}, {81, 5360}, {98, 11672}, {99, 2491}, {110, 3569}, {111, 9155}, {112, 684}, {114, 32654}, {184, 297}, {187, 5968}, {206, 34138}, {216, 19189}, {230, 34157}, {248, 2967}, {249, 44114}, {250, 41172}, {290, 9419}, {385, 14251}, {394, 34854}, {512, 2421}, {523, 14966}, {560, 46238}, {577, 6530}, {604, 44694}, {647, 4230}, {648, 39469}, {661, 23997}, {667, 42717}, {669, 2396}, {694, 36213}, {729, 6786}, {868, 23357}, {877, 3049}, {1297, 9475}, {1355, 15628}, {1495, 35910}, {1513, 40799}, {1576, 2799}, {1691, 40810}, {1821, 42075}, {1910, 23996}, {1964, 3405}, {1971, 40804}, {1974, 6393}, {1976, 36790}, {2393, 36823}, {2420, 32112}, {2422, 15631}, {2493, 40083}, {2715, 41167}, {3051, 20022}, {3117, 8840}, {3265, 34859}, {3284, 35908}, {3329, 39684}, {3447, 34349}, {4558, 17994}, {5000, 41197}, {5001, 41196}, {5111, 18873}, {5191, 46787}, {5467, 8430}, {5976, 9468}, {6037, 33569}, {8779, 39265}, {8882, 44716}, {9247, 40703}, {14533, 39569}, {14575, 44132}, {14601, 32458}, {14642, 44704}, {14998, 42743}, {16230, 32661}, {17462, 36051}, {17970, 39931}, {18024, 36425}, {23098, 41932}, {23611, 34536}, {32641, 42751}, {34238, 46888}, {34396, 46807}, {40866, 43112}, {41198, 44779}, {41199, 44778}
X(237) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 46273}, {2, 18024}, {6, 290}, {25, 16081}, {31, 1821}, {32, 98}, {48, 336}, {110, 43187}, {112, 22456}, {163, 36036}, {184, 287}, {206, 31636}, {232, 264}, {240, 1969}, {250, 41174}, {297, 18022}, {325, 1502}, {511, 76}, {512, 43665}, {560, 1910}, {571, 31635}, {577, 6394}, {669, 2395}, {684, 3267}, {868, 23962}, {1501, 1976}, {1513, 40822}, {1576, 2966}, {1691, 14382}, {1692, 14265}, {1755, 75}, {1923, 3404}, {1959, 561}, {1973, 36120}, {1974, 6531}, {1976, 34536}, {1977, 43920}, {2175, 15628}, {2211, 4}, {2396, 4609}, {2421, 670}, {2491, 523}, {2799, 44173}, {2967, 44132}, {3049, 879}, {3051, 20021}, {3289, 69}, {3405, 18833}, {3569, 850}, {4230, 6331}, {5191, 46786}, {5360, 321}, {5968, 18023}, {5976, 14603}, {6393, 40050}, {6530, 18027}, {6786, 30736}, {8789, 34238}, {9155, 3266}, {9233, 14601}, {9247, 293}, {9407, 35906}, {9417, 1}, {9418, 6}, {9419, 511}, {9420, 23878}, {9426, 2422}, {9427, 15630}, {9468, 36897}, {9475, 30737}, {11672, 325}, {12212, 39685}, {14251, 1916}, {14356, 20573}, {14567, 5967}, {14574, 2715}, {14575, 248}, {14585, 17974}, {14601, 41932}, {14602, 40820}, {14966, 99}, {17209, 310}, {17938, 39291}, {17994, 14618}, {19189, 276}, {19627, 14355}, {20022, 40016}, {20968, 11610}, {23098, 32458}, {23611, 36790}, {23996, 46238}, {23997, 799}, {32654, 40428}, {32661, 17932}, {32740, 9154}, {33875, 36822}, {34138, 40421}, {34157, 8781}, {34396, 46806}, {34854, 2052}, {34859, 107}, {36212, 305}, {36213, 3978}, {36823, 46140}, {39469, 525}, {39684, 42006}, {40373, 14600}, {40601, 14957}, {40810, 18896}, {41172, 339}, {41196, 42812}, {41197, 42811}, {41270, 95}, {42075, 1959}, {42702, 20336}, {42717, 6386}, {43034, 6063}, {43112, 46245}, {44114, 338}, {44132, 44161}, {44694, 28659}, {44716, 28706}, {44778, 41195}, {44779, 41194}, {46238, 1928}, {46272, 39058}
X(237) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3, 14096}, {2, 4, 37988}, {2, 20, 37190}, {2, 22, 7467}, {2, 11328, 37338}, {2, 14957, 21531}, {2, 37184, 3}, {2, 37465, 11328}, {2, 46518, 14957}, {3, 22, 46546}, {3, 23, 37916}, {3, 25, 3148}, {3, 381, 35934}, {3, 3148, 37457}, {3, 6638, 37188}, {3, 6660, 37183}, {3, 11328, 2}, {3, 15329, 45662}, {3, 20854, 6660}, {3, 20960, 27369}, {3, 21512, 6636}, {3, 27369, 11326}, {3, 37465, 37338}, {3, 37914, 23}, {3, 41266, 25}, {3, 44894, 441}, {3, 44895, 15000}, {3, 44896, 44887}, {4, 37114, 3}, {6, 160, 20775}, {23, 11145, 34008}, {23, 11146, 34009}, {23, 35296, 37183}, {23, 35298, 3}, {23, 37183, 6660}, {25, 3135, 418}, {25, 11325, 6620}, {25, 20885, 2}, {25, 41266, 20897}, {25, 41275, 37457}, {32, 184, 34396}, {32, 3117, 3051}, {50, 1576, 23200}, {50, 18374, 1576}, {141, 8266, 22062}, {159, 1609, 40947}, {186, 37937, 468}, {187, 1495, 5191}, {187, 3229, 8623}, {187, 5106, 3231}, {187, 21444, 32518}, {206, 571, 14575}, {216, 1843, 23635}, {297, 1513, 2450}, {297, 2450, 868}, {297, 4230, 15143}, {401, 419, 1316}, {418, 27369, 3148}, {419, 11676, 401}, {441, 468, 44887}, {441, 44887, 15000}, {441, 44894, 852}, {468, 44886, 852}, {851, 46513, 854}, {851, 46524, 14953}, {855, 861, 862}, {1113, 1114, 1316}, {1513, 5112, 868}, {1576, 18374, 9407}, {1576, 39231, 50}, {2450, 5112, 297}, {2454, 2455, 44345}, {2479, 2480, 10684}, {3001, 34990, 22087}, {3053, 15270, 682}, {3117, 41278, 32}, {3129, 3130, 23}, {3131, 3132, 6636}, {3135, 20960, 3148}, {3148, 20897, 25}, {3148, 41275, 3}, {3155, 3156, 22}, {3229, 8623, 3231}, {4230, 21525, 868}, {4230, 37123, 5112}, {5004, 5005, 5999}, {5106, 8623, 3229}, {6660, 20854, 23}, {6660, 37183, 37916}, {6660, 37914, 20854}, {7426, 37461, 45662}, {7437, 46501, 46579}, {7437, 46502, 46501}, {8553, 20987, 157}, {8598, 37927, 11634}, {9407, 23200, 1576}, {11007, 21536, 33314}, {11007, 33314, 46562}, {11063, 19596, 7669}, {11257, 40814, 39906}, {11328, 37184, 14096}, {14096, 37338, 2}, {14953, 37018, 46524}, {15244, 15245, 6636}, {15247, 15248, 25}, {18374, 39231, 23200}, {18773, 18774, 21444}, {20775, 40981, 6}, {20854, 35296, 37916}, {20897, 41275, 3148}, {21032, 21036, 20885}, {21522, 46513, 851}, {21536, 33314, 46561}, {34394, 34395, 14567}, {35222, 41328, 3589}, {35296, 37183, 3}, {35298, 37183, 35296}, {35298, 37914, 37916}, {37184, 37465, 2}, {41378, 41379, 184}, {42667, 42668, 5191}, {42789, 42790, 35474}, {44886, 44890, 468}, {44886, 44896, 44894}, {44894, 44896, 44895}, {46501, 46502, 46578}, {46561, 46562, 33314}, {46564, 46568, 857}, {46578, 46579, 46501}, {46600, 46601, 2}


X(238) = X(1)-LINE CONJUGATE OF X(37)

Trilinears       a2 - bc : b2 - ca : c2 - ab
Barycentrics  a3 - abc : b3 - abc : c3 - abc

X(238) is the perspector of triangle ABC and the tangential triangle of the conic that passes through A, B, C, and the pair P(8) and U(8) of bicentric points (see the notes just before X(1908). (Randy Hutson, 9/23/2011)

X(238) lies on these lines: 1,6   2,31   3,978   4,602   7,1471   8,983   10,82   21,256   36,513   40,1722   42,1621   43,55   47,499   56,87   57,1707   58,86   63,614   71,1244   100,899   105,291   106,898   162,415   190,726   212,497   239,740   241,1456   242,419   244,896   390,1253   459,1395   484,1739   516,673   517,1052   519,765   580,946   601,631   651,1458   662,1326   942,1046   987,1472   992,1009   993,995   1006,1064   1040,1711   1054,1155   1284,1428   1465,1758   1479,1714   1699,1754

X(238) = midpoint of X(1) and X(1279)
X(238) = reflection of X(1) in X(1297)
X(238) = isogonal conjugate of X(291)
X(238) = isotomic conjugate of X(334)
X(238) = anticomplement of X(3836)
X(238) = X(i)-Ceva conjugate of X(j) for these (i,j): (105,1), (292,171)
X(238) = X(659)-cross conjugate of X(3573)
X(238) = crosssum of X(i) and X(j) for these (i,j): (10,726), (42,672), (239,894)
X(238) = crossdifference of every pair of points on line X(37)X(513)
X(238) = X(i)-Hirst inverse of X(j) for these (i,j): (1,6), (43,55)
X(238) = X(1)-line conjugate of X(37)
X(238) = X(105)-aleph conjugate of X(238)
X(238) = X(i)-beth conjugate of X(j) for these (i,j): (21,238), (643,902), (644,238), (932,238)
X(238) = {X(1),X(9)}-harmonic conjugate of X(984)
X(238) = intersection of trilinear polars of PU(8)
X(238) = inverse-in-circumconic-centered-at-X(9) of X(6)
X(238) = crossdifference of PU(i) for these i: 6, 52, 53
X(238) = trilinear product of PU(134)
X(238) = X(6530)-of-excentral-triangle
X(238) = trilinear pole of line X(659)X(4435) (the perspectrix of ABC and Gemini triangle 33)
X(238) = perspector of Gemini triangle 34 and cross-triangle of ABC and Gemini triangle 34


X(239) = X(1)-LINE CONJUGATE OF X(42)

Trilinears    bc(a2 - bc) : :
Barycentrics    a2 - bc : :

X(239) is the perspector of triangle ABC and the tangential triangle of the conic that passes through A, B, C, and the pair P(6) and U(6) of bicentric points (see the notes just before X(1908). (Randy Hutson, 9/23/2011)

X(239) is the point of intersection of the following lines:
     X(1)X(2) = trilinear polar of X(190)
     trilinear polar of cevapoint{X(1), X(2)}, which is X(239)X(514)
     UV, where U = X(1)-Ceva-conjugate-of-(2) = X(192), and V = X(2)-Ceva-conjugate-of-X(1) = X(9)
(Randy Hutson, December 26, 2015)

X(239) lies on these lines: 1,2   6,75   7,193   9,192   44,190   57,330   63,194   81,274   83,213   86,1100   92,607   141,319   238,740   241,664   257,333   294,666   318,458   320,524   335,518   514,649   1043,1104

X(239) = reflection of X(i) in X(j) for these (i,j): (190,44), (320,1086)
X(239) = isogonal conjugate of X(292)
X(239) = isotomic conjugate of X(335)
X(239) = complement of X(6542)
X(239) = anticomplement of X(3912)
X(239) = crosspoint of X(256) and X(291)
X(239) = crosssum of X(i) and X(j) for these (i,j): (3,255), (212,219)
X(239) = crossdifference of every pair of points on line X(42)X(649)
X(239) = X(i)-Hirst inverse of X(j) for these (i,j): (171,238), (665,1015)
X(239) = X(1)-line conjugate of X(42)
X(239) = X(i)-beth conjugate of X(j) for these (i,j): (333,239), (645,44)
X(239) = perspector of conic {A,B,C,X(86),X(190)}
X(239) = inverse-in-Steiner-circumellipse of X(1)
X(239) = trilinear pole of line X(659)X(812)
X(239) = crossdifference of PU(8)
X(239) = intersection of trilinear polars of PU(6) (the 1st and 2nd bicentrics of the Lemoine axis)
X(239) = X(2)-Ceva conjugate of X(6651)
X(239) = trilinear pole of PU(134)
X(239) = trilinear square root of X(39044)
X(239) = barycentric square root of X(4366)


X(240) = X(1)-LINE CONJUGATE OF X(48)

Trilinears       sec A cos(A + ω) : sec B cos(B + ω) : sec C cos(C + ω)
Trilinears    (a^2 b^2 + a^2 c^2 - b^4 - c^4)/(a^2 - b^2 - c^2) : :
Barycentrics  tan A cos(A + ω) : tan B cos(B + ω) : tan C cos(C + ω)

X(240) lies on these lines: 1,19   4,256   38,92   63,1096   75,158   162,896   278,982   281,984   522,656   607,611   608,613

X(240) = isogonal conjugate of X(293)
X(240) = isotomic conjugate of X(336)
X(240) = crossdifference of every pair of points on line X(48)X(656)
X(240) = X(1)-Hirst inverse of X(19)
X(240) = X(1)-line conjugate of X(48)
X(240) = X(318)-beth conjugate of X(240)
X(240) = crossdifference of PU(22)
X(240) = X(2)-Ceva conjugate of X(39039)
X(240) = perspector of hyperbola {A,B,C,PU(23)}
X(240) = intersection of trilinear polars of P(23) and U(23)
X(240) = pole wrt polar circle of trilinear polar of X(1821)
X(240) = X(48)-isoconjugate (polar conjugate) of X(1821)


X(241) = X(1)-LINE CONJUGATE OF X(55)

Trilinears    cos4B/2 - [cos2(A/2)][cos2(B/2) +cos2(C/2)] + cos4(C/2) : :
Trilinears    (b^2 + c^2 - a b - a c)/(a - b - c) : :
Barycentrics    (b^2 + c^2)(1 - cos A) - a^2(cos B + cos C) : :

X(241) lies on these lines: 1,3   2,85   6,77   7,37   9,269   44,651   63,220   141,307   218,222   239,664   277,278   294,910   347,1108   514,650   960,1042

X(241) = isogonal conjugate of X(294)
X(241) = isotomic conjugate of X(36796)
X(241) = complement of X(30807)
X(241) = complement of polar conjugate of X(36122)
X(241) = anticomplement of X(34852)
X(241) = X(2)-Ceva conjugate of X(39063)
X(241) = crosssum of X(i) and X(j) for these (i,j): (6,910), (518,1376
X(241) = crossdifference of every pair of points on line X(55)X(650)
X(241) = X(1)-Hirst inverse of X(57)
X(241) = X(1)-line conjugate of X(55)
X(241) = X(i)-beth conjugate of X(j) for these (i,j): (2,241), (100,241), (1025,241), (1026,241)
X(241) = trilinear pole of line X(926)X(1362)
X(241) = X(237)-of-intouch-triangle
X(241) = perspector of hyperbola {A,B,C,PU(46)}
X(241) = crossdifference of PU(112)


X(242) = X(4)-LINE CONJUGATE OF X(71)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sec A)(sin2A - sin B sin C)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(242) lies on these lines: 4,9   25,92   28,261   29,257   34,87   162,422   238,419   278,459   915,929

X(242) = isogonal conjugate of X(295)
X(242) = isotomic conjugate of X(337)
X(242) = crossdifference of every pair of points on line X(71)X(1459)
X(242) = X(4)-Hirst inverse of X(19)
X(242) = X(4)-line conjugate of X(71)
X(242) = inverse-in-polar-circle of X(10)
X(242) = pole wrt polar circle of the line X(10)X(514)
X(242) = X(48)-isoconjugate (polar conjugate) of X(335)


X(243) = X(4)-LINE CONJUGATE OF X(73)

Trilinears    (sec A)(cos2A - cos B cos C) : :
Barycentrics    (a - b - c) (a^4 - a^2 (b^2 - b c + c^2) - b c (b - c)^2)/(a^2 - b^2 - c^2) : :

X(243) is the perspector of triangle ABC and the tangential triangle of the conic that passes through A, B, C, and the pair P(15) and U(15) of bicentric points (see the notes just before X(1908). (Randy Hutson, 9/23/2011)

X(243) is the point of intersection of the following lines:
     trilinear polars of P(15) and U(15)
     X(1)X(4)
     trilinear polar of cevapoint{X(1),X(4)}
     UV, where U = X(1)-Ceva-conjugate-of-(4) = X(1148), and V = X(4)-Ceva-conjugate-of-X(1) = X(46)
(Randy Hutson, December 26, 2015)

X(243) lies on the cubics K040, K683, K1185 and these lines: {1, 4}, {2, 1857}, {3, 158}, {11, 17923}, {20, 1118}, {21, 1896}, {27, 1859}, {29, 2646}, {36, 1784}, {46, 1148}, {55, 92}, {56, 1895}, {57, 43160}, {65, 412}, {90, 7049}, {105, 1309}, {107, 14192}, {108, 2723}, {132, 8229}, {162, 2361}, {196, 3474}, {212, 1957}, {240, 7004}, {242, 1283}, {281, 5218}, {296, 8764}, {318, 958}, {411, 821}, {415, 2652}, {425, 662}, {468, 42069}, {518, 1897}, {522, 652}, {648, 2651}, {653, 1155}, {851, 41500}, {917, 2222}, {920, 1075}, {929, 20624}, {1040, 1096}, {1324, 14017}, {1430, 1936}, {1447, 47212}, {1465, 36127}, {1503, 38357}, {1715, 7098}, {1788, 3176}, {1837, 5125}, {1854, 5786}, {1858, 2907}, {1861, 26013}, {1864, 37279}, {1871, 12711}, {1875, 37420}, {1882, 7513}, {1944, 15146}, {1948, 7360}, {2073, 14194}, {2202, 39032}, {2376, 2731}, {2586, 34592}, {2587, 34593}, {2659, 37142}, {3100, 14956}, {3147, 7040}, {3149, 47372}, {3326, 5057}, {3601, 39585}, {3612, 7531}, {3685, 36797}, {3812, 37278}, {4219, 40149}, {4459, 15150}, {5081, 44669}, {5174, 10950}, {5205, 6335}, {5219, 39531}, {5327, 41083}, {5794, 17555}, {6331, 14195}, {6336, 14190}, {6985, 20764}, {7017, 7081}, {7288, 40836}, {8748, 40937}, {8758, 43764}, {9371, 14571}, {10589, 17917}, {11502, 35994}, {13149, 14189}, {13411, 39574}, {14006, 41234}, {14202, 17983}, {15252, 51368}, {17605, 42387}, {22768, 37253}, {23207, 35981}, {23353, 26884}, {23711, 37769}, {24032, 36002}, {24929, 39529}, {36599, 38249}, {37737, 44225}

X(243) = reflection of X(51368) in X(15252)
X(243) = isogonal conjugate of X(296)
X(243) = incircle-inverse of X(40960)
X(243) = polar-circle-inverse of X(226)
X(243) = polar conjugate of X(1952)
X(243) = polar conjugate of the isotomic conjugate of X(1944)
X(243) = polar conjugate of the isogonal conjugate of X(1951)
X(243) = orthic-isogonal conjugate of X(41499)
X(243) = X(31)-complementary conjugate of X(39033)
X(243) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 39033}, {4, 41499}, {1937, 1940}, {8764, 1}, {15146, 1936}, {41207, 650}, {43764, 4}
X(243) = X(i)-cross conjugate of X(j) for these (i,j): {851, 1936}, {1951, 1944}
X(243) = cevapoint of X(415) and X(2659)
X(243) = crosspoint of X(i) and X(j) for these (i,j): {1, 2656}, {1937, 7105}
X(243) = crosssum of X(i) and X(j) for these (i,j): {1, 2655}, {3, 17975}, {1935, 1936}
X(243) = crossdifference of every pair of points on line {73, 652}
X(243) = X(i)-line conjugate of X(j) for these (i,j): {1, 73}, {522, 652}
X(243) = X(i)-Hirst inverse of X(j) for these (i,j): (1,4), (46,1148)
X(243) = perspector of conic {A,B,C,X(29),X(653),PU(15)}}
X(243) = perspector of conic {A,B,C,X(29),X(653),PU(15)}
X(243) = crossdifference of PU(16)
X(243) = pole wrt polar circle of the line X(226)X(522)
X(243) = X(48)-isoconjugate (polar conjugate) of X(1952)
X(243) = X(i)-isoconjugate of X(j) for these (i,j): {1, 296}, {2, 1949}, {3, 1937}, {6, 40843}, {48, 1952}, {63, 1945}, {73, 37142}, {647, 41206}, {822, 41207}, {1214, 2249}, {1409, 35145}, {1935, 1942}
X(243) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 39033}, {3, 296}, {3, 39037}, {9, 40843}, {63, 39032}, {69, 39035}, {75, 39036}, {307, 35075}, {1249, 1952}, {1937, 36103}, {1943, 39034}, {1945, 3162}, {1949, 32664}, {39052, 41206}
X(243) = barycentric product X(i)*X(j) for these {i,j}: {1, 1948}, {4, 1944}, {29, 8680}, {75, 2202}, {92, 1936}, {158, 6518}, {226, 15146}, {264, 1951}, {278, 7360}, {281, 5088}, {312, 1430}, {450, 7108}, {522, 1981}, {851, 31623}, {1172, 44150}, {2656, 39036}, {4391, 23353}, {7017, 26884}, {15418, 18344}, {17947, 41499}, {40843, 41500}, {42669, 44130}
X(243) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 40843}, {4, 1952}, {6, 296}, {19, 1937}, {25, 1945}, {29, 35145}, {31, 1949}, {107, 41207}, {162, 41206}, {450, 1943}, {851, 1214}, {1172, 37142}, {1430, 57}, {1936, 63}, {1944, 69}, {1948, 75}, {1951, 3}, {1981, 664}, {2202, 1}, {2299, 2249}, {5088, 348}, {6518, 326}, {7106, 1942}, {7360, 345}, {8680, 307}, {15146, 333}, {23353, 651}, {26884, 222}, {41368, 1940}, {41497, 1947}, {41499, 17950}, {41500, 1948}, {42669, 73}, {44096, 1950}, {44112, 1409}, {44150, 1231}, {51647, 1439}
X(243) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 2655, 8763}, {1, 51282, 4}, {3, 158, 1940}, {278, 44695, 497}, {1309, 16082, 14198}, {2635, 8763, 2655}, {2646, 42385, 29}, {3176, 37417, 1788}


X(244) = X(1)-LINE CONJUGATE OF X(100)

Trilinears    (b - c)2 : (c - a)2 : (a - b)2
Trilinears    [1 - cos(B - C)]sin2(A/2) : :
Trilinears    d(a,b,c) : : , where d(a,b,c) = (distance from A to Nagel line)2
Barycentrics    a(b - c)2 : :

Let O* be a circle with center X(3) and variable radius R*. Let La be the radical axis of O* and the A-excircle, and define Lb and Lc cyclically. Let A'=Lb∩Lc, B'=Lc∩La, C'=La∩Lb. Then A'B'C' is perspective to ABC, and the locus of the perspector as R* varies is the hyperbola {A,B,C,X(1),X(10)}}, which has center X(244). Also, X(244) lies in the inellipse centered at X(10), as well as the Hofstadter ellpse E(1/2), which is the incentral inellipse. (Randy Hutson, December 26, 2016)

Art from triangle centers: Let X = X(244). In a square, suppose that P is a randomly chosen point, and form a triangle using two vertices of the square and P as the third vertex. Then let P = X and repeat the procedure, several thousand times. For typical results, see X(244) art and X(244) art animated. For results using other triangle centers for X, see X(41) art, X(58) art, X(76) art, X(115) art, X(141) art, X(179) art, and X(255) art. (Contributed by Peter Kagey, February 8, 2023) Figure 1

X(244) lies on aforementioned ellipses and these lines: 1,88   2,38   11,867   31,57   34,1106   42,354   58,229   63,748   238,896   474,976   518,899   596,1089   665,866

X(244) = isogonal conjugate of X(765)
X(244) = isotomic conjugate of X(7035)
X(244) = anticomplement of X(24003)
X(244) = crosssum of circumcircle intercepts of line X(1)X(21)
X(244) = perspector of anti-Aquila and 2nd Sharygin triangles
X(244) = barycentric product of vertices of Garcia reflection triangle
X(244) = trilinear pole of line X(764)X(2087)
X(244) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,513), (75,514)
X(244) = crosspoint of X(1) and X(513)
X(244) = crosssum of X(i) and X(j) for these (i,j): (1,100), (31,101), (78,1331), (109,1420), (200,644), (651,1445), (678,1023), (756,1018)
X(244) = crossdifference of every pair of points on line X(100)X(101)
X(244) = X(1)-Hirst inverse of X(1054)
X(244) = X(1)-line conjugate of X(100)
X(244) = complement of X(3952)
X(244) = antipode of X(4738) in inellipse centered at X(10)
X(244) = reflection of X(4738) in X(10)
X(244) = bicentric difference of PU(34)
X(244) = PU(34)-harmonic conjugate of X(1635)
X(244) = tripolar centroid of X(1022)
X(244) = perspector of circumconic centered at X(661)
X(244) = center of circumconic that is locus of trilinear poles of lines passing through X(661)
X(244) = X(2)-Ceva conjugate of X(661)
X(244) = trilinear pole wrt incentral triangle of line X(1)X(6)
X(244) = intersection of tangents to Steiner inellipse at X(1015) and X(1086)
X(244) = crosspoint wrt medial triangle of X(1015) and X(1086)
X(244) = trilinear square of X(513)


X(245) = X(1)-LINE CONJUGATE OF X(110)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where
                        f(A,B,C) = csc2(C - A) + csc(C - B) [csc(C - A) -csc(B - A)] + csc2(A - B)

Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(245) lies on these lines: 1,60   115,125

X(245) = X(1)-line conjugate of X(110)


X(246) = X(3)-LINE CONJUGATE OF X(110)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where
                        f(A,B,C) = csc(B-A)[cos A csc(B - A) + cos C csc(B - C)] + csc(C - A) u(A,B,C),
                        u(A,B,C) = [cos A csc(C - A) + cos B csc(C - B)]

Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(246) lies on these lines: 3,74   115,125

X(246) = X(3)-line conjugate of X(110)


X(247) = X(4)-LINE CONJUGATE OF X(110)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where
                        f(A,B,C) = csc(B-A)[sec A csc(B - A) + sec C csc(B - C)] + csc(C - A) u(A,B,C),
                        u(A,B,C) = [sec A csc(C - A) + sec B csc(C - B)]

Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(247) lies on these lines: 4,110   115,125

X(247) = crossdifference of every pair of points on line X(110)X(686)
X(247) = X(4)-line conjugate of X(110)


X(248) = X(4)-LINE CONJUGATE OF X(132)

Trilinears       sin 2A sec(A + ω) : sin 2B sec(B + ω) : sin 2C sec(C + ω)
Barycentrics  sin A sin 2A sec(A + ω) : sin B sin 2B sec(B + ω) : sin C sin 2C sec(C + ω)
Barycentrics    a^2 (a^2 - b^2 - c^2)/(b^4 + c^4 - a^2 b^2 - a^2 c^2) : :

X(248) lies on these lines: 4,32   6,157   39,54   50,67   65,172   66,571   69,287   72,293   74,187   290,385   682,695

X(248) = isogonal conjugate of X(297)
X(248) = crosspoint of X(98) and X(287)
X(248) = crosssum of X(232) and X(511)
X(248) = crossdifference of every pair of points on line X(114)X(132)
X(248) = X(4)-line conjugate of X(132)
X(248) = trilinear pole of line X(184)X(647)
X(248) = crossdifference of every pair of points on line X(114)X(132)
X(248) = X(2)-Ceva conjugate of X(39085)
X(248) = X(92)-isoconjugate of X(511)
X(248) = perspector of conic {A,B,C,X(685),X(2065),X(2715),X(15407),X(17932)}}
X(248) = barycentric product X(3)*X(98)
X(248) = barycentric product of circumcircle intercepts of line X(3)X(525)

leftri

Centers X(249)-X(297)

rightri
Centers X(249)-X(297) are isogonal conjugates of previously listed centers.

X(249) = ISOGONAL CONJUGATE OF X(115)

Trilinears    (csc A)csc2(B - C) : (csc B)csc2(C - A) : (csc C)csc2(A - B)
Trilinears    a/(b2 - c2)2 : b/(c2 - a2)2 : c/(a2 - b2)2
Barycentrics    csc2(B - C) : csc2(C - A) : csc2(A - B)
Barycentrics    a^2*(a - b)^2*(a + b)^2*(a - c)^2*(a + c)^2 : :
X(249) = X[99] + 2 X[14999], 2 X[110] + X[691], X[110] + 2 X[9181], X[691] - 4 X[9181], 4 X[9181] + X[33803], 2 X[9218] + X[33803], 4 X[113] - X[44969], 2 X[187] + X[323], X[316] - 4 X[11064], X[399] + 2 X[38611], X[47288] + 2 X[53379], X[47288] - 4 X[53735], X[53379] + 2 X[53735], X[842] - 4 X[1511], 4 X[2030] - X[41617], X[3448] - 4 X[40544], 2 X[7472] + X[15342], 2 X[12121] + X[44972], X[12383] + 2 X[16188], 10 X[15034] - X[38680], 5 X[15040] - 2 X[38613], 3 X[21166] - 2 X[54248], 2 X[34153] + X[38953], 2 X[40112] + X[51224], X[45018] + 2 X[51405]

X(249) is the vertex conjugate of the foci of the inellipse that is the barycentric square of line X(2)X(6). The center of this inellipse is X(620), and its perspector is X(4590). (Randy Hutson, October 15, 2018)

X(249) is the trilinear pole of line X(110)X(351), which is the tangent to the circumcircle at X(110), and the locus of trilinear poles of tangents at P to hyperbola {A,B,C,X(6),P}}, as P moves on the Brocard axis. (Randy Hutson, October 15, 2018)

X(249) lies on ther cubics K579, K630, K941, K1307, the curve Q120, and these lines: {2, 14846}, {3, 14355}, {6, 33704}, {20, 61754}, {30, 57650}, {32, 18872}, {60, 3110}, {76, 41174}, {99, 525}, {110, 512}, {112, 10425}, {113, 44969}, {186, 250}, {187, 323}, {297, 316}, {315, 57504}, {394, 35910}, {399, 38611}, {476, 39448}, {524, 1691}, {530, 36211}, {531, 36210}, {593, 5170}, {598, 52940}, {648, 687}, {662, 14838}, {758, 57649}, {805, 827}, {826, 47288}, {842, 1511}, {843, 39689}, {849, 1110}, {1016, 6064}, {1078, 36952}, {1092, 40804}, {1101, 1326}, {1199, 11554}, {1384, 51927}, {1501, 14608}, {1509, 7340}, {1576, 32717}, {1692, 37784}, {1915, 31632}, {1931, 4567}, {1970, 7769}, {1993, 18879}, {1994, 15544}, {2030, 41617}, {2076, 23963}, {2080, 44221}, {2185, 4564}, {2407, 2411}, {2420, 2421}, {2502, 10630}, {2702, 6578}, {2703, 58982}, {2709, 11636}, {2713, 59039}, {3053, 14253}, {3111, 5012}, {3124, 57728}, {3202, 38527}, {3448, 35605}, {3566, 7472}, {3580, 37802}, {3800, 47290}, {4235, 14591}, {4556, 4591}, {4558, 32640}, {4576, 17708}, {4600, 27573}, {4611, 15631}, {5060, 52378}, {5118, 56980}, {5152, 57562}, {5640, 34154}, {5663, 38702}, {5994, 10409}, {5995, 10410}, {6593, 32741}, {6785, 13352}, {6787, 9306}, {7757, 52438}, {7771, 42313}, {7782, 9289}, {7953, 46970}, {8115, 39299}, {8116, 39298}, {9160, 10420}, {9217, 59801}, {9273, 40214}, {9494, 17938}, {10419, 15395}, {11634, 61213}, {12071, 60055}, {12074, 32694}, {12121, 44972}, {12383, 16188}, {13586, 19627}, {13754, 57651}, {14183, 35329}, {14184, 35330}, {14246, 33928}, {14270, 60610}, {14570, 15412}, {14587, 32762}, {14960, 17941}, {15034, 38680}, {15040, 38613}, {15388, 34138}, {16172, 57638}, {17939, 57157}, {17940, 50512}, {18314, 35139}, {18321, 61753}, {18593, 35049}, {18898, 39292}, {20806, 36823}, {21166, 54248}, {22259, 52697}, {23698, 54168}, {23872, 23896}, {23873, 23895}, {31850, 34148}, {32423, 57307}, {32609, 53793}, {34153, 38953}, {34537, 56976}, {34834, 60022}, {34968, 38861}, {35602, 39265}, {36066, 53971}, {36069, 53606}, {36472, 54453}, {39201, 43754}, {40112, 51224}, {40156, 52349}, {40157, 52348}, {40820, 47044}, {43187, 57082}, {45018, 51405}, {46276, 59803}, {50711, 58908}, {57216, 61206}, {57545, 59996}

X(249) = midpoint of X(i) and X(j) for these {i,j}: {110, 9218}, {691, 33803}
X(249) = reflection of X(i) in X(j) for these {i,j}: {691, 9218}, {3448, 35605}, {9218, 9181}, {33803, 110}, {35605, 40544}, {37784, 1692}, {38704, 15035}, {54453, 36472}
X(249) = isogonal conjugate of X(115)
X(249) = isotomic conjugate of X(338)
X(249) = circumcircle-inverse of X(14366)
X(249) = polar conjugate of X(2970)
X(249) = antigonal image of X(54453)
X(249) = reflection of X(9218) in the Brocard axis
X(249) = perspector of ABC and reflection of symmedial triangle in the Brocard axis
X(249) = Vu circlecevian point of PU(2)
X(249) = X(i)-Ceva conjugate of X(j) for these (i,j): {99, 14366}, {18020, 250}, {31614, 1576}, {45773, 5467}, {47443, 4558}, {55270, 99}, {59152, 110}
X(249) = X(i)-cross conjugate of X(j) for these (i,j): {2, 18315}, {3, 99}, {6, 110}, {15, 10409}, {16, 10410}, {32, 827}, {39, 7953}, {50, 10420}, {58, 6578}, {60, 52935}, {110, 59152}, {187, 691}, {323, 44769}, {394, 4558}, {511, 10425}, {566, 58975}, {571, 933}, {574, 12074}, {577, 59039}, {593, 4556}, {1030, 100}, {1151, 1306}, {1152, 1307}, {1154, 35139}, {1333, 58982}, {1350, 35575}, {1384, 11636}, {1501, 1576}, {1511, 10411}, {1576, 31614}, {1609, 107}, {1620, 53886}, {1691, 2715}, {1970, 59009}, {1986, 61188}, {1993, 648}, {2076, 805}, {2245, 6083}, {2305, 109}, {2420, 58979}, {2965, 59004}, {2979, 670}, {3003, 1304}, {3053, 112}, {3285, 36069}, {3286, 36066}, {3289, 43754}, {4252, 59065}, {4558, 47443}, {4570, 24041}, {5008, 58120}, {5012, 4577}, {5013, 907}, {5017, 26714}, {5023, 3565}, {5104, 2709}, {5116, 43357}, {5124, 34594}, {5201, 9150}, {5206, 53884}, {5210, 1296}, {5467, 45773}, {5585, 58091}, {5889, 6528}, {6101, 46139}, {6593, 5468}, {7054, 4636}, {8041, 1634}, {8266, 689}, {8553, 925}, {8573, 59038}, {8588, 58092}, {9605, 58121}, {9786, 43352}, {11063, 476}, {11064, 43755}, {11126, 23896}, {11127, 23895}, {11130, 17403}, {11131, 17402}, {11412, 46134}, {11422, 35138}, {12963, 39384}, {12968, 39383}, {13330, 59008}, {15109, 20189}, {15513, 58094}, {15655, 33638}, {15801, 33513}, {15815, 58116}, {18755, 101}, {19118, 57216}, {19780, 5995}, {19781, 5994}, {20675, 2702}, {20806, 4563}, {20976, 6}, {22133, 1331}, {22331, 58100}, {23061, 892}, {23130, 1332}, {23357, 250}, {24729, 53621}, {30435, 7954}, {32761, 39448}, {33863, 43076}, {34148, 18831}, {34396, 33514}, {34834, 14590}, {34990, 2}, {35296, 32697}, {36743, 8690}, {36744, 931}, {36790, 2421}, {37672, 46639}, {39689, 5467}, {40214, 662}, {41328, 58118}, {41336, 10423}, {41673, 69}, {43574, 18878}, {47406, 4230}, {51318, 56980}, {53095, 58090}, {54371, 1310}, {54439, 5649}, {55566, 54031}, {55567, 54030}, {56840, 1414}, {56915, 17938}, {59232, 58111}
X(249) = X(i)-isoconjugate of X(j) for these (i,j): {1, 115}, {2, 2643}, {4, 3708}, {6, 1109}, {9, 1365}, {10, 3125}, {11, 2171}, {12, 2170}, {19, 125}, {25, 20902}, {28, 21046}, {31, 338}, {32, 23994}, {37, 3120}, {38, 34294}, {42, 16732}, {48, 2970}, {57, 4092}, {63, 8754}, {65, 21044}, {75, 3124}, {79, 21824}, {81, 21043}, {82, 39691}, {86, 21833}, {91, 47421}, {92, 20975}, {100, 21131}, {136, 1820}, {158, 3269}, {163, 23105}, {181, 4858}, {201, 8735}, {210, 53545}, {213, 21207}, {225, 53560}, {226, 4516}, {240, 51404}, {244, 594}, {257, 21725}, {304, 2971}, {312, 61052}, {313, 3121}, {321, 3122}, {339, 1973}, {393, 2632}, {512, 1577}, {513, 4024}, {514, 4705}, {522, 57185}, {523, 661}, {560, 23962}, {561, 1084}, {647, 24006}, {649, 4036}, {656, 2501}, {662, 8029}, {667, 52623}, {669, 20948}, {690, 23894}, {693, 4079}, {756, 1086}, {762, 17205}, {764, 4103}, {798, 850}, {799, 22260}, {810, 14618}, {826, 55240}, {868, 1910}, {872, 23989}, {897, 1648}, {923, 52628}, {924, 55250}, {1015, 1089}, {1093, 37754}, {1096, 15526}, {1111, 1500}, {1146, 1254}, {1356, 28659}, {1427, 52335}, {1502, 4117}, {1783, 21134}, {1821, 44114}, {1824, 4466}, {1826, 18210}, {1924, 44173}, {1928, 9427}, {1930, 51906}, {1934, 2086}, {1953, 8901}, {1959, 51441}, {2084, 52618}, {2087, 4013}, {2088, 2166}, {2153, 30465}, {2154, 30468}, {2156, 53569}, {2159, 58261}, {2160, 21054}, {2167, 41221}, {2173, 12079}, {2181, 53576}, {2207, 17879}, {2250, 42759}, {2310, 6354}, {2321, 53540}, {2433, 36035}, {2489, 14208}, {2578, 39241}, {2579, 39240}, {2611, 8818}, {2616, 12077}, {2618, 2623}, {2624, 10412}, {2631, 18808}, {2642, 5466}, {2969, 3949}, {2972, 6520}, {3005, 18070}, {3119, 6046}, {3248, 28654}, {3261, 50487}, {3271, 6358}, {3375, 43968}, {3384, 43967}, {3668, 36197}, {3700, 4017}, {3709, 4077}, {3737, 55197}, {3942, 7140}, {4033, 8034}, {4041, 7178}, {4049, 4730}, {4064, 6591}, {4081, 7147}, {4086, 7180}, {4088, 55261}, {4120, 55244}, {4155, 4444}, {4551, 55195}, {4602, 23099}, {4931, 55246}, {4983, 31010}, {5489, 24019}, {6057, 53538}, {6070, 36151}, {6089, 35354}, {6328, 16562}, {6367, 47947}, {6388, 8769}, {6507, 62524}, {6521, 34980}, {6535, 16726}, {6545, 40521}, {6627, 13610}, {6757, 20982}, {6791, 55923}, {7004, 8736}, {7018, 21823}, {7063, 20567}, {7117, 56285}, {7148, 21138}, {7199, 58289}, {7649, 55232}, {8061, 58784}, {8288, 55927}, {8770, 17876}, {9396, 10278}, {10413, 51804}, {14086, 60551}, {15475, 32679}, {15630, 46238}, {16613, 43677}, {17094, 55206}, {17881, 60501}, {17924, 55230}, {18105, 62418}, {18344, 57243}, {19610, 20939}, {21124, 57162}, {21132, 21859}, {21141, 56193}, {21723, 40143}, {21832, 35352}, {21906, 46277}, {21950, 56174}, {21963, 56123}, {24018, 58757}, {24020, 36434}, {24041, 61339}, {30572, 61179}, {30591, 58294}, {33919, 36085}, {36120, 41172}, {37755, 42069}, {38362, 53010}, {39786, 43534}, {40364, 42068}, {40495, 53581}, {40608, 60245}, {42067, 52369}, {42666, 60074}, {44426, 55234}, {45775, 52940}, {46273, 58260}, {52355, 55208}, {52651, 53559}, {53527, 55238}, {55236, 57099}
X(249) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 338}, {3, 115}, {6, 125}, {9, 1109}, {110, 45801}, {115, 23105}, {135, 55278}, {141, 39691}, {187, 5099}, {206, 3124}, {478, 1365}, {511, 59805}, {1084, 8029}, {1147, 3269}, {1249, 2970}, {2482, 52628}, {3005, 61339}, {3162, 8754}, {3163, 58261}, {3284, 3258}, {5375, 4036}, {5452, 4092}, {5976, 62431}, {6337, 339}, {6338, 36793}, {6374, 23962}, {6376, 23994}, {6503, 15526}, {6505, 20902}, {6593, 1648}, {6626, 21207}, {6631, 52623}, {8054, 21131}, {8542, 8288}, {9428, 44173}, {11126, 43962}, {11127, 43961}, {11597, 2088}, {11672, 868}, {21905, 42344}, {22391, 20975}, {23357, 30715}, {31998, 850}, {32664, 2643}, {34116, 47421}, {34961, 3700}, {35071, 5489}, {36033, 3708}, {36830, 523}, {36896, 12079}, {37867, 2972}, {38988, 33919}, {38996, 22260}, {39006, 21134}, {39026, 4024}, {39052, 24006}, {39054, 1577}, {39062, 14618}, {39073, 57430}, {39085, 51404}, {40368, 1084}, {40369, 9427}, {40580, 30465}, {40581, 30468}, {40586, 21043}, {40588, 41221}, {40589, 3120}, {40591, 21046}, {40592, 16732}, {40596, 2501}, {40600, 21833}, {40601, 44114}, {40602, 21044}, {40604, 62551}, {45248, 1562}, {46094, 41172}, {52042, 62417}, {55047, 55273}, {56948, 6741}, {62452, 52618}, {62613, 41079}
X(249) = cevapoint of X(i) and X(j) for these (i,j): {2, 14570}, {3, 32661}, {6, 110}, {24, 112}, {69, 4576}, {99, 1078}, {100, 38871}, {101, 33771}, {109, 38864}, {143, 1625}, {163, 849}, {394, 4558}, {593, 4556}, {662, 2185}, {1379, 1380}, {1501, 1576}, {1511, 2420}, {1634, 8041}, {1986, 14591}, {2421, 36790}, {4636, 7054}, {5467, 39689}, {5546, 35193}, {8115, 8116}, {11130, 17403}, {11131, 17402}, {19118, 61206}, {21873, 61172}, {23357, 47390}, {35324, 36153}, {35603, 61208}, {41512, 56404}, {41616, 61207}, {51318, 56980}
X(249) = crosspoint of X(4590) and X(18020)
X(249) = crosssum of X(i) and X(j) for these (i,j): {2, 54104}, {512, 8574}, {523, 10278}, {1648, 23992}, {1649, 33906}, {2679, 41178}, {3124, 20975}, {8029, 61339}, {14443, 42344}, {21043, 21833}, {41172, 41181}, {41176, 41177}, {41179, 57462}, {41180, 57461}, {41182, 57463}, {57464, 57465}
X(249) = trilinear pole of line {110, 351}
X(249) = crossdifference of every pair of points on line {1648, 8029}
X(249) = barycentric product X(i)*X(j) for these {i,j}: {1, 24041}, {3, 18020}, {6, 4590}, {19, 62719}, {24, 57763}, {25, 47389}, {31, 24037}, {32, 34537}, {48, 46254}, {55, 7340}, {56, 6064}, {58, 4600}, {59, 261}, {60, 4998}, {69, 250}, {75, 1101}, {76, 23357}, {81, 4567}, {86, 4570}, {99, 110}, {100, 52935}, {101, 4610}, {112, 4563}, {143, 57764}, {162, 4592}, {163, 799}, {187, 52940}, {190, 4556}, {255, 23999}, {264, 47390}, {284, 4620}, {305, 57655}, {311, 14587}, {317, 44174}, {323, 39295}, {325, 57742}, {326, 24000}, {333, 52378}, {394, 23582}, {476, 10411}, {511, 57991}, {512, 31614}, {523, 59152}, {525, 47443}, {552, 6065}, {561, 23995}, {593, 1016}, {643, 1414}, {645, 4565}, {647, 55270}, {648, 4558}, {651, 4612}, {662, 662}, {664, 4636}, {670, 1576}, {690, 45773}, {691, 5468}, {692, 4623}, {757, 765}, {805, 17941}, {811, 4575}, {827, 4576}, {849, 7035}, {873, 1110}, {877, 43754}, {880, 17938}, {892, 5467}, {906, 55231}, {1078, 27867}, {1098, 7045}, {1252, 1509}, {1262, 7058}, {1275, 7054}, {1333, 4601}, {1415, 4631}, {1444, 5379}, {1501, 44168}, {1502, 23963}, {1634, 4577}, {1691, 39292}, {2149, 52379}, {2185, 4564}, {2396, 2715}, {2407, 44769}, {2421, 2966}, {2482, 34539}, {2709, 34245}, {3268, 58979}, {3289, 41174}, {3565, 57216}, {3573, 36066}, {3580, 18879}, {3926, 23964}, {3936, 9273}, {3964, 32230}, {4076, 7341}, {4143, 59153}, {4226, 10425}, {4230, 17932}, {4427, 6578}, {4559, 55196}, {4573, 5546}, {4585, 37140}, {4609, 14574}, {4611, 44766}, {4637, 7259}, {5118, 9150}, {5376, 30576}, {5649, 14999}, {6061, 59457}, {6148, 15395}, {6331, 32661}, {6516, 52914}, {6517, 52921}, {6753, 55277}, {7252, 55194}, {7953, 10330}, {8041, 57545}, {8115, 39299}, {8116, 39298}, {8673, 55272}, {9145, 35138}, {9146, 11636}, {9170, 9181}, {9217, 31632}, {9218, 37880}, {9274, 35550}, {10409, 35314}, {10410, 35315}, {10420, 61188}, {11130, 57580}, {11131, 57579}, {12074, 35356}, {13485, 14366}, {14570, 18315}, {14590, 60053}, {14966, 43187}, {15329, 18878}, {15388, 34254}, {15631, 41173}, {16237, 43755}, {16806, 55198}, {16807, 55200}, {17402, 23895}, {17403, 23896}, {17708, 52630}, {17929, 17944}, {17930, 17943}, {17931, 17942}, {17934, 17940}, {17935, 17939}, {18829, 56980}, {18831, 23181}, {20806, 44183}, {21906, 42370}, {23342, 32717}, {23889, 36085}, {23997, 36036}, {24039, 36142}, {32036, 52605}, {32037, 52606}, {32656, 55229}, {32660, 55233}, {32676, 55202}, {32692, 55252}, {32739, 52612}, {34072, 55239}, {35049, 56440}, {35137, 61211}, {35139, 52603}, {35278, 35575}, {35324, 55279}, {35342, 62535}, {35357, 42367}, {35602, 44181}, {36145, 55249}, {36212, 60179}, {36790, 57562}, {36841, 46639}, {37134, 56982}, {39689, 57552}, {42308, 51394}, {44717, 46103}, {51318, 57558}, {52608, 61206}, {53205, 62523}, {53332, 58982}, {53628, 57060}, {53633, 57249}, {55218, 61194}, {55268, 58796}, {57639, 57805}
X(249) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1109}, {2, 338}, {3, 125}, {4, 2970}, {6, 115}, {15, 30465}, {16, 30468}, {22, 53569}, {24, 136}, {25, 8754}, {30, 58261}, {31, 2643}, {32, 3124}, {35, 21054}, {39, 39691}, {42, 21043}, {48, 3708}, {50, 2088}, {51, 41221}, {54, 8901}, {55, 4092}, {56, 1365}, {58, 3120}, {59, 12}, {60, 11}, {63, 20902}, {69, 339}, {71, 21046}, {74, 12079}, {75, 23994}, {76, 23962}, {81, 16732}, {86, 21207}, {97, 53576}, {99, 850}, {100, 4036}, {101, 4024}, {110, 523}, {112, 2501}, {143, 137}, {162, 24006}, {163, 661}, {184, 20975}, {186, 35235}, {187, 1648}, {190, 52623}, {213, 21833}, {237, 44114}, {248, 51404}, {250, 4}, {251, 34294}, {255, 2632}, {261, 34387}, {284, 21044}, {323, 62551}, {325, 62431}, {326, 17879}, {351, 33919}, {394, 15526}, {476, 10412}, {511, 868}, {512, 8029}, {520, 5489}, {523, 23105}, {524, 52628}, {571, 47421}, {574, 8288}, {577, 3269}, {593, 1086}, {643, 4086}, {648, 14618}, {649, 21131}, {662, 1577}, {669, 22260}, {670, 44173}, {691, 5466}, {692, 4705}, {757, 1111}, {763, 16727}, {765, 1089}, {799, 20948}, {827, 58784}, {849, 244}, {859, 42759}, {892, 52632}, {895, 51258}, {906, 55232}, {1016, 28654}, {1078, 36901}, {1092, 2972}, {1098, 24026}, {1101, 1}, {1110, 756}, {1113, 39241}, {1114, 39240}, {1252, 594}, {1259, 7068}, {1262, 6354}, {1304, 18808}, {1331, 4064}, {1333, 3125}, {1379, 13722}, {1380, 13636}, {1384, 6791}, {1397, 61052}, {1408, 53540}, {1412, 53545}, {1414, 4077}, {1415, 57185}, {1437, 18210}, {1459, 21134}, {1501, 1084}, {1509, 23989}, {1511, 3258}, {1576, 512}, {1625, 12077}, {1634, 826}, {1707, 17876}, {1790, 4466}, {1804, 1367}, {1813, 57243}, {1917, 4117}, {1974, 2971}, {1976, 51441}, {1983, 2610}, {1986, 16221}, {2149, 2171}, {2150, 2170}, {2174, 21824}, {2185, 4858}, {2189, 8735}, {2193, 53560}, {2194, 4516}, {2206, 3122}, {2328, 52335}, {2407, 41079}, {2420, 1637}, {2421, 2799}, {2617, 2618}, {2709, 34246}, {2715, 2395}, {2966, 43665}, {2979, 53575}, {3053, 6388}, {3060, 34981}, {3124, 61339}, {3133, 55072}, {3233, 58263}, {3289, 41172}, {3447, 6328}, {3457, 30452}, {3458, 30453}, {3926, 36793}, {4100, 37754}, {4143, 23107}, {4184, 21045}, {4230, 16230}, {4556, 514}, {4558, 525}, {4559, 55197}, {4563, 3267}, {4564, 6358}, {4565, 7178}, {4567, 321}, {4570, 10}, {4575, 656}, {4576, 23285}, {4577, 52618}, {4590, 76}, {4591, 4049}, {4592, 14208}, {4599, 18070}, {4600, 313}, {4601, 27801}, {4610, 3261}, {4611, 33294}, {4612, 4391}, {4619, 4605}, {4620, 349}, {4623, 40495}, {4629, 31010}, {4630, 18105}, {4636, 522}, {4998, 34388}, {5012, 7668}, {5118, 9148}, {5191, 51428}, {5379, 41013}, {5467, 690}, {5468, 35522}, {5546, 3700}, {5562, 35442}, {5649, 14223}, {5663, 6070}, {5994, 20579}, {5995, 20578}, {6061, 4081}, {6064, 3596}, {6065, 6057}, {6066, 7064}, {6524, 62524}, {6578, 4608}, {6593, 5099}, {6753, 55278}, {7012, 56285}, {7054, 1146}, {7058, 23978}, {7115, 8736}, {7122, 21725}, {7252, 55195}, {7335, 61058}, {7339, 6046}, {7340, 6063}, {7341, 1358}, {7342, 1357}, {7492, 38361}, {7669, 58908}, {7953, 31065}, {8041, 15449}, {8673, 55273}, {9145, 3906}, {9155, 51429}, {9181, 8371}, {9218, 10278}, {9233, 9427}, {9268, 4013}, {9273, 24624}, {9274, 759}, {9418, 58260}, {9426, 23099}, {9448, 7063}, {9475, 57430}, {10316, 38356}, {10409, 62631}, {10410, 62632}, {10411, 3268}, {10420, 15328}, {10425, 62645}, {11063, 10413}, {11130, 43962}, {11131, 43961}, {11141, 43968}, {11142, 43967}, {11634, 9134}, {11636, 8599}, {11672, 59805}, {13198, 34978}, {13434, 8902}, {14060, 34953}, {14246, 10555}, {14366, 3448}, {14385, 56792}, {14559, 51479}, {14560, 15475}, {14567, 21906}, {14570, 18314}, {14574, 669}, {14586, 2623}, {14587, 54}, {14590, 44427}, {14591, 47230}, {14601, 15630}, {14602, 2086}, {14966, 3569}, {14999, 18312}, {15035, 3154}, {15107, 38393}, {15329, 55121}, {15384, 6526}, {15388, 13854}, {15395, 5627}, {15406, 32133}, {15460, 1313}, {15461, 1312}, {15462, 36189}, {15631, 62555}, {15742, 7141}, {15905, 1562}, {15958, 23286}, {16806, 55199}, {16807, 55201}, {17104, 2611}, {17402, 23870}, {17403, 23871}, {17938, 882}, {17939, 18015}, {17940, 18014}, {17941, 14295}, {17942, 18006}, {17943, 18004}, {17944, 18003}, {18020, 264}, {18315, 15412}, {18755, 6627}, {18829, 56981}, {18879, 2986}, {19118, 5139}, {19121, 53570}, {20806, 127}, {20976, 23991}, {21525, 58909}, {21784, 12071}, {21906, 42344}, {22115, 16186}, {22151, 62563}, {23181, 6368}, {23348, 18007}, {23357, 6}, {23582, 2052}, {23606, 34980}, {23963, 32}, {23964, 393}, {23975, 36434}, {23990, 1500}, {23995, 31}, {23999, 57806}, {24000, 158}, {24027, 1254}, {24037, 561}, {24041, 75}, {27867, 3613}, {31614, 670}, {32230, 1093}, {32583, 23288}, {32640, 2433}, {32656, 55230}, {32660, 55234}, {32661, 647}, {32662, 14582}, {32692, 55253}, {32696, 53149}, {32697, 60338}, {32713, 58757}, {32717, 60028}, {32729, 9178}, {32739, 4079}, {33628, 21950}, {33771, 55065}, {33803, 62663}, {33875, 52625}, {34072, 55240}, {34148, 53577}, {34396, 6784}, {34537, 1502}, {34539, 57539}, {35049, 43682}, {35193, 6741}, {35278, 30735}, {35324, 55280}, {35327, 6367}, {35357, 12073}, {35360, 23290}, {35602, 122}, {35603, 135}, {36134, 2616}, {36142, 23894}, {36145, 55250}, {36153, 11792}, {36208, 30460}, {36209, 30463}, {36790, 35088}, {36830, 45801}, {37140, 60074}, {39024, 31644}, {39292, 18896}, {39295, 94}, {39298, 2593}, {39299, 2592}, {39689, 23992}, {40049, 62339}, {40214, 8287}, {40373, 23216}, {40948, 57424}, {41174, 60199}, {41280, 1356}, {41616, 48317}, {41679, 57065}, {41937, 2207}, {42742, 55141}, {43574, 3134}, {43754, 879}, {43755, 15421}, {44162, 42068}, {44168, 40362}, {44174, 68}, {44179, 17881}, {44183, 43678}, {44717, 26942}, {44769, 2394}, {45773, 892}, {46127, 15359}, {46249, 47229}, {46254, 1969}, {46288, 51906}, {46639, 58759}, {46726, 36199}, {47053, 45147}, {47377, 47270}, {47389, 305}, {47390, 3}, {47406, 41181}, {47443, 648}, {51318, 35078}, {51394, 1650}, {51478, 9213}, {52003, 46658}, {52238, 57604}, {52378, 226}, {52432, 34338}, {52603, 526}, {52604, 51513}, {52605, 23872}, {52606, 23873}, {52613, 23616}, {52630, 9979}, {52699, 14120}, {52914, 44426}, {52915, 59932}, {52935, 693}, {52940, 18023}, {53273, 12075}, {53708, 62519}, {53760, 47004}, {53863, 38394}, {54274, 14443}, {54353, 4088}, {55270, 6331}, {56840, 8286}, {56894, 21710}, {56915, 41178}, {56934, 17886}, {56980, 804}, {57153, 44705}, {57562, 34536}, {57638, 32132}, {57639, 252}, {57655, 25}, {57742, 98}, {57763, 20563}, {57764, 57765}, {57991, 290}, {58796, 55269}, {58979, 476}, {58982, 4581}, {59004, 50946}, {59149, 4103}, {59152, 99}, {59153, 6529}, {59482, 21666}, {59495, 16177}, {59994, 62417}, {60053, 14592}, {60179, 16081}, {60605, 18039}, {61194, 55219}, {61198, 47138}, {61206, 2489}, {61207, 14273}, {61208, 6753}, {61209, 47236}, {61211, 7927}, {61213, 55122}, {61378, 24862}, {62194, 47430}, {62719, 304}
X(249) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {110, 9181, 691}, {662, 39054, 14838}, {2420, 2421, 52630}, {10411, 52630, 2421}, {53379, 53735, 47288}


X(250) = ISOGONAL CONJUGATE OF X(125)

Trilinears       (sec A)csc2(B - C) : (sec B)csc2(C - A) : (sec C)csc2(A - B)
                        = (a2sec A)/(b2 - c2)2 : (b2sec B)/(c2 - a2)2 : (c2sec C)/(a2 - b2)2

Barycentrics  (tan A)csc2(B - C) : (tan B)csc2(C - A) : (tan C)csc2(A - B)

X(250) is the trilinear pole of line X(110)X(112), which is the tangent to the MacBeath circumconic at X(110), and the locus of trilinear poles of tangents at P to conic {A,B,C,X(3),P}}, as P moves on the Brocard axis. (Randy Hutson, March 21, 2019)

X(250) lies on these lines: 23,232   107,687   110,520   112,691   186,249   325,340   476,933   523,648   827,935

X(250) = isogonal conjugate of X(125)
X(250) = isotomic conjugate of X(339)
X(250) = cevapoint of X(i) and X(j) for these (i,j): (3,110), (25,112), (162,270), (1113,1114)
X(250) = X(i)-cross conjugate of X(j) for these (i,j): (3,110), (22,99), (24,107), (25,112), (199,101)
X(250) = polar conjugate of X(338)
X(250) = circumcenter of reflection triangle of X(125)
X(250) = barycentric product X(99)*X(112)
X(250) = cevapoint of MacBeath circumconic intercepts of Brocard axis


X(251) = ISOGONAL CONJUGATE OF X(141)

Trilinears       a2csc(A + ω) : b2csc(B + ω) : c2csc(C + ω)
                        = a/(b2 + c2) : b/(c2 + a2) : c/(a2 + b2)

Barycentrics  a3csc(A + ω) : b3csc(B + ω) : c3csc(C + ω)

Let K be the symmedian point of ABC and let A' be the symmedian point of the triangle BCK; define B' and C' cyclically. The lines AA', BB', CC' concur in X(251). (Randy Hutson, 9/23/2011)

Let A5'B5'C5' be the 5th anti-Brocard triangle. The radical center of the circumcircles of BCA5', CAB5', ABC5' is X(251). (Randy Hutson, July 20, 2016)

X(251) lies on these lines: 2,32   6,22   37,82   110,694   112,427   184,263   308,385   609,614   689,699

X(251) = isogonal conjugate of X(141)
X(251) = complement of X(1369)
X(251) = anticomplement of X(21248)
X(251) = cevapoint of X(6) and X(32)
X(251) = X(i)-cross conjugate of X(j) for these (i,j): (6,83), (23,111), (523,112)
X(251) = isotomic conjugate of X(8024)
X(251) = similitude center of ABC and 1st orthosymmedial triangle
X(251) = pole wrt polar circle of trilinear polar of X(1235)
X(251) = X(48)-isoconjugate (polar conjugate) of X(1235)
X(251) = barycentric product of vertices of circummedial triangle
X(251) = perspector of ABC and cross-triangle of ABC and circummedial triangle
X(251) = Kosnita(X(6),X(6)) point
X(251) = homothetic center of 1st orthosymmedial and 1st anti-orthosymmedial triangles


X(252) = ISOGONAL CONJUGATE OF X(143)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where
                        f(A,B,C) = sec(B - C)/[1 - 2 cos(2A)]
Trilinears       h(A,B,C) : h(B,C,A) : h(C,A,B), where
                        h(A,B,C) = cos A sec(3A) sec(B - C) (Manol Iliev, 4/01/07)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C) f(C,A,B) Let A'B'C' be the orthic triangle. Let OA be the circle with center A passing through A'. Let A″ be the intersection, other than A', of OA and the nine-point circle. Define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(252). (Randy Hutson, June 7, 2019)

Let HA be the reflection of X(4) in the Euler line of BCX(4), and define HB and HC cyclically. The lines AHA, BHB, CHC concur in X(252). (Randy Hutson, June 7, 2019)

X(252) lies on these lines: 3,930   54,140   93,186

X(252) = isogonal conjugate of X(143)
X(252) = anticomplement of X(31376)
X(252) = X(5)-isoconjugate of X(2964)


X(253) = X(4)-CROSS CONJUGATE OF X(2)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (csc A)/(tan B + tan C - tan A)
                         = g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (csc2A)/(cos A - cos B cos C)
Barycentrics    g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = 1/(tan B + tan C - tan A)
Barycentrics    1/(3 a^4 - 2 a^2 (b^2 + c^2) - (b^2 - c^2)^2) : :
X(253) is the unique point whose complement is also its polar conjugate (X(1249)). (Randy Hutson, July 11, 2019)

X(253) is the perspector of ABC and the pedal triangle of X(64).

X(253) lies on the Lucas cubic and these lines: 2,1073   7,280   8,307   20,64   193,287   306,329   318,342   322,341

X(253) = isogonal conjugate of X(154)
X(253) = isotomic conjugate of X(20)
X(253) = cyclocevian conjugate of X(69)
X(253) = cevapoint of X(i) and X(j) for these (i,j): (4,459), (122,525)
X(253) = X(i)-cross conjugate of X(j) for these (i,j): (4,2), (122,525)
X(253) = anticomplement of X(1249)
X(253) = polar conjugate of X(1249)
X(253) = perspector of ABC and the reflection in X(1073) of the pedal triangle of X(1073)
X(253) = perspector of de Longchamps circle
X(253) = pole, wrt de Longchamps circle, of trilinear polar of X(69) (line X(441)X(525))


X(254) = X(3)-CROSS CONJUGATE OF X(4)

Trilinears    (sec A)/(cos2B + cos2C - cos2A) : :
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (tan A)/(cos2B + cos2C - cos2A)
Barycentrics    1/((a^2 - b^2 - c^2) (a^6 - 3 a^4 (b^2 + c^2) + a^2 (3 b^4 - 2 b^2 c^2 + 3 c^4) - (b^2 - c^2)^2 (b^2 + c^2))) : :
Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^6 - a^4*b^2 - a^2*b^4 + b^6 - 3*a^4*c^2 + 2*a^2*b^2*c^2 - 3*b^4*c^2 + 3*a^2*c^4 + 3*b^2*c^4 - c^6)*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 - a^4*c^2 + 2*a^2*b^2*c^2 + 3*b^4*c^2 - a^2*c^4 - 3*b^2*c^4 + c^6) : :

X(254) lies on the following curves: the circumconics {A,B,C,G,X(24)}}, {A,B,C,O,X(155)}}, {A,B,C,H,X(93)}}; the cubics K006, K045, K163, K364, K491, K519, K617, K919; Q066; and these lines: {2, 847}, {3, 33495}, {4, 155}, {20, 1300}, {24, 393}, {46, 225}, {68, 136}, {186, 18126}, {264, 3541}, {378, 1217}, {403, 6526}, {487, 24244}, {488, 24243}, {1092, 8754}, {1093, 3542}, {1147, 14593}, {1179, 7487}, {1594, 18855}, {1826, 17857}, {2970, 3548}, {3564, 20422}, {6344, 7505}, {8801, 15559}, {8883, 8884}, {8889, 18854}, {18560, 18850}

X(254) = reflection of X(33495) in X(3)
X(254) = isogonal conjugate of X(155)
X(254) = isotomic conjugate of isogonal conjugate of X(39109)
X(254) = anticomplement of X(34853)
X(254) = polar conjugate of X(6515)
X(254) = cyclocevian conjugate of X(13579)
X(254) = isogonal conjugate of the anticomplement of X(12359)
X(254) = isogonal conjugate of the complement of X(11411)
X(254) = isotomic conjugate of the anticomplement of X(2165)
X(254) = polar conjugate of the isotomic conjugate of X(6504)
X(254) = X(i)-cross conjugate of X(j) for these (i,j): {3, 4}, {2165, 2}
X(254) = X(i)-isoconjugate of X(j) for these (i,j): {1, 155}, {3, 920}, {19, 6503}, {48, 6515}, {63, 1609}, {184, 33808}, {255, 3542}, {454, 921}, {1725, 15478}
X(254) = cevapoint of X(i) and X(j) for these (i,j): {3, 15316}, {4, 3147}, {136, 523}, {647, 8754}
X(254) = trilinear pole of line {924, 2501}
X(254) = barycentric product X(i)*X(j) for these {i,j}: {4, 6504}, {92, 921}, {275, 8800}, {2052, 15316}, {2986, 16172}, {11547, 32132}, {13398, 14618}
X(254) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 6503}, {4, 6515}, {6, 155}, {19, 920}, {25, 1609}, {92, 33808}, {393, 3542}, {921, 63}, {1609, 454}, {6504, 69}, {8800, 343}, {8882, 8883}, {13398, 4558}, {14910, 15478}, {15316, 394}, {16172, 3580}, {16310, 27087}
X(254) = {X(8800),X(15316)}-harmonic conjugate of X(6504)


X(255) = ISOGONAL CONJUGATE OF X(158)

Trilinears    cos2A : cos2B : cos2C
Trilinears    1 + cos 2A : 1 + cos 2B : 1 + cos 2C
Trilinears    tan B tan C - 1 : :
Barycentrics   sin A cos2A : sin B cos2B : sin C cos2C

Let A'B'C' and A″B″C″ be the Lucas and Lucas(-1) central triangles. Let A* be the trilinear product A'*A″, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(255). (Randy Hutson, October 15, 2018)

Let A'B'C' and A″B″C″ be the Lucas and Lucas(-1) antipodal triangles. Let A* be the trilinear product A'*A″, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(255). (Randy Hutson, October 15, 2018)

For an artistic design generated by X(255), see X(244).

X(255) lies on these lines: 1,21   3,73   35,991   36,1106   40,109   48,563   55,601   56,602   57,580   91,1109   92,1087   158,775   162,1099   165,1103   200,271   201,1060   219,268   293,304   326,1102   411,651   498,750   499,748

X(255) = isogonal conjugate of X(158)
X(255) = isotomic conjugate of polar conjugate of X(48)
X(255) = X(i)-Ceva conjugate of X(j) for these (i,j): (63,48), (283,3)
X(255) = crosspoint of X(63) and X(326)
X(255) = crosssum of X(i) and X(j) for these (i,j): (1,290), (4,1068), (19,1096)
X(255) = X(i)-aleph conjugate of X(j) for these (i,j): (775,255), (1105,158)
X(255) = trilinear pole of line X(680)X(822)
X(255) = trilinear product X(1124)*X(1335)
X(255) = trilinear square of X(3)
X(255) = polar conjugate of X(6521)
X(255) = X(19)-isoconjugate of X(92)
X(255) = {X(1),X(31)}-harmonic conjugate of X(1497)
X(255) = {X(3074),X(3075)}-harmonic conjugate of X(2)


X(256) = 1st SHARYGIN POINT

Trilinears       1/(a2 + bc) : 1/(b2 + ca) : 1/(c2 + ab)
Barycentrics  a/(a2 + bc) : b/(b2 + ca) : c/(c2 + ab)

See the description at X(1281). The lines AD, BE, CF defined there concur in X(256).

X(256) lies on these lines: 1,511   3,987   4,240   7,982   8,192   9,43   21,238   37,694   40,989   55,983   84,988   104,1064   291,894   314,350   573,981

X(256) = isogonal conjugate of X(171)
X(256) = isotomic conjugate of X(1909)
X(256) = X(239)-cross conjugate of X(291)
X(256) = crosssum of X(43) and X(846)
X(256) = X(238)-Hirst inverse of X(904)
X(256) = cevapoint of PU(6)
X(256) = trilinear pole of line X(650)X(3250)
X(256) = crossdifference of every pair of points on line X(3287)X(3805) (the perspectrix of ABC and Gemini triangle 34)


X(257) = ISOGONAL CONJUGATE OF X(172)

Trilinears       1/(a3 + abc) : 1/(b3 + abc) : 1/(c3 + abc)
Barycentrics  1/(a2 + bc) : 1/(b2 + ca) : 1/(c2 + ab)

X(257) lies on these lines: 1,385   8,192   29,242   65,894   75,698   92,297   194,986   239,333   330,982   335,694

X(257) = isogonal conjugate of X(172)
X(257) = isotomic conjugate of X(894)
X(257) = X(350)-cross conjugate of X(335)
X(257) = X(239)-Hirst inverse of X(893)
X(257) = trilinear pole of line X(522)X(1491)
X(257) = cevapoint of PU(10)
X(257) = pole wrt polar circle of line X(2533)X(3287)
X(257) = X(48)-isoconjugate (polar conjugate) of X(7009)


X(258) = CONGRUENT INCIRCLES ISOSCELIZER POINT

Trilinears    1/(cos B/2 + cos C/2 - cos A/2) : :
Trilinears    1 + sin(B/2) + sin(C/2) - sin(A/2) : :
Trilinears    tan(A/2) - sec(A/2) : :
Trilinears    tan(B/2) - sec(B/2) : tan(C/2) - sec(C/2) : :
Trilinears    1/(b' + c' - a') : : , where A'B'C' is the excentral triangle
Trilinears    tan A'/2 : : , where A'B'C' is the excentral triangle
Trilinears    cot A' - csc A' : : , where A'B'C' is the excentral triangle
Trilinears    (distance from A to A-excircle) : :

In Yff's isoscelizer configuration, if X = X(258), then the isosceles triangles TA, TB, TC have congruent incircles.

If you have The Geometer's Sketchpad, you can view X(258).

X(258) lies on these lines: 1,164   57,173   259,289

X(258) = isogonal conjugate of X(173)
X(258) = X(259)-cross conjugate of X(1)
X(258) = X(366)-aleph conjugate of X(363)
X(258) = SS(a→a') of X(57), where A'B'C' is the excentral triangle (trilinear substitution)
X(258) = X(33)-of-excentral-triangle
X(258) = homothetic center of ABC and intangents triangle of excentral triangle
X(258) = insimilicenter of incircle and incircle of excentral triangle
X(258) = {X(1),X(164)}-harmonic conjugate of X(8078)
X(258) = perspector of ABC and the extouch triangle of the intouch triangle


X(259) = ISOGONAL CONJUGATE OF X(174)

Trilinears    cos A/2 : :
Trilinears    [a(b + c - a)]1/2 : :
Trilinears    sin A csc A/2 : :
Trilinears    sin A' : : , where A'B'C' is the excentral triangle
Trilinears    sin(∠BIC) : :
Barycentrics  sin A cos A/2 : :
Trilinears    (b + c - a) sin A/2 : :

X(259) lies on these lines: 1,168   258,289   260,266

X(259) = isogonal conjugate of X(174)
X(259) = X(i)-Ceva conjugate of X(j) for these (i,j): (174,266), (260,55)
X(259) = cevapoint of X(1) and X(503)
X(259) = crosspoint of X(i) and X(j) for these (i,j): (1,258), (174,188)
X(259) = crosssum of X(i) and X(j) for these (i,j): (1,173), (259,266)
X(259) = SS(A→A') of X(6), where A'B'C' is the excentral triangle
X(259) = trilinear square root of X(55)
X(259) = perspector of ABC and unary cofactor triangle of tangential mid-arc triangle
X(259) = excentral-to-ABC trilinear image of X(9)
X(259) = intouch-to-ABC trilinear image of X(7)


X(260) = ISOGONAL CONJUGATE OF X(177)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A/2)/(cos B/2 + cos C/2)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(260) lies on these lines: 1,3   259,266

X(260) = isogonal conjugate of X(177)
X(260) = cevapoint of X(55) and X(259)


X(261) = ISOTOMIC CONJUGATE OF X(12)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = [(csc A)(sec(B/2 - C/2))]2
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(261) lies on these lines: 2,593   9,645   21,314   28,242   58,86   75,99   272,310   284,332   317,406   319,502   552,873   572,662

X(261) = isogonal conjugate of X(181)
X(261) = isotomic conjugate of X(12)
X(261) = complement of anticomplementary conjugate of X(35614)
X(261) = X(873)-Ceva conjugate of X(1509)
X(261) = cevapoint of X(21) and X(333)
X(261) = polar conjugate of X(8736)
X(261) = trilinear pole of line X(3904)X(3910)


X(262) = ISOGONAL CONJUGATE OF X(182)

Trilinears    sec(A - ω) : sec(B - ω) : sec(C - ω)
Barycentrics    sin A sec(A - ω) : sin B sec(B - ω) : sin C sec(C - ω)
Barycentrics    1/(a^4 - a^2b^2 - a^2c^2 - 2b^2c^2) : :
Barycentrics    (a^2*b^2 - b^4 + 2*a^2*c^2 + b^2*c^2)*(2*a^2*b^2 + a^2*c^2 + b^2*c^2 - c^4)
X(262) = 4 X[2] - X[33706], 2 X[3] - 5 X[7786], X[3] - 4 X[11272], X[3] + 2 X[14881], X[4] + 2 X[39], 2 X[4] + X[11257], 4 X[5] - X[76], 2 X[5] + X[3095], X[20] - 4 X[13334], 4 X[39] - X[11257], X[76] + 2 X[3095], X[98] - 4 X[2023], 2 X[114] + X[1916], 2 X[119] + X[32454], 4 X[140] - X[9821], X[194] + 5 X[3091], X[194] + 2 X[6248], 2 X[355] + X[7976], 2 X[381] + X[7757], 3 X[381] - 2 X[22681], 3 X[381] + X[32519], 2 X[546] + X[32448], 5 X[631] - 2 X[5188], 5 X[631] - 8 X[6683], 2 X[946] + X[12782], X[1350] - 4 X[10007], 2 X[1352] + X[32451], 7 X[3090] - 4 X[3934], 7 X[3090] - X[12251], 5 X[3091] - 2 X[6248], X[3094] + 2 X[5480], X[3146] + 5 X[32522], 5 X[3618] - 2 X[13354], X[3627] + 2 X[32516], 4 X[3628] - X[32521], 7 X[3851] - X[13108], 11 X[3855] + 4 X[32450], 4 X[3934] - X[12251], 11 X[5056] - 5 X[31276], 8 X[5066] + X[11055], 13 X[5067] - 10 X[31239], 13 X[5068] - X[20081], 5 X[5071] - 2 X[9466], 11 X[5072] + X[32520], X[5188] - 4 X[6683], 4 X[5476] - X[22486], 2 X[6194] - 3 X[22712], 4 X[6194] - 3 X[33706], 2 X[6249] + X[32476], X[6309] - 4 X[7764], X[7709] + 2 X[22682], 3 X[7757] + 4 X[22681], 3 X[7757] - 2 X[32519], X[7759] + 2 X[32189], 2 X[7775] + X[13085], 5 X[7786] - 8 X[11272], 5 X[7786] + 4 X[14881], 5 X[7786] - X[22676], 5 X[7786] + 2 X[22728], X[7823] + 2 X[32152], 5 X[7921] + X[9863], 7 X[7989] - X[9902], 5 X[8227] - 2 X[12263], 2 X[8356] + X[34733], 8 X[9300] + X[14458], 7 X[9781] - 4 X[27375], 2 X[9880] + X[11152], X[11257] + 4 X[22682], 2 X[11272] + X[14881], 8 X[11272] - X[22676], 4 X[11272] + X[22728], 4 X[12102] + 5 X[32523], X[13325] - 4 X[14633], X[13326] - 4 X[14632], 2 X[14251] + X[14265], 3 X[14561] - X[31958], 3 X[14639] + X[32469], 4 X[14881] + X[22676], 4 X[15819] - 3 X[22712], 8 X[15819] - 3 X[33706], 2 X[22475] + X[22650], 4 X[22475] - X[22713], 2 X[22650] + X[22713], X[22676] + 2 X[22728], 2 X[22681] + 3 X[32447], 2 X[22681] + X[32519], 3 X[32447] - X[32519]

Let A'B'C' be the orthic triangle. X(262) is the radical center of the Brocard circles of AB'C', BC'A', CA'B'. (Randy Hutson, February 10, 2016)

Let Pa be the parabola with focus A and directrix BC. Let La be the polar of X(3) wrt Pa and define Lb and Lc cyclically. Let A' = Lb∩Lc, B' = Lc∩La, C' = La∩Lb. The lines AA', BB', CC' concur in X(262). Also, X(262) is also the isotomic conjugate, wrt A'B'C', of X(3).

Let A' be the apex of the isosceles triangle BA'C constructed outward on BC such that ∠A'BC = ∠A'CB = ω. Define B', C' cyclically. Let Ha be the orthocenter of BA'C, and define Hb, Hc cyclically. The lines AHa, BHb, CHc concur in X(262). (Randy Hutson, July 20, 2016)

Let A' be the apex of the isosceles triangle BA'C constructed intward on BC such that ∠A'BC = ∠A'CB = ω/2. Define B', C' cyclically. Let Oa be the circumcenter of BA'C, and define Ob, Oc cyclically. The lines AOa, BOb, COc concur in X(262). (Randy Hutson, July 20, 2016)

X(262) lies on the Kiepert circumhyperbola, the cubics K062, K263, K280, K300, K305, K353, K412, K422, K423, K581, K582, K677, K756, K790, K791, K802, K930, K939, K1012, and these lines: {1, 22475}, {2, 51}, {3, 83}, {4, 39}, {5, 76}, {6, 98}, {10, 2186}, {11, 12837}, {12, 12836}, {13, 383}, {14, 1080}, {17, 3104}, {18, 3105}, {20, 5395}, {23, 7578}, {25, 275}, {30, 598}, {32, 3406}, {54, 3202}, {55, 22556}, {56, 18971}, {67, 15182}, {94, 5169}, {95, 14495}, {114, 1916}, {119, 13109}, {140, 7846}, {147, 3818}, {182, 3329}, {183, 1351}, {187, 10788}, {194, 2996}, {226, 982}, {230, 7607}, {256, 10853}, {264, 2967}, {291, 10854}, {321, 3705}, {338, 18575}, {355, 7976}, {371, 10850}, {372, 10849}, {376, 18842}, {381, 671}, {384, 9737}, {385, 576}, {402, 22698}, {427, 2052}, {459, 3168}, {485, 3102}, {486, 3103}, {493, 22709}, {494, 22710}, {524, 11167}, {538, 3545}, {542, 10033}, {546, 32448}, {574, 11170}, {631, 5188}, {698, 9764}, {726, 3817}, {730, 5587}, {732, 9765}, {736, 7775}, {801, 5020}, {842, 1316}, {858, 16311}, {946, 12782}, {1003, 21166}, {1007, 18906}, {1131, 7000}, {1132, 7374}, {1151, 10841}, {1152, 10842}, {1180, 14773}, {1340, 6039}, {1341, 6040}, {1346, 2592}, {1347, 2593}, {1350, 10007}, {1352, 7774}, {1446, 7185}, {1503, 9300}, {1506, 3399}, {1513, 3094}, {1587, 14244}, {1588, 14229}, {1656, 7868}, {1670, 1676}, {1671, 1677}, {1699, 3097}, {1751, 7413}, {1975, 10983}, {1992, 11172}, {1995, 2986}, {2021, 7737}, {2080, 7771}, {2394, 3906}, {2452, 9139}, {2552, 13580}, {2553, 13581}, {2794, 7753}, {3054, 11668}, {3055, 11669}, {3068, 22720}, {3069, 22721}, {3070, 14238}, {3071, 14234}, {3090, 3934}, {3146, 18845}, {3148, 3425}, {3314, 24206}, {3398, 7878}, {3402, 6210}, {3424, 6776}, {3529, 18843}, {3557, 6178}, {3558, 6177}, {3574, 9290}, {3618, 13354}, {3627, 32516}, {3628, 32521}, {3845, 17503}, {3851, 13108}, {3855, 32450}, {4049, 28565}, {4518, 31395}, {5013, 8719}, {5024, 14485}, {5052, 7612}, {5055, 10302}, {5056, 31276}, {5066, 11055}, {5067, 31239}, {5068, 20081}, {5071, 9466}, {5072, 32520}, {5093, 14614}, {5094, 16080}, {5097, 7766}, {5102, 8667}, {5133, 5392}, {5145, 13478}, {5171, 7824}, {5309, 9302}, {5355, 11623}, {5466, 5996}, {5490, 33344}, {5491, 33345}, {5503, 5969}, {5597, 22668}, {5598, 22672}, {5603, 14839}, {5611, 5981}, {5615, 5980}, {5965, 7837}, {5976, 8781}, {5987, 19140}, {6036, 7806}, {6055, 8587}, {6114, 7684}, {6115, 7685}, {6179, 10104}, {6234, 8842}, {6249, 32476}, {6272, 10515}, {6273, 10514}, {6309, 7764}, {6417, 10845}, {6418, 10846}, {6419, 10847}, {6420, 10848}, {6421, 10837}, {6422, 10838}, {6425, 10843}, {6426, 10844}, {6504, 6997}, {6568, 13785}, {6569, 13665}, {6625, 7385}, {7378, 8796}, {7394, 13579}, {7395, 9917}, {7507, 12143}, {7533, 13582}, {7608, 31489}, {7703, 31127}, {7741, 10079}, {7759, 31982}, {7778, 24256}, {7779, 34507}, {7782, 18502}, {7787, 13335}, {7790, 15980}, {7792, 18583}, {7804, 18860}, {7807, 10256}, {7808, 30270}, {7823, 32152}, {7840, 11178}, {7857, 20576}, {7862, 18806}, {7875, 25555}, {7921, 9863}, {7926, 9996}, {7941, 9983}, {7951, 10063}, {7989, 9902}, {8227, 12263}, {8289, 32135}, {8356, 34733}, {8722, 15482}, {8926, 17795}, {9159, 10989}, {9180, 19912}, {9307, 9747}, {9605, 32467}, {9734, 13586}, {9773, 9880}, {9781, 27375}, {10348, 26316}, {10722, 15484}, {10783, 14243}, {10784, 14228}, {10895, 18982}, {10896, 13077}, {11059, 34087}, {11156, 14036}, {11159, 12117}, {11361, 23698}, {11477, 15271}, {11602, 16809}, {11603, 16808}, {11842, 12042}, {12102, 32523}, {12150, 34473}, {12233, 13380}, {13110, 26470}, {13325, 14633}, {13326, 14632}, {13576, 17756}, {13638, 22722}, {13758, 22723}, {14223, 23350}, {14251, 14265}, {14269, 33698}, {14355, 32716}, {14534, 19544}, {17749, 21554}, {18844, 33703}, {20021, 30499}, {23235, 31859}

X(262) = midpoint of X(i) and X(j) for these {i,j}: {1, 22650}, {3, 22728}, {4, 7709}, {39, 22682}, {381, 32447}, {1699, 3097}, {1916, 9772}, {3095, 7697}, {3106, 22694}, {3107, 22693}, {22699, 22700}, {33434, 33435}
X(262) = reflection of X(i) in X(j) for these {i,j}: {1, 22475}, {4, 22682}, {76, 7697}, {376, 21163}, {6194, 15819}, {7697, 5}, {7709, 39}, {7757, 32447}, {9772, 114}, {11257, 7709}, {22676, 3}, {22677, 11261}, {22684, 33463}, {22686, 33462}, {22697, 10}, {22698, 402}, {22712, 2}, {22713, 1}, {22714, 33478}, {22715, 33479}, {22728, 14881}, {22731, 22729}, {22732, 22730}, {33706, 22712}
X(262) = isogonal conjugate of X(182)
X(262) = isotomic conjugate of X(183)
X(262) = complement of X(6194)
X(262) = anticomplement of X(15819)
X(262) = isogonal conjugate of the anticomplement of X(24206)
X(262) = isogonal conjugate of the complement of X(1352)
X(262) = isotomic conjugate of the anticomplement of X(3815)
X(262) = isotomic conjugate of the complement of X(7774)
X(262) = isotomic conjugate of the isogonal conjugate of X(263)
X(262) = isogonal conjugate of the isotomic conjugate of X(327)
X(262) = polar conjugate of X(458)
X(262) = psi-transform of X(22735)
X(262) = antigonal conjugate of isotomic conjugate of X(39099)
X(262) = perspector of ABC and Artzt triangle
X(262) = X(i)-cross conjugate of X(j) for these (i,j): {1513, 98}, {3094, 76}, {3815, 2}, {5480, 4}, {9993, 14458}
X(262) = X(i)-isoconjugate of X(j) for these (i,j): {1, 182}, {31, 183}, {32, 3403}, {48, 458}, {63, 10311}, {75, 34396}, {82, 14096}, {163, 23878}, {255, 33971}, {560, 20023}, {662, 3288}, {1933, 8842}, {6784, 24041}
X(262) = cevapoint of X(i) and X(j) for these (i,j): {2, 7774}, {3, 19139}, {6, 3148}, {11, 1491}, {1689, 1690}, {2525, 12037}, {3124, 17415}
X(262) = crosssum of X(3288) and X(6784)
X(262) = radical center of (Brocard circle reflected in BC, CA, and AB)
X(262) = pole wrt polar circle of trilinear polar of X(458)
X(262) = X(48)-isoconjugate (polar conjugate) of X(458)
X(262) = trilinear pole of line X(523)X(3569)
X(262) = pole of Lemoine axis wrt orthoptic circle of the Steiner inellipse (a.k.a. {circumcircle, nine-point circle}-inverter)
X(262) = perspector of orthoptic circle of the Steiner inellipse (a.k.a. {circumcircle, nine-point circle}-inverter)
X(262) = perspector of ABC and 2nd Neuberg triangle
X(262) = trilinear product of vertices of 2nd Neuberg triangle
X(262) = centroid of X(4)PU(1)
X(262) = Cundy-Parry Phi transform of X(83)
X(262) = Cundy-Parry Psi transform of X(39)
X(262) = barycentric product X(i)*X(j) for these {i,j}: {6, 327}, {75, 2186}, {76, 263}, {561, 3402}, {850, 26714}, {2799, 6037}
X(262) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 183}, {4, 458}, {6, 182}, {25, 10311}, {32, 34396}, {39, 14096}, {75, 3403}, {76, 20023}, {141, 14994}, {263, 6}, {327, 76}, {393, 33971}, {512, 3288}, {523, 23878}, {1916, 8842}, {2186, 1}, {2491, 9420}, {3124, 6784}, {3402, 31}, {3815, 15819}, {6037, 2966}, {7735, 9755}, {26714, 110}, {32716, 2715}
X(262) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 6194, 15819}, {2, 9748, 9752}, {2, 9752, 9754}, {2, 14853, 9753}, {2, 22503, 22677}, {3, 10796, 3972}, {3, 11272, 7786}, {4, 39, 11257}, {4, 7736, 9744}, {5, 3095, 76}, {6, 9756, 9755}, {6, 13860, 98}, {39, 31701, 22708}, {39, 31702, 22707}, {114, 19130, 13862}, {194, 3091, 6248}, {381, 11163, 6054}, {485, 3102, 19090}, {486, 3103, 19089}, {1513, 5480, 9993}, {2544, 2545, 7736}, {3090, 12251, 3934}, {3329, 5999, 182}, {3815, 5480, 1513}, {5188, 6683, 631}, {6194, 15819, 22712}, {7777, 13862, 114}, {9737, 10358, 384}, {9748, 9752, 9753}, {9752, 14853, 9748}, {9753, 9754, 9752}, {9755, 9756, 98}, {9755, 13860, 9756}, {10839, 10840, 3}, {11272, 14881, 3}, {19063, 19064, 6}, {21445, 22521, 32}, {22475, 22650, 22713}, {22726, 22727, 31958}, {22729, 22730, 1}, {22731, 22732, 22713}, {31701, 31702, 5475}


X(263) = ISOGONAL CONJUGATE OF X(183)

Trilinears    a2sec(A - ω) : :
Barycentrics    a3sec(A - ω) : :

Let V = U(2)-of-pedal-triangle-of-P(1), and let W = P(2)-of-pedal-triangle-of-U(1). Then X(263) = trilinear pole of VW. (Randy Hutson, December 26, 2015)

Let A1B1C1 and A2B2C2 be the pedal triangles of PU(1). Then X(263) is the radical center of the circumcircles of AA1A2, BB1B2, CC1C2. (Randy Hutson, July 31 2018)

X(263) lies on these lines: 2,51   6,160   69,308   184,251

X(263) = isogonal conjugate of X(183)
X(263) = isotomic conjugate of X(20023)


X(264) = ISOTOMIC CONJUGATE OF CIRCUMCENTER

Trilinears    csc A csc 2A : csc B csc 2B : csc C csc 2C
Trilinears    sec A csc2A : sec B csc2B : sec C csc2C
Trilinears    tan A csc(A - ω) : tan B csc(B - ω) : tan C csc(C - ω)
Trilinears    sec A + cot A csc A : :
Barycentrics    csc 2A : csc 2B : csc 2C
Barycentrics    1/[a2(a2 - b2 - c2)] : :
Barycentrics    tan A + cot A : :
Barycentrics    tan A - cot B - cot C + cot ω : :

Five constructions by Randy Hutson, January 29, 2015:

(1) Let A'B'C' be the tangential triangle. Let A″ be the pole, wrt the polar circle, of line B'C', and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X(264).

(2) Let A'B'C' be the symmedial triangle. Let A″ be the pole, wrt the polar circle, of line B'C', and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X(264).

(3) Let A'B'C' be the circumsymmedial triangle. Let A″ be the pole, wrt the polar circle, of line B'C', and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X(264).

(4) Let A'B'C' be the Lucas(t) central triangle (for any t). Let A″ be the pole, wrt the polar circle, of line B'C', and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X(264).

(5) X(264) is the trilinear pole of the line X(297)X(525). This line is the isotomic conjugate of the MacBeath circumconic, which is the isogonal conjugate of the orthic axis. The line is also the polar of X(6) wrt the polar circle, and the radical axis of the polar and orthosymmedial circles, and the polar conjugate of the circumcircle)

Let A' be the trilinear product of the vertices of the A-anti-altimedial triangle, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(264). (Randy Hutson, November 2, 2017)

X(264) lies on these cubics: K045, K146, K183, K184, K208, K257, K276, K342a, K342b, K494, K504, K647, K674, K675, K677, K718

X(264) lies on these lines: {2,216}, {3,95}, {4,69}, {5,1093}, {6,287}, {9,1948}, {22,1629}, {24,1078}, {25,183}, {33,350}, {34,1909}, {53,141}, {57,1947}, {75,225}, {85,309}, {92,306}, {93,1273}, {98,3425}, {99,378}, {107,1995}, {112,2367}, {157,1485}, {186,7771}, {193,3087}, {250,1316}, {253,3091}, {254,3541}, {262,2967}, {274,475}, {275,1993}, {281,344}, {298,472}, {299,473}, {300,302}, {301,303}, {305,325}, {310,4196}, {319,5081}, {320,7282}, {328,6344}, {339,381}, {379,823}, {384,1968}, {401,577}, {419,1974}, {450,5651}, {491,1585}, {492,1586}, {524,6748}, {623,6116}, {624,6117}, {801,2063}, {811,5136}, {847,1594}, {850,7703}, {1007,6340}, {1043,7513}, {1105,1593}, {1217,3088}, {1225,7809}, {1238,7796}, {1249,3618}, {1309,2861}, {1441,2476}, {1595,3933}, {1726,7094}, {1785,4357}, {1896,2478}, {1897,4360}, {1969,3262}, {1990,3589}, {2207,7770}, {2419,3267}, {2453,3447}, {2897,6840}, {2970,5094}, {3148,6394}, {3168,5943}, {3199,3934}, {3520,7782}, {3575,7750}, {3629,6749}, {3785,7487}, {5064,7788}, {5117,6374}, {5523,7790}, {6103,7806}, {6240,7802}, {6524,7392}, {6525,7398}, {6756,7767}, {7378,8024}, {7507,7773}, {7576,7811}

X(264) = reflection of X(3164) in X(216)
X(264) = isogonal conjugate of X(184)
X(264) = isotomic conjugate of X(3)
X(264) = complement of X(3164)
X(264) = anticomplement of X(216)
X(264) = X(264) = X(i)-Ceva conjugate of X(j) for these (i,j): (276,2), (1969,7017), (6528,850)
X(264) = cevapoint of X(i) and X(j) for these (i,j): (2,4), (5,324), (6,157), (92,318), (273,342), (338,523), (491,492)
X(264) = X(i)-cross conjugate of X(j) for these (i,)}: (2,76), (3,5392), (4,2052), (5,2), (30,94), (92,331), (235,459), (318,7017), (339,850), (427,4), (442,321), (523,648), (850,6528), (858,671), (1312,2593), (1313,2592), (1368,2996), (1441,75), (1591,5490), (1592,5491), (1594,275), (2072,2986), (2450,98), (2967,297), (2968,4391), (2971,2501), (2972,525), (3007,903), (3134,2394), (3136,10), (3141,4049), (3142,226), (3143,5466), (5133,83), (5169,598), (6530,6330), (6563,99)
X(264) = X(1988)-complementary conjugate of X(10)
X(264) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (54,6360), (92,2888), (95,4329), (275,8), (276,6327), (933,4560), (2148,3164), (2167,20), (2190,2)
X(264) = antipode of X(1972) in hyperbola {}A,B,C,X(2),X(69)}}
X(264) = pole of Lemoine axis wrt polar circle
X(264) = X(48)-isoconjugate (polar conjugate) of X(6)
X(264) = polar-circle inverse of X(5167)
X(264) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,324,2052), (2,3164,216), (4,69,317), (4,1235,76), (4,3186,1843), (53,141,297), (69,311,76), (69,317,340), (273,318,75), (311,3260,69)
X(264) = Brianchon point (perspector) of the MacBeath inconic
X(264) = X(i)-isoconjugate of X(j) for these {i,j}: {1,184}, {3,31}, {6,48}, {19,577}, {25,255}, {28,4055}, {32,63}, {33,7335}, {34,6056}, {41,222}, {42,1437}, {47,2351}, {51,2169}, {55,603}, {56,212}, {58,228}, {69,560}, {71,1333}, {72,2206}, {73,2194}, {77,2175}, {78,1397}, {81,2200}, {97,2179}, {109,1946}, {110,810}, {112,822}, {163,647}, {172,7116}, {213,1790}, {216,2148}, {217,2167}, {219,604}, {220,7099}, {221,2188}, {237,293}, {248,1755}, {251,4020}, {268,2199}, {283,1402}, {284,1409}, {295,2210}, {304,1501}, {305,1917}, {326,1974}, {393,4100}, {394,1973}, {418,2190}, {512,4575}, {563,2165}, {571,1820}, {607,7125}, {608,2289}, {649,906}, {652,1415}, {656,1576}, {662,3049}, {667,1331}, {669,4592}, {692,1459}, {798,4558}, {849,3690}, {895,922}, {904,3955}, {923,3292}, {1092,1096}, {1106,1260}, {1110,3937}, {1176,1964}, {1253,7053}, {1259,1395}, {1332,1919}, {1399,8606}, {1400,2193}, {1407,1802}, {1408,2318}, {1410,2328}, {1433,2187}, {1444,1918}, {1472,7085}, {1473,7084}, {1474,3990}, {1797,2251}, {1798,3725}, {1799,1923}, {1804,2212}, {1813,3063}, {1910,3289}, {1911,7193}, {1914,2196}, {1924,4563}, {1949,1951}, {1950,7107}, {1980,4561}, {2149,7117}, {2150,2197}, {2159,3284}, {2192,7114}, {2203,3682}, {2207,6507}, {2208,7078}, {2300,2359}, {7011,7118}, {7015,7122}
X(264) = X(i)-beth conjugate of X(j) for these (i,j): (264,273), (811,7), (3596,322)
X(264) = trilinear pole of the line (297,525)
X(264) = barycentric product X(i)*X(j) for these {i,j}: {1,1969}, {4,76}, {5,276}, {7,7017}, {8,331}, {19,561}, {25,1502}, {27,313}, {29,349}, {69,2052}, {75,92}, {83,1235}, {85,318}, {93,7769}, {94,340}, {95,324}, {107,3267}, {158,304}, {273,312}, {275,311}, {278,3596}, {281,6063}, {286,321}, {290,297}, {300,470}, {301,471}, {305,393}, {308,427}, {310,1826}, {317,5392}, {326,6521}, {327,458}, {341,1847}, {523,6331}, {525,6528}, {648,850}, {670,2501}, {683,5254}, {693,6335}, {811,1577}, {847,7763}, {1016,2973}, {1088,7101}, {1093,3926}, {1231,1896}, {1240,1848}, {1509,7141}, {1824,6385}, {1897,3261}, {1928,1973}, {1978,7649}, {2489,4609}, {2970,4590}, {3064,4572}, {3114,5117}, {3264,6336}, {6344,7799}, {6386,6591}
X(264) = trilinear product of PU(20) (see Tables: Bicentric Pairs)
X(264) = trilinear product X(i)*X(j) for these {i,j}: {2,92}, {4,75}, {6,1969}, {7,318}, {8,273}, {9,331}, {10,286}, {19,76}, {25,561}, {27,321}, {28,313}, {29,1441}, {33,6063}, {34,3596}, {57,7017}, {63,2052}, {69,158}, {82,1235}, {85,281}, {91,317}, {162,850}, {225,314}, {240,290}, {242,334}, {253,1895}, {274,1826}, {276,1953}, {278,312}, {279,7101}, {280,342}, {297,1821}, {304,393}, {305,1096}, {307,1896}, {309,7952}, {310,1824}, {311,2190}, {324,2167}, {326,1093}, {336,6530}, {340,2166}, {341,1119}, {346,1847}, {347,7020}, {349,1172}, {394,6521}, {419,1934}, {427,3112}, {514,6335}, {523,811}, {525,823}, {648,1577}, {653,4391}, {656,6528}, {661,6331}, {668,7649}, {693,1897}, {757,7141}, {765,2973}, {799,2501}, {873,7140}, {1088,7046}, {1118,3718}, {1240,1829}, {1446,2322}, {1494,1784}, {1502,1973}, {1748,5392}, {1783,3261}, {1857,7182}, {1861,2481}, {1928,1974}, {1947,7108}, {1948,1952}, {1978,6591}, {2333,6385}, {2489,4602}, {2580,2592}, {2581,2593}, {2969,7035}, {2997,5125}, {3064,4554}, {3113,5117}, {3926,6520}, {4358,6336}, {5342,5936}, {7009,7018}
X(264) = barycentric quotient X(i)/X(j) for these (i,j): (1,48), (2,3), (4,6), (5,216), (6,184), (7,222), (8,219), (9,212), (10,71), (19,31), (25,32), (27,58), (29,284), (33,41), (37,228), (51,217), (63,255), (69,394), (94,265), (95,97), (98,248), (107,112), (162,163), (196,221), (216,418), (232,237), (304,326), (311,343), (445,500)


X(265) = REFLECTION OF X(3) IN X(125)

Trilinears    sin 2A csc 3A : :
Trilinears    1/(4 cos A - sec A) : :
Trilinears    csc(A + π/3) - csc(A - π/3) : :
Trilinears    (cos A)/(1 - 4 cos^2 A) : :
Barycentrics    sin A sin 2A csc 3A : :
Barycentrics    (a^2 - b^2 - c^2)/[(a^2 - b^2 - c^2)^2 - b^2c^2] : :
Tripolars    a^2((a^2 - b^2 - c^2)^2 - b^2 c^2) : :
X(265) = 3 X[2] - 5 X[15081],3 X[2] - 4 X[20304],3 X[3] - 4 X[6699],2 X[3] - 5 X[15027],2 X[3] - 3 X[15061],3 X[3] - 2 X[16163],5 X[3] - 8 X[20397],3 X[4] - X[146],3 X[4] - 2 X[1539],3 X[4] + X[12317],3 X[5] - 2 X[10272],4 X[5] - 3 X[14643],2 X[5] - 3 X[14644],4 X[5] - X[23236],X[20] - 4 X[20379],3 X[51] - 2 X[11557],X[67] + 2 X[32273],X[74] - 3 X[9140],3 X[74] - 2 X[14677],2 X[74] - 3 X[20126],2 X[98] - 3 X[14849],2 X[99] - 3 X[14850],3 X[110] - 4 X[10272],X[110] - 4 X[11801],2 X[110] - 3 X[14643],X[110] - 3 X[14644],2 X[113] - 3 X[381],4 X[113] - 3 X[5655],3 X[113] - 2 X[6053],3 X[125] - 2 X[6699],4 X[125] - X[12121],2 X[125] + X[12902],4 X[125] - 5 X[15027],4 X[125] - 3 X[15061],3 X[125] - X[16163],5 X[125] - 4 X[20397],4 X[140] - 3 X[15035],4 X[140] - 5 X[15059],X[146] + 3 X[3448],2 X[146] - 3 X[7728],X[146] - 6 X[10113],3 X[381] - X[399],9 X[381] - 4 X[6053],3 X[381] - 4 X[7687],X[382] + 2 X[16003],2 X[399] - 3 X[5655],3 X[399] - 4 X[6053],X[399] - 4 X[7687],X[476] - 3 X[5627],2 X[476] - 3 X[14993],2 X[477] - 3 X[14851]

Let P = X(74), H = X(4), H' =H-of-BCP, H'' = H-of-CAP, and H''' = H-of ABP. Then X(265) is the circumcenter of the cyclic quadrilateral HH'H''H'''. (Randy Hutson, 9/23/2011)

Let A' be the reflection in line BC of the A-vertex of the tangential triangle, and define B' and C' cyclically. The circumcircles of A'BC, B'CA, and C'AB concur in X(265). Let A'' be the reflection in line BC of the A-vertex of the anticevian triangle of X(5), and define B'' and C'' cyclically. The circumcircles of AB''C'', BC''A'', CA''B'' concur in X(265). (Randy Hutson, August 26, 2014)

Let A*B*C* be the Kosnita triangle. Let A' be the orthopole of line B*C*, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(265). (Randy Hutson, August 26, 2014)

Let A'B'C' be the X(3)-Fuhrmann triangle. Let A'' be the reflection of A in line B'C', and define B'' and C'' cyclically. Then A''B''C'' is inversely similar to ABC, with similtude center X(265), and A''B''C'' is perspective to ABC with persepctor X(74). (Randy Hutson, August 26, 2014)

Let A'B'C' be the reflection triangle. Let L be the line through A' parallel to the Euler line, and define M and N cyclically. Let L' be the reflection of L in sideline BC, and define M' and N' cyclically. The lines L',M', N' concur in X(265). (Randy Hutson, August 26, 2014)

Let A'B'C' be the reflection triangle. Let A″ be the trilinear pole of line B'C', and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X(5). Let A* be the trilinear pole, wrt A'B'C', of line B″C″, and define B*, C* cyclically. The lines AA*, BB*, CC* concur in X(265). (Randy Hutson, July 20, 2016)

Let A' be the point such that triangle A'BC is directly similar to the orthic triangle, and define B', C' cyclically. The lines AA', BB', CC' concur in X(265). If 'inversely' is substituted for 'directly', the lines concur in X(3). (Randy Hutson, July 20, 2016)

Let A' be the isogonal conjugate of A wrt the A-altimedial triangle, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(265). (Randy Hutson, November 2, 2017)

Let La be the line through A parallel to the Euler line of the A-altimedial triangle, and define Lb and Lc cyclically. Lines La, Lb, Lc concur in X(265). (Randy Hutson, November 2, 2017)

Let AA1A2, BB1B2, CC1C2 be the circumcircle-inscribed equilateral triangles used in the construction of the Trinh triangle. Let A' be the crosssum of A1 and A2, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(265). (Randy Hutson, October 8, 2019)

Let OA be the circle centered at the A-vertex of the orthocentroidal triangle and passing through A; define OB and OC cyclically. X(265) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

X(265) lies on the Jerabek circumhyperbola, the Johnson circumconic (see K714), the curves K025, K060, K112, K255, K275, K300, K301, K339, K427, K449, K464, K481, K494, K497, K513, K530, K595, K596, K597, K611, K638, K668, K669, K724, K885, K929, K930, K942, Q106, Q110, Q114, Q125, and also on these lines: {2,1511}, {3,125}, {4,94}, {5,49}, {6,13}, {10,12778}, {11,10091}, {12,10088}, {20,11270}, {25,12140}, {30,74}, {32,12201}, {51,11557}, {52,6145}, {55,12334}, {56,18968}, {64,382}, {65,79}, {66,2781}, {67,511}, {68,7723}, {69,328}, {70,6243}, {71,12661}, {73,1807}, {98,14849}, {99,14850}, {111,14846}, {140,15035}, {148,18331}, {155,15317}, {156,3047}, {182,19151}, {184,10254}, {185,3521}, {186,30522}, {195,10112}, {248,10317}, {290,316}, {300,621}, {301,622}, {355,6757}, {376,20421}, {389,11562}, {402,12790}, {403,10540}, {427,15472}, {477,14851}, {493,12894}, {494,12895}, {515,13605}, {517,10693}, {518,10100}, {520,14220}, {523,15453}, {524,5505}, {526,10412}, {539,1568}, {541,3426}, {546,1173}, {547,13392}, {548,15057}, {549,15051}, {550,15055}, {576,18432}, {590,10819}, {611,32297}, {613,32243}, {615,10820}, {631,20396}, {632,15020}, {690,6321}, {759,13743}, {879,9517}, {895,3564}, {952,6742}, {974,1899}, {1117,13582}, {1147,10255}, {1154,3153}, {1175,6841}, {1176,5622}, {1177,1503}, {1204,18565}, {1312,15461}, {1313,15460}, {1351,23049}, {1352,2854}, {1353,32234}, {1411,7073}, {1469,32308}, {1478,3028}, {1479,3024}, {1531,11692}, {1594,12370}, {1614,13406}, {1656,5972}, {1657,3532}, {1658,12289}, {1853,2935}, {1885,18433}, {1994,2914}, {2070,13289}, {2072,5504}, {2574,10750}, {2575,10751}, {2772,10741}, {2773,10747}, {2774,10739}, {2775,15521}, {2776,15522}, {2779,10740}, {2780,22338}, {2782,9513}, {2836,10743}, {2842,10744}, {2850,10746}, {2888,11591}, {2930,10516}, {2937,11750}, {2948,5587}, {3043,7577}, {3056,32307}, {3060,7731}, {3068,13915}, {3069,13979}, {3090,15088}, {3091,5609}, {3146,12244}, {3154,14934}, {3410,15060}, {3519,5562}, {3520,13561}, {3526,6723}, {3527,3843}, {3530,15036}, {3531,14269}, {3541,15114}, {3543,11738}, {3545,9143}, {3548,18466}, {3549,18945}, {3567,15102}, {3574,11536}, {3583,7727}, {3589,15462}, {3627,10721}, {3628,15025}, {3629,15093}, {3657,8674}, {3839,14491}, {3845,10706}, {3851,16534}, {5050,15118}, {5055,5642}, {5072,15046}, {5073,10990}, {5076,22334}, {5079,15039}, {5093,5095}, {5097,15432}, {5102,16176}, {5446,11572}, {5447,26861}, {5448,15002}, {5462,16223}, {5480,9970}, {5576,12241}, {5597,12466}, {5598,12467}, {5621,12083}, {5648,11178}, {5886,11720}, {5889,12281}, {5890,12270}, {5891,13622}, {5900,13391}, {5944,12254}, {5946,11561}, {5987,13862}, {6000,10688}, {6146,10024}, {6214,7733}, {6215,7732}, {6221,8994}, {6284,10065}, {6334,9033}, {6398,13969}, {6413,12891}, {6414,12892}, {6528,8795}, {6593,14561}, {6639,19467}, {6640,12118}, {6643,13416}, {6776,25320}, {6798,31392}, {7354,10081}, {7387,13171}, {7422,31127}, {7426,15362}, {7507,19504}, {7512,13470}, {7517,10117}, {7528,15465}, {7547,12161}, {7579,32235}, {7583,19111}, {7584,19110}, {7689,18562}, {7693,13364}, {7699,11422}, {7978,22791}, {8044,14616}, {8200,13208}, {8207,13209}, {8220,13215}, {8221,13216}, {8571,14585}, {8612,18416}, {8673,18557}, {8976,8998}, {9818,12168}, {9826,12099}, {9833,15647}, {9919,18534}, {9955,11699}, {9996,13210}, {10095,22804}, {10097,14582}, {10152,18507}, {10201,20773}, {10246,11735}, {10263,15101}, {10293,14915}, {10575,17855}, {10663,18468}, {10664,18470}, {10752,21850}, {10767,22938}, {10796,13193}, {10942,13217}, {10943,13218}, {11061,14853}, {11138,11582}, {11139,11581}, {11250,23294}, {11251,15395}, {11262,32369}, {11411,15077}, {11442,12825}, {11449,11704}, {11457,17854}, {11459,12273}, {11499,13204}, {11563,14157}, {11564,18572}, {11585,22808}, {11694,15699}, {11709,18481}, {11723,18493}, {11746,14542}, {11807,15321}, {11818,12824}, {11898,32275}, {12038,16665}, {12111,12284}, {12162,22466}, {12165,18386}, {12192,14880}, {12278,26917}, {12359,18442}, {12429,15316}, {12584,24206}, {12773,26700}, {12811,15029}, {12828,15473}, {12888,18455}, {13160,13353}, {13201,17711}, {13291,14695}, {13292,23047}, {13293,20299}, {13340,14791}, {13352,15131}, {13403,14130}, {13598,18555}, {13603,15687}, {13951,13990}, {14128,18368}, {14216,31725}, {14452,19658}, {14487,14893}, {14639,15342}, {14708,18912}, {14859,19552}, {14919,20123}, {14940,32171}, {15021,15704}, {15042,15693}, {15136,18441}, {15357,23698}, {15538,15550}, {15740,16270}, {16010,25330}, {16168,17511}, {17040,18537}, {18281,25487}, {18376,18434}, {18414,19484}, {18415,19485}, {18462,19482}, {18463,19483}, {18489,32255}, {18570,23293}, {18909,31371}, {19402,32353}, {22586,22758}, {23329,25564}, {25335,32271}, {29012,32305}.

X(265) = midpoint of X(i) and X(j) for these {i,j}: {3, 12902}, {4, 3448}, {74, 10733}, {146, 12317}, {148, 18331}, {382, 10620}, {1351, 32306}, {3146, 12244}, {5889, 12281}, {6321, 15545}, {7731, 15100}, {10263, 15101}, {10721, 15054}, {12111, 12284}, {12293, 12302}, {12295, 16003}, {12803, 12804}, {13213, 13214}, {14508, 14989}, {21649, 21650}

X(265) = reflection of X(i) in X(j) for these {i,j}: {3, 125}, {4, 10113}, {5, 11801}, {20, 12041}, {52, 11800}, {54, 11804}, {74, 10264}, {98, 15535}, {110, 5}, {113, 7687}, {146, 1539}, {182, 20301}, {185, 11806}, {382, 12295}, {399, 113}, {477, 16340}, {1352, 32274}, {1511, 20304}, {1657, 16111}, {1986, 12236}, {3581, 3580}, {5504, 23306}, {5648, 11178}, {5655, 381}, {6102, 13358}, {7722, 6102}, {7723, 15738}, {7728, 4}, {7978, 22791}, {9833, 15647}, {9970, 5480}, {10114, 19481}, {10540, 403}, {10575, 17855}, {10620, 16003}, {10706, 3845}, {10721, 3627}, {10752, 21850}, {10767, 22938}, {11562, 389}, {11579, 25328}, {11699, 9955}, {11898, 32275}, {12041, 20379}, {12121, 3}, {12308, 15063}, {12368, 18480}, {12383, 1511}, {12584, 24206}, {12778, 10}, {12790, 402}, {12893, 5449}, {13293, 20299}, {13417, 5446}, {14157, 11563}, {14559, 14356}, {14643, 14644}, {14683, 5609}, {14934, 3154}, {14982, 3818}, {14989, 21269}, {14993, 5627}, {15131, 23325}, {16111, 20417}, {16163, 6699}, {18332, 115}, {18403, 13851}, {18436, 7723}, {18439, 12292}, {18481, 11709}, {19140, 19130}, {19506, 18383}, {20126, 9140}, {20127, 74}, {22115, 2072}, {22584, 21650}, {23236, 110}, {24981, 16534}, {25711, 11746}, {30714, 5972}, {32233, 182}, {32234, 1353}, {32609, 23515}

X(265) = isogonal conjugate of X(186)
X(265) = isotomic conjugate of X(340)
X(265) = complement of X(12383)
X(265) = anticomplement of X(1511)
X(265) = circumcircle-inverse of X(5961)
X(265) = nine-point-circle-inverse of X(15367)
X(265) = polar-circle-inverse of X(1986)
X(265) = circumcircle-of-anticomplementary-triangle-inverse of X(12317)
X(265) = polar conjugate of X(14165)
X(265) = antigonal image of X(3)
X(265) = syngonal conjugate of X(5)
X(265) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {2159, 18301}, {2166, 146}, {2349, 1272}, {5627, 8}, {11079, 6360}, {15395, 6758}
X(265) = X(i)-Ceva conjugate of X(j) for these (i,j): {94, 1989}, {1141, 5961}, {12028, 3}, {14254, 14993}, {18817, 18883}
X(265) = X(i)-cross conjugate of X(j) for these (i,j): {3, 15392}, {1531, 4846}, {3284, 2}, {13754, 3}, {13851, 4}, {14391, 648}, {18403, 3521}, {21649, 5504}, {21650, 74}
X(265) = cevapoint of X(i) and X(j) for these (i,j): {4, 6761}, {5, 30}, {125, 9033}, {184, 9380}, {520, 1650}, {621, 622}, {686, 20975}
X(265) = crosspoint of X(i) and X(j) for these (i,j): {94, 328}, {1494, 2986}
X(265) = crosssum of X(i) and X(j) for these (i,j): {3, 2931}, {526, 16186}, {1495, 3003}, {2088, 14270}
X(265) = trilinear pole of line {216, 647}
X(265) = crossdifference of every pair of points on line {526, 2081}
X(265) = X(8)-beth conjugate of X(12778)
X(265) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 186}, {1021, 2624}
X(265) = X(10412)-line conjugate of X(526)
X(265) = X(i)-vertex conjugate of X(j) for these (i,j): {3, 11744}, {186, 6344}, {1177, 5504}

X(265) = X(i)-isoconjugate of X(j) for these (i,j): {1, 186}, {4, 6149}, {19, 323}, {31, 340}, {35, 1870}, {36, 6198}, {47, 5962}, {48, 14165}, {50, 92}, {112, 32679}, {158, 22115}, {162, 526}, {240, 14355}, {275, 2290}, {320, 14975}, {470, 2152}, {471, 2151}, {562, 2964}, {648, 2624}, {661, 14590}, {811, 14270}, {860, 17104}, {1154, 2190}, {1399, 5081}, {1464, 11107}, {1577, 14591}, {1784, 14385}, {1969, 19627}, {1973, 7799}, {2148, 14918}, {2159, 14920}, {2166, 3043}, {2167, 11062}, {2174, 17923}, {2361, 7282}, {2594, 17515}, {2605, 4242}, {3268, 32676}, {8552, 24019}, {16186, 24000}

X(265) = barycentric product X(i)*X(j) for these {i,j}: {3, 94}, {6, 328}, {63, 2166}, {68, 18883}, {69, 1989}, {99, 14582}, {110, 14592}, {184, 20573}, {287, 14356}, {305, 11060}, {311, 11077}, {343, 1141}, {394, 6344}, {476, 525}, {577, 18817}, {656, 32680}, {850, 32662}, {1304, 18557}, {1807, 30690}, {2410, 14220}, {3260, 11079}, {3267, 14560}, {3519, 30529}, {3580, 12028}, {3926, 18384}, {4558, 10412}, {4563, 15475}, {5392, 5961}, {5627, 11064}, {7100, 18359}, {7799, 14595}, {10217, 11092}, {10218, 11078}, {14208, 32678}, {14254, 14919}, {14559, 14977}, {16077, 18558}

X(265) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 340}, {3, 323}, {4, 14165}, {5, 14918}, {6, 186}, {13, 471}, {14, 470}, {30, 14920}, {48, 6149}, {50, 3043}, {51, 11062}, {69, 7799}, {79, 17923}, {94, 264}, {110, 14590}, {184, 50}, {216, 1154}, {248, 14355}, {328, 76}, {343, 1273}, {476, 648}, {520, 8552}, {525, 3268}, {577, 22115}, {647, 526}, {656, 32679}, {810, 2624}, {1141, 275}, {1576, 14591}, {1807, 3219}, {1989, 4}, {2006, 7282}, {2160, 1870}, {2161, 6198}, {2165, 5962}, {2166, 92}, {2341, 11107}, {2437, 7480}, {2963, 562}, {3003, 1986}, {3049, 14270}, {3269, 16186}, {3284, 1511}, {3457, 8740}, {3458, 8739}, {4558, 10411}, {5158, 3581}, {5627, 16080}, {5961, 1993}, {6344, 2052}, {7100, 3218}, {7110, 5081}, {8014, 23714}, {8015, 23715}, {8606, 2323}, {8818, 860}, {9033, 5664}, {9380, 11597}, {10097, 9213}, {10217, 11078}, {10218, 11092}, {10412, 14618}, {11060, 25}, {11063, 2914}, {11064, 6148}, {11077, 54}, {11079, 74}, {11083, 10632}, {11088, 10633}, {12028, 2986}, {14220, 2411}, {14356, 297}, {14559, 4235}, {14560, 112}, {14575, 19627}, {14582, 523}, {14583, 1990}, {14592, 850}, {14595, 1989}, {15392, 13582}, {15451, 2081}, {15475, 2501}, {18384, 393}, {18479, 5158}, {18558, 9033}, {18817, 18027}, {18877, 14385}, {18883, 317}, {20573, 18022}, {20975, 2088}, {23968, 7473}, {30529, 32002}, {32662, 110}, {32678, 162}, {32680, 811}

X(265) = Johnson-circumconic antipode of X(110)
X(265) = perspector of ABC and 2nd isogonal triangle of X(4)
X(265) = perspector of ABC and Ehrmann side-triangle
X(265) = perspector, wrt Ehrmann side-triangle, of Ehrmann conic
X(265) = pole of line X(3)X(523) (the line of the Ehrmann cross-triangle) wrt the Ehrmann conic
X(265) = homothetic center of orthic triangle and cross-triangle of Ehrmann side- and Ehrmann vertex-triangles

X(265) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 12383, 1511}, {2, 15081, 20304}, {3, 125, 15061}, {4, 146, 1539}, {4, 12317, 146}, {4, 18300, 18576}, {4, 32140, 18439}, {5, 110, 14643}, {5, 11801, 14644}, {5, 12022, 567}, {5, 14516, 18350}, {13, 14, 1989}, {52, 18383, 31724}, {52, 31724, 15800}, {68, 18404, 18436}, {74, 9140, 10264}, {74, 10264, 20126}, {79, 80, 2166}, {94, 18300, 4}, {98, 15535, 14849}, {110, 12228, 11597}, {110, 14644, 5}, {113, 399, 5655}, {113, 7687, 381}, {113, 10111, 18445}, {113, 19481, 15087}, {125, 12902, 12121}, {125, 16163, 6699}, {125, 21659, 32607}, {146, 1539, 7728}, {146, 3448, 12317}, {381, 399, 113}, {381, 15087, 18388}, {399, 19456, 18445}, {477, 16340, 14851}, {567, 11597, 12228}, {568, 18430, 4}, {1511, 20304, 2}, {1656, 32609, 5972}, {1657, 15041, 16111}, {1986, 12236, 568}, {2931, 19457, 3}, {3060, 15100, 7731}, {3448, 10113, 7728}, {3583, 7727, 12374}, {3585, 19470, 12373}, {5449, 21659, 3}, {5622, 19138, 19129}, {5889, 18394, 18377}, {5972, 23515, 1656}, {5972, 30714, 32609}, {6102, 18379, 4}, {6344, 23956, 4}, {6564, 6565, 9220}, {6699, 16163, 3}, {7687, 10114, 18388}, {8836, 8838, 18883}, {9140, 10733, 74}, {10114, 18388, 12227}, {10264, 10733, 20127}, {11746, 25711, 16222}, {12121, 15027, 15061}, {12121, 15061, 3}, {12317, 18932, 18917}, {12319, 18933, 18531}, {12383, 15081, 2}, {12893, 32607, 3}, {13851, 21649, 19479}, {14094, 15044, 546}, {14643, 23236, 110}, {14852, 18396, 3}, {15025, 15034, 3628}, {15027, 15061, 125}, {15035, 15059, 140}, {16111, 20417, 15041}, {16770, 16771, 94}, {17838, 18451, 399}, {18390, 18474, 381}, {19051, 19052, 6}, {20126, 20127, 74}, {23515, 30714, 5972}.


X(266) = ISOGONAL CONJUGATE OF X(188)

Trilinears    sin A/2 : sin B/2 : sin C/2
Trilinears    [a/(b + c - a)]1/2 : :
Trilinears    sin A sec A/2 : :
Trilinears    cos A' : :, where A'B'C' is the excentral triangle
Trilinears    |X(1)A'| : |X(1)B'| : |X(1)C'|, where A'B'C' is the excentral triangle
Trilinears    cos(angle BIC) : cos(angle CIA) : cos(angle AIB)
Barycentrics    sin A sin A/2 : :

The trilinear polar of X(266) passes through X(6729). (Randy Hutson, October 15, 2018)

X(266) lies on these lines:1,164   56,289   174,188   259,260   361,978

X(266) = isogonal conjugate of X(188)
X(266) = eigencenter of cevian triangle of X(174)
X(266) = eigencenter of anticevian triangle of X(259)
X(266) = X(174)-Ceva conjugate of X(259)
X(266) = cevapoint of X(1) and X(361)
X(266) = X(6)-cross conjugate of X(289)
X(266) = crosspoint of X(1) and X(505)
X(266) = crosssum of X(1) and X(164)
X(266) = SS(A→A') of X(3), where A'B'C' is the excentral triangle
X(266) = trilinear square root of X(56)
X(266) = excentral-to-ABC trilinear image of X(40)
X(266) = intouch-to-ABC trilinear image of X(1)


X(267) = ISOGONAL CONJUGATE OF X(191)

Trilinears    1/[b3 + c3 - a3 + (b + c - a)(bc + ca + ab)] : :

X(267) lies on these lines: 1,229   10,191   35,37

X(267) = reflection of X(1) in X(229)
X(267) = isogonal conjugate of X(191)
X(267) = cevapoint of X(58) and X(501)
X(267) = X(58)-cross conjugate of X(1)
X(267) = isotomic conjugate of X(20932)
X(267) = trilinear pole of line X(661)X(2605)


X(268) = ISOGONAL CONJUGATE OF X(196)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A cot A/2)/(-1 - cos A + cos B + cos C)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(268) lies on these lines: 3,9   21,280   219,255   220,577   222,1073   281,1012

X(268) = isogonal conjugate of X(196)
X(268) = X(i)-cross conjugate of X(j) for these (i,j): (48,219), (55,3)
X(268) = crosssum of X(19) and X(207)
X(268) = X(92)-isoconjugate of X(221)


X(269) = ISOGONAL CONJUGATE OF X(200)

Trilinears    tan2A/2 : tan2B/2 : tan2C/2
Trilinears    [a2 - (b - c)2]2 : :
Trilinears    1 - sec^2(A/2) : :
Trilinears    (1 - cos A)/(1 + cos A) : :
Trilinears    (1 - sec A)/(1 + sec A) : :
Barycentrics    sin A tan2A/2 : :

X(269) is the vertex conjugate of the foci of the inellipse that is the trilinear square of the Gergonne line. The center of this inellipse is X(11019), and the Brianchon point (perspector) is X(1088). (Randy Hutson, October 15, 2018)

Let A1B1C1 be Gemini triangle 1. Let A' be the center of conic {A,B,C,B1,C1}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(269). (Randy Hutson, January 15, 2019)

X(269) lies on these lines: 1,7   3,939   6,57   9,241   46,1103   56,738   69,200   86,1088   106,934   142,948   273,1111   292,1020   307,936   320,326   479,614

X(269) = isogonal conjugate of X(200)
X(269) = isotomic conjugate of X(341)
X(269) = X(279)-Ceva conjugate of X(57)
X(269) = X(56)-cross conjugate of X(57)
X(269) = crosspoint of X(279) and X(479)
X(269) = crosssum of X(220) and X(480)
X(269) = {X(1),X(7)}-harmonic conjugate of X(4328)
X(269) = polar conjugate of X(7101)
X(269) = X(92)-isoconjugate of X(1802)
X(269) = trilinear square of X(57)


X(270) = ISOGONAL CONJUGATE OF X(201)

Trilinears       (sec A)/[1 + cos(B - C)] : (sec B)/[1 + cos(C - A)] : (sec C)/[1 + cos(A - B)]
Barycentrics  (tan A)/[1 + cos(B - C)] : (tan B)/[1 + cos(C - A)] : (tan C)/[1 + cos(A - B)]

X(270) lies on these lines: 4,162   27,58   28,60   29,283   759,933

X(270) = isogonal conjugate of X(201)
X(270) = X(250)-Ceva conjugate of X(162)
X(270) = cevapoint of X(28) and X(58)
X(270) = X(58)-cross conjugate of X(60)
X(270) = pole wrt polar circle of trilinear polar of X(6358) (line X(4036)X(4064))
X(270) = polar conjugate of X(6358)


X(271) = ISOGONAL CONJUGATE OF X(208)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cot A cot A/2)/(-1 - cos A + cos B + cos C)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(271) lies on these lines: 2,1034   8,20   78,394   200,255   282,283

X(271) = isogonal conjugate of X(208)
X(271) = isotomic conjugate of X(342)
X(271) = X(i)-cross conjugate of X(j) for these (i,j): (3,78), (9,63)


X(272) = ISOGONAL CONJUGATE OF X(209)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where
                        f(A,B,C) = 1/[(sin A + sin(A - B) + sin(A - C))(sin B + sin C)]

Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(272) lies on these lines: 2,284   7,58   21,75   28,273   60,86   261,310   1014,1088

X(272) = isogonal conjugate of X(209)
X(272) = X(3)-cross conjugate of X(81)


X(273) = ISOGONAL CONJUGATE OF X(212)

Trilinears       sec A sec2(A/2) : sec B sec2(B/2) : sec C sec2(C/2)
                        = (1- sec A)csc2A : (1 - sec B)csc2B : (1 - sec C)csc2C

Barycentrics  tan A sec2(A/2) : tan B sec2(B/2) : tan C sec2(C/2)

X(273) lies on these lines: 2,92   4,7   19,653   27,57   28,272   29,34   53,1086   75,225   78,322   108,675   226,469   269,1111   317,320   458,894

X(273) = isogonal conjugate of X(212)
X(273) = isotomic conjugate of X(78)
X(273) = X(i)-Ceva conjugate of X(j) for these (i,j): (264,342), (286,7), (331,92)
X(273) = cevapoint of X(i) and X(j) for these (i,j): (4,278), (34,57)
X(273) = X(i)-cross conjugate of X(j) for these (i,j): (4,92), (57,85), (225,278)
X(273) = trilinear pole of line X(514)X(3064) (polar of X(9) wrt polar circle)
X(273) = pole wrt polar circle of trilinear polar of X(9) (line X(650)X(663))
X(273) = polar conjugate of X(9)


X(274) = ISOGONAL CONJUGATE OF X(213)

Trilinears    b2c2/(b + c) : :
Trilinears    [a csc(A - ω)]/(b + c) : :
Barycentrics    bc/(b + c) : ca/(c + a) : ab/(a + b)

Let Ab, Ac, Bc, Ba, Ca, Cb be as in the construction of the Conway circle; see http://mathworld.wolfram.com/ConwayCircle.html. Let Oa be the circumcircle of ABaCa, and define Ob and Oc cyclically. Then X(274) is the radical center of Oa, Ob, Oc. see also X(8) and X(21). (Randy Hutson, April 9, 2016)

X(274) lies on these lines: 1,75   2,39   7,959   10,291   21,99   28,242   57,85   58,870   69,443   81,239   88,799   110,767   183,474   213,894   264,475   278,331   315,377   325,442   961,1014

X(274) = isogonal conjugate of X(213)
X(274) = isotomic conjugate of X(37)
X(274) = complement of X(1655)
X(274) = X(310)-Ceva conjugate of X(314)
X(274) = cevapoint of X(i) and X(j) for these (i,j): (2,75), (85,348), (86,333)
X(274) = X(i)-cross conjugate of X(j) for these (i,j): (2,86), (75,310), (81,286), (333,314)
X(274) = crossdifference of every pair of points on line X(669)X(798)
X(274) = trilinear pole of line X(320)X(350) (anticomplement of antiorthic axis)
X(274) = pole wrt polar circle of trilinear polar of X(1824)
X(274) = X(48)-isoconjugate (polar conjugate) of X(1824)
X(274) = trilinear product of vertices of Gemini triangle 1
X(274) = trilinear product of vertices of Gemini triangle 2
X(274) = trilinear product of vertices of Gemini triangle 3
X(274) = trilinear product of vertices of Gemini triangle 4


X(275) = CEVAPOINT OF ORTHOCENTER AND SYMMEDIAN POINT

Trilinears       csc 2A sec(B - C) : csc 2B sec(C - A) : csc 2C sec(A - B)
Trilinears    sec(A + T) : :, T as at X(389)
Barycentrics  sec A sec(B - C) : sec B sec(C - A) : sec C sec(A - B)
Barycentrics    1/((a^2 - b^2 - c^2) (a^2 (b^2 + c^2) - (b^2 - c^2)^2)) : :

Let A'B'C' be the circumorthic triangle. Let A″ be the trilinear pole of line B'C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(275). (Randy Hutson, June 7, 2019)

X(275) lies on these lines: 2,95   4,54   13,472   14,473   17,471   18,470   25,262   51,107   53,288   76,276   83,297   94,324   98,427

X(275) = isogonal conjugate of X(216)
X(275) = isotomic conjugate of X(343)
X(275) = X(276)-Ceva conjugate of X(95)
X(275) = cevapoint of X(4) and X(6)
X(275) = X(i)-cross conjugate of X(j) for these (i,j): (6,54), (54,95)
X(275) = crosssum of X(217) and X(418)
X(275) = trilinear pole of line X(186)X(523) (the polar of X(4) wrt the circumcircle, and the polar of X(5) wrt polar circle)
X(275) = crosspoint wrt tangential triangle of X(4) and X(6)
X(275) = intersection of tangents at X(4) and X(6) to bianticevian conic of X(4) and X(6)
X(275) = pole wrt polar circle of trilinear polar of X(5) (line X(2081)X(2600))
X(275) = polar conjugate of X(5)
X(275) = X(163)-isoconjugate of X(6368)
X(275) = barycentric product of circumcircle intercepts of line X(340)X(520)


X(276) = ISOGONAL CONJUGATE OF X(217)

Trilinears       csc3A sec A sec(B - C) : csc3B sec B sec(C - A) : csc3C sec C sec(A - B)
Barycentrics  csc2A sec A sec(B - C) : csc2B sec B sec(C - A) : csc2C sec C sec(A - B)

X(276) lies on these lines: 3,95   4,327   54,290   76,275   97,401

X(276) = isogonal conjugate of X(217)
X(276) = isotomic conjugate of X(216)
X(276) = cevapoint of X(i) and X(j) for these (i,j): (2,264), (95,275)
X(276) = X(i)-cross conjugate of X(j) for these (i,j): (2,95), (401,290) X(276) = trilinear pole of line X(216)X(647)
X(276) = inverse-in-Kiepert-hyperbola of X(1989)
X(276) = {X(13),X(14)}-harmonic conjugate of X(1989)
X(276) = X(92)-isoconjugate of X(50)
X(276) = Hofstadter -2 point
X(276) = trilinear pole of line X(340)X(520) (the isotomic conjugate of the Johnson circumconic, and the polar of X(51) wrt polar circle)
X(276) = polar conjugate of X(51)


X(277) = ISOGONAL CONJUGATE OF X(218)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where
                        f(A,B,C) = [sec2(A/2)]/[- cos4A/2 + cos4B/2 + cos4C/2]
Barycentrics    1/(a^2 + b^2 + c^2 - 2 a b - 2 a c) : :

X(277) lies on these lines: 1,142   3,105   7,218   57,169   220,1086   241,278   942,1002

X(277) = isogonal conjugate of X(218)
X(277) = isotomic conjugate of X(344)
X(277) = X(55)-cross conjugate of X(7)
X(277) = trilinear pole of de Longchamps line of intouch triangle


X(278) = ISOGONAL CONJUGATE OF X(219)

Trilinears    sec A tan A/2 : sec B tan B/2 : sec C tan C/2
Trilinears    csc A - 2 csc 2A : csc B - 2 csc 2B : csc C - 2 csc 2C
Trilinears    (1 - sec A)/a : (1 - sec B)/b : (1 - sec C)/c
Trilinears    bc/[(b + c - a)(b2 + c2 - a2)]

Barycentrics    tan A tan A/2 : tan B tan B/2 : tan C tan C/2
Barycentrics    1 - sec A : 1 - sec B : 1 - sec C

Let A'B'C' be the extouch triangle. Let A″ be the pole, wrt the polar circle, of line B'C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(278). (Randy Hutson, September 14, 2016)

Let A'B'C' be the 2nd extouch triangle. Let A″ be the pole, wrt the polar circle, of line B'C', and define B″ and C″ cyclically. The triangle A'B'C' is homothetic to ABC at X(278). (Randy Hutson, September 14, 2016)

Let Ha be the hyperbola passing through A, and with foci at B and C. This is called the A-Soddy hyperbola in (Paul Yiu, Introduction to the Geometry of the Triangle, 2002; 12.4 The Soddy hyperbolas, p. 143). Let La be the polar of X(3) with respect to Ha. Define Lb and Lc cyclically. Let A' = Lb∩Lc, B' = Lc∩La, C' = La∩Lb. Triangle A'B'C' is homothetic to ABC, and the center of homothety is X(278). (Randy Hutson, September 14, 2016)

X(278) lies on these lines: 1,4   2,92   7,27   19,57   25,105   28,56   65,387   88,653   109,917   219,329   240,982   241,277   242,459   274,331   354,955   393,1108   412,962   443,1038   614,1096

X(278) = isogonal conjugate of X(219)
X(278) = isotomic conjugate of X(345)
X(278) = X(i)-Ceva conjugate of X(j) for these (i,j): (27,57), (92,196), (273,4), (331,7)
X(278) = cevapoint of X(19) and X(34)
X(278) = X(i)-cross conjugate of X(j) for these (i,j): (19,4), (56,7), (225,273)
X(278) = trilinear pole of line X(513)X(1835) (the polar of X(8) wrt polar circle, and inverse-in-polar-circle of Fuhrmann circle)
X(278) = pole wrt polar circle of trilinear polar of X(8) (line X(522)X(650))
X(278) = X(48)-isoconjugate (polar conjugate) of X(8)
X(278) = {X(2),X(92)}-harmonic conjugate of X(281)
X(278) = X(19)-isoconjugate of X(1259)
X(278) = X(92)-isoconjugate of X(6056)
X(278) = vertex conjugate of foci of inconic that is the polar conjugate of the isogonal conjugate of the incircle
X(278) = perspector of Gemini triangle 38 and cross-triangle of ABC and Gemini triangle 38
X(278) = trilinear pole of line X(513)X(1835) (the perspectrix of ABC and Gemini triangle 37)
X(278) = Brianchon point (perspector) of inconic that is the polar conjugate of the isotomic conjugate of the incircle


X(279) = ISOGONAL CONJUGATE OF X(220)

Trilinears    csc A tan2A/2 : csc B tan2B/2 : csc C tan2C/2
Barycentrics    tan2A/2 : tan2B/2 : tan2C/2
Barycentrics    (a + b - c)^2*(a - b + c)^2 : :
X(279) = 2*s^2*X[1] - (r + 4 R)^2*X[7]

X(279) is the Brianchon point (perspector) of the inellipse that is the barycentric square of the Gergonne line. The center of the inellipse is X(4000). (Randy Hutson, October 15, 2018)

Let A1B1C1 and A2B2C2 be the 1st and 2nd Conway triangles. Let A' be the barycentric product A1*A2, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(279). (Randy Hutson, July 11, 2019)

An exarc circle is a circle tangent to two sides of a triangle ABC and externally tangent to the circumcircle of ABC. The point whose distance to the sides BC, CA, AB are proportional to the respective radii of the exarc circles is X(279). The point having distances to the sides proportional to the radii of the inarc circles is X(7). See Martin Lukarevski, "Exarc radii and the Finsler-Hadwiger inequality", The Mathematical Gazette 106, issue 565, March 2022, pp. 138-143.

X(279) lies on the circumconic {A, B, C, X(1), X(2)}}, the cubics K041, K360, K605, K623, K1011, K1044, K1189, and these lines: {1, 7}, {2, 85}, {4, 1565}, {6, 1170}, {8, 7273}, {10, 31994}, {11, 15511}, {28, 1014}, {56, 105}, {57, 479}, {65, 1002}, {69, 1257}, {75, 1219}, {76, 32017}, {81, 1407}, {88, 658}, {100, 2377}, {109, 2369}, {144, 220}, {145, 664}, {165, 3599}, {196, 37378}, {218, 651}, {222, 2982}, {226, 5308}, {253, 7219}, {273, 1440}, {274, 16713}, {277, 8732}, {278, 1847}, {280, 17880}, {291, 1254}, {294, 4209}, {304, 346}, {307, 5232}, {329, 25930}, {330, 7209}, {331, 13149}, {345, 21605}, {388, 1390}, {394, 9965}, {519, 25718}, {553, 39948}, {555, 16015}, {604, 28079}, {644, 28981}, {668, 6552}, {764, 43930}, {938, 34059}, {942, 955}, {959, 24471}, {961, 6046}, {985, 1106}, {1068, 38461}, {1086, 4626}, {1111, 3086}, {1146, 10405}, {1214, 8813}, {1224, 19855}, {1255, 6354}, {1275, 5376}, {1319, 24796}, {1400, 39970}, {1406, 38877}, {1418, 4000}, {1432, 41777}, {1439, 18732}, {1445, 16572}, {1447, 5265}, {1461, 2224}, {1462, 16502}, {1617, 40154}, {1788, 43037}, {1997, 33780}, {2006, 43047}, {2340, 36854}, {2355, 42382}, {2401, 24002}, {2724, 24016}, {2898, 5274}, {3008, 5435}, {3091, 17181}, {3146, 4872}, {3210, 39696}, {3218, 7183}, {3227, 4569}, {3241, 25716}, {3323, 18343}, {3361, 10521}, {3474, 30623}, {3476, 24797}, {3485, 4059}, {3501, 6168}, {3616, 32086}, {3617, 33298}, {3623, 32007}, {3673, 14986}, {3676, 21132}, {3752, 44794}, {3767, 4403}, {3875, 9797}, {3911, 39963}, {3926, 20924}, {4099, 4552}, {4188, 6516}, {4452, 6553}, {4454, 40862}, {4488, 10029}, {4554, 18135}, {4617, 34051}, {4644, 6610}, {5226, 29571}, {5252, 24798}, {5261, 7179}, {5281, 9446}, {5526, 41563}, {5905, 34401}, {6063, 30710}, {6356, 37179}, {6603, 20059}, {6743, 32099}, {6904, 41826}, {6995, 39732}, {7080, 30806}, {7132, 7175}, {7147, 43071}, {7181, 7288}, {7182, 20911}, {7270, 10513}, {7371, 10490}, {7674, 8271}, {8232, 16601}, {8809, 10429}, {9305, 28071}, {9316, 9441}, {9442, 43750}, {9445, 31526}, {10025, 26658}, {10529, 38468}, {10939, 31391}, {11019, 31527}, {11433, 20211}, {12447, 17272}, {14878, 40293}, {15634, 18328}, {16705, 37870}, {17080, 37597}, {17084, 30571}, {17213, 28074}, {17863, 33673}, {18624, 40940}, {18625, 37887}, {19605, 32446}, {19789, 35058}, {20070, 38866}, {20111, 40868}, {20247, 35312}, {20925, 32834}, {23978, 44190}, {23983, 34403}, {24181, 30379}, {24858, 41803}, {25723, 38314}, {26125, 32009}, {27253, 40779}, {30545, 39703}, {30699, 39694}, {30719, 37626}, {32020, 46406}, {32624, 36152}, {32836, 33939}, {32841, 32851}, {33935, 40892}, {34018, 36838}, {34578, 37771}, {35094, 35158}, {36079, 41905}, {36603, 36621}, {37579, 38900}, {37758, 40014}, {43928, 43932}

X(279) = midpoint of X(25718) and X(32003)
X(279) = reflection of X(30695) in X(6554)
X(279) = isogonal conjugate of X(220)
X(279) = isotomic conjugate of X(346)
X(279) = complement of X(30695)
X(279) = anticomplement of X(6554)
X(279) = anticomplement of the isotomic conjugate of X(30705)
X(279) = complement of the isotomic conjugate of X(42483)
X(279) = isogonal conjugate of the anticomplement of X(21258)
X(279) = isogonal conjugate of the complement of X(6604)
X(279) = isotomic conjugate of the anticomplement of X(4000)
X(279) = isotomic conjugate of the complement of X(4452)
X(279) = isotomic conjugate of the isogonal conjugate of X(1407)
X(279) = isotomic conjugate of the polar conjugate of X(1119)
X(279) = polar conjugate of the isotomic conjugate of X(7056)
X(279) = polar conjugate of the isogonal conjugate of X(7053)
X(279) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1037, 329}, {7084, 30695}, {7131, 3436}, {8269, 20295}, {8817, 21286}, {30705, 6327}
X(279) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 17113}, {2125, 1329}, {8917, 141}, {42483, 2887}
X(279) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 17113}, {7, 9533}, {85, 14256}, {1088, 7}, {1275, 658}, {1434, 7177}, {4569, 43932}, {4626, 3676}, {4637, 17096}, {10509, 57}, {13149, 24002}, {17093, 8732}, {23062, 479}, {23586, 4626}, {30705, 2}
X(279) = X(i)-cross conjugate of X(j) for these (i,j): {6, 40154}, {19, 39732}, {57, 7}, {269, 479}, {278, 1440}, {514, 36118}, {1086, 3676}, {1407, 1119}, {1418, 57}, {1427, 269}, {2260, 39734}, {3668, 1088}, {3669, 934}, {3676, 4626}, {3752, 19604}, {3942, 7192}, {4000, 2}, {6084, 927}, {7053, 7056}, {7216, 4566}, {11051, 10307}, {14596, 7371}, {14936, 513}, {18725, 189}, {23653, 6}, {24177, 75}, {30719, 664}, {37642, 30712}, {40133, 1}, {40940, 86}, {42754, 6548}, {43044, 43736}, {43049, 651}, {43064, 34056}, {43932, 4569}, {45227, 1170}
X(279) = cevapoint of X(i) and X(j) for these (i,j): {1, 16572}, {2, 4452}, {6, 1617}, {7, 5435}, {57, 269}, {513, 14936}, {514, 1565}, {1086, 3676}, {1358, 3669}, {1407, 7053}, {1427, 3668}
X(279) = crosspoint of X(i) and X(j) for these (i,j): {2, 42483}, {7, 36620}, {658, 1275}, {1088, 23062}, {4626, 23586}
X(279) = crosssum of X(i) and X(j) for these (i,j): {1, 170}, {2, 46706}, {6, 1615}, {657, 14936}, {1253, 6602}, {2310, 6608}, {3119, 4171}, {4105, 35508}, {8012, 8551}
X(279) = trilinear pole of line {513, 676}
X(279) = crossdifference of every pair of points on line {657, 4105}
X(279) = X(23618)-aleph conjugate of X(144)
X(279) = X(i)-beth conjugate of X(j) for these (i,j): {99, 6604}, {274, 348}, {1414, 56}
X(279) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 220}, {513, 657}
X(279) = trilinear pole of line X(513)X(676) (the radical axis of circumcircle and incircle)
X(279) = {X(175),X(176)}-harmonic conjugate of X(390)
X(279) = medial-isotomic conjugate of X(17113)
X(279) = barycentric square of X(7)
X(279) = X(i)-isoconjugate of X(j) for these (i,j): {1, 220}, {2, 1253}, {3, 7079}, {4, 1802}, {6, 200}, {7, 6602}, {8, 41}, {9, 55}, {19, 1260}, {21, 1334}, {25, 3692}, {31, 346}, {32, 341}, {33, 219}, {37, 2328}, {40, 7367}, {42, 2287}, {48, 7046}, {56, 728}, {57, 480}, {58, 4515}, {59, 3119}, {63, 7071}, {71, 4183}, {72, 2332}, {75, 14827}, {78, 607}, {84, 7368}, {100, 657}, {101, 3900}, {109, 4130}, {110, 4171}, {184, 7101}, {190, 8641}, {210, 284}, {212, 281}, {213, 1043}, {228, 2322}, {259, 6726}, {268, 40971}, {282, 7074}, {294, 2340}, {312, 2175}, {345, 2212}, {512, 7259}, {604, 5423}, {643, 3709}, {644, 663}, {649, 4578}, {650, 3939}, {651, 4105}, {662, 4524}, {667, 6558}, {669, 7258}, {672, 28071}, {677, 46392}, {692, 3239}, {756, 7054}, {765, 14936}, {798, 7256}, {872, 7058}, {901, 14427}, {949, 28043}, {1018, 21789}, {1021, 4557}, {1037, 28070}, {1098, 1500}, {1110, 1146}, {1172, 2318}, {1174, 3059}, {1212, 10482}, {1252, 2310}, {1261, 2347}, {1262, 24010}, {1265, 1973}, {1275, 24012}, {1333, 4082}, {1395, 30681}, {1397, 30693}, {1415, 4163}, {1792, 2333}, {1824, 2327}, {1857, 2289}, {2053, 3208}, {2149, 4081}, {2150, 6057}, {2170, 6065}, {2171, 6061}, {2185, 7064}, {2192, 2324}, {2194, 2321}, {2195, 3693}, {2204, 3710}, {2223, 6559}, {2258, 3713}, {2293, 6605}, {2299, 3694}, {2311, 4433}, {2316, 3689}, {2326, 3690}, {2338, 41339}, {2343, 10382}, {2344, 4517}, {2346, 8012}, {2361, 36910}, {2364, 3711}, {2389, 26722}, {3022, 4564}, {3063, 3699}, {3445, 4936}, {3596, 9447}, {3683, 33635}, {3684, 7077}, {3719, 6059}, {3975, 18265}, {4041, 5546}, {4069, 7252}, {4148, 34067}, {4258, 4866}, {4319, 7123}, {4397, 32739}, {4512, 34820}, {4513, 9439}, {4528, 32665}, {4546, 34080}, {4570, 36197}, {4587, 18344}, {4814, 5549}, {4827, 8694}, {4845, 6603}, {4858, 6066}, {4895, 5548}, {5526, 42064}, {6554, 7084}, {6555, 38266}, {6745, 18889}, {7045, 35508}, {7070, 30457}, {7080, 7118}, {7162, 32561}, {8551, 21453}, {9448, 28659}, {23970, 24027}, {23990, 24026}, {36627, 38293}, {36628, 38285}, {36802, 46388}
X(279) = barycentric product X(i)*X(j) for these {i,j}: {1, 1088}, {4, 7056}, {7, 7}, {8, 479}, {9, 23062}, {12, 552}, {34, 7182}, {56, 6063}, {57, 85}, {63, 1847}, {69, 1119}, {75, 269}, {76, 1407}, {77, 273}, {81, 1446}, {86, 3668}, {92, 7177}, {142, 10509}, {174, 555}, {189, 14256}, {196, 34400}, {222, 331}, {226, 1434}, {241, 34018}, {261, 6046}, {264, 7053}, {274, 1427}, {277, 17093}, {278, 348}, {281, 30682}, {286, 1439}, {304, 1435}, {305, 1398}, {310, 1042}, {312, 738}, {347, 1440}, {349, 1412}, {354, 42311}, {513, 4569}, {514, 658}, {522, 4626}, {523, 4616}, {561, 1106}, {604, 20567}, {649, 46406}, {650, 36838}, {651, 24002}, {661, 4635}, {664, 3676}, {668, 43932}, {670, 7250}, {693, 934}, {799, 7216}, {870, 7204}, {873, 1254}, {883, 43930}, {905, 13149}, {927, 43042}, {1014, 1441}, {1020, 7199}, {1086, 1275}, {1111, 7045}, {1118, 7055}, {1146, 23586}, {1231, 1396}, {1262, 23989}, {1358, 4998}, {1365, 7340}, {1397, 41283}, {1414, 4077}, {1418, 31618}, {1422, 40702}, {1423, 7209}, {1431, 7205}, {1432, 7196}, {1443, 18815}, {1447, 7233}, {1461, 3261}, {1462, 40704}, {1509, 6354}, {1577, 4637}, {1969, 7099}, {2006, 17078}, {2310, 24011}, {2400, 23973}, {2481, 34855}, {3160, 36620}, {3596, 7023}, {3669, 4554}, {4000, 30705}, {4017, 4625}, {4025, 36118}, {4076, 41292}, {4146, 7371}, {4306, 15467}, {4391, 4617}, {4552, 17096}, {4566, 7192}, {4572, 43924}, {4573, 7178}, {4619, 23100}, {4624, 30723}, {4860, 18810}, {5435, 27818}, {6049, 16078}, {6604, 40154}, {6611, 44190}, {6614, 35519}, {7002, 7022}, {7143, 18021}, {7153, 30545}, {7176, 7249}, {7195, 8817}, {7197, 30479}, {7339, 34387}, {7341, 34388}, {7366, 28659}, {8809, 33673}, {9533, 10405}, {10481, 21453}, {13436, 13459}, {13437, 13453}, {15413, 32714}, {15728, 38468}, {17106, 44186}, {17107, 21609}, {17113, 42483}, {17758, 33765}, {18886, 21456}, {19604, 39126}, {20618, 46103}, {23971, 23978}, {24013, 24026}, {24471, 31643}, {27475, 42309}, {30379, 43762}, {31526, 43750}, {34056, 37780}, {34521, 34522}, {34578, 37757}, {38859, 40216}, {41280, 41287}, {41281, 41289}, {41284, 41285}, {41286, 41290}, {43983, 44794}
X(279) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 200}, {2, 346}, {3, 1260}, {4, 7046}, {6, 220}, {7, 8}, {8, 5423}, {9, 728}, {10, 4082}, {11, 4081}, {12, 6057}, {19, 7079}, {25, 7071}, {27, 2322}, {28, 4183}, {31, 1253}, {32, 14827}, {34, 33}, {37, 4515}, {41, 6602}, {48, 1802}, {55, 480}, {56, 55}, {57, 9}, {58, 2328}, {59, 6065}, {60, 6061}, {63, 3692}, {65, 210}, {69, 1265}, {73, 2318}, {75, 341}, {77, 78}, {81, 2287}, {85, 312}, {86, 1043}, {92, 7101}, {99, 7256}, {100, 4578}, {105, 28071}, {109, 3939}, {145, 6555}, {174, 6731}, {181, 7064}, {190, 6558}, {198, 7368}, {208, 40971}, {221, 7074}, {222, 219}, {223, 2324}, {226, 2321}, {241, 3693}, {244, 2310}, {266, 6726}, {269, 1}, {273, 318}, {278, 281}, {307, 3710}, {312, 30693}, {331, 7017}, {345, 30681}, {347, 7080}, {348, 345}, {349, 30713}, {354, 3059}, {388, 3974}, {390, 28057}, {479, 7}, {497, 4012}, {512, 4524}, {513, 3900}, {514, 3239}, {522, 4163}, {552, 261}, {553, 3686}, {555, 556}, {593, 7054}, {603, 212}, {604, 41}, {608, 607}, {614, 4319}, {649, 657}, {650, 4130}, {651, 644}, {658, 190}, {661, 4171}, {662, 7259}, {663, 4105}, {664, 3699}, {667, 8641}, {673, 6559}, {693, 4397}, {738, 57}, {757, 1098}, {799, 7258}, {812, 4148}, {900, 4528}, {927, 36802}, {934, 100}, {940, 3713}, {982, 4073}, {1014, 21}, {1015, 14936}, {1019, 1021}, {1020, 1018}, {1042, 42}, {1086, 1146}, {1088, 75}, {1106, 31}, {1111, 24026}, {1118, 1857}, {1119, 4}, {1122, 3057}, {1146, 23970}, {1170, 6605}, {1212, 45791}, {1214, 3694}, {1254, 756}, {1262, 1252}, {1275, 1016}, {1284, 4433}, {1317, 4152}, {1319, 3689}, {1323, 6745}, {1354, 6062}, {1355, 7062}, {1356, 7063}, {1357, 3271}, {1358, 11}, {1363, 7065}, {1365, 4092}, {1366, 7067}, {1367, 7068}, {1393, 7069}, {1394, 7070}, {1395, 2212}, {1396, 1172}, {1397, 2175}, {1398, 25}, {1400, 1334}, {1401, 3688}, {1407, 6}, {1408, 2194}, {1410, 228}, {1412, 284}, {1413, 2192}, {1414, 643}, {1416, 2195}, {1418, 1212}, {1420, 3158}, {1422, 282}, {1423, 3208}, {1424, 7075}, {1425, 3690}, {1426, 1824}, {1427, 37}, {1429, 3684}, {1434, 333}, {1435, 19}, {1436, 7367}, {1439, 72}, {1440, 280}, {1441, 3701}, {1442, 4420}, {1443, 4511}, {1444, 1792}, {1446, 321}, {1447, 3685}, {1456, 41339}, {1458, 2340}, {1461, 101}, {1462, 294}, {1467, 10382}, {1469, 4517}, {1474, 2332}, {1475, 8012}, {1476, 1261}, {1509, 7058}, {1565, 2968}, {1617, 6600}, {1635, 14427}, {1743, 4936}, {1790, 2327}, {1804, 1259}, {1813, 4587}, {1847, 92}, {1851, 1863}, {2006, 36910}, {2082, 28070}, {2091, 42017}, {2099, 3711}, {2170, 3119}, {2263, 28043}, {2310, 24010}, {2488, 6607}, {2969, 42069}, {2973, 21666}, {3125, 36197}, {3212, 27538}, {3213, 7156}, {3271, 3022}, {3321, 6068}, {3338, 42012}, {3361, 4512}, {3598, 390}, {3600, 7172}, {3649, 4046}, {3660, 15733}, {3663, 6736}, {3664, 6737}, {3665, 3703}, {3666, 3965}, {3667, 4546}, {3668, 10}, {3669, 650}, {3671, 4061}, {3674, 3687}, {3676, 522}, {3733, 21789}, {3756, 4953}, {3911, 2325}, {3937, 3270}, {3942, 34591}, {3945, 20007}, {3982, 4060}, {4000, 6554}, {4017, 4041}, {4031, 3707}, {4032, 4095}, {4059, 3706}, {4077, 4086}, {4146, 7027}, {4306, 3190}, {4320, 612}, {4328, 4882}, {4331, 28118}, {4341, 3811}, {4350, 3870}, {4367, 4477}, {4369, 4529}, {4373, 6556}, {4452, 6552}, {4551, 4069}, {4552, 30730}, {4554, 646}, {4565, 5546}, {4566, 3952}, {4569, 668}, {4573, 645}, {4605, 4103}, {4616, 99}, {4617, 651}, {4625, 7257}, {4626, 664}, {4635, 799}, {4637, 662}, {4654, 4007}, {4790, 4827}, {4860, 42014}, {4955, 4113}, {4977, 4990}, {4998, 4076}, {5088, 7360}, {5173, 40659}, {5219, 4873}, {5221, 3715}, {5228, 37658}, {5435, 3161}, {5573, 4907}, {6046, 12}, {6049, 15519}, {6063, 3596}, {6180, 4513}, {6354, 594}, {6356, 3695}, {6357, 7359}, {6359, 7283}, {6516, 4571}, {6545, 42462}, {6609, 1604}, {6610, 6603}, {6611, 198}, {6612, 1436}, {6613, 8706}, {6614, 109}, {7023, 56}, {7045, 765}, {7053, 3}, {7055, 1264}, {7056, 69}, {7099, 48}, {7103, 7102}, {7125, 2289}, {7143, 181}, {7147, 2171}, {7153, 2319}, {7175, 2329}, {7176, 7081}, {7177, 63}, {7178, 3700}, {7179, 3790}, {7180, 3709}, {7181, 3712}, {7182, 3718}, {7183, 3719}, {7185, 3705}, {7192, 7253}, {7195, 497}, {7196, 17787}, {7197, 388}, {7198, 4030}, {7203, 3737}, {7204, 984}, {7209, 27424}, {7210, 4123}, {7214, 4157}, {7216, 661}, {7217, 4178}, {7233, 4518}, {7248, 3056}, {7249, 4451}, {7250, 512}, {7251, 4548}, {7254, 23090}, {7271, 4853}, {7273, 7322}, {7314, 6058}, {7316, 5547}, {7318, 36626}, {7335, 6056}, {7336, 5532}, {7337, 6059}, {7338, 6060}, {7339, 59}, {7340, 6064}, {7341, 60}, {7365, 2345}, {7366, 604}, {7370, 259}, {7371, 188}, {8712, 40137}, {8809, 44692}, {8810, 8805}, {9436, 3717}, {9533, 144}, {10004, 29616}, {10030, 3975}, {10481, 4847}, {10509, 32008}, {10521, 40998}, {13149, 6335}, {13436, 13458}, {13437, 13454}, {13438, 13456}, {13453, 13425}, {13459, 13426}, {13460, 13427}, {14027, 4542}, {14189, 28058}, {14256, 329}, {14413, 14392}, {14522, 23244}, {14596, 16016}, {14936, 35508}, {15413, 15416}, {15419, 15411}, {15728, 34894}, {16079, 33963}, {16502, 30706}, {16572, 24771}, {16609, 3985}, {16662, 30413}, {16663, 30412}, {16888, 4136}, {17078, 32851}, {17079, 28808}, {17090, 4903}, {17092, 25082}, {17093, 344}, {17095, 42033}, {17096, 4560}, {17106, 165}, {17113, 30695}, {17114, 23638}, {17925, 17926}, {18033, 4087}, {18623, 27382}, {19604, 3680}, {20229, 8551}, {20567, 28659}, {20617, 14973}, {20618, 26942}, {21132, 23615}, {21314, 5231}, {21454, 391}, {22464, 6735}, {23062, 85}, {23586, 1275}, {23971, 1262}, {23973, 2398}, {23979, 23990}, {23989, 23978}, {24002, 4391}, {24013, 7045}, {24015, 42719}, {24016, 677}, {24027, 1110}, {24471, 960}, {24796, 4863}, {27818, 6557}, {28017, 2082}, {30493, 44707}, {30545, 4110}, {30617, 30615}, {30623, 30620}, {30682, 348}, {30691, 30692}, {30705, 30701}, {30719, 4521}, {30723, 4765}, {30724, 4976}, {30725, 1639}, {31598, 2899}, {31605, 44448}, {32636, 3683}, {32668, 36039}, {32714, 1783}, {33765, 17277}, {34018, 36796}, {34056, 41798}, {34400, 44189}, {34489, 2900}, {34855, 518}, {35012, 41215}, {36118, 1897}, {36419, 36421}, {36570, 40968}, {36621, 38255}, {36838, 4554}, {36908, 8804}, {37566, 1864}, {37755, 3949}, {37757, 17264}, {38254, 36625}, {38374, 38357}, {38459, 3935}, {38859, 1621}, {39126, 44720}, {39771, 4543}, {39793, 4111}, {40153, 46889}, {40154, 6601}, {40223, 16389}, {40617, 4534}, {40719, 3886}, {40933, 3198}, {40961, 40965}, {41003, 3704}, {41280, 9448}, {41282, 215}, {41283, 40363}, {41287, 44159}, {41292, 1358}, {41351, 40872}, {41353, 1026}, {41777, 3061}, {42290, 40779}, {42309, 4384}, {43035, 40869}, {43037, 4009}, {43041, 3716}, {43052, 4944}, {43923, 18344}, {43924, 663}, {43930, 885}, {43932, 513}, {44696, 44695}, {46406, 1978}
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 269, 4350}, {1, 1323, 3160}, {1, 4350, 38459}, {1, 10481, 7}, {1, 21314, 10481}, {2, 20089, 30694}, {2, 30694, 27541}, {2, 30695, 6554}, {2, 43983, 85}, {7, 77, 3945}, {7, 347, 3672}, {7, 3160, 1}, {7, 3188, 20}, {7, 7176, 3600}, {7, 14189, 390}, {7, 22464, 4346}, {7, 36640, 3663}, {7, 41352, 4310}, {7, 41354, 4307}, {56, 1358, 7195}, {56, 7195, 3598}, {57, 479, 9533}, {57, 738, 7177}, {57, 18623, 37666}, {57, 34497, 1475}, {57, 43035, 5222}, {85, 348, 2}, {85, 1088, 1446}, {85, 17078, 348}, {85, 17079, 43983}, {85, 40702, 26563}, {175, 176, 390}, {241, 948, 2}, {269, 3668, 7}, {304, 32830, 346}, {348, 17079, 85}, {481, 482, 4312}, {664, 6604, 145}, {934, 38859, 56}, {1088, 17093, 2}, {1088, 37757, 1996}, {1323, 10481, 1}, {1323, 20121, 31721}, {1323, 21314, 7}, {1418, 40133, 45227}, {1427, 7365, 2}, {1447, 17081, 5265}, {1996, 17093, 37757}, {1996, 37757, 2}, {3476, 24797, 30617}, {3600, 7176, 7268}, {3616, 32086, 40719}, {3663, 7271, 7}, {3665, 7223, 388}, {5088, 17170, 20}, {6046, 7023, 7197}, {7056, 33765, 21454}, {7176, 7185, 7}, {7177, 14256, 9533}, {9312, 9436, 8}, {16662, 16663, 57}, {17014, 21454, 5228}, {17078, 17079, 2}, {17092, 37800, 8732}, {18886, 21456, 555}, {21314, 43186, 5543}, {26563, 37780, 40702}, {30701, 32034, 346}, {43064, 45227, 40133}


X(280) = X(1)-CROSS CONJUGATE OF X(8)

Trilinears    (csc2A/2)/(-1 - cos A + cos B + cos C)

Let Ea be the ellipse passing through A, and with foci at B and C (the A-Soddy ellipse). Let La be the polar of X(20) with respect to Ea. Define Lb and Lc cyclically. Let A' = Lb∩Lc, B' = Lc∩La, C' = La∩Lb. The lines AA', BB', CC' concur in X(280). (Randy Hutson, September 14, 2016)

X(280) lies on these lines: 2,318   7,253   8,20   21,268   75,309   78,282   285,1043   341,345

X(280) = isogonal conjugate of X(221)
X(280) = isotomic conjugate of X(347)
X(280) = X(309)-Ceva conjugate of X(189)
X(280) = cevapoint of X(1) and X(84)
X(280) = X(i)-cross conjugate of X(j) for these (i,j): (1,8), (281,2), (282,189)
X(280) = pole wrt polar circle of trilinear polar of X(196)
X(280) = X(48)-isoconjugate (polar conjugate) of X(196)
X(280) = trilinear pole of line X(521)X(3239) (the radical axis of circumcircle and Mandart circle, and the Monge line of the nine-point circles of the A-, B- and C-extouch triangles)


X(281) = X(37)-CROSS CONJUGATE OF X(9)

Trilinears    sec A cot A/2 : sec B cot B/2 : sec C cot C/2
Trilinears    csc A + 2 csc 2A : csc B + 2 csc 2B : csc C + 2 csc 2C
Trilinears    (1 + sec A)/a : (1 + sec B)/b : (1 + sec C)/c
Trilinears    (b + c - a) sec A : :
Barycentrics    tan A cot A/2 : tan B cot B/2 : tan C cot C/2
Barycentrics    1 + sec A : 1 + sec B : 1 + sec C
Barycentrics    (b + c - a)/(b^2 + c^2 - a^2) : :

X(281): Let A'B'C' be the intouch triangle. Let A″ be the pole, wrt the polar circle, of line B'C', and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X(281). (Randy Hutson, September 14, 2016)

Let A'B'C' be the 3rd extouch triangle. Let A″ be the pole, wrt the polar circle, of line B'C', and define B″, C″ cyclically. The triangle A'B'C' is homothetic to ABC at X(281). (Randy Hutson, September 14, 2016)

Let A'B'C' be the 4th extouch triangle. Let A″ be the pole, wrt the polar circle, of line B'C', and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X(281). (Randy Hutson, September 14, 2016)

Let A'B'C' be the 5th extouch triangle. Let A″ be the pole, wrt the polar circle, of line B'C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(281). (Randy Hutson, September 14, 2016)

Let Ea be the ellipse passing through A, and with foci at B and C (the A-Soddy ellipse). Let La be the polar of X(3) with respect to Ea. Define Lb and Lc cyclically. Let A' = Lb∩Lc, B' = Lc∩La, C' = La∩Lb. Triangle A'B'C' is homothetic to ABC, and the center of homothety is X(281). (Randy Hutson, September 14, 2016)

X(281) lies on these lines: 1,282   2,92   4,9   6,1146   7,653   8,29   28,958   33,200   37,158   45,53   48,944   100,1013   189,222   196,226   220,594   240,984   264,344   268,1012   318,346   380,950   451,1068   515,610   612,1096

X(281) = isogonal conjugate of X(222)
X(281) = isotomic conjugate of X(348)
X(281) = complement of X(347)
X(281) = X(i)-Ceva conjugate of X(j) for these (i,j): (29,33), (92,4)
X(281) = X(i)-cross conjugate of X(j) for these (i,j): (33,4), (37,9), (55,8)
X(281) = crosspoint of X(i) and X(j) for these (i,j): (2,280), (92,318)
X(281) = crosssum of X(i) and X(j) for these (i,j): (6,221), (48,603), (73,1409), (652,1364)
X(281) = {X(2),X(92)}-harmonic conjugate of X(278)
X(281) = pole, wrt polar circle, of Gergonne line
X(281) = X(48)-isoconjugate (polar conjugate) of X(7)
X(281) = crossdifference of every pair of points on line X(1459)X(1946)
X(281) = trilinear pole of line X(3064)X(3700) (the polar of X(7) wrt polar circle)
X(281) = perspector of Gemini triangle 37 and cross-triangle of ABC and Gemini triangle 37
X(281) = trilinear pole of line X(3064)X(3700) (the perspectrix of ABC and Gemini triangle 38)


X(282) = X(6)-CROSS CONJUGATE OF X(9)

Trilinears    (cot A/2)/(-1 - cos A + cos B + cos C) : :

X(282) lies on the Thomson cubic and these lines: 1,281   2,77   3,9   4,3351   6,3341   19,102   48,947   57,3343   78,280   200,219   271,283   380,1036

X(282) = isogonal conjugate of X(223)
X(282) = isotomic conjugate of isogonal conjugate of X(7118)
X(282) = complement of X(5932)
X(282) = anticomplement of X(20206)
X(282) = X(189)-Ceva conjugate of X(84)
X(282) = X(i)-cross conjugate of X(j) for these (i,j): (6,9), (33,1)
X(282) = crosspoint of X(189) and X(280)
X(282) = crosssum of X(i) and X(j) for these (i,j): (6,1035), (198,221)
X(282) = perspector of ABC and antipedal triangle of X(1490)
X(282) = perspector of pedal and anticevian triangles of X(84)
X(282) = perspector of ABC and medial triangle of pedal triangle of X(3345)
X(282) = perspector of circumconic centered at X(3341)
X(282) = center of circumconic that is locus of trilinear poles of lines passing through X(3341)
X(282) = X(2)-Ceva conjugate of X(3341)
X(282) = polar conjugate of X(342)


X(283) = X(3)-CROSS CONJUGATE OF X(21)

Trilinears    (cos A)/(cos B + cos C) : (cos B)/(cos C + cos A) : (cos C)/(cos A + cos B)
Barycentrics    (sin 2A)/(cos B + cos C) : (sin 2B)/(cos C + cos A) : (sin 2C)/(cos A + cos B)

X(283) lies on these lines: 1,21   2,580   3,49   29,270   60,284   77,603   78,212   86,307   102,110   271,282   474,582   643,1043   859,945   1010,1065

X(283) = isogonal conjugate of X(225)
X(283) = anticomplement of complementary conjugate of X(34851)
X(283) = X(333)-Ceva conjugate of X(284)
X(283) = cevapoint of X(i) and X(j) for these (i,j): (3,255), (212,219)
X(283) = X(3)-cross conjugate of X(21)
X(283) = crosspoint of X(332) and X(333)
X(283) = X(92)-isoconjugate of X(1400)


X(284) = X(55)-CROSS CONJUGATE OF X(21)

Trilinears    (sin A)/(cos B + cos C) : (sin B)/(cos C + cos A) : (sin C)/(cos A + cos B)
Trilinears    a (b + c - a)/(b + c) : :
Barycentrics    a2/(cos B + cos C) : b2/(cos C + cos A) : c2/(cos A + cos B)
X(284) = s*X(3) + (r + 2R)*cot(ω)*X(6)

X(284) lies on these lines: 1,19   2,272   3,6   9,21   27,226   29,950   35,71   37,101   55,219   57,77   60,283   73,951   86,142   102,112   109,296   163,909   198,859   261,332   405,965   501,942   515,1065

X(284) = isogonal conjugate of X(226)
X(284) = isotomic conjugate of X(349)
X(284) = anticomplement of X(17052)
X(284) = inverse-in-Brocard-circle of X(579)
X(284) = X(i)-Ceva conjugate of X(j) for these (i,j): (81,58), (333,283)
X(284) = cevapoint of X(i) and X(j) for these (i,j): (6,48), (41,55)
X(284) = X(55)-cross conjugate of X(21)
X(284) = crosspoint of X(i) and X(j) for these (i,j): (21,81), (29,333)
X(284) = crosssum of X(i) and X(j) for these (i,j): (37,65), (73,1400)
X(284) = crossdifference of every pair of points on line X(523)X(656)
X(284) = trilinear pole of line X(652)X(663)
X(284) = X(92)-isoconjugate of X(73)
X(284) = X(1577)-isoconjugate of X(109)
X(284) = perspector of ABC and unary cofactor triangle of Gemini triangle 10
X(284) = perspector of ABC and unary cofactor triangle of isogonal triangle of X(1) (a.k.a. reflection triangle of X(1))


X(285) = X(58)-CROSS CONJUGATE OF X(21)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where
                        f(A,B,C) = 1/[(cos B + cos C)(-1 - cos A + cos B + cos C)]

Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(285) lies on these lines: 21,84   29,81   271,282   280,1043

X(285) = isogonal conjugate of X(227)
X(285) = complement of anticomplementary conjugate of X(20220)
X(285) = X(58)-cross conjugate of X(21)


X(286) = X(4)-CROSS CONJUGATE OF X(27)

Trilinears       (csc 2A)/(sin B + sin C) : (csc 2B)/(sin C + sin A) : (csc 2C)/(sin A + sin B)
Barycentrics  (sec A)/(sin B + sin C) : (sec B)/(sin C + sin A) : (sec C)/(sin A + sin B)

X(286) lies on these lines: 4,69   7,331   19,27   28,242   29,34   99,915   112,767   158,969   322,1043

X(286) = isogonal conjugate of X(228)
X(286) = isotomic conjugate of X(72)
X(286) = cevapoint of X(i) and X(j) for these (i,j): (4,92), (7,273), (27,29), (28,81)
X(286) = X(i)-cross conjugate of X(j) for these (i,j): (4,27), (7,86), (81,274)
X(286) = trilinear pole of line X(693)X(905) (the polar of X(37) wrt polar circle, and the perspectrix of Gemini triangles 1 and 2)
X(286) = pole wrt polar circle of trilinear polar of X(37) (line X(512)X(661))
X(286) = polar conjugate of X(37)
X(286) = crossdifference of every pair of points on line X(810)X(3049)
X(286) = trilinear product X(2)*X(27)


X(287) = X(2)-HIRST INVERSE OF X(98)

Trilinears    cot A sec(A + ω) : cot B sec(B + ω) : cot C sec(C + ω)
Barycentrics    cos A sec(A + ω) : cos B sec(B + ω) : cos C sec(C + ω)

X(287) lies on the MacBeath circumconic and these lines: 2,98   6,264   69,248   83,217   95,141   185,384   193,253   293,306   297,685   305,394   401,511   651,894   879,895

X(287) = reflection of X(648) in X(6)
X(287) = isogonal conjugate of X(232)
X(287) = isotomic conjugate of X(297)
X(287) = X(290)-Ceva conjugate of X(98)
X(287) = cevapoint of X(2) and X(401)
X(287) = X(248)-cross conjugate of X(98)
X(287) = X(2)-Hirst inverse of X(98)
X(287) = trilinear pole of PU(37) (line X(3)X(525))
X(287) = antipode of X(69) in hyperbola {A,B,C,X(2),X(69)}
X(287) = MacBeath circumconic antipode of X(648)
X(287) = X(92)-isoconjugate of X(237)
X(287) = pole wrt polar circle of trilinear polar of X(6530)
X(287) = X(48)-isoconjugate (polar conjugate) of X(6530)
X(287) = X(63)-isoconjugate of X(34854)
X(287) = inverse-in-Steiner-circumellipse of X(98)


X(288) = CEVAPOINT OF X(6) AND X(54)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = [sec(B - C)]/[b cos(C - A) + c cos(B - A)]
Barycentrics    a/(b sec(A - B) + c sec(A - C)) : :

X(288) lies on these lines: 51,54   53,275   97,216

X(288) = isogonal conjugate of X(233)
X(288) = cevapoint of X(6) and X(54)
X(288) = trilinear pole of line X(1157)X(1510)


X(289) = ISOGONAL CONJUGATE OF X(236)

Trilinears        (sin A/2)/(cos B/2 + cos C/2 - cos A/2) : :
Trilinears        [1 - sin(A/2)] tan(A/2) : :     (M. Iliev, 4/12/2007)
Trilinears    a/(1 + sin(A/2)) : :

X(289) lies on these lines: 1,363   56,266   258,259

X(289) = isogonal conjugate of X(236)
X(289) = X(6)-cross conjugate of X(266)
X(289) = crosssum of X(1) and X(363)


X(290) = ISOGONAL CONJUGATE OF X(237)

Trilinears    csc2A sec(A + ω) : :
Barycentrics    csc A sec(A + ω) : :
Barycentrics    b^2 c^2/(b^4 + c^4 - a^2 b^2 - a^2 c^2) : :

If you have The Geometer's Sketchpad, you can view the following dynamic sketch:

X(290).

X(290) lies on the Steiner circumellipse and these lines: 2,327   3,76   6,264   54,276   66,317   67,340   68,315   69,670   71,190   72,668   73,336   248,385   265,316   308,311   892,895

X(290) = isogonal conjugate of X(237)
X(290) = isotomic conjugate of X(511)
X(290) = complement of X(39355)
X(290) = X(2)-Ceva conjugate of X(39058)
X(290) = cevapoint of X(i) and X(j) for these (i,j): (2,511), (98,287)
X(290) = X(i)-cross conjugate of X(j) for these (i,j): (385,308), (401,276), (511,2)
X(290) = point of intersection, other than A, B, C, of Steiner circumellipse and Jerabek hyperbola
X(290) = trilinear pole of line X(2)X(647) (the line through the polar conjugates of PU(39))
X(290) = pole wrt polar circle of trilinear polar of X(232)
X(290) = X(48)-isoconjugate (polar conjugate) of X(232)
X(290) = X(6)-isoconjugate of X(1755)
X(290) = crossdifference of PU(89)
X(290) = areal center of cevian triangles of PU(37)
X(290) = areal center of cevian triangles of PU(45)
X(290) = perspector of hyperbola {A,B,C,X(22456),PU(177)}}
X(290) = Steiner-circumellipse-X(3)-antipode of X(99)
X(290) = Steiner-circumellipse-X(4)-antipode of X(6528)
X(290) = Steiner-circumellipse-X(6)-antipode of X(648)


X(291) = 2nd SHARYGIN POINT

Trilinears    1/(a2 - bc) : 1/(b2 - ca) : 1/(c2 - ab)
Barycentrics  a/(a2 - bc) : b/(b2 - ca) : c/(c2 - ab)

See the description at X(1281). The lines AD', BE', CF' defined there concur in X(291).

X(291) lies on these lines: 1,39   2,38   6,985   8,330   10,274   42,81   43,57   88,660   105,238   256,894   337,986   350,726   659,897   876,891

X(291) = reflection of X(i) in X(j) for these (i,j): (1,1015), (668,10)
X(291) = isogonal conjugate of X(238)
X(291) = isotomic conjugate of X(350)
X(291) = X(2)-Ceva conjugate of X(36906)
X(291) = X(i)-cross conjugate of X(j) for these (i,j): (239,256), (518,1)
X(291) = X(i)-Hirst inverse of X(j) for these (i,j): (1,292), (2,335)
X(291) = trilinear pole of PU(i) for these i: 6, 52, 53
X(291) = antipode of X(1) in hyperbola {A,B,C,X(1),X(2)}
X(291) = X(19)-isoconjugate of X(20769)
X(291) = point of intersection, other than A, B, C, of 1st and 2nd bicentrics of the circumcircle
X(291) = crossdifference of every pair of points on line X(659)X(4435) (the perspectrix of ABC and Gemini triangle 33)


X(292) = X(1)-HIRST INVERSE OF X(291)

Trilinears       a/(a2 - bc) : b/(b2 - ca) : c/(c2 - ab)
Barycentrics  a2/(a2 - bc) : b2/(b2 - ca) : c2/(c2 - ab)

Let A33B33C33 be Gemini triangle 33. Let A' be the perspector of conic {A,B,C,B33,C33}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(292). (Randy Hutson, January 15, 2019)

X(292) lies on these lines: 1,39   2,334   6,869   9,87   37,86   44,660   58,101   106,813   171,893   269,1020   659,665

X(292) = isogonal conjugate of X(239)
X(292) = isotomic conjugate of X(1921)
X(292) = complement of X(20345)
X(292) = anticomplement of X(20333)
X(292) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,9470), (335,292), (813,3572)
X(292) = cevapoint of X(171) and X(238)
X(292) = crossdifference of every pair of points on line X(659)X(812)
X(292) = crossdifference of PU(134)
X(292) = X(1)-Hirst inverse of X(291)
X(292) = trilinear pole of line X(42)X(649)
X(292) = point of intersection, other than A, B, C, of 1st and 2nd bicentrics of the Steiner circumellipse
X(292) = perspector of hyperbola {A,B,C,X(660),X(813)}}
X(292) = perspector of ABC and unary cofactor triangle of obverse triangle of X(1)
X(292) = perspector of ABC and unary cofactor triangle of trilinear obverse triangle of X(2)
X(292) = trilinear square root of isogonal conjugate of X(39044)


X(293) = ISOGONAL CONJUGATE OF X(240)

Trilinears       cos A sec(A + ω) : cos B sec(B + ω) : cos C sec(C + ω)
Barycentrics  sin 2A sec(A + ω) : sin 2B sec(B + ω) : sin 2C sec(C + ω)

X(293) lies on these lines: 1,163   31,92   72,248   98,109   255,304   287,306

X(293) = isogonal conjugate of X(240)

X(293) = trilinear pole of line X(48)X(656)
X(293) = X(92)-isoconjugate of X(1755)


X(294) = ISOGONAL CONJUGATE OF X(241)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)/(b2 + c2 - ab - ac)

X(294) is the perspector of the anticevian triangle of X(105) and the unary cofactor triangle of the intangents triangle. (Randy Hutson, January 15, 2019)

X(294) lies on these lines: 1,41   2,949   4,218   6,7   8,220   19,1041   84,580   104,919   239,666   241,910   314,645

X(294) = isogonal conjugate of X(241)
X(294) = isotomic conjugate of X(40704)
X(294) = anticomplement of X(17060)
X(294) = crosssum of X(i) and X(j) for these (i,j): (672,1458), (910,1279)
X(294) = crossdifference of every pair of points on line X(926)X(1362)
X(294) = X(1)-Hirst inverse of X(105)
X(294) = trilinear pole of line X(55)X(650)
X(294) = perspector of conic {A,B,C,PU(96)}
X(294) = intersection of trilinear polars of P(96) and U(96)


X(295) = ISOGONAL CONJUGATE OF X(242)

Trilinears       (cos A)/(a2 - bc) : (cos B)/(b2 - ca) : (cos C)/(c2 - ab)
Barycentrics  (sin 2A)/(a2 - bc) : (sin 2B)/(b2 - ca) : (sin 2C)/(c2 - ab)

X(295) lies on these lines: 27,335   43,57   58,101   72,337   103,813   150,334   875,926   876,928

X(295) = isogonal conjugate of X(242)
X(295) = X(335)-Ceva conjugate of X(292)
X(295) = crosspoint of X(335) and X(337)
X(295) = trilinear pole of line X(71)X(1459)
X(295) = X(92)-isoconjugate of X(1914)


X(296) = ISOGONAL CONJUGATE OF X(243)

Barycentrics    a^2*(a + b - c)*(a - b + c)*(a^2 - b^2 - c^2)*(a^2*b^2 - b^4 + a^3*c - a*b^2*c - 2*a^2*c^2 + b^2*c^2 + a*c^3)*(a^3*b - 2*a^2*b^2 + a*b^3 + a^2*c^2 - a*b*c^2 + b^2*c^2 - c^4) : :
Barycentrics    (sin 2A)/(cos2A - cos B cos C) : :
Trilinears    (cos A)/[cos2A - cos B cos C] : :

X(296) lies on the conic {A,B,C,X(1),X(3)}}, the cubics K040 and K1185, and on these lines: {1, 185}, {3, 820}, {4, 41500}, {29, 65}, {46, 2636}, {77, 1364}, {78, 7066}, {109, 284}, {219, 1949}, {243, 8764}, {283, 1813}, {416, 2651}, {518, 14198}, {851, 2655}, {916, 1807}, {949, 19350}, {1069, 20764}, {1155, 23707}, {1935, 7016}, {2263, 2818}, {3466, 5018}, {8677, 23696}, {10570, 15556}, {17973, 17975}, {38248, 38284}, {40442, 40946}

X(296) = isogonal conjugate of X(243)
X(296) = isotomic conjugate of the polar conjugate of X(1945)
X(296) = isogonal conjugate of the polar conjugate of X(1952)
X(296) = X(i)-Ceva conjugate of X(j) for these (i,j): {1952, 1945}, {37142, 1937}
X(296) = X(i)-cross conjugate of X(j) for these (i,j): {1936, 7016}, {8763, 1}
X(296) = X(i)-isoconjugate of X(j) for these (i,j): {1, 243}, {2, 2202}, {4, 1936}, {6, 1948}, {8, 1430}, {19, 1944}, {29, 851}, {33, 5088}, {34, 7360}, {65, 15146}, {92, 1951}, {296, 41500}, {318, 26884}, {393, 6518}, {450, 7105}, {522, 23353}, {650, 1981}, {1172, 8680}, {2299, 44150}, {2322, 51647}, {2648, 41499}, {7016, 41497}, {31623, 42669}, {44112, 44130}
X(296) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 243}, {6, 1944}, {9, 1948}, {226, 44150}, {1936, 36033}, {1951, 22391}, {2202, 32664}, {7360, 11517}, {15146, 40602}
X(296) = cevapoint of X(i) and X(j) for these (i,j): {1, 2655}, {3, 17975}, {1935, 1936}
X(296) = crosssum of X(415) and X(2659)
X(296) = trilinear pole of line {73, 652}
X(296) = barycentric product X(i)*X(j) for these {i,j}: {1, 40843}, {3, 1952}, {63, 1937}, {69, 1945}, {73, 35145}, {75, 1949}, {307, 2249}, {520, 41207}, {656, 41206}, {1214, 37142}, {1942, 1943}
X(296) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1948}, {3, 1944}, {6, 243}, {31, 2202}, {48, 1936}, {73, 8680}, {109, 1981}, {184, 1951}, {219, 7360}, {222, 5088}, {255, 6518}, {284, 15146}, {604, 1430}, {1214, 44150}, {1409, 851}, {1410, 51647}, {1415, 23353}, {1937, 92}, {1942, 7108}, {1945, 4}, {1949, 1}, {1950, 450}, {1952, 264}, {2202, 41500}, {2249, 29}, {2655, 39036}, {6516, 15418}, {7120, 41497}, {17966, 41499}, {17975, 39035}, {35145, 44130}, {37142, 31623}, {40843, 75}, {41206, 811}, {41207, 6528}


X(297) = X(2)-HIRST INVERSE OF X(4)

Trilinears    csc 2A cos(A + ω) : csc 2B cos(B + ω) : csc 2C cos(C + ω)
Barycentrics    sec A cos(A + ω) : sec B cos(B + ω) : sec C cos(C + ω)
Barycentrics    tan A - cot ω : :
Barycentrics    SB*SC(-SA^2 + SB*SC) : :
Barycentrics    (b^4 + c^4 - a^2 b^2 - a^2 c^2)/(b^2 + c^2 - a^2) : :
X(297) = X(340) + 2X(1990)

As a point on the Euler line, X(297) has Shinagawa coefficients (EF + F2, - S2).

X(297) lies on these lines: {2, 3}, {6, 317}, {53, 141}, {69, 393}, {76, 343}, {83, 275}, {92, 257}, {107, 2710}, {112, 2857}, {193, 1249}, {230, 2966}, {232, 325}, {239, 5081}, {249, 316}, {273, 3662}, {286, 1865}, {287, 685}, {290, 1987}, {315, 394}, {318, 3661}, {324, 1235}, {340, 524}, {459, 2996}, {525, 850}, {530, 6111}, {531, 6110}, {623, 6117}, {624, 6116}, {626, 3199}, {1515, 1533}, {1654, 2322}, {1785, 3912}, {1915, 1970}, {3087, 3618}, {3978, 6331}, {3981, 5254}, {5032, 5702}

X(297) = midpoint of X(340) and X(648)
X(297) = reflection of X(i) in X(j) for these (i,j): (401,441), (648,1990), (2966,230)
X(297) = isogonal conjugate of X(248)
X(297) = isotomic conjugate of X(287)
X(297) = inverse-in-orthocentroidal-circle of X(458)
X(297) = complement of X(401)
X(297) = anticomplement of X(441)
X(297) = X(i)-Ceva conjugate of X(j) for these (i,j): (264,2967), (6330,2)
X(297) = cevapoint of X(232) and X(511)
X(297) = X(i)-cross conjugate of X(j) for these (i,j): (511,325), (2967,264), (3569,4230)
X(297) = crossdifference of every pair of points on line X(184)X(647)
X(297) = X(i)-Hirst inverse of X(j) for (i,j) = (2,4), (193,1249)
X(297) = X(i)-complementary conjugate of X(j) for these (i,j): (1953,129), (1956,141), (1972,2887), (1987,10)
X(297) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (1297,4329), (6330,6327}
X(297) = perspector of conic {A,B,C,PU(45)}
X(297) = intersection of trilinear polars of P(45) and U(45)
X(297) = trilinear pole of line X(114)X(132) (the polar of X(98) wrt polar circle)
X(297) = pole wrt polar circle of trilinear polar of X(98) (line X(6)X(523))
X(297) = X(48)-isoconjugate (polar conjugate) of X(98)
X(297) = inverse-in-polar-circle of X(1316)
X(297) = inverse-in-Steiner-circumellipse of X(4)
X(297) = {X(2479),X(2480)}-harmonic conjugate of X(4)
X(297) = inverse-in-Steiner-inellipse of X(5)
X(297) = {X(2454),X(2455)}-harmonic conjugate of X(5)
X(297) = midpoint of polar conjugates of PU(4)
X(297) = X(i)-isoconjugate of X(j) for these (i,j): {1,248}, {3,1910}, {6,293}, {31,287}, {32,336}, {48,98}, {63,1976}, {163,879}, {184,1821}, {656,2715}, {662,878}, {685,822}, {810,2966}, {1176,3404}, {1973,6394}, {2395,4575}, {2422,4592}
X(297) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,4,458), (2,401,441), (4,419,460), (4,420,419), (4,5117,427), (53,141,264), (237,868,2450), (237,2450,1513), (419,420,468), (460,468,419), (470,471,468), (472,473,428), (868,5112,1513), (1585,1586,25), (2450,5112,237), (2454,2455,5), (2479,2480,4)




leftri

Centers 298- 350 are isotomic conjugates of previously listed centers.

rightri

X(298) = ISOTOMIC CONJUGATE OF 1st ISOGONIC CENTER

Trilinears    csc2A sin(A + π/3) : csc2B sin(B + π/3) : csc2C sin(C + π/3)
Barycentrics   csc A sin(A + π/3) : csc B sin(B + π/3) : csc C sin(C + π/3)
Barycentrics   S + sqrt(3)*SA : :

X(298) lies on these lines: 2,6   3,617   5,634   13,532   14,76   15,533   18,636   99,531   140,628   264,472   316,530   317,473   319,1082   340,470   381,622   511,1080

X(298) = midpoint of X(616) and X(621)
X(298) = reflection of X(i) in X(j) for these (i,j): (13,623), (15,618), (299,325), (385,395)
X(298) = isogonal conjugate of X(3457)
X(298) = isotomic conjugate of X(13)
X(298) = complement of X(3180)
X(298) = anticomplement of X(396)
X(298) = X(300)-Ceva conjugate of X(303)
X(298) = X(15)-cross conjugate of X(470)
X(298) = X(2)-Hirst inverse of X(299)
X(298) = pole wrt polar circle of trilinear polar of X(8737) (line X(462)X(2501))
X(298) = polar conjugate of X(8737)
X(298) = crosspoint of X(2) and X(616) wrt both the excentral and anticomplementary triangles


X(299) = ISOTOMIC CONJUGATE OF 2nd ISOGONIC CENTER

Trilinears    csc2A sin(A - π/3) : csc2B sin(B - π/3) : csc2C sin(C - π/3)
Barycentrics   csc A sin(A - π/3) : csc B sin(B - π/3) : csc C sin(C - π/3)
Barycentrics   S - sqrt(3)*SA : :

X(299) lies on these lines: 2,6   3,616   5,633   13,76   14,533   16,532   17,635   30,617   75,554   99,530   140,627   264,473   316,531   317,472   319,559   340,471   381,621   383,511

X(299) = midpoint of X(617) and X(622)
X(299) = reflection of X(i) in X(j) for these (i,j): (14,624), (16,619), (298,325), (385,396)
X(299) = isogonal conjugate of X(3458)
X(299) = isotomic conjugate of X(14)
X(299) = complement of X(3181)
X(299) = anticomplement of X(395)
X(299) = X(301)-Ceva conjugate of X(302)
X(299) = X(16)-cross conjugate of X(471)
X(299) = X(2)-Hirst inverse of X(298)
X(299) = pole wrt polar circle of trilinear polar of X(8738) (line X(463)X(2501))
X(299) = polar conjugate of X(8738)
X(299) = crosspoint of X(2) and X(617) wrt both the excentral and anticomplementary triangles


X(300) = ISOTOMIC CONJUGATE OF 1st ISODYNAMIC CENTER

Trilinears       csc2A csc(A + π/3) : csc2B csc(B + π/3) : csc2C csc(C + π/3)
Barycentrics  csc A csc(A + π/3) : csc B csc(B + π/3) : csc C csc(C + π/3)

X(300) lies on these lines: 2,94   13,76   264,302   265,621   303,311

X(300) = isogonal conjugate of X(34394)
X(300) = isotomic conjugate of X(15)
X(300) = cevapoint of X(i) and X(j) for these (i,j): {2, 621}, {298, 303}
X(300) = polar conjugate of X(8739)
X(300) = trilinear pole of line X(850)X(20578)
X(300) = X(94)-Hirst inverse of X(301)


X(301) = ISOTOMIC CONJUGATE OF 2nd ISODYNAMIC CENTER

Trilinears       csc2A csc(A - π/3) : csc2B csc(B - π/3) : csc2C csc(C - π/3)
Barycentrics  csc A csc(A - π/3) : csc B csc(B - π/3) : csc C csc(C - π/3)

X(301) lies on these lines: 2,94   14,76   264,303   265,622   302,311

X(301) = isogonal conjugate of X(34395)
X(301) = isotomic conjugate of X(16)
X(301) = cevapoint of X(i) and X(j) for these (i,j): {2, 622}, {299, 302}
X(301) = X(94)-Hirst inverse of X(300)
X(301) = polar conjugate of X(8740)
X(301) = trilinear pole of line X(850)X(20579)


X(302) = ISOTOMIC CONJUGATE OF 1st NAPOLEON POINT

Trilinears       csc2A sin(A + π/6) : csc2B sin(B + π/6) : csc2C sin(C + π/6)
Barycentrics  csc A sin(A + π/6) : csc B sin(B + π/6) : csc C sin(C + π/6)

If you have The Geometer's Sketchpad, you can view X(302).

X(302) lies on these lines: 2,6   3,621   5,622   14,99   16,316   18,76   61,629   140,633   264,300   301,311   317,470   381,616   549,617

X(302) = isotomic conjugate of X(17)
X(302) = X(301)-Ceva conjugate of X(299)
X(302) = X(61)-cross conjugate of X(473)
X(302) = polar conjugate of X(8741)
X(302) = crosspoint of X(2) and X(627) wrt both the excentral and anticomplementary triangles


X(303) = ISOTOMIC CONJUGATE OF 2nd NAPOLEON POINT

Trilinears       csc2A sin(A - π/6) : csc2B sin(B - π/6) : csc2C sin(C - π/6)
Barycentrics  csc A sin(A - π/6) : csc B sin(B - π/6) : csc C sin(C - π/6)

If you have The Geometer's Sketchpad, you can view X(303).

X(303) lies on these lines: 2,6   3,622   5,621   13,99   15,316   17,76   62,630   140,634   264,301   300,311   317,471   381,617   549,616

X(303) = isotomic conjugate of X(18)
X(303) = X(300)-Ceva conjugate of X(298)
X(303) = polar conjugate of X(8742)
X(303) = crosspoint of X(2) and X(628) wrt both the excentral and anticomplementary triangles
X(303) = X(62)-cross conjugate of X(472)


X(304) = ISOTOMIC CONJUGATE OF X(19)

Trilinears    (cot A)csc2A : (cot B)csc2B : (cot C)csc2C
Trilinears    cos A csc(A - ω) : cos B csc(B - ω) : cos C csc(C - ω)
Barycentrics    (cos A)csc2A : (cos B)csc2B : (cos C)csc2C

Let AaBaCa, AbBbCb, AcBcCc be the A-, B-, and C-anti-altimedial triangles, resp. X(304) is the trilinear product Ba*Ca*Cb*Ab*Ac*Bc. (Randy Hutson, November 2, 2017)

X(304) lies on these lines: 1,75   8,3263   63,1102   69,72   76,85   92,561   255,293   279,346   305,306   309,322   337,1565   341,668   345,348   662,2172   742,2176   799,2349   811,1895   1921,3061   1958,1973

X(304) = isogonal conjugate of X(1973)
X(304) = isotomic conjugate of X(19)
X(304) = complement of X(21216)
X(304) = anticomplement of X(16583)
X(304) = cevapoint of X(i) and X(j) for these (i,j): (63,326), (69,345), (312,322)
X(304) = X(i)-cross conjugate of X(j) for these (i,j): (63,75), (306,69)
X(304) = X(i)-isoconjugate of X(j) for these (i,j): (6,25), (48,1096), (92,560)
X(304) = polar conjugate of X(1096)
X(304) = trilinear product of vertices of Gemini triangle 35
X(304) = trilinear product of vertices of Gemini triangle 36


X(305) = ISOTOMIC CONJUGATE OF X(25)

Trilinears       b4c4cos A : c4a4cos B : a4b4cos C
                        = cot A csc(A - ω) : cot B csc(B - ω) : cot C csc(C - ω)

Barycentrics  b3c3cos A : c3a3cos B : a3b3cos C

X(305) = trilinear-pole-of-line-X(525)X(3267) = pole-with-respect-to-polar-circle-of trilinear-polar-of-X(2207) = X(48)-isoconjugate-of-X(2207) = X(92)-isoconjugate-of-X(1501)    Randy Hutson, August 15, 2013

X(305) lies on these lines: 2,39   22,99   25,683   95,183   264,325   287,394   304,306   311,1007   341,1088   350,614   561,1441

X(305) = isogonal conjugate of X(1974)
X(305) = isotomic conjugate of X(25)
X(305) = anticomplement of X(1196)
X(305) = X(63)-isoconjugate of X(36417)
X(305) = X(i)-cross conjugate of X(j) for these (i,j): (69,76), (339, (3267)
X(305) = cevapont of X(339) and X(3267)


X(306) = ISOTOMIC CONJUGATE OF X(27)

Trilinears       (b2c2)(b + c)cos A : (c2a2)(c + a)cos B : (a2b2)(a + b)cos C
Barycentrics  bc(b + c)cos A : ca(c + a)cos B : ab(a + b)cos C

X(306) lies on these lines: 1,2   27,1043   63,69   72,440   92,264   209,518   226,321   253,329   287,293   304,305   319,333

X(306) = isogonal conjugate of X(1474)
X(306) = isotomic conjugate of X(27)
X(306) = complement of X(3187)
X(306) = X(i)-Ceva conjugate of X(j) for these (i,j): (69, 72), (312,321), (313,10)
X(306) = X(i)-cross conjugate of X(j) for these (i,j): (71,10), (72,307), (440,2)
X(306) = crosspoint of X(i) and X(j) for these (i,j): (69,304), (312,345)
X(306) = crosssum of X(604) and X(608)
X(306) = trilinear pole of line X(525)X(656)
X(306) = pole wrt polar circle of trilinear polar of X(8747) (line X(649)X(7649))
X(306) = polar conjugate of X(8747)
X(306) = X(6)-isoconjugate of X(28)
X(306) = X(92)-isoconjugate of X(2206)


X(307) = ISOTOMIC CONJUGATE OF X(29)

Trilinears    b2c2(b + c)(cos A)/(b + c - a)
Barycentrics    (csc A) (sec B + sec C) : :

X(307) lies on these lines: 2,7   8,253   69,73   75,225   86,283   95,320   141,241   269,936   319,664   948,966

X(307) = isogonal conjugate of X(2299)
X(307) = isotomic conjugate of X(29)
X(307) = X(349)-Ceva conjugate of X(226)
X(307) = X(i)-cross conjugate of X(j) for these (i,j): (72,306), (73,226)
X(307) = crosspoint of X(69) and X(75)
X(307) = crosssum of X(25) and X(31)

X(307) = trilinear pole of line X(525)X(8611)
X(307) = pole wrt polar circle of trilinear polar of X(8748) (line X(663)X(3064))
X(307) = polar conjugate of X(8748)

X(308) = ISOTOMIC CONJUGATE OF X(39)

Trilinears    b3c3/(b2 + c2) : :
Trilinears    csc2A csc(A + ω) : :
Trilinears    [csc(A - ω)]/(b2 + c2) : :
Trilinears    csc(A + ω) + csc(A - ω) : :
Barycentrics    (b2c2)/(b2 + c2) : :

Barycentrics    csc A csc(A + ω) : :

X(308) lies on these lines: 2,702   6,76   25,183   42,313   69,263   111,689   141,670   251,385   290,311

X(308) = isogonal conjugate of X(3051)
X(308) = isotomic conjugate of X(39)
X(308) = anticomplement of isotomic conjugate of X(31622)
X(308) = anticomplement of polar conjugate of isogonal conjugate of X(23210)
X(308) = anticomplement of crosspoint of X(2) and X(39)
X(308) = anticomplement of crosssum of X(6) and X(83)
X(308) = cevapoint of X(2) and X(76)
X(308) = X(i)-cross conjugate of X(j) for these (i,j): (2,83), (385,290)
X(308) = trilinear pole of line X(316)X(512) (anticomplement of Lemoine axis)
X(308) = polar conjugate of X(1843)
X(308) = barycentric product X(76)*X(83)


X(309) = ISOTOMIC CONJUGATE OF X(40)

Trilinears    (csc2A)/(-1 - cos A + cos B + cos C) : :
Trilinears    b^2 c^2 (a^2 (a^2 - (b - c)^2)^2 - (b - c)^2 (a^2 - (b + c)^2)^2) : :

X(309) lies on these lines: 69,189   75,280   77,318   84,314   85,264   304,322

X(309) = isogonal conjugate of X(2187)
X(309) = isotomic conjugate of X(40)
X(309) = cevapoint of X(189) and X(280)
X(309) = X(i)-cross conjugate of X(j) for these (i,j): (7,75), (92,85)
X(309) = polar conjugate of X(2331)


X(310) = ISOTOMIC CONJUGATE OF X(42)

Trilinears       b3c3/(b + c) : c3a3/(c + a) : a3b3/(a + b)
Barycentrics  b2c2/(b + c) : c2a2/(c + a) : a2b2/(a + b)

X(310) lies on these lines: 2,39   7,314   38,75   86,350   99,675   261,272   321,335   333,673   670,903   871,982

X(310) = isogonal conjugate of X(1918)
X(310) = isotomic conjugate of X(42)
X(310) = complement of polar conjugate of isogonal conjugate of X(23176)
X(310) = cevapoint of X(i) and X(j) for these (i,j): (75,76), (274,314)
X(310) = X(75)-cross conjugate of X(274)
X(310) = polar conjugate of X(2333)
X(310) = trilinear product of vertices of Gemini triangle 23
X(310) = trilinear product of vertices of Gemini triangle 24
X(310) = trilinear product of vertices of Gemini triangle 25


X(311) = ISOTOMIC CONJUGATE OF X(54)

Trilinears    csc2A cos(B - C) : :
Barycentrics  csc A cos(B - C) : csc B cos(C - A) : csc C cos(A - B)

Let OAOBOC be the Kosnita triangle. Let A' be the trilinear pole of line OBOC, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(1994). Let A″ be the trilinear pole of line B'C', and define B″and C″ cyclically. The lines AA″, BB″, CC″ concur in X(311). (Randy Hutson, November 30, 2018)

X(311) lies on these lines: 2,570   4,69   22,157   53,324   95,99   141,338   290,308   300,303   301,302   305,1007

X(311) = isotomic conjugate of X(54)
X(311) = anticomplement of X(570)
X(311) = complement of polar conjugate of isogonal conjugate of X(23158)
X(311) = X(76)-Ceva conjugate of X(343)
X(311) = cevapoint of X(5) and X(343)
X(311) = X(5)-cross conjugate of X(324)
X(311) = pole wrt polar circle of trilinear polar of X(8882) (line X(512)X(2623))
X(311) = polar conjugate of X(8882)
X(311) = barycentric product X(5)*X(76)


X(312) = ISOTOMIC CONJUGATE OF X(57)

Trilinears    (b + c - a)b2c2 : (c + a - b)c2a2 : (a + b - c)a2b2
Trilinears    (1 + cos A)csc(A - ω) : (1 + cos B)csc(B - ω) : (1 + cos C)csc(C - ω)
Trilinears    (csc A)/(1 - cos A) : (csc B)/(1 - cos B) : (csc C)/(1 - cos C)     (M. Iliev, 4/12/07)
Trilinears    bc tan A' : ca tan B' : ab tan C', where A'B'C' is the excentral triangle
Barycentrics    bc(b + c - a) : ca(c + a - b) : ab(a + b - c)

Let A37B37C37 be Gemini triangle 37. Let A' be the perspector of conic {A,B,C,B37,C37}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(312). (Randy Hutson, January 15, 2019)

Let A39B39C39 be Gemini triangle 39. Let A' be the perspector of conic {A,B,C,B39,C39}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(312). (Randy Hutson, January 15, 2019)

X(312) lies on these lines: 1,1089   2,37   8,210   9,314   29,33   63,190   69,189   76,85   92,264   212,643   223,664   726,982   894,940   975,1010

X(312) = isogonal conjugate of X(604)
X(312) = isotomic conjugate of X(57)
X(312) = complement of X(3210)
X(312) = anticomplement of X(3752)
X(312) = X(i)-Ceva conjugate of X(j) for these (i,j): (76,75), (304,322), (314,8)
X(312) = cevapoint of X(i) and X(j) for these (i,j): (2,329), (8,346), (9,78), (306,321)
X(312) = X(i)-cross conjugate of X(j) for these (i,j): (8,75), (9,318), (306,345), (346,341)
X(312) = crosssum of X(i) and X(j) for these (i,j): (32,1397), (56,1403), (57,1424)
X(312) = polar conjugate of X(34)
X(312) = perspector of ABC and extraversion triangle of X(85)
X(312) = BSS(A→A') of X(4), where A'B'C' is the excentral triangle
X(312) = barycentric product of vertices of Gemini triangle 27
X(312) = perspector of ABC and cross-triangle of ABC and Gemini triangle 27
X(312) = trilinear pole of line X(522)X(3717) (the polar of X(34) wrt polar circle, and the radical axis of the circumcircles of the outer and inner Garcia triangles)


X(313) = ISOTOMIC CONJUGATE OF X(58)

Trilinears       (b + c)b3c3 : (c + a)c3a3 : (a + b)a3b3
                        = (b + c)csc(A - ω) : (c + a)csc(B - ω) : (a + b)csc(C - ω)

Barycentrics  (b + c)b2c2 : (c + a)c2a2 : (a + b)a2b2

Let A40B40C40 be Gemini triangle 40. Let A' be the center of conic {A,B,C,B40,C40}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(313). (Randy Hutson, January 15, 2019)

X(313) lies on these lines: 10,75   12,349   42,308   71,190   80,314   92,264   321,594   561,696

X(313) = isogonal conjugate of X(2206)
X(313) = isotomic conjugate of X(58)
X(313) = complement of X(17148)
X(313) = anticomplement of polar conjugate of isogonal conjugate of X(23197)
X(313) = X(76)-Ceva conjugate of X(321)
X(313) = cevapoint of X(10) and X(306)
X(313) = X(321)-cross conjugate of X(349)
X(313) = crosssum of X(32) and X(560)
X(313) = polar conjugate of X(1474)


X(314) = ISOTOMIC CONJUGATE OF X(65)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b2c2(b + c - a)/(b + c)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = bc(b + c - a)/(b + c)

Let D and E be the intersections of line X(1)X(3) with lines PU(3) and PU(6), respectively. Let D' and E' be the isogonal conjugates of D and E, respectively. Let D" and E" be the isotomic conjugates of D and E, respectively. Then X(314) = D'D" ∩E'E". (Randy Hutson, December 26, 2015)

Let A4B4C4 be the 4th Conway triangle. Let A' be the cevapoint of B4 and C4, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(314). (Randy Hutson, December 10, 2016)

Let A5B5C5 be the 5th Conway triangle. Let A' be the cevapoint of B5 and C5, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(314). (Randy Hutson, December 10, 2016)

X(314) lies on these lines: 1,75   2,941   4,69   6,981   7,310   9,312   21,261   29,1039   58,987   79,320   80,313   81,321   84,309   99,104   256,350   294,645

X(314) = isogonal conjugate of X(1402)
X(314) = isotomic conjugate of X(65)
X(314) = anticomplement of X(2092)
X(314) = X(310)-Ceva conjugate of X(274)
X(314) = cevapoint of X(i) and X(j) for these (i,j): (8,312), (69,75)
X(314) = X(i)-cross conjugate of X(j) for these (i,j): (8,333), (69,332), (333,274), (497,29)

X(314) = crosspoint of X(1) and X(1764) wrt excentral triangle
X(314) = crosspoint of X(1) and X(1764) wrt anticomplementary triangle
X(314) = pole wrt polar circle of trilinear polar of X(1880)
X(314) = X(48)-isoconjugate (polar conjugate) of X(1880)
X(314) = perspector of 4th Conway triangle and cross-triangle of ABC and 4th Conway triangle
X(314) = perspector of 5th Conway triangle and cross-triangle of ABC and 5th Conway triangle
X(314) = perspector of inverse-in-Conway-circle triangle and cross-triangle of ABC and inverse-in-Conway-circle triangle
X(314) = barycentric product X(99)*X(4391)

X(315) = ISOTOMIC CONJUGATE OF X(66)

Trilinears       bc(b4 + c4 - a4) : ca(c4 + a4 - b4) : ab(a4 + b4 - c4)
Barycentrics  b4 + c4 - a4 : c4 + a4 - b4 : a4 + b4 - c4

X(315) lies on these lines: 2,32   3,325   4,69   5,183   8,760   20,99   68,290   192,746   194,736   274,377   297,394   343,458   371,491   372,492   631,1007

X(315) = midpoint of X(637) and X(638)
X(315) = reflection of X(i) in X(j) for these (i,j): (32,626), (371,640, (372,639)
X(315) = isogonal conjugate of X(2353)
X(315) = isotomic conjugate of X(66)
X(315) = anticomplement of X(32)
X(315) = anticomplementary conjugate of X(194)
X(315) = X(i)-cross conjugate of X(j) for these (i,j): (206,2)


X(316) = DROUSSENT PIVOT

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b4 + c4 - a4 - b2c2)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = b4 + c4 - a4 - b2c2
Tripolars    a^2 Sqrt[2(b^2 + c^2) - a^2] : :

Let A'B'C' be the antipedal triangle of X(4) (the anticomplementary triangle). The circumcircles of AA'X(4), BB'X(4), CC'X(4) concur, other than X(4), in X(316). (Randy Hutson, June 27, 2018)

See Lucien Droussent, "Cubiques circulaires anallagmatiques par points réciproques ou isogonaux," Mathesis 62 (1953) 204-215.

X(316) lies on these lines: 2,187   4,69   15,303   16,302   30,99   115,385   148,538   183,381   249,297   265,290   298,530   299,531   376,1007   384,626   512,850   524,671   691,858

X(316) = midpoint of X(621) and X(622)
X(316) = reflection of X(i) in X(j) for these (i,j): (15,624), (16,623), (99,325), (385,115), (691,858)
X(316) = isogonal conjugate of X(3455)
X(316) = isotomic conjugate of X(67)
X(316) = anticomplement of X(187)
X(316) = crosssum of X(39) and X(187)
X(316) = reflection of X(99) in the polar of X(76)
X(316) = antigonal conjugate of X(23)
X(316) = reflection of X(99) in the de Longchamps line
X(316) = inverse-in-polar-circle of X(1843)
X(316) = trilinear pole of line X(2492)X(7664)
X(316) = pole wrt polar circle of trilinear polar of X(8791) (line X(512)X(1843))
X(316) = polar conjugate of X(8791)
X(316) = inverse-in-circumcircle of X(21395)
X(316) = intersection of Lemoine axes of 1st and 2nd Ehrmann circumscribing triangles
X(316) = intersection of Lemoine axes of anticevian triangles of PU(4)


X(317) = ISOTOMIC CONJUGATE OF X(68)

Trilinears    sec A cos 2A csc2A : :
Trilinears    bc cot 2A : ca cot 2B : ab cot 2C
Trilinears    sec A - cot A csc A : :
Barycentrics    cot 2A : cot 2B : cot 2C
Barycentrics   tan A cos 2A csc2A
Barycentrics   tan A - cot A : :
Barycentrics   tan A + cot B + cot C - cot ω : :
Barycentrics    (a^4 + b^4 + c^4 - 2 a^2 b^2 - 2 a^2 c^2) / (b^2 + c^2 - a^2) : :

X(317) lies on these lines: 2,95   4,69   6,297   25,325   53,524   66,290   141,458   183,427   193,393   261,406   273,320   298,473   299,472   302,470   303,471   318,319   1007,6353

X(317) = isogonal conjugate of X(2351)
X(317) = isotomic conjugate of X(68)
X(317) = anticomplement of X(577)
X(317) = cevapoint of X(52) and X(467)
X(317) = polar conjugate of X(2165)
X(317) = X(1721)-of-orthic-triangle if ABC is acute


X(318) = ISOTOMIC CONJUGATE OF X(77)

Trilinears    (1 + sec A)/a2 : :
Trilinears    sec A csc2A/2 : :
Barycentrics    (1 + sec A)/a : (1 + sec B)/b : (1 + sec C)/c
Barycentrics    b c (a - b - c)/(a^2 - b^2 - c^2) : :

Let A'B'C' be the mixtilinear incentral triangle. Let A″ be the pole, wrt the polar circle, of line B'C', and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X(318). (Randy Hutson, November 30, 2018)

X(318) lies on these lines: 2,280   4,8   10,158   29,33   53,594   63,412   75,225   77,309   108,404   200,1089   208,653   239,458   243,958   253,342   281,346   317,319   475,1068

X(318) = isogonal conjugate of X(603)
X(318) = isotomic conjugate of X(77)
X(318) = X(264)-Ceva conjugate of X(92)
X(318) = cevapoint of X(9) and X(33)
X(318) = X(i)-cross conjugate of X(j) for these (i,j): (9,312), (10,8), (281,92)
X(318) = trilinear pole of line X(3064)X(3239) (the radical axis of Mandart circle and excircles radical circle, and the polar of X(57) wrt polar circle)
X(318) = pole wrt polar circle of trilinear polar of X(57) (line X(513)X(663))
X(318) = polar conjugate of X(57)
X(318) = inverse-in-Fuhrmann-circle of X(5081)


X(319) = ISOTOMIC CONJUGATE OF X(79)

Trilinears    (1 + 2 cos A)/a2 : :
Barycentrics   ( 1 + 2 cos A)/a : :
Barycentrics    a^2 - b^2 - c^2 - b c : :
Barycentrics    b c + 2 SA : :

X(319) lies on these lines: 2,1100   7,8   10,86   80,313   141,239   171,757   200,326   261,502   298,1082   299,559   306,333   307,664   317,318   321,1029   344,391    524,594

X(319) = reflection of X(894) in X(594)
X(319) = isotomic conjugate of X(79)
X(319) = anticomplement of X(1100)
X(319) = complement of polar conjugate of isogonal conjugate of X(23165)
X(319) = trilinear pole of line X(4467)X(7265) (the perspectrix of ABC and Gemini triangle 25)


X(320) = ISOTOMIC CONJUGATE OF X(80)

Trilinears    (1 - 2 cos A)/a2 : :
Barycentrics  (1 - 2 cos A)/a : :
Barycentrics    a^2 - b^2 - c^2 + b c : :
Barycentrics    b c - 2 SA : :
X(320) = 3 X[1] - 2 X[49700], 4 X[49700] - 3 X[49709], 3 X[2] - 4 X[3834], 9 X[2] - 8 X[6687], 9 X[2] - 10 X[31243], 3 X[44] - 4 X[6687], X[44] - 3 X[31138], 3 X[44] - 5 X[31243], 3 X[3834] - 2 X[6687], 4 X[3834] - X[20072], 2 X[3834] - 3 X[31138], 6 X[3834] - 5 X[31243], 8 X[6687] - 3 X[20072], 4 X[6687] - 9 X[31138], 4 X[6687] - 5 X[31243], X[20072] - 6 X[31138], 3 X[20072] - 10 X[31243], 9 X[31138] - 5 X[31243], X[8] - 3 X[4645], 2 X[8] - 3 X[32850], 7 X[8] - 6 X[49694], 4 X[8] - 3 X[49698], 3 X[8] - 2 X[49702], 5 X[8] - 3 X[49707], 7 X[4645] - 2 X[49694], and many others

X(320) lies on the cubics K311, K636, and K660, and these lines: {1, 752}, {2, 44}, {6, 3662}, {7, 8}, {9, 17234}, {10, 4896}, {31, 29638}, {37, 6646}, {38, 32949}, {42, 24707}, {45, 17244}, {57, 4417}, {58, 86}, {63, 18134}, {72, 33943}, {76, 4911}, {77, 44179}, {79, 314}, {80, 20568}, {81, 17184}, {95, 307}, {141, 894}, {142, 3707}, {144, 344}, {145, 4346}, {150, 18159}, {171, 33064}, {183, 7179}, {190, 527}, {192, 4851}, {193, 3759}, {219, 28965}, {226, 14829}, {239, 524}, {241, 17950}, {244, 32843}, {256, 4022}, {264, 7282}, {269, 326}, {273, 317}, {291, 2228}, {298, 36669}, {299, 36668}, {306, 32939}, {309, 20570}, {312, 5905}, {313, 44139}, {315, 3673}, {318, 44134}, {321, 17483}, {325, 1447}, {329, 18141}, {333, 4001}, {334, 660}, {335, 742}, {340, 1835}, {345, 9965}, {346, 17240}, {350, 513}, {354, 4388}, {369, 3232}, {489, 31549}, {490, 31550}, {491, 32792}, {492, 32791}, {514, 30190}, {516, 4684}, {517, 49779}, {519, 679}, {536, 4440}, {537, 32847}, {540, 30117}, {545, 3943}, {553, 3687}, {573, 29747}, {583, 27678}, {594, 3631}, {597, 29630}, {599, 3661}, {662, 20769}, {664, 22464}, {668, 3264}, {670, 18891}, {674, 3888}, {726, 32846}, {740, 32857}, {750, 33065}, {758, 1227}, {765, 1275}, {896, 29632}, {908, 1797}, {940, 27184}, {942, 1330}, {960, 41874}, {966, 4751}, {982, 3764}, {1043, 4292}, {1088, 7055}, {1100, 17235}, {1104, 20077}, {1119, 32001}, {1150, 31019}, {1193, 33947}, {1203, 33955}, {1213, 17252}, {1215, 33085}, {1232, 34388}, {1267, 1270}, {1271, 5391}, {1278, 17299}, {1279, 3622}, {1439, 46752}, {1443, 1464}, {1444, 5303}, {1449, 17304}, {1478, 20925}, {1647, 27922}, {1654, 3739}, {1698, 1757}, {1730, 29790}, {1738, 34379}, {1743, 17282}, {1760, 7289}, {1764, 29788}, {1836, 4479}, {1931, 25536}, {1944, 7359}, {1959, 3942}, {1964, 7184}, {1992, 5222}, {1999, 3782}, {2234, 3783}, {2239, 37632}, {2245, 3218}, {2260, 28402}, {2308, 33123}, {2321, 17295}, {2323, 4585}, {2345, 3620}, {2481, 3254}, {2796, 4693}, {2887, 29861}, {2895, 4359}, {2911, 26657}, {2968, 40996}, {3008, 27191}, {3120, 32919}, {3187, 19796}, {3219, 18139}, {3241, 49699}, {3242, 50289}, {3244, 3663}, {3246, 3616}, {3260, 34387}, {3263, 8047}, {3296, 30479}, {3306, 5233}, {3509, 4070}, {3570, 24602}, {3589, 7277}, {3593, 32795}, {3595, 32796}, {3598, 37668}, {3618, 17370}, {3619, 5749}, {3623, 3672}, {3625, 50099}, {3626, 49713}, {3629, 17121}, {3630, 4405}, {3632, 50088}, {3633, 3875}, {3644, 17314}, {3666, 17778}, {3675, 24516}, {3679, 24693}, {3684, 4987}, {3685, 4966}, {3686, 24199}, {3693, 40868}, {3702, 14450}, {3705, 7788}, {3717, 5850}, {3720, 4683}, {3729, 4873}, {3741, 33097}, {3744, 20101}, {3751, 4429}, {3760, 4056}, {3761, 7272}, {3763, 17368}, {3769, 33144}, {3770, 20891}, {3771, 4650}, {3772, 37683}, {3775, 24342}, {3791, 33147}, {3821, 4649}, {3823, 5232}, {3824, 25446}, {3840, 33096}, {3873, 4514}, {3874, 5015}, {3882, 20367}, {3883, 5542}, {3886, 4312}, {3894, 4680}, {3896, 33102}, {3916, 25650}, {3923, 33087}, {3932, 5852}, {3946, 4982}, {3957, 4450}, {3980, 33084}, {3999, 5211}, {4009, 20947}, {4014, 6007}, {4021, 49696}, {4029, 29601}, {4033, 40875}, {4038, 4425}, {4062, 32845}, {4080, 51908}, {4089, 4867}, {4118, 18207}, {4273, 21997}, {4277, 4850}, {4285, 17012}, {4295, 4673}, {4310, 51192}, {4358, 8046}, {4361, 17363}, {4362, 33103}, {4364, 16826}, {4373, 20054}, {4384, 6173}, {4392, 33070}, {4393, 17301}, {4406, 37998}, {4407, 36531}, {4409, 28297}, {4418, 33081}, {4419, 4664}, {4422, 17266}, {4430, 5014}, {4432, 28558}, {4439, 24821}, {4445, 17118}, {4447, 21320}, {4452, 20014}, {4454, 29616}, {4465, 30967}, {4472, 29610}, {4473, 41310}, {4569, 35164}, {4640, 29839}, {4648, 4687}, {4654, 11679}, {4657, 17236}, {4659, 17294}, {4660, 49490}, {4665, 22165}, {4667, 16786}, {4672, 29637}, {4676, 24695}, {4679, 24482}, {4686, 17372}, {4688, 4690}, {4691, 4967}, {4697, 32783}, {4699, 17275}, {4700, 17067}, {4702, 28534}, {4703, 26102}, {4713, 31028}, {4722, 29850}, {4725, 20016}, {4740, 20055}, {4742, 5180}, {4749, 7191}, {4753, 25351}, {4766, 9318}, {4772, 28634}, {4799, 17027}, {4860, 26240}, {4892, 33140}, {4902, 17151}, {4908, 17487}, {4938, 14459}, {4945, 9326}, {4973, 7799}, {4996, 52437}, {5088, 10609}, {5231, 36278}, {5263, 49511}, {5278, 27186}, {5280, 17192}, {5308, 51488}, {5718, 24627}, {5739, 19804}, {5741, 27003}, {5744, 30828}, {5745, 41878}, {5750, 17307}, {5805, 48878}, {5839, 20080}, {5847, 24231}, {5902, 33934}, {5904, 33933}, {6163, 33864}, {6172, 29627}, {6224, 36917}, {6356, 41008}, {7058, 52361}, {7081, 37671}, {7113, 27950}, {7122, 18209}, {7185, 12635}, {7222, 48630}, {7227, 48635}, {7229, 48640}, {7231, 48636}, {7262, 29642}, {7291, 16568}, {7779, 33891}, {8822, 18650}, {9025, 20358}, {9055, 49752}, {9776, 14555}, {9791, 15569}, {10025, 51384}, {10446, 12699}, {10889, 41864}, {11160, 50077}, {11246, 32932}, {12586, 21280}, {13476, 17153}, {14616, 35156}, {15523, 32940}, {15533, 17119}, {15668, 17248}, {15988, 26573}, {16666, 17382}, {16669, 17356}, {16672, 24441}, {16704, 33129}, {16777, 17247}, {16815, 17330}, {16816, 50074}, {16817, 49716}, {16823, 25557}, {16884, 17323}, {16885, 17265}, {17011, 42045}, {17018, 32950}, {17026, 24694}, {17028, 30958}, {17045, 17324}, {17056, 38000}, {17062, 17739}, {17126, 33122}, {17132, 49765}, {17133, 49761}, {17135, 20292}, {17140, 20290}, {17144, 17753}, {17145, 21282}, {17152, 25303}, {17155, 32852}, {17165, 33078}, {17167, 29766}, {17170, 18156}, {17178, 26971}, {17179, 49997}, {17181, 36280}, {17202, 18166}, {17230, 17281}, {17231, 17280}, {17232, 17279}, {17238, 17303}, {17239, 28604}, {17242, 17262}, {17243, 17261}, {17245, 17260}, {17246, 17319}, {17251, 29576}, {17259, 17331}, {17267, 17339}, {17268, 17340}, {17269, 29577}, {17270, 25590}, {17278, 17349}, {17283, 17353}, {17284, 17354}, {17285, 17355}, {17293, 48634}, {17306, 17381}, {17318, 17389}, {17325, 17397}, {17326, 17398}, {17341, 26685}, {17342, 29579}, {17359, 29587}, {17383, 37677}, {17395, 29584}, {17399, 26626}, {17449, 23633}, {17499, 21240}, {17555, 42856}, {17579, 49687}, {17720, 37684}, {17731, 24194}, {17755, 27487}, {17762, 33865}, {17763, 32856}, {17776, 20078}, {17787, 18040}, {17789, 34377}, {17790, 52043}, {17863, 21287}, {17889, 32853}, {17923, 22128}, {18025, 35157}, {18041, 18161}, {18042, 18162}, {18049, 18727}, {18133, 20245}, {18145, 49999}, {18147, 18148}, {18151, 30807}, {18194, 25572}, {18480, 21276}, {18653, 30606}, {18690, 45797}, {18713, 18725}, {18714, 18726}, {18715, 18728}, {18716, 18729}, {18717, 18730}, {18718, 18731}, {18719, 18732}, {18720, 18733}, {18721, 18734}, {18722, 18735}, {18750, 26871}, {18816, 20566}, {18827, 35147}, {19742, 26724}, {19808, 32782}, {20017, 50106}, {20073, 29583}, {20086, 33150}, {20142, 25357}, {20335, 37686}, {20345, 25333}, {20349, 21221}, {20878, 23363}, {20886, 30690}, {20888, 33297}, {20920, 48380}, {21061, 29382}, {21290, 36919}, {21342, 29840}, {21356, 29611}, {21358, 29613}, {21873, 27705}, {22003, 22047}, {24165, 32861}, {24214, 33296}, {24215, 34063}, {24220, 29746}, {24248, 49470}, {24325, 33082}, {24330, 31027}, {24331, 50296}, {24352, 31038}, {24470, 41014}, {24512, 31004}, {24514, 30945}, {24589, 37656}, {24603, 31144}, {24628, 24685}, {24712, 30997}, {24725, 30942}, {24789, 37652}, {24833, 29331}, {24851, 35633}, {24864, 35962}, {25284, 28597}, {25384, 27495}, {25957, 32912}, {25959, 33114}, {26132, 37642}, {26223, 33172}, {26540, 26651}, {26580, 37633}, {26756, 27102}, {26772, 27017}, {27002, 37663}, {27396, 40905}, {27509, 28753}, {27526, 40892}, {28538, 50015}, {28580, 49763}, {28609, 30567}, {29570, 41312}, {29571, 50093}, {29572, 41313}, {29573, 49748}, {29575, 49742}, {29578, 49738}, {29586, 41311}, {29588, 50125}, {29596, 50115}, {29605, 50121}, {29607, 40480}, {29619, 50113}, {29620, 49737}, {29643, 36263}, {29649, 33101}, {29658, 36267}, {29660, 50300}, {29674, 32935}, {29687, 32938}, {30596, 44147}, {30598, 30712}, {30829, 31018}, {30854, 52457}, {30966, 30969}, {31017, 32779}, {31134, 33120}, {31145, 49703}, {31252, 51073}, {31289, 34595}, {31637, 36086}, {31993, 37653}, {32771, 33080}, {32774, 37685}, {32793, 32814}, {32801, 32808}, {32802, 32809}, {32803, 32805}, {32804, 32806}, {32858, 32933}, {32866, 42055}, {32915, 33098}, {32942, 41011}, {33076, 49479}, {33086, 46897}, {33095, 42057}, {33112, 46909}, {33133, 37639}, {33142, 48646}, {33149, 49488}, {33170, 48647}, {33869, 49452}, {33890, 44453}, {34255, 42034}, {34282, 50599}, {34920, 40438}, {35102, 49755}, {35160, 43762}, {36479, 51055}, {36480, 50301}, {36534, 47358}, {38989, 39044}, {40940, 41629}, {41138, 41141}, {45222, 50256}, {49508, 49521}

X(320) = midpoint of X(4440) and X(6542)
X(320) = reflection of X(i) in X(j) for these {i,j}: {2, 31138}, {44, 3834}, {190, 3912}, {238, 49676}, {239, 1086}, {1086, 7238}, {1266, 4887}, {1757, 3836}, {3257, 908}, {3685, 4966}, {4480, 2325}, {4693, 49764}, {4700, 17067}, {4753, 25351}, {4969, 4395}, {6542, 17374}, {17160, 1266}, {17264, 17297}, {17487, 4908}, {20072, 44}, {24715, 24692}, {24821, 4439}, {25048, 20358}, {32850, 4645}, {32922, 24231}, {49695, 49675}, {49698, 32850}, {49704, 4864}, {49708, 3244}, {49709, 1}, {49710, 1125}, {49711, 3664}, {49712, 10}, {49713, 3626}, {49714, 8}, {49715, 75}
X(320) = isogonal conjugate of X(6187)
X(320) = isotomic conjugate of X(80)
X(320) = complement of X(20072)
X(320) = anticomplement of X(44)
X(320) = anticomplement of the isogonal conjugate of X(88)
X(320) = anticomplement of the isotomic conjugate of X(20568)
X(320) = isotomic conjugate of the anticomplement of X(214)
X(320) = isotomic conjugate of the complement of X(6224)
X(320) = isotomic conjugate of the isogonal conjugate of X(36)
X(320) = isogonal conjugate of the isotomic conjugate of X(40075)
X(320) = isotomic conjugate of the polar conjugate of X(17923)
X(320) = polar conjugate of the isogonal conjugate of X(22128)
X(320) = anticomplementary isogonal conjugate of X(30578)
X(320) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 30578}, {2, 21290}, {6, 17487}, {56, 30577}, {58, 30579}, {88, 8}, {101, 44009}, {106, 2}, {649, 39349}, {679, 320}, {901, 514}, {903, 69}, {1022, 149}, {1168, 17484}, {1318, 908}, {1320, 329}, {1417, 3210}, {1797, 20}, {2226, 519}, {2316, 144}, {2403, 34548}, {3257, 513}, {4049, 3448}, {4080, 1330}, {4555, 20295}, {4591, 523}, {4615, 512}, {4618, 21297}, {4622, 7192}, {4634, 17217}, {4638, 900}, {4674, 2895}, {4945, 21291}, {4997, 3436}, {5376, 3952}, {5548, 4468}, {6336, 4}, {6548, 150}, {8752, 193}, {9268, 190}, {9456, 192}, {10428, 3218}, {20568, 6327}, {23345, 4440}, {23838, 37781}, {27922, 20345}, {31227, 42020}, {32659, 3164}, {32665, 17494}, {32686, 47773}, {32719, 21225}, {34230, 20533}, {36058, 6360}, {36125, 5905}, {36814, 20355}, {39414, 6548}, {40215, 6224}, {40833, 21283}, {46150, 2896}, {52031, 153}
X(320) = X(i)-Ceva conjugate of X(j) for these (i,j): {7, 41873}, {4615, 514}, {18816, 75}, {20568, 2}, {20924, 32851}, {35156, 693}
X(320) = X(i)-cross conjugate of X(j) for these (i,j): {36, 17923}, {214, 2}, {758, 3218}, {3218, 17078}, {3738, 4585}, {3936, 20924}, {4089, 4453}, {4511, 32851}
X(320) = X(i)-isoconjugate of X(j) for these (i,j): {1, 6187}, {6, 2161}, {19, 52431}, {25, 1807}, {31, 80}, {32, 18359}, {37, 34079}, {41, 2006}, {42, 759}, {55, 1411}, {56, 52371}, {58, 34857}, {106, 40172}, {181, 52380}, {213, 24624}, {265, 14975}, {560, 20566}, {604, 36910}, {650, 32675}, {655, 3063}, {663, 2222}, {667, 51562}, {798, 47318}, {902, 1168}, {1397, 52409}, {1400, 2341}, {1402, 6740}, {1911, 36815}, {1918, 14616}, {1919, 36804}, {1973, 52351}, {1989, 2174}, {2175, 18815}, {2194, 52383}, {2206, 15065}, {2212, 52392}, {2299, 52391}, {3219, 11060}, {3271, 52377}, {3457, 46077}, {3458, 46073}, {4024, 32671}, {4079, 37140}, {4705, 36069}, {6198, 52153}, {9273, 21833}, {9274, 21043}, {11073, 42624}, {11075, 19297}, {18384, 52408}, {34535, 52426}
X(320) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 40612}, {1, 52371}, {2, 80}, {3, 6187}, {6, 40584}, {6, 52431}, {9, 2161}, {10, 34857}, {10, 51583}, {35, 40604}, {36, 20989}, {37, 35069}, {42, 34586}, {55, 35204}, {214, 40172}, {223, 1411}, {226, 52391}, {320, 20072}, {333, 36927}, {484, 3218}, {517, 908}, {519, 3936}, {650, 35128}, {655, 10001}, {663, 38984}, {759, 40592}, {1168, 40594}, {1214, 52383}, {1639, 4530}, {1647, 3960}, {1807, 6505}, {2006, 3160}, {2174, 34544}, {2183, 2245}, {2323, 17796}, {2341, 40582}, {2361, 6149}, {3161, 36910}, {3814, 20962}, {4705, 38982}, {5664, 21054}, {6337, 52351}, {6374, 20566}, {6376, 18359}, {6626, 24624}, {6631, 51562}, {6651, 36815}, {6740, 40605}, {9296, 36804}, {10015, 35015}, {13999, 18344}, {14584, 52659}, {14616, 34021}, {15065, 40603}, {16585, 45926}, {18815, 40593}, {31998, 47318}, {32851, 36926}, {33136, 49758}, {34079, 40589}, {36914, 36920}, {40624, 52356}, {46974, 51361}
X(320) = cevapoint of X(i) and X(j) for these (i,j): {2, 6224}, {7, 36918}, {8, 30578}, {36, 22128}, {519, 908}, {758, 3936}, {3218, 4511}, {4089, 4453}
X(320) = crosspoint of X(i) and X(j) for these (i,j): {75, 40716}, {86, 903}, {4555, 4998}
X(320) = crosssum of X(i) and X(j) for these (i,j): {42, 902}, {1960, 3271}
X(320) = trilinear pole of line {3904, 3960}
X(320) = crossdifference of every pair of points on line {213, 3063}
X(320) = X(320) = barycentric product X(i)*X(j) for these {i,j}: {1, 20924}, {6, 40075}, {7, 32851}, {8, 17078}, {36, 76}, {69, 17923}, {75, 3218}, {79, 7799}, {81, 35550}, {85, 4511}, {86, 3936}, {88, 1227}, {99, 4707}, {190, 4453}, {214, 20568}, {264, 22128}, {274, 758}, {304, 1870}, {305, 52413}, {310, 2245}, {312, 1443}, {314, 18593}, {323, 20565}, {333, 41804}, {334, 27950}, {340, 52381}, {348, 5081}, {561, 7113}, {654, 4572}, {664, 3904}, {668, 3960}, {670, 21828}, {693, 4585}, {860, 17206}, {873, 4053}, {903, 51583}, {1016, 4089}, {1231, 17515}, {1464, 28660}, {1502, 52434}, {1969, 52407}, {1983, 40495}, {2323, 6063}, {2361, 20567}, {2610, 4623}, {3264, 40215}, {3724, 6385}, {3738, 4554}, {4242, 15413}, {4358, 52553}, {4597, 23884}, {4610, 6370}, {4631, 51663}, {4867, 20569}, {4881, 40014}, {4973, 32018}, {4997, 41801}, {6386, 21758}, {16586, 18816}, {27757, 39704}, {27836, 43290}, {28659, 52440}, {30608, 36589}, {36923, 40833}, {40612, 40716}, {41283, 52426}, {42666, 52612}
X(320) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2161}, {2, 80}, {3, 52431}, {6, 6187}, {7, 2006}, {8, 36910}, {9, 52371}, {21, 2341}, {36, 6}, {37, 34857}, {44, 40172}, {57, 1411}, {58, 34079}, {63, 1807}, {69, 52351}, {75, 18359}, {76, 20566}, {79, 1989}, {81, 759}, {85, 18815}, {86, 24624}, {88, 1168}, {99, 47318}, {109, 32675}, {190, 51562}, {214, 44}, {215, 52426}, {226, 52383}, {239, 36815}, {274, 14616}, {312, 52409}, {319, 41226}, {321, 15065}, {323, 35}, {333, 6740}, {340, 52412}, {348, 52392}, {651, 2222}, {654, 663}, {664, 655}, {668, 36804}, {758, 37}, {860, 1826}, {1214, 52391}, {1227, 4358}, {1443, 57}, {1464, 1400}, {1812, 1793}, {1835, 1880}, {1845, 14571}, {1870, 19}, {1983, 692}, {2185, 52380}, {2245, 42}, {2323, 55}, {2361, 41}, {2610, 4705}, {3065, 11075}, {3179, 11072}, {3218, 1}, {3268, 7265}, {3336, 11069}, {3724, 213}, {3738, 650}, {3792, 2276}, {3904, 522}, {3911, 14584}, {3936, 10}, {3960, 513}, {4053, 756}, {4089, 1086}, {4242, 1783}, {4282, 2194}, {4358, 51975}, {4391, 52356}, {4453, 514}, {4511, 9}, {4554, 35174}, {4556, 36069}, {4564, 52377}, {4572, 46405}, {4585, 100}, {4707, 523}, {4736, 4053}, {4867, 45}, {4880, 16777}, {4881, 1743}, {4973, 1100}, {4996, 2323}, {4997, 36590}, {5081, 281}, {5239, 19551}, {5240, 7126}, {5249, 45926}, {5357, 42624}, {6126, 19297}, {6149, 2174}, {6186, 11060}, {6370, 4024}, {7113, 31}, {7799, 319}, {8125, 1128}, {8126, 10215}, {8648, 3063}, {11570, 8609}, {11700, 2182}, {13486, 32678}, {16586, 517}, {16696, 46160}, {16944, 9456}, {17078, 7}, {17080, 34242}, {17455, 902}, {17515, 1172}, {17923, 4}, {18593, 65}, {18815, 34535}, {19619, 34431}, {20565, 94}, {20924, 75}, {21758, 667}, {21828, 512}, {22128, 3}, {22379, 22383}, {22464, 52212}, {23884, 4777}, {24781, 51834}, {27757, 3679}, {27950, 238}, {30578, 36909}, {30690, 2166}, {32851, 8}, {33129, 38938}, {34234, 40437}, {34544, 2361}, {34586, 2183}, {35204, 17796}, {35550, 321}, {36589, 5219}, {36913, 36920}, {36923, 4908}, {37772, 33655}, {37773, 7052}, {39149, 21353}, {39152, 2154}, {39153, 2153}, {40075, 76}, {40215, 106}, {40584, 20989}, {40605, 36927}, {40612, 484}, {40988, 21805}, {41225, 11073}, {41801, 3911}, {41804, 226}, {41873, 37759}, {42666, 4079}, {42701, 3678}, {44113, 2333}, {44428, 3064}, {46398, 35015}, {51402, 4530}, {51583, 519}, {52059, 52434}, {52368, 16548}, {52381, 265}, {52407, 48}, {52413, 25}, {52414, 6198}, {52426, 2175}, {52427, 607}, {52434, 32}, {52440, 604}, {52553, 88}
X(320) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 4389, 17320}, {1, 4655, 24723}, {1, 17274, 4389}, {2, 4643, 17256}, {2, 4644, 3758}, {2, 4741, 4643}, {2, 20072, 44}, {2, 32859, 33066}, {6, 3662, 16706}, {6, 7232, 3662}, {6, 17290, 17367}, {7, 8, 42697}, {7, 69, 75}, {7, 75, 7321}, {7, 17361, 319}, {7, 21296, 69}, {7, 32099, 31995}, {8, 69, 17360}, {8, 17360, 319}, {8, 42697, 75}, {8, 49714, 49698}, {9, 17234, 17263}, {9, 17298, 17234}, {10, 4896, 50116}, {31, 33069, 33124}, {37, 6646, 17258}, {37, 17300, 17317}, {37, 17345, 6646}, {37, 17376, 17300}, {38, 32949, 33073}, {42, 33067, 33068}, {44, 3834, 2}, {44, 28362, 27637}, {44, 31138, 3834}, {44, 31243, 6687}, {45, 17313, 17244}, {63, 18134, 33116}, {69, 75, 319}, {69, 7321, 5564}, {69, 21296, 17361}, {69, 42696, 32099}, {69, 42697, 8}, {75, 319, 5564}, {75, 17360, 8}, {75, 17361, 69}, {75, 17791, 3262}, {75, 46749, 1441}, {80, 32097, 20568}, {80, 32109, 32032}, {81, 17184, 19786}, {86, 4357, 17322}, {86, 17273, 4357}, {89, 30991, 2}, {141, 894, 17289}, {141, 17365, 894}, {141, 17369, 17292}, {142, 4416, 17277}, {144, 344, 17336}, {144, 4869, 344}, {145, 4346, 50101}, {171, 33064, 33126}, {190, 3912, 17264}, {190, 17297, 3912}, {192, 4851, 17315}, {192, 17375, 4851}, {193, 4000, 3759}, {239, 1086, 37756}, {319, 7321, 75}, {319, 49715, 49698}, {322, 39126, 75}, {329, 18141, 18743}, {344, 4869, 17241}, {350, 30941, 41851}, {594, 3631, 17287}, {594, 7228, 17116}, {599, 4363, 3661}, {894, 17288, 141}, {894, 17292, 17369}, {903, 9460, 36594}, {903, 17160, 1266}, {982, 32946, 33071}, {1086, 4969, 4395}, {1100, 17235, 17302}, {1125, 49710, 238}, {1266, 4887, 903}, {1278, 17373, 17299}, {1443, 36589, 41804}, {1443, 41804, 17078}, {1449, 17304, 17380}, {1654, 26806, 3739}, {1743, 17282, 17352}, {1944, 26932, 37774}, {2325, 4480, 190}, {2345, 3620, 17228}, {2887, 32913, 33121}, {2895, 4359, 4886}, {2895, 26842, 4359}, {3187, 33146, 19796}, {3218, 3936, 32851}, {3218, 27757, 51583}, {3244, 49708, 49695}, {3262, 30806, 17791}, {3589, 7277, 17120}, {3589, 48632, 17291}, {3619, 5749, 17371}, {3629, 17366, 17121}, {3629, 48631, 17366}, {3630, 7263, 17362}, {3631, 7228, 594}, {3644, 17386, 17314}, {3661, 50128, 4363}, {3662, 17364, 6}, {3662, 17367, 17290}, {3663, 3879, 4360}, {3664, 4357, 86}, {3664, 17273, 17322}, {3729, 17296, 17233}, {3739, 17344, 1654}, {3758, 17227, 2}, {3759, 48629, 4000}, {3834, 6687, 31243}, {3873, 6327, 4514}, {3875, 4862, 4398}, {3912, 4480, 2325}, {3936, 51583, 27757}, {3945, 17321, 17394}, {4001, 5249, 333}, {4357, 49711, 238}, {4361, 40341, 17363}, {4364, 17392, 16826}, {4389, 17378, 1}, {4395, 4969, 239}, {4398, 17377, 3875}, {4407, 50299, 36531}, {4419, 17316, 4664}, {4445, 17118, 48628}, {4454, 29616, 50107}, {4643, 4675, 2}, {4648, 17257, 4687}, {4664, 17387, 17316}, {4667, 17023, 46922}, {4667, 50092, 17023}, {4670, 17237, 2}, {4675, 4741, 17256}, {4687, 17329, 17257}, {4699, 17343, 17275}, {4700, 17067, 41140}, {4751, 17328, 966}, {4851, 17276, 192}, {4888, 17272, 10436}, {5224, 10436, 28653}, {6646, 17300, 37}, {6646, 17376, 17317}, {6687, 31243, 2}, {7232, 17364, 16706}, {7263, 17362, 17117}, {7277, 48632, 3589}, {7768, 33940, 5015}, {10436, 17272, 5224}, {15668, 17253, 17248}, {16777, 17255, 17247}, {16826, 17254, 4364}, {16884, 17323, 17396}, {16885, 17265, 17338}, {17023, 50092, 17305}, {17116, 17287, 594}, {17120, 17291, 3589}, {17139, 30941, 30939}, {17140, 20290, 33075}, {17231, 17351, 17280}, {17232, 17350, 17279}, {17234, 17347, 9}, {17236, 17379, 4657}, {17241, 17336, 344}, {17243, 17334, 17261}, {17244, 17333, 45}, {17245, 17332, 17260}, {17246, 17390, 17319}, {17247, 17391, 16777}, {17249, 17394, 17321}, {17250, 39704, 41847}, {17250, 41847, 2}, {17258, 17317, 37}, {17261, 17312, 17243}, {17262, 17311, 17242}, {17274, 17378, 17320}, {17276, 17375, 17315}, {17280, 31300, 17351}, {17284, 50127, 17354}, {17288, 17365, 17289}, {17290, 17367, 16706}, {17292, 17369, 17289}, {17298, 17347, 17263}, {17300, 17345, 17258}, {17302, 20090, 1100}, {17305, 46922, 17023}, {17330, 34824, 16815}, {17331, 27147, 17259}, {17345, 17376, 37}, {17353, 21255, 17283}, {17363, 48627, 4361}, {17368, 48633, 3763}, {17371, 48638, 3619}, {17483, 32863, 321}, {17491, 29824, 5057}, {17778, 26840, 3666}, {20347, 30941, 350}, {20568, 32032, 80}, {21281, 36854, 24524}, {21356, 29611, 48639}, {22165, 49727, 29615}, {24893, 25658, 2}, {25957, 32912, 33118}, {26768, 26816, 2}, {26975, 27106, 2}, {27036, 27159, 2}, {27265, 27315, 2}, {27757, 51583, 32851}, {29601, 50090, 4029}, {30807, 37788, 18151}, {30939, 39995, 350}, {30946, 30962, 30963}, {31151, 49712, 10}, {31995, 32099, 42696}, {31995, 42696, 75}, {32097, 32109, 80}, {32850, 49714, 8}, {32850, 49715, 5564}, {32858, 32933, 42033}, {36589, 41801, 17078}, {36928, 36929, 36920}, {41801, 41804, 1443}, {49479, 50304, 33076}, {49511, 50307, 5263}, {49675, 49708, 3244}, {49676, 49710, 1125}, {49676, 49711, 86}


X(321) = ISOTOMIC CONJUGATE OF X(81)

Trilinears    (b + c)b2c2 : :
Trilinears    a(b + c)csc(A - ω) : :
Barycentrics   bc(b + c) : :
Barycentrics   |AP(1)| + |AU(1)| : :

Let A26B26C26 be Gemini triangle 26. Let A' be the perspector of conic {A,B,C,B26,C26}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(321). (Randy Hutson, January 15, 2019)

X(321) lies on the Kiepert hyperbola and these lines: 1,964   2,37   4,8   10,756   38,726   76,561  81,314   83,213   98,100   190,333   226,306   310,335   313,594   319,1029   668,671   693,824

X(321) = reflection of X(42) in X(1215)
X(321) = isogonal conjugate of X(1333)
X(321) = isotomic conjugate of X(81)
X(321) = anticomplement of X(3666)
X(321) = X(i)-Ceva conjugate of X(j) for (i,j) = (75,10), (76,313), (312,306)
X(321) = cevapoint of X(37) and X(72)
X(321) = X(442)-cross conjugate of X(264)
X(321) = crosspoint of X(i) and X(j) for these (i,j): (75,76), (313,349)
X(321) = crosssum of X(31) and X(32)
X(321) = crossdifference of every pair of points on line X(667)X(838)
X(321) = homothetic center of ABC and inverse of n(Medial)*n(Incentral) triangle
X(321) = pole wrt polar circle of trilinear polar of X(28) (line X(513)X(1430))
X(321) = polar conjugate of X(28)
X(321) = Danneels point of X(75)
X(321) = inverse-in-Fuhrmann-circle of X(5016)
X(321) = barycentric product X(100)*X(850)
X(321) = perspector of ABC and cross-triangle of Gemini triangles 13 and 14
X(321) = perspector of ABC and cross-triangle of ABC and Gemini triangle 13
X(321) = perspector of ABC and cross-triangle of ABC and Gemini triangle 14
X(321) = perspector of ABC and cross-triangle of ABC and Gemini triangle 20
X(321) = barycentric product of vertices of Gemini triangle 16
X(321) = barycentric product of vertices of Gemini triangle 20
X(321) = perspector of Gemini triangle 22 and cross-triangle of ABC and Gemini triangle 22
X(321) = trilinear pole of line X(523)X(1577) (the perspectrix of ABC and Gemini triangles 21 and 27)


X(322) = ISOTOMIC CONJUGATE OF X(84)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (-1 - cos A + cos B + cos C)csc2A
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (-1 - cos A + cos B + cos C)csc A

X(322) lies on these lines: 2,1108   7,8   78,273   92,264   227,347   253,341   286,1043   304,309   326,664

X(322) = isogonal conjugate of X(2208)
X(322) = isotomic conjugate of X(84)
X(322) = complement of polar conjugate of isogonal conjugate of X(23168)
X(322) = anticomplement of X(1108)
X(322) = X(304)-Ceva conjugate of X(312)
X(322) = X(347)-cross conjugate of X(75)
X(322) = polar conjugate of X(7129)


X(323) = ISOTOMIC CONJUGATE OF X(94)

Trilinears    sin 3A csc2A : sin 3B csc2B : sin 3C csc2C
Trilinears    4 sin A - 3 csc A :
Barycentrics    sin 3A csc A : sin 3B csc B : sin 3C csc C
Barycentrics    a^2[(a^2 - b^2 - c^2)^2 - b^2 c^2] : :
Barycentrics    directed distance of A to Hatzipolakis axis : :

Let A'B'C' be the Trinh triangle. Let A″ be the trilinear pole of line B'C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(323). (Randy Hutson, October 13, 2015)

X(323) lies on these lines: 2,6   20,155   23,110   30,146   140,195   187,249   401,525

X(323) = reflection of X(23) in X(110)
X(323) = isogonal conjugate of X(1989)
X(323) = isotomic conjugate of X(94)
X(323) = complement of X(37779)
X(323) = anticomplement of X(3580)
X(323) = X(340)-Ceva conjugate of X(186)
X(323) = cevapoint of X(6) and X(399)
X(323) = X(50)-cross conjugate of X(186)
X(323) = crosssum of X(395) and X(396)
X(323) = crosssum of X(36298) and X(36299)
X(323) = crossdifference of every pair of points on line X(51)X(512)
X(323) = crosspoint of X(6) and X(399) wrt both the excentral and tangential triangles
X(323) = inverse-in-MacBeath-circumconic of X(2)
X(323) = orthocentroidal-to-ABC similarity image of X(110)
X(323) = 4th-Brocard-to-circumsymmedial similarity image of X(110)
X(323) = X(13192)-of-circumsymmedial-triangle
X(323) = Vu circlecevian point V(X(15),X(16))
X(323) = intersection of the tangent to hyperbola {A,B,C,X(6),X(13),X(16)}} at X(16) and the tangent to hyperbola {A,B,C,X(6),X(14),X(15)}} at X(15)


X(324) = ISOTOMIC CONJUGATE OF X(97)

Trilinears       bc sec A cos(B - C) : ca sec B cos(C - A) : ab sec C cos(A - B)
Barycentrics  sec A cos(B - C) : sec B cos(C - A) : sec C cos(A - B)

X(324) lies on these lines: 2,216   4,52   53,311   94,275   110,436   143,565

X(324) = isotomic conjugate of X(97)
X(324) = X(264)-Ceva conjugate of X(5)
X(324) = cevapoint of X(i) and X(j) for these (i,j): (5,53), (52,216)
X(324) = X(5)-cross conjugate of X(311)
X(324) = trilinear pole of polar of X(54) wrt polar circle
X(324) = pole wrt polar circle of trilinear polar of X(54) (line X(50)X(647))
X(324) = X(48)-isoconjugate (polar conjugate) of X(54)
X(324) = Danneels point of X(264)


X(325) = X(2)-HIRST INVERSE OF X(69)

Trilinears    cos(A+ω) csc2A : :
Trilinears    bc(b4 + c4 - a2b2 - a2c2) : :
Barycentrics    cot A - tan ω : :
Barycentrics    b4 + c4 - a2b2 - a2c2 : :
Barycentrics    SA^2 - SB*SC : :
Barycentrics    Cot[A]^2 - Cot[B]*Cot[C] : :
Barycentrics    Cos[A + w]*Csc[A] : :
X(325) = 3 X[2] + X[7779], 3 X[2] - 5 X[7925], 5 X[2] - 3 X[8859], 9 X[2] - 2 X[15480], 2 X[2] - 3 X[41133], X[2] + 3 X[41136], 7 X[2] - 6 X[41139], 5 X[2] - X[44367], 3 X[2] - 4 X[44377], 9 X[2] - 8 X[44381], 5 X[2] - 4 X[44401], 9 X[2] - X[50248], 6 X[2] - X[50251], 3 X[2] + 2 X[50771], 9 X[2] - 4 X[50774], 2 X[230] + X[7779], 2 X[230] + 3 X[7840], and many others

Let La be the line through A parallel to the Lemoine axis, and define Lb and Lc cyclically. Let Ma be the reflection of BC in La, and define Mb and Mc cyclically. Let A' = Mb∩Mc, and define B' and C' cyclically. The triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. Let A″B″C″ be the reflection of A'B'C' in the Lemoine axis. The triangle A″B″C″ is homothetic to ABC, with center of homothety X(325); see Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, November 30, 2015)

X(325) lies on the curves K357, K718, K738, K741, K777, K778, K779, K953, K1023, Q101, Q166, and these lines: {2, 6}, {3, 315}, {4, 1975}, {5, 76}, {8, 17084}, {10, 50254}, {11, 350}, {12, 1909}, {20, 6337}, {22, 160}, {23, 3447}, {25, 317}, {30, 99}, {32, 3788}, {39, 626}, {51, 4121}, {58, 56731}, {74, 2855}, {75, 2886}, {83, 7819}, {85, 15844}, {95, 1799}, {98, 2065}, {100, 2856}, {110, 2857}, {111, 2858}, {114, 511}, {115, 538}, {116, 30109}, {120, 33677}, {126, 9151}, {127, 14961}, {132, 44704}, {140, 1078}, {147, 1503}, {148, 14041}, {172, 26686}, {182, 37450}, {187, 620}, {190, 17747}, {194, 5025}, {232, 297}, {235, 54412}, {237, 20022}, {250, 340}, {253, 6340}, {257, 21965}, {264, 305}, {274, 442}, {286, 37362}, {287, 41932}, {290, 3978}, {304, 17181}, {308, 45093}, {310, 3136}, {311, 1238}, {313, 44411}, {314, 37360}, {319, 7081}, {320, 1447}, {322, 20935}, {328, 18019}, {332, 2893}, {334, 19567}, {337, 17789}, {339, 1236}, {376, 11181}, {381, 11185}, {383, 622}, {384, 7745}, {386, 56732}, {401, 39359}, {403, 44146}, {420, 59651}, {428, 16276}, {441, 2966}, {487, 45406}, {488, 45407}, {489, 12306}, {490, 12305}, {519, 15903}, {523, 684}, {525, 46245}, {532, 6108}, {533, 6109}, {540, 6629}, {542, 5939}, {543, 8352}, {549, 7771}, {550, 7782}, {574, 7761}, {594, 30179}, {612, 55392}, {614, 55391}, {621, 1080}, {623, 6115}, {624, 6114}, {631, 3785}, {633, 37463}, {634, 37464}, {637, 6289}, {638, 6290}, {639, 3103}, {640, 3102}, {648, 16318}, {664, 35149}, {668, 17757}, {670, 46142}, {671, 5503}, {672, 4766}, {698, 1916}, {732, 2023}, {740, 5988}, {758, 5977}, {868, 2396}, {877, 35908}, {892, 1494}, {903, 46143}, {1003, 7737}, {1030, 56772}, {1043, 7379}, {1086, 33891}, {1107, 26558}, {1125, 50250}, {1146, 49755}, {1193, 24995}, {1228, 37983}, {1232, 37990}, {1235, 1594}, {1272, 10989}, {1281, 17768}, {1285, 33191}, {1312, 15164}, {1313, 15165}, {1329, 6376}, {1330, 6998}, {1350, 37182}, {1351, 9753}, {1352, 13860}, {1368, 6374}, {1369, 15246}, {1370, 20477}, {1384, 11288}, {1444, 4220}, {1506, 3934}, {1509, 49743}, {1529, 14944}, {1552, 16077}, {1565, 20924}, {1575, 20541}, {1596, 58782}, {1609, 42406}, {1625, 54074}, {1655, 17669}, {1656, 32832}, {1691, 10352}, {1834, 33296}, {1914, 26629}, {1930, 30171}, {1971, 58354}, {2001, 19156}, {2071, 5866}, {2076, 59695}, {2080, 15561}, {2271, 56563}, {2275, 26561}, {2276, 26590}, {2373, 10420}, {2387, 14962}, {2418, 38951}, {2476, 34284}, {2482, 3849}, {2493, 34827}, {2548, 7770}, {2549, 7841}, {2782, 15980}, {2794, 18860}, {2799, 8430}, {2896, 7824}, {2967, 6530}, {2996, 32980}, {3053, 16925}, {3090, 32828}, {3091, 32830}, {3096, 7786}, {3137, 55254}, {3138, 55256}, {3139, 55258}, {3140, 55260}, {3141, 55262}, {3142, 28660}, {3146, 32841}, {3233, 6148}, {3291, 23991}, {3292, 47200}, {3363, 59780}, {3407, 10334}, {3454, 16887}, {3491, 40951}, {3523, 32835}, {3525, 32839}, {3530, 9751}, {3545, 32836}, {3552, 7823}, {3616, 50247}, {3663, 49554}, {3665, 33930}, {3670, 17211}, {3695, 33939}, {3703, 33931}, {3704, 17762}, {3734, 5475}, {3760, 7741}, {3761, 7951}, {3767, 7754}, {3770, 50036}, {3813, 17144}, {3814, 6381}, {3816, 30963}, {3818, 58851}, {3829, 4479}, {3832, 32840}, {3836, 40533}, {3845, 48913}, {3846, 4357}, {3850, 15031}, {3875, 33141}, {3912, 26012}, {3932, 4518}, {3943, 33889}, {3944, 49518}, {3948, 26019}, {3972, 7812}, {4027, 12830}, {4071, 24318}, {4087, 51411}, {4136, 17760}, {4187, 18140}, {4193, 18135}, {4202, 27162}, {4230, 52486}, {4252, 56733}, {4360, 29840}, {4441, 11680}, {4554, 16090}, {4558, 10313}, {4561, 16086}, {4609, 18901}, {4611, 10317}, {4643, 36405}, {4872, 32851}, {4966, 41851}, {4987, 24628}, {5000, 44780}, {5001, 44781}, {5007, 6680}, {5008, 41750}, {5013, 7784}, {5016, 27187}, {5021, 56562}, {5023, 32964}, {5024, 11287}, {5026, 53499}, {5041, 7829}, {5051, 16705}, {5055, 53127}, {5056, 32834}, {5067, 32838}, {5071, 46951}, {5077, 11165}, {5094, 36207}, {5107, 14645}, {5111, 36849}, {5112, 9155}, {5124, 56771}, {5149, 5162}, {5159, 16315}, {5167, 55005}, {5169, 9464}, {5189, 47245}, {5215, 22247}, {5277, 17694}, {5280, 30103}, {5286, 7851}, {5299, 30104}, {5305, 7760}, {5309, 7798}, {5319, 33218}, {5355, 7817}, {5480, 13862}, {5523, 41676}, {5613, 51018}, {5617, 51016}, {5921, 53015}, {5965, 6036}, {5969, 53505}, {5980, 11133}, {5981, 11132}, {5982, 22508}, {5983, 22506}, {5992, 28530}, {6007, 51464}, {6031, 54087}, {6038, 25046}, {6039, 51899}, {6040, 51898}, {6103, 56021}, {6179, 7857}, {6292, 6683}, {6331, 16089}, {6333, 32112}, {6353, 32001}, {6392, 32972}, {6527, 7396}, {6528, 51385}, {6626, 19312}, {6636, 44180}, {6655, 7783}, {6661, 7753}, {6676, 33651}, {6677, 22468}, {6693, 17200}, {6720, 52950}, {6722, 31275}, {7000, 12323}, {7124, 27516}, {7200, 21057}, {7270, 17095}, {7374, 12322}, {7380, 10449}, {7386, 22263}, {7394, 45795}, {7467, 20775}, {7473, 47150}, {7495, 26233}, {7603, 9466}, {7618, 35955}, {7667, 16275}, {7738, 32974}, {7739, 33219}, {7746, 7751}, {7747, 7816}, {7748, 7781}, {7749, 7780}, {7755, 7805}, {7756, 7842}, {7757, 7790}, {7765, 7861}, {7772, 7834}, {7787, 7892}, {7793, 7893}, {7797, 7839}, {7800, 7879}, {7803, 7866}, {7808, 7822}, {7810, 7848}, {7815, 7854}, {7827, 7919}, {7830, 7873}, {7831, 7883}, {7833, 7898}, {7846, 7878}, {7847, 7911}, {7856, 7894}, {7859, 7944}, {7864, 7933}, {7865, 15482}, {7876, 7938}, {7889, 7915}, {7904, 7929}, {7920, 7932}, {7928, 33021}, {7935, 53096}, {8149, 32452}, {8229, 17139}, {8290, 9866}, {8355, 9166}, {8368, 12150}, {8588, 47101}, {8591, 8597}, {8703, 9774}, {8716, 33017}, {8822, 37443}, {8840, 9418}, {8878, 16951}, {8889, 32000}, {9066, 32730}, {9080, 9184}, {9164, 51541}, {9187, 17948}, {9189, 33915}, {9477, 17949}, {9496, 35971}, {9711, 16284}, {9737, 54393}, {9749, 14538}, {9750, 14539}, {9755, 22525}, {9756, 15069}, {9760, 50858}, {9762, 50855}, {9769, 32244}, {9772, 22503}, {9818, 22241}, {9855, 52695}, {9867, 33340}, {9868, 33341}, {9877, 50639}, {9939, 33274}, {9993, 21850}, {9996, 37345}, {10008, 58883}, {10011, 34380}, {10295, 46987}, {10302, 54509}, {10311, 27377}, {10316, 28697}, {10350, 34870}, {10436, 33111}, {10446, 51612}, {10583, 14043}, {11007, 40877}, {11056, 37454}, {11059, 30739}, {11128, 44219}, {11164, 23334}, {11281, 16823}, {11286, 15484}, {11318, 22253}, {11361, 53418}, {11585, 41009}, {11606, 35005}, {11610, 51454}, {11645, 14928}, {12040, 55164}, {12251, 37446}, {12607, 24524}, {13160, 26166}, {13186, 33919}, {13449, 23698}, {13481, 31074}, {13586, 14712}, {13881, 32961}, {14001, 53033}, {14023, 33233}, {14061, 14568}, {14063, 44518}, {14221, 36170}, {14376, 28695}, {14387, 59258}, {14482, 33196}, {14558, 21395}, {14601, 31635}, {14711, 39601}, {14881, 44230}, {14941, 16083}, {14994, 24206}, {15013, 54075}, {15122, 46633}, {15300, 32479}, {15559, 44142}, {15815, 32965}, {15819, 40107}, {15888, 25303}, {16041, 43448}, {16043, 31400}, {16044, 17128}, {16052, 16712}, {16084, 52608}, {16094, 16101}, {16096, 44326}, {16589, 33034}, {16600, 30170}, {16696, 21245}, {16720, 16886}, {16830, 41879}, {16891, 20966}, {16921, 31276}, {17030, 31466}, {17046, 29960}, {17062, 30038}, {17103, 49745}, {17129, 32967}, {17143, 24390}, {17192, 24786}, {17321, 50255}, {17394, 29634}, {17416, 35087}, {17533, 18145}, {17759, 21956}, {17907, 45141}, {18018, 20563}, {18025, 30790}, {18152, 47513}, {18167, 23639}, {18424, 18546}, {18584, 33005}, {18738, 21596}, {18755, 56561}, {18806, 46313}, {18816, 35147}, {18823, 35179}, {18827, 20337}, {18840, 32957}, {19130, 44422}, {19179, 34386}, {19576, 44347}, {19577, 47298}, {19599, 56376}, {19758, 56994}, {19839, 40071}, {20023, 37988}, {20081, 32966}, {20088, 33225}, {20258, 20528}, {20459, 29990}, {20487, 41318}, {20532, 35080}, {20888, 25639}, {20923, 21239}, {21031, 25280}, {21213, 55551}, {21281, 33298}, {21485, 36744}, {21536, 59567}, {21993, 33863}, {22510, 40335}, {22511, 40334}, {22664, 50641}, {22712, 37451}, {23115, 28405}, {23350, 34765}, {23967, 44578}, {24241, 29671}, {24256, 53484}, {24284, 52038}, {24291, 37717}, {24345, 29857}, {24348, 29639}, {24467, 55470}, {24471, 25135}, {24598, 37096}, {25353, 49516}, {25466, 31997}, {25645, 33953}, {26085, 33830}, {26099, 33819}, {26235, 44148}, {26601, 40773}, {26921, 55469}, {27020, 31460}, {27088, 41134}, {27092, 27515}, {27109, 33839}, {27269, 33046}, {27376, 28728}, {28710, 39575}, {29181, 40236}, {30172, 33942}, {30435, 32954}, {30471, 42942}, {30472, 42943}, {30476, 47229}, {30716, 47177}, {30741, 36223}, {30745, 47242}, {30748, 36227}, {30771, 40995}, {30787, 36239}, {30789, 53379}, {31068, 40511}, {31076, 31088}, {31084, 31130}, {31125, 31132}, {31127, 53348}, {31274, 58448}, {31404, 32968}, {31415, 44543}, {31492, 33258}, {32113, 47557}, {32190, 44772}, {32220, 47550}, {32419, 48728}, {32421, 48729}, {32449, 51848}, {32850, 50441}, {32870, 46936}, {32871, 55864}, {32876, 33703}, {32879, 50689}, {32881, 50693}, {32883, 52718}, {32896, 41099}, {32897, 46935}, {32997, 44519}, {33000, 44535}, {33006, 34505}, {33008, 53095}, {33207, 44541}, {33297, 41014}, {33799, 46517}, {34016, 37047}, {34119, 51857}, {34209, 35139}, {34245, 45662}, {34370, 46787}, {34393, 35154}, {34885, 44224}, {35142, 36898}, {35152, 54987}, {35153, 53647}, {35684, 42060}, {35685, 42009}, {36163, 59227}, {36166, 45772}, {36183, 44155}, {36426, 40887}, {36743, 56773}, {36790, 46807}, {36890, 52488}, {36953, 52898}, {37049, 50156}, {37159, 50177}, {37353, 45090}, {37439, 40022}, {37532, 55416}, {37760, 47246}, {37911, 47240}, {37981, 44138}, {38225, 38750}, {38743, 47618}, {38748, 47113}, {39387, 39648}, {39388, 39679}, {40017, 56171}, {40032, 59756}, {40410, 57852}, {40413, 57800}, {40810, 44114}, {40884, 51372}, {42010, 43535}, {42087, 59539}, {42088, 59540}, {42147, 59541}, {42148, 59542}, {42703, 46238}, {42717, 59734}, {42811, 47612}, {42812, 47613}, {43118, 45508}, {43119, 45509}, {43618, 44678}, {44099, 46096}, {44280, 47000}, {44882, 59552}, {45404, 45459}, {45405, 45458}, {45444, 45714}, {45445, 45713}, {45460, 45471}, {45461, 45470}, {45476, 49330}, {45477, 49329}, {46228, 52979}, {46992, 47313}, {46998, 47244}, {47455, 47561}, {49351, 53480}, {49352, 53479}, {49367, 53512}, {49368, 53515}, {51161, 53431}, {51162, 53443}, {52090, 53765}, {52284, 52710}, {52772, 54215}, {53266, 53347}, {53367, 57607}, {53417, 56023}, {54731, 54841}

X(325) = midpoint of X(i) and X(j) for these {i,j}: {2, 7840}, {69, 39099}, {99, 316}, {115, 7813}, {147, 5999}, {187, 7845}, {230, 50771}, {298, 299}, {385, 7779}, {401, 39359}, {1916, 9865}, {3978, 18829}, {5207, 12215}, {5978, 5979}, {5982, 22508}, {5983, 22506}, {5988, 49544}, {6033, 35002}, {7697, 44775}, {7799, 7809}, {8290, 9866}, {8591, 8597}, {9772, 22503}, {9867, 33340}, {9868, 33341}, {31173, 39785}, {40888, 44363}, {44352, 44370}, {44361, 44362}, {44364, 44365}, {46517, 47154}, {50567, 51396}, {51387, 51388}, {51395, 51401}
reflection of X(i) in X(j) for these {i,j}: {2, 22110}, {6, 44380}, {8, 50772}, {23, 16320}, {98, 56370}, {99, 6390}, {115, 625}, {148, 53419}, {187, 620}, {230, 44377}, {297, 35088}, {385, 230}, {395, 44383}, {396, 44382}, {671, 37350}, {1513, 114}, {1971, 59706}, {2076, 59695}, {2080, 37459}, {2966, 441}, {5976, 51373}, {6393, 51371}, {6781, 32456}, {7426, 46986}, {7779, 50771}, {8352, 31173}, {8598, 2482}, {10295, 46987}, {14999, 11064}, {15480, 50774}, {15993, 141}, {16092, 47097}, {16315, 5159}, {21445, 10256}, {22329, 2}, {32113, 47557}, {32220, 47550}, {36166, 47570}, {44369, 69}, {44375, 44389}, {44392, 44390}, {44394, 44391}, {46633, 15122}, {47229, 30476}, {47286, 115}, {47313, 46992}, {50247, 50776}, {50248, 15480}, {50250, 50775}, {50251, 385}, {50252, 44379}, {50254, 10}, {50567, 51397}, {50774, 44381}, {50775, 1125}, {51224, 27088}, {51374, 6393}, {51412, 51426}, {51427, 59571}, {51438, 50567}, {51439, 51386}, {51440, 51439}, {51441, 21531}, {52038, 24284}, {53475, 5031}, {53499, 5026}, {54996, 18860}, {59634, 7799}
X(325) = isogonal conjugate of X(1976)
X(325) = isotomic conjugate of X(98)
X(325) = complement of X(385)
X(325) = anticomplement of X(230)
X(325) = circumcircle-inverse of X(54088)
X(325) = orthoptic-circle-of-Steiner-circumellipe-inverse of X(38940)
X(325) = orthoptic-circle-of-Steiner-inellipe-inverse of X(5108)
X(325) = polar conjugate of X(6531)
X(325) = antitomic image of X(297)
X(325) = anticomplement of the isogonal conjugate of X(2987)
X(325) = complement of the isogonal conjugate of X(694)
X(325) = anticomplement of the isotomic conjugate of X(8781)
X(325) = complement of the isotomic conjugate of X(1916)
X(325) = isotomic conjugate of the anticomplement of X(114)
X(325) = isotomic conjugate of the complement of X(147)
X(325) = isotomic conjugate of the isogonal conjugate of X(511)
X(325) = isotomic conjugate of the polar conjugate of X(297)
X(325) = isogonal conjugate of the polar conjugate of X(44132)
X(325) = polar conjugate of the isotomic conjugate of X(6393)
X(325) = polar conjugate of the isogonal conjugate of X(36212)
X(325) = medial-isogonal conjugate of X(39080)
X(325) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1821, 56688}, {2987, 8}, {3563, 5905}, {8773, 69}, {8781, 6327}, {10425, 7192}, {32654, 192}, {32697, 7253}, {35142, 21270}, {35364, 21221}, {36051, 2}, {36105, 850}, {42065, 6360}, {43705, 4329}, {56109, 329}
X(325) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 39080}, {6, 19563}, {31, 5976}, {256, 20333}, {257, 20542}, {292, 51575}, {661, 2679}, {694, 10}, {733, 1215}, {798, 35078}, {805, 4369}, {881, 16592}, {882, 8287}, {893, 17793}, {904, 17755}, {1581, 141}, {1755, 46840}, {1911, 59509}, {1916, 2887}, {1927, 39}, {1934, 626}, {1965, 39082}, {1967, 2}, {2084, 39079}, {3572, 40608}, {3903, 27854}, {8789, 16584}, {9468, 37}, {14251, 16591}, {14946, 18905}, {14970, 21238}, {17938, 14838}, {17970, 1214}, {17980, 226}, {18829, 42327}, {18872, 16597}, {18896, 21235}, {34238, 16609}, {36214, 18589}, {37134, 512}, {40729, 35068}, {43763, 3934}, {52651, 45162}, {56978, 21249}
X(325) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 5976}, {76, 36790}, {287, 1975}, {297, 51374}, {877, 6333}, {2396, 2799}, {3978, 698}, {8781, 2}, {10425, 6563}, {17941, 9479}, {18024, 76}, {18829, 523}, {18896, 141}, {20022, 511}, {35140, 69}, {35142, 52091}, {36214, 52636}, {43187, 525}, {44132, 297}, {51370, 1959}, {57861, 290}, {57991, 99}
X(325) = X(i)-isoconjugate of X(j) for these (i,j): {1, 1976}, {6, 1910}, {19, 248}, {25, 293}, {31, 98}, {32, 1821}, {48, 6531}, {63, 57260}, {75, 14601}, {82, 51869}, {92, 14600}, {162, 878}, {163, 2395}, {184, 36120}, {251, 3404}, {287, 1973}, {290, 560}, {336, 1974}, {512, 36084}, {604, 15628}, {647, 36104}, {656, 32696}, {661, 2715}, {662, 2422}, {669, 36036}, {685, 810}, {798, 2966}, {822, 20031}, {879, 32676}, {922, 9154}, {923, 5967}, {1096, 17974}, {1101, 51441}, {1110, 43920}, {1501, 46273}, {1580, 34238}, {1755, 41932}, {1917, 18024}, {1924, 43187}, {1927, 14382}, {1933, 36897}, {1967, 40820}, {2065, 8772}, {2156, 11610}, {2159, 35906}, {2186, 51542}, {2643, 57742}, {3288, 36132}, {3402, 46806}, {4575, 53149}, {9247, 16081}, {9417, 34536}, {15391, 56828}, {15630, 24041}, {20021, 46289}, {32540, 43761}, {36051, 51820}, {36142, 52038}, {47388, 57653}
X(325) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 98}, {3, 1976}, {6, 248}, {9, 1910}, {39, 20021}, {114, 51820}, {115, 2395}, {125, 878}, {132, 25}, {136, 53149}, {141, 51869}, {206, 14601}, {232, 16318}, {297, 41204}, {325, 385}, {441, 1503}, {511, 237}, {514, 43920}, {523, 51441}, {647, 51404}, {868, 55122}, {1084, 2422}, {1249, 6531}, {1959, 1756}, {2482, 5967}, {2491, 2086}, {2679, 669}, {2799, 868}, {3005, 15630}, {3150, 53265}, {3161, 15628}, {3162, 57260}, {3163, 35906}, {5976, 2}, {6337, 287}, {6338, 6394}, {6374, 290}, {6376, 1821}, {6393, 19599}, {6503, 17974}, {6505, 293}, {8290, 40820}, {8623, 1691}, {9428, 43187}, {11672, 6}, {15526, 879}, {15595, 34156}, {16609, 1284}, {22391, 14600}, {23967, 34369}, {23976, 51963}, {23992, 52038}, {31998, 2966}, {33569, 59804}, {34834, 52451}, {35073, 36822}, {35088, 523}, {36212, 3564}, {36790, 52006}, {36830, 2715}, {36899, 41932}, {36901, 43665}, {38970, 2501}, {38987, 512}, {39000, 647}, {39009, 3288}, {39039, 19}, {39040, 1}, {39052, 36104}, {39054, 36084}, {39058, 34536}, {39061, 9154}, {39062, 685}, {39073, 42671}, {39080, 32540}, {39081, 32545}, {39092, 34238}, {40585, 3404}, {40596, 32696}, {40601, 32}, {40604, 14355}, {41167, 20975}, {41172, 3569}, {46094, 184}, {47648, 694}, {48316, 2451}, {50440, 55}, {50567, 47047}, {51389, 34810}, {51580, 46806}, {52032, 53174}, {52878, 40981}, {55071, 14270}, {55267, 115}, {59734, 2238}
X(325) = cevapoint of X(i) and X(j) for these (i,j): {2, 147}, {69, 57009}, {297, 44704}, {441, 3564}, {511, 36212}, {868, 2799}, {1959, 44694}, {3569, 58260}, {3978, 44137}, {23098, 36790}
X(325) = X(i)-cross conjugate of X(j) for these (i,j): (114,2), (511,297)
X(325) = crossdifference of every pair of points on line X(32)X(512)
X(325) = X(2)-Hirst inverse of X(69)
X(325) = {X(2),X(69)}-harmonic conjugate of X(183)
X(325) = perspector of hyperbola {A,B,C,X(99),PU(37)}}
X(325) = intersection of trilinear polars of X(99), P(37), and U(37)
X(325) = crosspoint of X(2) and X(147) wrt both the excentral and anticomplementary triangles
X(325) = trilinear pole of line X(2799)X(3569)
X(325) = X(115)-of-1st-anti-Brocard-triangle
X(325) = X(114)-of-anti-McCay-triangle
X(325) = intersection of Simson line of X(99) (line X(114)X(325)) and trilinear polar of X(99) (line X(2)X(6))
X(325) = pole wrt polar circle of trilinear polar of X(6531) (line X(25)X(669))
X(325) = X(48)-isoconjugate (polar conjugate) of X(6531)
X(325) = polar conjugate of isogonal conjugate of X(36212)
X(325) = X(i)-line conjugate of X(j) for these (i,j): {114, 46627}, {3788, 32}
X(325) = barycentric product X(i)*X(j) for these {i,j}: {1, 46238}, {3, 44132}, {4, 6393}, {10, 51370}, {63, 40703}, {69, 297}, {75, 1959}, {76, 511}, {81, 42703}, {83, 51371}, {85, 44694}, {94, 51383}, {98, 32458}, {99, 2799}, {114, 8781}, {141, 20022}, {183, 46807}, {232, 305}, {237, 1502}, {240, 304}, {262, 51373}, {264, 36212}, {276, 44716}, {290, 36790}, {313, 17209}, {315, 34138}, {321, 51369}, {523, 2396}, {525, 877}, {561, 1755}, {598, 51397}, {648, 6333}, {670, 3569}, {671, 50567}, {684, 6331}, {693, 42717}, {850, 2421}, {868, 4590}, {1236, 36823}, {1494, 51389}, {1513, 40824}, {1916, 5976}, {1928, 9417}, {1930, 3405}, {1978, 53521}, {2052, 51386}, {2211, 40050}, {2450, 42407}, {2491, 4609}, {2967, 57799}, {2996, 51374}, {3260, 35910}, {3266, 5968}, {3267, 4230}, {3289, 18022}, {3314, 8840}, {3596, 43034}, {3926, 6530}, {3978, 40810}, {4563, 16230}, {5360, 6385}, {5392, 51439}, {5485, 51438}, {6063, 59734}, {6394, 36426}, {6786, 34087}, {7799, 14356}, {8024, 51862}, {9155, 18023}, {9418, 40362}, {9464, 52692}, {10302, 51396}, {11140, 51440}, {11672, 18024}, {14251, 14603}, {14501, 57575}, {14502, 57576}, {14966, 44173}, {14999, 34765}, {15595, 35140}, {15631, 43665}, {17994, 52608}, {18896, 36213}, {19189, 28706}, {20023, 51543}, {20948, 23997}, {23098, 57541}, {23996, 46273}, {27818, 44728}, {28659, 51651}, {30736, 52765}, {32014, 51417}, {32833, 56925}, {34386, 39569}, {34403, 44704}, {34537, 44114}, {35088, 57991}, {36892, 47286}, {39931, 40708}, {40017, 50440}, {40162, 51427}, {40364, 57653}, {40706, 51387}, {40707, 51388}, {40804, 44137}, {40831, 51412}, {41167, 43187}, {41198, 44781}, {41199, 44780}, {42702, 57796}, {44168, 58260}, {46235, 46236}, {51429, 52940}, {51481, 52091}, {52617, 58070}
X(325) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1910}, {2, 98}, {3, 248}, {4, 6531}, {6, 1976}, {8, 15628}, {22, 11610}, {25, 57260}, {30, 35906}, {32, 14601}, {38, 3404}, {39, 51869}, {63, 293}, {69, 287}, {75, 1821}, {76, 290}, {92, 36120}, {98, 41932}, {99, 2966}, {107, 20031}, {110, 2715}, {112, 32696}, {114, 230}, {115, 51441}, {125, 51404}, {132, 16318}, {141, 20021}, and many others
X(325) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 6, 7792}, {2, 69, 183}, {2, 183, 37688}, {2, 193, 7735}, {2, 385, 230}, {2, 1654, 26244}, {2, 2895, 26243}, {2, 3314, 141}, {2, 3329, 3589}, {2, 7736, 11174}, {2, 7766, 7806}, {2, 7774, 6}, {2, 7777, 3815}, {2, 7779, 385}, {2, 7788, 37671}, {2, 7837, 5306}, {2, 7897, 3314}, {2, 7925, 44377}, {2, 8859, 44401}, {2, 9740, 23055}, {2, 9766, 41624}, {2, 9770, 11163}, {2, 10513, 15589}, {2, 15589, 34229}, {2, 16990, 15271}, {2, 17004, 3054}, {2, 17005, 3055}, {2, 17008, 37637}, {2, 20080, 37667}, {2, 22110, 41133}, {2, 31034, 26282}, {2, 31089, 1211}, {2, 31090, 1213}, {2, 31120, 5718}, and many others


X(326) = ISOTOMIC CONJUGATE OF X(158)

Trilinears    cot2A : :
Trilinears    b2 + c2 - S2 : :
Trilinears    (distance from A to orthic axis)2 : :
Barycentrics    csc A - sin A : :

X(326) lies on these lines: 1,75   48,63   69,73   200,319   255,1102   269,320   322,664   610,662

X(326) = isogonal conjugate of X(1096)
X(326) = isotomic conjugate of X(158)
X(326) = X(i)-Ceva conjugate of X(j) for these (i,j): (304,63), (332,69)
X(326) = X(255)-cross conjugate of X(63)
X(326) = trilinear square of X(63)
X(326) = pole wrt polar circle of trilinear polar of X(6520)
X(326) = X(48)-isoconjugate (polar conjugate) of X(6520)
X(326) = X(92)-isoconjugate of X(1973)
X(326) = trilinear pole of line X(822)X(4131)


X(327) = ISOTOMIC CONJUGATE OF X(182)

Trilinears       csc2A sec(A - ω) : csc2B sec(B - ω) : csc2C sec(C - ω)
                        = sin A csc(2A - 2 ω): sin B csc(2B - 2 ω) : sin C csc(2C - 2 ω)

Barycentrics  csc A sec(A - ω) : csc B sec(B - ω) : csc C sec(C - ω)

X(327) lies on these lines: 2,290   4,276   5,76   53,141   69,263   95,160

X(327) = isogonal conjugate of X(34396)
X(327) = isotomic conjugate of X(182)
X(327) = cevapoint of X(2) and X(1352)
X(327) = trilinear pole of line X(850)X(2525)


X(328) = ISOTOMIC CONJUGATE OF X(186)

Trilinears       cot A csc 3A : cot B csc 3B : cot C csc 3C
Barycentrics  cos A csc 3A : cos B csc 3B : cos C csc 3C

X(328) lies on these lines: 2,94   69,265   95,99

X(328) = isogonal conjugate of X(34397)
X(328) = isotomic conjugate of X(186)
X(328) = X(265)-cross conjugate of X(94)
X(328) = cevapoint of X(i) and X(j) for these (i,j): {2, 3153}, {125, 6334}, {311, 3260}, {525, 16186}
X(328) = trilinear pole of line X(343)X(525)


X(329)  = ISOTOMIC CONJUGATE OF X(189)

Trilinears    ( -1 - cos A + cos B + cos C)(csc A) : :
Barycentrics    -1 - cos A + cos B + cos C : :
Barycentrics    a^3 + a^2(b + c) - a(b + c)^2 - (b - c)^2(b + c) : :
X(329) = 3X(4) - 4X(10) = 2X(10) - 3X(40) = 3X(20) - X(145) = 2X(1) - 3X(376)

X(329) lies on the Lucas cubic and these lines: {1,452}, {2,7}, {3,2096}, {4,8}, {5,2095}, {6,4415}, {10,2093}, {11,5825}, {20,78}, {69,189}, {75,14555}, {100,972}, {165,6745}, {193,1999}, {198,1817}, {220,948}, {440,3161}, {971,10430}, {5687,6361}, {5817,7956}.

X(329) = isogonal conjugate of X(1436)
X(329) = isotomic conjugate of X(189)
X(329) = cyclocevian conjugate of X(1034)
X(329) = anticomplement of X(57)
X(329) = anticomplementary conjugate of X(7)
X(329) = X(i)-Ceva conjugate of X(j) for (i,j) = (69,8), (312,2)
X(329) = X(i)-cross conjugate of X(j) for these (i,j): (40,347), (223,2)
X(329) = perspector of triangle ABC and the pedal triangle of X(1490)
X(329) = X(25)-of-2nd-extouch-triangle
X(329) = perspector of 2nd extouch triangle and anticevian triangle of X(8)
X(329) = perspector of ABC and the reflection in X(9) of the pedal triangle of X(9)
X(329) = inverse-in-Fuhrmann-circle of X(5175)
X(329) = orthologic center of 2nd extouch triangle and 1st (or 2nd) mixtilinear triangle
X(329) = trilinear pole of line X(6129)X(8058)
X(329) = perspector of 2nd extouch triangle and cross-triangle of ABC and 2nd extouch triangle


X(330) = ISOTOMIC CONJUGATE OF X(192)

Trilinears       bc/(ab + ac - bc) : ca/(bc + ba - ca) : ab/(ca + cb - ab)
Barycentrics  1/(ab + ac - bc) : 1/(bc + ba - ca) : 1/(ca + cb - ab)

X(330) lies on these lines: 1,87   2,1107   8,291   56,385   57,239   76,1015   105,932   145,1002   193,959   257,982

X(330) = isogonal conjugate of X(2176)
X(330) = isotomic conjugate of X(192)
X(330) = complement of X(21219)
X(330) = anticomplement of X(6376)
X(330) = X(87)-Ceva conjugate of X(2)
X(330) = X(75)-cross conjugate of X(2)
X(330) = cyclocevian conjugate of X(7357)
X(330) = polar conjugate of isogonal conjugate of X(23086)
X(330) = X(19)-isoconjugate of X(20760)
X(330) = perspector of ABC and inverse of n(Incentral)*n(Medial)
X(330) = trilinear pole of line X(513)X(3716) (complement of antiorthic axis, and perspectrix of Gemini triangles 3 and 6)


X(331) = ISOTOMIC CONJUGATE OF X(219)

Trilinears       sec2(A/2) csc(2A) : sec2(B/2) csc(2B) : sec2(C/2) csc(2C)
                        = (1 - sec A)csc(A - ω) : (1 - sec B)csc(B - ω) : (1 - sec C)csc(C - ω)

Barycentrics  sec2(A/2) sec A : sec2(B/2) sec B : sec2(C/2) sec C

X(331) is the Brianchon point (perspector) of the inconic that is the polar conjugate of the isogonal conjugate of the incircle. (Randy Hutson, October 15, 2018)

X(331) lies on these lines: 4,150   7,286   34,870   75,225   85,92   108,767   274,278

X(331) = isogonal conjugate of polar conjugate of isotomic conjugate of X(6056)
X(331) = isotomic conjugate of X(219)
X(331) = cevapoint of X(i) and X(j) for these (i,j): (7,278), (92,273)
X(331) = X(92)-cross conjugate of X(264)
X(331) = X(19)-isoconjugate of X(6056)
X(331) = trilinear pole of polar of X(55) wrt polar circle
X(331) = pole wrt polar circle of trilinear polar of X(55) (line X(657)X(663))
X(331) = polar conjugate of X(55)


X(332) = ISOTOMIC CONJUGATE OF X(225)

Trilinears         (cot A csc A)/(cos B + cos C) : (cot B csc B)/(cos C + cos A) : (cot C csc C)/(cos A + cos B)
Barycentrics  (cot A)/(cos B + cos C) : (cot B)/(cos C + cos A) : (cot C)/(cos A + cos B)
Barycentrics    (b^2 + c^2 - a^2) (b + c - a)/(b + c) : :

X(332) lies on these lines: 1,75   3,69   21,1036   99,102   219,345   261,284   1014,1037

X(332) = isogonal conjugate of polar conjugate of X(28660)
X(332) = isotomic conjugate of X(225)
X(332) = cevapoint of X(i) and X(j) for these (i,j): (69,326), (78,345)
X(332) = X(i)-cross conjugate of X(j) for these (i,j): (69,314), (283,333)
X(332) = trilinear pole of line X(652)X(6332)


X(333) = CEVAPOINT OF X(8) AND X(9)

Trilinears       bc(b + c - a)/(b + c) : ca(c + a - b)/(c + a) : ab(a + b - c)/(a + b)
Barycentrics  (b + c - a)/(b + c) : (c + a - b)/(c + a) : (a + b - c)/(a + b)

Let A27B27C27 be Gemini triangle 27. Let A' be the perspector of conic {A,B,C,B27,C27}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(333). (Randy Hutson, January 15, 2019)

X(333) lies on these lines: 2,6   8,21   9,312   10,58   19,27   29,270   57,85   190,321   239,257   261,284   306,319   310,673   662,909   740,846   859,956   1021,1024

X(333) = isogonal conjugate of X(1400)
X(333) = isotomic conjugate of X(226)
X(333) = X(i)-Ceva conjugate of X(j) for these (i,j): (261,21), (274,86)
X(333) = cevapoint of X(i) and X(j) for these (i,j): (2,63), (8,9), (283,284)
X(333) = X(i)-cross conjugate of X(j) for these (i,j): (8,314), (9,21), (21,86), (283,332), (284,29)
X(333) = crosspoint of X(274) and X(314)
X(333) = crosssum of X(213) and X(1402)
X(333) = crossdifference of every pair of points on line X(512)X(810)
X(333) = trilinear pole of line X(522)X(663) (the perspectrix of ABC and Gemini triangle 1)
X(333) = polar conjugate of X(225)
X(333) = crosspoint of X(2) and X(63) wrt both the excentral and anticomplementary triangles
X(333) = trilinear product X(2)*X(21)
X(333) = trilinear product of Feuerbach hyperbola intercepts of line X(2)X(6)
X(333) = perspector of Gemini triangle 2 and cross-triangle of ABC and Gemini triangle 2
X(333) = barycentric product of vertices of Gemini triangle 28
X(333) = perspector of ABC and cross-triangle of ABC and Gemini triangle 28


X(334) = ISOTOMIC CONJUGATE OF X(238)

Trilinears       b2c2/(a2 - bc) : c2a2/(b2 - ca) : a2b2/(c2 - ab)
Barycentrics  bc/(a2 - bc) : ca/(b2 - ca) : ab/(c2 - ab) Let A31B31C31 be Gemini triangle 31. Let A' be the perspector of conic {A,B,C,B31,C31}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(334). (Randy Hutson, January 15, 2019)

X(334) lies on these lines: 2,292   10,274   12,85   75,141   76,1089   150,295   320,660   741,839   767,813

X(334) = isogonal conjugate of X(2210)
X(334) = isotomic conjugate of X(238)
X(334) = X(75)-Hirst inverse of X(335)
X(334) = polar conjugate of X(2201)
X(334) = complement of X(30667)


X(335) = ISOTOMIC CONJUGATE OF X(239)

Trilinears       bc/(a2 - bc) : ca/(b2 - ca) : ab/(c2 - ab)
Barycentrics  1/(a2 - bc) : 1/(b2 - ca) : 1/(c2 - ab)

X(335) lies on these lines: 1,384   2,38   7,192   27,295   37,86   75,141   76,871   239,518   257,694   310,321   320,742   536,903   675,813   741,835   876,900

X(335) = reflection of X(i) in X(j) for these (i,j): (75,1086), (190,37)
X(335) = isogonal conjugate of X(1914)
X(335) = isotomic conjugate of X(239)
X(335) = complement of X(33888)
X(335) = cevapoint of X(i) and X(j) for these (i,j): (37,518), (292,295)
X(335) = X(i)-cross conjugate of X(j) for these (i,j): (295,337), (350,257)
X(335) = X(i)-Hirst inverse of X(j) for these (i,j): (2,291), (75,334), (292,894)
X(335) = trilinear pole of line X(10)X(514)
X(335) = pole wrt polar circle of trilinear polar of X(242)
X(335) = X(48)-isoconjugate (polar conjugate) of X(242)


X(336) = ISOTOMIC CONJUGATE OF X(240)

Trilinears       csc A cot A sec(A + ω) : csc B cot B sec(B + ω) : csc C cot C sec(C + ω)
Barycentrics  cot A sec(A + ω) : cot B sec(B + ω) : cot C sec(C + ω)

X(336) lies on these lines: 1,811   48,75   73,290   255,293

X(336) = isotomic conjugate of X(240)


X(337) = ISOTOMIC CONJUGATE OF X(242)

Trilinears       (csc A cot A)/(a2 - bc) : (csc B cot B)/(b2 - ca) : (csc C cot C)/(c2 - ab)
Barycentrics  (cot A)/(a2 - bc) : (cot B)/(b2 - ca) : (cot C)/(c2 - ab)

X(337) lies on these lines: 12,85   37,86   72,295   201,348   291,986

X(337) = isotomic conjugate of X(242)
X(337) = X(295)-cross conjugate of X(335)


X(338) = CEVAPOINT OF X(115) AND X(125)

Trilinears       (b2 - c2)2/a3 : (c2 - a2)2/b3 : (a2 - b2)2/c3
                        = csc A sin2(B - C) : csc B sin2(C - A) : csc C sin2(A - B)

Barycentrics  (b2 - c2)2/a2 : (c2 - a2)2/b2 : (a2 - b2)2/c2
                        = sin2(B - C) : sin2(C - A) : sin2(A - B)

X(338) lies on the inconic with perspector X(2052).

X(338) lies on these lines: 2,94   4,67   6,264   50,401   76,599   115,127   125,136   141,311

X(338) = isogonal conjugate of X(23357)
X(338) = isotomic conjugate of X(249)
X(338) = anticomplement of X(34990)
X(338) = X(264)-Ceva conjugate of X(523)
X(338) = cevapoint of X(115) and X(125)
X(338) = X(125)-cross conjugate of X(339)
X(338) = pole wrt polar circle of trilinear polar of X(250) (line X(110)X(112))
X(338) = polar conjugate of X(250)
X(338) = X(6)-isoconjugate of X(1101)


X(339) = ISOTOMIC CONJUGATE OF X(250)

Trilinears       (b2 - c2)2(cos A)/a4 : (c2 - a2)2(cos B)/b4 : (a2 - b2)2(cos C)/c4
                        = csc A cot A sin2(B - C) : csc B cot B sin2(C - A) : csc C cot C sin2(A - B)

Barycentrics  (b2 - c2)2(cos A)/a3 : (c2 - a2)2(cos B)/b3 : (a2 - b2)2(cos C)/c3
                        = cot A sin2(B - C) : cot B sin2(C - A) : cot C sin2(A - B)
Barycentrics    b^2 c^2 (b^2 - c^2)^2 (b^2 + c^2 - a^2) : :

X(339) lies on the MacBeath inconic and these lines: 3,76   69,265   115,127   264,381

X(339) = isotomic conjugate of X(250)
X(339) = X(i)-Ceva conjugate of X(j) for these (i,j): (76,525), (305,3267)
X(339) = X(125)-cross conjugate of X(338)
X(339) = X(25)-isoconjugate of X(1101)
X(339) = crosspoint of X(305) and X(3267)


X(340) = ISOTOMIC CONJUGATE OF X(265)

Trilinears    sec A sin 3A csc3A : :
Barycentrics    sec A sin 3A csc2A : :
Barycentrics    ((a^2 - b^2 - c^2)^2 - b^2 c^2)/(b^2 + c^2 - a^2) : :

X(340) lies on these lines: 4,69   67,290   95,140   250,325   297,524   298,470   299,471   447,540    458,599   520,850

X(340) = reflection of X(648) in X(297)
X(340) = isotomic conjugate of X(265)
X(340) = anticomplement of X(3284)
X(340) = cevapoint of X(186) and X(323)
X(340) = pole wrt polar circle of trilinear polar of X(1989) (line X(51)X(512))
X(340) = polar conjugate of X(1989)
X(340) = crossdifference of every pair of points on line X(217)X(3049)


X(341) = ISOTOMIC CONJUGATE OF X(269)

Trilinears       b2c2(b + c - a)2 : c2a2(c + a - b)2 : a2b2(a + b - c)2
                        = csc4A/2 : csc4B/2 : csc4C/2

Barycentrics  bc(b + c - a)2 : ca(c + a - b)2 : ab(a + b - c)2

X(341) lies on these lines: 1,1050   8,210   10,75   40,190   200,1043   253,322   280,345   304,668   305,1088

X(341) = isogonal conjugate of X(1106)
X(341) = isotomic conjugate of X(269)
X(341) = complement of X(17480)
X(341) = anticomplement of polar conjugate of isogonal conjugate of X(23222)
X(341) = X(346)-cross conjugate of X(312)
X(341) = polar conjugate of X(1435)
X(341) = trilinear square of X(8)


X(342) = ISOTOMIC CONJUGATE OF X(271)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (csc 2A tan A/2)(1 + cos A - cos B - cos C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sec A tan A/2)(1 + cos A - cos B - cos C)

X(342) lies on these lines: 4,7   9,653   85,264   92,226   108,1005   196,329   253,318   393,948

X(342) = isogonal conjugate of X(2188)
X(342) = isotomic conjugate of X(271)
X(342) = X(i)-Ceva conjugate of X(j) for these (i,j): (85,92), (264,273)
X(342) = cevapoint of X(208) and X(223)
X(342) = polar conjugate of X(282)


X(343) = ISOTOMIC CONJUGATE OF X(275)

Trilinears       cot A cos(B - C) : cot B cos(C - A) : cot C cos(A - B)
Barycentrics  cos A cos(B - C) : cos B cos(C - A) : cos C cos(A - B)
Barycentrics    tan B tan C + 1 : :
Barycentrics    (b^2 + c^2 - a^2) (a^2 (b^2 + c^2) - (b^2 - c^2)^2) : :

X(343) is the barycentric multiplier for the Johnson circumconic. (The barycentric product of X(343) and the circumcircle is the Johnson circumconic.) (Randy Hutson, August 19, 2019)

X(343) lies on these lines: 2,6   3,68   5,51   22,161   53,311   76,297   140,569   315,458   427,511   470,634   471,633   472,621   473,622

X(343) = isogonal conjugate of X(8882)
X(343) = isotomic conjugate of X(275)
X(343) = complement of X(1993)
X(343) = X(i)-Ceva conjugate of X(j) for these (i,j): (76,311), (311,5)
X(343) = X(216)-cross conjugate of X(5)
X(343) = crosspoint of X(69) and X(76)
X(343) = crosssum of X(i) and X(j) for these (i,j): (6,571), (25,32)
X(343) = pole wrt polar circle of trilinear polar of X(8884) (line X(421)X(2501))
X(343) = polar conjugate of X(8884)
X(343) = crosspoint of X(6) and X(2917) wrt both the excentral and tangential triangles
X(343) = X(7580)-of-orthic-triangle if ABC is acute


X(344) = ISOTOMIC CONJUGATE OF X(277)

Trilinears    (csc2A/2)[cos4(B/2) + cos4(C/2) - cos4(A/2)] : :
Barycentrics    a^2 + b^2 + c^2 - 2 a b - 2 a c : :

X(344) lies on these lines: 2,37   7,190   8,480   9,69   44,193   45,141   144,320   264,281   319,391

X(344) = isotomic conjugate of X(277)
X(344) = trilinear pole of polar, wrt AC-incircle, of X(7) (line X(3309)X(4468))
X(344) = anticomplement of X(17278)


X(345) = ISOTOMIC CONJUGATE OF X(278)

Trilinears    (csc A)/(1 - sec A) : (csc B)/(1 - sec B) : (csc C)/(1 - sec C)

Trilinears    bc(b + c - a)(b2 + c2 - a2) : :
Barycentrics  1/(1 - sec A) : 1/(1 - sec B) : 1/(1 - sec C)

Let A35B35C35 be Gemini triangle 35. Let A' be the perspector of conic {A,B,C,B35,C35}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(345). (Randy Hutson, January 15, 2019)

X(345) lies on these lines: 2,37   8,21   22,100   57,728   63,69   78,1040   190,329   219,332   280,341   304,348   498,1089

X(345) = isogonal conjugate of X(608)
X(345) = isotomic conjugate of X(278)
X(345) = X(i)-Ceva conjugate of X(j) for these (i,j): (304,69), (332,78)
X(345) = X(i)-cross conjugate of X(j) for these (i,j): (78,69), (219,8), (306,312)
X(345) = complement of X(30699)
X(345) = anticomplement of X(3772)
X(345) = polar conjugate of X(1118)


X(346) = ISOTOMIC CONJUGATE OF X(279)

Trilinears    bc(b + c - a)2 : :
Trilinears    cos(A/2) csc3(A/2) : :
Barycentrics   (b + c - a)2 : :
Barycentrics   (1 + cos A)(1 - cos A) : :
X(346) = 2 X[2] + 19 X[345] = X[1743] + 2 X[4072] = 17 X[4686] - 2 X[4688] = 19 X[4688] - 4 X[4788] = 5 X[37] - 14 X[4821]

The cevian triangle of X(346) is perspective to the Ayme triangle; see X(3610).

Let A' be the point where BC cuts the radical axis of the circumcircle of ABC and its A-excircle, and define B' and C' cyclically. Then A', B', C' are collinear on the tripolar of X(346). (César Lozada, October 8, 2019)

X(346) lies on the cubics K605 and K697 and these lines: {1, 1219}, {2, 37}, {4, 3695}, {6, 145}, {7, 3729}, {8, 9}, {10, 3731}, {19, 3610}, {20, 1766}, {44, 3621}, {45, 594}, {55, 3974}, {69, 144}, {78, 280}, {100, 198}, {101, 2370}, {141, 4419}, {142, 4659}, {193, 6542}, {200, 4082}, {210, 11997}, {219, 644}, {220, 1043}, {253, 306}, {279, 304}, {281, 318}, {341, 3965}, {347, 4552}, {387, 2901}, {480, 4012}, {497, 3703}, {519, 1743}, {527, 4488}, {545, 7232}, {573, 1018}, {646, 3596}, {672, 10453}, {726, 4310}, {894, 3945}, {941, 1500}, {962, 10445}, {1023, 6790}, {1086, 4373}, {1089, 3085}, {1100, 3623}, {1125, 4098}, {1146, 6558}, {1212, 4673}, {1249, 1897}, {1260, 2322}, {1376, 1696}, {1400, 3501}, {1441, 8232}, {1449, 3241}, {1698, 3986}, {1761, 4427}, {1901, 3936}, {2178, 4188}, {2256, 5782}, {2264, 3189}, {2268, 2329}, {2285, 3600}, {2298, 4195}, {2303, 11115}, {2310, 4073}, {2550, 3932}, {2551, 3704}, {2899, 8165}, {2951, 9950}, {3008, 4402}, {3039, 8168}, {3062, 10324}, {3146, 7270}, {3217, 3684}, {3219, 3719}, {3244, 4898}, {3247, 3616}, {3294, 9534}, {3416, 5698}, {3452, 8055}, {3618, 4360}, {3619, 4389}, {3620, 6646}, {3632, 3973}, {3633, 4856}, {3661, 5232}, {3662, 4346}, {3679, 4058}, {3705, 5274}, {3712, 5218}, {3715, 4046}, {3730, 10449}, {3767, 7230}, {3771, 4135}, {3836, 7613}, {3871, 4254}, {3875, 5222}, {3923, 4307}, {3949, 3952}, {3969, 5739}, {3970, 11036}, {3975, 4110}, {3977, 5744}, {3985, 4451}, {3991, 4385}, {3996, 6605}, {4030, 10385}, {4066, 10198}, {4123, 9539}, {4130, 4397}, {4171, 4529}, {4222, 5687}, {4293, 5525}, {4294, 7206}, {4361, 4422}, {4363, 4648}, {4366, 4473}, {4384, 4431}, {4416, 6172}, {4470, 7227}, {4644, 4851}, {4675, 7222}, {4872, 10513}, {4876, 7155}, {4936, 6737}, {5257, 9780}, {5273, 11679}, {5281, 7081}, {5308, 7229}, {5430, 7027}, {5552, 7110}, {6556, 6736}, {7991, 10443}, {8557, 12649}, {8609, 10529}, {9025, 9309}

X(346) = reflection of X(i) in X(j) for these {i,j}: {8, 4901}, {4452, 4000}, {5838, 9}
X(346) = isogonal conjugate of X(1407)
X(346) = isotomic conjugate of X(279)
X(346) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 6552}, {8, 6555}, {304, 10327}, {312, 8}, {341, 5423}, {345, 7080}, {646, 4397}, {1016, 3699}, {1043, 200}, {3699, 4163}, {4076, 4578}, {6557, 6556}, {7258, 7253}
X(346) = X(i)-cross conjugate of X(j) for these (i,j): {200, 8}, {220, 7046}, {728, 5423}, {1260, 1265}, {2310, 7253}, {2321, 7101}, {3239, 6558}, {3694, 3692}, {3965, 2287}, {4081, 4397}, {4082, 341}, {4130, 4578}, {4163, 3699}, {4515, 200}, {4529, 7258}, {4907, 7}, {4953, 522}, {5423, 6556}, {6554, 2}
X(346) = cevapoint of X(i) and X(j) for these (i,j): {1, 10860}, {6, 1604}, {9, 2324}, {37, 8804}, {200, 728}, {220, 1260}, {650, 4534}, {1146, 3239}, {2310, 4171}, {2321, 3694}, {4081, 4130}, {4082, 4515}
X(346) = crosspoint of X(i) and X(j) for these (i,j): {2, 6553}, {8, 6557}, {312, 341}, {646, 4076}, {1016, 3699}
X(346) = crosssum of X(i) and X(j) for these (i,j): {6, 1616}, {604, 1106}, {649, 3937}
X(346) = crossdifference of every pair of points on line {667, 6363, 7250}
X(346) = complement of X(4452)
X(346) = anticomplement of X(4000)
X(346) = pole wrt polar circle of trilinear polar of X(1119)
X(346) = X(48)-isoconjugate (polar conjugate) of X(1119)
X(346) = centroid of the set consisting of the interiors (with or without boundaries) of the 3 Soddy circles
X(346) = polar conjugate of X(1119)
X(346) = X(i)-beth conjugate of X(j) for these (i,j): {190, 347}, {346, 9}, {644, 573}, {645, 144}, {646, 346}, {1043, 390}, {6558, 346}, {7258, 3596}, {7259, 219}
X(346) = X(i)-gimel conjugate of X(j) for these (i,j): {4462, 346}, {8712, 346}
X(346) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 1407}, {905, 649}
X(346) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1037, 7}, {7084, 2}, {7123, 8}, {7131, 3434}
X(346) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 6552}, {2137, 2886}, {6553, 2887}
X(346) = X(i)-Hirst inverse of X(j) for these (i,j): {8, 3717}, {3975, 4110}
X(346) = trilinear pole of line {3239, 3900, 4148, 4163, 4524, 4528, 4546, 4990}
X(346) = barycentric square of X(8)
X(346) = isoconjugate of X(j) and X(j) for these (i,j): {1, 1407}, {2, 1106}, {3, 1435}, {4, 7099}, {6, 269}, {7, 604}, {8, 7366}, {9, 7023}, {19, 7053}, {25, 7177}, {27, 1410}, {31, 279}, {32, 1088}, {34, 222}, {40, 6612}, {41, 479}, {48, 1119}, {55, 738}, {56, 57}, {58, 1427}, {60, 7147}, {63, 1398}, {65, 1412}, {73, 1396}, {77, 608}, {81, 1042}, {84, 6611}, {85, 1397}, {109, 3669}, {110, 7216}, {184, 1847}, {221, 1422}, {223, 1413}, {226, 1408}, {241, 1416}, {244, 1262}, {266, 7370}, {278, 603}, {348, 1395}, {512, 4637}, {513, 1461}, {593, 1254}, {649, 934}, {650, 6614}, {658, 667}, {662, 7250}, {663, 4617}, {669, 4635}, {764, 4619}, {798, 4616}, {849, 6354}, {1014, 1400}, {1015, 7045}, {1020, 3733}, {1118, 7125}, {1122, 3451}, {1275, 3248}, {1333, 3668}, {1357, 4564}, {1358, 2149}, {1402, 1434}, {1403, 7153}, {1414, 7180}, {1415, 3676}, {1417, 3911}, {1426, 1790}, {1431, 7175}, {1439, 1474}, {1440, 2199}, {1446, 2206}, {1458, 1462}, {1472, 7365}, {1919, 4569}, {1973, 7056}, {2150, 6046}, {2170, 7339}, {2171, 7341}, {2185, 7143}, {2221, 4320}, {3063, 4626}, {3937, 7128}, {4017, 4565}, {4559, 7203}, {6358, 7342}, {7132, 7248}, {7183, 7337}
X(346) = barycentric product X(i)*X(j) for these {i,j}: {1, 341}, {4, 1265}, {7, 5423}, {8, 8}, {9, 312}, {10, 1043}, {11, 4076}, {21, 3701}, {29, 3710}, {33, 3718}, {55, 3596}, {63, 7101}, {69, 7046}, {75, 200}, {76, 220}, {78, 318}, {85, 728}, {86, 4082}, {92, 3692}, {100, 4397}, {145, 6556}, {179, 400}, {188, 7027}, {190, 3239}, {210, 314}, {219, 7017}, {261, 6057}, {264, 1260}, {274, 4515}, {280, 7080}, {281, 345}, {304, 7079}, {305, 7071}, {306, 2322}, {313, 2328}, {321, 2287}, {333, 2321}, {480, 6063}, {514, 6558}, {522, 3699}, {523, 7256}, {556, 6731}, {561, 1253}, {594, 7058}, {643, 4086}, {644, 4391}, {645, 3700}, {646, 650}, {657, 1978}, {661, 7258}, {664, 4163}, {668, 3900}, {670, 4524}, {693, 4578}, {799, 4171}, {1016, 1146}, {1021, 4033}, {1089, 1098}, {1222, 6736}, {1229, 6605}, {1264, 1857}, {1320, 4723}, {1577, 7259}, {1639, 4582}, {1802, 1969}, {2310, 7035}, {2319, 4110}, {2325, 4997}, {3161, 6557}, {3685, 4518}, {3686, 4102}, {3912, 6559}, {3952, 7253}, {3975, 4876}, {4012, 8817}, {4041, 7257}, {4073, 7033}, {4081, 4998}, {4087, 7077}, {4092, 6064}, {4105, 4572}, {4130, 4554}, {4148, 4562}, {4373, 6555}, {4451, 7081}, {4528, 4555}, {4673, 4866}, {4990, 6540}, {6386, 8641}, {6552, 6553}
X(346) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 269}, {2, 279}, {3, 7053}, {4, 1119}, {6, 1407}, {7, 479}, {8, 7}, {9, 57}, {10, 3668}, {11, 1358}, {12, 6046}, {19, 1435}, {21, 1014}, {25, 1398}, {31, 1106}, {33, 34}, {37, 1427}, {41, 604}, {42, 1042}, {48, 7099}, {55, 56}, {56, 7023}, {57, 738}, {59, 7339}, {60, 7341}, {63, 7177}, {69, 7056}, {72, 1439}, {75, 1088}, {78, 77}, {92, 1847}, {99, 4616}, {100, 934}, {101, 1461}, {109, 6614}, {144, 9533}, {181, 7143}, {188, 7371}, {190, 658}, {198, 6611}, {200, 1}, {210, 65}, {212, 603}, {219, 222}, {220, 6}, {228, 1410}, {259, 7370}, {261, 552}, {280, 1440}, {281, 278}, {282, 1422}, {284, 1412}, {294, 1462}, {312, 85}, {318, 273}, {321, 1446}, {333, 1434}, {341, 75}, {345, 348}, {388, 7197}, {390, 3598}, {480, 55}, {497, 7195}, {512, 7250}, {522, 3676}, {556, 555}, {594, 6354}, {604, 7366}, {607, 608}, {612, 4320}, {643, 1414}, {644, 651}, {645, 4573}, {646, 4554}, {650, 3669}, {651, 4617}, {657, 649}, {661, 7216}, {662, 4637}, {664, 4626}, {668, 4569}, {728, 9}, {756, 1254}, {765, 7045}, {799, 4635}, {984, 7204}, {1016, 1275}, {1018, 1020}, {1021, 1019}, {1043, 86}, {1098, 757}, {1146, 1086}, {1172, 1396}, {1212, 1418}, {1252, 1262}, {1253, 31}, {1259, 1804}, {1260, 3}, {1261, 1476}, {1264, 7055}, {1265, 69}, {1334, 1400}, {1436, 6612}, {1604, 6609}, {1792, 1444}, {1802, 48}, {1824, 1426}, {1857, 1118}, {1863, 1851}, {2171, 7147}, {2175, 1397}, {2192, 1413}, {2194, 1408}, {2195, 1416}, {2212, 1395}, {2287, 81}, {2289, 7125}, {2310, 244}, {2318, 73}, {2319, 7153}, {2321, 226}, {2322, 27}, {2324, 223}, {2325, 3911}, {2327, 1790}, {2328, 58}, {2329, 7175}, {2332, 1474}, {2340, 1458}, {2345, 7365}, {2968, 1565}, {3022, 3271}, {3056, 7248}, {3057, 1122}, {3059, 354}, {3119, 2170}, {3158, 1420}, {3161, 5435}, {3190, 4306}, {3208, 1423}, {3239, 514}, {3270, 3937}, {3271, 1357}, {3596, 6063}, {3684, 1429}, {3685, 1447}, {3686, 553}, {3687, 3674}, {3688, 1401}, {3689, 1319}, {3690, 1425}, {3692, 63}, {3693, 241}, {3694, 1214}, {3695, 6356}, {3699, 664}, {3700, 7178}, {3701, 1441}, {3703, 3665}, {3705, 7185}, {3706, 4059}, {3707, 4031}, {3709, 7180}, {3710, 307}, {3711, 2099}, {3712, 7181}, {3713, 940}, {3715, 5221}, {3717, 9436}, {3718, 7182}, {3719, 7183}, {3737, 7203}, {3790, 7179}, {3811, 4341}, {3870, 4350}, {3900, 513}, {3939, 109}, {3952, 4566}, {3965, 3666}, {3974, 388}, {3975, 10030}, {4007, 4654}, {4012, 497}, {4030, 7198}, {4041, 4017}, {4046, 3649}, {4060, 3982}, {4061, 3671}, {4069, 4551}, {4073, 982}, {4076, 4998}, {4081, 11}, {4082, 10}, {4086, 4077}, {4092, 1365}, {4095, 4032}, {4103, 4605}, {4105, 663}, {4113, 4955}, {4123, 7210}, {4130, 650}, {4148, 812}, {4152, 1317}, {4157, 7214}, {4163, 522}, {4171, 661}, {4178, 7217}, {4183, 28}, {4319, 614}, {4397, 693}, {4420, 1442}, {4433, 1284}, {4451, 7249}, {4477, 4367}, {4511, 1443}, {4512, 3361}, {4513, 6180}, {4515, 37}, {4517, 1469}, {4518, 7233}, {4524, 512}, {4528, 900}, {4529, 4369}, {4546, 3667}, {4548, 7251}, {4571, 6516}, {4578, 100}, {4587, 1813}, {4827, 4790}, {4847, 10481}, {4853, 7271}, {4873, 5219}, {4882, 4328}, {4907, 5573}, {4936, 1743}, {4953, 3756}, {4990, 4977}, {5423, 8}, {5532, 7336}, {5546, 4565}, {5547, 7316}, {6056, 7335}, {6057, 12}, {6058, 7314}, {6059, 7337}, {6060, 7338}, {6061, 60}, {6062, 1354}, {6064, 7340}, {6065, 59}, {6068, 3321}, {6335, 13149}, {6552, 4452}, {6554, 4000}, {6555, 145}, {6556, 4373}, {6558, 190}, {6559, 673}, {6600, 1617}, {6602, 41}, {6603, 6610}, {6605, 1170}, {6607, 2488}, {6726, 266}, {6731, 174}, {6736, 3663}, {6737, 3664}, {6745, 1323}, {7017, 331}, {7027, 4146}, {7046, 4}, {7054, 593}, {7058, 1509}, {7062, 1355}, {7063, 1356}, {7064, 181}, {7065, 1363}, {7067, 1366}, {7068, 1367}, {7069, 1393}, {7070, 1394}, {7071, 25}, {7074, 221}, {7075, 1424}, {7079, 19}, {7080, 347}, {7081, 7176}, {7101, 92}, {7102, 7103}, {7156, 3213}, {7172, 3600}, {7253, 7192}, {7256, 99}, {7257, 4625}, {7258, 799}, {7259, 662}, {7283, 6359}, {7322, 7273}, {7359, 6357}, {7360, 5088}, {7367, 1436}, {7368, 198}, {8012, 1475}, {8641, 667}, {8706, 6613}, {8805, 8810}, {10382, 1467}


X(347) = ISOTOMIC CONJUGATE OF X(280)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (-1 - cos A + cos B + cos C) sec2(A/2)
Barycentrics  af(A,B,C) : bf(B,C,A) : cf(C,A,B)

Let Ha be the hyperbola passing through A, and with foci at B and C. This is called the A-Soddy hyperbola in (Paul Yiu, Introduction to the Geometry of the Triangle, 2002; 12.4 The Soddy hyperbolas, p. 143). Let La be the polar of X(20) with respect to Ha. Define Lb, Lc cyclically. La, Lb, Lc concur in X(347). (Randy Hutson, November 30, 2018)

X(347) lies on these lines: 1,7   2,92   8,253   34,452   37,948   69,664   75,280   144,219   223,329   227,322   241,1108   573,1020

X(347) = isogonal conjugate of X(2192)
X(347) = isotomic conjugate of X(280)
X(347) = complement of polar conjugate of X(38268)
X(347) = anticomplement of X(281)
X(347) = X(i)-Ceva conjugate of X(j) for these (i,j): (75,7), (348,2)
X(347) = cevapoint of X(40) and X(223)
X(347) = X(i)-cross conjugate of X(j) for these (i,j): (40,329), (221,196), (227,223)
X(347) = crosspoint of X(75) and X(322)
X(347) = polar conjugate of X(7003)
X(347) = {X(175),X(176)}-harmonic conjugate of X(20)


X(348) = ISOTOMIC CONJUGATE OF X(281)

Trilinears    cot A sec2(A/2) : :
Trilinears    sec2(C/2)
Trilinears    (csc A)/(1 + sec A) : :
Barycentrics  1/(1 + sec A) : 1/(1 + sec B) : 1/(1 + sec C)
Barycentrics    (b^2 + c^2 - a^2)/(b + c - a) : :

X(348) is the Brianchon point (perspector) of the inconic that is the isotomic conjugate of the polar conjugate of the incircle. The center of the inconic is X(17073). (Randy Hutson, November 30, 2018)

Let A36B36C36 be Gemini triangle 36. Let A' be the perspector of conic {A,B,C,B36,C36}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(348). (Randy Hutson, January 15, 2019)

X(348) lies on these lines: 2,85   7,21   8,664   69,73   75,280   150,944   201,337   274,278   304,345   499,1111

X(348) = isogonal conjugate of X(607)
X(348) = isotomic conjugate of X(281)
X(348) = complement of X(30694)
X(348) = anticomplement of polar conjugate of X(34398)
X(348) = X(274)-Ceva conjugate of X(85)
X(348) = cevapoint of X(i) and X(j) for these (i,j): (2,347), (63,77)
X(348) = X(222)-cross conjugate of X(7)

X(348) = polar conjugate of X(1857)
X(348) = trilinear pole of line X(4025)X(4131)

X(349) = ISOTOMIC CONJUGATE OF X(284)

Trilinears       (cos B + cos C)csc3A : (cos C + cos A)csc3B : (cos A + cos B) csc3C
                        = (cos B + cos C)csc(A - ω) : (cos C + cos A)csc(B - ω) : (cos A + cos B)csc(C - ω)

Barycentrics  (cos B + cos C)csc2A : (cos C + cos A)csc2B : (cos A + cos B)(csc C/2)2

X(349) lies on these lines: 12,313   73,290   75,225   76,85

X(349) = isotomic conjugate of X(284)
X(349) = cevapoint of X(226) and X(307)
X(349) = X(321)-cross conjugate of X(313)
X(349) = polar conjugate of X(2299)


X(350) = X(2)-HIRST INVERSE OF X(75)

Trilinears       (a2 - bc)b2c2 : (b2 - ca)c2a2 : (c2 - ab)a2b2
Barycentrics  bc(a2 - bc) : ca(b2 - ca) : ab(c2 - ab)

X(350) is the perspector of triangle ABC and the tangential triangle of the conic that passes through A, B, C, and the pair P(10) and U(10) of bicentric points (see the notes just before X(1908). (Randy Hutson, 9/23/2011)

X(350) lies on these lines: 1,76   2,37   11,325   33,264   36,99   42,308   55,183   69,497   86,310   172,384   190,672   256,314   291,726   305,614   320,513   447,811   519,668   538,1015   889,903

X(350) = isogonal conjugate of X(1911)
X(350) = isotomic conjugate of X(291)
X(350) = complement of X(17759)
X(350) = anticomplement of X(1575)
X(350) = anticomplementary conjugate of X(20335)
X(350) = crosspoint of X(257) and X(335)
X(350) = crossdifference of every pair of points on line X(213)X(667)
X(350) = X(2)-Hirst inverse of X(75)
X(350) = X(2)-Ceva conjugate of X(39028)
X(350) = perspector of hyperbola {A,B,C,X(274),X(668),X(874),PU(10)}}
X(350) = intersection of trilinear polars of P(10) and U(10)


X(351) = CENTER OF THE PARRY CIRCLE

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 - c2)(b2 + c2 - 2a2)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2(b2 - c2)(b2 + c2 - 2a2)

X(351) is the center of the Parry circle introduced in TCCT (Art. 8.13) as the circle that passes through X(i) for I = 2, 15, 16, 23, 110, 111, 352, 353.

Let OA be the circle centered at the A-vertex of the 1st anti-Parry triangle and passing through A; define OB and OC cyclically. X(351) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

Let OA be the circle centered at the A-vertex of the 2nd anti-Parry triangle and passing through A; define OB and OC cyclically. X(351) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

Let S be the set of these 9 circles: circumcircle, Brocard circle, ninepoint circle of tangential triangle, McCay circumcircle, Lucas inner circle, Lucas circles radical circle, outer Montesdeoca-Lemoine circle, inner Montesdeoca-Lemoine circle, Parry isodynamic circle. Let S' be any one of the 84 subsets of S that consist of 3 circles. The radical center of S' is X(351). (Peter Moses, March 4, 2023)

Taken in pairs (i.e. {{S', radical axis of S'}...}, we have the following radical axes:

{{circumcircle, Brocard circle},{Lemoine axis},{187, 237, 351, 512, 647, 649, 663, 665, 667, 669, ...}}
{{circumcircle, ninepoint circle of tangential triangle},{},{110, 351, 526, 684, 1576, 1624, 1634, 2421, 4556, 4558, ...}}
{{circumcircle, McCay circumcircle},{},{351}}
{{circumcircle, Lucas inner circle},{Lemoine axis},{187, 237, 351, 512, 647, 649, 663, 665, 667, 669, ...}}
{{circumcircle, Lucas circles radical circle},{Lemoine axis},{187, 237, 351, 512, 647, 649, 663, 665, 667, 669, ...}}
{{circumcircle, outer Montesdeoca-Lemoine circle},{Lemoine axis},{187, 237, 351, 512, 647, 649, 663, 665, 667, 669, ...}}
{{circumcircle, inner Montesdeoca-Lemoine circle},{Lemoine axis},{187, 237, 351, 512, 647, 649, 663, 665, 667, 669, ...}}
{{circumcircle, Parry isodynamic circle},{},{6, 351, 9023, 9188, 48450}}
{{Brocard circle, ninepoint circle of tangential triangle},{},{351, 3265}}
{{Brocard circle, McCay circumcircle},{},{351, 10166, 11171}}
{{Brocard circle, Lucas inner circle},{Lemoine axis},{187, 237, 351, 512, 647, 649, 663, 665, 667, 669, ...}}
{{Brocard circle, Lucas circles radical circle},{Lemoine axis},{187, 237, 351, 512, 647, 649, 663, 665, 667, 669, ...}}
{{Brocard circle, outer Montesdeoca-Lemoine circle},{Lemoine axis},{187, 237, 351, 512, 647, 649, 663, 665, 667, 669, ...}}
{{Brocard circle, inner Montesdeoca-Lemoine circle},{Lemoine axis},{187, 237, 351, 512, 647, 649, 663, 665, 667, 669, ...}}
{{Brocard circle, Parry isodynamic circle},{},{351, 574}}
{{ninepoint circle of tangential triangle, McCay circumcircle},{},{351}}
{{ninepoint circle of tangential triangle, Lucas inner circle},{},{351}}
{{ninepoint circle of tangential triangle, Lucas circles radical circle},{},{351}}
{{ninepoint circle of tangential triangle, outer Montesdeoca-Lemoine circle},{},{351}}
{{ninepoint circle of tangential triangle, inner Montesdeoca-Lemoine circle},{},{351}}
{{ninepoint circle of tangential triangle, Parry isodynamic circle},{},{351}}
{{McCay circumcircle, Lucas inner circle},{},{351}}
{{McCay circumcircle, Lucas circles radical circle},{},{351}}
{{McCay circumcircle, outer Montesdeoca-Lemoine circle},{},{351}}
{{McCay circumcircle, inner Montesdeoca-Lemoine circle},{},{351}}
{{McCay circumcircle, Parry isodynamic circle},{},{351}}
{{Lucas inner circle, Lucas circles radical circle},{Lemoine axis},{187, 237, 351, 512, 647, 649, 663, 665, 667, 669, ...}}
{{Lucas inner circle, outer Montesdeoca-Lemoine circle},{Lemoine axis},{187, 237, 351, 512, 647, 649, 663, 665, 667, 669, ...}}
{{Lucas inner circle, inner Montesdeoca-Lemoine circle},{Lemoine axis},{187, 237, 351, 512, 647, 649, 663, 665, 667, 669, ...}}
{{Lucas inner circle, Parry isodynamic circle},{},{351, 6468}}
{{Lucas circles radical circle, outer Montesdeoca-Lemoine circle},{Lemoine axis},{187, 237, 351, 512, 647, 649, 663, 665, 667, 669, ...}}
{{Lucas circles radical circle, inner Montesdeoca-Lemoine circle},{Lemoine axis},{187, 237, 351, 512, 647, 649, 663, 665, 667, 669, ...}}
{{Lucas circles radical circle, Parry isodynamic circle},{},{351, 6221}}
{{outer Montesdeoca-Lemoine circle, inner Montesdeoca-Lemoine circle},{Lemoine axis},{187, 237, 351, 512, 647, 649, 663, 665, 667, 669, ...}}
{{outer Montesdeoca-Lemoine circle, Parry isodynamic circle},{},{351, 50663}}
{{inner Montesdeoca-Lemoine circle, Parry isodynamic circle},{},{351, 50662}}
(Peter Moses, March 4, 2023)

X(351) lies on the cubic K978 and these lines: {2, 804}, {3, 2780}, {6, 9023}, {15, 9162}, {16, 9163}, {23, 9213}, {25, 14998}, {30, 19912}, {37, 17989}, {42, 17990}, {51, 14397}, {110, 526}, {111, 2492}, {184, 686}, {187, 237}, {251, 17997}, {308, 17995}, {352, 9212}, {353, 13242}, {498, 31502}, {513, 9810}, {514, 23817}, {520, 13302}, {522, 49628}, {523, 4108}, {542, 36255}, {650, 50538}, {670, 23356}, {690, 1641}, {691, 34539}, {694, 881}, {740, 4763}, {812, 10180}, {850, 44451}, {865, 888}, {878, 1976}, {900, 9978}, {924, 14696}, {1304, 32696}, {1400, 17992}, {1499, 9127}, {1510, 13315}, {1635, 1962}, {1636, 23611}, {1637, 8029}, {2054, 17991}, {2395, 46316}, {2444, 14908}, {2514, 21006}, {2793, 6055}, {2799, 11123}, {2854, 40282}, {2872, 7600}, {2881, 9157}, {3268, 9479}, {3455, 14339}, {3566, 39904}, {3900, 13254}, {4455, 8034}, {4728, 53034}, {4809, 6370}, {4893, 9279}, {5020, 44817}, {5054, 16235}, {5466, 8587}, {5652, 16508}, {5663, 19902}, {5926, 9128}, {5996, 25423}, {6091, 34519}, {6131, 7665}, {6353, 47206}, {6368, 32407}, {6562, 6587}, {7664, 22105}, {8429, 35901}, {8672, 47826}, {8675, 46953}, {8704, 31772}, {8723, 33972}, {9009, 17415}, {9150, 18829}, {9158, 16171}, {9186, 9192}, {9194, 27550}, {9195, 27551}, {9200, 13305}, {9201, 13304}, {9429, 14406}, {9517, 45082}, {10166, 11171}, {10567, 20976}, {11205, 14403}, {11215, 31986}, {11616, 11620}, {13223, 13224}, {13263, 13264}, {13284, 13285}, {13290, 13291}, {13308, 13309}, {13316, 13319}, {13317, 13320}, {13718, 13719}, {13841, 13842}, {14084, 14830}, {14272, 47139}, {14277, 18310}, {14315, 48024}, {14316, 50545}, {14337, 14338}, {14698, 45147}, {16156, 16157}, {17999, 21448}, {19901, 33962}, {22984, 22985}, {23035, 23036}, {24809, 24810}, {27798, 45675}, {27811, 47776}, {28374, 29198}, {31176, 32472}, {31953, 47200}, {36213, 38366}, {39232, 46276}, {41079, 47173}, {41300, 47128}, {44427, 47217}, {44889, 46130}, {46127, 46131}, {51335, 52743}

X(351) = midpoint of X(i) and X(j) for these {i,j}: {2, 9147}, {15, 9162}, {16, 9163}, {23, 9213}, {110, 9138}, {111, 9156}, {352, 9212}, {353, 13242}, {647, 8644}, {669, 17414}, {1635, 1962}, {3288, 11186}, {3569, 9135}, {4108, 36900}, {5027, 9208}, {5466, 9485}, {5638, 5639}, {9123, 9185}, {9126, 11615}, {9131, 9979}, {9157, 13114}, {9200, 13305}, {9201, 13304}, {9810, 9811}, {9978, 9980}, {11123, 14420}, {11616, 11622}, {13223, 13224}, {13250, 13251}, {13254, 13255}, {13263, 13264}, {13284, 13285}, {13290, 13291}, {13302, 13303}, {13306, 13307}, {13308, 13309}, {13315, 13318}, {13316, 13319}, {13317, 13320}, {13718, 13719}, {13841, 13842}, {14337, 14338}, {14339, 14340}, {14406, 23610}, {14610, 32193}, {16156, 16157}, {22733, 22734}, {22888, 22889}, {22933, 22934}, {22984, 22985}, {23035, 23036}, {24809, 24810}, {32312, 32313}, {32407, 32408}, {39904, 39905}, {40282, 40283}, {49628, 49629}
X(351) = reflection of X(i) in X(j) for these {i,j}: {2, 11176}, {3, 9126}, {6, 9188}, {669, 8644}, {684, 34291}, {1649, 9125}, {2502, 39527}, {3005, 17414}, {3268, 10190}, {3569, 9208}, {4108, 45317}, {8029, 1637}, {8371, 9189}, {8644, 8651}, {9131, 14610}, {9134, 44564}, {9135, 5027}, {9148, 2}, {9175, 11621}, {9178, 2492}, {9208, 5113}, {9979, 32193}, {11123, 45687}, {11183, 45680}, {11186, 50550}, {11205, 14403}, {11622, 11620}, {13291, 14697}, {14277, 18310}, {14398, 14428}, {14424, 14417}, {17414, 647}, {27798, 45675}, {31986, 32194}, {34290, 45336}, {34291, 6132}, {42663, 9135}, {44420, 39526}
X(351) = isogonal conjugate of X(892)
X(351) = complement of anticomplementary conjugate of X(39356)
X(351) = X(2)-Ceva conjugate of X(38988)
X(351) = perspector of hyperbola {A,B,C,X(6),X(187)}}, which is the locus of the barycentric product of circumcircle-X(690)-antipodes
X(351) = crosspoint of X(110) and X(111)
X(351) = crosssum of X(i) and X(j) for these (i,j): (2,690), (523,524), (850,1236)
X(351) = crossdifference of every pair of points on line X(2)X(99)
X(351) = circumcircle-inverse of X(9129)
X(351) = Parry-isodynamic-circle-inverse of X(187)
X(351) = X(i)-line conjugate of X(j) for these (i,j): {187, 5106}, {690, 2482}, {804, 2}, {2492, 111}, {2793, 9172}, {3455, 41272}, {6131, 7665}, {22105, 7664}
X(351) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 38988}, {25322, 21253}. X(351) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 38988}, {110, 39689}, {111, 3124}, {187, 21906}, {468, 1648}, {598, 35507}, {691, 6}, {827, 6593}, {1177, 3269}, {1383, 38369}, {1576, 47426}, {2489, 21905}, {3455, 20975}, {5467, 187}, {9178, 512}, {10511, 8288}, {14357, 23992}, {14419, 2642}, {22105, 690}, {41498, 14444}, {45773, 20976}
X(351) = trilinear pole of line {21906, 35507}
X(351) = Lemoine-axis-intercept of trilinear polar of X(512)
X(351) = centroid of the triangle X(3)X(5607)X(5608)
X(351) = tripolar centroid of X(6)
X(351) = tripolar centroid of X(14608)
X(351) = X(9227)-anticomplementary conjugate of X(21294)
X(351) = centroid of Lemoine axis intercepts with sidelines of ABC
X(351) = X(351)-of-circumsymmedial-triangle
X(351) = intersection of tangents to Steiner inellipse at X(1084) and X(2482)
X(351) = crosspoint wrt medial triangle of X(1084) and X(2482)
X(351) = vertex conjugate of PU(62)
X(351) = radical center of (circumcircle, Brocard circle, McCay circumcircle)
X(351) = harmonic center of circles {X(14),X(15),X(16)}} and {X(13),X(15),X(16)}}
X(351) = bicentric difference of PU(i) for these i: 62, 63, 64, 65, 66, 67
X(351) = PU(62)-harmonic conjugate of X(6)
X(351) = PU(63)-harmonic conjugate of X(3)
X(351) = PU(64)-harmonic conjugate of X(1)
X(351) = PU(65)-harmonic conjugate of X(5)
X(351) = PU(66)-harmonic conjugate of X(10)
X(351) = PU(67)-harmonic conjugate of X(39)
X(351) = X(3)-of-1st-Parry-triangle
X(351) = X(3)-of-2nd-Parry-triangle
X(351) = X(3)-of-3rd-Parry-triangle
X(351) = inverse-in-Parry-isodynamic-circle of X(187); see X(2)
X(351) = pole of Brocard axis wrt Parry isodynamic circle
X(351) = bicentric sum of PU(105)
X(351) = centroid of (degenerate) cross-triangle of 1st and 3rd Parry triangles
X(351) = centroid of centers of A-, B- and C-Apollonian circles
X(351) = Lemoine axis (or line PU(2)) intercept of line connecting P(2)-Ceva conjugate of U(2) and U(2)-Ceva conjugate of P(2)
X(351) = X(i)-isoconjugate of X(j) for these (i,j): {1, 892}, {2, 36085}, {75, 691}, {76, 36142}, {86, 5380}, {99, 897}, {110, 46277}, {111, 799}, {162, 30786}, {163, 18023}, {561, 32729}, {661, 52940}, {662, 671}, {670, 923}, {811, 895}, {1101, 52632}, {1109, 45773}, {3112, 36827}, {4563, 36128}, {4575, 46111}, {4590, 23894}, {4592, 17983}, {4593, 46154}, {4599, 31125}, {4602, 32740}, {4622, 52747}, {4625, 5547}, {5466, 24041}, {5968, 36036}, {6331, 36060}, {7035, 43926}, {7257, 7316}, {8773, 52035}, {9170, 17955}, {9178, 24037}, {10097, 46254}, {10630, 24039}, {11059, 36045}, {14210, 34574}, {14728, 17467}, {36133, 52756}, {37204, 41272}, {37216, 52141}
X(351) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 892}, {115, 18023}, {125, 30786}, {136, 46111}, {206, 691}, {244, 46277}, {512, 9178}, {523, 52632}, {690, 35522}, {1084, 671}, {1560, 6331}, {1648, 3266}, {1649, 850}, {2482, 670}, {2679, 5968}, {3005, 5466}, {3124, 31125}, {5099, 52551}, {5139, 17983}, {6593, 99}, {15477, 34574}, {17413, 42008}, {17423, 895}, {21905, 523}, {21906, 47286}, {23992, 76}, {31654, 11059}, {32664, 36085}, {34452, 36827}, {36830, 52940}, {38986, 897}, {38988, 2}, {38996, 111}, {39010, 52756}, {39072, 52035}, {40368, 32729}, {40600, 5380}, {48317, 264}, {52881, 52608}
X(351) = barycentric product X(i)*X(j) for these {i,j}: {1, 2642}, {3, 14273}, {6, 690}, {25, 14417}, {32, 35522}, {37, 14419}, {39, 22105}, {42, 4750}, {50, 51479}, {99, 21906}, {110, 1648}, {111, 1649}, {115, 5467}, {187, 523}, {249, 33919}, {251, 14424}, {468, 647}, {511, 52038}, {512, 524}, {513, 21839}, {525, 44102}, {574, 23287}, {649, 4062}, {661, 896}, {667, 42713}, {669, 3266}, {691, 23992}, {694, 11183}, {798, 14210}, {804, 18872}, {843, 33921}, {850, 14567}, {882, 5026}, {888, 14608}, {922, 1577}, {935, 47415}, {1379, 46462}, {1380, 46463}, {1400, 14432}, {1576, 52628}, {1637, 9717}, {1974, 45807}, {1989, 44814}, {2088, 14559}, {2395, 9155}, {2422, 50567}, {2433, 5642}, {2434, 6791}, {2482, 9178}, {2489, 6390}, {2491, 52145}, {2492, 14357}, {2501, 3292}, {2502, 34763}, {2623, 41586}, {2643, 23889}, {2682, 44769}, {2715, 51429}, {2793, 51927}, {3005, 52898}, {3049, 44146}, {3121, 42721}, {3124, 5468}, {3284, 52475}, {3455, 18311}, {3457, 9204}, {3458, 9205}, {3569, 5967}, {3709, 7181}, {3712, 7180}, {4041, 51653}, {4079, 6629}, {4235, 20975}, {4705, 16702}, {5095, 10097}, {5191, 50942}, {5466, 39689}, {5477, 35364}, {5638, 52723}, {5639, 52722}, {6041, 52094}, {6137, 52039}, {6138, 52040}, {7813, 18105}, {8371, 48450}, {8664, 31068}, {9125, 21448}, {9148, 41309}, {9171, 51226}, {10630, 33915}, {11060, 45808}, {14270, 43084}, {14398, 36890}, {14404, 52757}, {14407, 52759}, {14444, 34574}, {14606, 45330}, {14618, 23200}, {14998, 45662}, {16741, 50487}, {17414, 51541}, {20382, 32583}, {21905, 41909}, {23894, 42081}, {28625, 30595}, {28658, 30605}, {32740, 52629}, {34212, 35282}, {34539, 46049}, {37778, 39201}, {39785, 46001}, {42665, 51823}, {45680, 52660}
X(351) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 892}, {31, 36085}, {32, 691}, {110, 52940}, {115, 52632}, {187, 99}, {213, 5380}, {468, 6331}, {512, 671}, {523, 18023}, {524, 670}, {560, 36142}, {647, 30786}, {661, 46277}, {669, 111}, {688, 46154}, {690, 76}, {798, 897}, {887, 14609}, {888, 52756}, {896, 799}, {922, 662}, {1084, 9178}, {1501, 32729}, {1648, 850}, {1649, 3266}, {1692, 52035}, {1924, 923}, {1977, 43926}, {2422, 9154}, {2489, 17983}, {2491, 5968}, {2492, 52551}, {2501, 46111}, {2502, 34760}, {2642, 75}, {2682, 41079}, {3005, 31125}, {3049, 895}, {3051, 36827}, {3124, 5466}, {3266, 4609}, {3292, 4563}, {4062, 1978}, {4750, 310}, {5026, 880}, {5191, 50941}, {5467, 4590}, {5468, 34537}, {5967, 43187}, {6041, 16092}, {6390, 52608}, {6629, 52612}, {8644, 52141}, {9125, 11059}, {9155, 2396}, {9171, 17948}, {9426, 32740}, {9429, 36821}, {9494, 41272}, {11183, 3978}, {14210, 4602}, {14273, 264}, {14398, 9214}, {14407, 52747}, {14417, 305}, {14419, 274}, {14424, 8024}, {14428, 52758}, {14432, 28660}, {14443, 52628}, {14444, 52629}, {14567, 110}, {14608, 886}, {16702, 4623}, {17414, 42008}, {18311, 40074}, {18872, 18829}, {19627, 51478}, {20975, 14977}, {21839, 668}, {21905, 47286}, {21906, 523}, {22105, 308}, {23200, 4558}, {23287, 40826}, {23357, 45773}, {23889, 24037}, {23992, 35522}, {32729, 34539}, {32740, 34574}, {33915, 36792}, {33919, 338}, {33921, 45809}, {35522, 1502}, {39689, 5468}, {41309, 9150}, {42081, 24039}, {42344, 23105}, {42663, 52450}, {42713, 6386}, {44102, 648}, {44814, 7799}, {45680, 41259}, {45807, 40050}, {46001, 18818}, {48450, 9170}, {51479, 20573}, {51653, 4625}, {51927, 46144}, {52038, 290}, {52628, 44173}, {52898, 689}
X(351) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {25, 47230, 17994}, {110, 9216, 9145}, {110, 15329, 42743}, {111, 9178, 17993}, {111, 9215, 9142}, {647, 669, 3005}, {647, 8651, 669}, {647, 42659, 42665}, {667, 42653, 42661}, {669, 3005, 8664}, {669, 3231, 42652}, {669, 8665, 3804}, {1649, 14424, 14417}, {2491, 6041, 14398}, {2492, 11631, 17993}, {3569, 5027, 42663}, {3569, 5191, 9409}, {4108, 15724, 45317}, {4455, 21828, 8034}, {5027, 5106, 887}, {5027, 5113, 3569}, {5027, 14270, 5191}, {5029, 5075, 5040}, {5029, 5168, 1960}, {5040, 5163, 890}, {5191, 21731, 42663}, {6137, 6138, 3569}, {6140, 14270, 21731}, {9123, 9131, 14610}, {9123, 9979, 9131}, {9129, 9130, 3}, {9131, 9185, 9979}, {9134, 9189, 44564}, {9134, 44564, 8371}, {9147, 11176, 9148}, {9173, 9174, 8371}, {9185, 9979, 32193}, {11215, 31986, 32194}, {14270, 21731, 9409}, {15724, 36900, 4108}, {42659, 42665, 9409}, {42667, 42668, 669}, {52722, 52723, 11183}


X(352) = INVERSE-IN-CIRCUMCIRCLE OF X(353)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(-a4 - b4 - c4 - 5b2c2 + 4a2b2 + 4a2c2)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(352) lies on the Parry circle and these lines: 2,6   3,353   23,2502   110,187   111,511

X(352) = reflection of X(843) in X(187)
X(352) = isogonal conjugate of X(6094)
X(352) = inverse-in-circumcircle of X(353)
X(352) = crossdifference of every pair of points on line X(373)X(512)
X(352) = X(23)-of-circumsymmedial-triangle
X(352) = inverse-in-{circumcircle, nine-point circle}-inverter of X(3815)
X(352) = orthocentroidal-to-ABC similarity image of X(111)
X(352) = 4th-Brocard-to-circumsymmedial similarity image of X(111)
X(352) = X(2709)-of-1st-Parry-triangle
X(352) = X(843)-of-2nd-Parry-triangle
X(352) = X(110)-of-3rd-Parry-triangle
X(352) = inverse-in-Parry-isodynamic-circle of X(23); see X(2)
X(352) = radical trace of circumcircles of Artzt and anti-Artzt triangles
X(352) = trilinear pole, wrt 3rd Parry triangle, of Lemoine axis


X(353) = INVERSE-IN-BROCARD-CIRCLE OF X(111)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(4a4 - 2b4 - 2c4 - b2c2 - 4a2b2 -4a2c2)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

Let AaBaCa, AbBbCb, AcBcCc be the A-, B- and C-anti-altimedial triangles. Let A' be the trilinear product Aa*Ab*Ac, and define B' and C' cyclically. Triangle A'B'C' is the anticomplementary triangle of the 1st Brocard triangle, and is similar to the anti-orthocentroidal triangle, with similitude center X(353). (Randy Hutson, November 2, 2017)

X(353) lies on the Parry circle and these lines: 3,352   6,23   110,574   111,182   187,3117

X(353) = inverse-in-circumcircle of X(352)
X(353) = inverse-in-Brocard-circle of X(111)
X(353) = centroid of circumsymmedial triangle
X(353) = circumcevian isogonal conjugate of X(6)
X(353) = X(5640)-of-4th-anti-Brocard triangle
X(353) = isogonal conjugate, wrt 2nd Brocard triangle, of X(2)
X(353) = inverse-in-Parry-isodynamic-circle of X(2); see X(2)


X(354) = WEILL POINT

Trilinears    (b - c)2 - ab - ac : :
Trilinears    2 + cos B + cos C : :
Trilinears    cos^2(B/2) + cos^2(C/2) : :
Trilinears    b(cot B/2) + c(cot C/2) : :
Barycentrics  a[(b - c)2 - ab - ac] : :

X(354) is the perspector of the intangents triangle and the triangle QAQBQC described at X(3598). (Peter Moses, Nov. 4, 2010)

Let A', B', C' be the inverse-in-{circumcircle, incircle}-inverter of A, B, C. Let A″B″C″ be the tangential triangle of A'B'C'. Then A″B″C″ is perspective to ABC at X(354). (Randy Hutson, December 26, 2015)

William Gallatly, The Modern Geometry of the Triangle, 2nd edition, Hodgson, London, 1913, page 16.

Let Ab be the intersection, other than B, of circle {X(1),B,C}} and line AB. Let Ac be the intersection, other than C, of circle {X(1),B,C}} and line AC. Define Bc, Ba, Ca, Cb cyclically. X(354) is the centroid of AbAcBcBaCaCb. Note that the lines AbAc, BcBa, CaCb bound the intangents triangle. (Randy Hutson, December 26, 2015)

X(354) lies on these lines: 1,3   2,210   6,374   7,479   11,118   37,38   42,244   44,748   48,584   63,1001   81,105   142,3059   278,955   373,375   388,938   392,551   516,553   1418,2293

X(354) = isogonal conjugate of X(2346)
X(354) = complement of X(3681)
X(354) = anticomplement of X(3740)
X(354) = inverse-in-incircle of X(1155)
X(354) = reflection of X(i) in X(j) for these (i,j): (210,2), (392,551)
X(354) = X(101)-Ceva conjugate of X(513)
X(354) = crosspoint of X(1) and X(7)
X(354) = crosssum of X(1) and X(55)
X(354) = X(2)-of-intouch-triangle
X(354) = homothetic center of intouch triangle and inverse-in-incircle triangle
X(354) = centroid of inverse-in-incircle triangle
X(354) = pole of antiorthic axis wrt incircle
X(354) = {X(1),X(40)}-harmonic conjugate of X(3303)
X(354) = {X(1),X(65)}-harmonic conjugate of X(3057)
X(354) = {X(3513),X(3514)}-harmonic conjugate of X(65)
X(354) = centroid of the six intersections of the Conway circle and the sidelines of ABC
X(354) = centroid of the six intersections of the sidelines of ABC and the antiparallels to sidelines through X(1)
X(354) = inverse-in-{circumcircle, incircle}-inverter of X(36)
X(354) = cevian isogonal conjugate of X(7) = intouch isogonal conjugate of X(7)
X(354) = bicentric sum of PU(94)
X(354) = PU(94)-harmonic conjugate of X(650)
X(354) = X(381)-of-incircle-circles-triangle
X(354) = {X(1),X(57)}-harmonic conjugate of X(55)
X(354) = centroid of the six intersections of the Adams' circle and the sidelines of ABC


X(355) = FUHRMANN CENTER

Trilinears    a cos A - (b + c)cos(B - C) : :
Trilinears    bc(b + c)[a2(b2 + c2) - (b2 - c2)2] - a3bc(b2 + c2 - a2) : :   (Michel Garitte, 4/3/03)
Trilinears    cos B + cos C - 2 cos B cos C - 1 : :
Trilinears    r - R cos(B - C) : :

X(355) = the center of the Fuhrmann circle, defined as the circumcircle of the Fuhrmann triangle A″B″C″, where A″ is obtained as follows: let A' be the midpoint of the circumcircle-arc having endpoints B and C and not containing A; then A″ is the reflection of A' in line BC. Vertices B″ and C″ are obtained cyclically. (Other constructions of A'', and hence the Fuhrmann triangle, follow: (1) Let IA be the reflection of X(1) in BC; then A'' is the circumcenter of IABC. (2) Let JA be the reflection of the A-excenter in BC; then A'' is the circumcenter of JABC.

X(355) is the homothetic center of the triangles formed by internal and external tangents of circumcircles of BCX(4), CAX(4), ABX(4). Equivalently, the incenter of the trianglar hull of circumcircles of BCX(4), CAX(4), ABX(4). (Randy Hutson, December 2, 2017)

Let Na = X(5)-of-BCX(1), Nb = X(5)-of-CAX(1), Nc = X(5)- of-ABX(1). Then X(355) = X(20)-of-NaNbNc. (Randy Hutson, December 2, 2017)

Let A'B'C' be the anticomplementary triangle. Then X(355) is the radical center of the incircles of A'BC, B'CA, C'AB. (Randy Hutson, December 2, 2017)

Let A'B'C' be the outer Garcia triangle and A″B″C″ the inner Garcia triangle. Let A* be the isogonal conjugate, wrt A'B'C', of A″, and define B* and C* cyclically. The lines A'A*, B'B*, C'C* concur in X(355); see also X(952). (Randy Hutson, December 2, 2017)

Ross Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, Mathematical Association of America, 1995. Chapter 6: The Fuhrmann Circle.

Let OA be the circle centered at the A-vertex of the Yff contact triangle and passing through A; define OB and OC cyclically. X(355) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

X(355) lies on these lines: 1,5   2,944   3,10   4,8   30,40   65,68   85,150   104,404   165,550   381,519   382,516   388,942   938,1056

X(355) = midpoint of X(4) and X(8)
X(355) = isogonal conjugate of X(3417)
X(355) = reflection of X(i) in X(j) for these (i,j): (1,5), (3,10), (944,1385), (1482,946)
X(355) = anticomplement of X(1385)
X(355) = complement of X(944)
X(355) = inverse-in-Feuerbach-hyperbola of X(1837)
X(355) = {X(1),X(80)}-harmonic conjugate of X(1837)
X(355) = outer-Garcia-isogonal conjugate of X(40)
X(355) = X(3)-of-outer-Garcia-triangle
X(355) = X(6265)-of-inner-Garcia-triangle
X(355) = X(6102)-of-excentral-triangle
X(355) = inner-Garcia-to-outer-Garcia similarity image of X(3)
X(355) = X(12844)-of-orthic-triangle if ABC is acute
X(355) = endo-homothetic center of Ehrmann side-triangle and 4th anti-Euler triangle; the homothetic center is X(12111)
X(355) = reflection of X(12699) in X(4)
X(355) = {X(1),X(1837)}-harmonic conjugate of X(5722)
X(355) = Ursa-minor-to-Ursa-major similarity image of X(1)
X(355) = {X(1),X(5)}-harmonic conjugate of X(5886)


X(356) = MORLEY CENTER

Trilinears    cos A/3 + 2 cos B/3 cos C/3 : :
Trilinears    = cos(B/3 - C/3) + sqrt(3)sin(A/3 + π/3) (M. Stevanovic, 12/25/2007)
X(356) = X(3276) + X(3277) - X(3)

X(356) is the centroid of the Morley equilateral triangle. For a discussion of the theorem and extensive list of references, see

C. O. Oakley and J. C. Baker, "The Morley trisector theorem," American Mathematical Monthly 85 (1978) 737-745.

For a sketch of the Morley cubic and list of centers on it, including X(356), X(357), X(358), see Bernard Gibert's site.

See also Dao Thanh Oai, Some new equilateral triangles in a plane geometry.

Let A'B'C' be the 1st Morley triangle. Let A″ be the cevapoint of B' and C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(356). (Randy Hutson, September 14, 2016)

If you have The Geometer's Sketchpad, you can view X(356).

For a biographical sketch, including details about Morley's famous theorem on angle trisectors, with history and references, see

Frank Morley (1860-1937) geometer.

X(356) lies on these lines: 2,5455   3,3278   357,358   1134,1135   3605,5456.

X(356) = isogonal conjugate of X(3605)
X(356) = orthology center of 1st Morley adjunct triangle and 1st Morley triangle


X(357) = 1st MORLEY-TAYLOR-MARR CENTER

Trilinears       sec A/3 : sec B/3 : sec C/3
Barycentrics  sin A sec A/3 : sin B sec B/3 : sin C sec C/3

X(357) is the perspector of Morley triangle and ABC, and also the Hofstadter 1/3 point. See

F. Glanville Taylor and W. L. Marr, "The six trisectors of each of the angles of a triangle," Proceedings of the Edinburgh Mathematical Society 33 (1913-14) 119-131, especially item 9, page 127.

If you have The Geometer's Sketchpad, you can view X(357).

X(357) lies on these lines: 356,358   1134,3275   3280,5454

X(357) = isogonal conjugate of X(358)
X(357) = {X(1135),X(3272)}-harmonic conjugate of X(3603)
X(357) = trilinear product of vertices of 1st Morley triangle


X(358) = 2nd MORLEY-TAYLOR-MARR CENTER

Trilinears    cos A/3 : cos B/3 : cos C/3
Barycentrics    sin A cos A/3 : sin B cos B/3 : sin C cos C/3

X(358) is the perspector of the adjunct Morley triangle and ABC, and also the Hofstadter 2/3 point.

If you have The Geometer's Sketchpad, you can view X(358).

X(358) lies on these lines: 356,357   16,1135

X(358) = isogonal conjugate of X(357)
X(358) = trilinear product of vertices of 1st Morley adjunct triangle


X(359) = HOFSTADTER ONE POINT

Trilinears    a/A : b/B : c/C
Barycentrics    a2/A : b2/B : c2/C

This point is the limit as r approaches 1 of the Hofstadter r point. See X(360) for details.

If you have The Geometer's Sketchpad, you can view X(359) and X(360) and Hofstadter Triangles. These sketches include the Hofstadter ellipse (actually a family of ellipses, indexed by a parameter r) introduced (February 4, 2005) by Peter J. C. Moses. The ellipse E(r) is given for 0 < r < 1 by the following equation in trilinears:

x2 + y2 + z2 + yz(D + 1/D) + zx(E + 1/E) + xy(F + 1/F) = 0, where D = cos A - sin A cot rA, E = cos B - sin B cot rB, F = cos C - sin C cot rC.

The Hofstadter ellipse E(1/2), given by x2 + y2 + z2 - 2yz - 2xz - 2xy = 0, has center X(37) and passes through X(i) for these i: 244, 678, 2310, 2632, 2638, 2643, 3248, 4094, 4117, 10501, 23063, 24012.

Taking the limit of E(r) as r tends to 0 gives information about the circumellipse, E(0) (which is also E(1)):

Equation:     ayz/A + bzx/B + cxy/C = 0
Center:     X(5945) = (a/A)(b2/B + c2/C - a2/A) : (b/B)(c2/C + a2/A - b2/B) : (c/C)(a2/A + b2/B - c2/C)
Intersection with circumcircle (other than A, B, C):     X(3067) = a/[A(B - C)] : b/[B(C - A)] : c/[C(A - B)]

The Hofstadter ellipse E(1/2) is also the incentral inellipse, centered at X(37), and touching ABC at the traces of X(1). The ellipse also passes through X(4094), and is the incentral isotomic conjugate of line X(512)X(4895). (Randy Hutson, October 15, 2018)

See Hofstadter Ellipse at MathWorld.

See also the preamble just before X(42074).

X(359) lies on this line: {6,9036}

X(359) = isogonal conjugate of X(360)
X(359) = X(2)-Ceva conjugate of X(5945)
X(359) = X(i)-isoconjugate of X(j) for these (i,j): {1, 360}, {2, 1049}
X(359) = barycentric product X(i)*X(j) for these {i,j}: {1, 1077}, {1028, 1049}
X(359) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 360}, {31, 1049}, {1077, 75}
X(359) = perspector of Hofstadter 0-ellipse


X(360) = HOFSTADTER ZERO POINT

Trilinears    A/a : B/b : C/c
Barycentrics    A : B : C
X(360) = A*[A] + B*[B] + C*[C], where A, B, C are the angles and [A], [B], [C] are the vertices

This point is obtained as a limit of perspectors. Let r denote a real number, but not 0 or 1. Using vertex B as a pivot, swing line BC toward vertex A through angle rB and swing line BC about C through angle rC. Let A(r) be the point in which the two swung lines meet. Obtain B(r) and C(r) cyclically. Triangle A(r)B(r)C(r) is the r-Hofstadter triangle; its perspector with ABC, called the Hofstadter r point, is the point given by trilinears

(sin rA)/sin(A - rA) : (sin rB)/sin(B - rB) : (sin rC)/sin(C - rC).

The limit of this point as r approaches 0 is X(360). The two Hofstadter points, X(359) and X(360) are examples of transcendental triangle centers, since they have no trilinear or barycentric representation using only algebraic functions of a,b,c (or sin A, sin B, sin C).

Clark Kimberling, "Hofstadter points," Nieuw Archief voor Wiskunde 12 (1994) 109-114.

Conjecture and corollary (Randy Hutson, August 10, 2014):
If r is an integer other than 0, 1, or 2, then the inverse-in-circumcircle of the Hofstadter r point is the Hofstadter (2-r) point;
thus, since the isogonal conjugate of the Hofstadter r point is the Hofstadter (1-r) point, if r is not -1, 0 or 1,
then the antigonal image of the Hofstadter r point is the Hofstadter -r point.

X(1) = Hofstadter 1/2 point = antigonal image of X(80)
X(3) = Hofstadter 2 point = antigonal image of X(265)
X(4) = Hofstadter -1 point
X(35) = Hofstadter 3/2 point
X(79) = Hofstadter -1/2 point = antigonal image of X(3065)
X(186) = Hofstadter 3 point = antigonal image of X(5962)
X(265) = Hofstadter -2 point = antigonal image of X(3)
X(357) = Hofstadter 1/3 point
X(358) = Hofstadter 2/3 point
X(359) = Hofstadter 0 point
X(360) = Hofstadter 1 point
X(1127) = Hofstadter 1/4 point
X(1129) = Hofstadter 3/4 point
X(5457) = Hofstadter -1/3 point
X(5458) = Hofstadter -2/3 point
X(5961) = Hofstadter 4 point = antigonal image of X(5964)
X(5962) = Hofstadter -3 point = antigonal image of X(186)
X(5963) = Hofstadter 5 point
X(5964) = Hofstadter -4 point = antigonal image of X(5961)

X(360) lies on this line: {2,1115}

X(360) = isogonal conjugate of X(359)
X(360) = anticomplement of X(1115)
X(360) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {7021, 8}, {7041, 69}
X(360) = X(i)-isoconjugate of X(j) for these (i,j): {1, 359}, {6, 1077}
X(360) = barycentric product X(75)*X(1049)
X(360) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1077}, {6, 359}, {1049, 1}, {1077, 1028}, {1085, 1049}


X(361) = X(266)-CEVA CONJUGATE OF X(1)

Trilinears       csc B/2 + csc C/2 - csc A/2 : csc C/2 + csc A/2 - csc B/2 : csc A/2 + csc B/2 - csc C/2
Barycentrics  f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sin A)(csc B/2 + csc C/2 - csc A/2)

The isoscelizer equation au(X) = bv(X) = cw(X) has solution X = X(361).

X(361) lies on these lines: 1,188   164,503   266,978

X(361) = X(266)-Ceva conjugate of X(1)


X(362) = CONGRUENT CIRCUMCIRCLES ISOSCELIZER POINT

Trilinears    b cos B/2 + c cos C/2 - a cos A/2
Barycentrics   Sin[A]*(-(Cos[A/2]*Sin[A]) + Cos[B/2]*Sin[B] + Cos[C/2]*Sin[C]) : :

The isoscelizer equations u(X)/a = v(X)/b = w(X)/c have solution X = X(362).

If you have The Geometer's Sketchpad, you can view X(362).

X(362) lies on these lines: {1, 60539}, {9, 2090}, {57, 234}, {164, 845}, {4146, 16551}

X(362) = X(31)-of-excentral triangle
X(362) = excentral-isogonal conjugate of X(844)
X(362) = X(4146)-Ceva conjugate of X(1)
X(362) = X(259)-Dao conjugate of X(188)


X(363) = EQUAL PERIMETERS ISOSCELIZER POINT

Trilinears    b/(1 + sin B/2) + c/(1 + sin C/2) - a/(1 + sin A/2) : :
Barycentrics   Sin[A]*(-(Sin[A]/(1 + Sin[A/2])) + Sin[B]/(1 + Sin[B/2]) + Sin[C]/(1 + Sin[C/2])) : :

If X = X(363), the isoscelizer triangles of X have equal perimeters.

X(363) is the homothetic center of the excentral triangle and the inner Hutson triangle. A construction of the latter follows. The internal bisector of angle A meets the A-excircle in two points. Let PA be the point closer to line BC and let QA be the other point. Define PB and PC cyclically, and define QB and QC cyclically. Let LA be the line tangent to the A-excircle at PA, and define LB and LC cyclically. Let MA be the line tangent to the A-excircle at QA, and define MB and MC cyclically. The inner Hutson triangle is the triangle A'B'C' given by A' = LB∩LC, B' = LC∩LA, C' = LA∩LB; the outer Hutson triangle is given by A'' = MB∩MC, B'' = MC∩MA, C' = MA∩MB. (Based on a description of A'B'C' by Randy Hutson, September 23, 2011)

Peter Moses (November 10, 2011) found trilinears for A'B'C' and A''B''C''. As these are central triangles, trilinears for A' and A'' suffice:

A' = aUA(a2 + b2 + c2 - 2bc - 2ca - 2ab) + bUB(b + c - a)(c + a - b) + cUC(b + c - a)(a + b - c)
     : aUA(a2 + b2 - 3c2 + 2bc + 2ca - 2ab) + UB(c - a)(-a + b + c)(a - b + c) - cUC(-a + b + c)(a + b - c)
     : aUA(a2 - 3b2 + c2 + 2bc - 2ca + 2ab) - bUB(-a + b + c)(a - b + c) + UC(b - a)(-a + b + c)(a + b - c)

A'' = aUA(a2 + b2 + c2 - 2bc - 2ca - 2ab) - bUB(b + c - a)(c + a - b) - cUC(b + c - a)(a + b - c)
     : aUA(a2 + b2 - 3c2 + 2bc + 2ca - 2ab) - UB(c - a)(-a + b + c)(a - b + c) + cUC(-a + b + c)(a + b - c)
     : aUA(a2 - 3b2 + c2 + 2bc - 2ca + 2ab) + bUB(-a + b + c)(a - b + c) - UC(b - a)(-a + b + c)(a + b - c),

where UA = sqrt[bc/((a - b + c)(a + b - c)] = (1/2)csc(A/2), and UB and UC are defined cyclically.

A'B'C' is perspective to the following triangles: intouch, hexyl,Yff, and the 1st and 2nd circumperp triangles. A''B''C'' is perspective to these: ABC, intouch, hexyl, Yff, and the 1st and 2nd circumperp triangles. A'B'C' is homothetic to A''B''C'', which is perspective to the excentral triangle at X(168). (Peter Moses, 11/10/11)

If you have The Geometer's Sketchpad, you can view X(363).

Let UVW and U'V'W' be the tangential-midarc triangles introduced in the preamble just before X(8075). The inner and outer Hutson triangles introduced at X(363) are the excircles-version of UVW and U'V'W'. (Randy Hutson and César Lozada, August 29, 2015)

X(363) lies on these lines: {1, 289}, {2, 9783}, {3, 8111}, {9, 5934}, {10, 12574}, {40, 164}, {55, 17607}, {57, 8113}, {63, 11685}, {165, 166}, {173, 3973}, {191, 16135}, {223, 34037}, {258, 6732}, {846, 8391}, {1376, 17621}, {1445, 8385}, {1490, 12673}, {1697, 8390}, {1698, 8380}, {1699, 8377}, {1721, 12719}, {1764, 11892}, {1768, 13260}, {2136, 12633}, {3333, 11039}, {3659, 55175}, {4882, 9847}, {5541, 12733}, {6326, 12759}, {8078, 8133}, {8231, 11922}, {8580, 11856}, {8915, 35897}, {10233, 12523}, {10234, 55174}, {11044, 58777}, {12396, 12886}, {12514, 12561}, {12565, 12707}, {12658, 12851}, {12659, 12878}, {12660, 12882}

X(363) = X(57)-of-excentral-triangle
X(363) = homothetic center of ABC and orthic triangle of inner Hutson triangle X(363) = excentral isogonal conjugate of X(168)
X(363) = X(236)-Ceva conjugate of X(1)
X(363) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 9805, 11527}, {2, 9783, 21618}, {3, 12488, 8111}, {40, 52797, 164}, {165, 845, 168}, {165, 8140, 8107}, {8107, 11222, 8140}, {8109, 9805, 1}, {11854, 12880, 8113}, {22993, 45702, 5934}


X(364) = WABASH CENTER (EQUAL AREAS ISOSCELIZER POINT)

Trilinears    b1/2 + c1/2 - a1/2 : :
Barycentrics   a*(-Sqrt[a] + Sqrt[b] + Sqrt[c]) : :

If X = X(364), the isoscelizer triangles T(X,a), T(X,b), T(X,c) have equal areas.

The isoscelizer equations H(A,X)D(A,X) = H(B,X)D(B,X) = H(C,X)D(C,X) have solution X = X(364). (Peter Yff, 4/9/99)

If you have The Geometer's Sketchpad, you can view X(364).

X(364) lies on the cubics K351 and K577 and these lines: {1, 365}, {2, 55321}, {9, 366}, {55, 55326}, {57, 40378}, {165, 4166}, {239, 40383}, {1743, 2068}, {20673, 20695}

X(364) = X(366)-Ceva conjugate of X(1)
X(364) = X(2)-isoconjugate of X(60552)
X(364) = X(i)-Dao conjugate of X(j) for these (i,j): {366, 18297}, {32664, 60552}
X(364) = cevapoint of X(365) and X(8832)
X(364) = barycentric product X(i)*X(j) for these {i,j}: {1, 20534}, {75, 20673}, {86, 20695}, {92, 20763}, {366, 40374}
X(364) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 60552}, {20534, 75}, {20673, 1}, {20695, 10}, {20763, 63}, {40374, 18297}
X(364) = {X(365),X(367)}-harmonic conjugate of X(1)


X(365) = SQUARE ROOT POINT

Trilinears       a1/2 : b1/2 : c1/2
Barycentrics  a3/2 : b3/2 : c3/2

For a construction of X(365), see the note at X(2), which provides for a construction barycentric square roots which one can easily extend to a construction for trilinear square roots.

X(365) lies on the Kiepert circumhyperbola of the excentral triangle, the cubics K131, K145, K155, K760, K1253, K1273, and these lines: {1, 364}, {2, 20334}, {6, 2118}, {43, 2068}, {55, 8832}, {75, 20631}, {100, 59461}, {292, 2146}, {366, 4179}, {1030, 60556}, {1386, 59440}, {2110, 2119}, {2144, 2147}, {4166, 18753}, {4640, 59462}, {21780, 60553}

X(365) = isogonal conjugate of X(366)
X(365) = complement of X(20346)
X(365) = anticomplement of X(20334)
X(365) = crosssum of X(1) and X(364)
X(365) = trilinear square root of X(6)
X(365) = isogonal conjugate of the anticomplement of X(20527)
X(365) = isogonal conjugate of the complement of X(20534)
X(365) = isogonal conjugate of the isotomic conjugate of X(18297)
X(365) = X(i)-Ceva conjugate of X(j) for these (i,j): {238, 2146}, {364, 8832}, {366, 4166}
X(365) = X(i)-isoconjugate of X(j) for these (i,j): {1, 366}, {2, 365}, {6, 18297}, {7, 4166}, {57, 4182}, {75, 18753}, {81, 4179}, {86, 60548}, {174, 60534}, {188, 509}, {259, 508}, {266, 55336}, {507, 7025}, {556, 60538}, {4146, 60530}
X(365) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 366}, {9, 18297}, {206, 18753}, {365, 20346}, {5452, 4182}, {18753, 20604}, {20543, 20592}, {32664, 365}, {40374, 75}, {40586, 4179}, {40600, 60548}
X(365) = cevapoint of X(i) and X(j) for these (i,j): {1, 40375}, {6, 20673}
X(365) = barycentric product X(i)*X(j) for these {i,j}: {1, 366}, {6, 18297}, {7, 4166}, {57, 4182}, {75, 18753}, {81, 4179}, {86, 60548}, {174, 60534}, {188, 509}, {259, 508}, {266, 55336}, {507, 7025}, {556, 60538}, {4146, 60530}
X(365) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 18297}, {6, 366}, {32, 18753}, {41, 4166}, {42, 4179}, {55, 4182}, {213, 60548}, {259, 55336}, {266, 508}, {366, 75}, {509, 4146}, {4166, 8}, {4179, 321}, {4182, 312}, {18297, 76}, {18753, 1}, {20664, 40378}, {20673, 40374}, {52866, 367}, {55326, 55322}, {58996, 55321}, {60530, 188}, {60534, 556}, {60538, 174}, {60539, 60534}, {60540, 39131}, {60542, 6724}, {60548, 10}
X(365) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 364, 367}, {2, 20346, 20334}, {75, 20645, 20631}, {366, 4182, 4179}, {18753, 60548, 4166}, {20673, 60552, 55}


X(366) = ISOGONAL CONJUGATE OF X(365)

Trilinears       a-1/2 : b-1/2 : c-1/2
Barycentrics   a1/2 : b1/2 : c1/2

See the note at X(365).

X(366) lies on the Jerabek circumhyperbola of the excentral triangle, the cubics on K101, K132, K202, K308, K323, K328, K332, K363, K977, K1080, K1081, K1082, the curve Q027, and these lines: {1, 4180}, {2, 367}, {6, 2068}, {9, 364}, {10, 59460}, {75, 20434}, {190, 59459}, {192, 40383}, {365, 4179}, {551, 59438}, {1631, 20469}, {18297, 60548}

X(366) = reflection of X(20534) in X(20527)
X(366) = isogonal conjugate of X(365)
X(366) = isotomic conjugate of X(18297)
X(366) = complement of X(20534)
X(366) = anticomplement of X(20527)
X(366) = complement of the isogonal conjugate of X(60552)
X(366) = isogonal conjugate of the anticomplement of X(20334)
X(366) = isogonal conjugate of the complement of X(20346)
X(366) = isotomic conjugate of the anticomplement of X(40378)
X(366) = isotomic conjugate of the complement of X(40383)
X(366) = isotomic conjugate of the isogonal conjugate of X(18753)
X(366) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 40374}, {60552, 10}
X(366) = X(367)-cross conjugate of X(1)
X(366) = trilinear square root of X(2)
X(366) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 40374}, {7, 56707}, {18297, 4182}
X(366) = X(i)-isoconjugate of X(j) for these (i,j): {1, 365}, {2, 18753}, {6, 366}, {31, 18297}, {56, 4182}, {57, 4166}, {58, 4179}, {81, 60548}, {174, 60530}, {188, 60538}, {259, 509}, {266, 60534}, {508, 60539}, {6727, 60537}, {40374, 60552}
X(366) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 4182}, {2, 18297}, {3, 365}, {9, 366}, {10, 4179}, {236, 55336}, {365, 510}, {366, 20534}, {5452, 4166}, {15495, 508}, {18753, 20874}, {20334, 20357}, {32664, 18753}, {40374, 2}, {40586, 60548}
X(366) = cevapoint of X(i) and X(j) for these (i,j): {1, 364}, {2, 40383}, {6, 20469}, {365, 4166}, {367, 40378}, {4179, 60548}
X(366) = barycentric product X(i)*X(j) for these {i,j}: {1, 18297}, {7, 4182}, {75, 365}, {76, 18753}, {85, 4166}, {86, 4179}, {174, 55336}, {188, 508}, {274, 60548}, {509, 556}, {4146, 60534}
X(366) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 18297}, {6, 365}, {9, 4182}, {31, 18753}, {37, 4179}, {42, 60548}, {55, 4166}, {174, 508}, {188, 55336}, {259, 60534}, {266, 509}, {364, 40374}, {365, 1}, {367, 40378}, {508, 4146}, {509, 174}, {4166, 9}, {4179, 10}, {4180, 4181}, {4182, 8}, {18297, 75}, {18753, 6}, {20779, 20751}, {40374, 20534}, {40378, 20527}, {52865, 52866}, {52866, 20664}, {55321, 55322}, {55325, 55321}, {55326, 55325}, {55336, 556}, {55374, 55373}, {58996, 55326}, {60530, 259}, {60531, 60535}, {60532, 60536}, {60533, 60537}, {60534, 188}, {60535, 39131}, {60537, 6724}, {60538, 266}, {60539, 60530}, {60540, 60531}, {60541, 60532}, {60542, 60533}, {60546, 60544}, {60547, 60545}, {60548, 37}
X(366) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 40378, 40374}, {2, 20534, 20527}, {75, 20447, 20434}, {365, 4179, 4182}, {2068, 2069, 6}


X(367) = CROSSPOINT OF X(1) AND X(366)

Trilinears       b1/2 + c1/2 : c1/2 + a1/2 : a1/2 + b1/2
Barycentrics  a(b1/2 + c1/2) : b(c1/2 + a1/2) : c(a1/2 + b1/2)

X(367) lies on the circumconic {{A,B,C,X(1),X(2)}} and these lines: 1,364   {1, 364}, {2, 366}, {57, 8832}, {81, 52865}, {88, 55325}, {105, 58996}, {330, 40383}, {3227, 55322}, {36805, 55373}

X(367) = isogonal conjugate of X(61143)
X(367) = isotomic conjugate of X(61144)
X(367) = crosspoint of X(1) and X(366)
X(367) = crosssum of X(1) and X(365)
X(367) = Danneels point of X(366)
X(367) = isotomic conjugate of the isogonal conjugate of X(52865)
X(367) = X(366)-Ceva conjugate of X(40378)
X(367) = X(i)-isoconjugate of X(j) for these (i,j): {367, 59461}, {20664, 59459}
X(367) = X(i)-Dao conjugate of X(j) for these (i,j): {20527, 18297}, {40378, 75}
X(367) = X(1)-line conjugate of X(2069)
X(367) = barycentric product X(i)*X(j) for these {i,j}: {1, 20527}, {57, 4181}, {75, 20664}, {76, 52865}, {86, 20682}, {92, 20751}, {366, 40378}, {513, 55322}, {514, 55325}, {693, 58996}, {3669, 55373}
X(367) = barycentric quotient X(i)/X(j) for these {i,j}: {4181, 312}, {20527, 75}, {20664, 1}, {20682, 10}, {20751, 63}, {40378, 18297}, {52865, 6}, {52866, 365}, {55322, 668}, {55325, 190}, {55373, 646}, {58996, 100}
X(367) = {X(1),X(364)}-harmonic conjugate of X(365)


X(368) = EQUI-BROCARD CENTER

Trilinears       (reasonable trilinears are sought)
Barycentrics  (reasonable barycentrics are sought)

The center X for which the triangle XBC, XCA, XAB have equal Brocard angles. Peter Yff proved that X(368) lies on the self-isogonal conjugate cubic with trilinear equation f(a,b,c)u + f(b,c,a)v + f(c,a,b)w = 0, where f(a,b,c) = bc(b2 - c2) and, for variable x : y : z, the cubics u, v, w are given by u(x,y,z) = x(y2 + z2), v = u(y,z,x), w = u(z,x,y).

Cyril Parry proved that X(368) lies on the anticomplement of the Kiepert hyperbola, this anticomplement being given by the trilinear equation a2(b2 - c2)x2 + b2(c2 - a2)y2 + c2(a2 - b2)z2 = 0.

If you have The Geometer's Sketchpad, you can view X(368) and X(368) With Curves.


X(369) = 1st TRISECTED PERIMETER POINT

Trilinears       x : y : z (see below)
Barycentrics  ax : by : cz

If you have The Geometer's Sketchpad, you can view X(369).

There exist points A', B', C' on segments BC, CA, AB, respectively, such that AB' + AC' = BC' + BA' = CA' + CB' = (a + b + c)/3, and the lines AA', BB', CC' concur in X(369). Near the end of the 20th century, Yff found trilinears for X(369) in terms of the unique real root, r, of the cubic polynomial

2t3 - 3(a + b + c)t2 + (a2 + b2 + c2 + 8bc + 8ca + 8ab)t - (cb2 + ac2 + ba2 + 5bc2 + 5ca2 + 5ab2 + 9abc),

as follows: x = bc(r - c + a)(r - a + b). Here x(a,c,b) ≠ x(a,b,c), so that y and z are not obtained from x by cyclically permutating a,b,c. At the geometry conference held at Miami University of Ohio, October 2, 2004, Yff, proved that X(369) is also given by x1 : y1 : z1 where y1 : z1 are given by cyclic permutations of a,b,c, in x1, where

x1 = bc[r2 - (2c + a)r + (- a2 + b2 + 2c2 + 2bc + 3ca + 2ab].

His presentation included a proof that there is only one point for which AB' + AC' = BC' + BA' = CA' + CB' .


X(370) = EQUILATERAL CEVIAN TRIANGLE POINT

Trilinears       (see below)
Barycentrics  (see below)

A point P is an equilateral cevian triangle point if the cevian triangle of P is equilateral. Jiang Huanxin introduced this notion in 1997.

Jean-Pierre Ehrmann notes (11/6/02) that the normalized barycentric coordinates (x,y,z) of X(370) are the unique solution of this system:

y(1 - y)SB + z(1 - z)SC = x(1 + x)F
z(1 - z)SC + x(1 - x)SA = y(1 + y)F
x(1 - x)SA + y(1 - y)SB = z(1 + z)F
x + y + z = 1,

where SA = (b2 + c2 - a2)/2; SB, SC are defined cyclically, F = [2 area(ABC)]/sqrt(3), and x>0, y>0, z>0.

Jiang Huanxin and David Goering, Problem 10358* and Solution, "Equilateral cevian triangles," American Mathematical Monthly 104 (1997) 567-570 [proposed 1994].

Chris van Tienhoven has computed the coordinates of the point. See Anthrakitis

X(370) lies on the Neuberg cubic.


X(371) = KENMOTU POINT (CONGRUENT SQUARES POINT)

Trilinears   cos(A - π/4) : :
Trilinears   cos A + sin A : :
Trilinears   a(SA + S) : :
Trilinears   a(b2 + c2 - a2 + 2S) : :
Barycentrics  sin A cos(A - π/4) : :
X(371) = LA/RA + LB/RB + LC/RC - X(3)/R, where LA, LB, LC are the centers of the Lucas circles, and RA, RB, RC their radii (Randy Hutson, July 23, 2015)
X(371) = LA/RA + LB/RB + LC/RC - LR/RR, where LR and RR are the center and radius of the Lucas radical circle (Randy Hutson, July 23, 2015)
X(371) = 3 X[2] + X[43134], 2 X[639] + X[43134], 3 X[6] - 4 X[44482], 3 X[6] - 2 X[44502], X[1152] - 3 X[12963], 2 X[1504] + X[40274], X[6396] - 4 X[9675], 2 X[6398] - 5 X[41410], X[11825] - 4 X[43120], X[11825] + 2 X[45489], 3 X[35840] - 8 X[44482], 3 X[35840] - 4 X[44502], 2 X[43120] + X[45489], X[3070] - 3 X[32787], 2 X[3070] - 3 X[35822], X[3070] + 3 X[41945], 2 X[3070] + X[42266], 3 X[3070] - X[42272], 5 X[3070] + 3 X[43210], X[3070] + 6 X[52047], 2 X[7583] - 3 X[32787], 4 X[7583] - X[35820], 4 X[7583] - 3 X[35822], 2 X[7583] + 3 X[41945], 2 X[7583] + X[42258], 4 X[7583] + X[42266], 6 X[7583] - X[42272], 10 X[7583] + 3 X[43210], X[7583] + 3 X[52047], 6 X[32787] - X[35820], 3 X[32787] + X[42258], 6 X[32787] + X[42266], 9 X[32787] - X[42272], 5 X[32787] + X[43210], X[32787] + 2 X[52047], X[35820] - 3 X[35822], X[35820] + 6 X[41945], X[35820] + 2 X[42258], 3 X[35820] - 2 X[42272], 5 X[35820] + 6 X[43210], X[35820] + 12 X[52047], X[35822] + 2 X[41945], 3 X[35822] + 2 X[42258], 3 X[35822] + X[42266], 9 X[35822] - 2 X[42272], 5 X[35822] + 2 X[43210], and many others

There exist three congruent squares U, V, W positioned in ABC as follows: U has opposing vertices on segments AB and AC; V has opposing vertices on segments BC and BA; W has opposing vertices on segments CA and CB, and there is a single point common to U, V, W. The common point, X(371), may have first been published in Kenmotu's Collection of Sangaku Problems in 1840, indicating that its first appearance may have been anonymously inscribed on a wooden board hung up in a Japanese shrine or temple. (The Kenmotu configuration uses only half-squares; i.e., isosceles right triangles). Trilinears were found by John Rigby.

The edgelength of the three squares is 21/2abc/(a2 + b2 + c2 + 4σ), where σ = area(ABC). (Edward Brisse, 2/12/2000)

X(371) is the internal center of similitude of the circumcircle and the 2nd Lemoine circle (cosine circle) (Peter J. C. Moses, 5/9/03). Also, X(371) is the internal center of similitude of the Gallatly circle (defined just before X(2007)) and the 1st Lemoine circle (Peter J. C. Moses, 9/10/2003).

X(371) is the perspector of several pairs of triangles associated with Lucas circles. Some of these triangles are defined at MathWorld; e.g. Lucas Central Triangle, and others are Lucas(L:W) configurations. The latter are generalizations of configurations associated with Lucas circles, in which the squares are replaced by rectangles of length-to-width ratio L:W, with length on the corresponding sideline of ABC. A negative ratio indicates that the rectangles are directed inward; e.g. Lucas(-1:1) indicates inward-directed squares, whereas Lucas(1:1) indicates the classical case of outward-directed squares. These generalizations and the following properties of X(371) were contributed by Randy Hutson, 9/23/2011. See also X(372) and X(1084).

Hidetoshi Fukagawa and John F. Rigby, Traditional Japanese Mathematics Problems of the 18th and 19th Centuries, SCT Publishing, Singapore, 2002. Reviewed, together with the Fukagawa and Pedoe book, Japanese Tempe Geometry Problems: San Gaku, by Clark Kimberling in The Mathematical Intelligencer 28, no. 1 (Winter 2006) 61-63.

Floor van Lamoen, Vierkanten in een driehoik: 3. Congruente vierkanten

Tony Rothman, with the cooperation of Hidetoshi Fukagawa, Japanese Temple Geometry (feature article in Scientific American)

Let A'B'C' be the Lucas central triangle. Let A″ be the cevapoint of B' and C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(371). (Randy Hutson, July 23, 2015)

If you have The Geometer's Sketchpad, you can view Kenmotu Point.

X(371) X(371) lies on the cubics K006, K233, K250, K337, K415, K424a, K633, K897, K1197, K1257, the curve Q168, and these lines: {1, 1702}, {2, 486}, {3, 6}, {4, 485}, {5, 590}, {8, 35843}, {10, 13912}, {11, 9661}, {12, 9646}, {13, 3391}, {14, 3366}, {17, 3367}, {18, 3392}, {20, 1587}, {22, 9683}, {24, 5413}, {25, 493}, {30, 3070}, {35, 3301}, {36, 3299}, {40, 5415}, {46, 2362}, {51, 3156}, {54, 6414}, {55, 1335}, {56, 1124}, {64, 6415}, {67, 19397}, {68, 35837}, {69, 11292}, {74, 19111}, {76, 35867}, {79, 35855}, {80, 35853}, {81, 16441}, {83, 22718}, {84, 35845}, {86, 21909}, {90, 7133}, {98, 13885}, {99, 19056}, {100, 19082}, {104, 19113}, {110, 7598}, {112, 19094}, {113, 8998}, {114, 6289}, {115, 8980}, {119, 13922}, {125, 8994}, {127, 13918}, {132, 13923}, {140, 615}, {155, 8909}, {157, 30398}, {165, 1703}, {184, 3155}, {185, 21640}, {186, 10881}, {193, 488}, {194, 22716}, {212, 3077}, {230, 49029}, {235, 13884}, {238, 8225}, {254, 41516}, {256, 8331}, {262, 10850}, {265, 35835}, {291, 8347}, {315, 491}, {355, 13911}, {376, 6460}, {378, 11474}, {381, 8976}, {382, 13665}, {394, 1583}, {395, 15765}, {396, 18585}, {397, 14814}, {398, 14813}, {399, 44592}, {402, 35791}, {427, 8280}, {474, 31473}, {489, 7389}, {490, 32421}, {492, 641}, {494, 35805}, {498, 44622}, {499, 44624}, {515, 13883}, {517, 7969}, {542, 32291}, {546, 13925}, {547, 43211}, {549, 13966}, {550, 19117}, {588, 3060}, {601, 606}, {602, 605}, {603, 3076}, {626, 11314}, {631, 3069}, {632, 32790}, {638, 11294}, {671, 35699}, {736, 49352}, {754, 1991}, {760, 45714}, {936, 31438}, {940, 16433}, {944, 19066}, {946, 8983}, {952, 35842}, {958, 1378}, {971, 35844}, {997, 30556}, {999, 3297}, {1092, 9686}, {1131, 1327}, {1132, 5056}, {1147, 9676}, {1154, 12965}, {1181, 19355}, {1296, 11835}, {1297, 19115}, {1328, 3316}, {1344, 2466}, {1345, 2465}, {1352, 37343}, {1376, 1377}, {1385, 7968}, {1478, 13905}, {1479, 13904}, {1482, 35810}, {1490, 19068}, {1498, 17819}, {1503, 36709}, {1511, 49269}, {1584, 5407}, {1585, 8968}, {1589, 11433}, {1590, 11427}, {1593, 3093}, {1594, 26951}, {1598, 13889}, {1599, 1993}, {1600, 5419}, {1614, 26886}, {1656, 8253}, {1657, 18512}, {1658, 11266}, {1672, 2008}, {1673, 2007}, {1676, 2010}, {1677, 2009}, {1680, 2014}, {1681, 2013}, {1699, 13888}, {1700, 2018}, {1701, 2017}, {1707, 6213}, {1708, 13388}, {1742, 31544}, {1743, 32555}, {1843, 19358}, {1870, 26948}, {1916, 8315}, {2015, 2472}, {2016, 2471}, {2041, 40693}, {2042, 40694}, {2043, 10653}, {2044, 10654}, {2045, 42149}, {2046, 42152}, {2077, 26465}, {2242, 31471}, {2544, 2547}, {2545, 2546}, {2548, 31463}, {2777, 13287}, {2782, 35824}, {2794, 13749}, {3008, 31541}, {3090, 23273}, {3091, 8972}, {3100, 9631}, {3146, 22644}, {3167, 8912}, {3235, 3238}, {3236, 3237}, {3295, 3298}, {3317, 3533}, {3357, 49251}, {3387, 42783}, {3388, 42784}, {3425, 8989}, {3428, 19014}, {3515, 5411}, {3522, 9543}, {3523, 7586}, {3524, 19053}, {3525, 13939}, {3526, 8252}, {3528, 9693}, {3529, 23267}, {3530, 35256}, {3534, 43339}, {3563, 39384}, {3564, 32497}, {3567, 26894}, {3574, 8995}, {3576, 9615}, {3579, 49227}, {3590, 12819}, {3591, 43559}, {3618, 11291}, {3627, 42225}, {3628, 18762}, {3629, 40288}, {3664, 31540}, {3788, 11315}, {3832, 23263}, {3843, 45384}, {3845, 42417}, {3859, 41948}, {3972, 35939}, {4293, 31408}, {4297, 49548}, {4317, 31475}, {4383, 16432}, {4769, 45445}, {4845, 30335}, {5054, 13847}, {5066, 42606}, {5204, 18995}, {5217, 19037}, {5254, 12960}, {5286, 21737}, {5318, 42176}, {5321, 35740}, {5334, 35732}, {5335, 42172}, {5339, 42279}, {5340, 42278}, {5393, 31562}, {5416, 10902}, {5417, 5446}, {5432, 9648}, {5433, 9663}, {5461, 13663}, {5473, 19074}, {5474, 19076}, {5475, 31481}, {5478, 13917}, {5479, 13916}, {5480, 13910}, {5587, 13893}, {5591, 7375}, {5597, 35781}, {5598, 35779}, {5603, 13902}, {5640, 35300}, {5657, 19065}, {5663, 12375}, {5690, 49233}, {5860, 6216}, {5861, 26619}, {5870, 7374}, {5881, 31440}, {5889, 26912}, {5890, 6458}, {5989, 8304}, {6000, 11241}, {6102, 44612}, {6145, 35859}, {6179, 6312}, {6197, 26952}, {6198, 9632}, {6201, 8975}, {6202, 7000}, {6222, 9756}, {6228, 44365}, {6241, 11462}, {6245, 8987}, {6246, 8988}, {6247, 8991}, {6248, 8992}, {6249, 8993}, {6251, 13921}, {6281, 13650}, {6284, 9660}, {6304, 33352}, {6305, 33350}, {6401, 11977}, {6402, 11976}, {6406, 45840}, {6416, 14528}, {6515, 11090}, {6525, 22838}, {6642, 10963}, {6680, 11316}, {6684, 13936}, {6699, 46689}, {6750, 8955}, {6759, 10533}, {6771, 49209}, {6774, 49211}, {6776, 21736}, {6811, 9744}, {6813, 9753}, {7160, 35863}, {7354, 9647}, {7387, 35776}, {7488, 11418}, {7494, 18290}, {7592, 26920}, {7599, 7601}, {7691, 19096}, {7709, 32471}, {7737, 31411}, {7738, 12123}, {7759, 51395}, {7762, 35685}, {7773, 32432}, {7834, 11313}, {7851, 32433}, {7887, 32435}, {7981, 45719}, {7987, 9585}, {8149, 49351}, {8196, 13890}, {8203, 13891}, {8212, 13899}, {8213, 13900}, {8277, 19005}, {8301, 8336}, {8305, 8308}, {8306, 8317}, {8320, 8326}, {8321, 8324}, {8322, 8333}, {8337, 8340}, {8338, 8349}, {8703, 41946}, {8825, 10237}, {8884, 19183}, {8908, 10282}, {8943, 15805}, {8946, 14248}, {8967, 12147}, {8978, 13389}, {9441, 31545}, {9584, 16192}, {9593, 31427}, {9618, 35242}, {9620, 31437}, {9649, 15326}, {9662, 15338}, {9692, 15717}, {9694, 17928}, {9695, 10323}, {9709, 31485}, {9751, 19091}, {9752, 9757}, {9880, 13908}, {9922, 45400}, {9927, 13909}, {9932, 12425}, {9993, 13892}, {10109, 43435}, {10113, 13915}, {10133, 11402}, {10164, 13975}, {10165, 13971}, {10246, 44636}, {10266, 35871}, {10267, 19049}, {10269, 19047}, {10299, 43510}, {10303, 13941}, {10304, 43511}, {10310, 19000}, {10525, 35796}, {10526, 35798}, {10531, 13906}, {10532, 13907}, {10606, 19087}, {10610, 49257}, {10665, 13754}, {10669, 12979}, {10673, 35806}, {10679, 35816}, {10680, 35818}, {10799, 12840}, {10817, 15055}, {10820, 15035}, {10840, 10843}, {10841, 10852}, {10893, 13895}, {10894, 13896}, {10895, 13897}, {10896, 13898}, {11001, 42414}, {11012, 26464}, {11202, 11242}, {11248, 19048}, {11249, 19050}, {11251, 35790}, {11252, 35778}, {11253, 35780}, {11257, 19090}, {11414, 19006}, {11447, 12111}, {11448, 11449}, {11463, 11464}, {11496, 13887}, {11539, 43887}, {11542, 42251}, {11543, 35738}, {11812, 43212}, {11822, 19008}, {11823, 19010}, {11826, 19024}, {11827, 19026}, {11828, 19032}, {11829, 19034}, {11833, 38698}, {11836, 38716}, {11897, 13894}, {11941, 11967}, {11942, 11963}, {11943, 11970}, {11944, 11965}, {11959, 11981}, {11961, 11980}, {12041, 49217}, {12042, 49213}, {12100, 52046}, {12110, 22722}, {12117, 19058}, {12118, 19062}, {12119, 19078}, {12120, 19086}, {12121, 19052}, {12122, 19092}, {12124, 19103}, {12160, 51946}, {12164, 19492}, {12256, 14912}, {12297, 22646}, {12359, 49225}, {12424, 35836}, {12514, 30557}, {12556, 19098}, {12584, 32292}, {12590, 19588}, {12599, 13914}, {12600, 13919}, {12601, 13881}, {12619, 49241}, {12788, 49078}, {12818, 14241}, {12835, 12841}, {12836, 12838}, {12837, 12839}, {12891, 17702}, {12892, 12893}, {12961, 13045}, {12966, 12972}, {12967, 12973}, {13047, 13049}, {13048, 13050}, {13134, 26518}, {13135, 26517}, {13288, 13289}, {13367, 21641}, {13630, 43867}, {13644, 45440}, {13666, 22541}, {13674, 45023}, {13687, 13920}, {13712, 26288}, {13748, 36656}, {13758, 45577}, {13786, 19100}, {13807, 13848}, {13835, 26620}, {13938, 22723}, {13943, 19409}, {13947, 31423}, {13958, 31500}, {13961, 15720}, {13967, 38737}, {13969, 38727}, {13973, 26446}, {13976, 38133}, {13977, 21154}, {13979, 34128}, {13980, 23328}, {13983, 15819}, {13989, 38748}, {13990, 38793}, {13991, 38760}, {14226, 43343}, {14233, 15846}, {14561, 37342}, {14893, 41952}, {14996, 21565}, {14997, 21568}, {15022, 43792}, {15061, 19051}, {15141, 19384}, {15640, 43342}, {15686, 43209}, {15690, 42418}, {15692, 42523}, {15702, 43255}, {15704, 42226}, {15712, 41961}, {15716, 43258}, {15723, 42579}, {15764, 43229}, {15803, 51842}, {16029, 16035}, {16030, 16034}, {16113, 19080}, {16125, 16148}, {16202, 44646}, {16203, 44644}, {16419, 34515}, {16440, 32911}, {16809, 35730}, {16962, 36469}, {16963, 36452}, {16964, 42238}, {16965, 42237}, {17277, 21992}, {17809, 21097}, {17818, 19022}, {17820, 17821}, {18400, 32384}, {18539, 22625}, {18581, 42246}, {18582, 42247}, {18909, 18923}, {18916, 26873}, {18924, 18925}, {18980, 19436}, {18981, 19438}, {19042, 31987}, {19055, 34473}, {19064, 22676}, {19067, 52027}, {19070, 22890}, {19072, 22843}, {19073, 21156}, {19075, 21157}, {19081, 38693}, {19084, 22951}, {19089, 22712}, {19093, 38717}, {19102, 26516}, {19108, 21166}, {19112, 34474}, {19114, 38699}, {19128, 26925}, {19184, 19185}, {19356, 19357}, {19360, 44589}, {19385, 19388}, {19386, 19399}, {19437, 19441}, {19439, 19440}, {19442, 42022}, {19548, 36586}, {19708, 42524}, {19925, 49618}, {21445, 33370}, {21492, 37680}, {21545, 37682}, {21547, 37674}, {21548, 37679}, {21553, 37633}, {21555, 37687}, {21566, 37685}, {22466, 35861}, {22521, 33435}, {22537, 22554}, {22591, 45078}, {22682, 22720}, {22753, 22763}, {22831, 22876}, {22832, 22921}, {22833, 22976}, {22960, 35860}, {22961, 22962}, {23046, 42639}, {23269, 31414}, {23311, 32491}, {23358, 32385}, {23698, 39823}, {24244, 49390}, {24813, 24819}, {24828, 24842}, {25406, 39875}, {26290, 26385}, {26291, 26409}, {26292, 26460}, {26293, 26461}, {26294, 26462}, {26295, 26463}, {26326, 45365}, {26327, 45368}, {26328, 45607}, {26329, 45605}, {26331, 49027}, {26332, 45650}, {26333, 45652}, {26398, 44583}, {26422, 44585}, {26451, 44611}, {26456, 46453}, {26459, 37561}, {26487, 44621}, {26492, 44619}, {26498, 45596}, {26507, 45598}, {26521, 44597}, {26877, 26930}, {26878, 26940}, {26879, 26950}, {30428, 45841}, {31687, 31689}, {31688, 31691}, {32170, 32171}, {32233, 32253}, {32274, 32303}, {32330, 32343}, {32369, 32399}, {32494, 38110}, {33346, 33452}, {33348, 33454}, {33366, 33438}, {33368, 33436}, {33371, 37334}, {33372, 33430}, {33393, 34508}, {33394, 34509}, {33699, 43340}, {33813, 49267}, {33814, 48715}, {34089, 43571}, {34091, 43569}, {34200, 52048}, {34862, 49235}, {35404, 42572}, {35434, 43385}, {35698, 49214}, {35753, 35848}, {35828, 35880}, {35838, 35868}, {35846, 35850}, {35852, 49240}, {35854, 49242}, {35862, 49248}, {35870, 49258}, {35872, 49260}, {35874, 49262}, {36436, 36446}, {36437, 36450}, {36438, 41944}, {36454, 36464}, {36455, 36468}, {36456, 41943}, {36474, 36492}, {36477, 36555}, {36489, 36553}, {36510, 36585}, {36526, 36549}, {36557, 36581}, {36655, 45441}, {36711, 36990}, {36998, 45407}, {37567, 38235}, {38602, 48701}, {38608, 49271}, {38624, 49219}, {39824, 39825}, {39853, 39854}, {40108, 49231}, {41490, 45421}, {41947, 42640}, {41957, 43409}, {41959, 41970}, {41965, 43317}, {42085, 42249}, {42086, 42248}, {42093, 42206}, {42094, 42204}, {42101, 42210}, {42102, 42208}, {42117, 42253}, {42118, 42252}, {42133, 42186}, {42134, 42184}, {42155, 51925}, {42159, 42187}, {42160, 42190}, {42161, 42189}, {42162, 42188}, {42163, 42211}, {42164, 42214}, {42165, 42213}, {42166, 42212}, {42168, 42201}, {42170, 42199}, {42227, 42999}, {42229, 42998}, {42557, 45385}, {42569, 43318}, {42573, 47599}, {42575, 43791}, {43321, 44245}, {43336, 43788}, {43337, 43383}, {43376, 43507}, {43377, 46935}, {43412, 43517}, {43432, 50690}, {43504, 43520}, {43505, 43513}, {43508, 43516}, {43560, 43570}, {43561, 43568}, {43601, 43826}, {43831, 43863}, {44394, 49356}, {44582, 45357}, {44584, 45359}, {44601, 45620}, {44603, 45621}, {44628, 45623}, {44630, 45624}, {44648, 49103}, {45375, 45542}, {45422, 45586}, {45424, 45584}, {45426, 45715}, {45438, 48660}, {45544, 45574}, {45546, 49347}, {45554, 49355}, {45572, 45713}, {45595, 45599}, {45597, 45601}, {45860, 45870}, {48906, 49229}, {49087, 49318}, {49102, 49215}, {49105, 49237}, {49106, 49239}, {49107, 49243}, {49108, 49245}, {49109, 49247}, {49110, 49249}, {49111, 49253}, {49112, 49255}, {49113, 49259}, {49114, 49261}, {49115, 49263}, {49116, 49265}, {49947, 51924}

X(371) = midpoint of X(i) and X(j) for these {i,j}: {3, 45489}, {4, 26441}, {637, 43134}, {638, 20065}, {1151, 12962}, {3070, 42258}, {7969, 49226}, {12375, 35826}, {12964, 49250}, {32787, 41945}, {35610, 35641}, {35820, 42266}, {35824, 35878}, {35828, 35880}, {35856, 35882}, {35949, 45420}, {48700, 48714}, {49212, 49266}, {49216, 49268}, {49218, 49270}
X(371) = reflection of X(i) in X(j) for these {i,j}: {3, 43120}, {315, 640}, {372, 32}, {637, 639}, {3070, 7583}, {10962, 42866}, {11825, 3}, {12375, 49268}, {12965, 49256}, {35610, 49226}, {35641, 7969}, {35698, 49214}, {35753, 49208}, {35820, 3070}, {35822, 32787}, {35824, 49212}, {35826, 49216}, {35828, 49218}, {35830, 49220}, {35832, 44647}, {35834, 49222}, {35836, 49224}, {35838, 49230}, {35840, 6}, {35842, 49232}, {35844, 49234}, {35846, 49236}, {35848, 49238}, {35850, 49210}, {35852, 49240}, {35854, 49242}, {35856, 48700}, {35858, 49244}, {35860, 49246}, {35862, 49248}, {35864, 49250}, {35866, 49252}, {35868, 49254}, {35870, 49258}, {35872, 49260}, {35874, 49262}, {35876, 49264}, {35878, 49266}, {35880, 49270}, {35882, 48714}, {39893, 49228}, {41945, 52047}, {42009, 641}, {42266, 42258}, {44486, 575}, {44502, 44482}, {49222, 46688}, {49226, 31439}, {49601, 13883}
X(371) = isogonal conjugate of X(485)
X(371) = isotomic conjugate of X(34391)
X(371) = complement of X(637)
X(371) = anticomplement of X(639)
X(371) = circumcircle-inverse of X(2459)
X(371) = Brocard-circle-inverse of X(372)
X(371) = 1st-Lemoine-circle-inverse of X(2461)
X(371) = Schoutte-circle-inverse of X(6200)
X(371) = 2nd-Brocard-circle-inverse of X(3103)
X(371) = Lucas-circles-radical-circle-inverse of X(2460)
X(371) = isogonal conjugate of the anticomplement of X(641)
X(371) = isogonal conjugate of the complement of X(488)
X(371) = isogonal conjugate of the isotomic conjugate of X(492)
X(371) = isotomic conjugate of the polar conjugate of X(5413)
X(371) = isogonal conjugate of the polar conjugate of X(1585)
X(371) = polar conjugate of the isotomic conjugate of X(5408)
X(371) = polar conjugate of the isogonal conjugate of X(8911)
X(371) = Thomson-isogonal conjugate of X(13712)
X(371) = orthic-isogonal conjugate of X(372)
X(371) = psi-transform of X(7599)
X(371) = X(31)-complementary conjugate of X(10962)
X(371) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 10962}, {4, 372}, {492, 5408}, {588, 6}, {1585, 5413}, {3068, 8854}, {5412, 8855}, {5417, 6420}, {18819, 8577}, {26912, 23248}
X(371) = X(i)-cross conjugate of X(j) for these (i,j): {1147, 372}, {8911, 5408}, {12239, 4}
X(371) = X(i)-isoconjugate of X(j) for these (i,j): {1, 485}, {7, 13455}, {19, 11090}, {31, 34391}, {63, 41515}, {75, 8577}, {91, 372}, {92, 6413}, {486, 3377}, {1577, 39383}, {1586, 1820}, {1953, 16032}, {8769, 8944}, {19218, 24246}
X(371) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 10962}, {2, 34391}, {3, 485}, {6, 11090}, {69, 5409}, {206, 8577}, {371, 637}, {372, 34116}, {577, 5409}, {1504, 42009}, {3162, 41515}, {5392, 24245}, {5408, 13441}, {6413, 22391}, {10960, 13439}
X(371) = cevapoint of X(i) and X(j) for these (i,j): {3, 8909}, {6, 10132}, {3311, 8826}
X(371) = crosspoint of X(i) and X(j) for these (i,j): {4, 13429}, {492, 1585}, {3387, 3388}
X(371) = crosssum of X(i) and X(j) for these (i,j): {3, 10665}, {6, 3155}, {590, 8035}, {3385, 3386}, {6413, 8577}
X(371) = crossdifference of every pair of points on line {523, 17431}
X(371) = perspector of ABC and the Lucas tangents triangle
X(371) = perspector of the Lucas central triangle and the anticevian triangle of X(6)
X(371) = perspector of the Lucas inner triangle and Lucas(-1:1) tangents triangle
X(371) = perspector of the Lucas(4:3) central triangle and the circumcevian triangle of X(6)
X(371) = perspector of the Lucas central triangle and the cevian triangle of X(588)
X(371) = radical center of the Lucas(2:1) circles
X(371) = X(481)-of-Lucas-central-triangle
X(371) = perspector of ABC and 2nd Lucas(-1) secondary tangents triangle
X(371) = perspector of tangential triangle and Lucas secondary central triangle
X(371) = perspector of Lucas inner tangential triangle and Lucas(-1) central triangle
X(371) = inverse-in-Lucas-radical-circle of X(2460)
X(371) = exsimilicenter of circumcircle and Lucas radical circle
X(371) = barycentric product X(i)*X(j) for these {i,j}: {3, 1585}, {4, 5408}, {6, 492}, {24, 11091}, {32, 45805}, {52, 16037}, {69, 5413}, {264, 8911}, {317, 6414}, {372, 13428}, {485, 1599}, {486, 1993}, {491, 44193}, {493, 39387}, {571, 34392}, {588, 641}, {1124, 13457}, {1306, 14325}, {1586, 10666}, {5409, 13429}, {5412, 13430}, {6563, 39384}, {7763, 8576}, {9723, 41516}, {11547, 26922}
X(371) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 34391}, {3, 11090}, {6, 485}, {24, 1586}, {25, 41515}, {32, 8577}, {41, 13455}, {54, 16032}, {184, 6413}, {372, 13439}, {486, 5392}, {492, 76}, {571, 372}, {1147, 5409}, {1576, 39383}, {1585, 264}, {1599, 492}, {1993, 491}, {3053, 8944}, {5062, 8035}, {5408, 69}, {5409, 13441}, {5411, 1321}, {5412, 13440}, {5413, 4}, {6414, 68}, {7763, 45806}, {8576, 2165}, {8911, 3}, {8950, 493}, {10132, 24246}, {10666, 11091}, {10962, 637}, {11091, 20563}, {13428, 34392}, {16037, 34385}, {26920, 10665}, {32568, 45472}, {39384, 925}, {41411, 21463}, {41516, 847}, {44077, 5412}, {44193, 486}, {45805, 1502}
X(371) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1702, 35775}, {1, 2066, 35808}, {1, 2067, 35768}, {1, 6204, 8953}, {1, 35642, 35811}, {1, 35775, 35642}, {1, 45527, 45587}, {1, 45529, 45585}, {1, 45531, 45716}, {1, 45641, 35819}, {1, 45643, 35817}, {1, 45716, 45573}, {2, 486, 10577}, {2, 637, 639}, {2, 1588, 486}, {2, 9540, 5418}, {2, 43134, 637}, {2, 45509, 642}, {3, 6, 372}, {3, 182, 45552}, {3, 372, 6396}, {3, 1151, 6200}, {3, 1160, 12305}, {3, 1161, 1350}, {3, 1351, 9733}, {3, 1504, 35840}, {3, 3102, 45565}, {3, 3103, 40275}, {3, 3311, 6}, {3, 3312, 1152}, {3, 3592, 6419}, {3, 5050, 43118}, {3, 5093, 12314}, {3, 6199, 3311}, {3, 6221, 1151}, {3, 6395, 6450}, {3, 6398, 6410}, {3, 6407, 6449}, {3, 6408, 6452}, {3, 6417, 3312}, {3, 6418, 6398}, {3, 6419, 6420}, {3, 6420, 6454}, {3, 6422, 3103}, {3, 6423, 2459}, {3, 6425, 6453}, {3, 6427, 3594}, {3, 6428, 6426}, {3, 6429, 6484}, {3, 6431, 35770}, {3, 6432, 6481}, {3, 6445, 6455}, {3, 6446, 6497}, {3, 6447, 6425}, {3, 6449, 6409}, {3, 6450, 6412}, {3, 6455, 6411}, {3, 6474, 9690}, {3, 6480, 6486}, {3, 6500, 6395}, {3, 9691, 6445}, {3, 9732, 11824}, {3, 9733, 45498}, {3, 9738, 45499}, {3, 10897, 11513}, {3, 11916, 1160}, {3, 12313, 9732}, {3, 18457, 10897}, {3, 26348, 5085}, {3, 30435, 39679}, {3, 43119, 45553}, {3, 45410, 43121}, {3, 45411, 182}, {3, 45488, 9739}, {3, 45578, 43141}, {3, 45579, 9738}, {4, 485, 6564}, {4, 3068, 485}, {4, 5412, 35764}, {4, 6459, 6561}, {4, 6561, 35821}, {4, 6564, 35786}, {4, 10880, 5412}, {4, 13886, 31412}, {4, 31412, 42269}, {4, 42228, 42232}, {4, 42230, 42234}, {4, 45511, 48467}, {4, 45524, 48735}, {4, 46621, 6413}, {5, 590, 10576}, {5, 3071, 6565}, {5, 8981, 590}, {5, 31454, 35812}, {5, 42215, 3071}, {5, 48773, 45555}, {6, 39, 45513}, {6, 372, 6420}, {6, 1151, 3}, {6, 1152, 3312}, {6, 1160, 35794}, {6, 1161, 35792}, {6, 3053, 6423}, {6, 3311, 6419}, {6, 3312, 35770}, {6, 3592, 3311}, {6, 3594, 6418}, {6, 5013, 6421}, {6, 5058, 45514}, {6, 5085, 19146}, {6, 5210, 8376}, {6, 6200, 6396}, {6, 6221, 6200}, {6, 6395, 6436}, {6, 6409, 1152}, {6, 6410, 3594}, {6, 6411, 6398}, {6, 6412, 6395}, {6, 6419, 35771}, {6, 6422, 45512}, {6, 6423, 45515}, {6, 6425, 1151}, {6, 6426, 6432}, {6, 6429, 6409}, {6, 6431, 6417}, {6, 6432, 6428}, {6, 6433, 6412}, {6, 6437, 6221}, {6, 6451, 6481}, {6, 6468, 6411}, {6, 6469, 6442}, {6, 6470, 6431}, {6, 6488, 6497}, {6, 6567, 45499}, {6, 8414, 43119}, {6, 9601, 15815}, {6, 9675, 41410}, {6, 9732, 3103}, {6, 9733, 45462}, {6, 11477, 9974}, {6, 12962, 1504}, {6, 12963, 32}, {6, 12968, 5062}, {6, 45579, 45564}, {10, 48765, 45547}, {10, 49602, 35789}, {11, 18965, 9661}, {12, 13901, 9646}, {14, 35731, 18586}, {15, 16, 6200}, {15, 61, 372}, {15, 62, 3389}, {15, 3364, 3}, {15, 35739, 5238}, {16, 61, 3364}, {16, 62, 372}, {16, 3389, 3}, {20, 1587, 6560}, {20, 6560, 42267}, {20, 7585, 1587}, {20, 9541, 42260}, {20, 42522, 7585}, {20, 43512, 9541}, {25, 493, 8956}, {25, 3092, 35765}, {32, 1504, 6}, {32, 9675, 12963}, {32, 12963, 41410}, {32, 40274, 6396}, {32, 45489, 35840}, {35, 3301, 5414}, {36, 3299, 6502}, {39, 182, 372}, {39, 1151, 45499}, {39, 5058, 6}, {39, 6567, 9738}, and many others


X(372) = {X(3),X(6)}-HARMONIC CONJUGATE OF X(371)

Trilinears    cos(A + π/4) : :
Trilinears    cos A - sin A : :
Trilinears   a(SA - S) : :
Trilinears   a(b2 + c2 - a2 - 2S) : :
Barycentrics   sin A cos(A + π/4) : :
X(372) = 3 X[2] + X[43133], 2 X[640] + X[43133], 3 X[6] - 4 X[44481], 3 X[6] - 2 X[44501], X[1151] - 3 X[12968], 2 X[1505] + X[40275], 2 X[6221] - 5 X[41411], X[11824] - 4 X[43121], X[11824] + 2 X[45488], 3 X[35841] - 8 X[44481], 3 X[35841] - 4 X[44501], 2 X[43121] + X[45488], X[3071] - 3 X[32788], 2 X[3071] - 3 X[35823], X[3071] + 3 X[41946], 2 X[3071] + X[42267], 3 X[3071] - X[42271], 5 X[3071] + 3 X[43209], X[3071] + 6 X[52048], 2 X[7584] - 3 X[32788], 4 X[7584] - X[35821], 4 X[7584] - 3 X[35823], 2 X[7584] + 3 X[41946], 2 X[7584] + X[42259], 4 X[7584] + X[42267], 6 X[7584] - X[42271], 10 X[7584] + 3 X[43209], X[7584] + 3 X[52048], 6 X[32788] - X[35821], 3 X[32788] + X[42259], 6 X[32788] + X[42267], 9 X[32788] - X[42271], 5 X[32788] + X[43209], X[32788] + 2 X[52048], X[35821] - 3 X[35823], X[35821] + 6 X[41946], X[35821] + 2 X[42259], 3 X[35821] - 2 X[42271], 5 X[35821] + 6 X[43209], X[35821] + 12 X[52048], X[35823] + 2 X[41946], 3 X[35823] + 2 X[42259], 3 X[35823] + X[42267], 9 X[35823] - 2 X[42271], 5 X[35823] + 2 X[43209], X[35823] + 4 X[52048], 3 X[41946] - X[42259], 6 X[41946] - X[42267], 9 X[41946] + X[42271], 5 X[41946] - X[43209], 3 X[42259] + X[42271], 5 X[42259] - 3 X[43209], X[42259] - 6 X[52048], 3 X[42267] + 2 X[42271], 5 X[42267] - 6 X[43209], X[42267] - 12 X[52048], 5 X[42271] + 9 X[43209], X[42271] + 18 X[52048], X[43209] - 10 X[52048], X[489] - 3 X[35948], X[489] + 3 X[45421], 2 X[7968] + X[35611], X[35642] + 2 X[49227], X[35857] + 2 X[48715], X[35883] + 2 X[48701], X[35825] + 2 X[49267], X[35879] + 2 X[49213], X[12376] + 2 X[49217], X[35827] + 2 X[49269], 3 X[11242] - X[12970], 6 X[11242] + X[35865], 3 X[11242] + X[49251], 2 X[12970] + X[35865], X[35835] - 4 X[46689], X[35829] + 2 X[49271], X[35881] + 2 X[49219]

X(372) is the external center of similitude of the circumcircle and the 2nd Lemoine circle (cosine circle) (Peter J. C. Moses, 5/9/03). Also, X(372) is the external center of similitude of the Gallatly circle (defined just before X(2007)) and the 1st Lemoine circle (Peter J. C. Moses, 9/10/03).

Let A'B'C' be the Lucas(-1) central triangle. Let A″ be the cevapoint of B' and C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(372). (Randy Hutson, July 23, 2015)

If you have The Geometer's Sketchpad, you can view 2nd Kenmotu Point.

X(372) lies on the cubics K006, K233, K250, K337, K415, K424b, K633, K897, K1197, K1257, the curve Q168, and these lines: {1, 1703}, {2, 485}, {3, 6}, {4, 486}, {5, 615}, {8, 35842}, {10, 13975}, {11, 13977}, {12, 13958}, {13, 3392}, {14, 3367}, {17, 3366}, {18, 3391}, {20, 1588}, {24, 5412}, {25, 494}, {30, 3071}, {35, 2066}, {36, 2067}, {40, 5416}, {46, 8942}, {51, 3155}, {54, 6413}, {55, 1124}, {56, 1335}, {64, 6416}, {67, 19396}, {68, 35836}, {69, 11291}, {74, 19110}, {76, 35866}, {79, 35854}, {80, 35852}, {81, 16440}, {83, 22716}, {84, 35844}, {86, 21992}, {90, 42013}, {98, 13938}, {99, 19055}, {100, 19081}, {104, 19112}, {110, 7599}, {112, 19093}, {113, 13990}, {114, 6290}, {115, 13967}, {119, 13991}, {125, 13969}, {127, 13985}, {132, 13992}, {140, 590}, {155, 8943}, {157, 30399}, {165, 1702}, {184, 3156}, {185, 21641}, {186, 10880}, {193, 487}, {194, 22718}, {212, 3076}, {230, 49028}, {235, 13937}, {238, 31546}, {254, 41515}, {256, 8330}, {262, 10849}, {265, 35834}, {291, 8346}, {315, 492}, {355, 13973}, {376, 6459}, {378, 11473}, {381, 13847}, {382, 13785}, {394, 1584}, {395, 18585}, {396, 15765}, {397, 14813}, {398, 14814}, {399, 44593}, {402, 35790}, {405, 31473}, {427, 8281}, {489, 32419}, {490, 7388}, {491, 642}, {493, 35804}, {498, 31472}, {499, 44623}, {515, 13936}, {517, 7968}, {542, 32292}, {546, 13993}, {547, 43212}, {549, 8981}, {550, 19116}, {589, 3060}, {591, 754}, {601, 605}, {602, 606}, {603, 3077}, {626, 11313}, {631, 3068}, {632, 32789}, {637, 11293}, {671, 35698}, {736, 49351}, {760, 45713}, {940, 16432}, {944, 19065}, {946, 13971}, {952, 35843}, {958, 1377}, {971, 35845}, {993, 31453}, {997, 30557}, {999, 3298}, {1092, 8963}, {1131, 5056}, {1132, 1328}, {1147, 10665}, {1154, 12971}, {1181, 19356}, {1199, 26891}, {1296, 11836}, {1297, 19114}, {1327, 3317}, {1344, 2465}, {1345, 2466}, {1352, 37342}, {1376, 1378}, {1385, 7969}, {1478, 13963}, {1479, 13962}, {1482, 35811}, {1490, 19067}, {1498, 17820}, {1503, 36714}, {1511, 49268}, {1583, 5406}, {1586, 11547}, {1589, 11427}, {1590, 11433}, {1593, 3092}, {1594, 26950}, {1598, 13943}, {1599, 5417}, {1600, 1993}, {1656, 8252}, {1657, 18510}, {1658, 11265}, {1672, 2007}, {1673, 2008}, {1676, 2009}, {1677, 2010}, {1680, 2013}, {1681, 2014}, {1699, 13942}, {1700, 2017}, {1701, 2018}, {1707, 6212}, {1708, 8953}, {1742, 31545}, {1743, 32556}, {1843, 19359}, {1916, 8314}, {2015, 2471}, {2016, 2472}, {2041, 40694}, {2042, 40693}, {2043, 10654}, {2044, 10653}, {2045, 42152}, {2046, 42149}, {2077, 26459}, {2099, 38235}, {2544, 2546}, {2545, 2547}, {2777, 13288}, {2782, 35825}, {2794, 13748}, {3008, 31540}, {3085, 31408}, {3090, 23267}, {3091, 13941}, {3146, 22615}, {3167, 13068}, {3235, 3237}, {3236, 3238}, {3295, 3297}, {3316, 3533}, {3357, 49250}, {3373, 42783}, {3374, 42784}, {3428, 19013}, {3515, 5410}, {3518, 26886}, {3522, 9541}, {3523, 7585}, {3524, 19054}, {3525, 13886}, {3526, 8253}, {3528, 9681}, {3529, 23273}, {3530, 31454}, {3534, 43338}, {3563, 39383}, {3564, 32494}, {3567, 26916}, {3574, 13986}, {3576, 18991}, {3579, 49226}, {3590, 43558}, {3591, 12818}, {3618, 11292}, {3627, 42226}, {3628, 18538}, {3629, 40289}, {3664, 31541}, {3788, 11316}, {3832, 23253}, {3843, 45385}, {3845, 42418}, {3859, 41947}, {3972, 35938}, {4297, 49547}, {4383, 16433}, {4769, 45444}, {4845, 30336}, {5054, 13846}, {5066, 42607}, {5067, 31414}, {5204, 18996}, {5217, 19038}, {5254, 12967}, {5286, 44595}, {5318, 42175}, {5321, 42177}, {5334, 42173}, {5335, 35732}, {5339, 42278}, {5340, 42279}, {5405, 31561}, {5415, 10902}, {5419, 5446}, {5432, 9646}, {5433, 9661}, {5461, 13783}, {5473, 19073}, {5474, 19075}, {5478, 13982}, {5479, 13981}, {5480, 13972}, {5587, 13947}, {5590, 7376}, {5597, 35778}, {5598, 35780}, {5603, 13959}, {5640, 35299}, {5657, 19066}, {5663, 12376}, {5690, 49232}, {5860, 26620}, {5861, 6397}, {5871, 7000}, {5889, 26922}, {5890, 6457}, {5989, 8305}, {6000, 11242}, {6102, 44613}, {6145, 35858}, {6179, 6316}, {6201, 7374}, {6202, 13949}, {6229, 44364}, {6241, 11463}, {6245, 13974}, {6246, 13976}, {6247, 13980}, {6248, 13983}, {6249, 13984}, {6250, 13880}, {6278, 13771}, {6284, 19029}, {6291, 45841}, {6300, 33351}, {6301, 33353}, {6399, 9756}, {6401, 11975}, {6402, 11978}, {6415, 14528}, {6515, 11091}, {6525, 22839}, {6642, 10961}, {6680, 11315}, {6684, 13883}, {6699, 46688}, {6750, 13960}, {6759, 10534}, {6771, 49208}, {6774, 49210}, {6776, 39893}, {6811, 9753}, {6813, 9744}, {7160, 35862}, {7354, 19027}, {7387, 35777}, {7488, 11417}, {7494, 18289}, {7592, 8911}, {7598, 7602}, {7691, 19095}, {7709, 32470}, {7738, 12124}, {7759, 51401}, {7762, 35684}, {7773, 32435}, {7834, 11314}, {7851, 32436}, {7887, 32432}, {7980, 45720}, {7987, 9583}, {8149, 49352}, {8196, 13944}, {8203, 13945}, {8212, 13956}, {8213, 13957}, {8276, 19006}, {8301, 8337}, {8304, 8309}, {8307, 8316}, {8320, 8325}, {8321, 8327}, {8323, 8332}, {8336, 8341}, {8339, 8348}, {8703, 41945}, {8884, 19184}, {8909, 47391}, {8939, 15805}, {8948, 14248}, {8962, 43650}, {8964, 41588}, {8972, 10303}, {8980, 38737}, {8983, 10165}, {8988, 38133}, {8991, 23328}, {8992, 15819}, {8994, 38727}, {8997, 38748}, {8998, 38793}, {9441, 31544}, {9582, 16192}, {9616, 35242}, {9647, 15326}, {9660, 15338}, {9680, 15717}, {9682, 17928}, {9683, 10323}, {9751, 19092}, {9752, 9758}, {9880, 13968}, {9921, 45401}, {9927, 13970}, {9932, 12424}, {9993, 13946}, {10109, 43434}, {10113, 13979}, {10132, 11402}, {10164, 13912}, {10246, 44635}, {10266, 35870}, {10267, 19050}, {10269, 19048}, {10282, 10533}, {10299, 43509}, {10304, 43512}, {10310, 18999}, {10525, 35797}, {10526, 35799}, {10531, 13964}, {10532, 13965}, {10606, 19088}, {10610, 49256}, {10666, 13754}, {10669, 35807}, {10673, 12978}, {10679, 35817}, {10680, 35819}, {10799, 12841}, {10818, 15055}, {10819, 15035}, {10839, 10844}, {10842, 10851}, {10893, 13952}, {10894, 13953}, {10895, 13954}, {10896, 13955}, {11001, 42413}, {11012, 26458}, {11202, 11241}, {11248, 19047}, {11249, 19049}, {11251, 35791}, {11252, 35781}, {11253, 35779}, {11257, 19089}, {11414, 19005}, {11447, 11449}, {11448, 12111}, {11462, 11464}, {11496, 13940}, {11539, 43888}, {11542, 35738}, {11543, 42252}, {11812, 43211}, {11822, 19007}, {11823, 19009}, {11826, 19023}, {11827, 19025}, {11828, 19031}, {11829, 19033}, {11834, 38698}, {11835, 38716}, {11897, 13948}, {11941, 11964}, {11942, 11969}, {11943, 11966}, {11944, 11968}, {11960, 11979}, {11962, 11982}, {12041, 49216}, {12042, 49212}, {12100, 52045}, {12110, 22723}, {12117, 19057}, {12118, 19061}, {12119, 19077}, {12120, 19085}, {12121, 19051}, {12122, 19091}, {12123, 19104}, {12160, 51905}, {12164, 19493}, {12257, 14912}, {12296, 22617}, {12359, 49224}, {12425, 35837}, {12514, 30556}, {12556, 19097}, {12584, 32291}, {12591, 19588}, {12599, 13978}, {12600, 13987}, {12602, 13881}, {12619, 49240}, {12787, 49079}, {12819, 14226}, {12835, 12840}, {12836, 12839}, {12837, 12838}, {12891, 12893}, {12892, 17702}, {12960, 12972}, {12961, 12973}, {12966, 13048}, {13045, 13049}, {13046, 13050}, {13132, 26523}, {13133, 26522}, {13287, 13289}, {13367, 21640}, {13388, 45126}, {13630, 43868}, {13638, 45576}, {13666, 19099}, {13687, 13988}, {13712, 26619}, {13749, 36655}, {13763, 45441}, {13786, 19101}, {13794, 45024}, {13807, 13849}, {13835, 26289}, {13885, 22722}, {13889, 19408}, {13893, 31423}, {13901, 31499}, {13903, 15720}, {13911, 26446}, {13913, 21154}, {13915, 34128}, {13922, 38760}, {14230, 15847}, {14241, 43342}, {14561, 37343}, {14853, 21736}, {14893, 41951}, {14996, 21568}, {14997, 21565}, {15022, 43791}, {15043, 26912}, {15061, 19052}, {15141, 19385}, {15640, 43343}, {15686, 43210}, {15690, 42417}, {15692, 42522}, {15702, 43254}, {15704, 42225}, {15712, 41962}, {15716, 43259}, {15723, 42578}, {15764, 43228}, {15803, 51841}, {16029, 16030}, {16034, 16035}, {16113, 19079}, {16125, 16149}, {16202, 44645}, {16203, 44643}, {16419, 34516}, {16441, 32911}, {16962, 36453}, {16963, 36470}, {16964, 42236}, {16965, 42235}, {17277, 21909}, {17818, 19021}, {17819, 17821}, {18400, 32385}, {18581, 42248}, {18582, 42249}, {18909, 18924}, {18916, 26945}, {18923, 18925}, {18980, 19439}, {18981, 19437}, {19041, 31988}, {19056, 34473}, {19063, 22676}, {19068, 52027}, {19069, 22843}, {19071, 22890}, {19074, 21156}, {19076, 21157}, {19082, 38693}, {19083, 22951}, {19090, 22712}, {19094, 38717}, {19105, 26521}, {19109, 21166}, {19113, 34474}, {19115, 38699}, {19183, 19185}, {19355, 19357}, {19360, 44588}, {19384, 19389}, {19387, 19398}, {19436, 19440}, {19438, 19441}, {19548, 36587}, {19708, 42525}, {19843, 31413}, {19925, 49619}, {21445, 33371}, {21492, 37633}, {21547, 37679}, {21548, 37674}, {21550, 37682}, {21552, 37687}, {21553, 37680}, {21567, 37685}, {22466, 35860}, {22521, 33434}, {22536, 22553}, {22592, 45079}, {22596, 26438}, {22682, 22721}, {22753, 22764}, {22831, 22877}, {22832, 22922}, {22833, 22977}, {22960, 22962}, {22961, 35861}, {23046, 42640}, {23275, 33703}, {23312, 32490}, {23358, 32384}, {23698, 39824}, {24243, 49387}, {24813, 24818}, {24828, 24843}, {25406, 39876}, {26290, 26384}, {26291, 26408}, {26292, 26454}, {26293, 26455}, {26294, 26456}, {26295, 26457}, {26326, 45366}, {26327, 45367}, {26328, 45606}, {26329, 45608}, {26330, 49026}, {26332, 45651}, {26333, 45653}, {26363, 31484}, {26398, 44582}, {26422, 44584}, {26451, 44610}, {26463, 46453}, {26465, 37561}, {26487, 44620}, {26492, 44618}, {26498, 45597}, {26507, 45595}, {26516, 44594}, {26879, 26951}, {30427, 45840}, {31400, 31403}, {31401, 31411}, {31421, 31427}, {31422, 31437}, {31424, 31438}, {31425, 31440}, {31439, 31663}, {31448, 31459}, {31449, 31464}, {31450, 31465}, {31451, 31471}, {31452, 31475}, {31455, 31481}, {31456, 31482}, {31457, 31483}, {31458, 31486}, {31687, 31692}, {31688, 31690}, {32169, 32171}, {32233, 32252}, {32274, 32304}, {32330, 32342}, {32369, 32400}, {32497, 38110}, {33347, 33453}, {33349, 33455}, {33367, 33437}, {33369, 33439}, {33370, 37334}, {33373, 33431}, {33392, 34509}, {33395, 34508}, {33699, 43341}, {33813, 49266}, {33814, 48714}, {34089, 43568}, {34091, 43570}, {34200, 52047}, {34862, 49234}, {35404, 42573}, {35434, 43384}, {35699, 49215}, {35733, 42990}, {35742, 47864}, {35754, 35847}, {35829, 35881}, {35839, 35869}, {35849, 35851}, {35853, 49241}, {35855, 49243}, {35863, 49249}, {35871, 49259}, {35873, 49261}, {35875, 49263}, {36436, 36447}, {36437, 36449}, {36438, 41943}, {36454, 36465}, {36455, 36467}, {36456, 41944}, {36474, 36491}, {36477, 36556}, {36489, 36552}, {36510, 36584}, {36526, 36550}, {36557, 36582}, {36656, 45440}, {36712, 36990}, {36998, 45406}, {38602, 48700}, {38608, 49270}, {38624, 49218}, {39823, 39825}, {39852, 39854}, {40108, 49230}, {41491, 45420}, {41948, 42639}, {41958, 43410}, {41960, 41969}, {41966, 43316}, {42022, 45414}, {42085, 42247}, {42086, 42246}, {42093, 42205}, {42094, 42203}, {42101, 42209}, {42102, 42207}, {42117, 42251}, {42118, 42250}, {42133, 42185}, {42134, 42183}, {42154, 51924}, {42159, 42189}, {42160, 42188}, {42161, 42187}, {42162, 42190}, {42163, 42213}, {42164, 42212}, {42165, 42211}, {42166, 42214}, {42167, 42202}, {42169, 42200}, {42228, 42999}, {42230, 42998}, {42558, 45384}, {42568, 43319}, {42572, 47599}, {42574, 43792}, {43320, 44245}, {43336, 43382}, {43337, 43787}, {43376, 46935}, {43377, 43508}, {43411, 43518}, {43433, 50690}, {43503, 43519}, {43506, 43514}, {43507, 43515}, {43560, 43569}, {43561, 43571}, {43601, 43825}, {43831, 43864}, {44392, 49355}, {44583, 45360}, {44585, 45358}, {44600, 45620}, {44602, 45621}, {44627, 45623}, {44629, 45624}, {44647, 49104}, {45376, 45543}, {45423, 45587}, {45425, 45585}, {45427, 45716}, {45439, 48659}, {45545, 45575}, {45547, 49348}, {45555, 49356}, {45573, 45714}, {45596, 45600}, {45598, 45602}, {48906, 49228}, {49086, 49317}, {49102, 49214}, {49105, 49236}, {49106, 49238}, {49107, 49242}, {49108, 49244}, {49109, 49246}, {49110, 49248}, {49111, 49252}, {49112, 49254}, {49113, 49258}, {49114, 49260}, {49115, 49262}, {49116, 49264}, {49948, 51925}

X(372) = midpoint of X(i) and X(j) for these {i,j}: {3, 45488}, {4, 8982}, {637, 20065}, {638, 43133}, {1152, 12969}, {3071, 42259}, {7968, 49227}, {12376, 35827}, {12970, 49251}, {32788, 41946}, {35611, 35642}, {35821, 42267}, {35825, 35879}, {35829, 35881}, {35857, 35883}, {35948, 45421}, {48701, 48715}, {49213, 49267}, {49217, 49269}, {49219, 49271}
X(372) = reflection of X(i) in X(j) for these {i,j}: {3, 43121}, {315, 639}, {371, 32}, {638, 640}, {3071, 7584}, {10960, 42864}, {11824, 3}, {12376, 49269}, {12971, 49257}, {35611, 49227}, {35642, 7968}, {35699, 49215}, {35754, 49209}, {35821, 3071}, {35823, 32788}, {35825, 49213}, {35827, 49217}, {35829, 49219}, {35831, 49221}, {35833, 44648}, {35835, 49223}, {35837, 49225}, {35839, 49231}, {35841, 6}, {35843, 49233}, {35845, 49235}, {35847, 49239}, {35849, 49237}, {35851, 49211}, {35853, 49241}, {35855, 49243}, {35857, 48701}, {35859, 49245}, {35861, 49247}, {35863, 49249}, {35865, 49251}, {35867, 49253}, {35869, 49255}, {35871, 49259}, {35873, 49261}, {35875, 49263}, {35877, 49265}, {35879, 49267}, {35881, 49271}, {35883, 48715}, {39894, 49229}, {41946, 52048}, {42060, 642}, {42267, 42259}, {44485, 575}, {44501, 44481}, {49223, 46689}, {49602, 13936}
X(372) = isogonal conjugate of X(486)
X(372) = isotomic conjugate of X(34392)
X(372) = complement of X(638)
X(372) = anticomplement of X(640)
X(372) = X(372) = circumcircle-inverse of X(2460
X(372) = Brocard-circle-inverse of X(371)
X(372) = 1st-Lemoine-circle-inverse of X(2462)
X(372) = Schoutte-circle-inverse of X(6396)
X(372) = 2nd-Brocard-circle-inverse of X(3102)
X(372) = isogonal conjugate of the anticomplement of X(642)
X(372) = isogonal conjugate of the complement of X(487)
X(372) = isogonal conjugate of the isotomic conjugate of X(491)
X(372) = isotomic conjugate of the polar conjugate of X(5412)
X(372) = isogonal conjugate of the polar conjugate of X(1586)
X(372) = polar conjugate of the isotomic conjugate of X(5409)
X(372) = polar conjugate of the isogonal conjugate of X(26920)
X(372) = Thomson-isogonal conjugate of X(13835)
X(372) = orthic-isogonal conjugate of X(371)
X(372) = psi-transform of X(7598)
X(372) = X(31)-complementary conjugate of X(10960)
X(372) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 10960}, {4, 371}, {491, 5409}, {589, 6}, {1586, 5412}, {3069, 8855}, {5413, 8854}, {5419, 6419}, {13439, 26875}, {18820, 8576}
X(372) = X(i)-cross conjugate of X(j) for these (i,j): {1147, 371}, {12240, 4}, {26920, 5409}
X(372) = X(i)-isoconjugate of X(j) for these (i,j): {1, 486}, {19, 11091}, {31, 34392}, {63, 41516}, {75, 8576}, {91, 371}, {92, 6414}, {158, 26922}, {485, 3378}, {1577, 39384}, {1585, 1820}, {1953, 16037}, {8769, 8940}, {19217, 24245}
X(372) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 10960}, {2, 34392}, {3, 486}, {6, 11091}, {69, 5408}, {206, 8576}, {371, 34116}, {372, 638}, {577, 5408}, {1147, 26922}, {1505, 42060}, {3162, 41516}, {5392, 24246}, {5409, 13430}, {6414, 22391}, {10962, 13428}
X(372) = cevapoint of X(6) and X(10133)
X(372) = crosspoint of X(i) and X(j) for these (i,j): {4, 13440}, {491, 1586}, {3373, 3374}
X(372) = crosssum of X(i) and X(j) for these (i,j): {3, 10666}, {6, 3156}, {615, 8036}, {3371, 3372}, {6414, 8576}
X(372) = crossdifference of every pair of points on line {523, 17432}
X(372) = perspector of ABC and the Lucas(-1:1) tangents triangle
X(372) = perspector of the tangential triangle and the Lucas(-1) central triangle
X(372) = perspector of the Lucas(-1:1) inner tangential triangle and the Lucas central triangle
X(372) = perspector of the Lucas(-4:3) central triangle and the circumcevian triangle of X(6)
X(372) = perspector of the Lucas(-1:1) central triangle and cevian triangle of X(589)
X(372) = radical center of the Lucas(-2:1) circles
X(372) = perspector of ABC and 2nd Lucas secondary tangents triangle
X(372) = perspector of tangential triangle and Lucas(-1) secondary central triangle
X(372) = perspector of Lucas(-1) inner triangle and Lucas tangents triangle
X(372) = inverse-in-2nd-Brocard-circle of X(3102)
X(372) = inverse-in-Lucas(-1)-radical-circle of X(2459)
X(372) = perspector of ABC and the free vertices of the Kenmotu squares (described at X(371))
X(372) = insimilicenter of circumcircle and Lucas(-1) radical circle
X(372) = barycentric product of circumcircle intercepts of inner Vecten circle
X(372) = barycentric product X(i)*X(j) for these {i,j}: {3, 1586}, {4, 5409}, {6, 491}, {24, 11090}, {32, 45806}, {52, 16032}, {56, 13461}, {69, 5412}, {264, 26920}, {317, 6413}, {371, 13439}, {485, 1993}, {486, 1600}, {492, 44192}, {494, 39388}, {571, 34391}, {589, 642}, {1307, 14326}, {1585, 10665}, {5408, 13440}, {5413, 13441}, {6563, 39383}, {7763, 8577}, {9723, 41515}
X(372) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 34392}, {3, 11091}, {6, 486}, {24, 1585}, {25, 41516}, {32, 8576}, {54, 16037}, {184, 6414}, {371, 13428}, {485, 5392}, {491, 76}, {571, 371}, {577, 26922}, {1147, 5408}, {1576, 39384}, {1586, 264}, {1600, 491}, {1993, 492}, {3053, 8940}, {5058, 8036}, {5408, 13430}, {5409, 69}, {5410, 1322}, {5412, 4}, {5413, 13429}, {6413, 68}, {7763, 45805}, {8577, 2165}, {8911, 10666}, {10133, 24245}, {10665, 11090}, {10960, 638}, {11090, 20563}, {13439, 34391}, {13461, 3596}, {16032, 34385}, {26875, 637}, {26920, 3}, {32575, 45473}, {39383, 925}, {41410, 21464}, {41515, 847}, {44077, 5413}, {44192, 485}, {45806, 1502}
X(372) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1703, 35774}, {1, 5414, 35809}, {1, 6502, 35769}, {1, 35641, 35810}, {1, 35774, 35641}, {1, 45526, 45586}, {1, 45528, 45584}, {1, 45530, 45715}, {1, 45640, 35818}, {1, 45642, 35816}, {1, 45715, 45572}, {2, 485, 10576}, {2, 638, 640}, {2, 1587, 485}, {2, 13935, 5420}, {2, 43133, 638}, {2, 45508, 641}, {3, 6, 371}, {3, 182, 45553}, {3, 371, 6200}, {3, 1152, 6396}, {3, 1160, 1350}, {3, 1161, 12306}, {3, 1351, 9732}, {3, 1505, 35841}, {3, 3102, 40274}, {3, 3103, 45564}, {3, 3311, 1151}, {3, 3312, 6}, {3, 3594, 6420}, {3, 5050, 43119}, {3, 5093, 12313}, {3, 6199, 6449}, {3, 6221, 6409}, {3, 6395, 3312}, {3, 6398, 1152}, {3, 6407, 6451}, {3, 6408, 6450}, {3, 6417, 6221}, {3, 6418, 3311}, {3, 6419, 6453}, {3, 6420, 6419}, {3, 6421, 3102}, {3, 6424, 2460}, {3, 6426, 6454}, {3, 6427, 6425}, {3, 6428, 3592}, {3, 6430, 6485}, {3, 6431, 6480}, {3, 6432, 35771}, {3, 6445, 6496}, {3, 6446, 6456}, {3, 6448, 6426}, {3, 6449, 6411}, {3, 6450, 6410}, {3, 6456, 6412}, {3, 6475, 43415}, {3, 6481, 6487}, {3, 6501, 6199}, {3, 9732, 45499}, {3, 9733, 11825}, {3, 9739, 45498}, {3, 10898, 11514}, {3, 11917, 1161}, {3, 12314, 9733}, {3, 18459, 10898}, {3, 26341, 5085}, {3, 30435, 39648}, {3, 43118, 45552}, {3, 45410, 182}, {3, 45411, 43120}, {3, 45489, 9738}, {3, 45578, 9739}, {3, 45579, 43144}, {4, 486, 6565}, {4, 3069, 486}, {4, 5413, 35765}, {4, 6460, 6560}, {4, 6560, 35820}, {4, 6565, 35787}, {4, 10881, 5413}, {4, 13939, 42561}, {4, 42227, 42231}, {4, 42229, 42233}, {4, 42561, 42268}, {4, 45510, 48466}, {4, 45525, 48734}, {4, 46622, 6414}, {5, 615, 10577}, {5, 3070, 6564}, {5, 13966, 615}, {5, 42216, 3070}, {5, 48772, 45554}, {6, 39, 45512}, {6, 371, 6419}, {6, 1151, 3311}, {6, 1152, 3}, {6, 1160, 35793}, {6, 1161, 35795}, {6, 3053, 6424}, {6, 3311, 35771}, {6, 3312, 6420}, {6, 3592, 6417}, {6, 3594, 3312}, {6, 5013, 6422}, {6, 5062, 45515}, {6, 5085, 19145}, {6, 5210, 8375}, {6, 6199, 6435}, {6, 6396, 6200}, {6, 6398, 6396}, {6, 6409, 3592}, {6, 6410, 1151}, {6, 6411, 6199}, {6, 6412, 6221}, {6, 6420, 35770}, {6, 6421, 45513}, {6, 6424, 45514}, {6, 6425, 6431}, {6, 6426, 1152}, {6, 6430, 6410}, {6, 6431, 6427}, {6, 6432, 6418}, {6, 6434, 6411}, {6, 6438, 6398}, {6, 6452, 6480}, {6, 6468, 6441}, {6, 6469, 6412}, {6, 6471, 6432}, {6, 6489, 6496}, {6, 6566, 45498}, {6, 8406, 43118}, {6, 9732, 45463}, {6, 9733, 3102}, {6, 11477, 9975}, {6, 12963, 5058}, {6, 12968, 32}, {6, 12969, 1505}, {6, 36748, 26868}, {6, 45578, 45565}, {10, 48764, 45546}, {10, 49601, 35788}, {15, 16, 6396}, {15, 61, 371}, {15, 62, 3390}, {15, 3365, 3}, {16, 61, 3365}, {16, 62, 371}, {16, 3390, 3}, {16, 51728, 11481}, {20, 1588, 6561}, {20, 6561, 42266}, {20, 7586, 1588}, {20, 42523, 7586}, {25, 3093, 35764}, {32, 1505, 6}, {32, 12968, 41411}, {32, 40275, 6200}, {32, 45488, 35841}, {35, 3299, 2066}, {36, 3301, 2067}, {39, 182, 371}, {39, 1152, 45498}, {39, 5062, 6}, {39, 6566, 9739}, {39, 9739, 45565}, {40, 18992, 35775}, {40, 35775, 35610}, {50, 568, 371}, {52, 571, 371}, {55, 1124, 35808}, {55, 18995, 1124}, {56, 1335, 35768}, {56, 19037, 1335}, {58, 573, 371}, {61, 62, 6420}, and many more


X(373) = CENTROID OF THE PEDAL TRIANGLE OF THE CENTROID

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2bc + ac cos C + ab cos B
                        = g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a(b4 + c4 - a2b2 - a2c2 - 6b2c2)
Barycentrics  h(a,b,c) : h(b,c,a) : h(c,a,b), where h(a,b,c) = 2abc + ca2cos C + ba2cos B

X(373) = 2*X(2) + X(51)

X(373) lies on these lines: 2,51   5,113   110,575   181,748   216,852   354,375   597,2854

X(373) = isogonal conjugate of X(11169)
X(373) = isotomic conjugate of polar conjugate of X(33842)
X(373) = crossdifference of every pair of points on line X(352)X(1499)
X(373) = {X(2),X(51)}-harmonic conjugate of X(3917)


X(374) = CENTROID OF THE PEDAL TRIANGLE OF X(9)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2b + 2c - 3a + (c + a)cos C + (b + a)cos B
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(374) lies on these lines: 6,354   9,517   44,65   51,210   966,3740


X(375) = CENTROID OF THE PEDAL TRIANGLE OF X(10)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2bc(b + c) + ca(c + a)cos C + ab(a + b)cos B
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(375) lies on these lines: 44,181   51,210   354,373

X(375) = midpoint of X(51) and X(210)


X(376) = CENTROID OF THE ANTIPEDAL TRIANGLE OF X(2)

Trilinears    5 cos A - cos(B - C) : 5 cos B - cos(C - A) : 5 cos C - cos(A - B)
Trilinears    2 cos A - cos B cos C : 2 cos B - cos C cos A : 2 cos C - cos A cos B
Trilinears    3 cos A - sin B sin C : 3 cos B - sin C sin A : 3 cos C - sin A sin B
Trilinears    sec A - 2 sec B sec C : sec B - 2 sec C sec A : sec C - 2 sec A sec B
Trilinears    (csc A)(5 sin 2A - sin 2B - sin 2C) : :
Trilinears    a/(b sec B + c sec C) - 2 cos A : :
Barycentrics    5 sin 2A - sin 2B - sin 2C : :
Barycentrics    2*SA*a^2 - SB*SC : :
Barycentrics    5 a^4 - 4 a^2 (b^2 + c^2) - (b^2 - c^2)^2 : :

As a point on the Euler line, X(376) has Shinagawa coefficients (2, -3).

Let Pa be the parabola with focus A and directrix BC. Let La be the polar of X(3) wrt Pa. Define Lb, Lc cyclically. Let A' = Lb∩Lc, B' = Lc∩La, C' = La∩Lb. X(376) = X(69) of triangle A'B'C'. (Randy Hutson, July 20, 2016)

Let Ma be the polar of X(4) wrt the circle centered at A and passing through X(2), and define Mb, Mc cyclically. (Note: X(4) is the perspector of any circle centered at a vertex of ABC.) Let A″ = Mb∩Mc, and define B″, C″ cyclically. Triangle A″B″C″ is homothetic to ABC, and its orthocenter is X(376). (Randy Hutson, July 20, 2016)

Let Aa, Ab, Ac be the centers of the inverse-in-A-excircle of lines BC, CA, AB, resp. Let A' be the point of concurrence of lines AAa, BAb, CAc. Define B', C' cyclically. Triangle A'B'C' is perspective to the excentral triangle at X(376). (Randy Hutson, July 20, 2016)

X(376) lies on these lines: 1,553   2,3   35,388   36,497   40,519   55,1056   56,1058   69,74   98,543   103,544   104,528   110,541   112,577   165,515   316,1007   390,999   476,841   477,691   487,490   488,489   516,551   524,1350

X(376) is the {X(3),X(20)}-harmonic conjugate of X(4). For a list of other harmonic conjugates of X(376), click Tables at the top of this page.

X(376) = midpoint of X(2) and X(20)
X(376) = reflection of X(i) in X(j) for these (i,j): (2,3), (4,2), (381,549)
X(376) = isogonal conjugate of X(3426)
X(376) = isotomic conjugate of X(36889)
X(376) = circumcircle-inverse of X(7464)
X(376) = orthocentroidal-circle-inverse of X(3545)
X(376) = nine-point-circle-inverse of complement of X(37952)
X(376) = complement of X(3543)
X(376) = anticomplement of X(381)
X(376) = X(51)-of-hexyl-triangle
X(376) = centroid of circumcevian triangle of X(30)
X(376) = antipedal isogonal conjugate of X(2)
X(376) = {X(3),X(4)}-harmonic conjugate of X(631)
X(376) = Thomson-isogonal conjugate of X(6)
X(376) = Lucas-isogonal conjugate of X(69)
X(376) = X(2)-of-1st-anti-tri-squares-triangle
X(376) = X(2)-of-2nd-anti-tri-squares-triangle
X(376) = homothetic center of 1st and 2nd anti-tri-squares triangles
X(376) = insimilicenter of circumcircles of ABC and anti-Euler triangle; the exsimilicenter is X(4)
X(376) = {X(2043),X(2044)}-harmonic conjugate of X(20)
X(376) = trisector nearest X(3) of segment X(3)X(20)
X(376) = trisector nearest X(20) of segment X(4)X(20)
X(376) = Ehrmann-mid-to-Johnson similarity image of X(2)
X(376) = Euler line intercept, other than X(4), of conic {X(4),X(13),X(14),X(15),X(16)}}


X(377) = EULER LINE INTERCEPT OF LINE X(7)X(8)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b4 + c4 - a4 - 2b2c2 - 2abc(a + b + c))
                        = g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = cos A +(cos A + cos B + cos C) cos B cos C
Barycentrics  h(a,b,c) : h(b,c,a) : h(c,a,b), where h(a,b,c) = b4 + c4 - a4 - 2b2c2 - 2abc(a + b + c)

As a point on the Euler line, X(377) has Shinagawa coefficients (abc*$a, 2S2).

X(377) lies on these lines: {1, 224}, {2, 3}, {7, 8}, {10, 46}, {12, 1259}, {56, 2886}, {58, 1714}, {72, 5905}, {77, 5930}, {78, 226}, {79, 5692}, {81, 387}, {84, 5587}, {100, 3085}, {142, 950}, {145, 1056}, {149, 1058}, {171, 5230}, {200, 5290}, {225, 1038}, {274, 315}, {278, 4296}, {283, 1754}, {318, 1947}, {329, 3876}, {348, 3188}, {355, 1071}, {394, 3193}, {495, 5687}, {497, 2646}, {498, 3822}, {516, 5250}, {527, 3951}, {528, 3303}, {667,1155}, {908, 936}, {938, 5175}, {940, 1834}, {942, 3419}, {958, 3925}, {960, 1836}, {962, 3877}, {965, 1901}, {966, 2245}, {993, 3841}, {1001, 6284}, {1060, 1068}, {1125, 1479}, {1155, 2551}, {1159, 3621}, {1210, 3306}, {1220, 4429}, {1329, 4413}, {1330, 5739}, {1454, 1788}, {1621, 4294}, {1698, 3585}, {1765, 5816}, {1771, 3561}, {1837, 3812}, {1935, 3215}, {2096, 5818}, {2182, 5749}, {2287, 5746}, {2327, 5747}, {2345, 5279}, {2549, 5283}, {2893, 5738}, {2975, 4293}, {3086, 5253}, {3189, 3475}, {3304, 3813}, {3361, 5231}, {3421, 3617}, {3476, 4861}, {3485, 4511}, {3583, 3624}, {3618, 5135}, {3679, 5270}, {3710, 3729}, {3767, 5277}, {3869, 4295}, {3873, 5178}, {3897, 5731}, {3916, 5791}, {4000, 5262}, {4255, 5718}, {4298, 4847}, {4302, 5248}, {4312, 5785}, {4359, 5016}, {4652, 5745}, {4999, 5204}, {5217, 6690}, {5219, 5438}, {5225, 5550}, {5254, 5275}, {5256, 5717}, {5276, 5286}, {5334, 5367}, {5335, 5362}, {5439, 5722}, {5691, 5732}, {5715, 6282}, {5927, 6259}

X(377) is the {X(3),X(20)}-harmonic conjugate of X(21). For a list of other harmonic conjugates of X(377), click Tables at the top of this page.

X(377) = anticomplement of X(405)
X(377) = {X(7),X(8)}-harmonic conjugate of X(3868)


X(378) = REFLECTION OF X(22) IN X(3)

Trilinears       sec A + 2 cos A : sec B + 2 cos B : sec C + 2 cos C
Barycentrics  tan A + sin 2A : tan B + sin 2B : tan C + sin 2C : :
Barycentrics   a^2 (a^4 + b^4 + c^4 - 2 a^2 b^2 - 2 a^2 c^2 + 4 b^2 c^2)/(b^2 + c^2 - a^2) : :
X(378) = 6 X(2) + (J^2 + 3) X(3) = 2 X(3) - X(22)

As a point on the Euler line, X(378) has Shinagawa coefficients (2F, E - 2F).

X(378) lies on these lines: 1,1063   2,3   6,74   33,36   34,35   54,64   99,264   185,578   232,574   477,935   847,1105

X(378) is the {X(3),X(4)}-harmonic conjugate of X(24). For a list of other harmonic conjugates of X(378), click Tables at the top of this page.

X(378) = reflection of X(i) in X(j) for these (i,j): (4,427), (22,3)
X(378) = isogonal conjugate of X(4846)
X(378) = anticomplement of X(15760)
X(378) = inverse-in-orthocentroidal-circle of X(403)
X(378) = harmonic center of circumcircle and polar circle
X(378) = polar conjugate of X(34289)
X(378) = endo-homothetic center of X(4)-anti-altimedial and anti-orthocentroidal triangles


X(379) = EULER LINE INTERCEPT OF LINE X(6)X(7)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[a5 + (b + c)(bca2 - (bc + ca + ab)(b - c)2)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a5 + (b + c)(bca2 - (bc + ca + ab)(b - c)2)

As a point on the Euler line, X(379) has Shinagawa coefficients ($a*SBSC$, $a$S2).

X(379) lies on these lines: 2,3   6,7   41,226   63,169   264,823

X(379) = complement of X(31015)
X(379) = anticomplement of X(30810)
X(379) = inverse-in-orthocentroidal-circle of X(857)
X(379) = crossdifference of every pair of points on line X(647)X(926)


X(380) = INTERSECTION OF LINES X(1)X(19) AND X(9)X(55)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)[3a3 + (b + c)(3a2 + (b - c)2 + a(b + c))]
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(380) lies on these lines: 1,19   6,40   9,55   165,579   223,608   281,950   282,1036


X(381) = MIDPOINT OF X(2) AND X(4)

Trilinears    2 cos(B - C) - cos A : :
Trilinears    cos A + 4 cos B cos C : :
Trilinears    3 cos A - 4 sin B sin C : :
Trilinears    (csc A)(2 tan A + tan B + tan C) + 2 sec A : :
Trilinears    3 cos B cos C + sin B sin C : :
Barycentrics   a(cos A + 4 cos B cos C) : :
Barycentrics    a^4 - 2b^4 - 2c^4 + a^2b^2 + a^2c^2 + 4b^2c^2 : :
Barycentrics    S^2 + 3 SB SC : :
Barycentrics    a^2*SA + 4*SB*SC : :
X381) = X(2) + X(4) = X(3) + 2 X(4)
X381) = X[1] - 4 X[9955], X[1] + 2 X[18480], X[1] + 5 X[18492], 2 X[1] - 5 X[18493], 2 X[1] + X[18525], 4 X[1] - X[18526], X[1] - 5 X[30308], 7 X[1] - 4 X[32900], X[1] - 3 X[38021], 5 X[2] - X[20], 5 X[2] - 4 X[140], 4 X[2] + X[382], X[2] + 4 X[546], 3 X[2] - 4 X[547], 11 X[2] - 4 X[548], 7 X[2] - 2 X[550], 7 X[2] - 5 X[631], 11 X[2] - 10 X[632], 4 X[2] - 5 X[1656], 8 X[2] - X[1657], 5 X[2] - 7 X[3090], X[2] - 5 X[3091], 7 X[2] + X[3146], 13 X[2] - 5 X[3522], 11 X[2] - 7 X[3523], 5 X[2] - 3 X[3524], 13 X[2] - 11 X[3525], 8 X[2] - 7 X[3526], 17 X[2] - 7 X[3528], 11 X[2] - X[3529], 13 X[2] - 8 X[3530], and many others

As a point on the Euler line, X(381) has Shinagawa coefficients (1,3).

Let A' be the reflection of X(3) in A, and define B'and C' cyclically. Let A'' be the reflection of X(3) in BC, and define B'' and C'' cyclically. The lines A'A'', B'B'', C'C'' concur in X(381).

X(381) is the point QA-P14 (Centroid of the Morley Triangle) of quadrangle ABCX(2) (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/42-qa-p14.html)

Let Pa be the parabola with focus A and directrix BC. Let La be the polar of X(3) wrt Pa. Define Lb and Lc cyclically. Let A' = Lb∩Lc, B' = Lc∩La, C' = La∩Lb. The symmedian point of triangle A'B'C' is X(381). Analogously, let La be the trilinear polar of A wrt BCX(3), and define Lb and Lc cyclically. Let A'=Lb∩Lc, B'=Lc∩La, C'=La∩Lb. Equivalently, A'B'C' is the cevian triangle of X(3) wrt the cevian triangle of X(3). Also, let A″ be the reflection of A' in BC, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(381). Finally, let A' be the orthocenter of BCX(6), and define B' and C' cyclically; then X(381) is the centroid of A'B'C'. (Randy Hutson, September 5, 2015)

Let Sa be the similitude center of the orthocentroidal triangle and the A-altimedial triangle. Define Sb and Sc cyclically. The lines ASa, BSb, CSc concur in X(381). (Randy Hutson, December 10, 2016)

X(381) lies on the McCay circumcircle, the curves K300, K358, K504, K649, K762, K833, K887, K911, Q164, and these lines: {1, 2463}, {2, 3}, {6, 13}, {7, 38073}, {8, 8148}, {9, 18482}, {10, 2467}, {11, 999}, {12, 1479}, {15, 16644}, {16, 16645}, {17, 12817}, {18, 12816}, {19, 18453}, {32, 13881}, {33, 9642}, {34, 18447}, {35, 12953}, {36, 12943}, {39, 2015}, {40, 7989}, {46, 7082}, {49, 578}, {51, 568}, {52, 5907}, {53, 18437}, {54, 156}, {55, 3583}, {56, 3582}, {57, 18540}, {61, 5339}, {62, 5340}, {64, 3521}, {65, 10826}, {67, 32271}, {68, 3527}, {69, 3531}, {74, 1539}, {76, 7773}, {79, 5221}, {80, 2099}, {83, 7851}, {84, 22792}, {90, 1454}, {98, 598}, {99, 22515}, {100, 22938}, {103, 38768}, {104, 22799}, {109, 38780}, {110, 7699}, {111, 6032}, {112, 19163}, {114, 543}, {116, 10741}, {117, 10747}, {118, 544}, {119, 528}, {120, 15521}, {121, 15522}, {122, 22337}, {123, 33566}, {124, 10740}, {125, 541}, {126, 22338}, {127, 133}, {128, 13512}, {130, 38594}, {131, 5139}, {132, 10749}, {137, 31656}, {141, 31670}, {142, 31672}, {143, 5876}, {145, 37705}, {146, 7693}, {147, 12243}, {148, 7777}, {149, 11698}, {153, 1484}, {154, 18376}, {155, 195}, {157, 11641}, {159, 18382}, {165, 11231}, {182, 2469}, {183, 316}, {184, 567}, {185, 5462}, {187, 37637}, {190, 24827}, {206, 34775}, {210, 517}, {216, 33842}, {220, 24045}, {226, 5722}, {230, 1384}, {233, 36751}, {262, 671}, {264, 339}, {275, 19176}, {298, 622}, {299, 621}, {302, 616}, {303, 617}, {315, 37671}, {325, 11185}, {355, 519}, {371, 8976}, {372, 13847}, {373, 5892}, {385, 34623}, {388, 496}, {389, 12162}, {390, 8164}, {393, 15851}, {394, 1568}, {395, 5318}, {396, 5321}, {397, 40694}, {398, 40693}, {485, 1328}, {486, 1327}, {487, 22809}, {488, 22810}, {493, 18520}, {494, 18522}, {495, 497}, {498, 3614}, {499, 5298}, {511, 599}, {512, 13240}, {513, 39483}, {514, 39489}, {515, 551}, {516, 3828}, {523, 16279}, {524, 1351}, {525, 39491}, {527, 2095}, {529, 3829}, {530, 5478}, {531, 5479}, {532, 5858}, {533, 5859}, {535, 11194}, {536, 20430}, {538, 3095}, {540, 7683}, {545, 24828}, {553, 1210}, {569, 6759}, {574, 7603}, {576, 15069}, {590, 6221}, {591, 6250}, {597, 1503}, {615, 6398}, {620, 38730}, {623, 3643}, {624, 3642}, {625, 3734}, {627, 33608}, {628, 33609}, {629, 33619}, {630, 33618}, {637, 32809}, {638, 32808}, {754, 6249}, {804, 19912}, {900, 39493}, {908, 3419}, {912, 5927}, {916, 31163}, {938, 5714}, {942, 1864}, {944, 5901}, {950, 11374}, {952, 3241}, {956, 5080}, {958, 25639}, {960, 16616}, {962, 5690}, {971, 6173}, {986, 5492}, {997, 5087}, {1001, 3822}, {1007, 6390}, {1056, 5274}, {1058, 5261}, {1060, 9817}, {1062, 9641}, {1073, 37873}, {1078, 11057}, {1092, 37495}, {1093, 14860}, {1112, 7723}, {1116, 32478}, {1125, 3653}, {1131, 6501}, {1132, 6500}, {1147, 11424}, {1151, 10576}, {1152, 10577}, {1154, 3060}, {1157, 16764}, {1159, 12019}, {1160, 7721}, {1161, 7720}, {1173, 45014}, {1181, 19361}, {1199, 43605}, {1209, 12307}, {1216, 37484}, {1217, 36609}, {1263, 16762}, {1285, 37689}, {1296, 38800}, {1297, 19160}, {1319, 23708}, {1329, 9709}, {1350, 14926}, {1353, 5032}, {1376, 3814}, {1385, 5691}, {1387, 3476}, {1388, 37735}, {1470, 13273}, {1480, 33104}, {1483, 10595}, {1490, 37615}, {1495, 14805}, {1498, 15047}, {1499, 8371}, {1506, 5013}, {1511, 10546}, {1514, 3426}, {1531, 3581}, {1533, 35237}, {1537, 19914}, {1565, 17079}, {1587, 6418}, {1588, 6417}, {1614, 13434}, {1698, 3579}, {1714, 9958}, {1737, 1836}, {1750, 18443}, {1770, 24914}, {1837, 12047}, {1843, 18438}, {1853, 5943}, {1870, 37729}, {1898, 13750}, {1974, 19129}, {1975, 7752}, {1986, 22584}, {1991, 6251}, {1992, 3564}, {1993, 15068}, {1994, 15052}, {2039, 38597}, {2040, 38596}, {2051, 9567}, {2080, 3849}, {2094, 5770}, {2096, 13226}, {2098, 37710}, {2393, 23049}, {2452, 9214}, {2453, 5099}, {2482, 15561}, {2548, 5254}, {2549, 3815}, {2550, 3820}, {2551, 31418}, {2646, 37692}, {2770, 32229}, {2771, 5902}, {2775, 14431}, {2777, 10606}, {2779, 15049}, {2780, 9148}, {2781, 16776}, {2794, 5461}, {2796, 24311}, {2797, 45319}, {2799, 44203}, {2800, 38161}, {2808, 10708}, {2818, 10709}, {2829, 23513}, {2883, 12315}, {2886, 9708}, {2888, 12316}, {2900, 5720}, {2917, 32365}, {2929, 22816}, {2930, 32273}, {2931, 19479}, {2935, 32743}, {2971, 23635}, {2979, 13391}, {2999, 18505}, {3023, 10054}, {3027, 10070}, {3052, 17734}, {3053, 7746}, {3054, 15655}, {3055, 43619}, {3057, 10827}, {3065, 16763}, {3068, 6199}, {3069, 6395}, {3085, 5225}, {3086, 5229}, {3087, 38292}, {3098, 3763}, {3106, 31702}, {3107, 31701}, {3120, 7986}, {3167, 5654}, {3258, 14685}, {3259, 38586}, {3297, 35800}, {3298, 35801}, {3303, 4857}, {3304, 5270}, {3309, 31149}, {3314, 43453}, {3316, 43567}, {3317, 43566}, {3336, 7701}, {3357, 5895}, {3358, 15239}, {3359, 11372}, {3361, 31776}, {3366, 42236}, {3367, 42238}, {3373, 12823}, {3388, 12822}, {3391, 42235}, {3392, 42237}, {3398, 7817}, {3411, 42533}, {3412, 42532}, {3413, 41880}, {3414, 41881}, {3424, 18842}, {3434, 17757}, {3436, 24390}, {3448, 11801}, {3459, 22335}, {3483, 16765}, {3485, 37730}, {3486, 37737}, {3487, 12433}, {3488, 5226}, {3567, 6102}, {3576, 7988}, {3586, 5219}, {3587, 7308}, {3589, 12017}, {3592, 8960}, {3616, 34773}, {3617, 38081}, {3618, 38079}, {3624, 13624}, {3632, 11278}, {3634, 31730}, {3652, 16125}, {3667, 39490}, {3695, 42032}, {3746, 9670}, {3767, 5306}, {3785, 32885}, {3793, 37667}, {3796, 44407}, {3824, 41854}, {3825, 25524}, {3847, 10200}, {3917, 10170}, {3925, 34618}, {3927, 6734}, {3928, 5789}, {3929, 5709}, {3933, 32816}, {3934, 7784}, {3944, 37717}, {3972, 10722}, {4252, 45939}, {4292, 37545}, {4293, 10589}, {4294, 10588}, {4297, 19883}, {4299, 5433}, {4301, 4669}, {4302, 5432}, {4370, 29243}, {4388, 5774}, {4413, 35238}, {4421, 11496}, {4428, 10197}, {4549, 32269}, {4664, 29010}, {4677, 7982}, {4745, 11362}, {4994, 19210}, {5012, 14157}, {5023, 7749}, {5031, 35456}, {5044, 37585}, {5045, 5290}, {5048, 37708}, {5085, 10168}, {5086, 5730}, {5090, 34713}, {5092, 7913}, {5095, 32272}, {5102, 5965}, {5103, 35458}, {5122, 31231}, {5123, 35460}, {5134, 42316}, {5158, 18487}, {5167, 18322}, {5203, 40809}, {5204, 10483}, {5206, 44535}, {5210, 6781}, {5215, 32414}, {5237, 42431}, {5238, 42432}, {5251, 31245}, {5252, 30384}, {5289, 11813}, {5292, 13408}, {5305, 43136}, {5325, 5791}, {5334, 11542}, {5335, 11543}, {5343, 42494}, {5344, 42495}, {5349, 42147}, {5350, 42148}, {5351, 42491}, {5352, 42490}, {5365, 42925}, {5366, 42924}, {5412, 18457}, {5413, 18459}, {5418, 6449}, {5420, 6450}, {5422, 11456}, {5437, 7171}, {5439, 13369}, {5440, 30852}, {5443, 34471}, {5446, 5562}, {5449, 12163}, {5459, 41022}, {5460, 41023}, {5463, 36765}, {5485, 14484}, {5510, 10744}, {5523, 45141}, {5533, 12763}, {5544, 11820}, {5563, 9657}, {5597, 18495}, {5598, 18497}, {5609, 11422}, {5627, 14254}, {5640, 5663}, {5642, 14643}, {5644, 5656}, {5646, 33544}, {5650, 36987}, {5651, 13857}, {5657, 9812}, {5687, 11681}, {5694, 37625}, {5704, 34753}, {5705, 31445}, {5706, 37509}, {5707, 36750}, {5715, 5777}, {5725, 24210}, {5726, 31393}, {5731, 28186}, {5732, 38093}, {5752, 15488}, {5762, 5817}, {5787, 6260}, {5794, 21616}, {5816, 17330}, {5840, 6174}, {5841, 31157}, {5842, 38109}, {5843, 38137}, {5844, 31145}, {5860, 6201}, {5861, 6202}, {5864, 21359}, {5865, 21360}, {5878, 5893}, {5881, 10222}, {5884, 31871}, {5885, 15071}, {5887, 7686}, {5898, 18427}, {5913, 21448}, {5944, 26882}, {5972, 12121}, {6002, 45665}, {6036, 14971}, {6090, 40112}, {6101, 11444}, {6108, 22513}, {6109, 22512}, {6114, 6775}, {6115, 6772}, {6145, 32364}, {6146, 19347}, {6152, 22815}, {6153, 43581}, {6200, 8253}, {6235, 6324}, {6241, 11439}, {6244, 7965}, {6245, 6259}, {6246, 6265}, {6253, 26487}, {6256, 16203}, {6278, 22485}, {6281, 22484}, {6291, 22811}, {6337, 32826}, {6361, 9780}, {6391, 45088}, {6396, 8252}, {6406, 22812}, {6407, 9540}, {6408, 13935}, {6409, 42266}, {6410, 42267}, {6425, 35730}, {6426, 35813}, {6445, 9541}, {6446, 32786}, {6447, 31454}, {6448, 42418}, {6451, 32789}, {6452, 32790}, {6455, 42260}, {6456, 42261}, {6459, 8981}, {6460, 13966}, {6472, 43520}, {6473, 43519}, {6480, 42558}, {6481, 42557}, {6484, 42568}, {6485, 42569}, {6519, 9681}, {6522, 41964}, {6526, 13157}, {6666, 38067}, {6667, 38069}, {6668, 38070}, {6683, 7872}, {6688, 14855}, {6689, 32340}, {6696, 20427}, {6697, 34778}, {6699, 13202}, {6702, 12515}, {6704, 8725}, {6712, 38765}, {6713, 38753}, {6716, 23240}, {6718, 38777}, {6721, 9167}, {6722, 38739}, {6723, 16111}, {6748, 15905}, {6750, 36245}, {6770, 20252}, {6771, 22489}, {6773, 20253}, {6774, 22490}, {6776, 13674}, {6782, 41745}, {6783, 41746}, {6785, 6787}, {6788, 18326}, {6792, 18346}, {6795, 38393}, {7160, 22801}, {7578, 18300}, {7585, 23273}, {7586, 23267}, {7592, 32139}, {7607, 45103}, {7608, 17503}, {7618, 9771}, {7620, 9770}, {7622, 32479}, {7666, 45248}, {7678, 42884}, {7691, 13565}, {7710, 46034}, {7735, 18907}, {7736, 15048}, {7738, 31404}, {7750, 32832}, {7751, 7843}, {7754, 7785}, {7756, 15815}, {7761, 15271}, {7763, 32819}, {7764, 22665}, {7767, 32006}, {7772, 39593}, {7774, 22253}, {7790, 11174}, {7793, 19569}, {7798, 32457}, {7804, 7844}, {7806, 9862}, {7808, 7861}, {7810, 8556}, {7812, 14568}, {7814, 32821}, {7815, 7842}, {7816, 7862}, {7821, 17130}, {7823, 32151}, {7827, 39646}, {7828, 9873}, {7840, 32515}, {7845, 17131}, {7868, 7934}, {7879, 7885}, {7881, 7912}, {7883, 33706}, {7900, 17129}, {7914, 35248}, {7935, 31239}, {7941, 20081}, {7956, 34625}, {7967, 10283}, {7999, 10627}, {8068, 8069}, {8070, 8071}, {8074, 18327}, {8182, 15597}, {8192, 34668}, {8193, 34657}, {8196, 8200}, {8203, 8207}, {8212, 8220}, {8213, 8221}, {8254, 12254}, {8288, 45723}, {8541, 18449}, {8547, 20113}, {8567, 25563}, {8573, 9722}, {8584, 11482}, {8585, 39602}, {8589, 44541}, {8680, 25362}, {8704, 13377}, {8717, 22112}, {8722, 15810}, {8757, 23070}, {8760, 45320}, {8780, 23292}, {8859, 10788}, {8860, 38227}, {8939, 18414}, {8943, 18415}, {9115, 23006}, {9117, 23013}, {9143, 25147}, {9144, 11005}, {9172, 14666}, {9175, 14684}, {9178, 18309}, {9306, 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{10263, 11017}, {10265, 16112}, {10266, 22805}, {10269, 41698}, {10272, 12383}, {10274, 40276}, {10278, 16220}, {10282, 17845}, {10302, 14488}, {10306, 10525}, {10311, 10317}, {10418, 14653}, {10446, 17271}, {10526, 22770}, {10531, 10599}, {10532, 10598}, {10572, 11375}, {10574, 12006}, {10596, 32213}, {10597, 32214}, {10601, 11550}, {10625, 11793}, {10641, 18468}, {10642, 18470}, {10645, 42096}, {10646, 42097}, {10712, 28915}, {10717, 33962}, {10718, 20410}, {10721, 12041}, {10723, 11164}, {10724, 33814}, {10725, 38599}, {10726, 38600}, {10727, 31273}, {10728, 31272}, {10729, 38603}, {10730, 38604}, {10731, 38606}, {10732, 38607}, {10734, 14650}, {10735, 38608}, {10746, 25640}, {10752, 13169}, {10753, 11161}, {10886, 37620}, {10897, 35764}, {10898, 35765}, {10984, 37471}, {10985, 18472}, {10990, 20397}, {10991, 20398}, {10992, 20399}, {10993, 20400}, {11002, 13451}, {11004, 18387}, {11011, 37711}, {11160, 34380}, {11167, 14485}, {11170, 43535}, {11171, 37808}, {11173, 15993}, {11182, 21733}, {11187, 44420}, {11188, 14984}, {11202, 39242}, {11245, 45967}, {11257, 11272}, {11271, 11803}, {11376, 45287}, {11381, 27355}, {11396, 34729}, {11402, 12022}, {11425, 13403}, {11426, 12134}, {11432, 12233}, {11433, 18917}, {11438, 26958}, {11440, 43613}, {11441, 12161}, {11449, 43394}, {11451, 11455}, {11457, 12174}, {11477, 15533}, {11480, 16241}, {11481, 16242}, {11487, 11850}, {11488, 42117}, {11489, 42118}, {11557, 21650}, {11561, 12270}, {11562, 41671}, {11572, 13353}, {11576, 12606}, {11669, 33698}, {11671, 14072}, {11693, 38795}, {11694, 34153}, {11695, 46850}, {11699, 12407}, {11704, 11999}, {11723, 12898}, {11750, 13419}, {11805, 33565}, {12112, 15018}, {12115, 30283}, {12154, 41060}, {12155, 41061}, {12160, 41628}, {12236, 12825}, {12242, 15752}, {12261, 12368}, {12278, 15807}, {12279, 15028}, {12281, 38898}, {12284, 13358}, {12292, 14708}, {12301, 20302}, {12302, 33547}, {12313, 45861}, {12314, 45860}, {12322, 32811}, {12323, 32810}, {12350, 13183}, {12351, 13182}, 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{16228, 44923}, {16229, 44918}, {16230, 44921}, {16231, 44925}, {16264, 17907}, {16318, 41370}, {16336, 35449}, {16337, 19552}, {16466, 21935}, {16509, 23334}, {16534, 23236}, {16772, 42150}, {16773, 42151}, {16835, 43585}, {16960, 44016}, {16961, 44015}, {17078, 17181}, {17254, 29369}, {17310, 29331}, {17313, 24220}, {17500, 43917}, {17502, 28168}, {17508, 29323}, {17647, 25681}, {17810, 21243}, {17811, 37483}, {17813, 23048}, {17814, 36747}, {17821, 34785}, {18201, 31520}, {18230, 38082}, {18279, 32417}, {18310, 44206}, {18321, 31850}, {18395, 37567}, {18402, 38585}, {18432, 34117}, {18442, 46027}, {18462, 19446}, {18463, 19447}, {18549, 22836}, {18576, 18883}, {18809, 38595}, {18911, 32111}, {18912, 32140}, {18945, 31804}, {19132, 34776}, {19177, 19189}, {19357, 21659}, {19722, 31872}, {19765, 37693}, {20192, 44569}, {20401, 33520}, {20415, 41020}, {20416, 41021}, {20546, 35462}, {20576, 36998}, {20582, 29181}, {20584, 21230}, {21151, 38171}, {21153, 38318}, {21154, 38319}, {21356, 40330}, {21635, 33594}, {22235, 33603}, {22237, 33602}, {22240, 33885}, {22246, 37665}, {22247, 38731}, {22550, 44686}, {22573, 22579}, {22574, 22580}, {22691, 22707}, {22692, 22708}, {22712, 31168}, {22808, 22970}, {22833, 22955}, {22846, 36760}, {22847, 22861}, {22848, 22862}, {22891, 36759}, {22892, 22906}, {22893, 22907}, {22971, 37497}, {23017, 25178}, {23018, 25183}, {23019, 25184}, {23020, 25185}, {23021, 25186}, {23022, 25220}, {23023, 25173}, {23024, 25187}, {23025, 25188}, {23026, 25189}, {23027, 25190}, {23028, 25219}, {23071, 34048}, {23294, 43807}, {23302, 42085}, {23303, 42086}, {23878, 33752}, {24042, 34626}, {24441, 29069}, {24466, 38762}, {24931, 46975}, {24953, 35250}, {25175, 25223}, {25180, 25224}, {25406, 38110}, {25423, 44823}, {26098, 37715}, {26326, 26386}, {26327, 26410}, {26328, 26466}, {26329, 26467}, {26363, 35252}, {26364, 35251}, {28212, 38112}, {28228, 38127}, {28234, 34641}, {29317, 31884}, {29365, 44430}, {30143, 33858}, {30209, 31174}, {30212, 45686}, {30258, 39530}, {30311, 45043}, {30484, 32536}, {30522, 35264}, {31274, 38736}, {31383, 37649}, {31399, 43174}, {31423, 31663}, {31428, 31430}, {31441, 31443}, {31466, 31468}, {31472, 31474}, {31484, 31485}, {31488, 31490}, {31492, 31652}, {31655, 38598}, {31657, 36991}, {31748, 46673}, {31793, 31822}, {31803, 31870}, {31824, 32156}, {31827, 34795}, {31841, 38584}, {31860, 32223}, {32152, 34506}, {32196, 32338}, {32267, 41424}, {32369, 32379}, {32445, 41334}, {32533, 43908}, {32599, 37827}, {32612, 37001}, {32613, 36999}, {32822, 32831}, {32823, 32830}, {32825, 32896}, {32869, 37668}, {33106, 37716}, {33416, 42100}, {33417, 42099}, {33531, 34286}, {33540, 33542}, {33586, 37494}, {33592, 37702}, {33596, 34701}, {33604, 43207}, {33605, 43208}, {33606, 42521}, {33607, 42520}, {33884, 44324}, {33971, 37765}, {34126, 38693}, {34127, 34473}, {34169, 45143}, {34170, 41372}, {34175, 36822}, {34360, 35088}, {34573, 43621}, {34699, 37725}, {34719, 37622}, {34754, 43782}, {34755, 43781}, {34799, 45970}, {34836, 41244}, {34842, 38956}, {34845, 43919}, {34897, 40801}, {34945, 39524}, {35194, 37591}, {35514, 38092}, {35698, 35879}, {35699, 35878}, {35719, 42441}, {36441, 36460}, {36442, 36459}, {36451, 36461}, {36523, 38745}, {36742, 45931}, {36755, 40334}, {36756, 40335}, {36764, 36772}, {36836, 42157}, {36843, 42158}, {36971, 41700}, {37487, 44673}, {37512, 44519}, {37529, 42043}, {37699, 42042}, {37726, 38757}, {37809, 44401}, {37810, 38226}, {38025, 43161}, {38094, 43182}, {38104, 46684}, {38609, 44967}, {38611, 44969}, {38612, 44970}, {38613, 44972}, {38614, 44973}, {38615, 44976}, {38616, 44977}, {38617, 44979}, {38618, 44981}, {38619, 44983}, {38620, 44984}, {38621, 44985}, {38622, 44986}, {38623, 44987}, {38624, 44988}, {38625, 44992}, {38723, 38793}, {38727, 38788}, {38737, 38742}, {38797, 40556}, {38947, 45146}, {38952, 45145}, {39201, 39510}, {39529, 39531}, {39532, 44928}, {39533, 44931}, {39534, 44929}, {39535, 44927}, {39536, 44930}, {39561, 45730}, {39818, 39828}, {39847, 39857}, {40286, 45484}, {40287, 45485}, {40671, 41045}, {40672, 41044}, {40879, 46996}, {41095, 41097}, {41096, 41123}, {41116, 41126}, {41311, 46475}, {41714, 44439}, {41953, 43323}, {41954, 43322}, {41961, 41967}, {41962, 41968}, {41971, 42892}, {41972, 42893}, {41973, 42976}, {41974, 42977}, {42087, 42092}, {42088, 42089}, {42090, 42108}, {42091, 42109}, {42112, 42500}, {42113, 42501}, {42119, 42124}, {42120, 42121}, {42122, 42140}, {42123, 42141}, {42144, 42984}, {42145, 42985}, {42175, 42182}, {42176, 42180}, {42177, 42181}, {42178, 42179}, {42435, 43551}, {42436, 43550}, {42496, 42633}, {42497, 42634}, {42506, 42991}, {42507, 42990}, {42512, 42688}, {42513, 42689}, {42518, 42635}, {42519, 42636}, {42566, 43337}, {42567, 43336}, {42570, 43341}, {42571, 43340}, {42572, 43343}, {42573, 43342}, {42588, 43109}, {42589, 43108}, {42610, 43369}, {42611, 43368}, {42612, 42961}, {42613, 42960}, {42627, 43466}, {42628, 43465}, {42645, 42727}, {42646, 42728}, {42682, 43105}, {42683, 43106}, {42692, 42777}, {42693, 42778}, {42694, 42939}, {42695, 42938}, {42725, 46473}, {42726, 46476}, {42775, 42999}, {42776, 42998}, {42791, 42945}, {42792, 42944}, {42795, 42955}, {42796, 42954}, {42799, 43332}, {42800, 43333}, {42888, 43463}, {42889, 43464}, {42904, 43004}, {42905, 43005}, {42908, 42979}, {42909, 42978}, {42922, 42987}, {42923, 42986}, {43006, 43233}, {43007, 43232}, {43010, 43014}, {43011, 43015}, {43012, 43019}, {43013, 43018}, {43195, 43241}, {43196, 43240}, {43197, 43243}, {43198, 43242}, {43203, 43249}, {43204, 43248}, {43205, 43251}, {43206, 43250}, {43278, 43389}, {43292, 43549}, {43293, 43548}, {43364, 43777}, {43365, 43778}, {43415, 43507}, {43467, 43636}, {43468, 43637}, {43469, 43641}, {43470, 43642}, {43471, 43490}, {43472, 43489}, {43473, 43494}, {43474, 43493}, {43479, 43502}, {43480, 43501}, {43511, 43521}, {43515, 43525}, {43516, 43526}, {43536, 43561}, {43575, 45731}, {43592, 44801}, {43699, 44731}, {44202, 44564}, {44469, 45034}, {44666, 45879}, {44667, 45880}, {45811, 45821}, {46633, 46977}, {46634, 46979}, {46980, 46988}, {46982, 46986}

X(381) = midpoint of X(i) and X(j) for these {i,j}: {2, 4}, {3, 3830}, {5, 3845}, {13, 41042}, {14, 41043}, {20, 15682}, {51, 15030}, {114, 9880}, {140, 12101}, {147, 12243}, {154, 18405}, {265, 5655}, {355, 3656}, {376, 3543}, {382, 3534}, {428, 34664}, {546, 5066}, {547, 14893}, {549, 15687}, {550, 33699}, {551, 34648}, {568, 18435}, {598, 10033}, {671, 6054}, {1352, 20423}, {1513, 8352}, {1531, 32225}, {1551, 36196}, {1699, 5587}, {1992, 11180}, {3058, 34746}, {3060, 11459}, {3091, 41099}, {3146, 11001}, {3241, 34627}, {3426, 44750}, {3529, 15640}, {3545, 3839}, {3627, 8703}, {3654, 12699}, {3679, 31162}, {3818, 5476}, {3832, 41106}, {3843, 19709}, {3850, 3860}, {3853, 12100}, {3861, 10109}, {4301, 4669}, {4421, 34706}, {4677, 7982}, {5054, 38335}, {5055, 14269}, {5073, 15685}, {5076, 15693}, {5434, 34697}, {5562, 21969}, {5613, 25164}, {5617, 25154}, {5640, 16261}, {5656, 32064}, {5657, 9812}, {5890, 15305}, {5907, 21849}, {5913, 38951}, {5943, 46847}, {6033, 11632}, {6235, 6324}, {6248, 44422}, {6278, 22485}, {6281, 22484}, {6321, 8724}, {6785, 6787}, {7620, 9770}, {7710, 46034}, {7728, 20126}, {7775, 18546}, {7811, 34733}, {8597, 11676}, {9140, 10706}, {9144, 11005}, {9730, 16194}, {9760, 22575}, {9762, 22576}, {9766, 34505}, {9909, 34725}, {10127, 44804}, {10201, 18568}, {10707, 10711}, {10708, 10710}, {10709, 10716}, {10719, 10720}, {10723, 12117}, {10752, 13169}, {10753, 11161}, {11194, 34739}, {11235, 11236}, {11305, 41028}, {11306, 41029}, {11317, 13860}, {11455, 15072}, {11477, 15533}, {11812, 12102}, {12150, 34681}, {12355, 13188}, {14093, 35434}, {14163, 14214}, {14164, 14215}, {14892, 41987}, {15069, 15534}, {15679, 21669}, {15681, 15684}, {15686, 35404}, {15694, 35403}, {15718, 35401}, {16200, 37712}, {17578, 19708}, {18323, 44265}, {18377, 44278}, {18492, 30308}, {18552, 42853}, {18572, 44266}, {22491, 22492}, {23046, 38071}, {25175, 25223}, {25180, 25224}, {31145, 34631}, {31693, 41016}, {31694, 41017}, {31862, 31863}, {34621, 44442}, {36437, 36455}, {36490, 36730}, {36523, 38745}, {36551, 36729}, {36718, 36734}, {36719, 36733}, {36720, 36732}, {36721, 36731}, {36722, 36728}, {36723, 36726}, {36724, 36725}, {36990, 43273}, {38724, 38789}, {38732, 38743}, {41096, 41123}, {44262, 44288}, {44263, 44287}, {46977, 46985}, {46979, 46983}, {46980, 46988}, {46982, 46986}

X(381) = reflection of X(i) in X(j) for these {i,j}: {2, 5}, {3, 2}, {4, 3845}, {5, 5066}, {6, 5476}, {20, 8703}, {21, 44257}, {22, 44262}, {23, 44266}, {24, 44270}, {25, 44275}, {26, 44278}, {52, 21849}, {140, 10109}, {165, 11231}, {186, 44282}, {376, 549}, {378, 44287}, {382, 3830}, {383, 44289}, {399, 5655}, {546, 3860}, {547, 11737}, {548, 11812}, {549, 547}, {550, 12100}, {568, 51}, {599, 11178}, {1113, 13626}, {1114, 13627}, {1116, 39494}, {1351, 20423}, {1482, 3656}, {1656, 19709}, {1657, 3534}, {1853, 23325}, {1995, 39487}, {2979, 15067}, {3095, 44422}, {3146, 33699}, {3167, 5654}, {3522, 15713}, {3524, 15699}, {3529, 19710}, {3534, 3}, {3543, 15687}, {3545, 38071}, {3576, 11230}, {3581, 32225}, {3627, 12101}, {3654, 10}, {3655, 551}, {3656, 946}, {3830, 4}, {3839, 23046}, {3843, 41099}, {3845, 546}, {3851, 41106}, {3860, 3856}, {3917, 10170}, {4930, 34647}, {5050, 14561}, {5054, 5055}, {5055, 3545}, {5066, 3850}, {5068, 41990}, {5073, 15682}, {5077, 37242}, {5085, 38317}, {5093, 14853}, {5094, 39484}, {5476, 19130}, {5587, 38140}, {5603, 38034}, {5655, 113}, {5657, 38042}, {5731, 38028}, {5790, 5587}, {5817, 38139}, {5858, 34508}, {5859, 34509}, {5886, 3817}, {5890, 5946}, {5946, 13364}, {6054, 22566}, {6055, 5461}, {6243, 21969}, {6321, 9880}, {7540, 428}, {7576, 13490}, {7610, 7617}, {7615, 20112}, {7618, 9771}, {7967, 10283}, {8182, 15597}, {8371, 39492}, {8598, 37459}, {8703, 140}, {8724, 114}, {9730, 5943}, {9766, 7775}, {9979, 44204}, {10031, 19907}, {10032, 19919}, {10109, 12811}, {10164, 10172}, {10165, 10171}, {10168, 25565}, {10246, 5886}, {10247, 5603}, {10295, 18579}, {10304, 11539}, {10606, 23329}, {10620, 20126}, {10992, 36521}, {11001, 550}, {11057, 34510}, {11159, 35930}, {11165, 11184}, {11178, 25561}, {11179, 597}, {11184, 8176}, {11362, 4745}, {11459, 15060}, {11632, 115}, {11812, 35018}, {11911, 11897}, {12100, 3628}, {12101, 3861}, {12103, 15759}, {12117, 33813}, {12188, 11632}, {12355, 6321}, {12702, 3654}, {13102, 25164}, {13103, 25154}, {13188, 8724}, {13340, 3917}, {14070, 10201}, {14093, 15694}, {14269, 3839}, {14643, 36518}, {14651, 38229}, {14666, 9172}, {14830, 6055}, {14848, 38072}, {14853, 38136}, {14855, 16836}, {14995, 14356}, {15041, 15061}, {15055, 34128}, {15061, 23515}, {15533, 34507}, {15534, 576}, {15561, 36519}, {15678, 31649}, {15681, 376}, {15682, 3627}, {15683, 15686}, {15684, 3543}, {15685, 20}, {15686, 34200}, {15687, 14893}, {15688, 5054}, {15689, 3524}, {15690, 3530}, {15691, 14891}, {15693, 1656}, {15694, 5071}, {15695, 631}, {15696, 15693}, {15697, 15712}, {15699, 14892}, {15700, 15703}, {15701, 3090}, {15704, 15690}, {15713, 12812}, {15716, 5070}, {15722, 7486}, {15759, 16239}, {15980, 37350}, {16194, 46847}, {16220, 10278}, {16836, 6688}, {17525, 16617}, {17538, 15711}, {17800, 11001}, {17813, 23048}, {18332, 5465}, {18405, 18376}, {18435, 15030}, {18493, 30308}, {18559, 38322}, {18575, 39486}, {19140, 25566}, {19706, 6918}, {19708, 632}, {19709, 3091}, {19710, 548}, {20126, 125}, {20128, 1651}, {20423, 5480}, {21151, 38171}, {21153, 38318}, {21154, 38319}, {21733, 11182}, {21849, 10110}, {21969, 5446}, {23039, 5891}, {25154, 5478}, {25164, 5479}, {25406, 38110}, {26446, 10175}, {27088, 10011}, {28460, 15670}, {30232, 14215}, {30233, 14214}, {31961, 31840}, {32447, 262}, {32519, 32447}, {32609, 14643}, {33699, 3853}, {33923, 11540}, {34153, 11694}, {34200, 10124}, {34473, 34127}, {34505, 18546}, {34682, 12150}, {34698, 5434}, {34707, 4421}, {34718, 3679}, {34726, 9909}, {34734, 7811}, {34740, 11194}, {34745, 3058}, {34748, 3241}, {34810, 14995}, {35434, 35403}, {35489, 16532}, {35752, 16001}, {35932, 44223}, {36330, 16002}, {36448, 42281}, {36466, 42280}, {36490, 36722}, {36521, 20399}, {36718, 36719}, {36720, 36490}, {36721, 36551}, {36723, 36724}, {36726, 36725}, {36729, 36727}, {36730, 36728}, {36731, 36729}, {36732, 36730}, {36734, 36733}, {37348, 3363}, {37477, 13857}, {37922, 37943}, {37956, 46451}, {38107, 38150}, {38224, 23514}, {38335, 14269}, {38693, 34126}, {38723, 38793}, {38724, 14644}, {38731, 38748}, {38732, 14639}, {38742, 38737}, {38754, 21154}, {38788, 38727}, {40280, 373}, {40727, 7615}, {41042, 22796}, {41043, 22797}, {41099, 3858}, {41106, 3857}, {41984, 41986}, {42278, 36466}, {42279, 36448}, {42733, 39491}, {42830, 18552}, {43273, 182}, {44202, 44564}, {44245, 44580}, {44255, 6675}, {44257, 46028}, {44261, 6676}, {44262, 46029}, {44265, 468}, {44266, 44961}, {44268, 16238}, {44270, 44235}, {44273, 6677}, {44275, 46030}, {44278, 13406}, {44280, 44452}, {44282, 46031}, {44284, 8728}, {44287, 39504}, {44750, 4846}, {44903, 15691}, {45700, 3829}, {46633, 46977}, {46634, 46983}

X(381) = isogonal conjugate of X(3431)
X(381) = isotomic conjugate of isogonal conjugate of X(34417)
X(381) = complement of X(376)
X(381) = anticomplement of X(549)
X(381) = center of the orthocentroidal circle
X(381) = centroid of the Euler triangle
X(381) = center of equilateral triangle X(4)PU(5)
X(381) = center of the Vu pedal-centroidal circle of X(382)
X(381) = circumcircle-inverse of X(7575)
X(381) = nine-point-circle-inverse of X(11799)
X(381) = polar-circle-inverse of X(10295)
X(381) = orthoptic-circle-of-Steiner-inellipse-inverse of X(7426)
X(381) = orthoptic-circle-of-Steiner-circumellipse-inverse of X(37901)
X(381) = Stammler-circle-inverse of X(37924)
X(381) = Hutson-Parry-circle-inverse of X(9169)
X(381) = circle-O(PU(5))-inverse of X(4)k
X(381) = Kiepert-hyperbola-inverse of X(6)
X(381) = crossdifference of every pair of points on line X(526)X(647)
X(381) = polar conjugate of the isotomic conjugate of X(37638)
X(381) = polar conjugate of the isogonal conjugate of X(5158)
X(381) = Thomson isogonal conjugate of X(7712)
X(381) = orthic isogonal conjugate of X(40909)
X(381) = psi-transform of X(9140)
X(381) = X(14483)-anticomplementary conjugate of X(8)
X(381) = X(i)-complementary conjugate of X(j) for these (i,j): {3426, 10}, {9064, 8062}, {36889, 2887}
X(381) = X(i)-Ceva conjugate of X(j) for these (i,j): {4, 40909}, {1302, 523}, {4993, 5158}, {7578, 6}, {44135, 37638}, {46808, 18487}
X(381) = X(i)-cross conjugate of X(j) for these (i,j): {5158, 37638}, {18487, 46808}
X(381) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3431}, {48, 43530}, {255, 16263}, {2159, 46809}, {6149, 18316}
X(381) = cevapoint of X(5158) and X(34417)
X(381) = crosspoint of X(i) and X(j) for these (i,j): {2, 36889}, {264, 34289}
X(381) = crosssum of X(i) and X(j) for these (i,j): {6, 26864}, {184, 5063}, {1511, 10564}
X(381) = trilinear pole of line {18487, 32225}
X(381) = crossdifference of every pair of points on line {526, 647}
X(381) = X(8)-beth conjugate of X(3654)
X(381) = X(2)-daleth conjugate of X(44216)
X(381) = X(1)-zayin conjugate of X(3431)
X(381) = pole of the line X(2)X(6) wrt the circle {X(2),X(13),X(14),X(111),X(476)}}
X(381) = pole of the Napoleon axis wrt the Lester circle
X(381) = X(3)-of-4th-Brocard-triangle
X(381) = X(3)-of-orthocentroidal-triangle
X(381) = center of conic that is the locus of orthopoles of lines passing through X(2)
X(381) = harmonic center of circumcircle and nine-point circle
X(381) = centroid of antipedal triangle of X(2) wrt medial triangle
X(381) = homothetic center of orthic triangle and 2nd isogonal triangle of X(4); see X(36)
X(381) = pole of line X(2)X(6) wrt Hutson-Parry circle
X(381) = Artzt-to-McCay similarity image of X(98)
X(381) = intersection of Fermat axes of ABC and Artzt triangle
X(381) = centroid of maltitude quadrangle of quadrangle ABCX(2)
X(381) = homothetic center of X(5)-altimedial and X(4)-anti-altimedial triangles
X(381) = X(3576)-of-orthic-triangle if ABC is acute
X(381) = trisector nearest X(5) of segment X(4)X(5)
X(381) = {X(3),X(4)}-harmonic conjugate of X(382)
X(381) = {X(2043),X(2044)}-harmonic conjugate of X(5)
X(381) = homothetic center of ABC and Ehrmann mid-triangle
X(381) = homothetic center of Ehrmann vertex-triangle and tangential triangle
X(381) = homothetic center of Ehrmann side-triangle and orthic triangle
X(381) = Johnson-to-Ehrmann-mid similarity image of X(2)

X(381) = barycentric product X(i)*X(j) for these {i,j}: {4, 37638}, {5, 4993}, {6, 44135}, {30, 46808}, {76, 34417}, {92, 18477}, {94, 3581}, {264, 5158}, {671, 32225}, {1494, 18487}, {1502, 34416}, {1531, 16080}, {2349, 18486}, {2996, 21970}, {4550, 34289}, {14165, 18478}, {14314, 46456}

X(381) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 43530}, {6, 3431}, {30, 46809}, {393, 16263}, {1531, 11064}, {1989, 18316}, {3581, 323}, {4550, 15066}, {4993, 95}, {5158, 3}, {8749, 22455}, {14314, 8552}, {14533, 46091}, {15362, 44555}, {18477, 63}, {18485, 18487}, {18486, 14206}, {18487, 30}, {21970, 193}, {32225, 524}, {34416, 32}, {34417, 6}, {37638, 69}, {40909, 37645}, {44135, 76}, {46808, 1494}

X(381) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 9955, 18493}, {1, 10895, 9654}, {1, 10896, 9669}, {1, 18480, 18525}, {1, 18492, 18480}, {1, 18525, 18526}, {1, 18527, 18530}, {1, 18529, 18528}, {1, 18542, 18545}, {1, 18544, 18543}, {1, 30308, 38021}, {1, 45630, 18544}, {1, 45631, 18542}, {2, 3, 5054}, {2, 5, 5055}, {2, 20, 3524}, {2, 376, 549}, {2, 382, 15688}, {2, 384, 33220}, {2, 428, 9909}, {2, 452, 17561}, {2, 546, 14269}, {2, 547, 15703}, {2, 549, 15694}, {2, 550, 15707}, {2, 631, 11539}, {2, 858, 32216}, {2, 1003, 11288}, {2, 1370, 43957}, {2, 1513, 40248}, {2, 1657, 15706}, {2, 3090, 15699}, {2, 3091, 3545}, {2, 3146, 10304}, {2, 3522, 15708}, {2, 3523, 15709}, {2, 3524, 140}, {2, 3529, 17504}, {2, 3534, 15693}, {2, 3543, 376}, {2, 3545, 5}, {2, 3627, 15689}, {2, 3830, 3534}, {2, 3832, 3839}, {2, 3839, 4}, {2, 3843, 38335}, {2, 3845, 3830}, {2, 3855, 38071}, {2, 5025, 33219}, {2, 5054, 3526}, {2, 5055, 1656}, {2, 5059, 15705}, {2, 5064, 34609}, {2, 5066, 19709}, {2, 5071, 547}, {2, 6175, 44217}, {2, 6756, 10245}, {2, 7576, 14070}, {2, 7714, 10154}, {2, 7841, 11287}, {2, 8352, 5077}, {2, 8370, 11286}, {2, 8597, 35955}, {2, 8703, 15701}, {2, 10304, 631}, {2, 11001, 12100}, {2, 11112, 16417}, {2, 11113, 16418}, {2, 11114, 16370}, {2, 11286, 33237}, {2, 11299, 11301}, {2, 11300, 11302}, {2, 11303, 11298}, {2, 11304, 11297}, {2, 11317, 11159}, {2, 11318, 33240}, {2, 11361, 1003}, {2, 11539, 46219}, {2, 12101, 15685}, {2, 14002, 37907}, {2, 14033, 8369}, {2, 14035, 33255}, {2, 14036, 8366}, {2, 14039, 8368}, {2, 14041, 7841}, {2, 14063, 33251}, {2, 14068, 33187}, {2, 14269, 382}, {2, 14893, 15684}, {2, 15640, 19708}, {2, 15681, 15700}, {2, 15682, 8703}, {2, 15683, 15692}, {2, 15684, 14093}, {2, 15685, 15716}, {2, 15686, 15718}, {2, 15687, 15681}, {2, 15688, 15720}, {2, 15690, 15722}, {2, 15692, 15702}, {2, 15694, 15723}, {2, 15697, 15719}, {2, 15698, 15713}, {2, 15699, 5070}, {2, 15702, 10124}, {2, 15705, 10303}, {2, 15708, 3525}, {2, 15709, 632}, {2, 15710, 14869}, {2, 16041, 33184}, {2, 16865, 15671}, {2, 17532, 17528}, {2, 17538, 41983}, {2, 17561, 6675}, {2, 17577, 17532}, {2, 17579, 16371}, {2, 17677, 11359}, {2, 18559, 18324}, {2, 18566, 18561}, {2, 18568, 18564}, {2, 19686, 33246}, {2, 19708, 11812}, {2, 23046, 3843}, {2, 24608, 1375}, {2, 26122, 25518}, {2, 26255, 468}, {2, 26786, 27055}, {2, 26833, 26994}, {2, 30775, 5159}, {2, 30922, 46531}, {2, 30976, 37370}, {2, 31048, 857}, {2, 31105, 858}, {2, 31133, 31152}, {2, 31156, 15670}, {2, 31789, 28451}, {2, 32986, 8359}, {2, 32996, 33278}, {2, 33006, 33228}, {2, 33007, 35297}, {2, 33013, 44543}, {2, 33016, 8370}, {2, 33017, 8356}, {2, 33187, 16925}, {2, 33192, 33008}, {2, 33219, 7866}, {2, 33220, 32954}, {2, 33228, 11318}, {2, 33251, 6656}, {2, 33255, 7807}, {2, 33264, 33273}, {2, 33265, 33274}, {2, 33272, 33215}, {2, 33278, 7791}, {2, 33285, 8360}, {2, 33699, 15695}, {2, 33703, 45759}, {2, 34938, 43934}, {2, 35927, 33216}, {2, 36436, 36457}, {2, 36439, 36456}, {2, 36445, 18585}, {2, 36454, 36439}, {2, 36457, 36438}, {2, 36463, 15765}, {2, 37038, 19279}, {2, 37077, 378}, {2, 37150, 19277}, {2, 37170, 37352}, {2, 37171, 37351}, {2, 37375, 17556}, {2, 38071, 3851}, {2, 38320, 3515}, {2, 38322, 37922}, {2, 38335, 1657}, {2, 41099, 3845}, {2, 41106, 5066}, {2, 43957, 16419}, {2, 44442, 10691}, {2, 44462, 46825}, {2, 44466, 46824}, {2, 44579, 11331}, {2, 46333, 15712}, {2, 46451, 7552}, {2, 46856, 36186}, {2, 46857, 36185}, {2, 46868, 44889}, {2, 46869, 44891}, {3, 4, 382}, {3, 5, 1656}, {3, 25, 2070}, {3, 376, 14093}, {3, 382, 1657}, {3, 547, 15723}, {3, 549, 15700}, {3, 1598, 7517}, {3, 1656, 3526}, {3, 1657, 15696}, {3, 2072, 30771}, {3, 3091, 5072}, {3, 3516, 35498}, {3, 3520, 35496}, {3, 3524, 15716}, {3, 3526, 15720}, {3, 3534, 15688}, {3, 3843, 4}, {3, 3845, 38335}, {3, 3851, 5}, {3, 5054, 15693}, {3, 5055, 2}, {3, 5070, 140}, {3, 5072, 5079}, {3, 5073, 20}, {3, 5899, 22}, {3, 6642, 43809}, {3, 6913, 7489}, {3, 6918, 45976}, {3, 7387, 13564}, {3, 7506, 45735}, {3, 7517, 2937}, {3, 7529, 7506}, {3, 7540, 34726}, {3, 7579, 5094}, {3, 13621, 24}, {3, 14269, 3830}, {3, 14893, 35434}, {3, 15681, 376}, {3, 15684, 15681}, {3, 15685, 15689}, {3, 15689, 8703}, {3, 15693, 15706}, {3, 15694, 549}, {3, 15695, 10304}, {3, 15701, 3524}, {3, 15703, 15694}, {3, 15707, 12100}, {3, 15718, 15692}, {3, 16418, 28443}, {3, 17800, 550}, {3, 18378, 26}, {3, 18531, 18536}, {3, 18534, 12083}, {3, 18535, 18534}, {3, 19709, 5055}, {3, 28453, 16370}, {3, 35400, 15686}, {3, 35401, 35400}, {3, 35403, 15684}, {3, 37234, 13743}, {3, 37411, 16117}, {3, 37922, 18324}, {3, 44457, 35243}, {3, 46219, 631}, {4, 5, 3}, {4, 20, 3627}, {4, 24, 12173}, {4, 25, 18494}, {4, 140, 5073}, {4, 186, 35480}, {4, 235, 1598}, {4, 376, 3543}, {4, 378, 44438}, {4, 382, 5076}, {4, 403, 25}, {4, 427, 1597}, {4, 442, 37411}, {4, 546, 3843}, {4, 547, 15681}, {4, 549, 15684}, {4, 631, 3146}, {4, 1346, 1344}, {4, 1347, 1345}, {4, 1532, 19541}, {4, 1594, 1593}, {4, 1596, 18535}, {4, 1656, 1657}, {4, 2043, 18587}, {4, 2044, 18586}, {4, 2476, 6985}, {4, 3088, 13488}, {4, 3089, 6756}, {4, 3090, 20}, {4, 3091, 5}, {4, 3146, 3853}, {4, 3153, 44288}, {4, 3520, 35490}, {4, 3524, 15682}, {4, 3525, 33703}, {4, 3529, 17578}, {4, 3541, 1885}, {4, 3542, 3575}, {4, 3543, 15687}, {4, 3544, 631}, {4, 3545, 2}, {4, 3547, 7553}, {4, 3628, 17800}, {4, 3830, 38335}, {4, 3832, 546}, {4, 3839, 3845}, {4, 3845, 14269}, {4, 3850, 3851}, {4, 3851, 1656}, {4, 3854, 3850}, {4, 3855, 3091}, {4, 5055, 3534}, {4, 5056, 550}, {4, 5066, 5055}, {4, 5067, 3529}, {4, 5068, 140}, {4, 5071, 376}, {4, 5072, 3526}, {4, 5079, 15696}, {4, 5084, 6851}, {4, 5133, 9818}, {4, 5169, 31861}, {4, 5187, 37356}, {4, 6622, 7487}, {4, 6623, 1596}, {4, 6804, 34938}, {4, 6816, 14790}, {4, 6824, 7491}, {4, 6826, 6923}, {4, 6828, 3560}, {4, 6829, 7580}, {4, 6830, 1012}, {4, 6834, 37468}, {4, 6835, 6917}, {4, 6837, 37290}, {4, 6841, 37234}, {4, 6843, 6907}, {4, 6844, 8727}, {4, 6846, 31789}, {4, 6848, 20420}, {4, 6849, 44229}, {4, 6854, 6925}, {4, 6855, 6868}, {4, 6856, 6869}, {4, 6864, 6850}, {4, 6866, 6841}, {4, 6867, 6842}, {4, 6871, 37406}, {4, 6873, 21}, {4, 6874, 411}, {4, 6879, 6938}, {4, 6893, 6928}, {4, 6896, 377}, {4, 6898, 6836}, {4, 6900, 2475}, {4, 6901, 37437}, {4, 6902, 6895}, {4, 6903, 37433}, {4, 6939, 6827}, {4, 6941, 3149}, {4, 6945, 6911}, {4, 6947, 10431}, {4, 6957, 6929}, {4, 6964, 31775}, {4, 6965, 6840}, {4, 6968, 1532}, {4, 6973, 6882}, {4, 6975, 37022}, {4, 6990, 405}, {4, 6997, 18420}, {4, 7384, 36663}, {4, 7385, 36685}, {4, 7394, 11818}, {4, 7399, 39568}, {4, 7404, 12605}, {4, 7486, 15704}, {4, 7505, 6240}, {4, 7547, 7507}, {4, 7577, 378}, {4, 8226, 6913}, {4, 10024, 7517}, {4, 10109, 15689}, {4, 10124, 35400}, {4, 10254, 2070}, {4, 10304, 33699}, {4, 11737, 15694}, {4, 12811, 5070}, {4, 13160, 7387}, {4, 13406, 18378}, {4, 13862, 35930}, {4, 14018, 46467}, {4, 14788, 11414}, {4, 14789, 12082}, {4, 14892, 15701}, {4, 14940, 34797}, {4, 15022, 548}, {4, 15559, 11403}, {4, 15681, 35434}, {4, 15687, 35403}, {4, 15692, 35404}, {4, 15699, 15685}, {4, 15709, 15640}, {4, 15760, 18534}, {4, 16044, 40279}, {4, 16072, 34609}, {4, 16868, 24}, {4, 17578, 12102}, {4, 18404, 31724}, {4, 18531, 31723}, {4, 18537, 18531}, {4, 19709, 5054}, {4, 21451, 45971}, {4, 34007, 44279}, {4, 34664, 34725}, {4, 34939, 37198}, {4, 35404, 35401}, {4, 35487, 3515}, {4, 35488, 37197}, {4, 36436, 36455}, {4, 36454, 36437}, {4, 36473, 6996}, {4, 36526, 36716}, {4, 36651, 13727}, {4, 36653, 36489}, {4, 36655, 36712}, {4, 36656, 36711}, {4, 36659, 36707}, {4, 36660, 36674}, {4, 36662, 36474}, {4, 36665, 21737}, {4, 36666, 36701}, {4, 36667, 36703}, {4, 36677, 6998}, {4, 37119, 18560}, {4, 37347, 12083}, {4, 37353, 7514}, {4, 37372, 7497}, {4, 37943, 18559}, {4, 37988, 32444}, {4, 37990, 35243}, {4, 38071, 19709}, {4, 41099, 3839}, {4, 41106, 3545}, {4, 44229, 37230}, {4, 44235, 13621}, {4, 44440, 44276}, {4, 44958, 10594}, {4, 46029, 5899}, {5, 20, 5070}, {5, 140, 3090}, {5, 235, 10024}, {5, 376, 15703}, {5, 382, 3526}, {5, 403, 10254}, {5, 427, 2072}, {5, 546, 4}, {5, 547, 5071}, {5, 548, 5067}, {5, 549, 547}, {5, 550, 3628}, {5, 632, 35018}, {5, 1532, 6980}, {5, 1594, 10255}, {5, 1595, 11585}, {5, 1596, 15760}, {5, 1597, 30771}, {5, 1656, 5079}, {5, 1885, 6640}, {5, 1907, 37452}, {5, 3091, 3851}, {5, 3146, 46219}, {5, 3530, 7486}, {5, 3543, 15694}, {5, 3545, 19709}, {5, 3575, 6639}, {5, 3627, 140}, {5, 3628, 5056}, {5, 3830, 5054}, {5, 3832, 3843}, {5, 3839, 3830}, {5, 3843, 382}, {5, 3850, 3091}, {5, 3851, 5072}, {5, 3853, 631}, {5, 3856, 3832}, {5, 3857, 3850}, {5, 3858, 546}, {5, 3859, 3855}, {5, 3860, 3839}, {5, 3861, 20}, {5, 5066, 3545}, {5, 5076, 15720}, {5, 5169, 7579}, {5, 6756, 3549}, {5, 6823, 7405}, {5, 6831, 6971}, {5, 6851, 16853}, {5, 6907, 6881}, {5, 6917, 6918}, {5, 6929, 6913}, {5, 7401, 11484}, {5, 7403, 5576}, {5, 7514, 7539}, {5, 7528, 7529}, {5, 7564, 7507}, {5, 8703, 15699}, {5, 8727, 6882}, {5, 11563, 46029}, {5, 11818, 25}, {5, 12101, 3524}, {5, 12102, 3523}, {5, 12362, 14786}, {5, 12811, 5068}, {5, 13163, 21451}, {5, 13488, 3548}, {5, 13490, 10201}, {5, 14269, 3534}, {5, 14893, 376}, {5, 15684, 15723}, {5, 15687, 549}, {5, 15699, 10109}, {5, 15760, 37347}, {5, 15761, 13160}, {5, 15765, 18586}, {5, 16198, 6643}, {5, 18377, 7503}, {5, 18420, 5020}, {5, 18567, 14118}, {5, 18569, 7395}, {5, 18585, 18587}, {5, 20420, 6863}, {5, 23046, 3845}, {5, 23047, 18404}, {5, 31724, 34864}, {5, 31789, 6861}, {5, 31830, 7505}, {5, 31861, 5094}, {5, 33332, 10224}, {5, 33699, 11539}, {5, 33923, 46936}, {5, 34664, 14787}, {5, 35018, 15022}, {5, 35403, 15700}, {5, 35404, 10124}, {5, 35738, 2046}, {5, 35930, 37071}, {5, 36437, 36456}, {5, 36455, 36438}, {5, 36654, 36530}, {5, 36716, 36527}, {5, 37230, 37251}, {5, 37281, 6944}, {5, 37290, 6862}, {5, 37349, 5899}, {5, 37356, 4187}, {5, 37406, 442}, {5, 38071, 5066}, {5, 38335, 15693}, {5, 39504, 7577}, {5, 40250, 13860}, {5, 40277, 11317}, {5, 40279, 7770}, {5, 41099, 14269}, {5, 41987, 15682}, {5, 41991, 3858}, {5, 44235, 16868}, {5, 44245, 46935}, {5, 44263, 6644}, {5, 44279, 17928}, {5, 44288, 7514}, {5, 44804, 44441}, {5, 44920, 18537}, {5, 45971, 14940}, {5, 46028, 6873}, {5, 46030, 403}, {6, 13, 42974}, {6, 14, 42975}, {6, 3818, 18440}, {6, 5475, 15484}, {6, 5476, 14848}, {6, 6564, 13665}, {6, 6565, 13785}, {6, 13665, 18512}, {6, 13785, 18510}, {6, 15928, 34810}, {6, 16808, 42128}, {6, 16809, 42125}, {6, 18440, 39899}, {6, 18445, 15087}, {6, 18451, 18445}, {6, 18509, 26336}, {6, 18511, 26346}, {6, 38072, 5476}, {6, 42125, 42816}, {6, 42128, 42815}, {6, 45438, 45375}, {6, 45439, 45376}, {8, 22791, 8148}, {9, 18482, 31671}, {10, 3654, 38066}, {10, 12699, 12702}, {10, 18483, 12699}, {11, 1478, 999}, {11, 5434, 10072}, {11, 10742, 12773}, {11, 18516, 18519}, {12, 1479, 3295}, {12, 3058, 10056}, {12, 18517, 18518}, {13, 14, 6}, {13, 16809, 14}, {13, 42974, 42815}, {14, 16808, 13}, {14, 42975, 42816}, {15, 36970, 42154}, {15, 37832, 16644}, {15, 42093, 42126}, {15, 42098, 42132}, {15, 42919, 42098}, {16, 36969, 42155}, {16, 37835, 16645}, {16, 42094, 42127}, {16, 42095, 42129}, {16, 42918, 42095}, {17, 16964, 22236}, {17, 41101, 16962}, {18, 16965, 22238}, {18, 41100, 16963}, {20, 140, 3}, {20, 376, 15691}, {20, 3090, 140}, {20, 3091, 5068}, {20, 3524, 8703}, {20, 3545, 10109}, {20, 3627, 5073}, {20, 5068, 3090}, {20, 8703, 15689}, {20, 15689, 3534}, {20, 15699, 15701}, {20, 15716, 15688}, {20, 21735, 44245}, {20, 41099, 41987}, {20, 44903, 15681}, {20, 45762, 35403}, {22, 7514, 3}, {22, 7530, 5899}, {22, 37353, 7539}, {23, 35921, 7502}, {24, 7526, 3}, {24, 13861, 13621}, {24, 18324, 37922}, {25, 9818, 3}, {25, 18386, 4}, {25, 18403, 382}, {25, 37954, 24}, {26, 7503, 3}, {26, 10594, 18378}, {32, 18500, 18503}, {32, 18502, 18501}, {32, 39565, 13881}, {33, 18455, 9642}, {33, 37697, 18455}, {34, 37696, 18447}, {35, 18514, 12953}, {36, 18513, 12943}, {39, 39563, 11648}, {40, 7989, 9956}, {51, 568, 13321}, {52, 5907, 18436}, {54, 156, 9704}, {54, 43865, 43835}, {55, 3583, 9668}, {55, 7951, 31479}, {55, 18407, 18499}, {55, 18491, 18524}, {56, 3585, 9655}, {56, 18761, 26321}, {61, 41121, 16267}, {61, 42156, 42988}, {61, 42814, 5339}, {61, 42972, 41108}, {62, 41122, 16268}, {62, 42153, 42989}, {62, 42813, 5340}, {62, 42973, 41107}, {65, 31937, 40266}, {68, 22660, 12164}, {68, 45089, 37493}, {69, 21850, 44456}, {74, 1539, 38790}, {76, 7773, 7776}, {76, 7809, 7788}, {80, 18393, 2099}, {98, 10796, 11842}, {98, 22505, 38744}, {99, 22515, 38733}, {104, 22799, 38756}, {110, 10113, 12902}, {111, 6032, 9745}, {113, 265, 399}, {113, 7687, 265}, {113, 18390, 18445}, {113, 18474, 18451}, {114, 6321, 13188}, {115, 5475, 6}, {115, 6033, 12188}, {115, 7753, 5309}, {115, 43457, 5475}, {116, 10741, 38574}, {117, 10747, 38579}, {118, 10739, 38572}, {119, 10738, 12331}, {119, 26333, 10679}, {120, 15521, 38589}, {121, 15522, 38590}, {122, 22337, 38591}, {123, 33566, 38592}, {124, 10740, 38573}, {125, 7728, 10620}, {125, 46686, 7728}, {126, 22338, 38593}, {127, 12918, 13115}, {132, 10749, 13310}, {133, 10745, 38577}, {137, 31656, 38587}, {140, 546, 3861}, {140, 549, 15721}, {140, 3090, 5070}, {140, 3524, 15701}, {140, 3627, 20}, {140, 3861, 3627}, {140, 5066, 14892}, {140, 8703, 3524}, {140, 10109, 15699}, {140, 12811, 5}, {140, 14891, 549}, {140, 14892, 10109}, {140, 15682, 15689}, {140, 15689, 15716}, {140, 15691, 14891}, {140, 15699, 2}, {140, 15701, 5054}, {140, 41987, 12101}, {140, 41990, 3545}, {140, 44245, 44682}, {141, 31670, 33878}, {143, 5876, 5889}, {143, 45958, 5876}, {146, 15081, 10264}, {147, 41135, 12243}, {148, 7777, 31859}, {155, 9927, 12429}, {155, 10982, 36749}, {155, 36749, 195}, {184, 13851, 18396}, {184, 46261, 10540}, {185, 5462, 37481}, {186, 7527, 18570}, {186, 13595, 12106}, {186, 18570, 3}, {186, 35480, 37196}, {226, 5722, 15934}, {230, 7737, 1384}, {235, 18404, 7517}, {235, 23047, 4}, {237, 35934, 3}, {262, 6054, 11163}, {265, 18388, 15087}, {355, 946, 1482}, {355, 1482, 12645}, {355, 10893, 11928}, {355, 10894, 11929}, {371, 8976, 13903}, {371, 35787, 23261}, {371, 42265, 8976}, {371, 45543, 45378}, {372, 13951, 13961}, {372, 35786, 23251}, {372, 42262, 13951}, {372, 45542, 45377}, {376, 547, 15694}, {376, 549, 3}, {376, 631, 15715}, {376, 3091, 11737}, {376, 3545, 5071}, {376, 3839, 14893}, {376, 3845, 35403}, {376, 5055, 15723}, {376, 5071, 2}, {376, 7426, 14070}, {376, 10124, 15718}, {376, 11737, 5055}, {376, 14093, 15688}, {376, 14893, 3830}, {376, 15681, 3534}, {376, 15683, 15686}, {376, 15687, 15684}, {376, 15691, 15689}, {376, 15692, 34200}, {376, 15694, 15700}, {376, 15702, 15692}, {376, 15703, 5054}, {376, 15715, 10304}, {376, 15721, 14891}, {376, 15723, 15693}, {376, 31180, 31152}, {376, 35403, 382}, {377, 4187, 16408}, {377, 5187, 4187}, {377, 37356, 3}, {378, 1995, 6644}, {378, 6644, 3}, {378, 7577, 5094}, {379, 30808, 31184}, {379, 31014, 30808}, {382, 1656, 3}, {382, 3526, 15696}, {382, 3851, 5079}, {382, 5054, 3534}, {382, 5055, 15693}, {382, 5072, 1656}, {382, 5079, 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{14093, 38335, 15684}, {14120, 36169, 11799}, {14130, 18369, 45735}, {14130, 45735, 3}, {14269, 15681, 15687}, {14269, 15684, 35403}, {14269, 15694, 3543}, {14269, 15695, 3853}, {14269, 15701, 3627}, {14269, 15703, 15684}, {14269, 19709, 3}, {14269, 41106, 5072}, {14269, 46219, 33699}, {14583, 39170, 14993}, {14643, 36518, 15046}, {14666, 38796, 9172}, {14709, 14710, 10226}, {14709, 20479, 3}, {14710, 20478, 3}, {14782, 14783, 3091}, {14784, 14785, 631}, {14787, 18404, 34664}, {14788, 37444, 7393}, {14789, 46450, 7485}, {14813, 14814, 3523}, {14830, 38224, 6055}, {14845, 16194, 9730}, {14865, 22467, 11250}, {14869, 15711, 41983}, {14869, 17538, 3}, {14869, 33923, 15717}, {14890, 15687, 35414}, {14891, 14893, 3627}, {14891, 15682, 15681}, {14891, 15689, 14093}, {14891, 15691, 8703}, {14891, 44903, 376}, {14891, 45762, 15687}, {14892, 14893, 14891}, {14892, 15682, 5070}, {14892, 15685, 1656}, {14892, 15691, 547}, {14892, 41988, 549}, {14892, 45762, 15691}, {14893, 15683, 35401}, {14893, 15684, 38335}, {14893, 15687, 4}, {14893, 15694, 382}, {14893, 19709, 15700}, {14893, 41987, 45762}, {14893, 45762, 3861}, {14940, 34797, 32534}, {15022, 15640, 15709}, {15022, 17578, 3523}, {15038, 15087, 6}, {15046, 32609, 14643}, {15068, 39522, 1993}, {15154, 15155, 37924}, {15335, 20030, 27868}, {15640, 15709, 548}, {15640, 19708, 19710}, {15646, 35473, 3}, {15670, 28460, 28443}, {15670, 31156, 16418}, {15681, 15685, 44903}, {15681, 15687, 382}, {15681, 15694, 3}, {15681, 15700, 15688}, {15681, 15703, 549}, {15681, 19709, 547}, {15681, 35401, 35404}, {15681, 35403, 3543}, {15682, 15699, 3}, {15682, 15701, 3534}, {15682, 15721, 15691}, {15682, 41990, 19709}, {15683, 15686, 15681}, {15683, 15692, 376}, {15683, 15702, 34200}, {15683, 35401, 382}, {15683, 35404, 35400}, {15684, 15687, 35434}, {15684, 15694, 376}, {15684, 15701, 15691}, {15684, 15703, 3}, {15684, 15718, 15686}, {15684, 15723, 15688}, {15684, 19709, 15703}, {15684, 35403, 3830}, {15685, 15689, 20}, {15685, 15701, 8703}, {15685, 15721, 14093}, {15686, 15687, 35404}, {15686, 15702, 3}, {15686, 15718, 14093}, {15686, 34200, 376}, {15687, 15691, 15682}, {15687, 15692, 35400}, {15687, 15699, 15691}, {15687, 15703, 3534}, {15687, 15723, 1657}, {15687, 19709, 15723}, {15687, 35403, 38335}, {15687, 35434, 35402}, {15687, 38071, 11737}, {15688, 15693, 3}, {15689, 15699, 5054}, {15689, 15701, 3}, {15689, 15703, 15721}, {15689, 19709, 3090}, {15689, 41990, 5072}, {15690, 15708, 3}, {15690, 15713, 15698}, {15690, 45759, 3522}, {15691, 15694, 15716}, {15691, 15699, 15721}, {15691, 15721, 3}, {15691, 41987, 41988}, {15691, 41988, 3627}, {15691, 44903, 20}, {15691, 45762, 12101}, {15692, 15702, 549}, {15692, 15718, 15700}, {15692, 34200, 3}, {15692, 35400, 3534}, {15692, 35404, 15681}, {15694, 15703, 2}, {15694, 15723, 3526}, {15694, 19709, 5071}, {15694, 35401, 15683}, {15695, 15707, 3}, {15695, 15714, 14093}, {15695, 46219, 15707}, {15696, 15720, 3}, {15697, 15705, 3528}, {15697, 15719, 15759}, {15697, 46333, 12103}, {15698, 15708, 3530}, {15698, 15713, 15722}, {15698, 45759, 3}, {15699, 15716, 3526}, {15699, 17504, 45758}, {15699, 23046, 41987}, {15699, 41990, 12811}, {15699, 44903, 549}, {15699, 45762, 3543}, {15700, 15703, 3526}, {15700, 15723, 5054}, {15701, 15716, 15693}, {15701, 19709, 10109}, {15701, 44903, 14093}, {15702, 34200, 15718}, {15702, 35408, 15695}, {15703, 15718, 10124}, {15703, 35400, 15692}, {15703, 35402, 15696}, {15703, 35403, 15681}, {15703, 35434, 15688}, {15704, 15713, 45759}, {15704, 45759, 15690}, {15705, 15719, 15712}, {15705, 15759, 3}, {15706, 15720, 15693}, {15707, 15715, 15700}, {15707, 17800, 15695}, {15707, 19709, 5056}, {15708, 33703, 15690}, {15709, 17578, 19710}, {15709, 19708, 3523}, {15709, 19710, 3}, {15710, 15711, 3}, {15710, 15717, 15711}, {15710, 46936, 11540}, {15711, 33923, 15710}, {15711, 41983, 15717}, {15712, 15759, 15705}, {15712, 16239, 10303}, {15712, 45757, 2}, {15713, 45759, 3530}, {15714, 15715, 3}, {15716, 15723, 15721}, {15716, 38335, 5073}, {15717, 17538, 33923}, {15717, 33923, 3}, {15717, 46936, 3533}, {15719, 41989, 5055}, {15719, 46333, 3528}, {15721, 35401, 35410}, {15721, 41987, 35403}, {15721, 41988, 15684}, {15723, 35434, 3534}, {15759, 41989, 45757}, {15759, 45757, 16239}, {15760, 18531, 3}, {15760, 31723, 12083}, {15761, 18569, 7387}, {15765, 18585, 4}, {15765, 36457, 36437}, {15765, 42280, 2043}, {15980, 37348, 3}, {16041, 32983, 2}, {16043, 32982, 8357}, {16044, 32993, 5025}, {16045, 33180, 8364}, {16045, 33292, 33180}, {16179, 16180, 1316}, {16239, 41989, 44904}, {16241, 19107, 36967}, {16241, 36967, 11480}, {16242, 19106, 36968}, {16242, 36968, 11481}, {16252, 41362, 9833}, {16267, 41108, 61}, {16267, 41121, 42156}, {16267, 42814, 41108}, {16268, 41107, 62}, {16268, 41122, 42153}, {16268, 42813, 41107}, {16626, 22832, 16629}, {16627, 22831, 16628}, {16644, 37832, 42132}, {16644, 42093, 42154}, {16644, 42098, 37832}, {16644, 42154, 15}, {16645, 37835, 42129}, {16645, 42094, 42155}, {16645, 42095, 37835}, {16645, 42155, 16}, {16772, 42164, 42150}, {16773, 42165, 42151}, {16808, 16809, 6}, {16808, 42125, 42815}, {16808, 42128, 42962}, {16809, 42125, 42963}, {16809, 42128, 42816}, {16853, 44284, 5054}, {16868, 18559, 37943}, {16868, 18566, 37922}, {16898, 33283, 8363}, {16922, 33256, 33004}, {16923, 19696, 33014}, {16924, 33251, 2}, {16925, 32963, 33249}, {16962, 16964, 41101}, {16962, 41101, 22236}, {16963, 16965, 41100}, {16963, 41100, 22238}, {16966, 19107, 11480}, {16966, 36967, 16241}, {16966, 43227, 19107}, {16967, 19106, 11481}, {16967, 36968, 16242}, {16967, 43226, 19106}, {17504, 19708, 3}, {17504, 19710, 548}, {17504, 35018, 2}, {17504, 45761, 15692}, {17532, 17556, 2}, {17532, 44217, 6175}, {17538, 46936, 14869}, {17549, 28461, 6914}, {17556, 17577, 17528}, {17566, 37256, 19537}, {17567, 37435, 17563}, {17577, 37375, 2}, {17578, 35018, 3}, {17685, 33045, 33036}, {17800, 46219, 3}, {17928, 35502, 12084}, {18281, 23410, 6642}, {18281, 38323, 3}, {18324, 18561, 3534}, {18325, 32216, 3534}, {18350, 37472, 1147}, {18357, 22791, 8}, {18358, 21850, 69}, {18369, 45735, 7506}, {18377, 18378, 382}, {18377, 44958, 18378}, {18388, 18390, 6}, {18388, 18474, 18445}, {18390, 18418, 113}, {18390, 18474, 265}, {18391, 39542, 1159}, {18403, 46030, 7545}, {18424, 43457, 6}, {18439, 37481, 185}, {18440, 45375, 26346}, {18440, 45376, 26336}, {18445, 18451, 399}, {18451, 18474, 18440}, {18480, 18493, 18526}, {18483, 38076, 3654}, {18488, 43817, 20299}, {18493, 18525, 1}, {18504, 43865, 9704}, {18509, 18511, 3818}, {18509, 19130, 18512}, {18510, 18512, 6}, {18510, 26336, 39899}, {18511, 19130, 18510}, {18512, 26346, 39899}, {18527, 18529, 18525}, {18528, 18530, 18526}, {18531, 18535, 382}, {18531, 31723, 7574}, {18534, 18536, 1657}, {18534, 31723, 382}, {18535, 18537, 18536}, {18536, 37347, 3526}, {18538, 23259, 6199}, {18538, 42215, 3068}, {18542, 18544, 18525}, {18543, 18545, 18526}, {18559, 18566, 3830}, {18559, 37943, 24}, {18560, 37119, 3516}, {18565, 37119, 35498}, {18566, 37922, 382}, {18566, 44270, 18559}, {18568, 44275, 7576}, {18572, 44961, 23}, {18581, 42106, 5318}, {18582, 42103, 5321}, {18583, 39884, 6776}, {18584, 31489, 7603}, {18584, 44526, 31489}, {18585, 36439, 36455}, {18585, 42281, 2044}, {18586, 18587, 3}, {18762, 23249, 6395}, {18762, 42216, 3069}, {18907, 43291, 7735}, {19130, 22796, 14}, {19130, 22797, 13}, {19130, 45438, 13785}, {19130, 45439, 13665}, {19546, 37331, 19550}, {19550, 37331, 3}, {19686, 33246, 1003}, {19687, 33249, 16925}, {19697, 33186, 14069}, {19709, 35403, 15694}, {19709, 38335, 3526}, {19710, 35018, 15709}, {20030, 27868, 28237}, {20192, 45303, 44569}, {20299, 22802, 64}, {20423, 22491, 20426}, {20423, 22492, 20425}, {20425, 20426, 1351}, {20957, 25641, 38580}, {21308, 39504, 1656}, {21475, 21476, 11340}, {21735, 41984, 15701}, {21735, 44682, 3}, {21735, 46935, 140}, {21869, 21898, 33062}, {22493, 22494, 15533}, {22495, 22496, 15534}, {22605, 22606, 6290}, {22615, 42582, 6449}, {22634, 22635, 6289}, {22644, 42583, 6450}, {22753, 22758, 22765}, {22796, 22797, 3818}, {22804, 43865, 18379}, {22833, 22955, 22979}, {23046, 41106, 3}, {23046, 41990, 3627}, {23251, 42262, 372}, {23261, 42265, 371}, {23302, 42085, 42116}, {23302, 42101, 42085}, {23302, 42940, 42942}, {23302, 43104, 42911}, {23303, 42086, 42115}, {23303, 42102, 42086}, {23303, 42941, 42943}, {23303, 43101, 42910}, {23323, 46030, 4}, {24828, 24833, 24844}, {26326, 26386, 45369}, {26327, 26410, 45370}, {26328, 26466, 45610}, {26329, 26467, 45609}, {26332, 26470, 10680}, {26333, 37820, 10738}, {26336, 26346, 18440}, {27124, 27177, 2}, {27406, 27536, 33305}, {28033, 28103, 33302}, {28067, 28140, 33306}, {28263, 28382, 859}, {28431, 28721, 441}, {28447, 28448, 37955}, {29413, 29465, 46497}, {29521, 29727, 46575}, {29565, 29780, 46574}, {29998, 30070, 46514}, {30308, 38021, 9955}, {30771, 31726, 1657}, {30775, 40132, 2}, {31133, 31152, 34609}, {31133, 31180, 31181}, {31140, 31141, 3679}, {31159, 31160, 3679}, {31236, 35480, 18570}, {31489, 44526, 574}, {31664, 40894, 3}, {31665, 40895, 3}, {31693, 31694, 2}, {31693, 37351, 37352}, {31693, 37352, 37170}, {31694, 37351, 37171}, {31694, 37352, 37351}, {31723, 37347, 3}, {31824, 32156, 34792}, {31841, 40100, 38584}, {31861, 44263, 44438}, {32006, 32828, 7767}, {32369, 32379, 32402}, {32460, 46859, 2}, {32461, 46858, 2}, {32488, 33269, 7388}, {32489, 33269, 7389}, {32785, 42225, 6445}, {32786, 42226, 6446}, {32789, 42275, 6451}, {32790, 42276, 6452}, {32951, 33198, 33185}, {32961, 33255, 2}, {32962, 32996, 7791}, {32962, 33278, 2}, {32963, 33187, 2}, {32964, 33280, 33250}, {32965, 33279, 19695}, {32966, 33018, 384}, {32968, 32974, 8362}, {32971, 32980, 14064}, {32972, 32979, 14001}, {32974, 32991, 32968}, {32975, 33238, 32990}, {32976, 33239, 32989}, {32981, 32988, 32970}, {32982, 32987, 16043}, {32992, 33229, 7791}, {32994, 33264, 2}, {32997, 33009, 33001}, {32998, 33280, 32964}, {32999, 33279, 32965}, {33002, 33019, 7824}, {33004, 33010, 16922}, {33005, 33017, 2}, {33006, 33016, 2}, {33008, 33192, 8353}, {33016, 33228, 11286}, {33028, 33057, 33034}, {33029, 33056, 33033}, {33030, 33046, 33035}, {33184, 37350, 16041}, {33215, 33272, 8354}, {33216, 35927, 27088}, {33237, 33240, 2}, {33244, 33270, 33000}, {33264, 33273, 35955}, {33332, 44279, 35502}, {33532, 40916, 3}, {33923, 35417, 14893}, {33923, 41983, 15711}, {34200, 35404, 15683}, {34417, 37638, 21970}, {34484, 35500, 7488}, {34551, 34552, 3530}, {34559, 34562, 3850}, {35255, 42225, 9541}, {35382, 35400, 3526}, {35400, 35401, 3543}, {35401, 35404, 35434}, {35403, 35434, 5076}, {35404, 35408, 3146}, {35481, 37118, 11410}, {35481, 44452, 3}, {35493, 37968, 3}, {35500, 37440, 3}, {35732, 42282, 3525}, {35738, 42282, 3}, {35800, 35803, 3297}, {35801, 35802, 3298}, {35822, 35823, 6}, {35948, 35949, 8598}, {36169, 36184, 36193}, {36436, 36439, 36438}, {36436, 36454, 2}, {36436, 36455, 18587}, {36437, 36454, 18586}, {36437, 36457, 3}, {36438, 36456, 2}, {36439, 36455, 3}, {36439, 36457, 2}, {36445, 36463, 3839}, {36445, 41106, 2043}, {36448, 36466, 19709}, {36452, 36470, 16645}, {36453, 36469, 16644}, {36454, 36457, 36456}, {36463, 41106, 2044}, {36490, 36728, 36732}, {36526, 36530, 36527}, {36526, 36654, 3}, {36530, 36716, 3}, {36551, 36727, 36731}, {36557, 36561, 36558}, {36652, 36661, 3}, {36652, 36687, 5}, {36660, 36673, 5}, {36661, 36686, 4}, {36683, 36694, 5}, {36686, 36687, 3}, {36709, 37342, 3}, {36714, 37343, 3}, {36722, 36730, 36720}, {36969, 37835, 16}, {36969, 42155, 42127}, {36969, 42918, 37835}, {36970, 37832, 15}, {36970, 42154, 42126}, {36970, 42919, 37832}, {37077, 44263, 3830}, {37144, 37145, 2049}, {37162, 37433, 6903}, {37170, 37171, 2}, {37172, 37173, 439}, {37196, 37453, 186}, {37230, 37234, 382}, {37332, 37333, 3}, {37340, 37341, 14001}, {37349, 37353, 22}, {37351, 37352, 2}, {37353, 44262, 5055}, {37439, 43957, 2}, {37458, 37942, 6353}, {37460, 38282, 37935}, {37517, 43150, 40341}, {37638, 40909, 3581}, {37640, 42142, 43403}, {37640, 43403, 11542}, {37641, 42139, 43404}, {37641, 43404, 11543}, {37984, 44920, 15760}, {38071, 41099, 3}, {38071, 41991, 3860}, {38072, 41042, 42975}, {38072, 41043, 42974}, {38141, 38142, 38034}, {39504, 44263, 378}, {39530, 44924, 30258}, {39565, 39590, 32}, {40250, 40277, 3845}, {40647, 46849, 11381}, {40693, 42159, 398}, {40693, 42921, 42166}, {40694, 42162, 397}, {40694, 42920, 42163}, {41016, 41017, 4}, {41099, 41106, 2}, {41099, 41990, 15685}, {41107, 42813, 42973}, {41107, 42973, 5340}, {41108, 42814, 42972}, {41108, 42972, 5339}, {41112, 41120, 43229}, {41113, 41119, 43228}, {41121, 42972, 61}, {41122, 42973, 62}, {41748, 41750, 15534}, {41984, 44245, 44580}, {41984, 44580, 140}, {41987, 41990, 3090}, {41989, 44904, 5}, {41990, 44903, 11737}, {41990, 45762, 5071}, {42085, 42114, 23302}, {42086, 42111, 23303}, {42089, 42105, 42088}, {42092, 42104, 42087}, {42093, 42098, 15}, {42093, 42154, 36970}, {42093, 42919, 42132}, {42094, 42095, 16}, {42094, 42155, 36969}, {42094, 42918, 42129}, {42095, 42155, 16645}, {42096, 43029, 10645}, {42097, 43028, 10646}, {42098, 42154, 16644}, {42101, 42114, 42116}, {42101, 42942, 42940}, {42101, 43104, 42942}, {42102, 42111, 42115}, {42102, 42943, 42941}, {42102, 43101, 42943}, {42103, 42110, 11485}, {42106, 42107, 11486}, {42111, 42115, 42951}, {42111, 42910, 43101}, {42114, 42116, 42950}, {42114, 42911, 43104}, {42117, 42146, 11488}, {42118, 42143, 11489}, {42121, 42137, 42120}, {42124, 42136, 42119}, {42125, 42128, 6}, {42125, 42974, 42975}, {42125, 42975, 14}, {42126, 42132, 15}, {42127, 42129, 16}, {42128, 42974, 13}, {42128, 42975, 42974}, {42147, 42598, 42152}, {42148, 42599, 42149}, {42149, 42161, 42148}, {42152, 42160, 42147}, {42159, 42921, 40693}, {42162, 42920, 40694}, {42258, 42582, 5418}, {42259, 42583, 5420}, {42268, 42273, 3311}, {42269, 42270, 3312}, {42274, 42284, 6398}, {42277, 42283, 6221}, {42278, 42279, 1656}, {42280, 42281, 3091}, {42431, 42489, 5237}, {42432, 42488, 5238}, {42474, 42626, 43029}, {42475, 42625, 43028}, {42568, 43339, 6484}, {42569, 43338, 6485}, {42789, 42790, 35473}, {42791, 43107, 42945}, {42792, 43100, 42944}, {42807, 42808, 3850}, {42815, 42816, 6}, {42815, 42962, 42128}, {42815, 42963, 42816}, {42816, 42962, 42815}, {42816, 42963, 42125}, {42830, 42853, 42831}, {42894, 42895, 6}, {42894, 43030, 43031}, {42895, 43031, 43030}, {42910, 42941, 42115}, {42911, 42940, 42116}, {42936, 43632, 5352}, {42937, 43633, 5351}, {42940, 42942, 42085}, {42941, 42943, 42086}, {42942, 43104, 23302}, {42943, 43101, 23303}, {42962, 42963, 6}, {42974, 42975, 6}, {43030, 43031, 6}, {43193, 43239, 5237}, {43194, 43238, 5238}, {43209, 43255, 6456}, {43210, 43254, 6455}, {43211, 43257, 6407}, {43212, 43256, 6408}, {43821, 43831, 43845}, {43893, 46450, 37949}, {43957, 44454, 3534}, {44211, 44218, 549}, {44212, 44218, 44214}, {44236, 44452, 37118}, {44245, 44682, 21735}, {44276, 44440, 18325}, {44288, 46029, 22}, {44441, 44804, 1597}, {44580, 44682, 3524}, {45375, 45376, 18440}, {45438, 45439, 3818}, {45544, 45554, 45578}, {45545, 45555, 45579}, {45630, 45631, 18480}, {45924, 45926, 45923}, {46470, 46471, 378}


X(382) = REFLECTION OF CIRCUMCENTER IN ORTHOCENTER

Trilinears    cos A - 4 cos B cos C : cos B - 4 cos C cos A : cos C - 4 cos A cosB
Trilinears    5 cos A - 4 sin B sin C : 5 cos B - 4 sin C sin A : 5 cos C - 4 sin A sin B
Trilinears    4 sec A - sec B sec C : :
Trilinears    4 cos A - cos B cos C - 3 sin B sin C : :
Trilinears    cos A + 6 cos B cos C - 2 sin B sin C : :
Trilinears    4 cos B cos C - 2 sin B sin C + cos(B - C) : :
Trilinears    5 cos A - cos(B - C) + cos B cos C - 3 sin B sin C : :
Barycentrics  a(cos A - 4 cos B cos C) : b(cos B - 4 cos C cos A) : c(cos C - 4 cos A cos B)
Barycentrics   3 a^4 - a^2 (b^2 + c^2) - 2 (b^2 - c^2)^2 : :
X(382) = 6 X(2) - 5 X(3) = X(3) - 2 X(4)

As a point on the Euler line, X(382) has Shinagawa coefficients (1,-5).

Let A'B'C' be the reflection triangle. Let A″ be the trilinear pole, wrt A'B'C', of line BC, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(382). (Randy Hutson, January 29, 2015)

X(382) lies on these lines: 2,3   64,265   155,399   185,568   195,1498   355,516   952,962

X(382) is the {X(5),X(20)}-harmonic conjugate of X(3). For a list of other harmonic conjugates of X(382), click Tables at the top of this page.

X(382) = reflection of X(i) in X(j) for these (i,j): (3,4), (20,5), (110,1539), (550,546), (3534,381)
X(382) = inverse-in-orthocentroidal-circle of X(546)
X(382) = complement of X(3529)
X(382) = anticomplement of X(550)
X(382) = Kosnita-to-tangential similarity image of X(4)
X(382) = homothetic center of Johnson triangle and mid-triangle of medial and anticomplementary triangles
X(382) = {X(2),X(3)}-harmonic conjugate of X(15720)
X(382) = {X(3),X(4)}-harmonic conjugate of X(381)
X(382) = Ehrmann-mid-to-ABC similarity image of X(3)
X(382) = homothetic center of Ehrmann vertex-triangle and anti-Hutson intouch triangle
X(382) = homothetic center of Ehrmann side-triangle and anti-excenters-incenter reflections triangle
X(382) = homothetic center of Ehrmann mid-triangle and ABC-X3 reflections triangle
X(382) = {X(4),X(24)}-harmonic conjugate of X(37197)
X(382) = {X(2),X(3529)}-harmonic conjugate of X(550)


X(383) = EULER LINE INTERCEPT OF LINE X(14)X(98)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where
                        f(A,B,C) = csc(B - C) [sin 2B cos(C - ω) sin(C + π/3) - sin 2C cos(B - ω) sin(B + π/3)]

Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(383) has Shinagawa coefficients (S, -31/2(E + F)).

X(383) lies on these lines: 2,3   13,262   14,98   183,621   299,511   325,622

X(383) = reflection of X(1080) in X(1513)
X(383) = inverse-in-orthocentroidal-circle of X(1080)
X(383) = orthoptic-circle-of-Steiner-inellipse-inverse of X(32460)


X(384) = THE CONWAY POINT: EULER LINE INTERCEPT OF LINE X(32)X(76)

Trilinears     bc(a4 + b2c2) : ca(b4 + c2a2) : ab(c4 + a2b2)
Barycentrics    a4 + b2c2 : b4 + c2a2 : c4 + a2b2        (Peter Moses, April 15, 2020)
X(384) = 3 csc2(ω) X(2) - 4 X(3) = 3 X(2) - 4 sin2(ω) X(3)       
X(384) = 3 X[2] + X[6658], 2 X[5007] - 3 X[12150], X[6655] - 6 X[6661], 2 X[6656] + X[6658], X[6656] - 3 X[6661], X[6658] + 6 X[6661], X[7760] - 3 X[12150], 2 X[7765] - 3 X[7827], 3 X[7827] - 4 X[7829], 4 X[7849] - 3 X[7883], 2 X[7873] - 3 X[7883], 3 X[22521] - 4 X[32134] and many others

As a point on the Euler line, X(384) has Shinagawa coefficients (E + F)2 - S2, 2S2).

Contributed by John Horton Conway, email, 1998.

Let A'B'C' be the 1st Brocard triangle. Let A″ be the cevapoint of B' and C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(384). (Randy Hutson, December 11, 2015)

Let La be the line through the 2nd Brocard circle intercepts (other than PU(1)) of lines AP(1) and AU(1); define Lb and Lc cyclically. Let A' = Lb∩Lc, B' = Lc∩La, C' = La∩Lb. The triangle A'B'C' is here introduced as the 6th Brocard triangle; A'B'C' is homothetic to the 1st Brocard triangle at X(384), homothetic to the 1st anti-Brocard triangle at X(3), and perspective to the 3rd Brocard triangle at X(384). The 2nd Brocard circle of ABC is the 1st Lemoine circle of A'B'C'. (Randy Hutson, December 26, 2015)

Let A'B'C' be the 1st Brocard-reflected triangle. Let A″ be the isogonal conjugate of A', and define B″ and C″ cyclically. The lines A'A″, B'B″, C'C″ concur in X(384). (Randy Hutson, June 7, 2019)

X(384) lies on these lines these conics: {A,B,C,X(4),X(384),X(1031),X(2998),X(3114),X(9230),X(14970)}} and {A,B,C,X(25),X(384),X(733),X(1915),X(3224),X(3407), X(11380),X(14370}}

X(384) lies on these cubics: K020, K322, K421, K532, K828

X(384) lies on the cubics K020, K322, K421, K532, K828, and these lines: {1, 335}, {2, 3}, {6, 194}, {11, 26686}, {12, 26629}, {31, 17033}, {32, 76}, {35, 27020}, {36, 26959}, {37, 20136}, {39, 83}, {41, 24514}, {58, 17034}, {61, 12204}, {62, 12205}, {69, 5017}, {75, 4426}, {98, 6248}, {100, 26752}, {110, 35399}, {112, 1235}, {115, 5152}, {141, 1031}, {146, 38653}, {147, 38654}, {148, 5254}, {149, 38655}, {150, 38656}, {153, 38657}, {171, 41240}, {172, 350}, {182, 10131}, {183, 3053}, {185, 287}, {187, 1078}, {192, 54416}, {193, 32830}, {217, 39355}, {218, 17350}, {230, 44530}, {238, 16827}, {239, 5247}, {251, 1241}, {257, 3496}, {262, 9737}, {264, 1968}, {274, 17000}, {302, 53441}, {303, 53429}, {308, 38834}, {315, 3314}, {316, 626}, {325, 7745}, {330, 16502}, {511, 10333}, {524, 34604}, {538, 5007}, {543, 7765}, {574, 7782}, {575, 5182}, {576, 22486}, {595, 40859}, {597, 8591}, {598, 7775}, {599, 9939}, {609, 3760}, {620, 1506}, {625, 7874}, {671, 7817}, {683, 57260}, {694, 695}, {710, 9230}, {726, 49545}, {730, 12194}, {732, 12206}, {754, 7768}, {894, 41239}, {904, 6196}, {910, 25994}, {925, 27367}, {958, 20172}, {993, 17030}, {1001, 20133}, {1089, 30167}, {1107, 20179}, {1153, 51237}, {1194, 16276}, {1285, 63046}, {1330, 54365}, {1352, 9863}, {1376, 26687}, {1468, 17027}, {1495, 35301}, {1503, 46311}, {1580, 24259}, {1627, 39998}, {1632, 29959}, {1655, 5276}, {1691, 24256}, {1724, 20142}, {1799, 8891}, {1843, 9229}, {1909, 1914}, {1915, 4074}, {1941, 62595}, {1971, 59530}, {2080, 49111}, {2134, 17398}, {2178, 26107}, {2220, 3770}, {2221, 3210}, {2241, 53680}, {2251, 4721}, {2482, 9698}, {2548, 7763}, {2549, 7803}, {2782, 3398}, {2794, 35385}, {2887, 30175}, {2975, 26801}, {2996, 62887}, {3051, 3499}, {3094, 10345}, {3095, 10334}, {3096, 7761}, {3117, 4159}, {3118, 10337}, {3172, 9308}, {3448, 38661}, {3491, 17970}, {3492, 19558}, {3493, 51244}, {3494, 3502}, {3495, 3503}, {3500, 3501}, {3583, 30103}, {3585, 30104}, {3589, 5116}, {3618, 7738}, {3642, 9988}, {3643, 9989}, {3662, 24549}, {3666, 41258}, {3670, 30111}, {3729, 54329}, {3761, 7031}, {3767, 7806}, {3785, 16990}, {3788, 5475}, {3797, 7283}, {3815, 44532}, {3818, 9873}, {3849, 7849}, {3923, 16822}, {3926, 7774}, {3933, 7762}, {4045, 7756}, {4116, 4172}, {4173, 61101}, {4251, 17499}, {4257, 29455}, {4279, 24267}, {4339, 17316}, {4352, 17379}, {4372, 33931}, {4376, 33930}, {4386, 6376}, {4427, 25248}, {4577, 14885}, {4590, 61497}, {4676, 40861}, {4680, 30149}, {5008, 7805}, {5013, 11174}, {5015, 30179}, {5023, 15271}, {5026, 5038}, {5039, 32451}, {5041, 32450}, {5088, 7187}, {5108, 40871}, {5132, 20148}, {5171, 22712}, {5172, 28771}, {5206, 7771}, {5248, 27255}, {5251, 16819}, {5255, 17752}, {5258, 16829}, {5259, 31996}, {5264, 30114}, {5275, 27269}, {5277, 16999}, {5280, 25264}, {5286, 7920}, {5291, 17143}, {5304, 6392}, {5305, 47286}, {5306, 19570}, {5309, 7856}, {5395, 32831}, {5403, 38720}, {5404, 38721}, {5480, 47619}, {5651, 35275}, {5687, 53675}, {5902, 30139}, {5903, 30136}, {5969, 12191}, {5976, 34870}, {6162, 40865}, {6194, 39656}, {6228, 9992}, {6229, 9991}, {6272, 10793}, {6273, 10792}, {6284, 26590}, {6287, 38741}, {6292, 6781}, {6308, 42006}, {6337, 7736}, {6390, 53489}, {6683, 32456}, {7354, 26561}, {7608, 54872}, {7618, 31450}, {7697, 10104}, {7709, 10359}, {7746, 7857}, {7748, 7790}, {7749, 17006}, {7753, 7764}, {7754, 7766}, {7755, 14568}, {7757, 7772}, {7758, 7837}, {7759, 7796}, {7767, 63044}, {7773, 7778}, {7776, 7881}, {7780, 9466}, {7784, 7868}, {7788, 7946}, {7798, 7894}, {7800, 7904}, {7809, 7821}, {7810, 51224}, {7811, 7854}, {7813, 7838}, {7818, 7860}, {7825, 7867}, {7842, 7853}, {7844, 7942}, {7845, 7895}, {7848, 32027}, {7850, 7896}, {7851, 7932}, {7852, 7861}, {7855, 7877}, {7862, 7940}, {7865, 7936}, {7871, 7903}, {7872, 7913}, {7879, 7929}, {7884, 7902}, {7886, 14061}, {7910, 7914}, {7916, 7949}, {7976, 10800}, {8177, 59232}, {8267, 34482}, {8289, 53765}, {8623, 60707}, {8667, 22331}, {8782, 10346}, {8790, 19573}, {8861, 51250}, {8864, 51248}, {8865, 51251}, {8868, 8875}, {8870, 14509}, {8992, 13885}, {9166, 47617}, {9300, 59546}, {9301, 32521}, {9484, 39938}, {9605, 31859}, {9606, 52695}, {9766, 32821}, {9902, 10789}, {9917, 10790}, {10027, 37588}, {10063, 10801}, {10079, 10802}, {10159, 31168}, {10311, 54412}, {10312, 44146}, {10313, 26214}, {10347, 46283}, {10353, 13188}, {10548, 63063}, {10754, 44499}, {10788, 12251}, {10791, 12782}, {10794, 12923}, {10795, 12933}, {10797, 12837}, {10798, 12836}, {10799, 13077}, {10803, 13109}, {10804, 13110}, {10958, 28925}, {10992, 25555}, {11164, 32480}, {11272, 33813}, {11364, 12263}, {11380, 12143}, {11490, 12338}, {11638, 19663}, {11837, 12474}, {11838, 12475}, {11839, 12794}, {11840, 12992}, {11841, 12993}, {11842, 13108}, {12039, 48539}, {12122, 14810}, {12154, 42991}, {12155, 42990}, {12156, 41750}, {12195, 14839}, {12835, 18982}, {13030, 13033}, {13032, 13034}, {13111, 47618}, {13219, 38663}, {13232, 13237}, {13330, 39099}, {13571, 32820}, {13684, 13686}, {13804, 13806}, {13938, 13983}, {14135, 34236}, {14360, 38662}, {14377, 24190}, {14382, 48452}, {14516, 40867}, {14535, 45017}, {14604, 44160}, {14630, 51492}, {14631, 51493}, {14762, 55801}, {14822, 20027}, {14880, 26316}, {14881, 18502}, {14962, 27375}, {14965, 63069}, {14994, 41413}, {15048, 20094}, {15482, 15515}, {15483, 60072}, {15484, 63021}, {15513, 31239}, {16080, 54551}, {16275, 21248}, {16500, 20144}, {16589, 16993}, {16689, 18091}, {16783, 20147}, {16946, 34283}, {16991, 26085}, {16997, 18135}, {16998, 34284}, {17004, 32832}, {17008, 32828}, {17023, 53590}, {17103, 24512}, {17729, 24170}, {18092, 41328}, {18393, 30120}, {18501, 48673}, {18755, 37678}, {18800, 33749}, {18899, 33786}, {18993, 19089}, {18994, 19090}, {19130, 52995}, {19568, 42037}, {19761, 19768}, {20105, 22253}, {20112, 51238}, {20180, 20963}, {20344, 38658}, {20992, 56332}, {21001, 53164}, {21290, 38659}, {21850, 35458}, {22398, 36511}, {22520, 22779}, {22521, 32134}, {22677, 22679}, {22687, 54297}, {22689, 54298}, {22736, 22748}, {22737, 22749}, {22760, 28934}, {24206, 32152}, {24279, 24502}, {24358, 33943}, {24726, 25364}, {24729, 40981}, {25406, 35423}, {25440, 27091}, {25957, 30174}, {26068, 32561}, {26164, 30737}, {26166, 44132}, {26379, 48515}, {26403, 48516}, {26427, 49424}, {26428, 49423}, {26429, 49082}, {26430, 49083}, {26431, 49187}, {26432, 49188}, {26558, 57288}, {26613, 34506}, {27374, 55005}, {28660, 45983}, {28677, 46227}, {29012, 35422}, {29433, 52680}, {30038, 53602}, {30135, 37571}, {30140, 37525}, {30168, 49480}, {31078, 52898}, {31128, 51999}, {31404, 32829}, {31492, 42849}, {31652, 44562}, {32445, 57275}, {32522, 63424}, {32816, 53033}, {32817, 63017}, {32822, 63045}, {32826, 43448}, {32834, 37667}, {32835, 63077}, {32836, 63093}, {32840, 63091}, {33695, 40866}, {33863, 37686}, {34186, 38660}, {34504, 52691}, {34511, 63028}, {34545, 43843}, {35060, 58500}, {35387, 40253}, {35432, 54189}, {35464, 51872}, {35766, 35866}, {35767, 35867}, {35971, 37888}, {36759, 42675}, {36760, 42674}, {38862, 39668}, {39875, 43133}, {39876, 43134}, {40107, 52994}, {41134, 62362}, {43527, 54540}, {43530, 54828}, {44162, 60694}, {44367, 59780}, {44519, 47355}, {44531, 63534}, {44586, 49252}, {44587, 49253}, {45402, 49351}, {45403, 49352}, {45504, 48768}, {45505, 48769}, {46285, 51848}, {46323, 53475}, {47287, 63633}, {49560, 49562}, {49789, 59236}, {50685, 63428}, {51427, 63569}, {51523, 58765}, {52713, 63048}, {53105, 62902}, {53109, 60231}, {54126, 60205}, {54127, 60204}, {54373, 56046}, {54824, 62899}, {54829, 60191}, {54899, 62951}, {55164, 55738}, {57259, 60601}, {63294, 63372}

X(384) = midpoint of X(i) and X(j) for these {i,j}: {2, 19686}, {6655, 6658}, {6656, 19687}, {19689, 19693}, {19696, 33256}
X(384) = reflection of X(i) in X(j) for these {i,j}: {1, 51710}, {2, 6661}, {3, 44224}, {4, 44230}, {5, 44237}, {6, 42421}, {20, 44251}, {6655, 6656}, {6656, 7819}, {6658, 19687}, {7470, 3}, {7760, 5007}, {7765, 7829}, {7768, 7794}, {7819, 19697}, {7873, 7849}, {7924, 2}, {7948, 19689}, {8357, 8364}, {11303, 37341}, {11304, 37340}, {19664, 19676}, {19688, 19678}, {19691, 19695}, {19694, 19692}, {19695, 8357}, {19696, 6658}, {24726, 25364}, {33256, 6655}, {37243, 5}, {37888, 35971}, {63038, 12150}
X(384) = isogonal conjugate of X(695)
X(384) = isotomic conjugate of X(9229)
X(384) = complement of X(6655)
X(384) = anticomplement of X(6656)
X(384) = antigonal image of X(37888)
X(384) = circumcircle-inverse of X(37896)
X(384) = orthocentroidal-circle-inverse of X(5025)
X(384) = polar-circle-inverse of X(46560)
X(384) = eigencenter of cevian triangle of X(694)
X(384) = eigencenter of anticevian triangle of X(385)
X(384) = antitomic image of X(16101)
X(384) = polar conjugate of X(37892)
X(384) = polar conjugate of X(37892)
X(384) = antigonal conjugate of X(37888)
X(384) = isogonal conjugate of the anticomplement of X(37890)
X(384) = isogonal conjugate of the complement of X(37889)
X(384) = isotomic conjugate of the anticomplement of X(37891)
X(384) = isotomic conjugate of the isogonal conjugate of X(1915)
X(384) = isogonal conjugate of the isotomic conjugate of X(9230)
X(384) = polar conjugate of the isotomic conjugate of X(37894)
X(384) = polar conjugate of the isogonal conjugate of X(37893)
X(384) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1241, 21275}, {35567, 21305}, {60125, 21270}
X(384) = X(31)-complementary conjugate of X(37895)
X(384) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 37895}, {694, 385}, {9230, 37894}, {14617, 3329}
X(384) = X(i)-cross conjugate of X(j) for these (i,j): {4074, 9230}, {6657, 36432}, {37891, 2}, {37893, 37894}
X(384) = X(i)-isoconjugate of X(j) for these (i,j): {1, 695}, {2, 9288}, {6, 9285}, {31, 9229}, {32, 9239}, {48, 37892}, {75, 51948}, {76, 9236}, {1580, 51982}, {1933, 40847}, {1966, 14946}, {1967, 54129}, {18833, 57503}
X(384) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 9229}, {3, 695}, {9, 9285}, {206, 51948}, {384, 6655}, {1249, 37892}, {6376, 9239}, {8290, 54129}, {9467, 14946}, {32664, 9288}, {35971, 523}, {37895, 2}, {39092, 51982}
X(384) = cevapoint of X(i) and X(j) for these (i,j): {1, 8866}, {3, 3499}, {4, 8863}, {32, 3492}, {39, 3491}, {76, 8790}, {83, 60602}, {194, 2896}, {695, 8874}, {1915, 37893}, {3493, 8871}, {3494, 8872}, {3496, 6196}, {3498, 60601}, {3501, 8865}, {3503, 8862}, {6657, 37891}, {8861, 8870}, {8864, 8873}, {8867, 8875}
X(384) = crosspoint of X(i) and X(j) for these (i,j): {83, 3115}, {39291, 41174}
X(384) = crosssum of X(i) and X(j) for these (i,j): {39, 3118}, {76, 24734}
X(384) = crossdifference of every pair of points on line {647, 3221}
X(384) = intersection of tangents at PU(1) to hyperbola {A,B,C,X(99),PU(1)}
X(384) = perspector of ABC and symmedial triangle of 1st Brocard triangle
X(384) = perspector of 1st and 3rd Brocard triangles
X(384) = X(4027)-of-6th-Brocard-triangle
X(384) = perspector of 1st Brocard triangle and cross-triangle of ABC and 1st Brocard triangle
X(384) = perspector of 3rd Brocard triangle and cross-triangle of ABC and 3rd Brocard triangle
X(384) = pole of Brocard axis wrt conic {X(13),X(14),X(15),X(16),X(76)}}
X(384) = homothetic center of 1st and 6th Brocard triangles
X(384) = endo-homothetic center of 1st and 6th anti-Brocard triangles
X(384) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1241, 21275}, {35567, 21305}
X(384) = X(9491)-vertex conjugate of X(19566)
X(384) = X(333)-beth conjugate of X(24610)
X(384) = X(1)-zayin conjugate of X(695)
X(384) = X(i)-Hirst inverse of X(j) for these (i,j): {2, 6660}, {4, 15145},{6, 19566},{1915, 16985},{1916, 9468}
X(384) = barycentric product X(i)*X(j) for these {i,j}: {1, 1965}, {4, 37894}, {6, 9230}, {31, 1925}, {75, 1582}, {76, 1915}, {83, 4074}, {264, 37893}, {305, 11380}, {385, 54130}, {561, 1932}, {782, 41209}, {1799, 12143}, {1916, 16985}, {1934, 51904}, {6660, 16101}, {9229, 36432}, {18896, 51320}, {19566, 22252}
barycentric quotient X(i)/X(j) for these {i,j}: {1, 9285}, {2, 9229}, {4, 37892}, {6, 695}, {31, 9288}, {32, 51948}, {75, 9239}, {385, 54129}, {560, 9236}, {694, 51982}, {1582, 1}, {1915, 6}, {1916, 40847}, {1925, 561}, {1932, 31}, {1965, 75}, {3618, 3866}, {4074, 141}, {6657, 37891}, {6660, 3505}, {9230, 76}, {9468, 14946}, {11380, 25}, {12143, 427}, {16985, 385}, {37891, 6656}, {37893, 3}, {37894, 69}, {37895, 6655}, {41209, 18828}, {41331, 57503}, {51320, 1691}, {51904, 1580}, {54130, 1916}
X(384) = {X(2),X(3)}-harmonic conjugate of X(7824)
X(384) = {X(2),X(4)}-harmonic conjugate of X(5025)
X(384) = {X(2),X(5)}-harmonic conjugate of X(32967)
X(384) = {X(2),X(20)}-harmonic conjugate of X(7791)
X(384) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3, 7824}, {2, 4, 5025}, {2, 5, 32967}, {2, 20, 7791}, {2, 21, 33047}, {2, 23, 26257}, {2, 405, 16912}, {2, 439, 3523}, {2, 452, 33029}, {2, 631, 33015}, {2, 857, 30846}, {2, 858, 30777}, {2, 1003, 13586}, {2, 1375, 31220}, {2, 2475, 33841}, {2, 2478, 33046}, {2, 3091, 32961}, {2, 3146, 32974}, {2, 3522, 32990}, {2, 3523, 33001}, {2, 3543, 33251}, {2, 3552, 3}, {2, 3832, 32972}, {2, 4189, 17684}, {2, 4195, 17688}, {2, 4209, 33825}, {2, 5025, 7901}, {2, 5046, 17669}, {2, 5056, 32998}, {2, 5059, 33025}, {2, 5068, 32988}, {2, 5189, 31107}, {2, 6655, 6656}, {2, 6656, 7948}, {2, 6658, 6655}, {2, 6919, 33053}, {2, 7791, 7876}, {2, 7807, 33245}, {2, 7819, 19694}, {2, 7876, 16897}, {2, 7892, 14043}, {2, 7901, 14047}, {2, 7907, 16923}, {2, 7933, 7866}, {2, 8370, 33013}, {2, 10303, 33003}, {2, 11106, 33040}, {2, 11159, 8597}, {2, 11319, 33816}, {2, 11321, 16911}, {2, 11361, 14041}, {2, 13586, 33273}, {2, 14001, 7892}, {2, 14002, 16055}, {2, 14031, 14035}, {2, 14033, 11361}, {2, 14034, 14042}, {2, 14035, 4}, {2, 14037, 14001}, {2, 14039, 14036}, {2, 14041, 14046}, {2, 14042, 14045}, {2, 14044, 33286}, {2, 14062, 33284}, {2, 14063, 14064}, {2, 14064, 14065}, {2, 14066, 33289}, {2, 14068, 14063}, {2, 14069, 14067}, {2, 14953, 24610}, {2, 14955, 30924}, {2, 14956, 30978}, {2, 16044, 5}, {2, 16895, 16896}, {2, 16898, 16895}, {2, 16899, 16901}, {2, 16900, 16902}, {2, 16903, 16899}, {2, 16904, 16900}, {2, 16905, 16907}, {2, 16906, 16908}, {2, 16909, 16905}, {2, 16910, 16906}, {2, 16913, 11321}, {2, 16914, 405}, {2, 16915, 16917}, {2, 16916, 16918}, {2, 16919, 16915}, {2, 16920, 16916}, {2, 16921, 16922}, {2, 16924, 16921}, {2, 16925, 7907}, {2, 16926, 16928}, {2, 16927, 16929}, {2, 16930, 16926}, {2, 16931, 16927}, {2, 16932, 16950}, {2, 16949, 16951}, {2, 16952, 16932}, {2, 16953, 16949}, {2, 16954, 16956}, {2, 16955, 16957}, {2, 16958, 16954}, {2, 16959, 16955}, {2, 17565, 17670}, {2, 17578, 33200}, {2, 17680, 17673}, {2, 17689, 19270}, {2, 17691, 33827}, {2, 17692, 21}, {2, 17693, 404}, {2, 17697, 33817}, {2, 19233, 19229}, {2, 19235, 19231}, {2, 19237, 19224}, {2, 19650, 19664}, {2, 19665, 19672}, {2, 19678, 19675}, {2, 19685, 19671}, {2, 19687, 33256}, {2, 19689, 7819}, {2, 19691, 19690}, {2, 19692, 19689}, {2, 19693, 6658}, {2, 19700, 19670}, {2, 26788, 27126}, {2, 26835, 27179}, {2, 32962, 3090}, {2, 32963, 32969}, {2, 32964, 631}, {2, 32965, 16043}, {2, 32966, 7887}, {2, 32971, 16924}, {2, 32972, 33248}, {2, 32973, 16925}, {2, 32979, 3091}, {2, 32980, 33199}, {2, 32981, 20}, {2, 32982, 33180}, {2, 32985, 33274}, {2, 32987, 32999}, {2, 32989, 33000}, {2, 32991, 5056}, {2, 32995, 32963}, {2, 32996, 33283}, {2, 33002, 1656}, {2, 33004, 11285}, {2, 33007, 7833}, {2, 33009, 5067}, {2, 33012, 32978}, {2, 33014, 33004}, {2, 33018, 32966}, {2, 33019, 7933}, {2, 33020, 32992}, {2, 33021, 8362}, {2, 33023, 33202}, {2, 33030, 442}, {2, 33056, 6856}, {2, 33058, 443}, {2, 33062, 474}, {2, 33187, 376}, {2, 33192, 33190}, {2, 33193, 32986}, {2, 33198, 16898}, {2, 33201, 32973}, {2, 33204, 3533}, {2, 33205, 32989}, {2, 33206, 3525}, {2, 33208, 33215}, {2, 33225, 7807}, {2, 33239, 33275}, {2, 33244, 32965}, {2, 33255, 33246}, {2, 33258, 32960}, {2, 33259, 140}, {2, 33260, 33021}, {2, 33261, 32975}, {2, 33262, 32977}, {2, 33264, 11287}, {2, 33265, 8356}, {2, 33266, 3524}, {2, 33269, 32968}, {2, 33270, 32976}, {2, 33277, 32955}, {2, 33278, 33223}, {2, 33283, 32951}, {2, 33336, 419}, {2, 33818, 33838}, {2, 33820, 33839}, {2, 33822, 17671}, {2, 33823, 33840}, {2, 33824, 17550}, {2, 33829, 33833}, {2, 33831, 4202}, {2, 33832, 16062}, {2, 35296, 60700}, {2, 35927, 33008}, {2, 35929, 6636}, {2, 40853, 41237}, {2, 62967, 63797}, {3, 4, 5999}, {3, 5, 37334}, {3, 1003, 3552}, {3, 3552, 13586}, {3, 7770, 2}, {3, 7824, 33273}, {3, 11285, 33004}, {3, 11286, 7770}, {3, 11356, 6656}, {3, 13586, 33276}, {3, 15013, 28723}, {3, 33235, 33014}, {3, 35930, 4}, {3, 37123, 7488}, {3, 54993, 3522}, {4, 5025, 14041}, {4, 7892, 7901}, {4, 7901, 14045}, {4, 11361, 14042}, {4, 14001, 2}, {4, 14031, 14034}, {4, 14033, 14035}, {4, 14035, 11361}, {4, 14036, 14043}, {4, 14037, 7892}, {4, 14039, 14001}, {4, 14041, 14044}, {4, 14043, 14046}, {4, 14047, 33291}, {4, 14063, 14062}, {4, 14064, 14063}, {4, 14065, 33289}, {4, 14067, 33284}, {4, 14068, 14066}, {4, 14069, 14064}, {4, 16041, 32996}, {4, 16924, 33300}, {4, 32951, 16041}, {4, 32952, 33285}, {4, 32953, 33292}, {4, 35925, 3}, {5, 631, 10486}, {5, 7807, 2}, {5, 8369, 7807}, {5, 8370, 16044}, {5, 16044, 33013}, {5, 33225, 33245}, {6, 194, 7839}, {6, 1975, 194}, {6, 4048, 12215}, {20, 7791, 7833}, {20, 7833, 33267}, {20, 16898, 7876}, {20, 32981, 33007}, {20, 33007, 33257}, {20, 33023, 33253}, {20, 33198, 2}, {20, 33202, 33023}, {20, 33257, 9855}, {21, 17686, 2}, {25, 11324, 2}, {32, 76, 385}, {32, 3734, 76}, {32, 7751, 6179}, {32, 17128, 17129}, {32, 17130, 7751}, {39, 83, 3329}, {39, 99, 7783}, {39, 7804, 83}, {39, 7816, 99}, {39, 10341, 38382}, {69, 20065, 7893}, {76, 385, 17129}, {76, 3734, 17128}, {76, 3972, 32}, {76, 6179, 7751}, {83, 99, 39}, {83, 7816, 7783}, {99, 5149, 8290}, {99, 7804, 3329}, {115, 6680, 7828}, {140, 32992, 2}, {140, 35297, 33259}, {141, 2076, 60702}, {141, 7750, 2896}, {148, 7797, 5254}, {148, 10583, 7797}, {183, 3053, 7793}, {187, 3934, 1078}, {194, 7787, 6}, {315, 3314, 7939}, {315, 7737, 7823}, {315, 7795, 3314}, {316, 626, 7885}, {316, 7820, 7931}, {316, 7832, 626}, {325, 7745, 7785}, {325, 7785, 7941}, {325, 7789, 7836}, {325, 7836, 7947}, {376, 16043, 32965}, {376, 16045, 16043}, {376, 32965, 33275}, {376, 33239, 33244}, {376, 33244, 33268}, {381, 7887, 32966}, {381, 32954, 7887}, {381, 33220, 2}, {381, 33242, 32954}, {382, 7841, 33019}, {382, 7866, 7841}, {382, 33019, 8597}, {382, 33217, 7933}, {382, 33237, 7866}, {385, 17128, 76}, {404, 17541, 2}, {405, 11321, 2}, {405, 16913, 16911}, {405, 19230, 19232}, {458, 37199, 1593}, {458, 37344, 2}, {546, 8361, 33228}, {546, 8368, 8361}, {546, 33228, 32993}, {550, 8356, 33260}, {550, 8362, 8356}, {550, 33250, 33265}, {574, 7808, 7786}, {598, 7870, 7775}, {620, 1506, 7769}, {625, 7874, 7899}, {626, 7747, 316}, {626, 7820, 7832}, {626, 7832, 7931}, {631, 32964, 33274}, {631, 32968, 2}, {631, 32985, 32964}, {694, 695, 14820}, {1003, 7770, 3}, {1003, 11285, 33235}, {1003, 11286, 2}, {1003, 11356, 10997}, {1008, 13723, 19312}, {1010, 17682, 17670}, {1010, 37035, 56985}, {1113, 1114, 37896}, {1352, 36998, 9863}, {1506, 7769, 17005}, {1656, 11288, 33233}, {1656, 33233, 2}, {1656, 44543, 33002}, {1657, 11287, 33234}, {1657, 33234, 33264}, {1915, 4074, 37894}, {1975, 7787, 7839}, {2043, 2044, 55008}, {2076, 24273, 141}, {2454, 2455, 44347}, {2479, 2480, 6660}, {2548, 7763, 7777}, {2549, 7803, 7864}, {2896, 14712, 7750}, {2896, 46226, 141}, {3090, 32970, 2}, {3090, 32983, 32962}, {3090, 33191, 32970}, {3091, 32979, 33016}, {3091, 33181, 2}, {3095, 10349, 39101}, {3096, 7761, 7928}, {3096, 7802, 7761}, {3146, 32974, 33017}, {3314, 7823, 315}, {3329, 7783, 39}, {3496, 3497, 19555}, {3496, 24291, 257}, {3500, 3501, 19591}, {3522, 32990, 33008}, {3522, 35927, 33254}, {3524, 32957, 32978}, {3524, 32978, 33012}, {3525, 32975, 2}, {3525, 33216, 33206}, {3528, 32960, 33215}, {3529, 32956, 32986}, {3529, 32986, 32997}, {3543, 32982, 33279}, {3543, 33180, 32982}, {3544, 33231, 32958}, {3545, 32969, 32963}, {3545, 33189, 32969}, {3545, 33224, 2}, {3552, 6655, 10997}, {3552, 7770, 7824}, {3552, 7824, 33276}, {3552, 33004, 33014}, {3552, 33014, 33235}, {3627, 33184, 33229}, {3627, 33185, 33184}, {3734, 3972, 385}, {3734, 7751, 17130}, {3788, 5475, 7752}, {3788, 7752, 7925}, {3832, 32972, 33006}, {3839, 33183, 33199}, {3839, 33199, 32980}, {3855, 32955, 32984}, {3855, 33197, 32955}, {3855, 33222, 33277}, {3926, 7774, 7906}, {3933, 7762, 7779}, {3933, 18907, 7762}, {4045, 7756, 7847}, {4045, 7889, 7859}, {4187, 17694, 2}, {4195, 4201, 1010}, {4195, 17691, 2}, {4195, 56990, 56986}, {4209, 17697, 2}, {4234, 17681, 16061}, {4235, 37125, 3520}, {4366, 6645, 1}, {5004, 5005, 7467}, {5007, 7760, 63038}, {5025, 7892, 2}, {5025, 7901, 14046}, {5025, 11361, 4}, {5025, 14001, 14043}, {5025, 14030, 14034}, {5025, 14032, 14035}, {5025, 14034, 11361}, {5025, 14035, 14042}, {5025, 14036, 7892}, {5025, 14038, 14001}, {5025, 14040, 14036}, {5025, 14041, 14045}, {5025, 14042, 14044}, {5025, 14043, 14047}, {5025, 14045, 33291}, {5025, 14046, 33286}, {5025, 14062, 14063}, {5025, 14063, 33289}, {5025, 14064, 33284}, {5025, 14065, 14064}, {5025, 14066, 14062}, {5025, 14067, 14065}, {5025, 16041, 33293}, {5025, 33283, 33288}, {5056, 32991, 33005}, {5056, 33203, 2}, {5059, 33025, 33272}, {5059, 33272, 33271}, {5067, 32977, 2}, {5071, 32959, 32976}, {5071, 32976, 33270}, {5071, 33236, 32959}, {5192, 33830, 2}, {5206, 7815, 7771}, {5254, 7792, 7797}, {5254, 32819, 148}, {5277, 18140, 16999}, {5286, 16989, 7920}, {5475, 7835, 7925}, {6179, 7751, 385}, {6196, 7346, 18272}, {6248, 13335, 98}, {6292, 6781, 7830}, {6292, 7830, 7831}, {6390, 53489, 63018}, {6655, 6656, 7924}, {6655, 6661, 19694}, {6655, 7819, 7948}, {6655, 10997, 7470}, {6655, 19665, 19666}, {6655, 19668, 19688}, {6655, 19678, 19664}, {6655, 19679, 19675}, {6655, 19686, 6658}, {6655, 19687, 19696}, {6655, 19689, 2}, {6655, 19690, 8357}, {6655, 19691, 19695}, {6655, 19692, 7819}, {6655, 19693, 19687}, {6656, 6658, 33256}, {6656, 6661, 7819}, {6656, 7819, 2}, {6656, 8357, 19690}, {6656, 19650, 19688}, {6656, 19668, 19664}, {6656, 19676, 19675}, {6656, 19686, 19696}, {6656, 19689, 19694}, {6656, 19695, 8357}, {6656, 19697, 19689}, {6658, 6661, 7948}, {6658, 7819, 7924}, {6658, 19676, 19688}, {6658, 19677, 19675}, {6658, 19679, 19664}, {6658, 19686, 19687}, {6658, 19689, 6656}, {6658, 19690, 19691}, {6658, 19692, 2}, {6658, 19693, 19686}, {6658, 19697, 19694}, {6661, 7819, 19689}, {6661, 19686, 7924}, {6661, 19687, 6656}, {6661, 19693, 33256}, {6661, 19697, 19692}, {6675, 33033, 2}, {6680, 7828, 16984}, {6683, 32456, 37512}, {6815, 26204, 2}, {6999, 37416, 23512}, {7399, 26205, 2}, {7737, 7795, 315}, {7745, 7789, 325}, {7745, 7836, 7941}, {7747, 7820, 626}, {7747, 7832, 7885}, {7748, 7834, 7790}, {7748, 7846, 7923}, {7751, 17130, 76}, {7752, 7835, 3788}, {7753, 7764, 7858}, {7753, 7863, 7764}, {7754, 30435, 7766}, {7756, 7889, 4045}, {7757, 7878, 7772}, {7759, 7796, 7840}, {7759, 7801, 7796}, {7760, 12150, 5007}, {7761, 7822, 3096}, {7762, 18907, 20088}, {7764, 7863, 7799}, {7765, 7829, 7827}, {7766, 20081, 7754}, {7770, 10997, 7948}, {7770, 15013, 26221}, {7770, 33235, 11285}, {7772, 7781, 7757}, {7773, 7778, 7912}, {7775, 7888, 7814}, {7776, 7881, 7897}, {7777, 7891, 7763}, {7779, 20088, 7762}, {7782, 7786, 574}, {7782, 60855, 7786}, {7784, 7868, 7938}, {7785, 7789, 7947}, {7785, 7836, 325}, {7786, 60855, 7808}, {7790, 7834, 7923}, {7790, 7846, 7834}, {7791, 16895, 16897}, {7791, 16898, 2}, {7791, 32981, 33257}, {7791, 33007, 20}, {7791, 33198, 16895}, {7791, 33253, 33023}, {7791, 33257, 33267}, {7792, 32819, 5254}, {7793, 31276, 183}, {7795, 7823, 7939}, {7796, 7812, 7759}, {7797, 10583, 7792}, {7799, 7858, 7764}, {7800, 14907, 7904}, {7801, 7812, 7840}, {7802, 7822, 7928}, {7804, 7816, 39}, {7807, 8369, 33225}, {7807, 8370, 5}, {7807, 16044, 32967}, {7809, 7909, 7821}, {7813, 7838, 7905}, {7814, 7870, 7888}, {7818, 7869, 7922}, {7819, 8357, 8364}, {7819, 10997, 7824}, {7819, 19668, 19675}, {7819, 19686, 33256}, {7819, 19687, 6655}, {7819, 19693, 19696}, {7819, 19697, 6661}, {7821, 7843, 7809}, {7821, 7880, 7909}, {7821, 14537, 7843}, {7824, 13586, 3}, {7825, 7867, 7934}, {7833, 7876, 7791}, {7833, 16895, 7876}, {7833, 16898, 16897}, {7833, 33007, 9855}, {7833, 33198, 16896}, {7833, 33257, 20}, {7841, 7866, 7933}, {7841, 33217, 7866}, {7841, 33237, 2}, {7842, 7853, 7911}, {7842, 7915, 7853}, {7843, 7880, 7821}, {7845, 7895, 7917}, {7847, 7859, 4045}, {7849, 7873, 7883}, {7852, 7861, 7919}, {7853, 7915, 7944}, {7859, 7889, 16987}, {7860, 7922, 7818}, {7864, 7875, 7803}, {7866, 11159, 382}, {7866, 33217, 2}, {7866, 33237, 33217}, {7867, 62203, 7825}, {7871, 7926, 7903}, {7872, 7913, 7918}, {7874, 39590, 625}, {7876, 16895, 2}, {7876, 16898, 16896}, {7876, 32981, 9855}, {7876, 33007, 33267}, {7876, 33257, 7833}, {7880, 14537, 7809}, {7885, 7931, 626}, {7886, 39565, 14061}, {7887, 32954, 2}, {7887, 33220, 32954}, {7892, 11361, 5025}, {7892, 14030, 14035}, {7892, 14032, 11361}, {7892, 14033, 14042}, {7892, 14034, 4}, {7892, 14035, 14041}, {7892, 14036, 14001}, {7892, 14038, 14036}, {7892, 14040, 14037}, {7892, 14041, 14047}, {7892, 14042, 14046}, {7892, 14062, 14065}, {7892, 14065, 14067}, {7892, 14066, 14064}, {7892, 14067, 14069}, {7892, 14068, 33284}, {7892, 33300, 16922}, {7897, 7900, 7776}, {7898, 7938, 7784}, {7901, 11361, 14044}, {7901, 14041, 5025}, {7901, 14042, 14041}, {7901, 14043, 2}, {7901, 14044, 33291}, {7901, 14045, 33286}, {7901, 33284, 14064}, {7901, 33289, 33284}, {7901, 33293, 33283}, {7903, 7908, 7871}, {7904, 16986, 7800}, {7906, 7921, 7774}, {7907, 16921, 2}, {7907, 16924, 16922}, {7907, 33246, 16925}, {7907, 33300, 7901}, {7910, 7937, 7935}, {7911, 7944, 7853}, {7912, 7945, 7778}, {7914, 7935, 7937}, {7918, 7943, 7913}, {7924, 7948, 6656}, {7924, 19694, 7948}, {7924, 19696, 33256}, {7924, 33256, 6655}, {7930, 7934, 7867}, {7933, 33019, 7841}, {7941, 7947, 325}, {7948, 19694, 2}, {7948, 19696, 6655}, {7948, 33256, 7924}, {8356, 8362, 33021}, {8356, 33250, 550}, {8357, 8364, 6656}, {8357, 19690, 7924}, {8357, 19695, 6655}, {8357, 19697, 19702}, {8357, 19702, 2}, {8362, 33250, 33260}, {8363, 33185, 2}, {8363, 33229, 33184}, {8364, 19687, 19691}, {8364, 19691, 7924}, {8364, 19695, 19690}, {8366, 11317, 11318}, {8366, 11318, 2}, {8368, 33228, 2}, {8369, 8370, 2}, {8369, 16044, 33245}, {8370, 33225, 32967}, {8370, 35954, 8369}, {9466, 35007, 7780}, {9737, 10358, 262}, {9855, 16896, 7791}, {9855, 33267, 20}, {10337, 10344, 10338}, {10340, 52083, 10339}, {10997, 11356, 7924}, {10998, 37336, 37334}, {10999, 11000, 6656}, {11001, 33247, 33209}, {11057, 47005, 7865}, {11104, 37027, 1009}, {11108, 33035, 2}, {11112, 17540, 17670}, {11112, 17670, 17565}, {11114, 17550, 33824}, {11159, 33217, 33019}, {11159, 33237, 7841}, {11164, 47352, 32480}, {11285, 33004, 7824}, {11285, 33235, 3}, {11288, 44543, 2}, {11293, 11294, 4}, {11299, 11303, 37333}, {11300, 11304, 37332}, {11321, 16914, 16912}, {11321, 19230, 19231}, {11328, 11335, 11338}, {11328, 11338, 2}, {11339, 16405, 2}, {11343, 41236, 2}, {11343, 50060, 41236}, {11353, 16412, 2}, {11361, 14001, 7901}, {11361, 14030, 14033}, {11361, 14032, 14034}, {11361, 14034, 14035}, {11361, 14036, 2}, {11361, 14037, 14043}, {11361, 14038, 7892}, {11361, 14040, 14001}, {11361, 14043, 14045}, {11361, 14062, 14066}, {11361, 14065, 14062}, {11361, 14066, 14068}, {11361, 14067, 14063}, {11361, 14069, 33289}, {13735, 17682, 33821}, {13740, 16060, 2}, {13740, 35916, 37148}, {13741, 33828, 2}, {14001, 14031, 11361}, {14001, 14032, 14042}, {14001, 14033, 4}, {14001, 14034, 14041}, {14001, 14035, 5025}, {14001, 14037, 14036}, {14001, 14039, 14037}, {14001, 14042, 14047}, {14001, 14063, 14067}, {14001, 14064, 14069}, {14001, 14068, 14065}, {14030, 14032, 14031}, {14030, 14034, 14032}, {14030, 14036, 11361}, {14030, 14037, 14042}, {14030, 14038, 4}, {14030, 14039, 14041}, {14030, 14040, 5025}, {14031, 14033, 14032}, {14031, 14035, 14033}, {14031, 14036, 14042}, {14031, 14037, 4}, {14031, 14038, 14041}, {14031, 14039, 5025}, {14031, 14040, 7901}, {14032, 14034, 14033}, {14032, 14036, 4}, {14032, 14037, 14041}, {14032, 14038, 5025}, {14032, 14039, 7901}, {14032, 14040, 2}, {14033, 14035, 14034}, {14033, 14036, 14041}, {14033, 14037, 5025}, {14033, 14038, 7901}, {14033, 14039, 2}, {14033, 14040, 14043}, {14033, 14069, 14068}, {14034, 14036, 5025}, {14034, 14037, 7901}, {14034, 14038, 2}, {14034, 14039, 14043}, {14034, 14040, 7892}, {14034, 14065, 14068}, {14034, 14067, 14066}, {14035, 14036, 7901}, {14035, 14037, 2}, {14035, 14038, 14043}, {14035, 14039, 7892}, {14035, 14043, 14044}, {14035, 14063, 14068}, {14035, 14064, 14066}, {14035, 14069, 14062}, {14036, 14038, 14037}, {14036, 14040, 14038}, {14036, 14062, 14069}, {14036, 14066, 14067}, {14037, 14039, 14038}, {14037, 14068, 14069}, {14038, 14040, 14039}, {14041, 14042, 4}, {14041, 14043, 7901}, {14041, 14046, 33291}, {14041, 14047, 33286}, {14041, 33284, 33289}, {14041, 33288, 16041}, {14041, 33289, 14063}, {14042, 14043, 5025}, {14042, 33284, 14062}, {14043, 33289, 14065}, {14044, 14045, 14041}, {14044, 14046, 14045}, {14044, 14047, 5025}, {14045, 14046, 5025}, {14045, 14047, 14046}, {14046, 14047, 7901}, {14061, 15031, 39565}, {14062, 14063, 14041}, {14062, 14064, 33289}, {14062, 14065, 5025}, {14062, 14066, 4}, {14062, 14067, 14064}, {14062, 14069, 7901}, {14062, 33284, 14045}, {14063, 14064, 5025}, {14063, 14065, 33284}, {14063, 14067, 7901}, {14063, 14068, 4}, {14063, 14069, 14065}, {14063, 33287, 33292}, {14063, 33289, 14045}, {14064, 14065, 7901}, {14064, 14066, 14041}, {14064, 14068, 14062}, {14064, 14069, 2}, {14064, 33284, 14046}, {14064, 33292, 33287}, {14065, 14066, 14063}, {14065, 14067, 2}, {14065, 14068, 14041}, {14065, 33289, 14046}, {14066, 14067, 5025}, {14066, 14068, 14042}, {14066, 14069, 33284}, {14066, 33289, 14044}, {14067, 14068, 33289}, {14067, 14069, 14043}, {14067, 33284, 14047}, {14068, 14069, 5025}, {14068, 33284, 14044}, {14069, 33289, 14047}, {14069, 33292, 32953}, {14712, 46226, 2896}, {14784, 14785, 37242}, {15682, 33223, 33278}, {16041, 32951, 33283}, {16041, 33283, 5025}, {16043, 16045, 2}, {16043, 33187, 33268}, {16043, 33239, 376}, {16043, 33244, 33275}, {16044, 33225, 2}, {16045, 33187, 33275}, {16045, 33239, 32965}, {16050, 50200, 2}, {16061, 17681, 2}, {16845, 33026, 2}, {16895, 32981, 33267}, {16895, 33257, 7791}, {16896, 16897, 2}, {16897, 33267, 7791}, {16898, 32981, 7833}, {16898, 33007, 7791}, {16899, 16900, 2}, {16899, 16904, 16902}, {16900, 16903, 16901}, {16901, 16902, 2}, {16903, 16904, 2}, {16905, 16906, 2}, {16905, 16910, 16908}, {16906, 16909, 16907}, {16907, 16908, 2}, {16909, 16910, 2}, {16911, 16912, 2}, {16913, 16914, 2}, {16914, 19235, 19232}, {16915, 16916, 2}, {16915, 16920, 16918}, {16916, 16919, 16917}, {16917, 16918, 2}, {16919, 16920, 2}, {16921, 16925, 16923}, {16921, 33246, 7907}, {16922, 16923, 2}, {16924, 16925, 2}, {16924, 32973, 7907}, {16924, 32999, 32987}, {16924, 33000, 32999}, {16924, 33201, 33246}, {16924, 33246, 16923}, {16924, 33255, 16925}, {16925, 32971, 16921}, {16925, 32973, 33246}, {16925, 32999, 33000}, {16925, 33000, 32989}, {16925, 33255, 32973}, {16926, 16927, 2}, {16926, 16931, 16929}, {16927, 16930, 16928}, {16928, 16929, 2}, {16930, 16931, 2}, {16932, 16949, 2}, {16932, 16953, 16951}, {16949, 16952, 16950}, {16950, 16951, 2}, {16952, 16953, 2}, {16954, 16955, 2}, {16954, 16959, 16957}, {16955, 16958, 16956}, {16956, 16957, 2}, {16958, 16959, 2}, {17538, 33226, 33207}, {17540, 17670, 2}, {17559, 33043, 2}, {17579, 33840, 33823}, {17682, 33821, 2}, {17686, 17692, 33047}, {17688, 33827, 2}, {17928, 26226, 2}, {18502, 35002, 14881}, {19226, 19234, 16912}, {19226, 19237, 19230}, {19227, 19228, 2}, {19227, 19230, 19224}, {19228, 19231, 19235}, {19229, 19233, 19224}, {19229, 19237, 19233}, {19231, 19237, 16912}, {19232, 19237, 405}, {19650, 19669, 19666}, {19650, 19671, 19685}, {19650, 19677, 19676}, {19650, 19678, 19668}, {19650, 19679, 19678}, {19650, 19689, 19675}, {19659, 19660, 8362}, {19665, 19670, 19675}, {19668, 19673, 19672}, {19668, 19676, 19678}, {19669, 19670, 2}, {19669, 19678, 19672}, {19670, 19673, 19694}, {19670, 19674, 19672}, {19671, 19685, 19664}, {19674, 19678, 19665}, {19674, 19700, 19675}, {19677, 19678, 19679}, {19677, 19686, 19664}, {19678, 19679, 19676}, {19679, 19687, 19688}, {19686, 19689, 6655}, {19686, 19692, 6656}, {19686, 19697, 7948}, {19687, 19689, 7924}, {19687, 19692, 7948}, {19687, 19697, 2}, {19687, 19702, 8357}, {19687, 44224, 10997}, {19689, 19691, 8364}, {19689, 19692, 6661}, {19690, 19691, 6655}, {19691, 19695, 33256}, {19691, 19702, 7948}, {19692, 19693, 6655}, {19693, 19697, 7924}, {19694, 19696, 7924}, {19694, 33256, 6656}, {19695, 19702, 8364}, {20094, 63020, 15048}, {25465, 25678, 2}, {26221, 35928, 7824}, {26996, 27057, 2}, {27281, 30000, 30846}, {27584, 27723, 30447}, {28723, 35928, 3}, {29415, 46499, 29729}, {29467, 46498, 29782}, {29523, 29567, 46497}, {31404, 32829, 63083}, {32476, 56789, 39}, {32951, 32996, 5025}, {32951, 33293, 14046}, {32953, 33292, 14064}, {32954, 33242, 33220}, {32955, 32984, 33277}, {32955, 33197, 33222}, {32955, 33222, 2}, {32957, 32978, 2}, {32959, 32976, 2}, {32960, 33215, 33258}, {32961, 33016, 3091}, {32963, 32995, 3545}, {32964, 32968, 33015}, {32964, 33269, 2}, {32965, 33187, 33244}, {32965, 33239, 33268}, {32965, 33244, 376}, {32966, 33018, 381}, {32967, 33013, 5}, {32967, 33245, 2}, {32968, 32985, 631}, {32969, 33189, 2}, {32969, 33224, 33189}, {32970, 32983, 3090}, {32971, 32973, 2}, {32971, 32989, 32987}, {32971, 33201, 16925}, {32971, 33205, 32999}, {32971, 33246, 16922}, {32971, 33255, 7907}, {32973, 32987, 32989}, {32973, 32989, 33205}, {32973, 33201, 33255}, {32975, 33216, 3525}, {32979, 33181, 32961}, {32981, 33198, 7791}, {32982, 33180, 33251}, {32983, 33191, 2}, {32984, 33197, 2}, {32984, 33222, 32955}, {32985, 33269, 33015}, {32985, 35951, 3552}, {32987, 32989, 2}, {32987, 32999, 16921}, {32987, 33205, 33000}, {32989, 33000, 7907}, {32989, 33205, 16925}, {32990, 35927, 3522}, {32991, 33203, 32998}, {32992, 35297, 140}, {32996, 33283, 16041}, {32996, 33288, 14045}, {32997, 33193, 3529}, {32998, 33005, 5056}, {32999, 33000, 2}, {32999, 33205, 7907}, {32999, 33255, 33205}, {33004, 33014, 3}, {33006, 33248, 32972}, {33007, 33198, 7876}, {33008, 33254, 3522}, {33009, 33262, 2}, {33013, 33245, 32967}, {33013, 35950, 33276}, {33014, 33235, 13586}, {33015, 33274, 631}, {33017, 33280, 3146}, {33020, 33259, 2}, {33021, 33260, 8356}, {33021, 33265, 33260}, {33023, 33024, 37336}, {33023, 33202, 7791}, {33023, 33253, 7833}, {33183, 33199, 2}, {33184, 33185, 8363}, {33187, 33244, 33239}, {33190, 33703, 33238}, {33198, 33257, 16897}, {33206, 33261, 2}, {33207, 33214, 17538}, {33209, 33263, 33247}, {33221, 33238, 33190}, {33233, 44543, 1656}, {33238, 33703, 33192}, {33251, 33279, 32982}, {33260, 33265, 550}, {33268, 33275, 376}, {33273, 33276, 3}, {33284, 33289, 5025}, {33285, 33290, 5025}, {33286, 33291, 5025}, {33287, 33292, 5025}, {33288, 33293, 5025}, {33816, 33826, 2}, {33817, 33825, 2}, {35925, 35930, 5999}, {35938, 35939, 35925}, {35952, 37186, 3}, {36156, 36165, 23}, {36432, 37891, 37895}, {37348, 37466, 37446}, {40858, 52083, 3051}, {41231, 52275, 2}, {42789, 42790, 35476}, {51876, 51878, 2076}, {57013, 57014, 15145}


X(385) = X(2)-HIRST INVERSE OF X(6)

Trilinears    bc(a4 - b2c2) : :
Barycentrics    cot B + cot C - cot A - tan ω : :

Contributed by John Horton Conway, 1998.

X(385) is the perspector of triangle ABC and the tangential triangle of the conic that passes through A, B, C, and the pair P(1) and U(1) of bicentric points (see the notes just before X(1908). (Randy Hutson, 9/23/2011)

Let A'B'C' be the 1st Brocard triangle. Let A″ be the trilinear pole of line B'C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(385). (Randy Hutson, November 30, 2015)

Let La be the line through A parallel to the Lemoine axis, and define Lb and Lc cyclically. Let Ma be the reflection of BC in La, and define Mb and Mc cyclically. Let A' = Mb∩Mc, and define B', C' cyclically. Triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. Let A″B″C″ be the reflection of A'B'C' in the Lemoine axis. The triangle A″B″C″ is homothetic to ABC, with center of homothety X(325) and centroid X(385); see Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, November 30, 2015)

X(385) lies on the anti-Brocard circle, the anti-McCay circumcircle, and these lines: {1, 257}, {2, 6}, {3, 194}, {4, 7823}, {5, 7762}, {15, 5980}, {16, 5981}, {20, 6392}, {21, 1655}, {22, 3164}, {23, 523}, {25, 2998}, {30, 148}, {32, 76}, {39, 1078}, {41, 17033}, {55, 192}, {56, 330}, {58, 17499}, {61, 22702}, {62, 22701}, {75, 4386}, {83, 3934}, {98, 511}, {99, 187}, {100, 17759}, {110, 9418}, {111, 892}, {112, 2868}, {114, 5965}, {115, 316}, {140, 13571}, {147, 1513}, {171, 894}, {172, 1909}, {182, 22712}, {190, 17735}, {232, 648}, {237, 3511}, {238, 17793}, {239, 1429}, {248, 290}, {251, 308}, {262, 576}, {264, 10311}, {274, 5277}, {287, 8779}, {297, 9473}, {310, 16956}, {315, 3767}, {335, 3509}, {339, 10317}, {340, 6103}, {350, 1914}, {383, 20426}, {419, 16985}, {512, 9879}, {513, 5990}, {514, 5991}, {518, 5985}, {532, 5979}, {533, 5978}, {543, 6781}, {546, 13111}, {574, 7757}, {575, 15819}, {609, 3761}, {614, 18194}, {620, 7799}, {623, 22511}, {624, 22510}, {625, 7809}, {626, 7755}, {664, 17966}, {668, 5291}, {671, 3849}, {694, 8840}, {698, 2076}, {706, 14871}, {710, 8783}, {730, 11364}, {732, 1691}, {740, 1281}, {858, 16315}, {895, 9769}, {903, 17969}, {1003, 1384}, {1080, 20425}, {1154, 15093}, {1160, 10845}, {1161, 10846}, {1194, 1799}, {1204, 9289}, {1235, 10312}, {1284, 17493}, {1285, 14033}, {1333, 3770}, {1351, 13860}, {1352, 9753}, {1383, 9462}, {1423, 2319}, {1492, 18262}, {1503, 5984}, {1506, 7838}, {1580, 4039}, {1627, 8024}, {1692, 10352}, {1757, 4518}, {1911, 3510}, {1975, 3053}, {2021, 15483}, {2023, 5111}, {2030, 5182}, {2080, 2782}, {2280, 17027}, {2329, 17752}, {2373, 9091}, {2393, 5986}, {2452, 9832}, {2459, 8317}, {2460, 8316}, {2482, 14148}, {2548, 7921}, {2549, 7833}, {2793, 10787}, {2854, 5987}, {2896, 5305}, {2975, 21226}, {2996, 3146}, {3091, 9748}, {3095, 10104}, {3096, 5346}, {3225, 3229}, {3226, 17962}, {3266, 4590}, {3304, 19318}, {3506, 19558}, {3507, 7166}, {3508, 7167}, {3550, 3729}, {3705, 17363}, {3747, 19580}, {3750, 17319}, {3760, 7031}, {3785, 5286}, {3788, 7796}, {3818, 9993}, {3830, 19569}, {3920, 4093}, {3926, 7891}, {3933, 7807}, {3978, 8623}, {3985, 6651}, {3996, 20056}, {4037, 4760}, {4045, 5355}, {4095, 17741}, {4195, 19761}, {4251, 17034}, {4352, 22267}, {4369, 17212}, {4426, 6376}, {4713, 21793}, {4754, 17103}, {4831, 6163}, {5008, 7804}, {5017, 18906}, {5023, 20105}, {5033, 9764}, {5041, 6683}, {5103, 9478}, {5104, 5939}, {5133, 8878}, {5152, 5162}, {5171, 11257}, {5188, 12203}, {5206, 7781}, {5210, 8716}, {5254, 6655}, {5309, 7761}, {5319, 7800}, {5368, 6292}, {5475, 7812}, {5503, 8587}, {5977, 8682}, {5988, 17770}, {5992, 17768}, {6031, 6322}, {6248, 12110}, {6309, 10131}, {6566, 8307}, {6567, 8306}, {6636, 8266}, {6653, 21956}, {6680, 7794}, {6795, 15915}, {7000, 12221}, {7075, 22370}, {7179, 17364}, {7191, 18170}, {7200, 7267}, {7374, 12222}, {7379, 20077}, {7426, 7665}, {7470, 9821}, {7697, 10796}, {7737, 11185}, {7745, 16044}, {7746, 7752}, {7747, 14042}, {7748, 7802}, {7749, 7764}, {7758, 7763}, {7765, 7830}, {7770, 7787}, {7772, 7786}, {7773, 7900}, {7775, 7926}, {7776, 7887}, {7784, 7851}, {7795, 7892}, {7801, 7835}, {7808, 7878}, {7814, 7862}, {7817, 7848}, {7818, 7844}, {7819, 10583}, {7821, 7882}, {7822, 7846}, {7825, 7860}, {7841, 7898}, {7849, 7852}, {7861, 7873}, {7865, 7884}, {7866, 7879}, {7867, 7896}, {7869, 7930}, {7870, 7908}, {7871, 7888}, {7872, 7910}, {7874, 7895}, {7881, 7945}, {7889, 16896}, {7902, 7918}, {7914, 7943}, {8300, 17031}, {8356, 15048}, {8370, 18907}, {8591, 8598}, {8592, 8593}, {8669, 17760}, {9477, 14970}, {9605, 11285}, {9751, 20190}, {9756, 11477}, {9772, 12177}, {9773, 9877}, {9867, 13926}, {9868, 13873}, {10063, 10802}, {10079, 10801}, {10256, 10303}, {10353, 13196}, {10488, 10811}, {10807, 11317}, {10989, 16092}, {11057, 11648}, {11286, 21309}, {12194, 12263}, {13449, 14639}, {14651, 15980}, {14693, 15561}, {14910, 18372}, {16508, 22561}, {16589, 16912}, {16678, 17148}, {16800, 21257}, {16916, 18135}, {16918, 18140}, {16957, 18152}, {17037, 18287}, {17737, 20553}, {17961, 18825}, {18144, 19670}, {18901, 19585}, {18902, 19571}, {18993, 19090}, {18994, 19089}, {20060, 20102}, {20179, 21264}, {20794, 20885}

X(385) = reflection of X(i) in X(j) for these (i,j): (99,187), (147,1513), (298,395), (299,396), (316,115), (325,230)
X(385) = isogonal conjugate of X(694)
X(385) = isotomic conjugate of X(1916)
X(385) = complement of X(7779)
X(385) = anticomplement of X(325)
X(385) = circumcircle-inverse of X(32531)
X(385) = X(i)-Ceva conjugate of X(j) for these (i,j): (98,2), (511,401), (694,384)
X(385) = crosspoint of X(290) and X(308)
X(385) = crosssum of X(i) in X(j) for these (i,j): (141,698), (384,385)
X(385) = crossdifference of every pair of points on line X(39)X(512)
X(385) = trilinear pole of PU(185)
X(385) = orthoptic-circle-of-Steiner-inellipe-inverse of X(32525)
X(385) = X(i)-Hirst inverse of X(j) for these (i,j): (2,6), (3,194), (171,894)
X(385) = {X(2),X(193)}-harmonic conjugate of X(7774)
X(385) = intersection of trilinear polars of P(1) and U(1)
X(385) = trilinear pole of line X(804)X(4107) (line is perspectrix of any pair of {ABC, 1st Brocard triangle, 3rd Brocard triangle}, and is also the Lemoine axis of the 1st Brocard triangle.)
X(385) = anticomplementary isotomic conjugate of X(147)
X(385) = crosspoint of X(6) and X(3511) wrt both the excentral and tangential triangles
X(385) = inverse-in-Steiner-circumellipse of X(6)
X(385) = X(99)-of-1st-anti-Brocard-triangle
X(385) = X(98)-of-anti-McCay-triangle
X(385) = barycentric product of PU(133)
X(385) = barycentric product X(239)*X(894)
X(385) = perspector of ABC and side-triangle of cevian triangles of PU(1)


X(386) = INVERSE-IN-BROCARD-CIRCLE OF X(58)

Trilinears    a(b2 + c2 + bc + ca + ab) : :
Trilinears    r cos A + s sin A : : , where s = semiperimeter, r = inradius
Trilinears    sin(A + U) : : , where cot U = cot(A/2) cot(B/2) cot(C/2)
Barycentrics    a2(b2 + c2 + bc + ca + ab) : :

X(386) is the external center of similitude of the circumcircle and Apollonius circle. The internal center is X(573). (Peter J. C. Moses, 8/22/03)

Let A' be the inverse-in-excircles-radical-circle of the midpoint of BC, and define B' and C' cyclically. Triangle A'B'C' is inscribed in the Apollonius circle and homothetic to the 2nd circumperp triangle at X(386). (Randy Hutson, November 18, 2015)

X(386) lies on these lines: {1, 2}, {3, 6}, {4, 2051}, {5, 1834}, {9, 10467}, {11, 9555}, {12, 9552}, {20, 9535}, {21, 1724}, {22, 9571}, {24, 9570}, {31, 35}, {34, 14018}, {36, 1468}, {37, 5044}, {38, 5904}, {40, 1064}, {41, 2172}, {55, 595}, {56, 181}, {57, 73}, {60, 3453}, {65, 994}, {69, 16887}, {72, 3666}, {81, 404}, {99, 3029}, {100, 3032}, {101, 3033}, {105, 3034}, {106, 2334}, {109, 11509}, {110, 3031}, {165, 1695}, {184, 3437}, {191, 4414}, {192, 3159}, {194, 17499}, {213, 2276}, {222, 1466}, {223, 1448}, {226, 23537}, {238, 5248}, {244, 18398}, {312, 2901}, {333, 19270}, {388, 4551}, {405, 4383}, {411, 1754}, {442, 5718}, {443, 5712}, {474, 940}, {514, 22090}, {517, 4646}, {518, 4719}, {549, 5453}, {590, 8959}, {601, 2077}, {602, 10902}, {603, 2003}, {631, 9568}, {727, 11490}, {741, 5213}, {748, 5259}, {758, 986}, {759, 6044}, {810, 14838}, {872, 984}, {939, 13404}, {942, 3752}, {946, 3755}, {956, 16302}, {958, 9564}, {960, 3931}, {982, 3874}, {987, 5150}, {988, 3751}, {990, 1490}, {993, 5247}, {999, 5399}, {1001, 16288}, {1010, 5331}, {1036, 8193}, {1040, 10393}, {1042, 3339}, {1043, 13740}, {1045, 3923}, {1046, 17596}, {1060, 8555}, {1066, 3333}, {1086, 6147}, {1100, 5956}, {1104, 16290}, {1107, 21904}, {1147, 9562}, {1191, 3295}, {1211, 13728}, {1245, 2258}, {1330, 4201}, {1376, 5711}, {1386, 5266}, {1420, 1450}, {1437, 20842}, {1449, 1818}, {1453, 3601}, {1457, 3340}, {1458, 3361}, {1464, 5221}, {1500, 2176}, {1575, 17750}, {1616, 6767}, {1738, 12609}, {1742, 12512}, {1745, 4292}, {1780, 20846}, {1871, 14571}, {2050, 5786}, {2140, 3487}, {2177, 3746}, {2194, 11334}, {2221, 7085}, {2238, 5283}, {2242, 21008}, {2251, 7296}, {2273, 7193}, {2274, 4649}, {2275, 20963}, {2280, 5299}, {2292, 5692}, {2293, 16469}, {2299, 14017}, {2300, 3781}, {2308, 5010}, {2309, 16468}, {2328, 11344}, {2332, 8743}, {2345, 10469}, {2536, 2538}, {2537, 2539}, {2635, 9579}, {2646, 16455}, {2650, 5902}, {2653, 3981}, {2654, 9581}, {2915, 5347}, {2933, 20986}, {3072, 6796}, {3091, 5400}, {3100, 9550}, {3120, 6127}, {3145, 5320}, {3149, 5706}, {3191, 22020}, {3194, 4219}, {3303, 16296}, {3454, 4417}, {3501, 3997}, {3553, 5720}, {3576, 9548}, {3589, 17698}, {3596, 4360}, {3597, 13478}, {3647, 7262}, {3664, 12436}, {3670, 3868}, {3702, 3896}, {3725, 3743}, {3731, 8951}, {3772, 11374}, {3780, 16975}, {3795, 4116}, {3869, 4424}, {3873, 3953}, {3878, 4868}, {3881, 3976}, {3889, 4694}, {3914, 12047}, {3916, 4641}, {3936, 4202}, {3945, 17580}, {3993, 14823}, {4065, 19582}, {4188, 17187}, {4205, 5743}, {4216, 10457}, {4224, 5358}, {4225, 20966}, {4303, 15803}, {4322, 13462}, {4340, 6904}, {4642, 5903}, {4648, 17582}, {4749, 16300}, {4878, 7174}, {4991, 18194}, {5051, 5741}, {5091, 16382}, {5134, 9598}, {5251, 10448}, {5255, 8715}, {5275, 16852}, {5278, 10458}, {5439, 16610}, {5563, 16474}, {5584, 10823}, {5640, 7419}, {5687, 5710}, {5707, 6911}, {5713, 6826}, {5719, 17366}, {5721, 6831}, {5725, 5794}, {5732, 10443}, {5736, 17189}, {5737, 19273}, {6198, 9551}, {6284, 9554}, {6748, 7546}, {7031, 21764}, {7354, 9553}, {7513, 8747}, {7951, 21935}, {8069, 16294}, {8071, 16295}, {8300, 20862}, {8728, 17056}, {9310, 16785}, {9371, 12711}, {9441, 12511}, {9575, 20606}, {10407, 10895}, {10441, 19513}, {10470, 21363}, {11110, 17277}, {11263, 17889}, {11343, 19758}, {11993, 17467}, {13161, 21077}, {13407, 23536}, {13725, 14555}, {14793, 16473}, {14996, 17572}, {14997, 16865}, {15488, 19540}, {15654, 23638}, {15934, 17054}, {16343, 19757}, {16347, 19742}, {16351, 19723}, {16352, 19725}, {16353, 19724}, {16434, 19782}, {16454, 19684}, {16457, 19744}, {16458, 19701}, {16476, 23629}, {16498, 17477}, {16549, 17756}, {16589, 16846}, {16844, 17259}, {16917, 20132}, {16948, 17549}, {17379, 18792}, {17674, 18139}, {18169, 19278}, {18178, 19550}, {18185, 19261}, {19284, 19717}, {19289, 19750}, {19290, 19722}, {19331, 19739}, {19336, 19738}, {19337, 19741}, {19523, 19728}, {19645, 19752}, {19761, 21477}, {23414, 23659}

X(386) is the {X(3),X(6)}-harmonic conjugate of X(58). For a list of other harmonic conjugates of X(386), click Tables at the top of this page.

X(386) = complement of X(10449)
X(386) = crosssum of X(6) in X(1011)
X(386) = crossdifference of every pair of points on line X(523)X(649)
X(386) = Brocard-circle-inverse of X(58)
X(386) = intersection of tangents at X(2) and X(6) to Thomson cubic K002
X(386) = intersection of Nagel line and Brocard axis


X(387) = INTERSECTION OF LINES X(1,2) AND X(4,6)

Trilinears    bc[-a4 + 2a2(a + b + c)2 + (b2 - c2)2]
Barycentrics    -a4 + 2a2(a + b + c)2 + (b2 - c2)2

X(387) lies on these lines: {1, 2}, {4, 6}, {7, 23537}, {20, 58}, {31, 4294}, {40, 579}, {46, 3101}, {55, 1612}, {57, 5930}, {65, 278}, {69, 16062}, {81, 377}, {193, 1330}, {219, 2551}, {230, 7410}, {333, 13725}, {341, 18147}, {346, 2901}, {376, 4252}, {390, 595}, {442, 5712}, {443, 940}, {452, 1724}, {497, 16466}, {580, 6987}, {581, 6908}, {631, 4255}, {942, 4000}, {950, 1453}, {966, 4205}, {990, 9799}, {999, 16415}, {1058, 1191}, {1104, 3488}, {1108, 4261}, {1126, 5261}, {1203, 1479}, {1214, 1788}, {1448, 18623}, {1449, 5717}, {1468, 4293}, {1617, 16453}, {1723, 12514}, {1743, 12572}, {1780, 6872}, {1838, 3914}, {1992, 17677}, {2047, 3068}, {2082, 7713}, {2271, 6998}, {2334, 15888}, {2345, 5295}, {2478, 16471}, {2550, 5711}, {3089, 3192}, {3189, 5266}, {3295, 16290}, {3339, 3668}, {3419, 5716}, {3487, 3772}, {3522, 4257}, {3523, 4256}, {3618, 13740}, {3695, 17314}, {3751, 13161}, {3767, 20970}, {3868, 4463}, {3874, 4310}, {3927, 4419}, {3945, 4208}, {4219, 8885}, {4251, 5304}, {4259, 18178}, {4260, 10441}, {4267, 19262}, {4356, 18249}, {4383, 5084}, {4648, 8728}, {4653, 17558}, {5021, 7738}, {5051, 5739}, {5082, 5710}, {5165, 6361}, {5264, 17784}, {5324, 13730}, {5396, 6825}, {5398, 6868}, {5587, 5747}, {5707, 6826}, {5713, 6843}, {5718, 6856}, {5814, 5839}, {6392, 17499}, {6857, 19765}, {7380, 7736}, {7682, 8282}, {7742, 16451}, {8676, 17922}, {9708, 16848}, {10480, 10822}, {10590, 21935}, {11109, 11427}, {11433, 17555}, {16704, 17676}, {17054, 17366}, {17685, 20158}, {19543, 22770}.

X(387) = crossdifference of every pair of points on line X(520)X(649)


X(388) = X(1)X(4)∩X(7)X(8)

Trilinears    bc[a2 + (b + c)2]/(b + c - a)
Trilinears    1 + cos B cos C : 1 + cos C cos A : 1 + cos A cos B
Barycentrics    [a2 + (b + c)2]/(b + c - a)
X(388) = 2(R/r)*X(1) + 3X(2) - 2X(3)

Let A″B″C″ be the reflection of the Mandart-incircle triangle in X(1). A″B″C″ is homothetic to the anticomplementary triangle at X(388). (Randy Hutson, September 14, 2016)

X(388) lies on these lines: 1,4   2,12   3,495   5,999   7,8   10,57   11,153   20,55   29,1037   35,376   36,498   79,1000   108,406   171,603   201,984   329,960   354,938   355,942   381,496   442,956   452,1001   612,1038   750,1106   1059,1067

X(388) is the {X(7),X(8)}-harmonic conjugate of X(65). For a list of other harmonic conjugates of X(388), click Tables at the top of this page.

X(388) = isogonal conjugate of X(1036)
X(388) = anticomplement of X(958)

X(388) = X(1181)-of-intouch triangle
X(388) = perspector of ABC and 5th extouch triangle
X(388) = homothetic center of anticomplementary triangle and cross-triangle of ABC and 1st Johnson-Yff triangle
X(388) = homothetic center of 1st Johnson-Yff triangle and cross-triangle of ABC and 1st Johnson-Yff triangle
X(388) = isotomic conjugate of X(30479)
X(388) = homothetic center of anti-inverse-in-incircle triangle and anti-tangential midarc triangle
X(388) = centroid of curvatures of anticomplementary circle and excircles

X(389) = CENTER OF THE TAYLOR CIRCLE

Trilinears    cos A - cos 2A cos(B - C) : cos B - cos 2B cos(C - A) : cos C - cos 2C cos(A - B) : :
Trilinears    cos A - sin A (tan A + tan B + tan C) : :
Trilinears    cos A (cot A cot B cot C) - sin A : :
Barycentrics    a[cos A - cos 2A cos(B - C)] : b[cos B - cos 2B cos(C - A)] : c[cos C - cos 2C cos(A - B)] : :
Barycentrics    a^2[a^6(b^2 + c^2) - 3a^4(b^4 + c^4) + 3a^2(b^2 - c^2)^2(b^2 + c^2) - (b^2 - c^2)^2(b^4 + c^4)] : :

If ABC is acute then X(389) is the Spieker center of the orthic triangle. Peter Yff reports (Sept. 19, 2001) that since X(389) is on the Brocard axis, there must exist T for which X(389) is sin(A+T) : sin(B+T) : sin(C+T), and that tan(T) = - cot A cot B cot C.

Let HA be the A-altitude of triangle ABC, and let A' be the midpoint of segment AHA. Let LA be the line through A' parallel to AO, where O denotes the circumcenter. Define LB and LC cyclically. The lines LA, LB, LC concur in X(389). (Construction by Alexei Myakishev, March 24, 2010.)

Let OA be the circle with center A tangent to line BC, and define OB and OC cyclically. X(389) is the radical center of the three circles. (Randy Hutson, 9/23/2011)

Let A'B'C' be the orthic triangle, let A'' be the orthocenter of AB'C', and define B'' and C'' cyclically. Triangle A''B''C'' is the medial triangle of the circumorthic triangle. The triangle A''B''C'' is homothetic to A'B'C', and the center of homothety is X(389). (Randy Hutson, 9/23/2011)

Let A'B'C' be the orthic triangle. Let Oa be the circle with center A tangent to line B'C', and define Ob and Oc cyclically. The radical circle of Oa, Ob, Oc is the Taylor circle, which is also the Spieker radical circle of the orthic triangle if ABC is acute. (Randy Hutson, December 2, 2017)

Let A'B'C' be the orthic triangle. X(389) is the radical center of the polar circles of AB'C', BC'A', CA'B'. (Randy Hutson, December 2, 2017)

A construction of X(389) is given at 24162. (Antreas Hatzipolakis, August 29, 2016)

Let A'B'C' be the midheight triangle. Let AB, AC be the orthogonal projections of A' on CA, AB, resp. Define BC, BA, CA, CB cyclically. Let A″ = CAAC∩ABBA, and define B″ and C″ cyclically. Triangle A″B″C″ is homothetic to ABC at X(6). The lines A'A″, B'B″, C'C″ concur in X(389), which is also X(3)-of-A″B″C″. (Randy Hutson, March 29, 2020)

If you have The Geometer's Sketchpad, you can view X(389).

X(389) lies on these lines: 3,6   4,51   24,184   30,143   54,186   115,129   217,232   517,950

X(389) = midpoint of X(i) and X(j) for these (i,j): (3,52), (4,185), (974,1112)
X(389) = reflection of X(1216) in X(140)
X(389) = complement of X(5562)
X(389) = anticomplement of X(11793)
X(389) = polar-circle-inverse of X(6761)
X(389) = Schoute-circle-inverse of X(22052)
X(389) = circle-{X(371),X(372),PU(1),PU(39)}}-inverse of X(577)
X(389) = {X(15),X(16)}-harmonic conjugate of X(22052)
X(389) = {X(371),X(372)}-harmonic conjugate of X(577)
X(389) = inverse-in-Brocard-circle of X(578)
X(389) = crosspoint of X(4) and X(54)
X(389) = crosssum of X(i) and X(j) for these (i,j): (3,5), (6,418)
X(389) = orthology center of half-altitude and orthic triangles
X(389) = complement of X(4) wrt orthic triangle
X(389) = X(20)-of-polar-triangle-of-complement-of-polar-circle
X(389) = X(10)-of-orthic-triangle if ABC is acute
X(389) = X(4)-of-1st-Hyacinth-triangle
X(389) = excentral-to-ABC functional image of X(10)
X(389) = {X(61),X(62)}-harmonic conjugate of X(3284)


X(390) = REFLECTION OF GERGONNE POINT IN INCENTER

Trilinears    bc(b + c - a)[3a2 + (b - c)2]
Barycentrics    (b + c - a)[3a2 + (b - c)2]
X(390) = 4(R/r)*X(1) - 3X(2) + 4X(3)

X(390) is the point in whIch the extended legs X(1)X(7) and X(8)X(9) of the trapezoid X(1)X(7)X(8)X(9) meet. (Randy Hutson, February 10, 2016)

Let A'B'C' be the extangents-to-intangents similarity image of ABC. A'B'C' is homothetic to ABC at X(55) and to the anticomplementary triangle at X(390). (Randy Hutson, June 7, 2019)

X(390) lies on these lines: 1,7   2,11   3,1058   4,495   8,9   30,1056   40,938   144,145   376,999   387,595   496,631   944,971   952,1000

X(390) = midpoint of X(144) and X(145)
X(390) = reflection of X(i) in X(j) for these (i,j): (7,1), (8,9)
X(390) = anticomplement of X(2550)
X(390) = crossdifference of every pair of points on line X(657)X(665)
X(390) = inverse-in-Feuerbach-hyperbola of X(2)
X(390) = {X(1),X(20)}-harmonic conjugate of X(3600)
X(390) = {X(175),X(176)}-harmonic conjugate of X(279)
X(390) = bicentric sum of PU(122)


X(391) = INTERSECTION OF LINES X(2,6) AND X(8,9)

Trilinears    bc(3a + b + c)(b + c - a) : :
Barycentrics    (3a + b + c)(b + c - a) : :
Barycentrics    a2 - s2 : b2 - s2 : c2 - s2

X(391) lies on these lines: 2,6   8,9   20,573   37,145   75,144   319,344

X(391) is the {X(8),X(9)}-harmonic conjugate of X(346). For a list of other harmonic conjugates of X(391), click Tables at the top of this page.

X(391) = crosssum of X(940) and X(4383)


X(392) =  X(1,6)∩X(10,11)

Trilinears    (b + c)(b2 + c2 - a2) + 4abc : :

X(392) lies on the Thomson-Gibert-Moses hyperbola and these lines: (pending)

X(392) = {X(1),X(9)}-harmonic conjugate of X(956)
X(392) = X(2)-of-X(1)-Brocard-triangle
X(392) = Thomson-isogonal conjugate of X(4220)


X(393) = X(25)-CROSS CONJUGATE OF X(4)

Trilinears    bc tan2A : ca tan2B : ab tan2C
Trilinears    sec A - csc B csc C : sec B - csc C csc A : sec C - csc A csc B
Trilinears    sec A tan A : :
Trilinears    1/(sin A - csc A) : :
Barycentrics    tan2A : tan2B : tan2C
Barycentrics    (tan A)/(cot B + cot C - cot ω) : :
Barycentrics    [a^4 - (b^2 - c^2)^2]^2 : :
Barycentrics    tan A - cot B - cot C : :

Let Ha be the foot of the A-altitude of ABC. Let Pa be the foot of the altitude from Ha to AB, and Qa the foot of the altitude from Ha to CA. Define Hb, Hc, Pb, Pc, Qb, Qc cyclically. Let A' be the trilinear pole of line PaQa, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(393). Lines PaQa, PbQb, PcQc are the antiparallels in the construction of the Taylor circle. (Randy Hutson, January 29, 2018)

Let A'B'C' and A″B″C″ be the Euler and anti-Euler triangles, resp. Let A* be the barycentric product A'*A'', and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(393). (Randy Hutson, January 29, 2018)

Let A'B'C' be the orthic triangle. Let A″ be the cevapoint of the (real or imaginary) circumcircle intercepts of line B'C'. Define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(393). (Randy Hutson, July 31 2018)

X(393) is the Brianchon point (perspector) of the inellipse that is the barycentric square of the orthic axis. The center of this inellipse is X(3767). (Randy Hutson, October 15, 2018)

X(393) lies on these lines: 1,836   2,216   4,6   19,208   20,577   24,254   25,1033   27,967   33,42   37,158   69,297   107,111   193,317   230,459   278,1108   342,948   394,837   800,1093   3146,3284

X(393) = isogonal conjugate of X(394)
X(393) = crosspoint of X(4) and X(459)
X(393) = X(25)-cross conjugate of X(4)
X(393) = crosssum of X(577) and X(1092)
X(393) = isotomic conjugate of X(3926)
X(393) = anticomplement of X(6389)
X(393) = cevapoint of X(5412) and X(5413)
X(393) = barycentric square of X(4)
X(393) = trilinear pole of line X(460)X(512) (the polar of X(69) wrt polar circle, and radical axis of nine-point circle and orthosymmedial circle)
X(393) = pole wrt polar circle of trilinear polar of X(69) (line X(441)X(525))
X(393) = polar conjugate of X(69)
X(393) = X(i)-isoconjugate of X(j) for these {i,j}: {1,394}, {31,3926}, {48,69}, {92,1092}
X(393) = vertex conjugate of the foci of the inconic with perspector X(2052)
X(393) = X(1743)-of-orthic-triangle if ABC is acute


X(394) = X(69)-CEVA CONJUGATE OF X(3)

Trilinears    cos A cot A : cos B cot B : cos C cot C
Trilinears    sin A - csc A : :
Trilinears    a cot2A : :
Barycentrics    cos2A : cos2B : cos2C
Barycentrics    1 - tan B tan C : :
Barycentrics    a^2(b^2 + c^2 - a^2)^2 : :

Let A' be the trilinear pole of the tangent to the circumcircle at the antipode of A, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(394). (Randy Hutson, November 18, 2015)

Let A'B'C' be the orthic triangle. Let A″ be the crosssum of the (real or imaginary) circumcircle intercepts of line B'C'. Define B″ andC″ cyclically. The lines AA″, BB″, CC″ concur in X(393). (Randy Hutson, July 31 2018)

X(394) lies on these lines: 2,6   3,49   20,1032   22,110   25,511   63,77   72,1060   76,275   78,271   287,305   297,315   329,651   393,837   399,541   470,633   471,634   472,622   473,621   493,1504   494,1505   611,612   613,614   1062,1069

X(394) = isogonal conjugate of X(393)
X(394) = isotomic conjugate of X(2052)
X(394) = complement of X(6515)
X(394) = crosssum of X(5412) and X(5413)
X(394) = crosspoint of X(6) and X(1498) wrt both the excentral and tangential triangles
X(394) = X(69)-Ceva conjugate of X(3)
X(394) = crosspoint of X(493) and X(494)
X(394) = crosspoint of X(11090) and X(11091)
X(394) = crosssum of X(4) and X(459)
X(394) = crossdifference of every pair of points on line X(460)X(512)
X(394) = intersection of tangents to hyperbola {A,B,C,X(2),X(6)}} at X(493) and X(494)
X(394) = Danneels point of X(69)
X(394) = pole wrt polar circle of trilinear polar of X(1093)
X(394) = X(48)-isoconjugate (polar conjugate) of X(1093)
X(394) = X(92)-isoconjugate of X(25)
X(394) = intersection of tangents at X(2) and X(20) to Lucas cubic K007
X(394) = barycentric square of X(63)


X(395) = MIDPOINT OF X(14) AND X(16)

Trilinears        cos(B - C) + 2 cos(A + π/3) : :

X(395) is the center of the (equilateral) pedal triangle of X(16), as well as the circumcenter of the pedal triangle of X(14).

Let F = X(14), the 2nd Fermat point. Let Ab be the point on CA and Ac the point on AB such that FAbAc is an equilateral triangle. Let A' be the midpoint of FAbAc, and define B' and C' cyclically. The triangle A'B'C', here named the 8th Fermat-Dao equilateral triangle, is perspective to ABC. The circumcircle of A'B'C' is the pedal circle of X(14) and (X16), and the centroid of A'B'C' is X(395). The perspector of ABC and A'B'C' is X(2), and the centroid of A'B'C' is X(395). See X(16247), X(16267), and X(16459). (Based on notes from Dao Thanh Dao, March 15, 2018)

X(395) is the pole of the Euler line wrt every conic passing through X(14), X(16), X(18) and X(62). (Randy Hutson, January 17, 2020)

You can view 7th and 8th Fermat-Dao equilateral triangles.

X(395) lies on these lines: 2,6   3,398   5,13   14,16   15,549   39,618   53,472   61,140   115,530   187,531   202,495   216,465   466,577   532,624   533,619

X(395) = midpoint of X(i) and X(j) for these (i,j): (14,16), (298,385)
X(395) = reflection of X(396) in X(230)
X(395) = isogonal conjugate of X(6151)
X(395) = complement of X(299)
X(395) = crosspoint of X(2) and X(14)
X(395) = crosssum of X(6) and X(16)
X(395) = crossdifference of every pair of points on line X(15)X(512)
X(395) = barycentric products X(13)*X(533), X(14)*X(619)
X(395) = {X(2),X(6)}-harmonic conjugate of X(396)
X(395) = polar conjugate of X(38427)
X(395) = perspector of circumconic centered at X(619)
X(395) = center of circumconic that is locus of trilinear poles of lines passing through X(619)
X(395) = X(2)-Ceva conjugate of X(619)


X(396) = MIDPOINT OF X(13) AND X(15)

Trilinears        cos(B - C) + 2 cos(A - π/3) : :

X(396) is the center of the (equilateral) pedal triangle of X(15), as well as the circumcenter of the pedal triangle of X(13).

X(396) is the centroid of the points Ab, Ac, Bc, Ba, Ca, Cb in the construction of the 3rd Fermat-Dao equilateral triangle; see X(16267). Also, X(396) is the center of the ellipse passing through these six points. (Randy Hutson, March 14, 2018)

Let F = X(13), the 1st Fermat point. Let Ab be the point on CA and Ac the point on AB such that FAbAc is an equilateral triangle. Let A' be the midpoint of FAbAc, and define B' and C' cyclically. The triangle A'B'C', here named the 7th Fermat-Dao equilateral triangle, is perspective to ABC. The circumcircle of A'B'C' is the pedal circle of X(13) and (X15), and the centroid of A'B'C' is X(396). The perspector of ABC and A'B'C' is X(2), and the centroid of A'B'C' is X(396).

X(396) is the pole of the Euler line wrt every conic passing through X(13), X(15), X(17) and X(61). (Randy Hutson, January 17, 2020)

If you have GeoGebra, you can view X(396).

X(396) lies on these lines: 2,6   3,397   5,14   13,15   16,549   39,619   53,473   62,140   115,531   187,530   203,495   216,466   465,577   532,618   533,623

X(396) = midpoint of X(i) and X(j) for these (i,j): (13,15), (299,385)
X(396) = reflection of X(395) in X(230)
X(396) = isogonal conjugate of X(2981)
X(396) = anticomplement of X(298)
X(396) = crosspoint of X(2) and X(13)
X(396) = crosssum of X(6) and X(15)
X(396) = crossdifference of every pair of points on line X(16)X(512)
X(396) = polar conjugate of X(38428)
X(396) = perspector of circumconic centered at X(618)
X(396) = center of circumconic that is locus of trilinear poles of lines passing through X(618)
X(396) = X(2)-Ceva conjugate of X(618)
X(396) = barycentric products X(14)*X(532), X(13)*X(618)
X(396) = {X(2),X(6)}-harmonic conjugate of X(395)


X(397) = CROSSPOINT OF X(4) AND X(17)

Trilinears    cos(B - C) - 2 cos(A + π/3) : :

X(397) is the pole of the Euler line wrt every conic passing through X(13), X(16), X(17) and X(62). (Randy Hutson, January 17, 2020)

X(397) lies on these lines: 3,396   4,6   5,13   14,546   15,550   16,17   30,61   51,462   141,634   184,463   202,496   524,633   532,635

X(397) is the {X(4),X(6)}-harmonic conjugate of X(398). For a list of other harmonic conjugates of X(397), click Tables at the top of this page.

X(397) = crosspoint of X(4) and X(17)
X(397) = crosssum of X(3) and X(61)


X(398) = CROSSPOINT OF X(4) AND X(18)

Trilinears    cos(B - C) - 2 cos(A - π/3) : :

X(398) is the pole of the Euler line wrt every conic passing through X(14), X(15), X(18) and X(61). (Randy Hutson, January 17, 2020)

X(398) lies on these lines: 3,395   4,6   5,14   13,546   15,18   16,550   30,62   51,463   141,633   184,462   203,496   524,634   533,636

X(398) is the {X(4),X(6)}-harmonic conjugate of X(397). For a list of other harmonic conjugates of X(398), click Tables at the top of this page.

X(398) = crosspoint of X(4) and X(18)
X(398) = crosssum of X(3) and X(62)


X(399) = PARRY REFLECTION POINT

Trilinears    5 cos A - 4 cos B cos C - 8 sin B sin C cos2A : :
Barycentrics   Barycentrics    a^2[a^8 - 4a^6(b^2 + c^2) + a^4(6b^4 + b^2c^2 + 6c^4) - a^2(4b^6 - b^4c^2 - b^2c^4 + 4 c^6) + (b^2 - c^2)^2(b^4 + 4b^2c^2 + c^4)] : :

Let L, M, N be lines through A, B, C, respectively, parallel to the Euler line. Let L' be the reflection of L in sideline BC, let M' be the reflection of M in sideline CA, and let N' be the reflection of N in sideline AB. The lines L', M', N' concur in X(399), as proved in

Cyril Parry, Problem 10637, American Mathematical Monthly 105 (1998) 68.

In Cosmin Pohoata, "On the Parry reflection point," Forum Geometricorum 8 (2008), 43-48 here, the following is proved:

Let A' be the reflection of vertex A in line BC, and define B', C' cyclically. Let AtBtCt be the tangential triangle of ABC. The circumcircles of the triangles AtB'C', A'BtC', A'B'Ct concur in X(399). Moreover, the circumcircles of triangles A'BtCt, AtB'Ct, AtBtC' concur in a point Q = X8157), here named the Parry-Pohoata point. Barycentric coordinates for Q, of degree 22 in a,b,c, were found by J. F. Garcia Captitán (Hyacinthos #15827, Nov. 19, 2007) and are included in Pohoata's article. Pohoata notes that the point Q lies on the circumcircle of the points X(3), X(4), X(399).

Let I, IA, IB, IC, denote the incenter and excenters of ABC. Lawrence Evans (Hyacinthos #6878) found that the circumcircles of the triangles IA'IA, IB'IB, IC'C concur in X(399).

The Pohoata article includes a proof that the circumcircles of the triangles A'IBIC, IAB'IC, IAIBC' also pass through X(399). Similar results involving the Fermat points, X(13) and X(14), are proved.

Pohoata reports that the following points are concyclic: X(13), X(16), X(110), X(399), X(1338), as are the points X(14), X(15), X(110), X(399), X(1337). Randy Hutson adds (Aug. 13, 2012) that the first of these circles also passes through X(2381), and the second, through X(2380).

X(399) is the point P on the line X(3)X(74) for which the P-Brocard triangle is perspective to ABC. (Randy Hutson, August 26, 2014)

Let AaBaCa, AbBbCb, AcBcCc be the A-, B- and C-anti-altimedial triangles. Let La be the line through Aa parallel to the Euler line of AaBaCa. Define Lb and Lc cyclically. The lines La, Lb, Lc concur in X(399). (Randy Hutson, November 2, 2017)

Let OA be the circle centered at the A-vertex of the Walsmith triangle and passing through A; define OB and OC cyclically. X(399) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

X(399) lies on the Neuberg cubic and these lines: 1,3065   3,74   4,195   6,13   30,146   155,382   394,541   1337,3441   1338,3440   3466,3483

X(399) = isogonal conjugate of X(1138)
X(399) = reflection of X(i) in X(j) for these (i,j): (3,110), (74,1511), (265,113)
X(399) = X(i)-Ceva conjugate of X(j) for these (i,j): (30,3), (323,6)
X(399) = inverse-in-circumcircle of X(1511)
X(399) = tangential isogonal conjugate of X(2070)
X(399) = antigonal image of X(1117)
X(399) = orthocentroidal-to-ABC similarity image of X(3)
X(399) = 4th-Brocard-to-circumsymmedial similarity image of X(3)
X(399) = X(80)-of-tangential-triangle if ABC is acute
X(399) = orthologic center of these triangles: reflection to 1st Hyacinth
X(399) = orthologic center of these triangles: reflection to AOA
X(399) = perspector of circlecevian triangles of X(15) and X(16)


X(400) = YFF-MALFATTI POINT

Trilinears       csc4(A/4) : csc4(B/4) : csc4(C/4)
Barycentrics  sin A csc4(A/4) : sin B csc4(B/4) : sin C csc4(C/4)

In 1997, Yff considered the configuration for the 1st Ajima-Malfatti point, X(179). He proved that the same tangencies are possible in another way if the circles are not required to lie inside ABC. With tangency points labeled as before, the lines AA', BB', CC' concur in X(400). If you have The Geometer's Sketchpad, you can view X(400).

leftri

Centers 401- 475, 2- 4, 20- 30, 376, 379, 381- 384 (and others) lie on the Euler line.

rightri

X(401) = BAILEY POINT

Trilinears    [sin 2B sin 2C - sin2(2A)](csc A) : :
Barycentrics    sin 2B sin 2C - sin2(2A) : :
Barycentrics    tan B + tan C - tan A - cot ω : :
Barycentrics    a^8 - 2 a^6 (b^2 + c^2) + a^4 (b^4 + b^2 c^2 + c^4) + b^2 c^2 (b^2 - c^2)^2 : :
Barycentrics    tan A - cot ω' : : , where ω' is the Brocard angle of the orthic triangle
X(401) = 3 SA SB SC SW X(2) - S^4 X(20)

As a point on the Euler line, X(401) has Shinagawa coefficients (EF + F2 - S2, 2S2).

X(401) lies on these lines: {2, 3}, {6, 3164}, {32, 40814}, {50, 338}, {51, 12110}, {81, 18667}, {94, 32662}, {95, 14767}, {97, 276}, {99, 36212}, {182, 42329}, {184, 11257}, {187, 41254}, {193, 6527}, {194, 1993}, {216, 36794}, {239, 10538}, {248, 290}, {253, 20080}, {264, 577}, {275, 9290}, {287, 511}, {317, 6389}, {323, 525}, {325, 39359}, {340, 15526}, {343, 7750}, {394, 1975}, {524, 39352}, {571, 41760}, {648, 3284}, {925, 2698}, {1073, 38256}, {1297, 6037}, {1503, 14721}, {1632, 2393}, {1899, 36998}, {1916, 15391}, {1968, 2052}, {1994, 7839}, {2782, 40870}, {2896, 34850}, {3087, 40680}, {3098, 42313}, {3100, 4366}, {3260, 4558}, {3289, 39355}, {3292, 23235}, {3329, 22240}, {3580, 14712}, {4296, 6645}, {5422, 7787}, {5480, 36988}, {5641, 35178}, {6515, 20065}, {6709, 36422}, {6748, 17035}, {7738, 11427}, {7925, 35088}, {8591, 40112}, {8859, 23967}, {9308, 15905}, {9512, 23200}, {9863, 11442}, {9873, 11550}, {9983, 18018}, {10329, 23292}, {10341, 10350}, {10420, 43654}, {11064, 15351}, {11078, 41997}, {11092, 41998}, {11596, 21639}, {12117, 13857}, {12215, 25332}, {13366, 32467}, {14570, 22151}, {14919, 16077}, {14920, 39008}, {14961, 41253}, {15066, 34360}, {15262, 41678}, {15595, 29012}, {15988, 18666}, {19571, 36790}, {19585, 32529}, {19924, 41145}, {21243, 32152}, {23582, 23590}, {23583, 37765}, {27377, 41005}, {30227, 37645}, {32428, 39081}, {34396, 39906}, {34545, 41334}, {35278, 42671}, {36748, 43980}, {38382, 39101}, {40897, 41008}, {41194, 41196}, {41195, 41197}

X(401) = reflection of X(297) in X(441)
X(401) = isogonal conjugate of X(1987)
X(401) = isotomic conjugate of X(1972)
X(401) = anticomplement of X(297)
X(401) = circumcircle-inverse of X(37918)
X(401) = perspector of conic {A,B,C,X(95),X(648)}}
X(401) = X(2)-Hirst inverse of X(3)
X(401) = inverse-in-Steiner-circumellipse of X(3)
X(401) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,39081), (287,2), (511,385)
X(401) = crosspoint of X(276) and X(290)
X(401) = crosssum of X(217) and X(237)
X(401) = crossdifference of every pair of points on line X(51)X(647)
X(401) = {X(2479),X(2480)}-harmonic conjugate of X(3)
X(401) = crossdifference of PU(157)


X(402) = ZEEMAN-GOSSARD PERSPECTOR

Barycentrics    (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*(a^8 - a^6*b^2 - 2*a^4*b^4 + 3*a^2*b^6 - b^8 - a^6*c^2 + 5*a^4*b^2*c^2 - 3*a^2*b^4*c^2 - b^6*c^2 - 2*a^4*c^4 - 3*a^2*b^2*c^4 + 4*b^4*c^4 + 3*a^2*c^6 - b^2*c^6 - c^8) : :       (Paul Yiu, February 20, 1999)

X(402) = X[1] - 3 X[11831],X[1] + 3 X[11852],5 X[2] + X[3081],3 X[2] + X[4240],5 X[2] - X[11050],3 X[2] - 5 X[15183],2 X[2] + X[34582],X[3] + 3 X[11911],X[3] - 3 X[26451],3 X[3] - X[35241],X[4] + 3 X[11845],X[4] - 3 X[11897],2 X[5] + X[15774],X[8] - 3 X[16210],X[8] + 3 X[16212],X[20] - 3 X[16190],X[145] - 3 X[16211],3 X[381] - X[18507],3 X[381] + X[18508],X[1650] + 3 X[1651],5 X[1650] + 3 X[3081],2 X[1650] - 3 X[11049],5 X[1650] - 3 X[11050],X[1650] - 5 X[15183],2 X[1650] + 3 X[34582],5 X[1651] - X[3081],3 X[1651] - X[4240],2 X[1651] + X[11049],5 X[1651] + X[11050],3 X[1651] + 5 X[15183],3 X[1651] + 2 X[15184],3 X[3081] - 5 X[4240],2 X[3081] + 5 X[11049],3 X[3081] + 25 X[15183],3 X[3081] + 10 X[15184],2 X[3081] - 5 X[34582],2 X[4240] + 3 X[11049],5 X[4240] + 3 X[11050],X[4240] + 5 X[15183],X[4240] + 2 X[15184],2 X[4240] - 3 X[34582],3 X[5055] + X[20128],5 X[11049] - 2 X[11050],3 X[11049] - 10 X[15183],3 X[11049] - 4 X[15184],3 X[11050] - 25 X[15183],3 X[11050] - 10 X[15184],2 X[11050] + 5 X[34582],X[11251] - 3 X[11911],X[11251] + 3 X[26451],3 X[11251] + X[35241],3 X[11831] + X[12438],3 X[11845] - X[12113],3 X[11852] - X[12438],3 X[11897] + X[12113],9 X[11911] + X[35241],X[12626] + 3 X[16210],X[12626] - 3 X[16212],5 X[15183] - 2 X[15184],10 X[15183] + 3 X[34582],4 X[15184] + 3 X[34582],9 X[26451] - X[35241]       (Peter Moses, November 26,2020)
As a point on the Euler line, X(402) has Shinagawa coefficients (5EF - 13F2 - S2, 3EF + 3F2 - S2).

In A History of Mathematics, Florian Cajori writes, "H. C. Gossard of the University of Oklahoma showed in 1916 that the three Euler lines of the triangles formed by the Euler line and the sides, taken by twos, of a given triangle, form a triangle . . . perspective with the given triangle and having the same Euler line." Let ABC be the given triangle and A'B'C' the Gossard triangle - that is, the triangle perspective with the given triangle and having the same Euler line. The lines AA', BB', CC' concur in X(402), named the Gosssard perspector by John Conway (1998).

Actually, X(402) dates back to an article by Christopher Zeeman in Wiskundige Opgaven 8 (1899-1902) 305. For details, see Paul Yiu's Hyacinthos message #7536 and others with Gossard in the subject line. (In ETC, the change of name from Gossard Perspector to Zeeman-Gossard Perspector was made on Oct. 15, 2003.) Further details are given by Wilson Stothers in Hyacinthos #8383, Oct. 21, 2003.

(From Angel Montesdeoca, November 25, 2020) Let P be a variable point on the Euler line and Q its isotomic conjugate. Let TP and TQ be the cevian triangles of P and Q, respectively. Let F be the finite fixed point of the affine transformation that carries TP onto TQ. The envelope of the line PF is a hyperbola (H) with center X(402), an asymptote of the Euler line. A barycentric equation for (H) follows:

cyclic sum [(-b^4+c^4+a^2 (b^2-c^2))^4 (-2 a^4+(b^2-c^2)^2+a^2 (b^2+c^2))^2x^2-2 (a^2-b^2)^2 (a^2-c^2)^2 (a^4-(b^2-c^2)^2)^2 (a^8-a^6 (b^2+c^2)+5 a^2 (b^2-c^2)^2 (b^2+c^2)+a^4 (-3 b^4+7 b^2 c^2-3 c^4)-(b^2-c^2)^2 (2 b^4+5 b^2 c^2+2 c^4))y z ] = 0.

X(402) lies on these lines: {1, 11831}, {2, 3}, {6, 11901}, {8, 12626}, {10, 11900}, {11, 11903}, {12, 11904}, {13, 12793}, {14, 12792}, {17, 22897}, {18, 22852}, {32, 11839}, {40, 12696}, {54, 12797}, {55, 11848}, {56, 18958}, {64, 12791}, {67, 32279}, {68, 12418}, {74, 12369}, {76, 12794}, {79, 16129}, {80, 12729}, {83, 12795}, {84, 12668}, {98, 12181}, {99, 13179}, {100, 13268}, {104, 12752}, {110, 13212}, {112, 13281}, {113, 38605}, {145, 16211}, {190, 24830}, {262, 22698}, {265, 12790}, {371, 35791}, {372, 35790}, {485, 12800}, {486, 12799}, {493, 11907}, {494, 11908}, {671, 12347}, {1297, 12796}, {1327, 13689}, {1328, 13809}, {3068, 13894}, {3069, 13948}, {5597, 11863}, {5598, 11864}, {5877, 26958}, {5972, 6130}, {6145, 32372}, {6776, 39886}, {7160, 12789}, {7740, 18279}, {10266, 12798}, {11657, 16186}, {13494, 13495}, {22466, 22943}, {25641, 38625}, {34297, 38794}

X(402) = midpoint of X(i) and X(j) for these {i,j}: {1, 12438}, {2, 1651}, {3, 11251}, {4, 12113}, {5, 32162}, {6, 12583}, {8, 12626}, {13, 12793}, {14, 12792}, {17, 22897}, {18, 22852}, {40, 12696}, {54, 12797}, {64, 12791}, {67, 32279}, {68, 12418}, {74, 12369}, {76, 12794}, {79, 16129}, {80, 12729}, {83, 12795}, {84, 12668}, {98, 12181}, {99, 13179}, {100, 13268}, {104, 12752}, {110, 13212}, {112, 13281}, {190, 24830}, {262, 22698}, {265, 12790}, {402, 402}, {485, 12800}, {486, 12799}, {671, 12347}, {1297, 12796}, {1327, 13689}, {1328, 13809}, {1650, 4240}, {3081, 11050}, {6145, 32372}, {6776, 39886}, {7160, 12789}, {7740, 18279}, {10266, 12798}, {11049, 34582}, {11831, 11852}, {11832, 11853}, {11839, 11885}, {11845, 11897}, {11848, 11909}, {11863, 26383}, {11864, 26407}, {11901, 19017}, {11902, 19018}, {11903, 11906}, {11904, 11905}, {11907, 26447}, {11908, 26448}, {11911, 26451}, {11912, 26452}, {11913, 26453}, {11914, 11915}, {13494, 13495}, {13894, 26449}, {13948, 26450}, {16210, 16212}, {18507, 18508}, {18958, 22755}, {22466, 22943}
X(402) = reflection of X(i) in X(j) for these {i,j}: {11049, 2}, {34582, 1651}, {1650, 15184}, {15774, 32162}
X(402) = complement of X(1650)
X(402) = anticomplement of X(15184)
X(402) = isotomic conjugate of the anticomplement of X(32750)
X(402) = X(i)-complementary conjugate of X(j) for these (i,j): {162, 16177}, {1304, 34846}, {15459, 21253}, {24000, 113}, {32640, 16595}, {32676, 39008}, {32695, 8287}, {32715, 16573}, {36034, 122}, {36117, 37985}, {36131, 15526}
X(402) = X(i)-Ceva conjugate of X(j) for these (i,j): {5972, 31945}, {9033, 30}, {13494, 523}
X(402) = X(32750)-cross conjugate of X(2)
X(402) = X(i)-isoconjugate of X(j) for these (i,j): {74, 9390}, {1304, 9392}, {2159, 15351}
X(402) = X(i)-reciprocal conjugate of X(j) for these (i,j): {30, 15351}, {1494, 39352}, {2173, 9390}, {2349, 2629}, {2631, 9392}, {15184, 32750}, {16077, 39062}, {34767, 38240}
X(402) = barycentric product X(i)*X(j) for these {i,j}: {30, 39352}, {2629, 14206}, {4240, 38240}, {9033, 39062}
X(402) = barycentric quotient X(i)/X(j) for these {i,j}: {30, 15351}, {2173, 9390}, {2629, 2349}, {2631, 9392}, {32750, 15184}, {38240, 34767}, {39062, 16077}, {39352, 1494}
X(402) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 11852, 12438}, {1, 26452, 11915}, {1, 26453, 11914}, {2, 1650, 15184}, {2, 4240, 1650}, {3, 11911, 11251}, {4, 11845, 12113}, {8, 16212, 12626}, {381, 18508, 18507}, {1650, 1651, 4240}, {1650, 15183, 2}, {1650, 15184, 11049}, {1651, 15183, 1650}, {4240, 15183, 15184}, {11251, 26451, 3}, {11831, 12438, 1}, {11897, 12113, 4}, {11901, 11902, 12583}, {11912, 11913, 1}, {11912, 12438, 11914}, {11913, 12438, 11915}, {12626, 16210, 8}, {19017, 19018, 6}, {26452, 26453, 12438}


X(403) = X(36) OF THE ORTHIC TRIANGLE

Trilinears    (sec A)(1 + cos 2B + cos 2C) : :
Barycentrics    (tan A)(1 + cos 2B + cos 2C) : :

As a point on the Euler line, X(403) has Shinagawa coefficients (2F, 2F - E).

X(403) is centroid of the triangle having vertices X(4), P(4), U(4). (Regarding the bicentric pair P(4) and U(4), see the notes just before X(1908)). (Randy Hutson, 9/23/2011)

Let A' be the reflection in BC of the A-vertex of the tangential triangle, and define B' and C' cyclically. The circumcircles of AB'C', BC'A', and CA'B' concur at the isogonal conjugate of X(403). Also, X(403) is the perspector of ABC and the reflection of the anticevian triangle of X(4) in the orthic axis (trilinear polar of X(4)). (Randy Hutson, July 23, 2015)

X(403) lies on these lines: 2,3   112,230   115,232   847,1093

X(403) = midpoint of X(4) and X(186)
X(403) = reflection of X(186) in X(468)
X(403) = isogonal conjugate of X(5504)
X(403) = complement of X(2071)
X(403) = anticomplement of X(10257)
X(403) = inverse-in-circumcircle of X(24)
X(403) = inverse-in-nine-point-circle of X(4)
X(403) = inverse-in-orthocentroidal-circle of X(378)
X(403) = anticomplementary-circle-inverse of X(34938)
X(403) = X(113)-cross conjugate of X(4)
X(403) = crossdifference of every pair of points on line X(577)X(647)
X(403) = X(36)-of-orthic-triangle if and only if ABC is acute
X(403) = X(186)-of Euler-triangle
X(403) = {X(4),X(5)}-harmonic conjugate of X(1594)
X(403) = inverse-in-polar-circle of X(3)
X(403) = inverse-in-{circumcircle, nine-point circle}}-inverter of X(25)
X(403) = reflection of X(186) in the orthic axis
X(403) = pole wrt polar circle of trilinear polar of X(2986)
X(403) = X(48)-isoconjugate (polar conjugate) of X(2986)
X(403) = perspector of the orthic triangle and the reflection of the Euler triangle in the Euler line
X(403) = radical trace of polar and second Droz-Farny circles
X(403) = inverse-in-second-Droz-Farny-circle of X(4)
X(403) = excentral-to-ABC functional image of X(36)
X(403) = Euler line intercept, other than X(403), of circle {X(403),X(858),PU(4)}


X(404) = {X(2),X(3)}-HARMONIC CONJUGATE OF X(21)

Barycentrics    a*(a^3 - a*b^2 + a*b*c + b^2*c - a*c^2 + b*c^2) : :
X(404) = 3 X[2] + X[37256], 3 X[2] - 4 X[52264], 5 X[631] - X[6903], 2 X[4187] + X[37256], X[5046] - 4 X[52264], 3 X[16370] - 5 X[19525], 2 X[37251] + X[37403], X[37256] + 4 X[52264], X[5330] - 4 X[17614], X[4420] + 2 X[32636]. 3*R*X[2] - 2*r*X[3].

As a point on the Euler line, X(404) has Shinagawa coefficients (abc*$a$ - 2S2, 2S2).

X(404) lies on these lines: {1, 88}, {2, 3}, {6, 19769}, {7, 1259}, {8, 56}, {9, 4652}, {10, 36}, {11, 6691}, {12, 3035}, {15, 5367}, {16, 5362}, {31, 978}, {32, 33854}, {35, 1125}, {38, 5293}, {39, 5276}, {40, 3877}, {41, 17754}, {42, 37607}, {43, 1468}, {46, 997}, {55, 3616}, {57, 78}, {58, 3216}, {60, 662}, {63, 936}, {65, 4511}, {69, 1014}, {72, 3218}, {73, 17074}, {75, 19850}, {79, 48698}, {80, 14800}, {81, 386}, {85, 6516}, {86, 5132}, {95, 1441}, {99, 18140}, {101, 16549}, {104, 355}, {105, 6012}, {108, 318}, {119, 18861}, {141, 5096}, {145, 999}, {149, 496}, {165, 5250}, {169, 26690}, {171, 1193}, {172, 1575}, {182, 15988}, {183, 34284}, {191, 5131}, {194, 16997}, {198, 5749}, {200, 3361}, {216, 40582}, {226, 27385}, {229, 25645}, {230, 44542}, {238, 27627}, {239, 37609}, {271, 37141}, {274, 1078}, {283, 13329}, {348, 41826}, {388, 1470}, {391, 5120}, {392, 3579}, {394, 36745}, {484, 3878}, {498, 14793}, {499, 11680}, {513, 38541}, {515, 24982}, {516, 41012}, {517, 5330}, {518, 4420}, {519, 5563}, {528, 37722}, {529, 21031}, {535, 4325}, {551, 3746}, {574, 5283}, {579, 2287}, {593, 5956}, {594, 21773}, {595, 49997}, {603, 651}, {612, 988}, {613, 25304}, {614, 11512}, {644, 3501}, {758, 3336}, {765, 20615}, {849, 3453}, {894, 22345}, {899, 5247}, {902, 28352}, {908, 4292}, {910, 33950}, {912, 26877}, {934, 9312}, {940, 4255}, {942, 5440}, {943, 27186}, {944, 5554}, {946, 2077}, {950, 6692}, {952, 37535}, {956, 3617}, {958, 4413}, {960, 1155}, {962, 10310}, {965, 37500}, {966, 36743}, {970, 3917}, {975, 28606}, {976, 982}, {990, 26669}, {992, 2305}, {993, 1698}, {995, 5264}, {1001, 5217}, {1030, 17398}, {1038, 17080}, {1042, 9364}, {1046, 5529}, {1055, 2329}, {1056, 10528}, {1058, 10586}, {1104, 16610}, {1124, 9679}, {1149, 37588}, {1150, 9534}, {1151, 31473}, {1153, 7621}, {1156, 15297}, {1191, 37540}, {1201, 5255}, {1213, 5124}, {1220, 23361}, {1260, 21454}, {1290, 2758}, {1292, 9083}, {1319, 4861}, {1329, 5080}, {1330, 5741}, {1333, 46838}, {1385, 3753}, {1388, 15813}, {1389, 46920}, {1420, 1706}, {1428, 17792}, {1437, 5012}, {1444, 5224}, {1445, 41228}, {1447, 20880}, {1465, 4296}, {1473, 26065}, {1475, 3684}, {1478, 11681}, {1479, 10200}, {1482, 45977}, {1483, 12331}, {1490, 11220}, {1574, 5291}, {1582, 28288}, {1603, 10829}, {1612, 26724}, {1617, 5265}, {1633, 4676}, {1652, 44689}, {1653, 44688}, {1655, 7783}, {1696, 3161}, {1724, 4257}, {1730, 3430}, {1737, 5086}, {1768, 31803}, {1770, 5057}, {1781, 25078}, {1792, 18134}, {1809, 34406}, {1834, 37634}, {1836, 25681}, {1914, 16604}, {1936, 22072}, {1975, 18135}, {1993, 36754}, {1994, 37509}, {2093, 11682}, {2108, 30649}, {2140, 25532}, {2178, 2345}, {2220, 39798}, {2223, 16823}, {2238, 33863}, {2275, 4386}, {2277, 2298}, {2292, 17596}, {2295, 21008}, {2303, 4261}, {2346, 38053}, {2352, 19804}, {2550, 7288}, {2646, 3812}, {2886, 5172}, {2893, 5740}, {2933, 4429}, {2979, 5752}, {3011, 24178}, {3054, 44517}, {3060, 37482}, {3075, 3562}, {3085, 8071}, {3086, 3434}, {3188, 40702}, {3193, 37530}, {3219, 3916}, {3220, 17353}, {3241, 3304}, {3244, 48696}, {3286, 17277}, {3294, 24047}, {3295, 3622}, {3303, 4421}, {3305, 31424}, {3333, 3870}, {3337, 3874}, {3338, 3811}, {3421, 20076}, {3436, 4293}, {3474, 11415}, {3485, 11509}, {3486, 11502}, {3487, 11517}, {3496, 39244}, {3509, 33299}, {3550, 3915}, {3555, 3935}, {3576, 3897}, {3582, 24387}, {3583, 3825}, {3585, 3814}, {3589, 4265}, {3600, 7080}, {3601, 5437}, {3614, 31235}, {3615, 25533}, {3618, 36740}, {3623, 7373}, {3624, 5010}, {3626, 5288}, {3632, 37587}, {3634, 5251}, {3635, 37602}, {3670, 30115}, {3673, 26229}, {3678, 4973}, {3679, 8666}, {3689, 34791}, {3695, 33168}, {3698, 37605}, {3701, 5205}, {3702, 32932}, {3705, 5300}, {3720, 37573}, {3724, 49598}, {3742, 37080}, {3744, 52541}, {3745, 4719}, {3749, 28011}, {3752, 5262}, {3785, 45962}, {3786, 16574}, {3813, 34612}, {3816, 6284}, {3819, 15489}, {3820, 20067}, {3822, 14792}, {3826, 24953}, {3833, 35016}, {3846, 28268}, {3880, 20323}, {3884, 11010}, {3890, 5119}, {3898, 37563}, {3911, 6734}, {3918, 51111}, {3920, 37592}, {3923, 25591}, {3924, 24174}, {3925, 4999}, {3928, 3951}, {3937, 29958}, {3938, 3976}, {3940, 23958}, {3957, 5045}, {3971, 8720}, {3980, 11688}, {3987, 15955}, {4004, 50194}, {4067, 4880}, {4084, 4867}, {4251, 35342}, {4252, 4383}, {4262, 16783}, {4276, 25526}, {4279, 18792}, {4297, 8582}, {4298, 6745}, {4301, 5537}, {4304, 9843}, {4315, 6736}, {4317, 34605}, {4358, 7283}, {4388, 27657}, {4391, 22091}, {4401, 28591}, {4417, 5323}, {4482, 29691}, {4512, 12511}, {4552, 51879}, {4640, 25917}, {4646, 17015}, {4671, 50044}, {4720, 10449}, {4723, 9369}, {4853, 13462}, {4911, 33864}, {4968, 7081}, {5013, 5275}, {5021, 37657}, {5022, 37658}, {5030, 16552}, {5061, 9565}, {5082, 10529}, {5088, 26563}, {5128, 15829}, {5156, 27642}, {5175, 5704}, {5178, 10916}, {5218, 26357}, {5221, 12635}, {5223, 53057}, {5241, 15447}, {5249, 12436}, {5252, 34880}, {5256, 37554}, {5259, 19862}, {5263, 20470}, {5266, 7191}, {5278, 19762}, {5289, 37567}, {5294, 7293}, {5297, 37599}, {5298, 49732}, {5326, 6668}, {5422, 36742}, {5425, 33815}, {5432, 25466}, {5434, 12607}, {5439, 24929}, {5450, 5587}, {5482, 18180}, {5535, 31806}, {5603, 11248}, {5650, 46623}, {5657, 11249}, {5690, 22765}, {5692, 11684}, {5698, 30295}, {5703, 9776}, {5718, 26131}, {5720, 12528}, {5722, 9963}, {5730, 36279}, {5731, 11500}, {5743, 26064}, {5746, 27395}, {5777, 17616}, {5784, 37787}, {5790, 32153}, {5794, 24914}, {5818, 22758}, {5880, 8543}, {5881, 38669}, {5884, 6326}, {5885, 22935}, {5886, 26285}, {5901, 11849}, {5902, 22836}, {5903, 30144}, {5927, 34862}, {5949, 15109}, {5985, 12042}, {6048, 9350}, {6147, 26842}, {6174, 15888}, {6224, 10950}, {6244, 20070}, {6265, 35004}, {6361, 35238}, {6600, 11038}, {6645, 26752}, {6667, 7173}, {6679, 28267}, {6681, 25639}, {6684, 11012}, {6690, 52793}, {6701, 38062}, {6713, 26470}, {6735, 10106}, {6762, 46917}, {7123, 9374}, {7176, 30806}, {7284, 32635}, {7292, 37589}, {7681, 11826}, {7686, 50371}, {7688, 24564}, {7702, 45393}, {7741, 10058}, {7742, 19843}, {7754, 17001}, {7793, 16998}, {7951, 51506}, {7967, 16203}, {7998, 26637}, {8143, 9978}, {8168, 20053}, {8257, 10394}, {8568, 32625}, {8648, 25380}, {8669, 24165}, {9436, 38859}, {9579, 30827}, {9581, 31190}, {9612, 30852}, {9657, 11236}, {9671, 34706}, {9708, 46933}, {9710, 41341}, {9711, 34606}, {9812, 25893}, {9850, 46677}, {9856, 17613}, {9859, 10396}, {9940, 18444}, {9945, 12433}, {9956, 23961}, {9961, 16209}, {10031, 37727}, {10072, 49719}, {10074, 12531}, {10107, 11011}, {10129, 37692}, {10165, 10902}, {10199, 10707}, {10246, 32141}, {10306, 24558}, {10428, 14513}, {10446, 24540}, {10448, 17124}, {10459, 37617}, {10572, 14803}, {10584, 10591}, {10595, 10679}, {10601, 36746}, {10609, 37730}, {10680, 12245}, {10698, 25413}, {10884, 37526}, {10914, 24928}, {11227, 40262}, {11230, 26086}, {11240, 33925}, {11246, 14450}, {11263, 35204}, {11281, 31660}, {11374, 31019}, {11495, 52653}, {11698, 35451}, {12019, 51636}, {12047, 20292}, {12512, 40998}, {12526, 53056}, {12527, 20103}, {12609, 39599}, {12632, 52804}, {12667, 30513}, {12702, 35272}, {12740, 25414}, {12746, 37717}, {12757, 17857}, {12773, 37705}, {12943, 31246}, {13205, 13463}, {13323, 43650}, {13384, 45036}, {13397, 40101}, {13528, 45776}, {13624, 35271}, {13751, 41541}, {13902, 44590}, {13959, 44591}, {14419, 16158}, {14804, 47033}, {14828, 17169}, {14882, 15950}, {14986, 17784}, {15018, 51340}, {15171, 20066}, {15299, 25722}, {15325, 24390}, {15338, 26127}, {15500, 17102}, {15625, 18613}, {16020, 26241}, {16133, 42843}, {16466, 17126}, {16468, 36646}, {16474, 50587}, {16483, 28370}, {16589, 37512}, {16691, 20475}, {16693, 23851}, {16706, 41230}, {16817, 24589}, {16830, 37575}, {16945, 31343}, {17011, 37594}, {17063, 28082}, {17103, 37678}, {17123, 28257}, {17127, 27625}, {17129, 40908}, {17282, 51687}, {17303, 38871}, {17349, 37507}, {17350, 23085}, {17355, 38869}, {17369, 19297}, {17379, 37502}, {17483, 24470}, {17619, 18480}, {17733, 32860}, {17757, 18990}, {17776, 27802}, {17825, 37501}, {17923, 41227}, {18048, 24349}, {18139, 25650}, {18141, 37538}, {18357, 26321}, {18391, 22766}, {18446, 37534}, {18465, 33852}, {18524, 34773}, {18755, 24512}, {18788, 35269}, {18999, 22764}, {19000, 22763}, {19065, 44606}, {19066, 44607}, {19684, 19763}, {19701, 19760}, {19716, 19764}, {19730, 19757}, {19732, 19759}, {19765, 37674}, {19784, 39582}, {19785, 19845}, {19786, 19842}, {19792, 19848}, {19806, 19840}, {19808, 19841}, {19822, 19844}, {19836, 37557}, {19851, 24620}, {19858, 39578}, {19872, 32633}, {19874, 52139}, {19878, 25542}, {20018, 37684}, {20244, 24203}, {20990, 32922}, {20999, 26073}, {21258, 26140}, {21740, 34339}, {22300, 50362}, {22344, 27064}, {22791, 35000}, {23206, 26223}, {23383, 26094}, {23537, 33133}, {23850, 24988}, {24271, 30819}, {24320, 26685}, {24328, 35578}, {24410, 40577}, {24440, 49487}, {24602, 29960}, {24703, 24954}, {24850, 25079}, {24883, 37646}, {24920, 42741}, {25082, 40131}, {25385, 30362}, {25441, 48843}, {25522, 41869}, {25881, 35263}, {25934, 37537}, {25954, 26531}, {26035, 26244}, {26040, 30478}, {26078, 39200}, {26102, 37574}, {26286, 26446}, {26363, 33108}, {26492, 37820}, {26582, 26686}, {26866, 42461}, {27005, 51862}, {27006, 51775}, {27065, 31445}, {27645, 30653}, {27665, 40432}, {28385, 41346}, {30143, 37571}, {30147, 37525}, {30818, 50054}, {31339, 32916}, {31419, 41345}, {31435, 35242}, {31938, 41547}, {32157, 45081}, {32537, 37829}, {32772, 35206}, {32863, 41014}, {32935, 52923}, {32939, 42705}, {32945, 50608}, {34124, 38603}, {34545, 36750}, {34701, 37723}, {37469, 37732}, {37492, 51171}, {37566, 37789}, {37612, 37700}, {37621, 38028}, {37656, 49716}, {37662, 49745}, {37687, 52680}, {38052, 52769}, {39199, 48243}, {39226, 48228}, {39476, 47794}, {40592, 52782}, {40980, 41806}, {41542, 52126}, {44408, 47793}, {46188, 46973}, {46635, 52200}, {47795, 48386}, {47796, 48387}, {48165, 48390}, {48204, 48382}, {48205, 48384}, {48207, 48389}, {48209, 48391}, {48246, 48383}, {50443, 53055}, {50843, 51525}

X(404) = midpoint of X(i) and X(j) for these {i,j}: {3, 37251}, {21, 35982}, {5046, 37256}
X(404) = reflection of X(i) in X(j) for these {i,j}: {1, 51714}, {4187, 52264}, {5046, 4187}, {37403, 3}
X(404) = complement of X(5046)
X(404) = anticomplement of X(4187)
X(404) = circumcircle-inverse of X(37919)
X(404) = orthocentroidal-circle-inverse of X(4193)
X(404) = isotomic conjugate of the isogonal conjugate of X(44085)
X(404) = isogonal conjugate of the isotomic conjugate of X(44139)
X(404) = X(270)-Ceva conjugate of X(3868)
X(404) = X(404) = X(i)-isoconjugate of X(j) for these (i,j): {56, 44040}, {3064, 40518}
X(i)-Dao conjugate of X(j) for these (i,j): {1, 44040}, {404, 5046}, {1459, 7004}, {3670, 3454}
X(404) = cevapoint of X(3) and X(3216)
X(404) = crossdifference of every pair of points on line {647, 1635}
X(404) = barycentric product X(i)*X(j) for these {i,j}: {1, 32939}, {6, 44139}, {28, 42705}, {76, 44085}, {100, 47796}, {190, 48281}, {651, 20293}, {4554, 48387}, {4564, 44311}, {21721, 52935}
X(404) = barycentric quotient X(i)/X(j) for these {i,j}: {9, 44040}, {20293, 4391}, {21721, 4036}, {32939, 75}, {36059, 40518}, {39006, 7004}, {42705, 20336}, {44085, 6}, {44139, 76}, {44311, 4858}, {47796, 693}, {48281, 514}, {48387, 650}
X(404) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 100, 3871}, {1, 1054, 24443}, {1, 14923, 1320}, {1, 25440, 100}, {2, 3, 21}, {2, 4, 4193}, {2, 20, 2478}, {2, 21, 5047}, {2, 377, 2476}, {2, 384, 17541}, {2, 405, 17536}, {2, 443, 4197}, {2, 474, 17531}, {2, 1004, 36002}, {2, 2475, 5}, {2, 2476, 7504}, {2, 3091, 6931}, {2, 3146, 6919}, {2, 3522, 452}, {2, 3523, 6910}, {2, 3552, 16916}, {2, 4188, 3}, {2, 4189, 405}, {2, 4190, 4}, {2, 4195, 5192}, {2, 4197, 31254}, {2, 4201, 5051}, {2, 5046, 4187}, {2, 5047, 17534}, {2, 5141, 1656}, {2, 5177, 6933}, {2, 6636, 37325}, {2, 6655, 17669}, {2, 6871, 3090}, {2, 6872, 5084}, {2, 6904, 377}, {2, 6921, 17566}, {2, 7791, 17550}, {2, 10304, 31156}, {2, 11108, 17546}, {2, 11112, 17577}, and many others


X(405) = EULER LINE INTERCEPT OF LINE X(1)X(6)

Trilinears    bc(a + b + c) + abc cos A : :
Trilinears    bcS + raSA : :
Barycentrics   b + c + a(1 + cos A) : :
Barycentrics   a*(a^3 - a*b^2 - 2*a*b*c - 2*b^2*c - a*c^2 - 2*b*c^2) : :

As a point on the Euler line, X(405) has Shinagawa coefficients (abc*$a$ + S2, -S2).

X(405) lies on these lines: {1, 6}, {2, 3}, {8, 943}, {10, 55}, {11, 19755}, {12, 10198}, {31, 5711}, {32, 5275}, {34, 1214}, {35, 1376}, {36, 3624}, {38, 19729}, {39, 19758}, {40, 1730}, {46, 3812}, {51, 5752}, {56, 226}, {57, 3916}, {58, 940}, {63, 942}, {65, 1708}, {75, 7283}, {76, 16992}, {78, 3305}, {84, 8726}, {100, 5175}, {104, 5811}, {142, 4292}, {144, 11036}, {145, 6767}, {171, 19730}, {183, 18140}, {191, 5902}, {192, 19851}, {194, 17000}, {198, 5257}, {200, 3697}, {210, 3811}, {212, 2654}, {222, 1935}, {241, 1448}, {274, 1975}, {283, 5398}, {284, 965}, {306, 5814}, {329, 999}, {333, 10449}, {355, 10267}, {386, 4383}, {388, 1617}, {390, 5082}, {480, 6743}, {495, 3436}, {496, 10527}, {497, 19843}, {498, 1329}, {499, 3816}, {511, 19782}, {516, 5584}, {517, 5250}, {519, 3303}, {528, 4309}, {535, 9657}, {551, 3304}, {572, 2360}, {579, 4877}, {580, 2328}, {595, 5710}, {612, 5266}, {614, 19724}, {748, 1193}, {756, 976}, {846, 986}, {908, 11374}, {920, 13750}, {936, 3601}, {938, 5273}, {946, 3428}, {952, 16202}, {962, 5759}, {966, 4254}, {968, 3931}, {970, 5943}, {971, 10884}, {978, 17123}, {988, 5272}, {990, 12689}, {997, 1864}, {1043, 9534}, {1046, 7262}, {1071, 7330}, {1175, 6061}, {1210, 5745}, {1259, 5722}, {1319, 9850}, {1330, 18134}, {1351, 15988}, {1377, 5414}, {1378, 2066}, {1385, 5777}, {1398, 6356}, {1437, 6176}, {1465, 19372}, {1466, 3911}, {1468, 3720}, {1470, 5433}, {1478, 7742}, {1479, 2886}, {1482, 3877}, {1486, 4026}, {1490, 3576}, {1573, 2241}, {1577, 21789}, {1612, 5716}, {1655, 7754}, {1696, 3986}, {1697, 9623}, {1706, 4002}, {1709, 9943}, {1712, 17102}, {1714, 1834}, {1722, 17594}, {1737, 11507}, {1746, 5786}, {1750, 7987}, {1770, 5880}, {1824, 9895}, {1836, 12609}, {1858, 7082}, {1901, 17398}, {1993, 22136}, {2067, 9678}, {2078, 9578}, {2093, 4004}, {2098, 3884}, {2099, 3878}, {2177, 3214}, {2238, 2271}, {2268, 5783}, {2280, 3691}, {2283, 19931}, {2292, 3924}, {2295, 14974}, {2346, 5686}, {2452, 13869}, {2550, 4294}, {2551, 3085}, {2893, 5224}, {2900, 3740}, {2932, 3035}, {3036, 10087}, {3052, 5264}, {3053, 5277}, {3158, 3921}, {3189, 6600}, {3216, 4255}, {3218, 5708}, {3219, 3868}, {3220, 17306}, {3244, 19750}, {3286, 15668}, {3306, 4652}, {3333, 10582}, {3338, 3742}, {3339, 5665}, {3434, 15171}, {3452, 13411}, {3556, 19869}, {3562, 22117}, {3600, 7677}, {3617, 3871}, {3618, 19766}, {3622, 7373}, {3626, 19751}, {3634, 4413}, {3635, 8162}, {3636, 19739}, {3647, 5221}, {3666, 19728}, {3670, 17054}, {3678, 3715}, {3679, 3746}, {3682, 14547}, {3685, 16824}, {3689, 3983}, {3711, 4015}, {3757, 4385}, {3763, 4265}, {3814, 5172}, {3820, 5552}, {3821, 12579}, {3822, 10895}, {3826, 4302}, {3841, 12953}, {3872, 9957}, {3876, 3940}, {3890, 4861}, {3897, 10246}, {3898, 22837}, {3915, 10459}, {3922, 5183}, {3923, 12567}, {3925, 6284}, {3929, 11518}, {3951, 11520}, {3962, 12559}, {4018, 11529}, {4256, 17749}, {4267, 5737}, {4295, 5698}, {4297, 7700}, {4304, 6666}, {4305, 10609}, {4306, 6180}, {4313, 5809}, {4340, 4648}, {4354, 9640}, {4391, 22160}, {4421, 19875}, {4533, 12260}, {4647, 5695}, {4666, 5045}, {4668, 8168}, {4673, 16821}, {4857, 11235}, {5080, 9654}, {5119, 5836}, {5120, 5746}, {5132, 17259}, {5204, 5267}, {5252, 11510}, {5253, 5550}, {5255, 8616}, {5263, 19853}, {5271, 5295}, {5282, 21808}, {5294, 7085}, {5316, 6700}, {5330, 10247}, {5362, 11485}, {5367, 11486}, {5426, 5506}, {5432, 10958}, {5437, 15803}, {5444, 15446}, {5450, 6260}, {5492, 7986}, {5534, 18908}, {5537, 9588}, {5554, 5690}, {5563, 11194}, {5587, 10902}, {5603, 5758}, {5640, 19771}, {5657, 10306}, {5691, 15931}, {5703, 18228}, {5705, 9581}, {5715, 8227}, {5731, 5817}, {5750, 8804}, {5766, 9785}, {5774, 17751}, {5779, 12528}, {5794, 10572}, {5812, 5886}, {5815, 10578}, {5818, 11491}, {5844, 12000}, {5901, 10680}, {5905, 6147}, {5934, 8109}, {5935, 8110}, {5985, 12188}, {6261, 12664}, {6265, 12691}, {6554, 15288}, {6598, 15175}, {6667, 10090}, {6684, 8582}, {6688, 15489}, {6738, 18249}, {6763, 18398}, {6764, 8236}, {6765, 10389}, {6796, 10175}, {7171, 17612}, {7587, 7593}, {7588, 8080}, {7713, 10319}, {7793, 16999}, {8053, 16828}, {8062, 23189}, {8077, 8079}, {8158, 21168}, {8225, 8233}, {8543, 12848}, {8983, 19014}, {9659, 14667}, {9669, 11680}, {9778, 11024}, {9812, 15911}, {9841, 10857}, {9956, 11499}, {10039, 11508}, {10056, 12607}, {10176, 18233}, {10283, 12001}, {10386, 20075}, {10392, 12447}, {10441, 17185}, {10585, 10592}, {10822, 21746}, {10827, 14798}, {10882, 10888}, {10966, 11376}, {11220, 12684}, {11372, 12565}, {11396, 21318}, {11502, 17606}, {12246, 21151}, {12330, 14647}, {12388, 12397}, {12433, 12649}, {12520, 12688}, {12521, 12692}, {12522, 12693}, {12523, 12694}, {12524, 12695}, {12527, 21620}, {12704, 13374}, {12739, 18254}, {13405, 18250}, {13883, 18999}, {13936, 19000}, {13971, 19013}, {15239, 21164}, {15808, 19746}, {16112, 17653}, {16173, 22560}, {16819, 20172}, {16826, 19719}, {16830, 23407}, {16994, 17128}, {16996, 17129}, {17194, 17811}, {17300, 20077}, {17303, 19857}, {17349, 20018}, {17478, 21761}, {17718, 21077}, {18357, 18518}, {18481, 18761}, {19786, 19844}, {19808, 19845}, {19812, 19841}, {19827, 19842}, {19836, 22654}, {19863, 23361}

X(405) is the {X(2),X(3)}-harmonic conjugate of X(474). For a list of harmonic conjugates of X(405), click Tables at the top of this page.

X(405) = complement of X(377)
X(405) = anticomplement of X(8728)
X(405) = crosssum of X(838) and X(1015)
X(405) = crossdifference of every pair of points on line X(513)X(647)
X(405) = inverse-in-orthocentroidal circle of X(442)
X(405) = {X(1),X(9)}-harmonic conjugate of X(72)


X(406) = EULER LINE INTERCEPT OF LINE X(10)X(33)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(a + b + c) + abc sec A
Barycentrics  b + c + a(1 + sec A) : c + a + b(1 + sec B) : a + b + c(1 + sec C)

As a point on the Euler line, X(406) has Shinagawa coefficients ($a$F, abc).

X(406) lies on these lines: 2,3   8,1061   10,33   37,158   92,1068   108,388   208,226   261,317

X(406) = anticomplement of X(34120)
X(406) = polar conjugate of isogonal conjugate of X(36744)
X(406) = inverse-in-orthocentroidal-circle of X(475)


X(407) = CROSSPOINT OF X(4) AND X(225)

Trilinears       (v + w)sec A : (w + u)sec B : (u + v)sec C, where
                        u : v : w = X(21); e.g., u = u(A,B,C) = 1/(cos B + cos C)

Barycentrics  (v + w) tan A : (w + u) tan B : (u + v) tan C

As a point on the Euler line, X(407) has Shinagawa coefficients (FS2, -($aSA$)2 - FS2).

X(407) lies on these lines: 2,3   12,228   65,225   117,136

X(407) = crosspoint of X(4) and X(225)
X(407) = crosssum of X(i) and X(j) for these (i,j): (3,283), (21,411)
X(407) = tangential-to-orthic similarity image of X(3145)


X(408) = EULER LINE INTERCEPT OF LINE X(73)X(228)

Trilinears       (v + w)cos A : (w + u)cos B : (u + v)cos C, where
                        u : v : w = X(29); e.g., u = u(A,B,C) = (sec A)/(cos B + cos C)

Barycentrics  (v + w)sin 2A : (w + u)sin 2B : (u + v)sin2C

As a point on the Euler line, X(408) has Shinagawa coefficients (($bcSBSC$)2, -($bcSBSC$)2 + EFS4).

X(408) lies on these lines: 2,3   73,228

X(408) = crosssum of X(29) and X(412)


X(409) = EULER X(21)-1st-SUBSTITUTION POINT

Trilinears    u2 + vw : v2 + wu : w2 + uv, where u : v : w = X(21); e.g., u = u(A,B,C) = 1/(cos B + cos C)
Barycentrics    a(u2 + vw) : b(v2 + wu) : c(w2 + uv)
Barycentrics    a*(a + b)*(a + c)*(a^4 - a^3*b + a*b^3 - b^4 - a^3*c + a^2*b*c + a*b^2*c + b^3*c + a*b*c^2 + 4*b^2*c^2 + a*c^3 + b*c^3 - c^4) : :

As a point on the Euler line, X(409) has Shinagawa coefficients ($aSA$[($aSA$)2 - $bcSBSC$] + 2abcFS2, -$aSA$[($aSA$)2 - $bcSBSC$] + abc[($aSA$)2 + $bcSBSC$ - 2FS2]).

X(409) lies on these lines: {1, 1247}, {2, 3}, {10, 37816}, {46, 51290}, {58, 5883}, {65, 1098}, {81, 3924}, {86, 40980}, {100, 21674}, {229, 2607}, {643, 5836}, {662, 2646}, {759, 1125}, {950, 19642}, {993, 47059}, {1054, 34882}, {1104, 2363}, {1178, 1201}, {1958, 10448}, {2648, 9398}, {2886, 52360}, {3304, 36224}, {3754, 5127}, {3897, 37793}, {4653, 27784}, {5260, 27714}, {7354, 52361}, {9275, 30143}, {15792, 35016}, {17104, 30147}, {24619, 26532}, {34195, 37783}

X(409) = isogonal conjugate of X(9399)
X(409) = X(2648)-Ceva conjugate of X(2651)
X(409) = X(i)-isoconjugate of X(j) for these (i,j): {1, 9399}, {65, 9398}
X(409) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 9399}, {9398, 40602}
X(409) = barycentric product X(333)*X(2647)
X(409) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 9399}, {284, 9398}, {2647, 226}
X(409) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 2475, 27687}, {2, 11101, 21}, {3, 11116, 35991}, {21, 29, 413}, {21, 13746, 15776}, {21, 17518, 13588}, {21, 17568, 1817}, {21, 35991, 3}, {21, 37294, 17549}, {65, 1098, 2651}, {405, 17512, 21}, {662, 40430, 2646}, {1010, 36011, 21}, {11102, 19259, 21}, {11110, 37227, 21}, {11110, 52244, 7483}, {13725, 14015, 21}, {13739, 37228, 21}, {16418, 37029, 21}, {16865, 37032, 21}


X(410) = EULER X(29)-1st-SUBSTITUTION POINT

Trilinears       u2 + vw : v2 + wu : w2 + uv, where
                        u : v : w = X(29); e.g., u = u(A,B,C) = (sec A)/(cos B + cos C)
Barycentrics  a(u2 + vw) : b(v2 + wu) : c(w2 + uv)
Barycentrics    (a + b)*(a + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^8*b^2 - 3*a^6*b^4 + 3*a^4*b^6 - a^2*b^8 + 3*a^8*b*c - a^7*b^2*c - 5*a^6*b^3*c + a^5*b^4*c + a^4*b^5*c + a^3*b^6*c + a^2*b^7*c - a*b^8*c + a^8*c^2 - a^7*b*c^2 - 3*a^6*b^2*c^2 + a^5*b^3*c^2 + 2*a^4*b^4*c^2 + a^3*b^5*c^2 + a^2*b^6*c^2 - a*b^7*c^2 - b^8*c^2 - 5*a^6*b*c^3 + a^5*b^2*c^3 + 8*a^4*b^3*c^3 - 2*a^3*b^4*c^3 - a^2*b^5*c^3 + a*b^6*c^3 - 2*b^7*c^3 - 3*a^6*c^4 + a^5*b*c^4 + 2*a^4*b^2*c^4 - 2*a^3*b^3*c^4 + a*b^5*c^4 + b^6*c^4 + a^4*b*c^5 + a^3*b^2*c^5 - a^2*b^3*c^5 + a*b^4*c^5 + 4*b^5*c^5 + 3*a^4*c^6 + a^3*b*c^6 + a^2*b^2*c^6 + a*b^3*c^6 + b^4*c^6 + a^2*b*c^7 - a*b^2*c^7 - 2*b^3*c^7 - a^2*c^8 - a*b*c^8 - b^2*c^8) : :

As a point on the Euler line, X(410) has Shinagawa coefficients ([S6 - (2E - F)FS4 + 3$bc$FS4 - 3$bcSBSC$FS2]F, -[S4 - 4(E - F)FS2 + 3$bc$FS2]FS2 + $bcSBSC$[S4 - 2(E - 2F)F S2 + abc$aSA$F]).

X(410) lies on these lines: 2,3   73,2659   823,2660  

X(410) lies on these lines: {1, 1248}, {2, 3}, {73, 2659}, {823, 2654}

X(410) = X(2656)-Ceva conjugate of X(2659)
X(410) = barycentric product X(2662)*X(31623)
X(410) = barycentric quotient X(2662)/X(1214)
X(410) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {21, 29, 414}
X(410) = crosspoint of PU(82)


X(411) = EULER X(21)-2nd-SUBSTITUTION POINT

Trilinears       (1/v)2 + (1/w)2 - (1/u)2 : (1/w)2 + (1/u)2 - (1/v)2 : (1/u)2 + (1/v)2 - (1/w)2, where
                        u : v : w = X(21); e.g., u = u(A,B,C) = 1/(cos B + cos C)
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = cos B cos C - (cos A + cos B + cos C)cos A
Barycentrics  a[(1/v)2 + (1/w)2 - (1/u)2] : b[(1/w)2 + (1/u)2 - (1/v)2] : c[(1/u)2 + (1/v)2 - (1/w)2]
Barycentrics    a*(a^6 - a^5*b - 2*a^4*b^2 + 2*a^3*b^3 + a^2*b^4 - a*b^5 - a^5*c - a^4*b*c + a*b^4*c + b^5*c - 2*a^4*c^2 + 2*a^2*b^2*c^2 + 2*a^3*c^3 - 2*b^3*c^3 + a^2*c^4 + a*b*c^4 - a*c^5 + b*c^5) : :
X(411) = 3 X[2] - 4 X[52265], 4 X[3] - 3 X[17549], 2 X[4] - 3 X[17577], 5 X[3091] - 6 X[17530], 5 X[3522] - 3 X[37299], 7 X[3523] - 6 X[37298], 4 X[6842] - 3 X[17577], X[6895] - 4 X[52265], 2 X[6906] - 3 X[17549]

As a point on the Euler line, X(411) has Shinagawa coefficients ($aSA$, -$aSA$ - abc).

X(411) lies on these lines: {1, 1254}, {2, 3}, {8, 3428}, {10, 44425}, {32, 40129}, {35, 516}, {36, 1210}, {40, 78}, {46, 12520}, {54, 34800}, {55, 962}, {56, 938}, {57, 10393}, {63, 1490}, {64, 34259}, {65, 45230}, {73, 1936}, {81, 581}, {84, 4652}, {90, 1156}, {104, 18481}, {108, 1895}, {145, 22770}, {165, 936}, {170, 2108}, {171, 4300}, {185, 970}, {191, 31803}, {198, 27382}, {212, 37694}, {225, 45231}, {243, 821}, {255, 651}, {273, 17134}, {285, 46021}, {329, 1259}, {386, 1754}, {388, 26357}, {391, 46012}, {496, 41345}, {497, 37579}, {515, 2975}, {517, 3871}, {573, 2287}, {580, 32911}, {774, 1758}, {934, 34059}, {942, 18444}, {943, 11374}, {944, 11249}, {946, 1621}, {950, 37583}, {965, 37499}, {971, 3916}, {991, 37522}, {993, 5691}, {1014, 5738}, {1035, 18623}, {1044, 9316}, {1064, 3072}, {1071, 3218}, {1125, 15931}, {1155, 1858}, {1158, 9961}, {1193, 37570}, {1290, 2695}, {1292, 1311}, {1376, 5584}, {1437, 34148}, {1445, 5732}, {1446, 5088}, {1476, 22767}, {1479, 36152}, {1617, 14986}, {1699, 5248}, {1742, 37603}, {1750, 31424}, {1764, 3430}, {1766, 27396}, {1768, 12845}, {1776, 1898}, {1780, 3216}, {1788, 11502}, {1792, 4417}, {1804, 5932}, {1861, 34851}, {1897, 20222}, {1935, 2635}, {2077, 5057}, {2078, 12053}, {2801, 6763}, {2829, 4996}, {2886, 6253}, {2897, 41005}, {2951, 5506}, {3035, 50031}, {3062, 51576}, {3075, 4303}, {3085, 40292}, {3086, 7677}, {3100, 17102}, {3219, 5777}, {3303, 5734}, {3304, 15933}, {3306, 8726}, {3341, 37141}, {3465, 44706}, {3474, 11509}, {3476, 10966}, {3561, 34035}, {3576, 5253}, {3579, 5887}, {3585, 14794}, {3616, 22753}, {3624, 38150}, {3647, 31871}, {3648, 9809}, {3652, 31828}, {3681, 17857}, {3746, 4301}, {3811, 41338}, {3817, 5259}, {3868, 5709}, {3873, 12704}, {3874, 5536}, {3876, 5720}, {3913, 34711}, {4255, 37537}, {4265, 29181}, {4293, 8071}, {4294, 8069}, {4295, 11507}, {4296, 46974}, {4299, 14793}, {4305, 22766}, {4316, 14792}, {4421, 34632}, {4511, 14110}, {4640, 12688}, {4855, 6282}, {5080, 11827}, {5096, 44882}, {5122, 31805}, {5131, 16767}, {5132, 5327}, {5172, 6284}, {5204, 5704}, {5218, 37601}, {5250, 10268}, {5251, 19925}, {5260, 5587}, {5267, 28164}, {5284, 8227}, {5288, 28236}, {5303, 5450}, {5396, 46441}, {5438, 37551}, {5440, 31793}, {5441, 14804}, {5443, 14799}, {5453, 45931}, {5493, 5537}, {5535, 5884}, {5603, 10267}, {5657, 11499}, {5687, 20007}, {5690, 18524}, {5693, 11684}, {5694, 16139}, {5698, 11495}, {5706, 19767}, {5715, 31266}, {5721, 24883}, {5736, 10446}, {5744, 9799}, {5752, 5889}, {5759, 11517}, {5812, 31053}, {5818, 18491}, {5819, 32561}, {5842, 15908}, {5907, 22076}, {5927, 31445}, {6198, 37565}, {6224, 22775}, {6361, 11248}, {6516, 40702}, {6684, 7688}, {6690, 31936}, {6700, 12512}, {7280, 10090}, {7288, 37578}, {7330, 21165}, {7354, 15844}, {7360, 52345}, {7967, 10680}, {7991, 8715}, {8273, 25524}, {8666, 38669}, {9342, 31423}, {9535, 19763}, {9778, 10310}, {9812, 11496}, {9940, 27003}, {10031, 12776}, {10058, 10724}, {10129, 41869}, {10164, 12617}, {10167, 37582}, {10171, 25542}, {10246, 45977}, {10306, 20070}, {10595, 16202}, {10860, 35242}, {11220, 41854}, {11424, 13323}, {11508, 30305}, {11531, 25439}, {11680, 48482}, {11849, 28174}, {12245, 20013}, {12699, 32613}, {12702, 32141}, {12705, 35258}, {13243, 24467}, {13369, 26877}, {13397, 32706}, {13464, 34486}, {14798, 30384}, {14829, 51978}, {15323, 29325}, {15489, 46850}, {15622, 16678}, {15852, 37539}, {16113, 48698}, {16193, 30284}, {17100, 24466}, {17613, 31663}, {17757, 31799}, {18219, 38399}, {18861, 38761}, {19716, 19769}, {19788, 19850}, {20171, 30273}, {21363, 51290}, {21635, 35204}, {22765, 34773}, {22791, 37621}, {23692, 40602}, {25406, 36741}, {26321, 28186}, {28146, 33862}, {29473, 37202}, {32760, 39599}, {34195, 37625}, {34461, 34462}, {35250, 37821}, {36740, 51212}, {37469, 48897}, {37562, 48363}, {37584, 37700}

X(411) = midpoint of X(20) and X(37437)
X(411) = reflection of X(i) in X(j) for these {i,j}: {1, 51717}, {4, 6842}, {2975, 11012}, {3871, 11491}, {6831, 52265}, {6895, 6831}, {6906, 3}, {34772, 33597}, {52367, 15908}
X(411) = complement of X(6895)
X(411) = anticomplement of X(6831)
X(411) = orthocentroidal-circle-inverse of X(6828)
X(411) = isotomic conjugate of the isogonal conjugate of X(44087)
X(411) = X(i)-Ceva conjugate of X(j) for these (i,j): {1098, 1}, {24032, 651}
X(411) = X(1630)-cross conjugate of X(34035)
X(411) = X(411)-Dao conjugate of X(6895)
X(411) = crosssum of X(661) and X(3270)
X(411) = ABC-to-excentral barycentric image of X(12)
X(411) = barycentric product X(i)*X(j) for these {i,j}: {8, 34035}, {75, 1630}, {76, 44087}, {92, 3561}
X(411) = barycentric quotient X(i)/X(j) for these {i,j}: {1630, 1}, {3561, 63}, {34035, 7}, {44087, 6}
X(411) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3, 6986}, {2, 4, 6828}, {2, 20, 6836}, {2, 3149, 6915}, {2, 3522, 37423}, {2, 4188, 37282}, {2, 6835, 6991}, {2, 6836, 6943}, {2, 6870, 6855}, {2, 6894, 5}, {2, 6895, 6831}, {2, 7538, 27378}, {2, 14953, 37389}, {2, 20846, 21}, {2, 35979, 404}, {2, 50695, 4}, {2, 50700, 6835}, {3, 4, 21}, {3, 5, 1006}, {3, 20, 6909}, {3, 382, 6914}, {3, 405, 37106}, {3, 474, 3523}, {3, 550, 37403}, {3, 1012, 4189}, {3, 2915, 7488}, {3, 3149, 2}, {3, 3560, 6875}, {3, 3651, 7411}, {3, 4192, 37399}, {3, 6905, 404}, {3, 6906, 17549}, {3, 6911, 631}, {3, 6924, 6940}, {3, 6942, 13587}, {3, 6985, 4}, {3, 7420, 4225}, {3, 7489, 5428}, {3, 7580, 20}, {3, 9122, 7520}, {3, 11479, 37246}, {3, 13743, 7508}, {3, 16117, 550}, {3, 19513, 19649}, {3, 19540, 13732}, {3, 19541, 405}, {3, 19543, 37431}, {3, 19548, 4220}, {3, 20420, 37306}, {3, 21669, 17574}, {3, 36002, 6912}, {3, 36558, 13731}, {3, 37034, 17928}, {3, 37251, 140}, {3, 37400, 37402}, {3, 37411, 1012}, {3, 37412, 37254}, {3, 37426, 3522}, {3, 45976, 549}, {3, 49127, 37328}, {4, 21, 6912}, {4, 376, 6868}, {4, 631, 6824}, {4, 2476, 7548}, {4, 3090, 6866}, {4, 6824, 10883}, {4, 6825, 2476}, {4, 6828, 52269}, {4, 6838, 6932}, {4, 6842, 17577}, {4, 6852, 6841}, {4, 6853, 5}, {4, 6855, 6870}, {4, 6856, 3091}, {4, 6857, 6837}, {4, 6868, 11114}, {4, 6874, 381}, {4, 6875, 3560}, {4, 6876, 3}, {4, 6942, 52270}, {4, 6985, 36002}, {4, 6988, 2}, {5, 1006, 5047}, {5, 28459, 6902}, {20, 6838, 4}, {20, 6960, 6840}, {20, 6962, 6943}, {20, 7580, 33557}, {20, 37421, 6925}, {21, 36002, 4}, {40, 6261, 3869}, {40, 6326, 31806}, {40, 6796, 100}, {40, 52026, 78}, {63, 1490, 12528}, {73, 1936, 3562}, {140, 6841, 6852}, {140, 6946, 17535}, {140, 37251, 6946}, {140, 37374, 6972}, {255, 1745, 651}, {376, 6927, 6865}, {376, 6942, 3}, {381, 21161, 16858}, {382, 6914, 21669}, {404, 7411, 3}, {404, 35976, 35977}, {405, 19541, 3091}, {431, 1885, 4}, {442, 20420, 6839}, {474, 16293, 2}, {474, 37284, 6857}, {546, 5428, 7489}, {550, 6922, 37428}, {550, 37406, 7491}, {580, 37732, 32911}, {581, 37530, 81}, {631, 6864, 2}, {631, 6911, 17531}, {946, 10902, 1621}, {1012, 37411, 3146}, {1071, 37623, 3218}, {1155, 1858, 7098}, {1158, 50528, 9961}, {1532, 31789, 5046}, {1532, 44238, 31789}, {1745, 51281, 255}, {2041, 2042, 6845}, {2478, 6848, 6945}, {2635, 22361, 1935}, {2915, 7515, 36018}, {3075, 4303, 17074}, {3086, 7742, 7677}, {3090, 6883, 17536}, {3091, 37106, 405}, {3145, 27622, 4224}, {3146, 4189, 1012}, {3146, 6871, 4}, {3149, 11344, 6835}, {3149, 20846, 6828}, {3428, 11500, 8}, {3522, 4188, 3}, {3522, 35986, 37426}, {3523, 6837, 6857}, {3523, 37105, 3}, {3545, 28466, 16861}, {3560, 6875, 21}, {3560, 6918, 6855}, {3627, 7508, 13743}, {3651, 6905, 3}, {3832, 16865, 6913}, {4186, 52271, 28376}, {4188, 35986, 3522}, {5004, 5005, 4224}, {5535, 16132, 5884}, {5709, 18446, 3868}, {6675, 8226, 6884}, {6825, 6869, 4}, {6826, 6889, 4197}, {6827, 6834, 4193}, {6830, 6863, 7504}, {6831, 52265, 2}, {6836, 6932, 52269}, {6836, 6962, 2}, {6840, 6960, 5}, {6848, 6987, 2478}, {6850, 6934, 17579}, {6851, 6954, 6833}, {6855, 6870, 6828}, {6857, 37284, 21}, {6865, 6927, 2}, {6865, 6942, 37301}, {6865, 37301, 6986}, {6868, 52270, 21}, {6876, 6985, 21}, {6878, 6896, 6887}, {6880, 6891, 17566}, {6880, 6899, 6891}, {6888, 37433, 8727}, {6903, 6949, 6882}, {6905, 6940, 6924}, {6906, 17577, 6912}, {6907, 37468, 2475}, {6908, 50701, 377}, {6909, 6915, 6943}, {6909, 6932, 6912}, {6909, 33557, 20}, {6910, 10431, 6847}, {6915, 6986, 2}, {6917, 6937, 6175}, {6919, 37313, 25875}, {6924, 6940, 404}, {6928, 6941, 37375}, {6953, 6992, 5084}, {6986, 36002, 52269}, {6988, 20846, 6986}, {6988, 50695, 6828}, {7411, 36002, 1005}, {7412, 37380, 29}, {7420, 7580, 37420}, {7466, 35981, 1005}, {7483, 8727, 6888}, {7491, 37406, 4}, {7513, 23512, 6836}, {11112, 37424, 37163}, {11344, 37229, 2}, {11499, 35239, 5657}, {11827, 18242, 5080}, {12511, 25440, 165}, {13738, 28077, 33849}, {13747, 50206, 2}, {14110, 37837, 4511}, {14782, 14783, 6861}, {16440, 16441, 1817}, {18481, 26286, 104}, {19520, 37240, 4208}, {27653, 35981, 21}, {28452, 37438, 6901}, {31793, 40262, 5440}, {31952, 37328, 37048}, {35986, 36003, 7411}, {37034, 37264, 11349}, {37229, 50700, 6915}, {37282, 37423, 6986}, {42789, 42790, 37117}


X(412) = EULER X(29)-2nd-SUBSTITUTION POINT

Trilinears       (1/v)2 + (1/w)2 - (1/u)2 : (1/w)2 + (1/u)2 - (1/v)2 : (1/u)2 + (1/v)2 - (1/w)2, where
                        u : v : w = X(29); e.g., u = u(A,B,C) = (sec A)/(cos B + cos C)
Barycentrics  a[(1/v)2 + (1/w)2 - (1/u)2] : b[(1/w)2 + (1/u)2 - (1/v)2] : c[(1/u)2 + (1/v)2 - (1/w)2]
Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^6 + a^5*b - 2*a^4*b^2 - 2*a^3*b^3 + a^2*b^4 + a*b^5 + a^5*c - a^4*b*c - a*b^4*c + b^5*c - 2*a^4*c^2 + 2*a^2*b^2*c^2 - 2*a^3*c^3 - 2*b^3*c^3 + a^2*c^4 - a*b*c^4 + a*c^5 + b*c^5) : :

As a point on the Euler line, X(412) has Shinagawa coefficients (FS2, -$bcSBSC$ - FS2).

X(412) lies on these lines: {1, 7138}, {2, 3}, {7, 7952}, {33, 10884}, {40, 92}, {46, 158}, {57, 1895}, {63, 318}, {64, 5786}, {65, 243}, {84, 7020}, {162, 580}, {165, 39585}, {185, 2659}, {208, 44697}, {225, 775}, {264, 8822}, {273, 7013}, {278, 962}, {515, 5174}, {516, 1838}, {572, 2326}, {579, 8748}, {820, 2655}, {938, 44695}, {946, 17923}, {1071, 1872}, {1118, 3474}, {1148, 36279}, {1155, 1940}, {1158, 1748}, {1445, 1712}, {1698, 39531}, {1715, 1896}, {1754, 8747}, {1784, 3336}, {1785, 3075}, {1788, 1857}, {1859, 9943}, {1861, 34831}, {1887, 10391}, {1897, 3868}, {1902, 7009}, {2322, 46011}, {3188, 36118}, {3362, 23707}, {3579, 39529}, {4313, 34231}, {5307, 11471}, {5435, 40836}, {5706, 41083}, {6198, 18444}, {6684, 39574}, {7282, 18650}, {16318, 45991}, {17606, 42387}, {45141, 45985}

X(412) = orthocentroida1-circle-inverse of X(52248)
X(412) = X(7045)-Ceva conjugate of X(653)
X(412) = X(24026)-Dao conjugate of X(44426)
X(412) = cevapoint of X(4) and X(1715)
X(412) = crosssum of X(822) and X(3270)
X(412) = barycentric product X(i)*X(j) for these {i,j}: {75, 38860}, {92, 3562}
X(412) = barycentric quotient X(i)/X(j) for these {i,j}: {3562, 63}, {38860, 1}
X(412) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 52248}, {3, 4, 29}, {3, 7524, 7531}, {3, 7567, 7572}, {4, 3144, 235}, {4, 4212, 37368}, {4, 4219, 7513}, {4, 5125, 7541}, {4, 5142, 7563}, {4, 7412, 14004}, {4, 7531, 7524}, {4, 7543, 15763}, {4, 7551, 44225}, {4, 7554, 7497}, {4, 37028, 7518}, {4, 37305, 11109}, {4, 37379, 4198}, {4, 37410, 4194}, {4, 37414, 17555}, {4, 37417, 2}, {20, 6838, 37419}, {20, 37437, 48890}, {27, 26003, 379}, {46, 158, 653}, {46, 51282, 158}, {140, 44225, 7551}, {407, 1885, 4}, {1013, 37235, 29}, {1155, 42385, 1940}, {3522, 7518, 37028}, {3559, 5125, 37279}, {6684, 39574, 52412}, {7497, 7554, 31903}, {7524, 7531, 29}, {7567, 30268, 3}, {14018, 37383, 37389}, {42789, 42790, 37115}


X(413) = EULER X(21)-3rd-SUBSTITUTION POINT

Trilinears         u3(v2 + w2 - vw) : v3(w2 + u2 - wu) : w3(u2 + v2 - uv), where
                        u : v : w = X(21); e.g., u = u(A,B,C) = 1/(cos B + cos C)
Barycentrics    au3(v2 + w2 - vw) : bv3(w2 + u2 - wu) : cw3(u2 + v2 - uv)
Barycentrics    a*(a + b)*(a - b - c)^3*(a + c)*(a^4 + a^3*b + a*b^3 + b^4 + a^3*c + a^2*b*c - a*b^2*c - b^3*c - a*b*c^2 + a*c^3 - b*c^3 + c^4) : :

As a point on the Euler line, X(413) has Shinagawa coefficients ($aSA$[($aSA$)2 - $bcSBSC$] - 2abcFS2, -($aSA$ - abc)[($aSA$)2 - $bcSBSC$] - 2[$bcSBSC$ + $aSA$F - abcF]S2).

X(413) lies on this line: 2,3

X(413) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {21, 29, 409}, {21, 13614, 13588}


X(414) = EULER X(29)-3rd-SUBSTITUTION POINT

Trilinears       u3(v2 + w2 - vw) : v3(w2 + u2 - wu) : w3(u2 + v2 - uv), where
                        u : v : w = X(29); e.g., u = u(A,B,C) = (sec A)1/(cos B + cos C)
Barycentrics  au3(v2 + w2 - vw) : bv3(w2 + u2 - wu) : cw3(u2 + v2 - uv)
Barycentrics    (a + b)*(a - b - c)^3*(a + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^8*b^2 + 2*a^7*b^3 - a^6*b^4 - 4*a^5*b^5 - a^4*b^6 + 2*a^3*b^7 + a^2*b^8 - a^8*b*c - a^7*b^2*c + 3*a^6*b^3*c + 3*a^5*b^4*c - 3*a^4*b^5*c - 3*a^3*b^6*c + a^2*b^7*c + a*b^8*c + a^8*c^2 - a^7*b*c^2 - 3*a^6*b^2*c^2 + a^5*b^3*c^2 + 4*a^4*b^4*c^2 + a^3*b^5*c^2 - 3*a^2*b^6*c^2 - a*b^7*c^2 + b^8*c^2 + 2*a^7*c^3 + 3*a^6*b*c^3 + a^5*b^2*c^3 - a^2*b^5*c^3 - 3*a*b^6*c^3 - 2*b^7*c^3 - a^6*c^4 + 3*a^5*b*c^4 + 4*a^4*b^2*c^4 + 4*a^2*b^4*c^4 + 3*a*b^5*c^4 - b^6*c^4 - 4*a^5*c^5 - 3*a^4*b*c^5 + a^3*b^2*c^5 - a^2*b^3*c^5 + 3*a*b^4*c^5 + 4*b^5*c^5 - a^4*c^6 - 3*a^3*b*c^6 - 3*a^2*b^2*c^6 - 3*a*b^3*c^6 - b^4*c^6 + 2*a^3*c^7 + a^2*b*c^7 - a*b^2*c^7 - 2*b^3*c^7 + a^2*c^8 + a*b*c^8 + b^2*c^8) : :

As a point on the Euler line, X(414) has Shinagawa coefficients ([(2E-F)F-S2]FS4 -$abSASB$F2S2 +$ab$F2S4, -[(3E-F)F-S2]FS4 -$ab(SASB)3$ +2$abSC3$FS2 -4$abSASB$F2S2 -2$abSC$[(E+F)2-2S2]FS2 -$ab$F2S4).

X(414) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {21, 29, 410}


X(415) = X(4)-HIRST INVERSE OF X(29)

Trilinears       (u2 - vw)sec A : (v2 - wu)sec B : (v2 - uv)sec C, where
                        u : v : w = X(21); e.g., u = u(A,B,C) = 1/(cos B + cos C)
Barycentrics  (u2 - vw)tan A : (v2 - wu)tan B : (w2 - uv)tan C
Barycentrics    (a + b)*(a + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^3 - 2*a^2*b + b^3 - 2*a^2*c + a*b*c + c^3) : :

As a point on the Euler line, X(415) has Shinagawa coefficients 3$bcSBSC$F + (E - 2F)FS2, -(E + F)*$bcSBSC$ + (E + F)FS2 - S4).

X(415) lies on these lines: {1, 2907}, {2, 3}, {107, 2708}, {158, 1247}, {162, 238}, {243, 2652}, {648, 23710}, {774, 26000}, {1098, 1935}, {1861, 36797}, {1937, 2659}, {2501, 17926}, {2651, 17950}, {4620, 18020}, {5993, 14192}, {6626, 30988}

X(415) = polar conjugate of X(11608)
X(415) = polar conjugate of the isotomic conjugate of X(40882)
X(415) = polar conjugate of the isogonal conjugate of X(5060)
X(415) = X(243)-Ceva conjugate of X(2659)
X(415) = X(i)-cross conjugate of X(j) for these (i,j): {1758, 2651}, {5060, 40882}
X(415) = X(i)-isoconjugate of X(j) for these (i,j): {3, 2652}, {48, 11608}, {65, 17973}, {73, 2648}, {656, 2701}, {810, 35154}, {1214, 17963}, {1409, 17947}, {6516, 18000}, {18013, 36059}
X(415) = X(i)-Dao conjugate of X(j) for these (i,j): {525, 35086}, {1214, 39055}, {1249, 11608}, {2652, 36103}, {2701, 40596}, {17973, 40602}, {18013, 20620}, {35154, 39062}
X(415) = cevapoint of X(1758) and X(17985)
X(415) = trilinear pole of line {2785, 41499}
X(415) = barycentric product X(i)*X(j) for these {i,j}: {4, 40882}, {29, 17950}, {92, 2651}, {264, 5060}, {333, 17985}, {648, 2785}, {1758, 31623}, {3064, 17933}, {5075, 6331}, {17942, 46110}, {17966, 44130}, {35145, 41499}
X(415) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 11608}, {19, 2652}, {29, 17947}, {112, 2701}, {284, 17973}, {648, 35154}, {1172, 2648}, {1758, 1214}, {2299, 17963}, {2651, 63}, {2785, 525}, {3064, 18013}, {5060, 3}, {5075, 647}, {17942, 1813}, {17950, 307}, {17966, 73}, {17975, 40152}, {17985, 226}, {40882, 69}, {41499, 8680}, {51643, 51640}
X(415) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 1982, 29}, {422, 468, 423}, {851, 52240, 416}, {860, 2074, 447}, {11110, 37383, 29}, {13739, 17555, 29}


X(416) = X(3)-HIRST INVERSE OF X(21)

Trilinears       (u2 - vw)cos A : (v2 - wu)cos B : (v2 - uv)cos C, where
                        u : v : w = X(29); e.g., u = u(A,B,C) = (sec A)/(cos B + cos C)
Barycentrics    (u2 - vw)sin(2A) : (v2 - wu)sin(2B) : (w2 - uv)sin(2C)
Barycentrics    a*(a + b)*(a + c)*(a^7*b^2 - a^6*b^3 - 2*a^5*b^4 + 2*a^4*b^5 + a^3*b^6 - a^2*b^7 + a^7*b*c - a^6*b^2*c - a^5*b^3*c + 2*a^4*b^4*c - a^3*b^5*c - a^2*b^6*c + a*b^7*c + a^7*c^2 - a^6*b*c^2 + a^5*b^2*c^2 - a^4*b^3*c^2 - a^3*b^4*c^2 + a^2*b^5*c^2 - a*b^6*c^2 + b^7*c^2 - a^6*c^3 - a^5*b*c^3 - a^4*b^2*c^3 + 2*a^3*b^3*c^3 + a^2*b^4*c^3 - a*b^5*c^3 + b^6*c^3 - 2*a^5*c^4 + 2*a^4*b*c^4 - a^3*b^2*c^4 + a^2*b^3*c^4 + 2*a*b^4*c^4 - 2*b^5*c^4 + 2*a^4*c^5 - a^3*b*c^5 + a^2*b^2*c^5 - a*b^3*c^5 - 2*b^4*c^5 + a^3*c^6 - a^2*b*c^6 - a*b^2*c^6 + b^3*c^6 - a^2*c^7 + a*b*c^7 + b^2*c^7) : :

As a point on the Euler line, X(416) has Shinagawa coefficients (abc[(E - 2F)FS4 + 2($aSA$)2F]S2 + $aSA$($bcSBSC$)2, -abc[S4 - (E+ F)FS2 + 2($aSA$)2F]S2 - $aSA$[3EFS4 + ($bcSBSC$)2]).

X(416) lies on these lines: {2, 3}, {255, 1248}, {296, 2651}, {820, 3562}, {1936, 2660}, {2659, 44354}, {23090, 32320}

X(416) = X(1936)-Ceva conjugate of X(2651)
X(2655)-cross conjugate of X(2659)
X(416) = X(i)-isoconjugate of X(j) for these (i,j): {4, 2660}, {65, 2656}
X(416) = X(i)-Dao conjugate of X(j) for these (i,j): {2656, 40602}, {2660, 36033}
X(416) = barycentric product X(i)*X(j) for these {i,j}: {21, 44354}, {63, 2659}, {333, 2655}
X(416) = barycentric quotient X(i)/X(j) for these {i,j}: {48, 2660}, {284, 2656}, {2655, 226}, {2659, 92}, {44354, 1441}
X(416) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 1982, 21}, {851, 52240, 415}


X(417) = X(3)-CEVA CONJUGATE OF X(185)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A)(sec2B + sec2C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin 2A)(sec2B + sec2C)
Barycentrics    a*(a + b)*(a + c)*(a^7*b^2 - a^6*b^3 - 2*a^5*b^4 + 2*a^4*b^5 + a^3*b^6 - a^2*b^7 + a^7*b*c - a^6*b^2*c - a^5*b^3*c + 2*a^4*b^4*c - a^3*b^5*c - a^2*b^6*c + a*b^7*c + a^7*c^2 - a^6*b*c^2 + a^5*b^2*c^2 - a^4*b^3*c^2 - a^3*b^4*c^2 + a^2*b^5*c^2 - a*b^6*c^2 + b^7*c^2 - a^6*c^3 - a^5*b*c^3 - a^4*b^2*c^3 + 2*a^3*b^3*c^3 + a^2*b^4*c^3 - a*b^5*c^3 + b^6*c^3 - 2*a^5*c^4 + 2*a^4*b*c^4 - a^3*b^2*c^4 + a^2*b^3*c^4 + 2*a*b^4*c^4 - 2*b^5*c^4 + 2*a^4*c^5 - a^3*b*c^5 + a^2*b^2*c^5 - a*b^3*c^5 - 2*b^4*c^5 + a^3*c^6 - a^2*b*c^6 - a*b^2*c^6 + b^3*c^6 - a^2*c^7 + a*b*c^7 + b^2*c^7) : :

X(417) lies on these lines: {2, 3}, {185, 6509}, {216, 51030}, {389, 13409}, {577, 14379}, {1092, 23606}, {1624, 2883}, {1942, 4558}, {2055, 43574}, {2972, 5562}, {3164, 45255}, {3917, 31388}, {5907, 44436}, {6389, 15653}, {6760, 14152}, {8763, 22341}, {9729, 46832}, {9730, 42441}, {10575, 40948}, {10607, 35602}, {11459, 14059}, {12359, 35442}, {13367, 20775}, {15043, 30258}, {15958, 19210}, {16035, 19166}, {22401, 47406}, {26900, 37526}

X(417) = isogonal conjugate of the polar conjugate of X(6509)
X(417) = X(i)-Ceva conjugate of X(j) for these (i,j): {3, 185}, {4558, 32320}
X(417) = X(i)-isoconjugate of X(j) for these (i,j): {4, 821}, {158, 1105}, {775, 1093}, {801, 6520}, {6521, 41890}
X(417) = X(i)-Dao conjugate of X(j) for these (i,j): {264, 13567}, {801, 37867}, {821, 36033}, {1093, 2883}, {1105, 1147}, {3269, 14618}
X(417) = crosspoint of X(3) and X(1092)
X(417) = crosssum of X(4) and X(1093)
X(417) = barycentric product X(i)*X(j) for these {i,j}: {3, 6509}, {63, 820}, {185, 394}, {255, 6508}, {577, 41005}, {774, 6507}, {800, 3964}, {1092, 13567}, {4100, 17858}, {5562, 19180}, {10607, 45199}, {14379, 45200}
X(417) = barycentric quotient X(i)/X(j) for these {i,j}: {48, 821}, {185, 2052}, {577, 1105}, {774, 6521}, {800, 1093}, {820, 92}, {1092, 801}, {1624, 15352}, {3964, 40830}, {4100, 775}, {6509, 264}, {16035, 8794}, {19180, 8795}, {23606, 41890}, {41005, 18027}
X(417) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3, 26897}, {3, 6617, 1593}, {3, 6638, 20}, {3, 15781, 3520}, {3, 26876, 34003}, {577, 14379, 43652}, {3524, 26876, 3}, {15717, 26874, 3}


X(418) = X(3)-CEVA-CONJUGATE OF X(216)

Trilinears    (cos A)(csc 2B + csc 2C) : :
Barycentrics    (sin 2A)(csc 2B + csc 2C) : :

As a point on the Euler line, X(418) has Shinagawa coefficients (F2 + S2,-(E + F)F - S2).

X(418) is the intersection of the isogonal conjugate of polar conjugate of Brocard axis (i.e., line X(184)X(418)) and the polar conjugate of isogonal conjugate of Brocard axis (i.e., line X(2)X(3)) (Randy Hutson, March 14, 2018)

X(418) lies on these lines: {2, 3}, {6, 26898}, {9, 26901}, {19, 26908}, {33, 26904}, {34, 26903}, {48, 23199}, {49, 19210}, {51, 216}, {52, 42441}, {53, 23607}, {54, 2055}, {57, 26900}, {97, 110}, {125, 50209}, {143, 46025}, {154, 160}, {157, 161}, {184, 577}, {212, 23198}, {217, 46394}, {268, 26867}, {275, 26902}, {324, 32428}, {343, 23181}, {476, 32439}, {511, 13409}, {682, 23209}, {1495, 22052}, {1614, 14152}, {1624, 10192}, {1629, 46760}, {1799, 6394}, {1974, 26899}, {2052, 42329}, {2972, 3917}, {3060, 26895}, {3284, 13366}, {3564, 23158}, {3567, 26896}, {3690, 35072}, {3819, 44436}, {5158, 15004}, {5562, 31388}, {5650, 46831}, {6146, 31381}, {6389, 22062}, {7011, 26866}, {7999, 14059}, {8266, 34828}, {8798, 31504}, {8884, 40448}, {8901, 35098}, {9407, 44078}, {9475, 40938}, {9967, 51335}, {10003, 12012}, {10184, 42862}, {10316, 34396}, {10979, 34417}, {11197, 39530}, {11402, 15905}, {11433, 26870}, {11442, 18437}, {11515, 44122}, {11516, 44083}, {13450, 51888}, {13567, 26905}, {14569, 42459}, {14855, 40948}, {15649, 17810}, {15653, 19467}, {16089, 46724}, {22075, 22391}, {22341, 22344}, {22401, 23210}, {23208, 42671}, {23613, 32320}, {23635, 47328}, {31353, 44088}, {31626, 38833}, {34983, 39201}, {34987, 37084}, {35225, 37813}, {35360, 42453}, {40681, 50645}, {40947, 41523}, {44711, 51243}, {44712, 51242}, {44716, 52032}, {46093, 52128}, {47409, 47426}, {51477, 52153}

X(418) = reflection of X(13409) in X(46832)
X(418) = isogonal conjugate of X(8795)
X(418) = circumcircle-inverse of X(41202)
X(418) = isotomic conjugate of the isogonal conjugate of X(44088)
X(418) = isogonal conjugate of the isotomic conjugate of X(5562)
X(418) = isotomic conjugate of the polar conjugate of X(217)
X(418) = isogonal conjugate of the polar conjugate of X(216)
X(418) = orthic-isogonal conjugate of X(31353)
X(418) = X(43679)-complementary conjugate of X(20305)
X(418) = X(i)-Ceva conjugate of X(j) for these (i,j): {3, 216}, {4, 31353}, {95, 41334}, {110, 32320}, {216, 217}, {14587, 32661}, {14941, 3289}, {23181, 17434}, {40448, 6}
X(418) = X(i)-cross conjugate of X(j) for these (i,j): {42556, 3}, {44088, 217}, {46394, 216}
X(418) = Danneels point of X(3)
X(418) = X(1764)-of-orthic-triangle if ABC is acute
X(418) = X(i)-isoconjugate of X(j) for these (i,j): {1, 8795}, {4, 40440}, {19, 276}, {63, 8794}, {75, 8884}, {92, 275}, {95, 158}, {97, 6521}, {264, 2190}, {661, 42405}, {799, 15422}, {821, 19166}, {822, 42401}, {823, 15412}, {1096, 34384}, {1577, 16813}, {1969, 8882}, {2052, 2167}, {2148, 18027}, {2616, 6528}, {3112, 19174}, {6520, 34386}, {8901, 23999}, {18831, 24006}, {36119, 43752}, {42300, 51315}
X(418) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 8795}, {5, 264}, {6, 276}, {95, 1147}, {130, 523}, {206, 8884}, {216, 18027}, {275, 22391}, {389, 45198}, {850, 2972}, {1511, 43752}, {2052, 40588}, {3162, 8794}, {6503, 34384}, {14618, 15450}, {14767, 42368}, {15422, 38996}, {16089, 52128}, {18022, 52032}, {19174, 34452}, {20572, 39171}, {21243, 23295}, {23962, 35441}, {34386, 37867}, {36033, 40440}, {36830, 42405}
X(418) = crosspoint of X(i) and X(j) for these (i,j): {3, 577}, {6, 22261}, {184, 2351}, {216, 5562}, {14587, 32661}
X(418) = crosssum of X(i) and X(j) for these (i,j): {2, 5889}, {4, 2052}, {264, 317}, {275, 8884}
X(418) = crossdifference of every pair of points on line {647, 14165}
X(418) = X(2)-line conjugate of X(44893)
X(418) = barycentric product X(i)*X(j) for these {i,j}: {3, 216}, {5, 577}, {6, 5562}, {32, 52347}, {48, 44706}, {51, 394}, {53, 1092}, {69, 217}, {71, 44709}, {76, 44088}, {95, 46394}, {99, 42293}, {110, 17434}, {184, 343}, {212, 44708}, {219, 30493}, {222, 44707}, {228, 16697}, {248, 44716}, {255, 1953}, {311, 14585}, {324, 23606}, {326, 2179}, {520, 1625}, {647, 23181}, {822, 2617}, {1154, 50433}, {1393, 2289}, {1568, 18877}, {1636, 36831}, {2181, 6507}, {2351, 52032}, {3199, 3964}, {3284, 44715}, {3926, 40981}, {3990, 18180}, {4055, 17167}, {4558, 15451}, {6368, 32661}, {6798, 45800}, {7069, 7125}, {7117, 44710}, {8798, 15905}, {9247, 18695}, {9380, 34900}, {10316, 41168}, {14213, 52430}, {14379, 42459}, {14570, 39201}, {14575, 28706}, {14576, 16391}, {14587, 39019}, {18315, 34983}, {18604, 21807}, {19210, 36412}, {20806, 27372}, {23357, 35442}, {23582, 41219}, {31388, 41891}, {31626, 32078}, {32320, 35360}, {36296, 44711}, {36297, 44712}, {40441, 42445}, {40799, 42353}, {41334, 42487}, {44713, 46112}, {44714, 46113}
X(418) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 276}, {5, 18027}, {6, 8795}, {25, 8794}, {32, 8884}, {48, 40440}, {51, 2052}, {107, 42401}, {110, 42405}, {184, 275}, {216, 264}, {217, 4}, {343, 18022}, {394, 34384}, {577, 95}, {669, 15422}, {1092, 34386}, {1576, 16813}, {1625, 6528}, {2179, 158}, {2181, 6521}, {3051, 19174}, {3199, 1093}, {3284, 43752}, {5562, 76}, {6528, 42369}, {6751, 41760}, {9247, 2190}, {14575, 8882}, {14585, 54}, {15451, 14618}, {17434, 850}, {23181, 6331}, {23606, 97}, {26880, 19188}, {27372, 43678}, {27374, 27376}, {28706, 44161}, {28724, 41488}, {30493, 331}, {31388, 26166}, {32078, 40684}, {32661, 18831}, {34983, 18314}, {35442, 23962}, {36433, 19210}, {39201, 15412}, {40981, 393}, {41219, 15526}, {42293, 523}, {42353, 40822}, {42556, 14767}, {44088, 6}, {44706, 1969}, {44707, 7017}, {44709, 44129}, {44716, 44132}, {46394, 5}, {50433, 46138}, {52347, 1502}, {52430, 2167}
X(418) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 6638, 852}, {2, 26874, 3}, {2, 30506, 5}, {3, 4, 26897}, {3, 25, 6641}, {3, 441, 14096}, {3, 6617, 7484}, {3, 6638, 2}, {3, 15329, 44891}, {3, 15781, 35921}, {3, 23246, 1590}, {3, 23256, 1589}, {3, 26874, 34003}, {3, 45842, 3520}, {4, 26876, 3}, {5, 13322, 4}, {6, 26909, 26898}, {25, 3135, 237}, {25, 26865, 3}, {51, 26907, 216}, {110, 39243, 97}, {184, 577, 23606}, {184, 23195, 20775}, {216, 26907, 32078}, {237, 3148, 27369}, {436, 8613, 41202}, {577, 26880, 184}, {1113, 1114, 41202}, {3129, 3130, 3518}, {3131, 3132, 186}, {3155, 3156, 24}, {3917, 6509, 2972}, {46600, 46601, 40853}


X(419) = X(4)-HIRST INVERSE OF X(25)

Trilinears       (u2 - vw)sec A : (v2 - wu)sec B : (v2 - uv)sec C, where
                        u : v : w = X(31); e.g., u = u(A,B,C) = a2
Barycentrics  (u2 - vw)tan A : (v2 - wu)tan B : (w2 - uv)tan C
Barycentrics    (a^2 - b*c)*(a^2 + b*c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) :

As a point on the Euler line, X(419) has Shinagawa coefficients (2(E + F)F,-(E + F)2 + S2).

X(419) lies on the cubics K252 and K785 and these lines: {2, 3}, {6, 3186}, {76, 9306}, {83, 3866}, {98, 42671}, {99, 19599}, {107, 2698}, {110, 51481}, {154, 39646}, {182, 43976}, {184, 40814}, {206, 41760}, {232, 6531}, {238, 242}, {264, 1974}, {275, 3399}, {290, 9418}, {316, 32223}, {338, 18374}, {385, 16985}, {394, 12251}, {648, 35146}, {685, 1976}, {1215, 1840}, {1235, 38830}, {1301, 48259}, {1304, 43654}, {1495, 41254}, {1576, 44375}, {1632, 3003}, {1843, 36794}, {2001, 44077}, {2052, 3406}, {2201, 19557}, {2207, 3224}, {2211, 41520}, {2501, 3288}, {2679, 40077}, {3167, 7754}, {3168, 8743}, {3229, 33874}, {3563, 6037}, {3978, 12215}, {4369, 22093}, {5254, 10192}, {6375, 51988}, {6530, 34130}, {6749, 41585}, {7760, 34986}, {7762, 41588}, {8623, 51324}, {8754, 37765}, {9307, 46432}, {9308, 19118}, {9407, 9512}, {10345, 11386}, {10359, 10601}, {11064, 43453}, {11596, 40135}, {12243, 35266}, {14602, 40820}, {17907, 41762}, {17941, 47736}, {18020, 34537}, {21445, 35278}, {27377, 41584}, {32085, 41884}, {32713, 37778}, {33630, 34208}, {35282, 38227}, {39089, 44090}, {40146, 43678}, {41194, 44778}, {41195, 44779}, {41253, 44084}, {41259, 51843}, {43696, 43721}, {44142, 46104}, {46151, 52418}, {51246, 51948}

X(419) = reflection of X(11596) in X(40135)
X(419) = isogonal conjugate of X(36214)
X(419) = isotomic conjugate of X(40708)
X(419) = orthocentroidal-circle-inverse of X(5117)
X(419) = polar-circle-inverse of X(11007)
X(419) = polar conjugate of X(1916)
X(419) = complement of the isogonal conjugate of X(43721)
X(419) = isotomic conjugate of the isogonal conjugate of X(44089)
X(419) = isogonal conjugate of the isotomic conjugate of X(17984)
X(419) = polar conjugate of the isotomic conjugate of X(385)
X(419) = polar conjugate of the isogonal conjugate of X(1691)
X(419) = X(43721)-complementary conjugate of X(10)
X(419) = X(i)-Ceva conjugate of X(j) for these (i,j): {232, 41204}, {6531, 4}, {17984, 385}, {41174, 112}, {47736, 39931}
X(419) = X(i)-cross conjugate of X(j) for these (i,j): {1691, 385}, {39931, 4}
X(419) = X(i)-isoconjugate of X(j) for these (i,j): {1, 36214}, {3, 1581}, {31, 40708}, {48, 1916}, {63, 694}, {69, 1967}, {75, 17970}, {184, 1934}, {256, 295}, {257, 2196}, {291, 7015}, {293, 40810}, {304, 9468}, {305, 1927}, {326, 17980}, {335, 7116}, {336, 14251}, {337, 904}, {647, 37134}, {656, 805}, {810, 18829}, {882, 4592}, {1911, 7019}, {1959, 15391}, {3917, 43763}, {4020, 14970}, {8789, 40364}, {9247, 18896}, {14208, 17938}
X(419) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 40708}, {3, 19576}, {3, 36214}, {48, 39031}, {63, 39043}, {69, 8290}, {132, 40810}, {206, 17970}, {304, 39044}, {325, 6393}, {525, 35078}, {694, 3162}, {805, 40596}, {882, 5139}, {1249, 1916}, {1581, 36103}, {2491, 41172}, {3917, 36213}, {6651, 7019}, {7015, 39029}, {8623, 36212}, {15259, 17980}, {18829, 39062}, {37134, 39052}, {39030, 40364}
X(419) = cevapoint of X(1691) and X(44089)
X(419) = crosspoint of X(i) and X(j) for these (i,j): {685, 18020}, {16081, 46104}
X(419) = crosssum of X(i) and X(j) for these (i,j): {684, 20975}, {3289, 20775}, {12215, 37894}
X(419) = trilinear pole of line {804, 12829}
X(419) = X(4)-Hirst inverse of X(25)
X(419) = pole wrt polar circle of trilinear polar of X(1916) (line X(141)X(523))
X(419) = crossdifference of every pair of points on line {647, 3917}
X(419) = barycentric product X(i)*X(j) for these {i,j}: {4, 385}, {6, 17984}, {19, 1966}, {25, 3978}, {27, 4039}, {76, 44089}, {92, 1580}, {98, 39931}, {107, 24284}, {112, 14295}, {172, 40717}, {230, 47736}, {232, 14382}, {239, 7009}, {242, 894}, {264, 1691}, {290, 51324}, {297, 40820}, {350, 7119}, {393, 12215}, {648, 804}, {732, 32085}, {862, 8033}, {880, 2489}, {1215, 31905}, {1783, 14296}, {1840, 33295}, {1874, 27958}, {1897, 4107}, {1909, 2201}, {1926, 1973}, {1933, 1969}, {1974, 14603}, {2501, 17941}, {2679, 41174}, {3186, 39927}, {4019, 34856}, {4032, 14024}, {4164, 6335}, {5026, 17983}, {5027, 6331}, {5976, 6531}, {8623, 46104}, {12829, 35142}, {14006, 16609}, {14602, 18022}, {16080, 51430}, {16081, 36213}, {16089, 32542}, {16985, 37892}, {18901, 44162}, {18902, 44161}, {36820, 37765}, {39495, 46456}
X(419) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 40708}, {4, 1916}, {6, 36214}, {19, 1581}, {25, 694}, {32, 17970}, {92, 1934}, {112, 805}, {162, 37134}, {172, 295}, {232, 40810}, {239, 7019}, {242, 257}, {264, 18896}, {385, 69}, {458, 8842}, {460, 47734}, {648, 18829}, {685, 39291}, {732, 3933}, {804, 525}, {894, 337}, {1580, 63}, {1691, 3}, {1840, 43534}, {1914, 7015}, {1926, 40364}, {1933, 48}, {1966, 304}, {1973, 1967}, {1974, 9468}, {1976, 15391}, {2086, 20975}, {2201, 256}, {2207, 17980}, {2210, 7116}, {2211, 14251}, {2489, 882}, {2679, 41172}, {3978, 305}, {4027, 12215}, {4039, 306}, {4107, 4025}, {4164, 905}, {5026, 6390}, {5027, 647}, {5976, 6393}, {6531, 36897}, {7009, 335}, {7119, 291}, {7122, 2196}, {8623, 3917}, {11183, 14417}, {11325, 47642}, {12215, 3926}, {12829, 3564}, {14006, 36800}, {14295, 3267}, {14296, 15413}, {14602, 184}, {14603, 40050}, {16985, 37894}, {17941, 4563}, {17980, 41517}, {17984, 76}, {18020, 39292}, {18022, 44160}, {18901, 40360}, {18902, 14575}, {24284, 3265}, {31905, 32010}, {32085, 14970}, {32542, 14941}, {32544, 8858}, {35325, 46161}, {36213, 36212}, {36820, 34897}, {37892, 40847}, {39495, 8552}, {39927, 43714}, {39931, 325}, {40717, 44187}, {40820, 287}, {42396, 41209}, {44089, 6}, {44102, 18872}, {44162, 8789}, {47736, 8781}, {50732, 51454}, {51320, 37893}, {51324, 511}, {51430, 11064}
X(419) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 5117}, {2, 3148, 37334}, {2, 6620, 4}, {2, 33336, 384}, {4, 186, 35474}, {4, 420, 297}, {4, 38282, 52283}, {22, 37190, 7470}, {22, 41238, 37190}, {23, 46571, 14957}, {25, 458, 4}, {25, 11325, 46505}, {29, 37389, 37362}, {237, 401, 11676}, {237, 1316, 401}, {297, 460, 4}, {297, 468, 420}, {421, 468, 450}, {460, 468, 297}, {462, 468, 470}, {463, 468, 471}, {468, 7473, 186}, {2450, 44887, 2}, {3144, 37055, 6998}, {5000, 5001, 5999}, {6660, 21531, 5999}, {14957, 46493, 23}, {14957, 46512, 46571}, {19128, 44145, 41204}, {21177, 46544, 15013}, {21531, 37906, 6660}, {37912, 46522, 15014}, {42789, 42790, 47620}, {44887, 50707, 2450}, {46493, 46512, 14957}


X(420) = X(4)-HIRST INVERSE OF X(427)

Trilinears       (u2 - vw)sec A : (v2 - wu)sec B : (v2 - uv)sec C, where
                        u : v : w = X(38); e.g., u = u(a,b,c) = b2 + c2
Barycentrics  (u2 - vw)tan A : (v2 - wu)tan B : (w2 - uv)tan C


Barycentrics    (a^2 - b*c)*(a^2 + b*c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) : :(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^4 + a^2*b^2 - b^4 + a^2*c^2 - b^2*c^2 - c^4) : :

As a point on the Euler line, X(420) has Shinagawa coefficients (4(E + F)F,-(E + F)2 - S2).

X(420) lies on these lines: {2, 3}, {112, 8623}, {232, 3229}, {240, 17927}, {340, 44102}, {685, 20021}, {1613, 8743}, {1843, 34236}, {1990, 38294}, {2207, 21001}, {2501, 9210}, {2896, 11380}, {3117, 39575}, {3186, 17907}, {3231, 8744}, {3819, 10357}, {3978, 44146}, {5207, 40876}, {6198, 40790}, {6688, 7859}, {7779, 44090}, {9208, 44427}, {9302, 16080}, {9862, 42671}, {12143, 46226}, {14165, 34854}, {16081, 46306}, {17414, 47217}, {18020, 20022}, {18371, 52418}, {19128, 36213}, {19558, 43696}, {26958, 39646}, {28408, 44443}, {32223, 43453}, {33873, 44084}, {39201, 44451}, {39231, 40559}, {40858, 41676}

X(420) = polar conjugate of X(11606)
X(420) = isotomic conjugate of the isogonal conjugate of X(44090)
X(420) = polar conjugate of the isotomic conjugate of X(7779)
X(420) = polar conjugate of the isogonal conjugate of X(2076)
X(420) = X(2076)-cross conjugate of X(7779)
X(420) = X(i)-isoconjugate of X(j) for these (i,j): {48, 11606}, {63, 46286}, {656, 46970}, {1176, 17957}
X(420) = X(i)-Dao conjugate of X(j) for these (i,j): {69, 39091}, {385, 12215}, {647, 39079}, {1249, 11606}, {3162, 46286}, {17949, 40938}, {40596, 46970}
X(420) = cevapoint of X(2076) and X(44090)
X(420) = trilinear pole of line {9479, 12830}
X(420) = crossdifference of every pair of points on line {647, 22352}
X(420) = barycentric product X(i)*X(j) for these {i,j}: {4, 7779}, {76, 44090}, {92, 17799}, {264, 2076}, {427, 40850}, {648, 9479}, {1235, 46228}, {5113, 6331}, {12830, 35142}, {18010, 41676}, {20883, 34054}
X(420) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 11606}, {25, 46286}, {112, 46970}, {427, 17949}, {2076, 3}, {5113, 647}, {7779, 69}, {8290, 12215}, {8864, 8858}, {9479, 525}, {12830, 3564}, {17442, 17957}, {17799, 63}, {18010, 4580}, {34054, 34055}, {40850, 1799}, {44090, 6}, {46228, 1176}
X(420) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {25, 5117, 4}, {25, 11331, 5117}, {297, 419, 4}, {297, 468, 419}, {403, 35474, 4}, {3535, 3536, 37187}, {5000, 5001, 40236}, {5112, 44887, 401}, {6353, 52283, 4}, {20854, 21536, 40236}


X(421) = X(4)-HIRST INVERSE OF X(24)

Trilinears       (u2 - vw)sec A : (v2 - wu)sec B : (v2 - uv)sec C, where
                         u : v : w = X(47); e.g., u = u(A,B,C) = cos 2A
Barycentrics  (u2 - vw)tan A : (v2 - wu)tan B : (w2 - uv)tan C
Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^8 - 2*a^6*b^2 + a^4*b^4 - 2*a^6*c^2 + a^4*b^2*c^2 - b^6*c^2 + a^4*c^4 + 2*b^4*c^4 - b^2*c^6) : :

As a point on the Euler line, X(421) has Shinagawa coefficients ((E + 4F)F,-(E + F)(E + 2F) + 2S2).

X(421) lies on these lines: {2, 3}, {107, 44084}, {110, 44145}, {136, 14165}, {275, 47328}, {338, 15139}, {847, 52432}, {1632, 47195}, {2052, 44077}, {2501, 2623}, {2970, 34397}, {6531, 14580}, {8794, 41271}, {11547, 14593}, {12133, 48364}, {18020, 44138}, {19128, 46106}, {43976, 44080}, {44089, 47202}, {44375, 51458}

X(421) = polar-circle-inverse of X(36190)
X(421) = polar conjugate of the isotomic conjugate of X(44375)
X(421) = crossdifference of every pair of points on line {647, 5562}
X(421) = barycentric product X(i)*X(j) for these {i,j}: {4, 44375}, {2052, 51458}
X(421) = barycentric quotient X(i)/X(j) for these {i,j}: {44375, 69}, {51458, 394}
X(421) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {419, 450, 468}, {2970, 34397, 41204}


X(422) = X(4)-HIRST INVERSE OF X(28)

Trilinears       (u2 - vw)sec A : (v2 - wu)sec B : (v2 - uv)sec C, where
                        u : v : w = X(58); e.g., u = u(a,b,c) = a/(b + c)
Barycentrics  (u2 - vw)tan A : (v2 - wu)tan B : (w2 - uv)tan C
Barycentrics    (a + b)*(a + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^3 + a*b*c - b^2*c - b*c^2) : :

As a point on the Euler line, X(422) has Shinagawa coefficients (4(E+F)2FS2-3$abSC$F+7$ab$(E+F)F, -2(E+F)[(E+F)2-S2]+$abSC$(E+F) -$ab$[3(E+F)2-2S2]).

X(422) lies on these lines: {2, 3}, {107, 2699}, {110, 48380}, {162, 242}, {648, 35155}, {685, 16082}, {1632, 8758}, {1824, 46103}, {2203, 31623}, {2501, 4581}, {4601, 5379}, {17790, 17977}

X(422) = polar conjugate of X(11611)
X(422) = polar conjugate of the isotomic conjugate of X(19623)
X(422) = polar conjugate of the isogonal conjugate of X(5006)
X(422) = X(5006)-cross conjugate of X(19623)
X(422) = X(i)-isoconjugate of X(j) for these (i,j): {10, 17971}, {48, 11611}, {71, 17946}, {72, 17954}, {73, 11609}, {306, 17961}, {656, 2703}, {810, 35147}, {1331, 18015}, {3682, 17981}, {4064, 17939}, {4561, 18002}
X(422) = X(i)-Dao conjugate of X(j) for these (i,j): {525, 35079}, {1249, 11611}, {2703, 40596}, {5521, 18015}, {35147, 39062}
X(422) = crossdifference of every pair of points on line {647, 22076}
X(422) = barycentric product X(i)*X(j) for these {i,j}: {4, 19623}, {19, 5209}, {27, 17763}, {28, 17790}, {81, 17987}, {264, 5006}, {286, 5291}, {648, 2787}, {5040, 6331}, {5061, 31623}, {6591, 17935}, {17924, 17944}
X(422) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 11611}, {28, 17946}, {112, 2703}, {648, 35147}, {1172, 11609}, {1333, 17971}, {1474, 17954}, {2203, 17961}, {2787, 525}, {5006, 3}, {5040, 647}, {5061, 1214}, {5209, 304}, {5291, 72}, {5317, 17981}, {6591, 18015}, {17763, 306}, {17790, 20336}, {17944, 1332}, {17977, 3998}, {17987, 321}, {19623, 69}
X(422) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {415, 423, 468}


X(423) = X(4)-HIRST INVERSE OF X(27)

Trilinears       (u2 - vw)sec A : (v2 - wu)sec B : (v2 - uv)sec C, where
                        u : v : w = X(81); e.g., u = u(a,b,c) = 1/(b + c)
Barycentrics  (u2 - vw)tan A : (v2 - wu)tan B : (w2 - uv)tan C
Barycentrics    (a + b)*(a + c)*(a^2 + b^2 - c^2)*(a^2 + a*b - b^2 + a*c - b*c - c^2)*(a^2 - b^2 + c^2) : :

As a point on the Euler line, X(423) has Shinagawa coefficients (2(E + F)F + 3$bc$F,-(E + F)2 - $bc$(E + F) + S2).

X(423) lies on these lines: {2, 3}, {19, 2905}, {86, 18161}, {92, 31997}, {107, 2700}, {110, 48381}, {648, 8756}, {2322, 27958}, {2501, 4608}, {4600, 18020}, {6542, 17927}, {7119, 52412}, {7282, 27691}, {11363, 27954}, {23864, 44451}, {24559, 25533}

X(423) = polar conjugate of X(11599)
X(423) = polar conjugate of the isotomic conjugate of X(17731)
X(423) = polar conjugate of the isogonal conjugate of X(1326)
X(423) = X(1326)-cross conjugate of X(17731)
X(423) = X(i)-isoconjugate of X(j) for these (i,j): {3, 9278}, {37, 17972}, {48, 11599}, {63, 2054}, {71, 1929}, {72, 17962}, {228, 6650}, {647, 37135}, {656, 2702}, {810, 35148}, {906, 18014}, {1332, 18001}, {1437, 6543}, {2200, 18032}, {3990, 17982}
X(423) = X(i)-Dao conjugate of X(j) for these (i,j): {63, 39042}, {72, 39041}, {306, 41841}, {525, 35080}, {1249, 11599}, {1326, 20813}, {2054, 3162}, {2702, 40596}, {4466, 27929}, {5190, 18014}, {9278, 36103}, {17972, 40589}, {20546, 20825}, {35148, 39062}, {37135, 39052}
X(423) = crosssum of X(3) and X(20766)
X(423) = crossdifference of every pair of points on line {647, 22080}
X(423) = barycentric product X(i)*X(j) for these {i,j}: {4, 17731}, {19, 52137}, {27, 6542}, {28, 20947}, {86, 17927}, {92, 1931}, {264, 1326}, {286, 1757}, {648, 2786}, {811, 9508}, {5029, 6331}, {6336, 31059}, {7649, 17934}, {17735, 44129}, {17943, 46107}
X(423) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 11599}, {19, 9278}, {25, 2054}, {27, 6650}, {28, 1929}, {58, 17972}, {112, 2702}, {162, 37135}, {286, 18032}, {648, 35148}, {1326, 3}, {1474, 17962}, {1757, 72}, {1826, 6543}, {1931, 63}, {2786, 525}, {2905, 39921}, {5029, 647}, {6541, 3695}, {6542, 306}, {7649, 18014}, {8747, 17982}, {9508, 656}, {17569, 40793}, {17731, 69}, {17735, 71}, {17927, 10}, {17934, 4561}, {17943, 1331}, {17976, 3682}, {18004, 4064}, {18266, 228}, {20693, 3949}, {20947, 20336}, {27929, 24459}, {28346, 51366}, {28602, 14429}, {31059, 3977}, {31905, 40725}, {52137, 304}
X(423) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 31904, 27}, {27, 28, 31913}, {27, 4248, 31912}, {27, 14013, 31914}, {27, 31912, 31915}, {28, 31909, 27}, {29, 14013, 27}, {422, 468, 415}, {1375, 3109, 448}, {4248, 31916, 31915}, {13739, 37448, 27}, {15149, 31905, 27}, {15149, 37168, 31905}, {31908, 31925, 27}, {31912, 31916, 27}


X(424) = X(4)-HIRST INVERSE OF X(451)

Trilinears       (u2 - vw)sec A : (v2 - wu)sec B : (v2 - uv)sec C, where
                         u : v : w = X(191); e.g., u = u(a,b,c) = (b + c - a)(bc + ca + ab) + b3 + c3 - a3
Barycentrics  (u2 - vw)tan A : (v2 - wu)tan B : (w2 - uv)tan C
Barycentrics    (b + c)*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(-(a^3*b) - a^2*b^2 + a*b^3 + b^4 - a^3*c - a^2*c^2 + a*c^3 + c^4) : :

As a point on the Euler line, X(424) has Shinagawa coefficients (4(E+F)2F-3$abSC$F+7$ab$(E+F)F, -4(E+F)S2+$abSC2$-$abSASB$ -$ab$[(E+F)2+3S2]).

X(424) lies on these lines: {2, 3}, {2501, 3700}

X(424) = polar conjugate of the isotomic conjugate of X(44396)
X(424) = polar conjugate of the isogonal conjugate of X(5164)
X(424) = X(5164)-cross conjugate of X(44396)
X(424) = X(7254)-Dao conjugate of X(41179)
X(424) = crossdifference of every pair of points on line {647, 1437}
X(424) = barycentric product X(i)*X(j) for these {i,j}: {4, 44396}, {264, 5164}
X(424) = barycentric quotient X(i)/X(j) for these {i,j}: {5164, 3}, {44396, 69}
X(424) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {}


X(425) = X(4)-HIRST INVERSE OF X(21)

Trilinears       (u2 - vw)sec A : (v2 - wu)sec B : (v2 - uv)sec C, where
                        u : v : w = X(283); e.g., u = u(A,B,C) = (cos A)/(cos B + cos C)
Barycentrics  (u2 - vw)tan A : (v2 - wu)tan B : (w2 - uv)tan C
Barycentrics    (a + b)*(a + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^6 - a^5*b - a^4*b^2 + a^3*b^3 - a^5*c + a^4*b*c + a^3*b^2*c - b^5*c - a^4*c^2 + a^3*b*c^2 + 2*a^2*b^2*c^2 - 2*a*b^3*c^2 + a^3*c^3 - 2*a*b^2*c^3 + 2*b^3*c^3 - b*c^5) : :

As a point on the Euler line, X(425) has Shinagawa coefficients ((E-2F)2FS2+$abSASB$(2E-7F)F +3$ab$F2S2, (E-2F)[(E+F)F- S2]S2 +3$abSASB$[3(E+F)F-2S2]-$ab$(E+F)FS2).

X(425) lies on these lines: {2, 3}, {107, 2707}, {243, 662}, {653, 2651}, {1098, 1940}, {2501, 23090}, {17923, 25533}, {18020, 23999}

X(425) = X(41349)-cross conjugate of X(23695)
X(425) = X(i)-isoconjugate of X(j) for these (i,j): {73, 43746}, {656, 2714}
X(425) = X(2714)-Dao conjugate of X(40596)
X(425) = barycentric product X(i)*X(j) for these {i,j}: {92, 23695}, {648, 2798}, {31623, 41349}
X(425) = barycentric quotient X(i)/X(j) for these {i,j}: {112, 2714}, {1172, 43746}, {2798, 525}, {23695, 63}, {41349, 1214}
X(425) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {}


X(426) = EULER X(19)-4th-SUBSTITUTION POINT

Trilinears       (v2 + w2)cos A : (w2 + u2)cos B : (u2 + v2)cos C, where
                        u : v : w = X(19); e.g., u = u(A,B,C) = tan A
Barycentrics  (v2 + w2)sin 2A : (w2 + u2)sin 2B : (u2 + v2)sin 2C
Barycentrics    (a + b)*(a + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^6 - a^5*b - a^4*b^2 + a^3*b^3 - a^5*c + a^4*b*c + a^3*b^2*c - b^5*c - a^4*c^2 + a^3*b*c^2 + 2*a^2*b^2*c^2 - 2*a*b^3*c^2 + a^3*c^3 - 2*a*b^2*c^3 + 2*b^3*c^3 - b*c^5) : :
Barycentrics    a^2*(a^2 - b^2 - c^2)^3*(a^4 + b^4 - 2*b^2*c^2 + c^4) : :

As a point on the Euler line, X(426) has Shinagawa coefficients ((2E + F)F - S2,-(E + F)F + S2).

X(426) lies on these lines: {2, 3}, {6, 13409}, {97, 7998}, {125, 2351}, {157, 1853}, {182, 46832}, {184, 6509}, {216, 43650}, {305, 6394}, {394, 2972}, {577, 3917}, {1073, 6090}, {1092, 16391}, {1899, 6389}, {3162, 9475}, {3964, 4176}, {5085, 26898}, {5422, 30258}, {5651, 46831}, {9306, 44436}, {11245, 41005}, {14059, 14152}, {22057, 22421}, {22352, 26880}, {23332, 37813}, {26905, 34828}, {36752, 42441}, {39243, 41462}, {40913, 43975}, {41204, 46717}, {47409, 47412}

X(426) = isogonal conjugate of the isotomic conjugate of X(44141)
X(426) = isotomic conjugate of the polar conjugate of X(39643)
X(426) = isogonal conjugate of the polar conjugate of X(6389)
X(426) = X(255)-complementary conjugate of X(37864)
X(426) = X(i)-Ceva conjugate of X(j) for these (i,j): {3, 40947}, {4563, 32320}, {6389, 39643}
X(426) = X(1096)-isoconjugate of X(34405)
X(426) = X(i)-Dao conjugate of X(j) for these (i,j): {264, 3767}, {1093, 6389}, {6503, 34405}, {6524, 14713}
X(426) = crosspoint of X(i) and X(j) for these (i,j): {3, 3964}, {6389, 44141}
X(426) = crosssum of X(4) and X(6524)
X(426) = barycentric product X(i)*X(j) for these {i,j}: {3, 6389}, {6, 44141}, {69, 39643}, {326, 2083}, {394, 1899}, {577, 41009}, {1092, 41760}, {3767, 3964}, {3926, 40947}, {4176, 42295}, {6507, 17871}, {6751, 34386}, {16391, 41770}
X(426) = barycentric quotient X(i)/X(j) for these {i,j}: {394, 34405}, {1632, 15352}, {1899, 2052}, {2083, 158}, {3767, 1093}, {3964, 42407}, {6389, 264}, {6751, 53}, {17871, 6521}, {39643, 4}, {40947, 393}, {41009, 18027}, {42295, 6524}, {44141, 76}
X(426) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3, 6641}, {3, 441, 3148}, {3, 6617, 25}, {3, 6638, 22}, {3, 7395, 26897}, {3, 15781, 378}, {3, 26865, 34003}, {25, 6617, 852}, {2972, 23606, 394}, {7386, 28412, 427}, {15246, 26874, 3}


X(427) = COMPLEMENT OF X(22)

Trilinears    sec A + cos(B - C)
Barycentrics    tan A + tan ω : :
Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(b^2 + c^2) : :
X(427) = 3 X[2] + X[7391], 9 X[2] - X[20062], X[2] + 3 X[31105], 3 X[2] - 5 X[31236], 5 X[2] - 3 X[47596], 3 X[4] - X[35480], 3 X[4] + X[35481], 3 X[5] - 4 X[13413], and many others

As a point on the Euler line, X(427) has Shinagawa coefficients (F,E + F).

Let LA be the line tangent to the nine-point circle at the midpoint of segment BC, and define LB and LC cyclically. The triangle formed by the lines LA, LB, LC is homothetic to the orthic triangle, and the center of homothety is X(427). (Randy Hutson, 9/23/2011)

Let (O) be the circumcircle, (N) the nine-point circle, and (IA, (IB, (IC the excircles of ABC. Let A' and A'' be the points of intersection of (O) and (IA. Let FA be the touchpoint of (N) and (IA, and let (KA) be the circle through A' and A'' that is internally tangent to (N); let LA be the touchpoint. Define FB, FC and LB, LC cyclically. The lines FALA, FBLB, FCLC concur ion X(427). (Tran Quang Hung ADGEOM #1458, August 5, 2014; see also #1459)

Let A'B'C' be the circummedial triangle. Let A″ be the pole, wrt the polar circle, of line B'C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(427). Moreover, X(427) is the Euler line intercept of radical axis of nine-point circle and every circle with center on orthic axis that is orthogonal to nine-point circle, and X(427) is the point in which the extended trapezoid legs (P(4),P(4)-Ceva conjugate of U(4)) and (U(4),U(4)-Ceva conjugate of P(4)) meet. Also, X(427) is the QA-P38 center (Montesdeoca-Hutson Point) of quadrangle ABCX(2). (Randy Hutson, October 13, 2015)

X(427) is the homothetic center of any pair of the following: polar triangle of nine-point circle, orthic axes triangle (see X(2501)), Yiu tangents triangle (see X(7495)). (Randy Hutson, August 19, 2019)

X(427) lies on the cubics K517, K533, K539, K823, K836, and these lines: {1, 5090}, {2, 3}, {6, 66}, {8, 11396}, {9, 21015}, {10, 1829}, {11, 33}, {12, 34}, {19, 3925}, {32, 47298}, {39, 12143}, {51, 125}, {52, 6746}, {53, 232}, {54, 31804}, {55, 11393}, {56, 11392}, {57, 1892}, {64, 14542}, {68, 36747}, {69, 12167}, {76, 8890}, {83, 11380}, {92, 2969}, {98, 275}, {107, 29011}, {110, 39884}, {112, 251}, {113, 12133}, {114, 136}, {115, 1196}, {116, 5185}, {119, 12138}, {120, 5521}, {122, 46831}, {126, 5139}, {127, 13166}, {135, 31842}, {137, 23319}, {141, 1843}, {143, 13561}, {154, 31383}, {155, 23307}, {182, 37649}, {183, 317}, {184, 1503}, {185, 3574}, {193, 8892}, {211, 27370}, {214, 12137}, {216, 26905}, {225, 26481}, {226, 1876}, {230, 571}, {238, 14975}, {242, 52412}, {262, 2052}, {264, 305}, {265, 15472}, {273, 7179}, {278, 1390}, {281, 1851}, {286, 37664}, {316, 33651}, {318, 3705}, {323, 3410}, {324, 2967}, {340, 37671}, {343, 511}, {371, 8280}, {372, 8281}, {373, 44079}, {385, 27377}, {388, 1398}, {389, 20299}, {393, 7736}, {394, 1352}, {395, 8739}, {396, 8740}, {459, 14484}, {485, 3093}, {486, 3092}, {487, 23309}, {488, 23310}, {495, 1870}, {496, 6198}, {497, 7071}, {498, 11398}, {499, 11399}, {515, 51707}, {524, 8541}, {542, 34986}, {578, 6146}, {590, 5412}, {597, 37875}, {608, 34261}, {615, 5413}, {618, 12142}, {619, 12141}, {625, 5140}, {629, 22482}, {630, 22481}, {641, 12148}, {642, 12147}, {648, 41624}, {748, 2212}, {750, 1395}, {804, 47230}, {899, 40976}, {908, 27409}, {930, 3563}, {940, 44105}, {946, 1902}, {958, 11391}, {973, 49108}, {1007, 19583}, {1031, 3329}, {1039, 1722}, {1086, 24163}, {1105, 14860}, {1125, 11363}, {1147, 12134}, {1172, 33854}, {1180, 5523}, {1181, 14216}, {1184, 3767}, {1194, 5254}, {1204, 6696}, {1209, 10625}, {1213, 44103}, {1217, 18855}, {1235, 3933}, {1241, 9230}, {1249, 37665}, {1291, 40118}, {1297, 20626}, {1329, 1828}, {1350, 43653}, {1351, 6515}, {1353, 1994}, {1369, 14929}, {1376, 11383}, {1401, 46152}, {1426, 15844}, {1447, 7282}, {1452, 24914}, {1474, 17398}, {1495, 10192}, {1506, 3199}, {1511, 12140}, {1568, 15030}, {1576, 39086}, {1611, 13881}, {1614, 16659}, {1627, 10312}, {1629, 14165}, {1698, 7713}, {1737, 1905}, {1753, 15908}, {1785, 24239}, {1799, 7750}, {1824, 1848}, {1826, 1841}, {1827, 21239}, {1838, 1867}, {1842, 30768}, {1865, 37661}, {1878, 3814}, {1890, 2355}, {1891, 25466}, {1897, 29840}, {1900, 25639}, {1916, 37892}, {1953, 21911}, {1968, 7745}, {1973, 29647}, {1974, 3589}, {1975, 34254}, {1986, 10264}, {1990, 9300}, {1992, 11405}, {1993, 3564}, {2023, 14715}, {2056, 11646}, {2183, 21912}, {2207, 2548}, {2211, 20965}, {2356, 3720}, {2453, 47177}, {2482, 12132}, {2486, 47232}, {2493, 34981}, {2501, 10278}, {2549, 15880}, {2550, 11406}, {2881, 47205}, {2883, 11381}, {2892, 32251}, {2904, 43588}, {2905, 17000}, {2971, 34336}, {2974, 34338}, {2979, 6403}, {3006, 41013}, {3011, 40950}, {3051, 20021}, {3054, 10985}, {3055, 14806}, {3060, 3580}, {3068, 5410}, {3069, 5411}, {3070, 11474}, {3071, 11473}, {3087, 7735}, {3096, 11386}, {3108, 41366}, {3167, 18440}, {3168, 51358}, {3172, 8879}, {3186, 3314}, {3192, 37662}, {3203, 27366}, {3258, 42426}, {3266, 44142}, {3291, 40326}, {3527, 45010}, {3567, 23294}, {3616, 7718}, {3618, 19118}, {3647, 16114}, {3703, 20883}, {3740, 41611}, {3757, 5174}, {3763, 7716}, {3796, 46264}, {3818, 9306}, {3819, 24206}, {3820, 29679}, {4383, 44086}, {4994, 8883}, {5012, 14389}, {5044, 41609}, {5081, 7081}, {5095, 8584}, {5097, 11225}, {5146, 29873}, {5155, 5268}, {5190, 5513}, {5203, 8770}, {5207, 37894}, {5249, 25365}, {5272, 7741}, {5275, 50036}, {5297, 10592}, {5304, 19041}, {5305, 5359}, {5306, 6103}, {5310, 6284}, {5318, 11476}, {5321, 11475}, {5322, 7354}, {5345, 10483}, {5422, 18583}, {5432, 52427}, {5446, 5449}, {5475, 9515}, {5490, 26373}, {5491, 26374}, {5552, 11400}, {5590, 11389}, {5591, 11388}, {5596, 19125}, {5599, 11384}, {5600, 11385}, {5640, 26913}, {5654, 18451}, {5690, 41722}, {5718, 44113}, {5743, 44092}, {5800, 44094}, {5889, 31802}, {5891, 51392}, {5913, 50718}, {5921, 15431}, {5943, 19130}, {5946, 52000}, {5966, 30248}, {5972, 16165}, {5996, 14618}, {6000, 18388}, {6033, 41253}, {6090, 14826}, {6108, 31687}, {6109, 31688}, {6114, 6117}, {6115, 6116}, {6152, 21230}, {6212, 16033}, {6213, 16027}, {6260, 12136}, {6288, 37495}, {6291, 23311}, {6292, 12144}, {6331, 8842}, {6340, 14248}, {6406, 23312}, {6467, 15583}, {6524, 10002}, {6525, 17830}, {6560, 18290}, {6561, 18289}, {6564, 8854}, {6565, 8855}, {6593, 32239}, {6697, 9969}, {6699, 15473}, {6759, 16655}, {6776, 11402}, {7009, 17923}, {7085, 50861}, {7292, 10593}, {7592, 11457}, {7603, 33842}, {7608, 39284}, {7649, 44316}, {7687, 32743}, {7699, 11455}, {7717, 18230}, {7722, 11804}, {7752, 47846}, {7753, 14581}, {7762, 8878}, {7774, 9308}, {7776, 40123}, {7778, 41762}, {7788, 44134}, {7789, 30747}, {7792, 36794}, {7840, 38294}, {8222, 11394}, {8223, 11395}, {8263, 11188}, {8428, 34866}, {8537, 41628}, {8550, 13366}, {8680, 25343}, {8753, 42008}, {8792, 41513}, {8796, 14494}, {8877, 10415}, {8968, 48467}, {8981, 10880}, {9134, 47236}, {9148, 17994}, {9157, 10735}, {9483, 42396}, {9544, 46818}, {9605, 41361}, {9732, 11091}, {9733, 11090}, {9744, 11547}, {9752, 35710}, {9766, 45921}, {9777, 11433}, {9781, 26917}, {9786, 26937}, {9792, 26954}, {9820, 10539}, {9833, 19357}, {9993, 43462}, {10110, 32767}, {10189, 41357}, {10190, 47627}, {10263, 34826}, {10282, 13419}, {10327, 17757}, {10516, 17811}, {10523, 15666}, {10527, 11401}, {10547, 16277}, {10550, 51862}, {10559, 51405}, {10576, 35764}, {10577, 35765}, {10601, 14561}, {10632, 42124}, {10633, 42121}, {10641, 23302}, {10642, 23303}, {10733, 32227}, {10881, 13966}, {10982, 39571}, {11174, 17907}, {11197, 44131}, {11206, 26864}, {11216, 47277}, {11408, 11488}, {11409, 11489}, {11411, 12160}, {11412, 33523}, {11424, 12241}, {11425, 19467}, {11430, 18400}, {11432, 18916}, {11435, 26957}, {11436, 26956}, {11438, 23329}, {11470, 15004}, {11472, 32123}, {11566, 40685}, {11572, 21659}, {11808, 14076}, {12022, 15033}, {12038, 45286}, {12058, 37511}, {12079, 35908}, {12139, 12864}, {12146, 13089}, {12161, 32140}, {12162, 18488}, {12165, 12317}, {12166, 12318}, {12168, 12319}, {12169, 12320}, {12170, 12321}, {12171, 12322}, {12172, 12323}, {12174, 12324}, {12175, 12325}, {12242, 14864}, {12295, 15432}, {12298, 45861}, {12299, 45860}, {12300, 20424}, {12618, 21062}, {12827, 14984}, {13007, 13025}, {13008, 13026}, {13051, 23313}, {13052, 23314}, {13148, 16003}, {13171, 13203}, {13292, 25738}, {13352, 18474}, {13367, 34782}, {13394, 29012}, {13403, 18383}, {13409, 23635}, {13428, 49029}, {13439, 49028}, {13562, 20806}, {13668, 13701}, {13788, 13821}, {13851, 23324}, {13857, 47354}, {14111, 35887}, {14156, 43586}, {14157, 16658}, {14376, 17407}, {14458, 43530}, {14492, 16080}, {14516, 34148}, {14583, 43090}, {14593, 40801}, {14627, 46443}, {14644, 43391}, {14767, 51412}, {14852, 44413}, {14918, 38429}, {14983, 45020}, {15038, 45967}, {15060, 51391}, {15066, 18358}, {15069, 37672}, {15116, 32246}, {15118, 41616}, {15152, 16252}, {15448, 44082}, {15449, 16102}, {15463, 32423}, {15484, 41370}, {15487, 46835}, {15523, 17442}, {15543, 15552}, {15544, 35605}, {15589, 32001}, {15819, 22480}, {16030, 19174}, {16188, 16221}, {16228, 47802}, {16229, 34964}, {16231, 39508}, {16276, 32819}, {16580, 40959}, {16607, 18636}, {16657, 18390}, {17451, 21717}, {17747, 24054}, {17810, 26958}, {17949, 32449}, {18287, 41925}, {18344, 21260}, {18356, 32358}, {18424, 24855}, {18475, 44407}, {18809, 46436}, {18853, 18854}, {18951, 37493}, {19119, 20079}, {19128, 38110}, {19347, 34780}, {19366, 26955}, {19404, 19420}, {19405, 19421}, {19446, 23298}, {19447, 23299}, {19459, 36851}, {19568, 47730}, {19724, 19755}, {19725, 19754}, {19784, 51686}, {19799, 19839}, {20185, 40120}, {20192, 45311}, {20306, 42448}, {20376, 52008}, {21072, 29016}, {21318, 41007}, {21637, 34774}, {21639, 23326}, {21663, 23328}, {21969, 41586}, {22240, 42459}, {22352, 44882}, {22483, 22966}, {22497, 22555}, {22750, 43614}, {22970, 23308}, {24321, 24686}, {26359, 26371}, {26360, 26372}, {26361, 26375}, {26362, 26376}, {26363, 26377}, {26364, 26378}, {26866, 26929}, {26867, 26939}, {26868, 26945}, {26892, 26932}, {26893, 26942}, {26894, 26950}, {26919, 26951}, {29667, 31419}, {30749, 40325}, {30786, 40413}, {31125, 41676}, {31127, 35360}, {31131, 44428}, {31406, 39575}, {31655, 48317}, {31670, 33586}, {31867, 35717}, {32000, 37668}, {32002, 37688}, {32046, 52432}, {32062, 51403}, {32085, 40425}, {32136, 45732}, {32184, 52003}, {32223, 48895}, {32240, 32255}, {32269, 48901}, {32332, 32391}, {32333, 32346}, {32337, 32341}, {32911, 44097}, {33547, 46686}, {34380, 45794}, {34417, 47296}, {34751, 44439}, {34781, 43841}, {34945, 41363}, {35540, 45211}, {35603, 36753}, {35718, 35729}, {35719, 35720}, {35723, 39565}, {35895, 35896}, {35971, 40377}, {36898, 38427}, {37472, 43595}, {37478, 44201}, {37650, 44100}, {37775, 42135}, {37776, 42138}, {38253, 43951}, {38396, 39879}, {38872, 44529}, {38970, 38975}, {39071, 47200}, {39511, 39533}, {39512, 51513}, {39532, 44432}, {39534, 48182}, {39662, 43448}, {39803, 39813}, {39832, 39842}, {39998, 44146}, {40149, 45964}, {40684, 44145}, {41171, 43574}, {41482, 51033}, {41603, 50649}, {41673, 41714}, {42353, 46832}, {42873, 51939}, {43817, 43823}, {44091, 51126}, {44426, 44429}, {45400, 45472}, {45401, 45473}, {45689, 47206}, {46147, 46151}, {46242, 46288}, {46261, 51425}, {47105, 51343}, {47392, 51385}

X(427) = midpoint of X(i) and X(j) for these {i,j}: {2, 31133}, {3, 31723}, {4, 378}, {22, 7391}, {112, 11605}, {184, 11550}, {1993, 11442}, {13352, 18474}, {15463, 44795}, {18570, 44288}, {24321, 24686}, {35480, 35481}
X(427) = reflection of X(i) in X(j) for these {i,j}: {1, 51718}, {3, 52262}, {5, 39504}, {6, 51744}, {22, 6676}, {184, 23292}, {235, 45179}, {343, 21243}, {1843, 51994}, {7502, 140}, {12083, 16618}, {15760, 5}, {16102, 15449}, {16165, 5972}, {16387, 5159}, {16789, 141}, {18570, 44236}, {19127, 3589}, {25337, 3628}, {34177, 6697}, {37478, 44201}, {37969, 468}, {44210, 2}, {44218, 44287}, {44239, 3}, {44249, 18570}, {44259, 16238}, {44260, 6677}, {44261, 549}, {44262, 547}, {44263, 546}, {44285, 44218}, {46029, 13413}, {51692, 1125}
X(427) = isogonal conjugate of X(1176)
X(427) = isotomic conjugate of X(1799)
X(427) = complement of X(22)
X(427) = anticomplement of X(6676)
X(427) = circumcircle-inverse of X(21284)
X(427) = nine-point-circle-inverse of X(468)
X(427) = orthocentroidal-circle-inverse of X(25)
X(427) = polar-circle-inverse of X(23)
X(427) = orthoptic-circle-of-the-Steiner-inellipse inverse of X(186)
X(427) = 1st-Droz-Farney-circle-inverse of X(45171)
X(427) = orthosymmedial-circle-inverse of X(1112)
X(427) = polar conjugate of X(83)
X(427) = antitomic image of X(16102)
X(427) = complement of the isogonal conjugate of X(66)
X(427) = complement of the isotomic conjugate of X(18018)
X(427) = isotomic conjugate of the complement of X(8878)
X(427) = isotomic conjugate of the isogonal conjugate of X(1843)
X(427) = isogonal conjugate of the isotomic conjugate of X(1235)
X(427) = isotomic conjugate of the polar conjugate of X(27376)
X(427) = polar conjugate of the isotomic conjugate of X(141)
X(427) = polar conjugate of the isogonal conjugate of X(39)
X(427) = medial-isogonal conjugate of X(206)
X(427) = orthic-isogonal conjugate of X(1843)
X(427) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 206}, {2, 21247}, {6, 16582}, {31, 40938}, {57, 17068}, {66, 10}, {1289, 8062}, {2156, 2}, {2353, 37}, {13854, 226}, {14376, 18589}, {15388, 16599}, {16277, 1215}, {18018, 2887}, {40146, 16584}, {40421, 21235}, {43678, 20305}, {44766, 4369}, {46244, 626}
X(427) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 40938}, {4, 1843}, {112, 523}, {264, 1235}, {1235, 141}, {6331, 2501}, {8791, 468}, {8889, 8893}, {16747, 20883}, {17171, 17442}, {27376, 41584}, {34129, 16318}, {46151, 826}
X(427) = X(i)-cross conjugate of X(j) for these (i,j): {39, 141}, {141, 47730}, {826, 46151}, {1843, 27376}, {2530, 46152}, {3005, 35325}, {21016, 20883}, {23642, 76}, {23666, 75}, {42442, 6}, {46026, 4}, {46654, 31067}
X(427) = X(i)-isoconjugate of X(j) for these (i,j): {1, 1176}, {3, 82}, {6, 34055}, {19, 28724}, {31, 1799}, {48, 83}, {63, 251}, {69, 46289}, {71, 52376}, {75, 10547}, {163, 4580}, {184, 3112}, {228, 52394}, {248, 3405}, {255, 32085}, {293, 51862}, {304, 46288}, {308, 9247}, {525, 34072}, {647, 4599}, {656, 827}, {810, 4577}, {822, 42396}, {905, 4628}, {906, 10566}, {1331, 18108}, {1437, 18082}, {1760, 46765}, {1790, 18098}, {2169, 17500}, {2172, 40404}, {2193, 18097}, {3049, 4593}, {4020, 52395}, {4574, 39179}, {4592, 18105}, {4630, 14208}, {10317, 37221}, {14575, 18833}, {18070, 32661}, {19611, 51508}, {22384, 36081}, {46104, 52430}
X(427) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 1799}, {2, 40938}, {3, 141}, {3, 1176}, {6, 28724}, {9, 34055}, {22, 427}, {25, 8793}, {39, 69}, {63, 40585}, {82, 36103}, {83, 1249}, {115, 4580}, {132, 51862}, {184, 34452}, {206, 10547}, {251, 3162}, {339, 3267}, {525, 15449}, {647, 3124}, {827, 40596}, {1194, 45201}, {1843, 15270}, {3405, 39039}, {3589, 7767}, {3933, 6665}, {3934, 22062}, {4577, 39062}, {4599, 39052}, {5139, 18105}, {5190, 10566}, {5521, 18108}, {6523, 32085}, {7664, 37804}, {8673, 47413}, {14363, 17500}, {18097, 47345}, {20775, 52042}, {21458, 50938}, {22105, 48317}, {23093, 44312}
X(427) = cevapoint of X(i) and X(j) for these (i,j): {2, 8878}, {25, 8792}, {39, 1843}, {115, 2514}, {3005, 39691}, {28666, 46026}
X(427) = crosspoint of X(i) and X(j) for these (i,j): {2, 18018}, {4, 264}, {16747, 17171}
X(427) = crosssum of X(i) and X(j) for these (i,j): {3, 184}, {6, 206}, {25, 51509}, {251, 51508}
X(427) = trilinear pole of line {826, 21108}
X(427) = crossdifference of every pair of points on line {647, 8673}
X(427) = X(2)-line conjugate of X(44894)
X(427) = X(4)-Hirst inverse of X(420)
X(427) = X(55) of orthic triangle if ABC is acute
X(427) = exsimilicenter of nine-point circle and incircle of orthic triangle if ABC is acute; the insimilicenter is X(235)
X(427) = intersection of tangents to nine-point circle at PU(4)
X(427) = pole of orthic axis wrt the nine-point circle
X(427) = pole wrt polar circle of trilinear polar of X(83) (line X(23)X(385))
X(427) = X(48)-isoconjugate (polar conjugate) of X(83)
X(427) = inverse-in-polar-circle of X(23)
X(427) = inverse-in-orthosymmedial-circle of X(1112)
X(427) = perspector of orthic and 5th Euler triangles
X(427) = radical trace of anticomplementary circle and tangential circle
X(427) = homothetic center of the medial triangle and the 2nd pedal triangle of X(4)
X(427) = perspector of 4th Brocard triangle and cross-triangle of ABC and 4th Brocard triangle
X(427) = perspector of ABC and cross-triangle of ABC and 5th Euler triangle
X(427) = harmonic center of nine-point circle and circle O(PU(4))
X(427) = orthic-isogonal conjugate of X(1843)
X(427) = intersection of tangents to Hatzipolakis-Lozada hyperbola at X(4) and X(6)
X(427) = crosspoint, wrt orthic triangle, of X(4) and X(6)
X(427) = excentral-to-ABC functional image of X(56)
X(427) = intersection of tangents to Walsmith rectangular hyperbola at X(113) and X(125)
X(427) = barycentric product X(i)*X(j) for these {i,j}: {1, 20883}, {4, 141}, {6, 1235}, {10, 17171}, {19, 1930}, {25, 8024}, {27, 15523}, {37, 16747}, {38, 92}, {39, 264}, {69, 27376}, {75, 17442}, {76, 1843}, {86, 21016}, {95, 27371}, {107, 2525}, {112, 23285}, {190, 21108}, {193, 47730}, {273, 33299}, {278, 3703}, {281, 3665}, {286, 3954}, {297, 20021}, {324, 16030}, {331, 3688}, {343, 19174}, {393, 3933}, {420, 17949}, {468, 31125}, {523, 41676}, {525, 46151}, {648, 826}, {653, 48278}, {811, 8061}, {850, 35325}, {1289, 23881}, {1401, 7017}, {1502, 27369}, {1634, 14618}, {1783, 48084}, {1824, 16703}, {1826, 16887}, {1897, 16892}, {1964, 1969}, {2052, 3917}, {2501, 4576}, {2528, 42396}, {2530, 6335}, {2592, 46166}, {2593, 46167}, {2996, 41584}, {3005, 6331}, {3051, 18022}, {3186, 42551}, {3313, 43678}, {3404, 40703}, {3613, 37125}, {3867, 18840}, {4074, 37892}, {4391, 46152}, {4553, 17924}, {4568, 7649}, {5094, 23297}, {5117, 14617}, {5485, 41585}, {5523, 46165}, {6531, 51371}, {7794, 32085}, {7813, 17983}, {8041, 46104}, {8362, 8801}, {9019, 46105}, {9229, 12143}, {10159, 46026}, {14376, 41375}, {16080, 51360}, {16102, 40889}, {16696, 41013}, {17980, 35540}, {18018, 40938}, {18020, 39691}, {18027, 20775}, {19568, 47847}, {21035, 44129}, {28666, 40425}, {31613, 42394}, {35140, 51434}, {39129, 41361}, {40416, 46508}, {41331, 44161}, {44132, 51869}, {44146, 46154}, {44427, 46155}, {46106, 46147}, {46107, 46148}, {46108, 46149}, {46109, 46150}, {46110, 46153}
X(427) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 34055}, {2, 1799}, {3, 28724}, {4, 83}, {6, 1176}, {19, 82}, {25, 251}, {27, 52394}, {28, 52376}, {32, 10547}, {38, 63}, {39, 3}, {53, 17500}, {66, 40404}, {92, 3112}, {107, 42396}, {112, 827}, {141, 69}, {162, 4599}, {225, 18097}, {232, 51862}, {240, 3405}, {264, 308}, {275, 39287}, {276, 41488}, {297, 20022}, {393, 32085}, {420, 40850}, {429, 27067}, {523, 4580}, {648, 4577}, {688, 3049}, {732, 12215}, {811, 4593}, {826, 525}, {1235, 76}, {1401, 222}, {1593, 26224}, {1634, 4558}, {1824, 18098}, {1826, 18082}, {1840, 18099}, {1843, 6}, {1883, 27005}, {1923, 9247}, {1930, 304}, {1964, 48}, {1969, 18833}, {1973, 46289}, {1974, 46288}, {2052, 46104}, {2084, 810}, {2353, 46765}, {2489, 18105}, {2525, 3265}, {2528, 2525}, {2530, 905}, {2971, 51906}, {3005, 647}, {3051, 184}, {3118, 4173}, {3162, 8793}, {3172, 51508}, {3313, 20806}, {3404, 293}, {3575, 10548}, {3665, 348}, {3688, 219}, {3703, 345}, {3787, 3167}, {3867, 3618}, {3914, 18084}, {3917, 394}, {3933, 3926}, {3954, 72}, {4020, 255}, {4074, 37894}, {4553, 1332}, {4568, 4561}, {4576, 4563}, {5064, 39668}, {5094, 10130}, {6292, 7767}, {6331, 689}, {6591, 18108}, {6995, 42037}, {7649, 10566}, {7794, 3933}, {7813, 6390}, {8024, 305}, {8041, 3917}, {8061, 656}, {8362, 3785}, {8711, 22159}, {8735, 18101}, {8750, 4628}, {8754, 34294}, {8791, 9076}, {9019, 22151}, {11105, 29534}, {11205, 22352}, {11325, 38834}, {12143, 384}, {13854, 16277}, {14273, 22105}, {14424, 14417}, {15523, 306}, {16030, 97}, {16318, 21458}, {16696, 1444}, {16747, 274}, {16887, 17206}, {16892, 4025}, {16893, 4121}, {17171, 86}, {17187, 1790}, {17407, 40357}, {17442, 1}, {17980, 733}, {18022, 40016}, {19118, 33632}, {19174, 275}, {19577, 51459}, {19595, 5596}, {19606, 15389}, {20021, 287}, {20775, 577}, {20883, 75}, {21016, 10}, {21035, 71}, {21108, 514}, {21123, 1459}, {21248, 45201}, {21336, 45220}, {21814, 228}, {23208, 10316}, {23285, 3267}, {23642, 11574}, {24006, 18070}, {27369, 32}, {27370, 41334}, {27371, 5}, {27373, 8743}, {27374, 217}, {27376, 4}, {28666, 6292}, {28667, 8788}, {29959, 41614}, {30489, 43697}, {31125, 30786}, {31390, 21637}, {32085, 52395}, {32676, 34072}, {33299, 78}, {35319, 23181}, {35325, 110}, {36794, 41296}, {37125, 1078}, {39284, 39289}, {39691, 125}, {40889, 16095}, {40936, 22061}, {40938, 22}, {40972, 212}, {41267, 2200}, {41272, 14908}, {41331, 14575}, {41375, 17907}, {41584, 193}, {41585, 1992}, {41676, 99}, {42551, 43714}, {44090, 46228}, {45211, 45210}, {46026, 3589}, {46147, 14919}, {46148, 1331}, {46149, 1814}, {46150, 1797}, {46151, 648}, {46152, 651}, {46153, 1813}, {46154, 895}, {46166, 8115}, {46167, 8116}, {46387, 22384}, {46508, 626}, {46509, 8023}, {47730, 2996}, {48084, 15413}, {48278, 6332}, {50521, 22383}, {51360, 11064}, {51371, 6393}, {51434, 1503}, {51869, 248}
X(427) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 5090, 12135}, {2, 3, 7499}, {2, 4, 25}, {2, 5, 37439}, {2, 20, 7494}, {2, 22, 6676}, {2, 25, 468}, {2, 468, 52297}, {2, 858, 1368}, {2, 1368, 30739}, {2, 1370, 3}, {2, 1995, 6677}, {2, 3091, 7392}, {2, 3146, 10565}, {2, 3830, 37904}, {2, 3832, 7398}, {2, 4232, 38282}, {2, 5064, 428}, {2, 5133, 5}, {2, 5169, 5133}, {2, 5189, 6636}, {2, 6353, 37453}, {2, 6636, 7495}, {2, 6995, 6353}, {2, 6997, 5020}, {2, 7378, 4}, {2, 7386, 7484}, {2, 7391, 22}, {2, 7392, 11284}, {2, 7394, 1995}, {2, 7396, 7386}, {2, 7398, 40132}, {2, 7408, 4232}, {2, 7409, 6995}, {2, 7485, 140}, {2, 7500, 7493}, {2, 7571, 3628}, {2, 8889, 5094}, {2, 14957, 7467}, {2, 16051, 31255}, {2, 16063, 7485}, {2, 16951, 7807}, {2, 24584, 1375}, {2, 26032, 37099}, {2, 26096, 25494}, {2, 26760, 27028}, {2, 26808, 26967}, {2, 28684, 28664}, {2, 30744, 5159}, {2, 30888, 46531}, {2, 30951, 37370}, {2, 31021, 857}, {2, 31074, 858}, {2, 31099, 1370}, {2, 31105, 31133}, {2, 31152, 43957}, {2, 31857, 31074}, {2, 34603, 10154}, {2, 34609, 7667}, {2, 37174, 37187}, {2, 37349, 13595}, {2, 37353, 37990}, {2, 37362, 37432}, {2, 37456, 4224}, {2, 37913, 52300}, {2, 38282, 52292}, {2, 46336, 16419}, {2, 47314, 8703}, {2, 52284, 8889}, {2, 52285, 10301}, {2, 52299, 52298}, {3, 4, 3575}, {3, 5, 7399}, {3, 25, 21213}, {3, 381, 18420}, {3, 1370, 7667}, {3, 5576, 5}, {3, 18494, 18533}, {3, 18533, 37931}, {3, 31099, 46517}, {3, 34609, 1370}, {3, 44441, 47090}, {4, 5, 235}, {4, 20, 12173}, {4, 24, 6756}, {4, 25, 428}, {4, 28, 37398}, {4, 186, 7576}, {4, 235, 1906}, {4, 381, 10151}, {4, 403, 1596}, {4, 406, 4186}, {4, 429, 1904}, {4, 442, 37376}, {4, 451, 4222}, {4, 458, 460}, {4, 468, 10301}, {4, 469, 430}, {4, 470, 462}, {4, 471, 463}, {4, 475, 4185}, {4, 631, 7487}, {4, 860, 1894}, {4, 1346, 1312}, {4, 1347, 1313}, {4, 1585, 52286}, {4, 1586, 52287}, {4, 1593, 1885}, {4, 1594, 5}, {4, 1595, 1907}, {4, 2476, 431}, {4, 3088, 1593}, {4, 3089, 5198}, {4, 3090, 3089}, {4, 3091, 37197}, {4, 3127, 32588}, {4, 3128, 32587}, {4, 3144, 37390}, {4, 3147, 37122}, {4, 3520, 6240}, {4, 3535, 52291}, {4, 3536, 5200}, {4, 3541, 3}, {4, 3542, 1598}, {4, 3545, 6623}, {4, 4194, 17516}, {4, 4196, 1889}, {4, 4200, 4214}, {4, 4212, 27}, {4, 4213, 14004}, {4, 5064, 52285}, {4, 5094, 468}, {4, 5117, 297}, {4, 5125, 407}, {4, 5136, 1884}, {4, 5142, 429}, {4, 5169, 37981}, {4, 6143, 3518}, {4, 6353, 6995}, {4, 7378, 5064}, {4, 7498, 28076}, {4, 7505, 10594}, {4, 7507, 23047}, {4, 7547, 546}, {4, 7577, 403}, {4, 7714, 7408}, {4, 8889, 2}, {4, 10018, 7715}, {4, 11109, 37226}, {4, 14865, 18560}, {4, 14940, 34484}, {4, 15559, 1595}, {4, 18533, 18494}, {4, 18560, 3627}, {4, 26020, 37432}, {4, 30100, 12225}, {4, 35473, 18559}, {4, 35475, 34797}, {4, 35481, 35480}, {4, 35482, 3520}, {4, 35488, 44226}, {4, 35490, 3853}, {4, 35502, 13488}, {4, 37074, 1529}, {4, 37117, 7511}, {4, 37118, 37458}, {4, 37119, 24}, {4, 37125, 27369}, {4, 37337, 11325}, {4, 37388, 37385}, {4, 37943, 52294}, {4, 38282, 7714}, {4, 44287, 44281}, {4, 44438, 13473}, {4, 44958, 44803}, {4, 52249, 52280}, {4, 52252, 28}, {4, 52280, 6755}, {4, 52284, 5094}, {4, 52288, 6620}, {4, 52290, 52301}, {4, 52295, 1594}, {4, 52296, 21841}, {4, 52299, 6353}, {5, 140, 7405}, {5, 858, 30739}, {5, 1368, 2}, {5, 1595, 4}, {5, 1596, 403}, {5, 1906, 45004}, {5, 1907, 1906}, {5, 3627, 15761}, {5, 3845, 46030}, {5, 6823, 13160}, {5, 13371, 11585}, {5, 15559, 1907}, {5, 15687, 11563}, {5, 15809, 25}, {5, 23335, 3}, {5, 44960, 16868}, {6, 66, 26926}, {6, 1853, 1899}, {6, 1899, 11245}, {20, 13160, 6823}, {22, 6676, 44210}, {22, 31133, 7391}, {22, 31236, 2}, {22, 37353, 46029}, {24, 37119, 140}, {25, 428, 10301}, {25, 5064, 4}, {25, 5094, 2}, {25, 7378, 52285}, {25, 37453, 6353}, {25, 52298, 37453}, {33, 5101, 1862}, {39, 27371, 27376}, {51, 125, 13567}, {53, 3815, 232}, {54, 34224, 31804}, {115, 1560, 44467}, {125, 132, 47202}, {140, 6756, 24}, {140, 10691, 7485}, {140, 11548, 2}, {140, 16198, 6756}, {140, 47315, 16063}, {140, 50138, 5}, {141, 1843, 41584}, {141, 3867, 1843}, {154, 36990, 31383}, {185, 3574, 12233}, {186, 7576, 37458}, {186, 37118, 549}, {230, 6748, 10311}, {235, 1907, 4}, {235, 30739, 468}, {235, 44960, 44995}, {235, 45003, 44996}, {264, 18022, 42394}, {264, 51843, 18022}, {275, 23295, 8901}, {343, 45303, 21243}, {378, 35480, 35481}, {378, 37970, 3520}, {378, 44269, 18570}, {378, 44274, 44281}, {378, 45179, 15760}, {381, 1597, 4}, {381, 2072, 5}, {381, 5020, 6997}, {381, 30771, 5020}, {382, 7493, 37899}, {382, 9909, 7500}, {393, 7736, 45141}, {403, 1594, 7577}, {403, 1596, 235}, {403, 7577, 5}, {428, 468, 25}, {428, 5094, 52297}, {428, 8889, 52293}, {428, 52285, 4}, {429, 1883, 4}, {442, 8226, 7522}, {465, 41034, 3132}, {466, 41035, 3131}, {468, 5094, 52293}, {468, 52285, 428}, {470, 471, 52289}, {470, 472, 16250}, {471, 473, 16249}, {472, 473, 52281}, {546, 5159, 1995}, {546, 10224, 5}, {546, 13488, 4}, {549, 37458, 186}, {578, 18381, 6146}, {590, 5412, 13884}, {615, 5413, 13937}, {631, 7487, 3515}, {631, 44831, 44837}, {858, 1594, 5094}, {858, 2072, 47097}, {858, 5133, 2}, {858, 5169, 5}, {858, 5576, 37454}, {858, 7378, 15809}, {858, 31099, 23335}, {858, 37981, 468}, {858, 46698, 1312}, {858, 46699, 1313}, {867, 37354, 8727}, {1113, 1114, 21284}, {1125, 49542, 11363}, {1204, 6696, 43903}, {1312, 1313, 468}, {1346, 1347, 5094}, {1346, 46699, 5}, {1347, 46698, 5}, {1368, 1595, 15809}, {1368, 5133, 37439}, {1370, 31099, 34609}, {1370, 34609, 46517}, {1513, 52280, 25}, {1585, 1586, 458}, {1593, 7507, 4}, {1593, 18386, 44438}, {1594, 1595, 235}, {1594, 15559, 4}, {1594, 15809, 37439}, {1595, 52284, 30739}, {1596, 16868, 45000}, {1597, 52284, 47097}, {1598, 1656, 3542}, {1656, 16419, 2}, {1656, 50137, 5}, {1799, 16275, 7750}, {1843, 46026, 3867}, {1848, 1861, 1824}, {1883, 5142, 1904}, {1885, 13473, 44438}, {1885, 23047, 4}, {1907, 37439, 428}, {1994, 3448, 45968}, {1994, 45968, 1353}, {1995, 30744, 2}, {2041, 2042, 9715}, {2052, 6530, 14569}, {2071, 38323, 44241}, {2450, 37988, 5}, {2476, 37983, 5}, {2479, 2480, 40889}, {2969, 7140, 92}, {2979, 37636, 48876}, {3060, 3580, 41588}, {3060, 7703, 23293}, {3060, 23293, 3580}, {3088, 7507, 1885}, {3091, 16051, 11284}, {3091, 37197, 10019}, {3127, 3128, 4}, {3146, 10565, 34608}, {3147, 37122, 3517}, {3148, 14003, 441}, {3153, 7527, 52069}, {3162, 13854, 16318}, {3258, 42426, 47223}, {3516, 12173, 20}, {3517, 3526, 3147}, {3518, 6143, 10018}, {3518, 7496, 37920}, {3520, 6240, 550}, {3547, 34938, 11414}, {3548, 7528, 6642}, {3567, 23294, 26879}, {3575, 7499, 21213}, {3575, 37454, 468}, {3575, 37931, 18533}, {3627, 45177, 235}, {3628, 10300, 40916}, {3628, 21841, 7505}, {3628, 50136, 5}, {3830, 7579, 10254}, {3830, 10254, 11799}, {3832, 30769, 40132}, {3839, 30775, 47597}, {3850, 44226, 35488}, {3850, 49673, 5}, {3851, 50143, 5}, {3853, 37897, 7519}, {4074, 40379, 141}, {4232, 7408, 7714}, {4232, 7714, 25}, {4232, 52292, 468}, {5000, 5001, 5}, {5002, 5003, 12225}, {5004, 5005, 7488}, {5020, 30771, 2}, {5055, 50135, 5}, {5064, 5094, 25}, {5064, 8889, 468}, {5064, 37439, 1906}, {5064, 52292, 7408}, {5064, 52298, 6995}, {5066, 50140, 5}, {5072, 50139, 5}, {5094, 7378, 428}, {5094, 37197, 11284}, {5094, 37453, 52298}, {5094, 37920, 6143}, {5094, 37981, 5}, {5094, 52298, 52299}, {5133, 15559, 5064}, {5133, 15809, 235}, {5133, 31074, 1368}, {5133, 37990, 37353}, {5159, 6677, 2}, {5169, 31074, 2}, {5169, 31101, 37353}, {5169, 31857, 858}, {5169, 40916, 50136}, {5169, 46336, 50137}, {5169, 52284, 1594}, {5189, 6636, 52397}, {5189, 7495, 550}, {5422, 18911, 45298}, {5446, 5449, 41587}, {5480, 13567, 51}, {5480, 23332, 13567}, {5576, 23335, 7399}, {6143, 10018, 632}, {6240, 7495, 21284}, {6247, 12233, 185}, {6353, 6995, 25}, {6353, 7577, 37990}, {6353, 8889, 52299}, {6353, 35481, 22}, {6353, 37453, 468}, {6353, 52299, 2}, {6623, 32216, 468}, {6636, 52397, 550}, {6639, 7517, 13383}, {6640, 7506, 16238}, {6640, 44259, 44907}, {6643, 7404, 7395}, {6644, 18281, 10257}, {6676, 13413, 37990}, {6696, 13568, 1204}, {6747, 42400, 53}, {6756, 16198, 4}, {6776, 11427, 11402}, {6995, 7378, 7409}, {6995, 7391, 35480}, {6995, 7409, 4}, {6995, 8889, 52298}, {6995, 37353, 403}, {6995, 52298, 468}, {6995, 52299, 37453}, {7378, 8889, 25}, {7378, 15809, 1907}, {7378, 52284, 2}, {7386, 7396, 31152}, {7386, 7484, 43957}, {7386, 28412, 426}, {7391, 31236, 6676}, {7392, 11403, 428}, {7392, 16051, 2}, {7394, 30744, 6677}, {7396, 7539, 43957}, {7398, 30769, 2}, {7400, 52398, 37198}, {7403, 11585, 5}, {7408, 38282, 25}, {7409, 8889, 37453}, {7409, 37453, 428}, {7409, 37990, 1596}, {7409, 52284, 52299}, {7409, 52299, 25}, {7418, 46591, 21284}, {7484, 7539, 2}, {7484, 31152, 7386}, {7485, 16063, 10691}, {7493, 7500, 9909}, {7494, 44442, 20}, {7495, 52397, 6636}, {7499, 7667, 3}, {7499, 37454, 2}, {7499, 46517, 7667}, {7500, 9909, 37899}, {7503, 37444, 12362}, {7505, 10594, 21841}, {7505, 52296, 3628}, {7507, 44438, 18386}, {7514, 31181, 14791}, {7526, 18569, 12605}, {7530, 10201, 37971}, {7536, 49132, 199}, {7539, 31152, 7484}, {7542, 7553, 26}, {7544, 17928, 9825}, {7547, 35502, 4}, {7558, 10323, 16197}, {7566, 17928, 7544}, {7569, 10323, 7558}, {7570, 37977, 14940}, {7571, 40916, 2}, {7576, 37118, 186}, {7577, 18533, 37347}, {7577, 31101, 52298}, {7577, 35480, 46029}, {7579, 11799, 5}, {7592, 11457, 18914}, {7667, 37454, 7499}, {7667, 46517, 1370}, {7714, 38282, 4232}, {8370, 35920, 15014}, {8889, 15559, 15809}, {8889, 15809, 30739}, {8889, 52285, 52297}, {8891, 21248, 141}, {9777, 26869, 11433}, {9786, 40686, 26937}, {9818, 18531, 34664}, {9825, 16196, 17928}, {9909, 18364, 6636}, {10109, 50134, 5}, {10151, 47097, 468}, {10295, 35473, 8703}, {10301, 52293, 468}, {10594, 52296, 7505}, {10691, 11548, 140}, {11056, 16275, 1799}, {11284, 31255, 2}, {11381, 43831, 2883}, {11403, 37197, 4}, {11410, 37196, 376}, {11427, 32064, 6776}, {11432, 26944, 18916}, {11433, 14853, 9777}, {11433, 23291, 26869}, {11548, 47315, 10691}, {11572, 21659, 41362}, {11737, 50142, 5}, {11818, 18281, 6644}, {12086, 34007, 52071}, {12100, 37934, 35472}, {12106, 44452, 44211}, {13371, 33332, 7403}, {13413, 46029, 5}, {13490, 44452, 12106}, {13567, 23332, 125}, {14130, 31724, 18563}, {14782, 14783, 7401}, {14784, 14785, 3547}, {14813, 14814, 34002}, {14826, 37669, 6090}, {14853, 23291, 11433}, {15033, 25739, 12022}, {15559, 52295, 5}, {16051, 37197, 468}, {16238, 32144, 6640}, {16252, 16621, 26883}, {16387, 44260, 44210}, {17111, 23304, 11}, {18386, 44438, 4}, {18420, 44441, 3}, {18494, 18533, 3575}, {18559, 35473, 10295}, {18560, 45177, 15761}, {18570, 44236, 44218}, {18570, 44249, 44285}, {18570, 44269, 44281}, {18570, 44274, 44269}, {18570, 44287, 44236}, {18583, 45298, 5422}, {19041, 19042, 40065}, {21213, 37454, 52297}, {21850, 41588, 3060}, {23336, 31830, 37814}, {24886, 24939, 2}, {25738, 36749, 13292}, {25985, 26020, 2}, {25985, 37362, 25}, {27098, 27150, 2}, {27386, 27510, 33305}, {28020, 28084, 33302}, {28048, 28126, 33306}, {28240, 28354, 859}, {28412, 28701, 441}, {29385, 29442, 46497}, {29498, 29701, 46575}, {29553, 29752, 46574}, {29970, 30040, 46514}, {30739, 37439, 2}, {31074, 37353, 31101}, {31099, 52284, 3541}, {31101, 37353, 2}, {31105, 52284, 378}, {31133, 31236, 22}, {31236, 35480, 37453}, {31670, 37638, 47582}, {31723, 52262, 44239}, {32587, 32588, 4}, {34221, 34222, 19161}, {34797, 35475, 35491}, {34797, 35491, 15704}, {35471, 35477, 548}, {35480, 52299, 6676}, {35487, 44803, 44958}, {35502, 52295, 10224}, {37353, 37990, 5}, {37452, 50137, 1656}, {37453, 52298, 2}, {37454, 46517, 3}, {37472, 44076, 43595}, {39504, 46029, 13413}, {42789, 42790, 35921}, {43957, 47311, 31152}, {44218, 44249, 18570}, {44236, 44288, 44249}, {44287, 44288, 18570}, {44637, 44638, 6748}, {44996, 45003, 45006}, {46517, 47097, 47090}, {46698, 46699, 37981}, {47090, 47310, 47340}, {47612, 47613, 11585}, {52286, 52287, 460}, {52293, 52297, 2}


X(428) = EULER X(38)-5th-SUBSTITUTION POINT

Trilinears    3 sec A - csc A tan ω : :
Trilinears    csc A - 3 sec A cot ω : :
Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(2*a^2 + b^2 + c^2) : :
X(428) = 5 X[2] - 4 X[7734], 3 X[2] - 4 X[10128], 7 X[2] - 8 X[13361], 4 X[2] - 3 X[43957], 4 X[4] - X[1885], 2 X[4] + X[3575], 5 X[4] + X[6240], X[4] + 2 X[6756], 5 X[4] - 2 X[13488], 3 X[4] + X[18559], 7 X[4] - X[18560], 11 X[4] + X[34797], 2 X[5] + X[7553], X[20] - 4 X[9825], 4 X[140] - 3 X[43934], X[382] + 2 X[31833], 2 X[546] + X[11819], 4 X[546] - X[12605], X[548] - 4 X[13163], 5 X[1656] - 8 X[23411], X[1885] + 2 X[3575], 5 X[1885] + 4 X[6240], X[1885] + 8 X[6756], X[1885] + 4 X[7576], 5 X[1885] - 8 X[13488], 3 X[1885] + 4 X[18559], 7 X[1885] - 4 X[18560], 11 X[1885] + 4 X[34797], 5 X[3091] - 2 X[12362], X[3146] + 2 X[31829], 5 X[3575] - 2 X[6240], X[3575] - 4 X[6756], 5 X[3575] + 4 X[13488], 3 X[3575] - 2 X[18559], 7 X[3575] + 2 X[18560], 11 X[3575] - 2 X[34797], X[3627] + 2 X[31830], 7 X[3832] - X[12225], 3 X[3839] - X[52069], 5 X[3843] - X[18564], 5 X[3858] - 2 X[52073], 4 X[3861] - X[52070], X[6240] - 10 X[6756], X[6240] - 5 X[7576], X[6240] + 2 X[13488], 3 X[6240] - 5 X[18559], 7 X[6240] + 5 X[18560], 11 X[6240] - 5 X[34797], and many others

As a point on the Euler line, X(428) has Shinagawa coefficients (F,-3E - 3F).

Let V = P(4)-Ceva conjugate of U(4)) and W = U(4)-Ceva conjugate of P(4); then V and W lie on the orthocentroidal circle, and with with PU(4) they are the vertices of a cyclic trapezoid. The midpoint of V and W is X(428). (Randy Hutson, December 26, 2015)

Let H = X(4) and let A'B'C' be the pedal triangle of H in the plane of a triangle ABC. Let
Bc = reflection of B' in HC', and define Ca and Ab cyclically;
Cb = reflection of C' in HB', and define Ac and Ba cyclically;
Ha = orthogonal projecton of A on BcCb, and define Hb and Hc cyclically.
Then X(428) = centroid of triangle HaHbHc. See Antreas Hatzipolakis and Peter Moses, Hyacinthos 27169

X(428) lies on these lines: {1, 51719}, {2, 3}, {6, 31166}, {11, 5322}, {12, 5310}, {19, 5101}, {33, 3058}, {34, 5434}, {51, 1503}, {53, 2980}, {98, 39284}, {107, 29316}, {125, 44106}, {132, 137}, {136, 44953}, {184, 5480}, {185, 11745}, {230, 1879}, {232, 5421}, {242, 7140}, {251, 5305}, {264, 37671}, {275, 14492}, {305, 32819}, {316, 45201}, {317, 7788}, {325, 16276}, {343, 3818}, {389, 16655}, {394, 31670}, {395, 10642}, {396, 10641}, {511, 22480}, {519, 1829}, {524, 1843}, {528, 1824}, {529, 1828}, {530, 12142}, {531, 12141}, {532, 22482}, {533, 22481}, {535, 1878}, {538, 12143}, {539, 5446}, {541, 12133}, {542, 1112}, {543, 5186}, {544, 5185}, {545, 24814}, {551, 11363}, {553, 1876}, {591, 45400}, {597, 1974}, {599, 7716}, {612, 6284}, {614, 7354}, {754, 12144}, {1162, 1163}, {1184, 7737}, {1192, 43903}, {1194, 7745}, {1196, 7747}, {1352, 33586}, {1495, 23292}, {1560, 45161}, {1629, 6530}, {1839, 5089}, {1848, 49736}, {1861, 2355}, {1892, 4654}, {1899, 17810}, {1902, 28194}, {1986, 11566}, {1991, 45401}, {1992, 12167}, {1993, 21850}, {1994, 46818}, {2052, 14458}, {2501, 25423}, {2969, 7009}, {3060, 3564}, {3087, 45141}, {3199, 7753}, {3241, 7718}, {3563, 20189}, {3567, 16659}, {3574, 16252}, {3589, 22352}, {3614, 7302}, {3679, 5090}, {3796, 14561}, {3819, 29317}, {3849, 5140}, {3917, 29181}, {3920, 15171}, {4421, 11383}, {4995, 52427}, {5012, 18583}, {5095, 41149}, {5130, 34606}, {5272, 10483}, {5309, 27376}, {5342, 50310}, {5345, 7741}, {5359, 18907}, {5370, 7173}, {5410, 19054}, {5411, 19053}, {5412, 32787}, {5413, 32788}, {5422, 48906}, {5476, 44077}, {5640, 45298}, {5651, 51163}, {5860, 11389}, {5861, 11388}, {5890, 16658}, {5943, 29012}, {6000, 16654}, {6090, 51538}, {6146, 10110}, {6152, 11817}, {6172, 7717}, {6198, 15170}, {6243, 31831}, {6515, 18440}, {6688, 29323}, {6759, 45089}, {6776, 9777}, {6800, 38136}, {7071, 10385}, {7191, 18990}, {7298, 7951}, {7738, 39951}, {7750, 40022}, {7767, 39998}, {7773, 34254}, {7811, 11386}, {7837, 27377}, {8280, 35786}, {8281, 35787}, {8541, 8584}, {8550, 15004}, {8667, 41762}, {8739, 43229}, {8740, 43228}, {8744, 8792}, {8780, 37645}, {8854, 35821}, {8855, 35820}, {8902, 42400}, {9143, 19504}, {9306, 48901}, {9530, 12145}, {9766, 45478}, {9781, 34224}, {9833, 10982}, {9969, 26926}, {10056, 11398}, {10072, 11399}, {10169, 47459}, {10192, 44082}, {10516, 43653}, {10601, 46264}, {10632, 42912}, {10633, 42913}, {11002, 45968}, {11064, 48895}, {11194, 22479}, {11206, 11402}, {11207, 11384}, {11208, 11385}, {11216, 47463}, {11235, 11390}, {11236, 11391}, {11237, 11392}, {11238, 11393}, {11239, 11400}, {11240, 11401}, {11380, 12150}, {11381, 13568}, {11394, 12152}, {11395, 12153}, {11406, 34607}, {11408, 37640}, {11409, 37641}, {11424, 34782}, {11427, 26864}, {11441, 31802}, {11442, 39884}, {11471, 34618}, {11473, 41945}, {11474, 41946}, {11475, 42942}, {11476, 42943}, {11550, 13567}, {11645, 44084}, {11743, 32332}, {12007, 34565}, {12099, 36201}, {12147, 32419}, {12148, 32421}, {12233, 26883}, {12242, 50414}, {12300, 31834}, {13142, 14516}, {13198, 51734}, {13846, 13884}, {13847, 13937}, {14389, 26881}, {14486, 14593}, {14495, 14569}, {14537, 33842}, {14583, 43089}, {14826, 51212}, {15058, 33523}, {15107, 18358}, {15172, 29815}, {15311, 32062}, {16317, 40326}, {16657, 18400}, {17171, 49738}, {17330, 44103}, {17811, 48910}, {17825, 48905}, {17984, 42394}, {18289, 42269}, {18290, 42268}, {18374, 51744}, {18488, 44158}, {18916, 34780}, {19124, 51737}, {19130, 37649}, {20625, 50938}, {21243, 32269}, {21668, 50668}, {22165, 41585}, {26371, 45696}, {26372, 45697}, {26373, 45699}, {26374, 45698}, {26377, 45700}, {26378, 45701}, {26869, 32064}, {26879, 38848}, {26958, 31860}, {29024, 40998}, {32085, 37765}, {32340, 52008}, {33522, 40330}, {34145, 47220}, {34288, 40144}, {35764, 35822}, {35765, 35823}, {37505, 45185}, {37648, 48884}, {37775, 42137}, {37776, 42136}, {39588, 50979}, {41490, 45502}, {41491, 45503}, {42426, 46439}, {43573, 43823}, {43650, 44882}, {51258, 52142}

X(428) = midpoint of X(i) and X(j) for these {i,j}: {2, 34603}, {4, 7576}, {376, 34613}, {381, 7540}, {551, 34633}, {3058, 34666}, {3241, 34668}, {3543, 38323}, {3679, 34657}, {3830, 38321}, {4421, 34655}, {5186, 12132}, {5434, 34653}, {5890, 16658}, {7811, 34661}, {9909, 34659}, {11194, 34663}, {12150, 34651}, {15682, 44458}, {15687, 38322}, {38320, 50687}
X(428) = reflection of X(i) in X(j) for these {i,j}: {1, 51719}, {3, 10127}, {6, 51745}, {549, 23410}, {3575, 7576}, {7576, 6756}, {7667, 2}, {10691, 10128}, {11245, 51}, {34614, 376}, {34634, 551}, {34650, 12150}, {34652, 5434}, {34654, 4421}, {34656, 3679}, {34658, 9909}, {34660, 7811}, {34662, 11194}, {34664, 381}, {34665, 3058}, {34667, 3241}, {52397, 10691}
X(428) = isogonal conjugate of X(41435)
X(428) = complement of X(52397)
X(428) = anticomplement of X(10691)
X(428) = circumcircle-inverse of X(37920)
X(428) = orthocentroidal-circle-inverse of X(5064)
X(428) = polar-circle-inverse of X(5189)
X(428) = orthoptic-circle-of-the-Steiner-inellipse inverse of X(37943)
X(428) = polar conjugate of X(10159)
X(428) = isotomic conjugate of the isogonal conjugate of X(44091)
X(428) = isogonal conjugate of the isotomic conjugate of X(44142)
X(428) = polar conjugate of the isotomic conjugate of X(3589)
X(428) = polar conjugate of the isogonal conjugate of X(5007)
X(428) = orthic-isogonal conjugate of X(46026)
X(428) = X(i)-Ceva conjugate of X(j) for these (i,j): {4, 46026}, {41676, 2489}, {42396, 2501}, {44142, 3589}
X(428) = X(i)-cross conjugate of X(j) for these (i,j): {5007, 3589}, {46026, 44142}
X(428) = X(i)-isoconjugate of X(j) for these (i,j): {1, 41435}, {48, 10159}, {63, 3108}, {656, 7953}, {810, 35137}, {4020, 40425}, {4575, 31065}, {31068, 36060}
X(428) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 41435}, {69, 6292}, {136, 31065}, {428, 52397}, {525, 15527}, {1249, 10159}, {1560, 31068}, {2525, 39691}, {3108, 3162}, {3589, 3933}, {4580, 51906}, {7953, 40596}, {35137, 39062}
X(428) = cevapoint of X(5007) and X(44091)
X(428) = crosspoint of X(4) and X(32085)
X(428) = crosssum of X(3) and X(3917)
X(428) = crossdifference of every pair of points on line {647, 22121}
X(428) = X(2)-of-anti-Ara-triangle
X(428) = X(354)-of-orthic-triangle if ABC is acute
X(428) = X(2) of 3rd pedal triangle of X(4)
X(428) = {X(2),X(4)}-harmonic conjugate of X(5064) X(428) = barycentric product X(i)*X(j) for these {i,j}: {4, 3589}, {6, 44142}, {25, 39998}, {76, 44091}, {83, 46026}, {92, 17469}, {264, 5007}, {278, 4030}, {281, 7198}, {286, 21802}, {393, 7767}, {648, 7927}, {1783, 48152}, {1824, 16707}, {1826, 17200}, {1897, 48101}, {2052, 22352}, {2501, 10330}, {6292, 32085}, {6331, 8664}, {11205, 46104}, {28666, 52395}
X(428) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 10159}, {6, 41435}, {25, 3108}, {112, 7953}, {468, 31068}, {648, 35137}, {2501, 31065}, {3589, 69}, {4030, 345}, {5007, 3}, {6292, 3933}, {7198, 348}, {7767, 3926}, {7927, 525}, {8664, 647}, {10330, 4563}, {11205, 3917}, {17200, 17206}, {17469, 63}, {21802, 72}, {22352, 394}, {28666, 7794}, {32085, 40425}, {39998, 305}, {44091, 6}, {44142, 76}, {46026, 141}, {48101, 4025}, {48152, 15413}
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 5064}, {2, 3543, 44442}, {2, 5064, 427}, {2, 6995, 7714}, {2, 7667, 43957}, {2, 7714, 25}, {2, 9909, 44210}, {2, 44442, 31152}, {2, 52397, 10691}, {3, 4, 1907}, {3, 6997, 37439}, {4, 20, 11403}, {4, 24, 1595}, {4, 25, 427}, {4, 28, 1883}, {4, 235, 23047}, {4, 427, 52285}, {4, 1594, 16198}, {4, 1596, 10151}, {4, 1598, 235}, {4, 3089, 7507}, {4, 3518, 15559}, {4, 3575, 1885}, {4, 4186, 1904}, {4, 4198, 4214}, {4, 4222, 429}, {4, 5198, 1906}, {4, 5200, 32587}, {4, 6240, 13488}, {4, 6353, 7378}, {4, 6623, 18386}, {4, 6756, 3575}, {4, 6995, 25}, {4, 7487, 1593}, {4, 7714, 2}, {4, 8889, 7409}, {4, 10301, 468}, {4, 10594, 5}, {4, 14004, 430}, {4, 18533, 1597}, {4, 19219, 3127}, {4, 34484, 1594}, {4, 37122, 3}, {4, 37390, 407}, {4, 37395, 37391}, {4, 44803, 44226}, {4, 52291, 32588}, {4, 52294, 403}, {4, 52301, 5094}, {5, 22, 7499}, {5, 7519, 37899}, {5, 17714, 34002}, {5, 37947, 25337}, {20, 7392, 7484}, {22, 7394, 5}, {22, 10594, 25}, {23, 546, 37454}, {23, 5133, 6676}, {23, 37349, 5133}, {25, 427, 468}, {25, 5064, 2}, {25, 5094, 6353}, {25, 6995, 10301}, {25, 7378, 52297}, {25, 7409, 52293}, {25, 11403, 7484}, {25, 37453, 4232}, {53, 10311, 16318}, {235, 23047, 10019}, {378, 37458, 37931}, {381, 9909, 2}, {382, 5020, 1370}, {382, 30739, 47095}, {403, 37969, 468}, {427, 10301, 25}, {427, 52293, 8889}, {427, 52297, 5094}, {468, 52285, 427}, {472, 473, 297}, {546, 6676, 5133}, {546, 11819, 12605}, {547, 21841, 37943}, {858, 13595, 6677}, {1113, 1114, 37920}, {1368, 3627, 7391}, {1368, 7391, 46517}, {1370, 5020, 30739}, {1585, 1586, 11331}, {1594, 34484, 21841}, {1594, 37943, 547}, {1595, 7715, 24}, {1598, 7517, 10594}, {1829, 49542, 12135}, {1907, 37439, 427}, {1995, 3627, 46517}, {1995, 7391, 1368}, {2043, 2044, 7395}, {2072, 7545, 44233}, {2937, 50137, 140}, {3091, 7494, 7539}, {3146, 7398, 7386}, {3518, 15559, 140}, {3567, 16659, 18914}, {3845, 15687, 18566}, {3845, 15809, 5064}, {3845, 44266, 5066}, {3850, 11548, 37353}, {3850, 37910, 7495}, {3861, 37897, 5169}, {4232, 7409, 8889}, {4232, 8889, 37453}, {4232, 52293, 468}, {5000, 5001, 140}, {5094, 6353, 52297}, {5094, 7378, 427}, {5133, 6676, 37454}, {5133, 37349, 546}, {5576, 18378, 13383}, {5899, 37347, 16618}, {6353, 7378, 5094}, {6353, 52297, 468}, {6353, 52301, 25}, {6636, 7533, 37990}, {6636, 37990, 140}, {6995, 7378, 52301}, {6995, 7394, 10594}, {6995, 7408, 4}, {6995, 7500, 37122}, {6997, 7500, 3}, {6997, 37122, 25}, {7378, 52301, 6353}, {7386, 7398, 11284}, {7387, 7528, 7399}, {7392, 11403, 427}, {7394, 7519, 22}, {7396, 40132, 31255}, {7409, 37453, 427}, {7485, 20062, 550}, {7495, 37353, 11548}, {7499, 37899, 22}, {7530, 11818, 15760}, {7530, 15760, 47093}, {7533, 37900, 140}, {7540, 34659, 34603}, {7714, 15809, 37904}, {8744, 34482, 8792}, {8889, 37453, 52293}, {10110, 13419, 6146}, {10128, 10691, 2}, {10691, 52397, 7667}, {11112, 11113, 11354}, {11206, 14853, 11402}, {11550, 34417, 13567}, {11745, 16621, 185}, {12088, 14788, 16197}, {13568, 16656, 11381}, {15687, 44212, 31133}, {15765, 18585, 7403}, {16198, 21841, 1594}, {17810, 36990, 1899}, {17984, 46104, 42394}, {18494, 18535, 4}, {18586, 18587, 14790}, {20405, 20406, 468}, {31133, 44212, 47097}, {33971, 52448, 14569}, {34658, 34664, 7667}, {34664, 44210, 43957}, {37118, 47485, 37935}, {37353, 37913, 7495}, {37900, 37990, 6636}, {37901, 37943, 21284}, {38282, 52284, 52298}, {39884, 41588, 11442}, {42789, 42790, 44832}, {44091, 46026, 3589}, {44266, 47313, 37904}, {44288, 46030, 10297}


X(429) = EULER X(58)-5th-SUBSTITUTION POINT

Trilinears       (v + w)sec A : (w + u)sec B : (u + v)sec C, where
                        u : v : w = X(58); e.g., u = u(a,b,c) = a/(b + c)
Barycentrics   (v + w)tan A : (w + u)tan B : (u + v)tan C
Barycentrics   (b + c)*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(a*b + b^2 + a*c + c^2) : :

As a point on the Euler line, X(429) has Shinagawa coefficients ($a$F,$a$(E + F) + 2abc).

X(429) lies on these on the cubic K253 and these lines: {1, 5130}, {2, 3}, {6, 20029}, {10, 1824}, {11, 1104}, {12, 37}, {19, 1213}, {33, 1834}, {34, 5155}, {56, 23304}, {65, 26955}, {92, 3948}, {108, 961}, {115, 20621}, {119, 136}, {120, 5139}, {125, 1902}, {135, 42423}, {208, 1892}, {226, 1426}, {264, 40828}, {281, 27040}, {318, 4451}, {343, 10441}, {495, 1068}, {607, 2238}, {956, 11401}, {958, 26377}, {960, 1211}, {1125, 40985}, {1329, 32777}, {1425, 51365}, {1452, 2245}, {1478, 27802}, {1698, 2960}, {1717, 1722}, {1785, 5530}, {1825, 40663}, {1827, 10395}, {1828, 3454}, {1835, 3649}, {1852, 52427}, {1856, 10958}, {1861, 1900}, {1862, 12019}, {1869, 3925}, {1870, 37737}, {1876, 18635}, {1878, 5087}, {1887, 8287}, {1890, 15254}, {1891, 11363}, {1896, 6530}, {1899, 5706}, {1905, 10974}, {2052, 3597}, {2354, 52087}, {2899, 11681}, {2901, 21072}, {2905, 2907}, {2970, 21664}, {2971, 34337}, {3193, 3564}, {3580, 41723}, {3695, 3701}, {4292, 26933}, {5089, 16589}, {5174, 16824}, {5307, 25466}, {5521, 31845}, {5730, 11396}, {5799, 13567}, {6198, 12135}, {6335, 17981}, {7079, 38930}, {7102, 7952}, {7283, 19839}, {7719, 46196}, {8192, 36844}, {9942, 12136}, {10896, 17111}, {11392, 17408}, {12259, 24474}, {15488, 21243}, {15888, 23710}, {15946, 32431}, {16221, 42422}, {16471, 44086}, {16747, 44146}, {20653, 21033}, {21016, 21029}, {21671, 40967}, {22345, 51414}, {24006, 31946}, {34332, 34338}, {40952, 44547}, {51870, 51879}

X(429) = midpoint of X(i) and X(j) for these {i,j}: {4, 7414}, {108, 39990}
X(429) = reflection of X(i) in X(j) for these {i,j}: {1, 51720}, {37361, 5}
X(429) = isogonal conjugate of X(1798)
X(429) = complement of X(16049)
X(429) = nine-point circle-inverse of X(37982)
X(429) = orthocentroidal-circle-inverse of X(4185)
X(429) = polar-circle-inverse of X(1325)
X(429) = polar conjugate of X(14534)
X(429) = complement of the isogonal conjugate of X(43703)
X(429) = isotomic conjugate of the isogonal conjugate of X(44092)
X(429) = polar conjugate of the isotomic conjugate of X(1211)
X(429) = polar conjugate of the isogonal conjugate of X(2092)
X(429) = orthic-isogonal conjugate of X(1829)
X(429) = X(i)-complementary conjugate of X(j) for these (i,j): {42, 478}, {3435, 1125}, {8048, 3741}, {34277, 21246}, {39167, 34851}, {40097, 8062}, {40454, 49598}, {42467, 3739}, {43703, 10}, {43742, 34831}, {46640, 4369}
X(429) = X(i)-Ceva conjugate of X(j) for these (i,j): {4, 1829}, {108, 523}, {6335, 2501}, {46878, 2292}
X(429) = X(2092)-cross conjugate of X(1211)
X(429) = X(i)-isoconjugate of X(j) for these (i,j): {1, 1798}, {3, 2363}, {48, 14534}, {58, 1791}, {63, 1169}, {81, 2359}, {163, 15420}, {283, 961}, {1220, 1437}, {1790, 2298}, {4575, 4581}, {7254, 36147}, {9247, 40827}, {23189, 36098}
X(429) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 960}, {3, 1798}, {10, 1791}, {69, 3666}, {115, 15420}, {136, 4581}, {429, 16049}, {905, 3125}, {1169, 3162}, {1211, 1444}, {1249, 14534}, {1790, 52087}, {1812, 2092}, {2359, 40586}, {2363, 36103}, {7254, 39015}, {23189, 38992}
X(429) = cevapoint of X(2092) and X(44092)
X(429) = crosspoint of X(i) and X(j) for these (i,j): {4, 41013}, {264, 40149}
X(429) = crosssum of X(i) and X(j) for these (i,j): {3, 1437}, {6, 52143}, {184, 2193}
X(429) = crossdifference of every pair of points on line {647, 23189}
X(429) = barycentric product X(i)*X(j) for these {i,j}: {4, 1211}, {10, 1848}, {19, 18697}, {25, 1228}, {27, 20653}, {33, 45196}, {76, 44092}, {92, 2292}, {225, 3687}, {226, 46878}, {264, 2092}, {273, 21033}, {278, 3704}, {281, 41003}, {286, 21810}, {313, 2354}, {321, 1829}, {331, 40966}, {349, 40976}, {427, 27067}, {960, 40149}, {1824, 20911}, {1826, 4357}, {1897, 21124}, {1969, 3725}, {2052, 22076}, {3666, 41013}, {3882, 24006}, {6331, 42661}, {6335, 50330}, {7140, 16705}, {7141, 40153}, {41609, 43675}
X(429) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 14534}, {6, 1798}, {19, 2363}, {25, 1169}, {37, 1791}, {42, 2359}, {264, 40827}, {523, 15420}, {960, 1812}, {1193, 1790}, {1211, 69}, {1228, 305}, {1824, 2298}, {1826, 1220}, {1829, 81}, {1848, 86}, {1880, 961}, {2092, 3}, {2269, 283}, {2292, 63}, {2300, 1437}, {2354, 58}, {2501, 4581}, {3004, 15419}, {3666, 1444}, {3687, 332}, {3704, 345}, {3725, 48}, {3882, 4592}, {3965, 1792}, {4357, 17206}, {6371, 7254}, {7140, 14624}, {18697, 304}, {20653, 306}, {20967, 2193}, {21033, 78}, {21124, 4025}, {21810, 72}, {22076, 394}, {22345, 18604}, {27067, 1799}, {40149, 31643}, {40966, 219}, {40976, 284}, {41003, 348}, {41013, 30710}, {41609, 40571}, {41611, 41610}, {42661, 647}, {44092, 6}, {45196, 7182}, {45218, 23086}, {46878, 333}, {50330, 905}, {52326, 23189}
X(429) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 4185}, {2, 2475, 17518}, {2, 7438, 468}, {4, 5, 37368}, {4, 24, 7511}, {4, 25, 37398}, {4, 29, 1884}, {4, 403, 15763}, {4, 406, 25}, {4, 427, 1883}, {4, 451, 28}, {4, 461, 28076}, {4, 475, 4214}, {4, 860, 407}, {4, 3088, 37391}, {4, 3089, 37387}, {4, 3144, 27}, {4, 3542, 7497}, {4, 4194, 4186}, {4, 4213, 29}, {4, 4219, 1885}, {4, 4222, 428}, {4, 5136, 37226}, {4, 5142, 427}, {4, 6353, 4198}, {4, 7412, 3575}, {4, 7559, 23047}, {4, 8889, 4200}, {4, 14018, 1889}, {4, 17555, 1894}, {5, 405, 37315}, {5, 47510, 37056}, {10, 39579, 1824}, {21, 4231, 20832}, {21, 37983, 37360}, {25, 37346, 37982}, {28, 451, 468}, {225, 1826, 1867}, {235, 27687, 37982}, {403, 1594, 45168}, {406, 860, 4205}, {407, 430, 4}, {427, 1904, 4}, {430, 4205, 37398}, {431, 442, 37982}, {440, 4205, 37225}, {442, 30446, 30444}, {442, 30447, 21530}, {442, 51557, 5}, {469, 17555, 4}, {851, 27685, 18641}, {1312, 1313, 37982}, {1848, 46878, 1829}, {1904, 5142, 1883}, {3136, 27687, 442}, {4198, 6353, 17523}, {4205, 37346, 442}, {4214, 5094, 475}, {5000, 5001, 37360}, {5064, 17516, 4}, {7563, 37420, 4}, {13442, 33305, 3145}, {21530, 30444, 442}, {25984, 26019, 14011}, {26120, 27531, 7413}, {27553, 37225, 440}, {29993, 46525, 26550}, {30446, 30447, 442}


X(430) = EULER X(81)-5th-SUBSTITUTION POINT

Trilinears        (v + w)sec A : (w + u)sec B : (u + v)sec C, where
                         u : v : w = X(81); e.g., u = u(a,b,c) = 1/(b + c)
Barycentrics  (v + w)tan A : (w + u)tan B : (u + v)tan C
Barycentrics   (b + c)*(2*a + b + c)*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2) : :

As a point on the Euler line, X(430) has Shinagawa coefficients (F,2$bc$ + E + F).

X(430) lies on these lines: {2, 3}, {11, 40956}, {33, 1865}, {53, 3192}, {55, 50036}, {115, 40984}, {118, 136}, {125, 20622}, {210, 594}, {226, 1893}, {343, 48902}, {661, 23615}, {1172, 44097}, {1211, 24703}, {1213, 1839}, {1230, 3702}, {1827, 1856}, {1848, 2969}, {1855, 38930}, {1859, 42069}, {1867, 2901}, {1897, 17982}, {2245, 7082}, {2328, 32431}, {2333, 40586}, {2970, 21665}, {3190, 24045}, {5130, 17156}, {5139, 5513}, {5155, 39594}, {8735, 40976}, {8754, 42070}, {15496, 46835}, {17056, 17605}, {21243, 48940}, {34335, 34338}, {38360, 50329}, {40954, 44546}

X(430) = orthocentroidal-circle-inverse of X(1889)
X(430) = polar-circle-inverse of X(5196)
X(430) = polar conjugate of X(32014)
X(430) = isogonal conjugate of the isotomic conjugate of X(44143)
X(430) = polar conjugate of the isotomic conjugate of X(1213)
X(430) = polar conjugate of the isogonal conjugate of X(20970)
X(430) = orthic-isogonal conjugate of X(1839)
X(430) = X(i)-Ceva conjugate of X(j) for these (i,j): {4, 1839}, {1897, 2501}, {26705, 523}, {44143, 1213}
X(430) = X(20970)-cross conjugate of X(1213)
X(430) = X(i)-isoconjugate of X(j) for these (i,j): {3, 40438}, {48, 32014}, {63, 1171}, {81, 1796}, {656, 6578}, {905, 4629}, {1126, 1444}, {1255, 1790}, {1268, 1437}, {1459, 4596}, {4558, 47947}, {4575, 4608}, {4592, 50344}, {4632, 22383}, {7254, 37212}, {17206, 28615}
X(430) = X(i)-Dao conjugate of X(j) for these (i,j): {69, 1125}, {136, 4608}, {1171, 3162}, {1213, 17206}, {1249, 32014}, {1444, 3647}, {1796, 40586}, {3120, 4025}, {4064, 21709}, {5139, 50344}, {6578, 40596}, {15419, 35076}, {36103, 40438}
X(430) = crosspoint of X(4) and X(1826)
X(430) = crosssum of X(3) and X(1790)
X(430) = crossdifference of every pair of points on line {647, 7254}
X(430) = tangential-to-orthic similarity image of X(199)
X(430) = barycentric product X(i)*X(j) for these {i,j}: {4, 1213}, {6, 44143}, {10, 1839}, {19, 4647}, {25, 1230}, {27, 8013}, {92, 1962}, {158, 3958}, {225, 3686}, {264, 20970}, {278, 4046}, {281, 3649}, {286, 21816}, {321, 2355}, {393, 41014}, {594, 31900}, {648, 6367}, {1100, 41013}, {1125, 1826}, {1269, 2333}, {1783, 30591}, {1824, 4359}, {1880, 3702}, {1897, 4988}, {2052, 22080}, {2501, 4427}, {3683, 40149}, {4103, 46542}, {4115, 7649}, {4983, 6335}, {6331, 8663}, {6531, 51417}, {7140, 8025}, {14618, 35327}, {24006, 35342}
X(430) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 32014}, {19, 40438}, {25, 1171}, {42, 1796}, {112, 6578}, {1100, 1444}, {1125, 17206}, {1213, 69}, {1230, 305}, {1783, 4596}, {1824, 1255}, {1826, 1268}, {1839, 86}, {1897, 4632}, {1962, 63}, {2308, 1790}, {2333, 1126}, {2355, 81}, {2489, 50344}, {2501, 4608}, {3649, 348}, {3683, 1812}, {3686, 332}, {3958, 326}, {4046, 345}, {4115, 4561}, {4427, 4563}, {4647, 304}, {4977, 15419}, {4983, 905}, {4988, 4025}, {4990, 15411}, {6367, 525}, {7140, 6539}, {8013, 306}, {8040, 4001}, {8663, 647}, {8750, 4629}, {20970, 3}, {21816, 72}, {22080, 394}, {23201, 18604}, {30591, 15413}, {31900, 1509}, {35327, 4558}, {35342, 4592}, {41013, 32018}, {41014, 3926}, {44143, 76}, {50512, 7254}, {51417, 6393}
X(430) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 1889}, {4, 24, 7546}, {4, 403, 15762}, {4, 429, 407}, {4, 451, 31902}, {4, 469, 427}, {4, 3542, 7534}, {4, 4207, 25}, {4, 4213, 27}, {4, 6353, 6994}, {4, 7414, 46467}, {4, 7513, 1885}, {4, 7563, 23047}, {4, 14004, 428}, {27, 4213, 468}, {429, 37398, 4205}, {1824, 1826, 7140}, {3136, 33329, 440}, {6818, 7377, 37439}, {11323, 37385, 13615}, {28044, 37385, 25}, {52286, 52287, 37226}


X(431) = EULER X(283)-5th-SUBSTITUTION POINT

Trilinears       (v + w)sec A : (w + u)sec B : (u + v)sec C, where
                        u : v : w = X(283); e.g., u = u(A,B,C) = (cos A)/(cos B + cos C)
Barycentrics  (v + w)tan A : (w + u)tan B : (u + v)tan C
Barycentrics   (b + c)*(2*a + b + c)*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2) : :(b + c)*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(a^4*b - 2*a^2*b^3 + b^5 + a^4*c + 2*a^3*b*c + 2*a^2*b^2*c - b^4*c + 2*a^2*b*c^2 - 2*a^2*c^3 - b*c^4 + c^5) : :

As a point on the Euler line, X(431) has Shinagawa coefficients ($aSA$F,- $aSA$(E - F) - 2abcF). X(431) lies on these lines: {2, 3}, {11, 40985}, {12, 1824}, {19, 50036}, {119, 135}, {136, 25640}, {225, 7363}, {388, 11401}, {1112, 12826}, {1478, 26377}, {1829, 12047}, {1834, 44113}, {1842, 5521}, {1848, 31936}, {2886, 5130}, {3485, 11396}, {3822, 39579}, {5086, 12135}, {5139, 20621}, {10572, 11363}, {10895, 11391}, {16178, 42422}, {21664, 34338}, {44092, 44545}

X(431) = midpoint of X(4) and X(31384)
X(431) = orthic-isogonal conjugate of X(1858)
X(431) = X(i)-Ceva conjugate of X(j) for these (i,j): {4, 1858}, {40097, 523}
X(431) = barycentric product X(i)*X(j) for these {i,j}: {19, 18692}, {225, 45206}, {1858, 40149}
X(431) = barycentric quotient X(i)/X(j) for these {i,j}: {1195, 283}, {1858, 1812}, {18692, 304}, {45206, 332}
X(431) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 24, 7491}, {4, 403, 6841}, {4, 411, 1885}, {4, 2476, 427}, {4, 3144, 3559}, {4, 3542, 3560}, {4, 6353, 6872}, {4, 6622, 6837}, {4, 6825, 1593}, {4, 6867, 7507}, {4, 6869, 44438}, {4, 7548, 23047}, {235, 37376, 25}, {429, 37982, 442}, {1596, 37406, 4}


X(432) = EULER X(155)-6th-SUBSTITUTION POINT

Trilinears       (v2 + w2)sec A : (w2 + u2)sec B : (u2 + v2)sec C, where
                         u : v : w = X(155); e.g., u = u(A,B,C) = (cos A)(cos2B + cos2C - cos2A)
Barycentrics    (v2 + w2)tan A : (w2 + u2)tan B : (u2 + v2)tan C
Barycentrics   (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^16*b^2 - 8*a^14*b^4 + 28*a^12*b^6 - 56*a^10*b^8 + 70*a^8*b^10 - 56*a^6*b^12 + 28*a^4*b^14 - 8*a^2*b^16 + b^18 + a^16*c^2 - 4*a^12*b^4*c^2 + 8*a^10*b^6*c^2 - 34*a^8*b^8*c^2 + 80*a^6*b^10*c^2 - 84*a^4*b^12*c^2 + 40*a^2*b^14*c^2 - 7*b^16*c^2 - 8*a^14*c^4 - 4*a^12*b^2*c^4 + 32*a^10*b^4*c^4 - 20*a^8*b^6*c^4 - 40*a^6*b^8*c^4 + 100*a^4*b^10*c^4 - 80*a^2*b^12*c^4 + 20*b^14*c^4 + 28*a^12*c^6 + 8*a^10*b^2*c^6 - 20*a^8*b^4*c^6 + 32*a^6*b^6*c^6 - 44*a^4*b^8*c^6 + 88*a^2*b^10*c^6 - 28*b^12*c^6 - 56*a^10*c^8 - 34*a^8*b^2*c^8 - 40*a^6*b^4*c^8 - 44*a^4*b^6*c^8 - 80*a^2*b^8*c^8 + 14*b^10*c^8 + 70*a^8*c^10 + 80*a^6*b^2*c^10 + 100*a^4*b^4*c^10 + 88*a^2*b^6*c^10 + 14*b^8*c^10 - 56*a^6*c^12 - 84*a^4*b^2*c^12 - 80*a^2*b^4*c^12 - 28*b^6*c^12 + 28*a^4*c^14 + 40*a^2*b^2*c^14 + 20*b^4*c^14 - 8*a^2*c^16 - 7*b^2*c^16 + c^18) : :

As a point on the Euler line, X(432) has Shinagawa coefficients ((E2 + 4EF - 4S2)F, -E3 - 3E2F - 4EF2 + 4(E - F)S2).

X(432) lies on these lines: {2, 3}, {135, 12359}


X(433) = EULER X(159)-6th-SUBSTITUTION POINT

Trilinears       (v2 + w2)sec A : (w2 + u2)sec B : (u2 + v2)sec C, where u : v : w = X(159)
Barycentrics  (v2 + w2)tan A : (w2 + u2)tan B : (u2 + v2)tan C
Barycentrics   (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^12*b^2 + 2*a^10*b^4 - a^8*b^6 - 4*a^6*b^8 - a^4*b^10 + 2*a^2*b^12 + b^14 + a^12*c^2 - 4*a^10*b^2*c^2 + a^8*b^4*c^2 + 8*a^6*b^6*c^2 - 5*a^4*b^8*c^2 - 4*a^2*b^10*c^2 + 3*b^12*c^2 + 2*a^10*c^4 + a^8*b^2*c^4 - 8*a^6*b^4*c^4 + 6*a^4*b^6*c^4 - 2*a^2*b^8*c^4 + b^10*c^4 - a^8*c^6 + 8*a^6*b^2*c^6 + 6*a^4*b^4*c^6 + 8*a^2*b^6*c^6 - 5*b^8*c^6 - 4*a^6*c^8 - 5*a^4*b^2*c^8 - 2*a^2*b^4*c^8 - 5*b^6*c^8 - a^4*c^10 - 4*a^2*b^2*c^10 + b^4*c^10 + 2*a^2*c^12 + 3*b^2*c^12 + c^14) : :

As a point on the Euler line, X(433) has Shinagawa coefficients (4(E + F)3F2 - E2FS2, (E + F)[4(E + F)3F - (E + 8F)ES2]).

X(433) lies on this line: 2,3


X(434) = EULER X(195)-6th-SUBSTITUTION POINT

Trilinears       (v2 + w2)sec A : (w2 + u2)sec B : (u2 + v2)sec C, where u : v : w = X(195)
Barycentrics  (v2 + w2)tan A : (w2 + u2)tan B : (u2 + v2)tan C
Barycentrics   (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^12*b^2 + 2*a^10*b^4 - a^8*b^6 - 4*a^6*b^8 - a^4*b^10 + 2*a^2*b^12 + b^14 + a^12*c^2 - 4*a^10*b^2*c^2 + a^8*b^4*c^2 + 8*a^6*b^6*c^2 - 5*a^4*b^8*c^2 - 4*a^2*b^10*c^2 + 3*b^12*c^2 + 2*a^10*c^4 + a^8*b^2*c^4 - 8*a^6*b^4*c^4 + 6*a^4*b^6*c^4 - 2*a^2*b^8*c^4 + b^10*c^4 - a^8*c^6 + 8*a^6*b^2*c^6 + 6*a^4*b^4*c^6 + 8*a^2*b^6*c^6 - 5*b^8*c^6 - 4*a^6*c^8 - 5*a^4*b^2*c^8 - 2*a^2*b^4*c^8 - 5*b^6*c^8 - a^4*c^10 - 4*a^2*b^2*c^10 + b^4*c^10 + 2*a^2*c^12 + 3*b^2*c^12 + c^14) : : (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^16*b^2 - 8*a^14*b^4 + 28*a^12*b^6 - 56*a^10*b^8 + 70*a^8*b^10 - 56*a^6*b^12 + 28*a^4*b^14 - 8*a^2*b^16 + b^18 + a^16*c^2 - 8*a^14*b^2*c^2 + 26*a^12*b^4*c^2 - 34*a^10*b^6*c^2 - 6*a^8*b^8*c^2 + 68*a^6*b^10*c^2 - 78*a^4*b^12*c^2 + 38*a^2*b^14*c^2 - 7*b^16*c^2 - 8*a^14*c^4 + 26*a^12*b^2*c^4 - 36*a^10*b^4*c^4 + 17*a^8*b^6*c^4 - 18*a^6*b^8*c^4 + 67*a^4*b^10*c^4 - 68*a^2*b^12*c^4 + 20*b^14*c^4 + 28*a^12*c^6 - 34*a^10*b^2*c^6 + 17*a^8*b^4*c^6 + 12*a^6*b^6*c^6 - 17*a^4*b^8*c^6 + 58*a^2*b^10*c^6 - 28*b^12*c^6 - 56*a^10*c^8 - 6*a^8*b^2*c^8 - 18*a^6*b^4*c^8 - 17*a^4*b^6*c^8 - 40*a^2*b^8*c^8 + 14*b^10*c^8 + 70*a^8*c^10 + 68*a^6*b^2*c^10 + 67*a^4*b^4*c^10 + 58*a^2*b^6*c^10 + 14*b^8*c^10 - 56*a^6*c^12 - 78*a^4*b^2*c^12 - 68*a^2*b^4*c^12 - 28*b^6*c^12 + 28*a^4*c^14 + 38*a^2*b^2*c^14 + 20*b^4*c^14 - 8*a^2*c^16 - 7*b^2*c^16 + c^18) : :

As a point on the Euler line, X(434) has Shinagawa coefficients (9E2F + 16EF2 - 64FS2, -9E3 - 23E2F -16EF2 + 32(E - 2F)S2)

X(434) lies on these lines: {2, 3}, {11576, 50479}


X(435) = EULER X(399)-6th-SUBSTITUTION POINT

Trilinears       (v2 + w2)sec A : (w2 + u2)sec B : (u2 + v2)sec C, where u : v : w = X(399)
Barycentrics  (v2 + w2)tan A : (w2 + u2)tan B : (u2 + v2)tan C
Barycentrics   (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^16*b^2 - 8*a^14*b^4 + 28*a^12*b^6 - 56*a^10*b^8 + 70*a^8*b^10 - 56*a^6*b^12 + 28*a^4*b^14 - 8*a^2*b^16 + b^18 + a^16*c^2 + 8*a^14*b^2*c^2 - 22*a^12*b^4*c^2 - 2*a^10*b^6*c^2 + 26*a^8*b^8*c^2 + 20*a^6*b^10*c^2 - 62*a^4*b^12*c^2 + 38*a^2*b^14*c^2 - 7*b^16*c^2 - 8*a^14*c^4 - 22*a^12*b^2*c^4 + 108*a^10*b^4*c^4 - 95*a^8*b^6*c^4 - 34*a^6*b^8*c^4 + 99*a^4*b^10*c^4 - 68*a^2*b^12*c^4 + 20*b^14*c^4 + 28*a^12*c^6 - 2*a^10*b^2*c^6 - 95*a^8*b^4*c^6 + 140*a^6*b^6*c^6 - 65*a^4*b^8*c^6 + 58*a^2*b^10*c^6 - 28*b^12*c^6 - 56*a^10*c^8 + 26*a^8*b^2*c^8 - 34*a^6*b^4*c^8 - 65*a^4*b^6*c^8 - 40*a^2*b^8*c^8 + 14*b^10*c^8 + 70*a^8*c^10 + 20*a^6*b^2*c^10 + 99*a^4*b^4*c^10 + 58*a^2*b^6*c^10 + 14*b^8*c^10 - 56*a^6*c^12 - 62*a^4*b^2*c^12 - 68*a^2*b^4*c^12 - 28*b^6*c^12 + 28*a^4*c^14 + 38*a^2*b^2*c^14 + 20*b^4*c^14 - 8*a^2*c^16 - 7*b^2*c^16 + c^18) : :

As a point on the Euler line, X(435) has Shinagawa coefficients (9E2F + 144EF2 - 64FS2, -9E3 + 89E2F - 128EF2 - 64F3 + 32(E - 2F)S2)

X(435) lies on this line: 2,3


X(436) = EULER X(48)-7th-SUBSTITUTION POINT

Trilinears       (u2 + vw)sec A : (v2 + wu)sec B : (w2 + uv)sec C, where
                        u : v : w = X(48); e.g., u(A,B,C) = sin 2A
Barycentrics  (u2 + vw)tan A : (v2 + wu)tan B : (w2 + uv)tan C
Barycentrics    tan A - tan ω' : :, where ω' is the Brocard angle of the orthic triangle
Barycentrics   (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^8 - 2*a^6*b^2 + a^4*b^4 - 2*a^6*c^2 + 3*a^4*b^2*c^2 - b^6*c^2 + a^4*c^4 + 2*b^4*c^4 - b^2*c^6) : :

As a point on the Euler line, X(436) has Shinagawa coefficients (2F2,-(E + F)F + S2).

X(436) lies on these lines: {2, 3}, {6, 1988}, {51, 107}, {53, 10192}, {54, 8794}, {110, 324}, {154, 33971}, {182, 15466}, {184, 2052}, {185, 51031}, {216, 40402}, {264, 9306}, {459, 18950}, {578, 1093}, {648, 34986}, {1075, 7592}, {1196, 6531}, {1199, 43840}, {1304, 32439}, {1495, 1629}, {1614, 44732}, {1941, 34148}, {1947, 7193}, {1948, 3955}, {1970, 27359}, {1994, 35360}, {3167, 9308}, {4994, 38848}, {5012, 46106}, {5890, 40664}, {5943, 36794}, {6344, 50463}, {6524, 11427}, {6525, 14853}, {6530, 23292}, {6747, 14165}, {6749, 45867}, {6761, 12022}, {8795, 26887}, {8884, 8887}, {10982, 45062}, {11245, 51358}, {11424, 14249}, {11455, 48364}, {14361, 14912}, {14363, 37505}, {14978, 18350}, {16657, 51385}, {19357, 41365}, {26880, 42329}, {27377, 41588}, {32002, 32223}, {35259, 41244}, {37894, 44132}, {38833, 39286}, {43650, 52147}

X(436) = orthocentroidal-circle-inverse of X(52249)
X(436) = polar conjugate of X(9290)
X(436) = isogonal conjugate of the isotomic conjugate of X(9291)
X(436) = polar conjugate of the isogonal conjugate of X(1970)
X(436) = X(i)-Ceva conjugate of X(j) for these (i,j): {1987, 41204}, {21449, 1970}, {40402, 4}
X(436) = X(i)-isoconjugate of X(j) for these (i,j): {3, 9251}, {48, 9290}, {656, 1303}
X(436) = X(i)-Dao conjugate of X(j) for these (i,j): {1249, 9290}, {1303, 40596}, {9251, 36103}
X(436) = crosssum of X(520) and X(41219)
X(436) = barycentric product X(i)*X(j) for these {i,j}: {1, 9252}, {5, 21449}, {6, 9291}, {92, 1954}, {95, 27359}, {112, 42331}, {264, 1970}
X(436) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 9290}, {19, 9251}, {112, 1303}, {1954, 63}, {1970, 3}, {9252, 75}, {9291, 76}, {21449, 95}, {27359, 5}, {42331, 3267}
X(436) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 52249}, {4, 3079, 7714}, {4, 52299, 6619}, {107, 275, 51}, {184, 2052, 41204}, {418, 41202, 8613}, {1495, 42400, 1629}, {1585, 1586, 52247}, {4240, 30506, 13595}, {6524, 11427, 41371}, {8887, 10282, 8884}


X(437) = EULER X(214)-7th-SUBSTITUTION POINT

Trilinears       (u2 + vw)sec A : (v2 + wu)sec B : (w2 + uv)sec C, where u : v : w = X(214)
Barycentrics  (u2 + vw)tan A : (v2 + wu)tan B : (w2 + uv)tan C
Barycentrics   (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(5*a^6 - 6*a^5*b - 8*a^4*b^2 + 11*a^3*b^3 - 5*a*b^5 + 3*b^6 - 6*a^5*c + 18*a^4*b*c - 5*a^3*b^2*c - 12*a^2*b^3*c + 12*a*b^4*c - 5*b^5*c - 8*a^4*c^2 - 5*a^3*b*c^2 + 21*a^2*b^2*c^2 - 11*a*b^3*c^2 - 2*b^4*c^2 + 11*a^3*c^3 - 12*a^2*b*c^3 - 11*a*b^2*c^3 + 12*b^3*c^3 + 12*a*b*c^4 - 2*b^2*c^4 - 5*a*c^5 - 5*b*c^5 + 3*c^6) : :

As a point on the Euler line, X(437) has Shinagawa coefficients (2(E+F)2F+10FS2+7$bcSA$F-5$ab$(E+F)F, (5E-16F)S2-$abSC2$ +41$bcSBSC$+$ab$[(E+F)2-19S2]).

X(437) lies on this line: 2,3


X(438) = EULER X(204)-8th-SUBSTITUTION POINT

Trilinears       (u2 + vw)csc A : (v2 + wu)csc B : (w2 + uv)csc C, where
                        u : v : w = X(204); e.g., u(A,B,C) = (tan A)(tan B + tan C - tan A)
Barycentrics  u2 + vw : v2 + wu : w2 + uv


Barycentrics   (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(5*a^12 - 7*a^10*b^2 - 8*a^8*b^4 + 18*a^6*b^6 - 11*a^4*b^8 + 5*a^2*b^10 - 2*b^12 - 7*a^10*c^2 + 26*a^8*b^2*c^2 - 18*a^6*b^4*c^2 - 16*a^4*b^6*c^2 + 17*a^2*b^8*c^2 - 2*b^10*c^2 - 8*a^8*c^4 - 18*a^6*b^2*c^4 + 54*a^4*b^4*c^4 - 22*a^2*b^6*c^4 - 6*b^8*c^4 + 18*a^6*c^6 - 16*a^4*b^2*c^6 - 22*a^2*b^4*c^6 + 20*b^6*c^6 - 11*a^4*c^8 + 17*a^2*b^2*c^8 - 6*b^4*c^8 + 5*a^2*c^10 - 2*b^2*c^10 - 2*c^12) : :

As a point on the Euler line, X(438) has Shinagawa coefficients (4(E + F)2F2 - (E + 3F)FS2, -[4(E - F)F - S2]S2)

X(438) lies on these lines: {2, 3}, {5895, 6330}, {35711, 42287}


X(439) = EULER X(193)-9th-SUBSTITUTION POINT

Trilinears       au2 : bv2 : cw2, where
                        u : v : w = X(193); e.g., u(A,B,C) = (csc A)(cot B + cot C - cot A)
Barycentrics  (au)2 : (bv)2 : (cw)2
Barycentrics   (3*a^2 - b^2 - c^2)^2 : :
X(439) = 3 X[2] - 4 X[32970]

As a point on the Euler line, X(439) has Shinagawa coefficients ((E + F) 2 - 4S2,4S2)

X(439) lies on these lines: {2, 3}, {32, 51170}, {69, 5023}, {99, 6392}, {115, 38259}, {187, 3926}, {193, 3053}, {230, 2996}, {620, 32816}, {754, 32825}, {1611, 18287}, {1975, 37667}, {1992, 22331}, {2482, 7758}, {3618, 15815}, {3619, 5585}, {3620, 5210}, {3767, 32456}, {3785, 5206}, {3815, 5395}, {3972, 31400}, {4366, 5265}, {4558, 40320}, {5007, 37809}, {5013, 51171}, {5032, 11147}, {5171, 39141}, {5281, 6645}, {5286, 7782}, {5304, 7783}, {5475, 18845}, {5921, 39647}, {6423, 6462}, {6424, 6463}, {6461, 40318}, {7603, 32884}, {7618, 7772}, {7737, 32829}, {7746, 32826}, {7751, 32824}, {7754, 46453}, {7759, 32837}, {7763, 7926}, {7767, 15655}, {7779, 32841}, {7780, 32836}, {7785, 32835}, {7793, 32830}, {7795, 15513}, {7800, 8588}, {7816, 21843}, {7821, 47102}, {7854, 8182}, {7857, 43448}, {7862, 43618}, {7886, 43619}, {7891, 37668}, {7922, 14907}, {9292, 51427}, {9742, 41400}, {13356, 14930}, {19118, 30558}, {20065, 32458}, {23357, 34161}, {26658, 35292}, {29585, 37552}, {34286, 36426}, {34511, 35007}, {39143, 44381}, {41139, 41895}

X(439) = reflection of X(32972) in X(32970)
X(439) = anticomplement of X(32972)
X(439) = orthocentroidal-circle-inverse of X(52250)
X(439) = X(6353)-Ceva conjugate of X(193)
X(439) = X(i)-isoconjugate of X(j) for these (i,j): {2996, 38252}, {8769, 8770}
X(439) = X(i)-Dao conjugate of X(j) for these (i,j): {69, 6340}, {115, 3566}, {2996, 51579}
X(439) = barycentric product X(i)*X(j) for these {i,j}: {193, 193}, {1707, 18156}, {4590, 15525}, {6337, 6353}, {10607, 21447}
X(439) = barycentric quotient X(i)/X(j) for these {i,j}: {193, 2996}, {1707, 8769}, {3053, 8770}, {3167, 6391}, {6337, 6340}, {6353, 34208}, {15525, 115}, {19118, 14248}, {41584, 47730}, {41588, 27364}
X(439) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 52250}, {2, 20, 32982}, {2, 3146, 32980}, {2, 3522, 33023}, {2, 3552, 32981}, {2, 5059, 14063}, {2, 6658, 3832}, {2, 6872, 33050}, {2, 14035, 32991}, {2, 14068, 5068}, {2, 17576, 33051}, {2, 17578, 32966}, {2, 19691, 33290}, {2, 21734, 32965}, {2, 32981, 32979}, {2, 32997, 33200}, {2, 33014, 3522}, {2, 33018, 15022}, {2, 33214, 33019}, {2, 33244, 3146}, {2, 33260, 33025}, {2, 50689, 32963}, {2, 50693, 6655}, {3, 7819, 33215}, {3, 8369, 16043}, {3, 14001, 32990}, {3, 32973, 2}, {3, 32985, 32973}, {3, 33242, 8359}, {4, 32989, 2}, {4, 33233, 32988}, {4, 33235, 35927}, {4, 35297, 32989}, {20, 16925, 2}, {20, 33199, 33017}, {20, 33203, 5025}, {140, 14033, 32987}, {140, 32987, 2}, {193, 51579, 6337}, {376, 7807, 32974}, {376, 32951, 33234}, {384, 3523, 2}, {548, 32954, 32986}, {550, 11288, 14064}, {550, 14064, 33272}, {631, 1003, 32971}, {631, 32971, 2}, {3053, 6337, 193}, {3053, 32459, 6337}, {3091, 7907, 2}, {3522, 33025, 33260}, {3522, 33205, 2}, {3528, 33191, 6656}, {3530, 11286, 32978}, {3534, 8361, 33238}, {3552, 16044, 33187}, {3552, 32964, 2}, {3552, 33259, 14035}, {5025, 13586, 33254}, {5025, 16925, 33203}, {5025, 33203, 2}, {5025, 33254, 20}, {5056, 33000, 2}, {5129, 33054, 2}, {6337, 32459, 51579}, {6655, 33208, 50693}, {6857, 33039, 2}, {7483, 33037, 2}, {7486, 16923, 2}, {7791, 33181, 2}, {7791, 33246, 33181}, {7791, 33276, 10304}, {7807, 32974, 2}, {7807, 33219, 33195}, {7807, 33234, 32951}, {7816, 21843, 32828}, {7824, 33198, 2}, {7824, 33255, 33198}, {7866, 8703, 33226}, {7887, 8598, 3529}, {7892, 33008, 33202}, {7892, 33202, 2}, {7901, 33253, 33210}, {7907, 14042, 32998}, {7907, 33007, 3091}, {9855, 33279, 49140}, {10299, 14039, 11285}, {10303, 16924, 2}, {10304, 33181, 7791}, {10304, 33246, 2}, {11001, 32955, 33229}, {11111, 17694, 33038}, {11288, 33272, 2}, {11291, 11292, 1656}, {11361, 33000, 5056}, {13586, 16925, 20}, {13586, 33245, 33268}, {14001, 32990, 2}, {14035, 32964, 33259}, {14035, 32991, 32979}, {14035, 33204, 33002}, {14035, 33259, 2}, {14037, 33004, 2}, {14042, 32998, 3091}, {14063, 33265, 5059}, {14069, 21735, 8356}, {15692, 33198, 7824}, {15692, 33255, 2}, {15696, 33184, 33247}, {15717, 33201, 2}, {16044, 33206, 2}, {16917, 17558, 2}, {16923, 33016, 7486}, {16924, 33274, 10303}, {16925, 33017, 33245}, {16925, 33254, 5025}, {16925, 33268, 33199}, {17538, 33189, 7841}, {17694, 33038, 2}, {19689, 33258, 2}, {19696, 33006, 50688}, {21495, 21511, 37269}, {21508, 21537, 11340}, {27088, 32985, 35287}, {32951, 33234, 32974}, {32959, 33703, 33228}, {32961, 33257, 3543}, {32964, 33187, 33206}, {32964, 33266, 3552}, {32965, 33225, 2}, {32966, 33193, 17578}, {32966, 33262, 2}, {32967, 33280, 3839}, {32970, 32972, 2}, {32973, 32990, 14001}, {32973, 35287, 3}, {32981, 32991, 14035}, {32985, 33227, 32990}, {32985, 35287, 2}, {32988, 32989, 33233}, {32988, 33233, 2}, {32989, 35927, 4}, {32998, 33007, 14042}, {33002, 33204, 2}, {33002, 33259, 33204}, {33014, 33205, 33023}, {33017, 33245, 33199}, {33017, 33268, 20}, {33019, 33214, 15683}, {33020, 33188, 2}, {33025, 33260, 33023}, {33186, 44245, 5077}, {33187, 33206, 16044}, {33193, 33262, 32966}, {33199, 33245, 2}, {33216, 33239, 5}, {33222, 33247, 33184}, {33224, 33226, 7866}, {33233, 33235, 33250}, {33233, 33250, 4}, {33235, 35297, 4}, {33245, 33268, 33017}, {33246, 33276, 7791}, {33250, 35297, 33233}, {33252, 33283, 33264}, {35297, 35927, 2}, {35303, 35304, 15685}, {37172, 37173, 381}, {37340, 37341, 15703}


X(440) = COMPLEMENT OF X(27)

Trilinears       bc(v + w) : ca(w + u) : ab(u + v), where
                        u : v : w = X(28); e.g., u(a,b,c) = (tan A)/(b + c)

Barycentrics  v + w : w + u : u + v

As a point on the Euler line, X(440) has Shinagawa coefficients ($bc$ + E - F,-$bc$ - E - F)

X(440) lies on these lines: 2,3   37,226   72,306   118,122   950,1104

X(440) = complement of X(27)
X(440) = complementary conjugate of X(34830)
X(440) = X(190)-Ceva conjugate of X(525)
X(440) = crosspoint of X(2) and X(306)
X(440) = crosssum of X(i) and X(j) for these (i,j): (6,1474), (284,579)


X(441) = COMPLEMENT OF X(297)

Trilinears       bc(v + w) : ca(w + u) : ab(u + v), where
                        u : v : w = X(240); e.g., u(A,B,C) = sec A cos(A + ω)
Barycentrics  v + w : w + u : u + v

As a point on the Euler line, X(441) has Shinagawa coefficients (2(E + F)F - S2, S2)

X(441) lies on these lines: 2,3   141,577   525,647

X(441) = midpoint of X(297) and X(401)
X(441) = isotomic conjugate of X(6330)
X(441) = complement of X(297)
X(441) = circumcircle-inverse of X(37921)
X(441) = crosssum of X(6) and X(232)
X(441) = crossdifference of every pair of points on line X(25)X(647)
X(441) = inverse-in-Steiner-inellipse of X(3)
X(441) = {X(2454),X(2455)}-harmonic conjugate of X(3)
X(441) = isogonal conjugate of polar conjugate of X(30737)


X(442) = COMPLEMENT OF SCHIFFLER POINT

Trilinears       bc(v + w) : ca(w + u) : ab(u + v), where
                        u : v : w = X(284); e.g., u(A,B,C) = (sin A)/(cos B + cos C)
Barycentrics  v + w : w + u : u + v

As a point on the Euler line, X(442) has Shinagawa coefficients ($aSA$ + abc,$aSA$ - abc)

Let IA, IB, IC be the excenters, let AB, AC be the projections of A onto IAIB and IAIC, respectively, and define BC, BA and CA, CB cyclically. The Euler lines of the four triangles ABC, AABAC, BBCBA, CCACB concur in X(442). (Jean-Pierre Ehrmann, 11/24/01)

In the plane of triangle ABC, let DEF denote the intouch triangle of ABC, and let
HA = orthocenter of IBC
MA = midpoint of segment BC
NA = midpoint of the arc BC which does not include A
SA = reflection of I in line DE,
and define cyclically the points HB, HC, MB, MC, NB, NC, and SB, SC. The lines SAMA, SBMB, SCAMC concur in X(442), and X(442) is the pole, with respect to the incircle, of the perspectrix of the triangles HAHBHC and NABC. (Dominik Burek, January 18 2012)

Let A'B'C' be the excentral triangle. X(442) is the radical center of the nine-point circles of A'BC, B'CA, C'AB. (Randy Hutson, September 14, 2016)

Let (Ja) be the A-excircle of the A-altimedial triangle, and define (Jb) and (Jc) cyclically. X(442) is the radical center of (Ja), (Jb), (Jc). (Randy Hutson, November 2, 2017)

X(442) lies on these lines: 2,3   8,495   9,46   10,12   11,214   100,943   115,120   119,125   274,325   388,956   392,946

X(442) = midpoint of X(79) and X(191)
X(442) = isogonal conjugate of X(1175)
X(442) = inverse-in-orthocentroidal-circle of X(405)
X(442) = complement of X(21)
X(442) = complementary conjugate of X(960)
X(442) = X(100)-Ceva conjugate of X(523)
X(442) = crosspoint of X(264) and X(321)
X(442) = crosssum of X(184) and X(1333)
X(442) = perspector of Feuerbach triangle and 2nd extouch triangle
X(442) = homothetic center of 4th Euler triangle and 2nd extouch triangle
X(442) = X(54)-of-2nd-extouch-triangle
X(442) = X(973)-of-excentral-triangle


X(443) = COMPLEMENT OF X(452)

Trilinears       bc(v + w) : ca(w + u) : ab(u + v), where
                        u : v : w = X(380)

Barycentrics  v + w : w + u : u + v

As a point on the Euler line, X(443) has Shinagawa coefficients (abc*$a$, S2).

X(443) lies on these lines: {1, 142}, {2, 3}, {6, 4340}, {7, 72}, {8, 942}, {9, 4292}, {10, 57}, {12, 1466}, {56, 3925}, {69, 274}, {78, 3487}, {141, 5800}, {153, 5789}, {226, 936}, {278, 1038}, {329, 5044}, {355, 5768}, {386, 5712}, {387, 940}, {392, 962}, {497, 1125}, {579, 966}, {610, 5750}, {750, 5230}, {908, 5714}, {938, 3419}, {946, 6282}, {948, 1448}, {956, 3600}, {958, 3826}, {960, 4295}, {965, 5746}, {997, 3485}, {1001, 4294}, {1058, 3434}, {1119, 1441}, {1210, 5437}, {1376, 3085}, {1453, 3008}, {1478, 1698}, {1479, 3624}, {1770, 5698}, {2093, 5837}, {2095, 5690}, {2886, 3086}, {2999, 5717}, {3333, 4847}, {3358, 5817}, {3436, 5744}, {3475, 3811}, {3587, 5250}, {3617, 5708}, {3618, 5138}, {3634, 5229}, {3697, 5815}, {3698, 5252}, {3812, 5794}, {3876, 5905}, {3911, 5705}, {3916, 5273}, {3940, 6147}, {4299, 5251}, {4302, 5259}, {4304, 5436}, {4317, 5258}, {4355, 5223}, {4423, 6284}, {4680, 6533}, {5080, 5122}, {5175, 5722}, {5219, 6700}, {5275, 5286}, {5440, 5703}, {5587, 6245}, {5657, 5709}, {5927, 6223}, {6256, 6705}

X(443) = complement of X(452)


X(444) = EULER LINE INTERCEPT OF LINE X(19)X(232)

Trilinears       (v + w)tan A : (w + u)tan B : (u + v)tan C, where
                        u : v : w = X(256); e.g., u(a,b,c) = 1/(a2 + bc)

Barycentrics  (v + w)(sin A tan A) : (w + u)(sin B tan B) : (u + v)(sin C tan C)

As a point on the Euler line, X(444) has Shinagawa coefficients ([$a$(E + F) + $aSA$ - abc]F,-(E + F)[$a$(E + F) + $aSA$ + abc]).

X(444) lies on these lines: 2,3   19,232


X(445) = EULER X(79)-10th-SUBSTITUTION POINT

Trilinears        (v + w)csc 2A : (w + u)csc 2B : (u + v)csc 2C, where
                         u : v : w = X(79); e.g., u(a,b,c) = 1/(1 + 2 cos A)
Barycentrics   (v + w)sec A : (w + u)sec B : (u + v)sec C
Barycentrics   (a^2 + b^2 - c^2)*(a^2 - b^2 - b*c - c^2)*(a^2 - b^2 + c^2)*(a^2*b - b^3 + a^2*c + 2*a*b*c + b^2*c + b*c^2 - c^3) : :

As a point on the Euler line, X(445) has Shinagawa coefficients (($aSA$ + 2abc)F,$a$S2).

X(445) lies on these lines: {2, 3}, {81, 18679}, {92, 30690}, {196, 51358}, {264, 18139}, {273, 27186}, {281, 3580}, {317, 5278}, {318, 32858}, {340, 3578}, {648, 42045}, {1844, 31938}, {1990, 37631}, {2322, 2895}, {3219, 7282}, {3936, 31623}, {8747, 26131}, {14165, 40214}, {14581, 50181}, {17907, 19684}, {18026, 40447}, {19742, 27377}, {35201, 40940}, {37635, 41083}, {45038, 48381}

X(445) = isotomic conjugate of the isogonal conjugate of X(44095)
X(445) = polar conjugate of the isogonal conjugate of X(500)
X(445) = X(i)-isoconjugate of X(j) for these (i,j): {1794, 2160}, {2259, 7100}, {2982, 8606}
X(445) = X(i)-Dao conjugate of X(j) for these (i,j): {63, 5249}, {7100, 18591}, {16585, 52381}, {40937, 52388}
X(445) = cevapoint of X(500) and X(44095)
X(445) = crosspoint of X(92) and X(52412)
X(445) = barycentric product X(i)*X(j) for these {i,j}: {75, 1844}, {76, 44095}, {92, 16585}, {264, 500}, {273, 31938}, {319, 1838}, {340, 45926}, {1841, 33939}, {1859, 52421}, {1865, 34016}, {5249, 52412}, {6734, 7282}
X(445) = barycentric quotient X(i)/X(j) for these {i,j}: {35, 1794}, {442, 52388}, {500, 3}, {942, 7100}, {1838, 79}, {1841, 2160}, {1844, 1}, {1859, 7073}, {1865, 8818}, {5249, 52381}, {6198, 943}, {14547, 8606}, {16585, 63}, {31938, 78}, {44095, 6}, {45926, 265}, {46883, 52375}, {52412, 40435}
X(445) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {297, 25986, 2}, {7282, 52412, 3219}


X(446) = CROSSPOINT OF X(98) AND X(511)

Trilinears       u(v2 + w2) : v(w2 + u2) : w(u2 + v2), where
                        u : v : w = X(98); e.g., u(A,B,C) = sec(A + ω)
Barycentrics  au(v2 + w2) : bv(w2 + u2) : cw(u2 + v2)

As a point on the Euler line, X(446) has Shinagawa coefficients ((E + F)3F - (E + F)(E - 2F)S2 + S4, (E - F)(E + F)3 - 2(E + F)2S2 - S4).

X(446) lies on this line: 2,3

X(446) = crosspoint of X(98) and X(511)
X(446) = crosssum of X(i) and X(j) for these (i,j): (98,511), (287,385)


X(447) = X(2)-HIRST INVERSE OF X(27)

Trilinears       bc(u2 - vw) : ca(v2 - wu) : ab(w2 - uv), where
                        u : v : w = X(28); e.g., u(a,b,c) = (tan A)/(b + c)

Barycentrics  u2 - vw : v2 - wu : w2 - uv

As a point on the Euler line, X(447) has Shinagawa coefficients (2(E+F)3FS2-2$abSC$(E+F)F +$ab$[4(E+F)2-S2]F, -(E+F)(2E-F)S2-S4 + 2($abSC$S2-$ab$(4E+F)S2).

X(447) lies on this line: 2,3   340,540   350,811   519,648

X(447) = X(2)-Hirst inverse of X(27)


X(448) = X(2)-HIRST INVERSE OF X(21)

Trilinears       bc(u2 - vw) : ca(v2 - wu) : ab(w2 - uv), where
                        u : v : w = X(284); e.g., u(A,B,C) = (sin A)/(cos B + cos C)

Barycentrics  u2 - vw : v2 - wu : w2 - uv

As a point on the Euler line, X(448) has Shinagawa coefficients ((E-2F)S4-2(E2-F2)FS2 -$abSASB$[4(E+F)F-3S2] +$ab$[2(E+F)FS2-S4], -(E-4F)S4 -5$abSASB$S2+$ab$S4).

X(448) lies on this line: 2,3

X(448) = X(2)-Hirst inverse of X(21)


X(449) = X(2)-HIRST INVERSE OF X(452)

Trilinears       bc(u2 - vw) : ca(v2 - wu) : ab(w2 - uv), where u : v : w = X(380)
Barycentrics  u2 - vw : v2 - wu : w2 - uv

As a point on the Euler line, X(449) has Shinagawa coefficients (2(E + F)F - abc*$a$ - S2,3abc*$a$ + S2).

X(449) lies on this line: 2,3

X(449) = X(2)-Hirst inverse of X(452)


X(450) = X(3)-HIRST INVERSE OF X(4)

Trilinears    (sec A)[cos4A - (cos B cos C)2] : :
Trilinears    (cos A)[sec4A - (sec B sec C)2] : :
Barycentrics    (tan A)[cos4A - (cos B cos C)2] : :
Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^4 - a^2*b^2 + a^2*b*c - b^3*c - a^2*c^2 + 2*b^2*c^2 - b*c^3)*(a^4 - a^2*b^2 - a^2*b*c + b^3*c - a^2*c^2 + 2*b^2*c^2 + b*c^3) : :

As a point on the Euler line, X(450) has Shinagawa coefficients ((E - 2F)F,(E + F)F - S2).

X(450) lies on these lines: {2, 3}, {107, 511}, {155, 1075}, {184, 15466}, {264, 5651}, {275, 5943}, {648, 3292}, {653, 17975}, {685, 17974}, {1069, 7049}, {1092, 1093}, {1148, 3157}, {1897, 17976}, {1935, 1940}, {1993, 3168}, {2052, 9306}, {2451, 2501}, {3260, 18020}, {3291, 6531}, {3462, 9820}, {5972, 14165}, {6090, 9308}, {6331, 12215}, {6335, 17977}, {6530, 11064}, {8062, 22382}, {8764, 17973}, {13346, 14249}.

X(450) = isogonal conjugate of X(1942)
X(450) = crossdifference of every pair of points on line X(185)X(647)
X(450) = X(3)-Hirst inverse of X(4)
X(450) = crossdifference of PU(17)
X(450) = perspector of hyperbola {A,B,C,PU(17)}
X(450) = intersection of trilinear polars of P(17) and U(17)
X(450) = X(2)-Ceva conjugate of X(39034)
X(450) = inverse-in-circumconic-centered-at-X(4) of X(25)


X(451) = X(4)-HIRST INVERSE OF X(424)

Trilinears       u sec A : v sec B : w sec C, where u : v : w = X(191)
Barycentrics  u tan A : v tan B : w tan C

As a point on the Euler line, X(451) has Shinagawa coefficients (2$a$F,abc).

X(451) lies on these lines: 2,3   12,108   281,1068

X(451) = X(4)-Hirst inverse of X(424)
X(451) = polar conjugate of X(1029)


X(452) = X(2)-HIRST INVERSE OF X(449)

Trilinears       u csc A : v csc B : w csc C, where u : v : w = X(380)
Barycentrics  u : v : w

As a point on the Euler line, X(452) has Shinagawa coefficients (abc*$a$ + S2,-2S2).

X(452) lies on these lines: 1,329   2,3   8,9   34,347   63,938   72,145   388,1001   392,944   497,958   956,1058

X(452) = isogonal conjugate of X(2213)
X(452) = anticomplement of X(443)
X(452) = X(2)-Hirst inverse of X(449)


X(453) = POINT ALSHAIN

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C)=(cos B + cos C - cos A)2/(cos B + cos C)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(453) has Shinagawa coefficients (2$aSBSC$+$aSA$(E+2F)-2$a$S2-2abcF, -$aSA$E+2$a$S2+abcE).

X(453) lies on these lines: 2,3   46,1800   1014,1454


X(454) = EULER X(155)-11th-SUBSTITUTION POINT

Trilinears       u2sec A : v2sec B : w2sec C, where
                        u : v : w = X(155); e.g., u(A,B,C) = (cos A)[cos2B + cos2C - cos2A]
Barycentrics  u2tan A : v2tan B : w2tan C
Barycentrics   a^2*(a^2 - b^2 - c^2)*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 - 3*a^4*c^2 - 2*a^2*b^2*c^2 + b^4*c^2 + 3*a^2*c^4 + b^2*c^4 - c^6)^2 : :

As a point on the Euler line, X(454) has Shinagawa coefficients (E(E + 4F) - 4S2,-E2 + 4S2).

X(454) lies on these lines: {2, 3}, {155, 6503}, {1609, 34853}, {3964, 44405}, {9908, 16391}, {9937, 23181}, {12164, 34333}, {15316, 47195}, {20975, 38260}

X(454) = X(3542)-Ceva conjugate of X(155)
X(454) = X(254)-isoconjugate of X(921)
X(454) = barycentric product X(i)*X(j) for these {i,j}: {155, 6515}, {1609, 40697}, {3542, 6503}
X(454) = barycentric quotient X(i)/X(j) for these {i,j}: {155, 6504}, {1609, 254}, {6515, 46746}
X(454) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6617, 3548}


X(455) = EULER X(159)-11th-SUBSTITUTION POINT

Trilinears       u2sec A : v2sec B : w2sec C, where u : v : w = X(159)
Barycentrics  u2tan A : v2tan B : w2tan C
Barycentrics   a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^6 + a^4*b^2 - a^2*b^4 - b^6 + a^4*c^2 - 2*a^2*b^2*c^2 + b^4*c^2 - a^2*c^4 + b^2*c^4 - c^6)^2 : :

As a point on the Euler line, X(455) has Shinagawa coefficients (4(E + F)3F2 - E2FS2, -4(E + F)4F + (E + F)(E + 4F)ES2).

X(455) lies on these lines: {2, 3}, {159, 17407}, {2386, 40144}

X(455) = X(1370)-Ceva conjugate of X(3162)
X(455) = X(39733)-isoconjugate of X(52041)
X(455) = X(25)-Dao conjugate of X(13575)
X(455) = barycentric product X(i)*X(j) for these {i,j}: {159, 41361}, {1370, 3162}, {23115, 41766}
X(455) = barycentric quotient X(i)/X(j) for these {i,j}: {3162, 13575}, {41361, 40009}
X(455) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {}


X(456) = EULER X(195)-11th-SUBSTITUTION POINT

Trilinears       u2sec A : v2sec B : w2sec C, where u : v : w = X(195)
Barycentrics  u2tan A : v2tan B : w2tan C
Barycentrics   a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 - 4*a^6*c^2 + 5*a^4*b^2*c^2 + a^2*b^4*c^2 - 2*b^6*c^2 + 6*a^4*c^4 + a^2*b^2*c^4 + 2*b^4*c^4 - 4*a^2*c^6 - 2*b^2*c^6 + c^8)^2 : :

As a point on the Euler line, X(456) has Shinagawa coefficients ((9E + 16F)EF2 - 64F2S2, -E2F2 + 16(E + 4F)FS2).

{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {15770, 15779, 3}


X(457) = EULER X(399)-11th-SUBSTITUTION POINT

Trilinears        u2sec A : v2sec B : w2sec C, where u : v : w = X(399)
Barycentrics  u2tan A : v2tan B : w2tan C
Barycentrics   a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 - 4*a^6*c^2 + a^4*b^2*c^2 + a^2*b^4*c^2 + 2*b^6*c^2 + 6*a^4*c^4 + a^2*b^2*c^4 - 6*b^4*c^4 - 4*a^2*c^6 + 2*b^2*c^6 + c^8)^2 : :

As a point on the Euler line, X(457) has Shinagawa coefficients (9(E + 16F)EF - 64FS2,-81E2F + 16(E + 4F)S2).

X(457) lies on these lines: {2, 3}, {14993, 52166}, {16240, 51345}, {37496, 47215}

X(457) = barycentric product X(1272)*X(52166)
X(457) = barycentric quotient X(52166)/X(1138)
X(457) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {15766, 15773, 3}


X(458) = EULER LINE INTERCEPT OF LINE X(76)X(275)

Trilinears       u csc 2A : v csc 2B : w csc 2C, where
                        u : v : w = X(182); e.g.; u(A,B,C) = cos(A - ω)
Barycentrics   u sec A : v sec B : w sec C
Barycentrics    tan A + cot ω : :
Barycentrics    sec A cos(A - ω) : :

As a point on the Euler line, X(458) has Shinagawa coefficients ((E + F)F, S2).

X(458) lies on these lines: 2,3   6,264   76,275   141,317   239,318   273,894   315,343   340,599

X(458) = inverse-in-orthocentroidal-circle of X(297)
X(458) = pole wrt polar circle of trilinear polar of X(262) (line X(523)X(3569))
X(458) = polar conjugate of X(262)


X(459) = X(253)-CEVA CONJUGATE OF X(4)

Trilinears    (sec A)/(tan B + tan C - tan A) : :
Trilinears    |A'A″| : :, where A'B'C', A″B″C″ are the cevian and anticevian triangles of X(4)
Barycentrics    1/(tan B tan C - 2) : :
Barycentrics    1/(SA(S2 - 2SBSC)) : :

Let A'B'C' be the midheight triangle. Let A″ be the pole, wrt the polar circle, of line B'C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(459). (Randy Hutson, June 7, 2019)

X(459) lies on the Kiepert hyperbola and these lines: {2,253}, {4,64}, {10,3176}, {25,3424}, {69,801}, {92,1446}, {96,3147}, {98,1301}, {154,5922}, {196,226}, {262,3168}, {297,2996}, {458,5395}, {485,3535}, {486,3536}, {1075,3090}, {1131,1585}, {1132,1586}, {1503,3079}, {5485,5523}

X(459) = isotomic conjugate of X(37669)
X(459) = X(253)-Ceva conjugate of X(4)
X(459) = X(2155)-complementary conjugate of X(3350)
X(459) = cevapoint of X(i) and X(j) for these {i,j}: {6,3515}, {125,2501}, {459,459}, {523,1562}
X(459) = X(i)-cross conjugate of X(j) for these (i,j): (64, 253), (235, 264), (393, 4), (1562, 523)
X(459) = polar conjugate of X(20)
X(459) = X(i)-isoconjugate of X(j) for these {i,j}: {3,610}, {20,48}, {63,154}, {204,394}, {219,1394}, {255,1249}, {326,3172}, {577,1895}, {1101,1562}, {1259,3213}, {1790,3198}, {2193,5930}
X(459) = {X(2),X(253)}-harmonic conjugate of X(1073)
X(459) = trilinear product X(i)*X(j) for these {i,j}: {4,2184}, {19,253}, {64,92}, {158,1073}, {264,2155}, {1301,1577}, {1880,5931}}
X(459) = barycentric product X(i)*X(j) for these {i,j}: {4,253}, {64,264}, {92,2184}, {225,5931}, {850,1301}, {1073,2052}, {1969,2155}
X(459) = barycentric quotient X(i)/X(j) for these {i,j}: {4,20}, {19,610}, {25,154}, {34,1394}, {64,3}, {115,1562}, {125,122}, {158,1895}, {225,5930}, {235,2883}, {253,69}, {393,1249}, {459,2}, {1073,394}, {1096,204}, {1301,110}, {1824,3198}, {1895,1097}, {2155,48}, {2184,63}, {2207,3172}, {3183,2060}, {5931,332}


X(460) = POINT ANTARES

Trilinears    (sec A)[a2(2a2 - b2 - c2) + (b2 - c2)2] : :

As a point on the Euler line, X(460) has Shinagawa coefficients ((E + F)F, -(E + F)2 + 2S2).

X(460) lies on this line: 2,3   53,1974   512,2501   685,2065

X(460) = crossdifference of every pair of points on line X(394)X(647)
X(460) = X(241)-of-orthic-triangle if ABC is acute
X(460) = pole wrt polar circle of trilinear polar of X(8781) (line X(69)X(523))
X(460) = polar conjugate of X(8781)
X(460) = barycentric product X(4)*X(230)


X(461) = EULER LINE INTERCEPT OF LINE X(33)X(200)

Trilinears       u tan A : v tan B : w tan C, where
                        u : v : w = X(391); e.g., u(a,b,c) = bc(3a + b + c)(b + c - a)

Barycentrics  u sin A tan A : v sin B tan B : w sin C tan C

As a point on the Euler line, X(461) has Shinagawa coefficients (2F,$bc$ - E - F).

X(461) lies on these lines: 2,3   33,200


X(462) = EULER LINE INTERCEPT OF LINE X(51)X(397)

Trilinears       u tan A : v tan B : w tan C, where
                        u : v : w = X(395); e.g., u(A,B,C) = cos(B - C) + 2 cos(A + π/3)

Barycentrics  u sin A tan A : v sin B tan B : w sin C tan C

As a point on the Euler line, X(462) has Shinagawa coefficients (31/2F, -31/2(E + F) + 2S).

X(462) lies on these lines: 2,3   51,397   184,398  


X(463) = EULER LINE INTERCEPT OF LINE X(51)X(398)

Trilinears       u tan A : v tan B : w tan C, where
                        u : v : w = X(396); e.g., u(A,B,C) = cos(B - C) + 2 cos(A - π/3)

Barycentrics  u sin A tan A : v sin B tan B : w sin C tan C

As a point on the Euler line, X(463) has Shinagawa coefficients (31/2F, -31/2(E + F) - 2S).

X(463) lies on these lines: 2,3   51,398   184,397


X(464) = EULER LINE INTERCEPT OF LINE X(63)X(69)

Trilinears       u cot A : v cot B : w cot C, where u : v : w = X(387)
Barycentrics  u cos A : v cos B : w cos C

As a point on the Euler line, X(464) has Shinagawa coefficients (E + $bc$, -E - F - $bc$).

X(464) lies on these lines: 2,3   63,69

X(464) is the {X(2),X(20)}-harmonic conjugate of X(27). For a list of other harmonic conjugates of X(464), click Tables at the top of this page.


X(465) = EULER LINE INTERCEPT OF LINE X(216)X(395)

Trilinears       u cot A : v cot B : w cot C, where
                        u : v : w = X(397); e.g., u(A,B,C) = cos(B - C) - 2 cos(A + π/3)

Barycentrics  u cos A : v cos B : w cos C

As a point on the Euler line, X(465) has Shinagawa coefficients (2F + 31/2S, -31/2S).

X(465) lies on these lines: 2,3   216,395   396,577

X(465) is the {X(2),X(3)}-harmonic conjugate of X(466). For a list of other harmonic conjugates of X(465), click Tables at the top of this page.

X(465) = complement of X(473)

X(465) = complementary conjugate of the complement of X(32585)

X(466) = EULER LINE INTERCEPT OF LINE X(216)X(396)

Trilinears       u cot A : v cot B : w cot C, where
                        u : v : w = X(398); e.g., u(A,B,C) = cos(B - C) - 2 cos(A - π/3)

Barycentrics  u cos A : v cos B : w cos C

As a point on the Euler line, X(466) has Shinagawa coefficients (2F - 31/2S, 31/2S).

X(466) lies on these lines: 2,3   216,396   395,577

X(466) is the {X(2),X(3)}-harmonic conjugate of X(465). For a list of other harmonic conjugates of X(466), click Tables at the top of this page.

X(446) = complement of X(472)
X(466) = complementary conjugate of the complement of X(32586)


X(467) = EULER LINE INTERCEPT OF LINE X(53)X(311)

Trilinears       u csc 2A : v csc 2B : w csc 2C, where
                        u : v : w = X(52); e.g., u(A,B,C) = cos 2A cos(B - C)

Barycentrics  u sec A : v sec B : w sec C

As a point on the Euler line, X(467) has Shinagawa coefficients ((E + 2F)F,-2S2).

X(467) lies on these lines: 2,3   53,311

X(467) = X(317)-Ceva conjugate of X(52)
X(467) = polar conjugate of X(96)


X(468) = X(2)-LINE CONJUGATE OF X(3)

Trilinears    (sec A)(cot B + cot C - 2 cot A) : :
Trilinears    sec A - 3 csc A tan ω : :
Trilinears    3 csc A - sec A cot ω : :
Barycentrics    (b^2 + c^2 - 2a^2)/(b^2 + c^2 - a^2) : :
X(468) = 3 X[2] + X[23], 9 X[2] - X[5189], 5 X[2] - X[10989], 15 X[2] + X[20063], 9 X[2] - 5 X[30745], 3 X[2] + 5 X[37760], 3 X[2] + 2 X[37897], 6 X[2] + X[37899], 9 X[2] + X[37900], 7 X[2] + X[37901], 2 X[2] + X[37904], X[2] + 3 X[37907], 5 X[2] + 3 X[37909], 9 X[2] + 2 X[37910], 3 X[2] - 4 X[37911], 6 X[2] - X[46517], 3 X[3] + X[18325], X[3] - 3 X[44214], X[4] + 3 X[186], X[4] - 3 X[403], 2 X[4] - 3 X[10151], 4 X[4] - 3 X[13473], 5 X[4] + 3 X[13619], 7 X[4] + 9 X[35489], 2 X[4] + 3 X[37931], X[4] + 2 X[37934], X[4] + 6 X[37935], X[4] - 6 X[37942], X[4] - 9 X[37943], 3 X[5] - X[18572], X[5] - 3 X[44282], X[20] - 5 X[37952], X[20] - 3 X[44280], X[23] + 2 X[5159], 3 X[23] + X[5189], X[23] - 3 X[7426], 5 X[23] + 3 X[10989], 5 X[23] - X[20063], 3 X[23] + 5 X[30745], X[23] - 5 X[37760], 3 X[23] - X[37900], 7 X[23] - 3 X[37901], 2 X[23] - 3 X[37904], X[23] - 9 X[37907], 5 X[23] - 9 X[37909], 3 X[23] - 2 X[37910], X[23] + 4 X[37911], 2 X[23] + X[46517], X[67] + 3 X[18374], 3 X[110] + X[41724], X[125] + 2 X[15448], X[140] + 3 X[10096], 4 X[140] - 3 X[10257], 2 X[140] + X[16619], 4 X[140] + 3 X[37971], X[140] - 3 X[44234], 2 X[140] - 3 X[44452], X[140] - 6 X[44900], 2 X[186] + X[10151], 3 X[186] - X[10295], 4 X[186] + X[13473], 5 X[186] - X[13619], 7 X[186] - 3 X[35489], 3 X[186] - 2 X[37934], X[186] + 2 X[37942], X[186] + 3 X[37943], 3 X[186] + 2 X[37984], 2 X[230] + X[16316], 3 X[250] + X[340], 3 X[381] - X[18323], 3 X[381] + 5 X[37958], 2 X[389] - 3 X[16227], 3 X[403] + X[10295], 4 X[403] - X[13473], 5 X[403] + X[13619], 7 X[403] + 3 X[35489], 2 X[403] + X[37931], 3 X[403] + 2 X[37934], X[403] + 2 X[37935], X[403] - 3 X[37943], 3 X[403] - 2 X[37984], 3 X[549] - X[37950], X[550] + 3 X[11563], X[550] - 3 X[15646], X[550] - 6 X[16531], X[550] - 9 X[16532], 3 X[597] - X[15826], 5 X[631] - X[7464], 5 X[632] + X[37967], X[842] + 3 X[38227], 3 X[858] - X[5189], X[858] + 3 X[7426], 5 X[858] - 3 X[10989], 5 X[858] + X[20063], 3 X[858] - 5 X[30745], X[858] + 5 X[37760], X[858] + 2 X[37897], 2 X[858] + X[37899], 3 X[858] + X[37900], 7 X[858] + 3 X[37901], 2 X[858] + 3 X[37904], X[858] + 9 X[37907], 5 X[858] + 9 X[37909], 3 X[858] + 2 X[37910], X[858] - 4 X[37911] As a point on the Euler line, X(468) has Shinagawa coefficients (3F,-E - F).

Let (Pa) be the A-power circle of the A-altimedial triangle, and define (Pb) and (Pc) cyclically. The radical center of (Pa), (Pb), (Pc) is X(4), and the harmonic traces of (Pa), (Pb), (Pc) are X(2) and X(468). (Randy Hutson, November 2, 2017)

Let P and P' be circumcircle antipodes. Let Q and Q' be the complements of P and P', resp. The rectangular hyperbola passing through P, P', Q, Q' has center X(468) for all P. (Randy Hutson, March 29, 2020)

X(468) lies on the Darboux quintic (Q071), the cubics K043, K164 ,K209, K217, K418, K452, K478, K533, K535, K600, K608, K698, K824, K869, K954, the GEOS circle, and on these lines: {2, 3}, {6, 5486}, {10, 11363}, {32, 40102}, {33, 5160}, {34, 5433}, {49, 13292}, {51, 23292}, {52, 9820}, {53, 3054}, {64, 43903}, {67, 1177}, {69, 6090}, {74, 10293}, {81, 44097}, {98, 685}, {99, 37803}, {105, 2766}, {107, 842}, {108, 2752}, {110, 3564}, {111, 935}, {112, 2770}, {113, 32110}, {114, 3233}, {115, 40350}, {125, 1495}, {126, 5203}, {132, 3258}, {133, 46436}, {136, 16188}, {141, 1974}, {154, 1899}, {171, 14975}, {182, 13394}, {183, 44134}, {184, 8550}, {185, 16252}, {187, 1560}, {206, 26926}, {230, 231}, {242, 2969}, {243, 42069}, {250, 325}, {262, 43530}, {264, 37688}, {275, 7608}, {305, 32820}, {323, 19504}, {343, 9306}, {373, 1843}, {385, 35511}, {389, 16227}, {393, 21448}, {394, 44492}, {395, 8740}, {396, 8739}, {459, 43537}, {476, 3563}, {477, 9064}, {498, 11399}, {499, 11398}, {511, 1112}, {512, 22264}, {524, 3292}, {525, 14697}, {542, 35266}, {590, 5413}, {597, 8541}, {612, 10149}, {615, 5412}, {620, 5186}, {648, 18823}, {653, 17985}, {691, 2374}, {748, 1395}, {750, 2212}, {879, 2433}, {899, 2356}, {908, 27421}, {930, 23096}, {940, 44086}, {942, 41609}, {973, 12242}, {1001, 11383}, {1078, 11380}, {1125, 1829}, {1147, 41587}, {1155, 40560}, {1164, 1165}, {1172, 37675}, {1196, 7755}, {1204, 2883}, {1211, 2203}, {1213, 1474}, {1235, 26235}, {1249, 37689}, {1289, 37801}, {1290, 15344}, {1297, 22239}, {1299, 16167}, {1300, 9060}, {1301, 2697}, {1302, 32710}, {1351, 21970}, {1352, 35259}, {1353, 37644}, {1384, 41370}, {1398, 7288}, {1452, 11375}, {1473, 20266}, {1493, 41598}, {1498, 26937}, {1514, 2777}, {1531, 36518}, {1533, 38727}, {1611, 3162}, {1614, 18914}, {1620, 5895}, {1648, 5967}, {1692, 6388}, {1698, 5090}, {1799, 40413}, {1824, 6690}, {1828, 6691}, {1848, 2355}, {1851, 17917}, {1853, 31383}, {1861, 1862}, {1870, 7292}, {1876, 3911}, {1878, 6681}, {1892, 5219}, {1897, 17927}, {1902, 6684}, {1976, 41175}, {1986, 10272}, {1993, 41588}, {2052, 7607}, {2080, 41253}, {2204, 5277}, {2211, 3231}, {2299, 17056}, {2373, 10423}, {2393, 12099}, {2452, 7735}, {2453, 37637}, {2687, 9107}, {2690, 9085}, {2758, 9088}, {2931, 32123}, {2967, 14920}, {2970, 38552}, {3043, 13392}, {3055, 6748}, {3066, 14561}, {3068, 5411}, {3069, 5410}, {3092, 5418}, {3093, 5420}, {3167, 6515}, {3186, 7806}, {3192, 37646}, {3199, 7749}, {3220, 26933}, {3266, 6390}, {3448, 35265}, {3532, 43695}, {3569, 32120}, {3574, 11745}, {3581, 14643}, {3616, 11396}, {3618, 12167}, {3624, 7713}, {3712, 42713}, {3720, 40976}, {3742, 41611}, {3815, 5063}, {3818, 45303}, {3917, 44079}, {3934, 12143}, {4383, 44105}, {4422, 24814}, {4648, 44100}, {4999, 40985}, {5012, 45298}, {5130, 24953}, {5139, 31655}, {5140, 6719}, {5185, 6710}, {5205, 26231}, {5218, 7071}, {5275, 45786}, {5285, 21015}, {5297, 6198}, {5304, 5702}, {5305, 9465}, {5306, 18361}, {5370, 15326}, {5446, 43823}, {5449, 12134}, {5461, 12132}, {5462, 6746}, {5476, 20192}, {5480, 34417}, {5520, 20621}, {5622, 40114}, {5640, 6403}, {5650, 12294}, {5654, 37489}, {5656, 18931}, {5891, 44201}, {5901, 41722}, {5943, 11649}, {6000, 15738}, {6036, 12131}, {6091, 41521}, {6108, 6111}, {6109, 6110}, {6118, 12148}, {6119, 12147}, {6146, 10282}, {6152, 8254}, {6225, 34469}, {6242, 22051}, {6247, 26883}, {6331, 17984}, {6335, 17987}, {6336, 17982}, {6467, 15585}, {6669, 12142}, {6670, 12141}, {6673, 22482}, {6674, 22481}, {6689, 11576}, {6696, 11381}, {6698, 32239}, {6699, 12133}, {6701, 16114}, {6702, 12137}, {6703, 44092}, {6704, 12144}, {6705, 12136}, {6707, 17171}, {6713, 12138}, {6720, 13166}, {6723, 29012}, {6776, 26864}, {6800, 18911}, {7009, 7140}, {7302, 15338}, {7583, 10881}, {7584, 10880}, {7665, 19577}, {7703, 32124}, {7718, 9780}, {7746, 27376}, {7747, 15820}, {7767, 26233}, {7768, 33651}, {7777, 27377}, {7789, 30749}, {7846, 11386}, {7857, 38526}, {7891, 30793}, {8263, 41614}, {8428, 14729}, {8537, 15019}, {8737, 11537}, {8738, 11549}, {8744, 11580}, {8753, 15899}, {8754, 9172}, {8770, 13854}, {8827, 46111}, {8854, 8960}, {8859, 38294}, {8901, 19189}, {9061, 10101}, {9084, 10098}, {9128, 34512}, {9164, 22110}, {9173, 39241}, {9174, 39240}, {9176, 43084}, {9185, 44427}, {9225, 15993}, {9308, 17008}, {9544, 45968}, {9707, 18912}, {9745, 18907}, {9777, 11427}, {10006, 18344}, {10102, 30247}, {10117, 32125}, {10163, 11594}, {10182, 11430}, {10185, 39284}, {10198, 26377}, {10200, 26378}, {10214, 11701}, {10416, 15398}, {10420, 40120}, {10422, 10424}, {10511, 46105}, {10535, 26956}, {10536, 26957}, {10539, 12359}, {10546, 18358}, {10564, 38793}, {10601, 44503}, {10619, 32391}, {10632, 11543}, {10633, 11542}, {10641, 23303}, {10642, 23302}, {10990, 15311}, {10991, 42671}, {11202, 18390}, {11206, 23291}, {11402, 11433}, {11408, 11489}, {11409, 11488}, {11424, 15873}, {11442, 35264}, {11459, 44683}, {11464, 12022}, {11547, 14569}, {11550, 23332}, {11645, 32267}, {11746, 44668}, {12007, 44109}, {12140, 20304}, {12145, 34841}, {12162, 44158}, {12165, 20125}, {12174, 18913}, {12241, 13367}, {12290, 43607}, {12370, 32171}, {12421, 33563}, {12827, 20772}, {12900, 15473}, {13142, 34148}, {13148, 13754}, {13399, 15152}, {13414, 44123}, {13415, 44124}, {13419, 32767}, {13474, 25563}, {13568, 43831}, {13869, 26228}, {14052, 34827}, {14457, 14528}, {14530, 26944}, {14560, 34310}, {15010, 21969}, {15106, 32247}, {15126, 36201}, {15128, 38885}, {15270, 15652}, {15271, 32224}, {15360, 23061}, {15433, 39662}, {15472, 37477}, {15526, 34147}, {15806, 16881}, {15904, 32126}, {16081, 46316}, {16103, 23992}, {16178, 31842}, {16655, 20299}, {16659, 23294}, {16823, 26259}, {17398, 44103}, {17409, 40320}, {17718, 44095}, {17811, 43653}, {17821, 19467}, {18350, 31831}, {18553, 21243}, {18916, 19347}, {18947, 37779}, {19121, 26156}, {19124, 22112}, {19357, 39571}, {20376, 32340}, {20725, 37853}, {20771, 46085}, {20774, 23234}, {20977, 41939}, {20987, 23300}, {22479, 25524}, {24206, 35283}, {26880, 26905}, {26881, 26913}, {26882, 26917}, {26884, 26932}, {26885, 26942}, {26886, 26950}, {26887, 26954}, {26888, 26955}, {28419, 37491}, {30714, 44665}, {30716, 40429}, {31489, 44658}, {32112, 32119}, {32114, 32127}, {32133, 38532}, {32821, 34254}, {34565, 41599}, {34573, 44091}, {34656, 37546}, {35268, 44882}, {35466, 44113}, {36990, 41424}, {37761, 38461}, {37762, 38462}, {37892, 43528}, {38110, 39588}, {40144, 40323}, {40347, 41336}, {40511, 44381}, {43461, 43462}, {43896, 44544}, {44125, 46167}, {44126, 46166}, {44899, 46184}

X(468) = midpoint of X(i) and X(j) for these {i,j}: midpoint of X(i) and X(j) for these {i,j}: {2, 7426}, {3, 11799}, {4, 10295}, {5, 7575}, {6, 32113}, {23, 858}, {69, 32220}, {74, 32111}, {110, 3580}, {113, 32110}, {125, 1495}, {140, 25338}, {141, 32217}, {186, 403}, {187, 5099}, {230, 16320}, {237, 36189}, {297, 7473}, {381, 44265}, {427, 37969}, {549, 44266}, {550, 44267}, {935, 5523}, {1316, 5112}, {1513, 36166}, {2070, 2072}, {2074, 37989}, {2770, 5913}, {2931, 32123}, {3292, 41586}, {3569, 32120}, {3589, 32218}, {3628, 44264}, {5000, 5001}, {5159, 37897}, {5189, 37900}, {5642, 32225}, {5972, 32223}, {6593, 8262}, {7703, 32124}, {8598, 36196}, {9128, 34512}, {10096, 44234}, {10117, 32125}, {10151, 37931}, {10257, 37971}, {11007, 37906}, {11064, 32269}, {11563, 15646}, {11657, 16319}, {14120, 36180}, {15122, 16619}, {15360, 40112}, {15904, 32126}, {16272, 16309}, {16303, 16334}, {16305, 16332}, {16315, 16316}, {16321, 16324}, {18571, 44961}, {21284, 37981}, {31726, 44246}, {32112, 32119}, {32114, 32127}, {32267, 45311}, {32460, 32461}, {34152, 43893}, {35266, 44569}, {36165, 37927}, {37899, 46517}, {37934, 37984}, {37935, 37942}, {37936, 37938}, {37974, 37975}

X(468) = reflection of X(i) in X(j) for these {i,j}: {4, 37984}, {23, 37897}, {186, 37935}, {403, 37942}, {858, 5159}, {1112, 44084}, {1495, 15448}, {2071, 16976}, {2072, 44911}, {2501, 41357}, {5095, 15471}, {5159, 37911}, {10151, 403}, {10257, 44452}, {10295, 37934}, {10297, 5}, {11064, 5972}, {12105, 44264}, {13473, 10151}, {15122, 140}, {15646, 16531}, {16103, 23992}, {16272, 16332}, {16303, 16324}, {16304, 16309}, {16309, 16305}, {16312, 16334}, {16315, 230}, {16316, 16320}, {16334, 16321}, {16387, 6676}, {16619, 25338}, {18571, 22249}, {20725, 37853}, {23323, 46031}, {32246, 35370}, {32269, 32223}, {36170, 10011}, {37899, 23}, {37900, 37910}, {37904, 7426}, {37931, 186}, {37987, 44334}, {44234, 44900}, {44452, 44234}, {46517, 858} X(468) = isogonal conjugate of X(895)
X(468) = isotomic conjugate of X(30786)
X(468) = complement of X(858)
X(468) = anticomplement of X(5159)

X(468) = circumcircle-inverse of X(25)
X(468) = nine-point-circle-inverse of X(427)
X(468) = nine-point-circle-of-medial triangle-inverse of X(2)
X(468) = orthocentroidal-circle-inverse of X(5094)
X(468) = Stevanovic-circle-inverse of X(5089)
X(468) = polar-circle-inverse of X(2)
X(468) = Dao-Moses-Telv-circle-inverse of X(6103)
X(468) = orthoptic-circle-of-Steiner-inellipse-inverse of X(4)
X(468) = orthoptic-circle-of-Steiner-circumellipe-inverse of X(3146)
X(468) = Moses-radical-circle-inverse of X(232)
X(468) = Moses-Parry-circle-inverse of X(44467)
X(468) = DeLongchamps-circle-inverse of X(7396)
X(468) = Stammler-circle-inverse of X(44454)
X(468) = {circumcircle, nine-point circle}-inverter-inverse of X(4)
X(468) = 1st-Droz-Farney-circle-inverse of X(44441)
X(468) = 2nd-Droz-Farney circle inverse of X(44438)
X(468) = circumcircle-of-anticomplementary-triangle-inverse of X(44442)
X(468) = circumcircle-of-inner-Napoleon-triangle-inverse of X(44459)
X(468) = circumcircle-of-outer-Napoleon-triangle-inverse of X(44463)

X(468) = X(2)-line conjugate of X(3)
X(468) = polar conjugate of X(671)
X(468) = polar conjugate of the isotomic conjugate of X(524)
X(468) = polar conjugate of the isogonal conjugate of X(187)
X(468) = cevapoint of X(i) and X(j) for these (i,j): {2, 7665}, {187, 44102}, {351, 1648}, {524, 32459}, {18374, 41336}
X(468) = crosspoint of X(i) and X(j) for these (i,j): {2, 2373}, {4, 17983}, {250, 10423}, {264, 46105}, {37778, 44146}
X(468) = crosssum of X(i) and X(j) for these (i,j): {3, 3292}, {6, 2393}, {184, 10317}, {8542, 10510}
X(468) = crossdifference of every pair of points on line X(3)X(647)
X(468) = X(i)-complementary conjugate of X(j) for these (i,j): 1, 15116}, {31, 1560}, {1177, 10}, {2373, 2887}, {10422, 4892}, {10423, 8062}, {18876, 18589}, {36095, 30476}, {37220, 626}, {46140, 21235}
X(468) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 1560}, {4, 5095}, {935, 523}, {2374, 25}, {4235, 14273}, {8791, 427}, {17983, 4}, {37765, 5523}, {41521, 46444}, {44146, 524
X(468) = X(i)-cross conjugate of X(j) for these (i,j): {187, 524}, {524, 5203}, {5095, 4}, {5099, 523}, {14273, 4235}, {14357, 41498}, {35282, 5967}, {44915, 671}}
X(468) = X(i)-Hirst inverse of X(j) for these (i,j): {523, 6103}, {1990, 16230}
X(468) = X(i)-vertex conjugate of X(j) for these (i,j): {23, 13574}, {25, 523}
X(468) = trilinear pole of line X(690)X(5095)
X(468) = medial-isogonal conjugate of X(15116)
X(468) = orthic-isogonal conjugate of X(5095)
X(468) = psi-transform of X(6776)
X(468) = X(1155)-of-orthic-triangle if ABC is acute
X(468) = X(5126)-of-circumorthic-triangle
X(468) = X(10295)-of-Euler-triangle
X(468) = X(10297)-of-Carnot-triangle
X(468) = X(10151)-of-Thomson-triangle
X(468) = X(5972)-of-anti-Brocard-triangle
X(468) = X(6719)-of-anti-McCay-triangle
X(468) = X(1319)-of-orthic-triangle-of-Thomson-triangle)
X(468) = X(23)-of-intouch-triangle-of-orthic-triangle
X(468) = centroid of ABCX(23)
X(468) = Kosnita(X(23),X(2)) point
X(468) = intersection of Euler line and orthic axis
X(468) = bicentric sum of PU(4)
X(468) = midpoint of PU(4)
X(468) = perspector of circumconic centered at X(1560)
X(468) = center of circumconic that is locus of trilinear poles of lines passing through X(1560)
X(468) = intersection of tangents to hyperbola {A,B,C,X(2),X(69)} at X(2) and X(2373)
X(468) = trilinear pole of line X(690)X(5095) (the perspectrix of ABC and 4th Brocard triangle)
X(468) = pole wrt polar circle of trilinear polar of X(671) (line X(2)X(523))
X(468) = X(48)-isoconjugate (polar conjugate) of X(671)
X(468) = radical trace of circumcircle and tangential circle
X(468) = radical trace of polar circle and {circumcircle, nine-point circle}-inverter
X(468) = radical trace of Moses-Parry circle and Moses radical circle
X(468) = excentral-to-ABC functional image of X(1155)
X(468) = center of Walsmith rectangular hyperbola

X(468) = X(i)-isoconjugate of X(j) for these {i,j}: {1, 895}, {2, 36060}, {3, 897}, {31, 30786}, {48, 671}, {63, 111}, {69, 923}, {75, 14908}, {77, 5547}, {78, 7316}, {163, 14977}, {184, 46277}, {255, 17983}, {293, 5968}, {304, 32740}, {326, 8753}, {394, 36128}, {525, 36142}, {647, 36085}, {656, 691}, {662, 10097}, {810, 892}, {896, 15398}, {1459, 5380}, {4558, 23894}, {4575, 5466}, {4592, 9178}, {6091, 8769}, {9213, 36061}, {9214, 35200}, {9247, 18023}, {14208, 32729}, {14209, 35188}, {18669, 41511}, {19626, 40364}, {34055, 46154}, {34158, 37220}

X(468) = barycentric product X(i)*X(j) for these {i,j}: {3, 37778}, {4, 524}, {6, 44146}, {19, 14210}, {25, 3266}, {27, 4062}, {28, 42713}, {76, 44102}, {92, 896}, {99, 14273}, {107, 14417}, {111, 34336}, {112, 35522}, {126, 2374}, {186, 43084}, {187, 264}, {193, 5203}, {275, 41586}, {278, 3712}, {281, 7181}, {286, 21839}, {297, 5967}, {351, 6331}, {393, 6390}, {428, 31068}, {523, 4235}, {648, 690}, {653, 14432}, {671, 5095}, {811, 2642}, {922, 1969}, {935, 18311}, {1309, 42760}, {1560, 2373}, {1648, 18020}, {1824, 16741}, {1826, 6629}, {1839, 31013}, {1897, 4750}, {1990, 36890}, {2052, 3292}, {2482, 17983}, {2501, 5468}, {2986, 12828}, {3793, 8801}, {5467, 14618}, {5477, 35142}, {5485, 15471}, {5486, 37855}, {5642, 16080}, {6330, 35282}, {6335, 14419}, {6591, 42721}, {6593, 46105}, {7664, 8791}, {7813, 32085}, {8753, 36792}, {9115, 38428}, {9117, 38427}, {9155, 16081}, {9204, 36306}, {9205, 36309}, {9717, 46106}, {10603, 24855}, {14052, 45291}, {14357, 37765}, {14424, 42396}, {14559, 44427}, {14567, 18022}, {16103, 40890}, {16702, 41013}, {17984, 18872}, {18027, 23200}, {22105, 41676}, {23889, 24006}, {24038, 36128}, {32225, 43530}, {32459, 34208}, {32713, 45807}, {39689, 46111}, {44814, 46456}

X(468) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 30786}, {4, 671}, {6, 895}, {19, 897}, {25, 111}, {31, 36060}, {32, 14908}, {92, 46277}, {111, 15398}, {112, 691}, {162, 36085}, {187, 3}, {232, 5968}, {264, 18023}, {351, 647}, {393, 17983}, {427, 31125}, {512, 10097}, {523, 14977}, {524, 69}, {607, 5547}, {608, 7316}, {648, 892}, {690, 525}, {896, 63}, {922, 48}, {1096, 36128}, {1177, 41511}, {1503, 36894}, {1560, 858}, {1648, 125}, {1649, 14417}, {1783, 5380}, {1843, 46154}, {1973, 923}, {1974, 32740}, {1990, 9214}, {2052, 46111}, {2207, 8753}, {2374, 44182}, {2482, 6390}, {2489, 9178}, {2501, 5466}, {2642, 656}, {3053, 6091}, {3266, 305}, {3292, 394}, {3712, 345}, {3793, 3785}, {4062, 306}, {4235, 99}, {4750, 4025}, {5026, 12215}, {5094, 42008}, {5095, 524}, {5140, 14263}, {5203, 2996}, {5467, 4558}, {5468, 4563}, {5477, 3564}, {5642, 11064}, {5967, 287}, {6103, 16092}, {6390, 3926}, {6531, 9154}, {6593, 22151}, {6629, 17206}, {7181, 348}, {7664, 37804}, {7813, 3933}, {8428, 19330}, {8541, 42007}, {8737, 36307}, {8738, 36310}, {8744, 14246}, {8749, 9139}, {8753, 10630}, {8791, 10415}, {9155, 36212}, {9717, 14919}, {11183, 24284}, {12828, 3580}, {14210, 304}, {14273, 523}, {14357, 34897}, {14417, 3265}, {14419, 905}, {14424, 2525}, {14432, 6332}, {14567, 184}, {15471, 1992}, {16702, 1444}, {17994, 8430}, {18872, 36214}, {21839, 72}, {21906, 20975}, {22105, 4580}, {23200, 577}, {23889, 4592}, {24855, 16051}, {27369, 41272}, {32225, 37638}, {32459, 6337}, {32676, 36142}, {34336, 3266}, {35282, 441}, {35325, 36827}, {35522, 3267}, {37778, 264}, {37855, 11185}, {39689, 3292}, {40890, 16093}, {41336, 39169}, {41586, 343}, {41616, 37784}, {41618, 41617}, {41911, 20977}, {42713, 20336}, {43084, 328}, {43925, 43926}, {44102, 6}, {44146, 76}, {44162, 19626}, {44467, 46783}, {44814, 8552}, {44915, 32257}, {45808, 45792}, {46522, 14609}

X(468) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3, 30739}, {2, 4, 5094}, {2, 5, 37454}, {2, 20, 16051}, {2, 22, 1368}, {2, 23, 858}, {2, 25, 427}, {2, 376, 32216}, {2, 858, 5159}, {2, 1113, 1312}, {2, 1114, 1313}, {2, 1370, 30771}, {2, 1995, 5}, {2, 2475, 30770}, {2, 3146, 30769}, {2, 3151, 30772}, {2, 3152, 30774}, {2, 3543, 30775}, {2, 4223, 37315}, {2, 4228, 37360}, {2, 4232, 4}, {2, 4233, 37362}, {2, 4239, 442}, {2, 5020, 37439}, {2, 5189, 30745}, {2, 6353, 25}, {2, 6655, 30777}, {2, 6676, 7499}, {2, 6872, 30776}, {2, 6995, 8889}, {2, 7386, 31255}, {2, 7391, 30744}, {2, 7392, 7539}, {2, 7394, 31236}, {2, 7417, 868}, {2, 7418, 3134}, {2, 7423, 3139}, {2, 7432, 3138}, {2, 7438, 429}, {2, 7439, 3137}, {2, 7448, 3141}, {2, 7449, 3142}, {2, 7453, 3136}, {2, 7458, 3140}, {2, 7471, 36170}, {2, 7493, 3}, {2, 7494, 7484}, {2, 7495, 140}, {2, 9832, 11007}, {2, 10154, 7667}, {2, 10565, 7386}, {2, 13595, 5133}, {2, 14002, 5169}, {2, 14790, 30803}, {2, 16387, 10257}, {2, 17522, 16067}, {2, 26252, 440}, {2, 26253, 21530}, {2, 26254, 18641}, {2, 26255, 381}, {2, 26256, 405}, {2, 26257, 6656}, {2, 26283, 11585}, {2, 26284, 13371}, {2, 32985, 11336}, {2, 34098, 37988}, {2, 35973, 25985}, {2, 36007, 37330}, {2, 36166, 3154}, {2, 36168, 14120}, {2, 37444, 30802}, {2, 37760, 23}, {2, 37777, 37981}, {2, 37897, 46517}, {2, 37907, 7426}, {2, 37909, 10989}, {2, 37913, 31101}, {2, 37962, 403}, {2, 37980, 2072}, {2, 37990, 11548}, {2, 38282, 37453}, {2, 40132, 11284}, {2, 44210, 43957}, {2, 46586, 867}, {2, 46589, 3143}, {3, 5, 34664}, {3, 235, 1885}, {3, 2070, 45171}, {3, 3542, 235}, {3, 7493, 44210}, {3, 30739, 43957}, {3, 37928, 37929}, {3, 37933, 186}, {3, 37973, 37928}, {4, 24, 37458}, {4, 25, 10301}, {4, 186, 10295}, {4, 403, 37984}, {4, 3147, 35486}, {4, 3523, 3516}, {4, 3524, 35483}, {4, 4232, 25}, {4, 5094, 427}, {4, 6353, 4232}, {4, 10018, 140}, {4, 10301, 428}, {4, 17506, 35491}, {4, 32534, 550}, {4, 35486, 3}, {4, 35487, 3850}, {4, 37458, 3575}, {4, 37460, 37196}, {4, 37984, 10151}, {5, 24, 3575}, {5, 1658, 12605}, {5, 3575, 23047}, {5, 10020, 7542}, {5, 34351, 3}, {5, 37458, 4}, {5, 38322, 546}, {5, 44212, 1995}, {10, 11363, 12135}, {20, 6622, 37197}, {20, 16051, 31152}, {20, 37952, 44280}, {22, 1368, 7667}, {22, 16063, 550}, {23, 186, 21284}, {23, 5159, 46517}, {23, 5189, 37900}, {23, 7426, 37897}, {23, 10989, 20063}, {23, 21284, 37969}, {23, 30745, 5189}, {23, 37760, 7426}, {23, 37777, 25}, {23, 37897, 37904}, {23, 37900, 37910}, {23, 37907, 37760}, {23, 37929, 37928}, {23, 37962, 37777}, {23, 37977, 186}, {23, 37978, 22}, {23, 37980, 37972}, {23, 40916, 37950}, {24, 186, 7575}, {24, 1995, 25}, {24, 3147, 34351}, {24, 7505, 5}, {24, 16868, 38322}, {25, 186, 37969}, {25, 427, 428}, {25, 5064, 6995}, {25, 5094, 4}, {25, 21284, 23}, {25, 30739, 1885}, {25, 32216, 44438}, {25, 37453, 2}, {25, 37454, 23047}, {25, 37920, 21284}, {27, 4213, 430}, {28, 451, 429}, {28, 7438, 25}, {29, 3144, 407}, {69, 19118, 46444}, {107, 14165, 6530}, {110, 19128, 34397}, {140, 6676, 7495}, {140, 7495, 7499}, {140, 10096, 25338}, {140, 13383, 16618}, {140, 15122, 10257}, {140, 16618, 3}, {140, 21841, 4}, {154, 26958, 1899}, {184, 13567, 11245}, {186, 419, 7473}, {186, 3542, 11799}, {186, 6353, 7426}, {186, 10295, 37934}, {186, 18533, 44265}, {186, 36191, 5112}, {186, 37777, 23}, {186, 37937, 237}, {186, 37942, 10151}, {186, 37943, 403}, {186, 37951, 2070}, {186, 37952, 15750}, {186, 37962, 25}, {186, 37970, 15646}, {186, 37977, 37920}, {186, 44893, 36189}, {186, 46619, 36180}, {230, 232, 16318}, {230, 1990, 6103}, {230, 2493, 16310}, {230, 10418, 16317}, {230, 16308, 16306}, {230, 16321, 16325}, {230, 16335, 16313}, {232, 3291, 14580}, {232, 6103, 1990}, {235, 30739, 427}, {235, 34664, 23047}, {237, 852, 44886}, {237, 44887, 441}, {242, 17923, 2969}, {297, 419, 460}, {376, 6623, 44438}, {403, 858, 37981}, {403, 1594, 45181}, {403, 7426, 25}, {403, 7473, 460}, {403, 7575, 3575}, {403, 10295, 4}, {403, 11799, 235}, {403, 37931, 13473}, {403, 37935, 37931}, {403, 37943, 37942}, {403, 37969, 428}, {403, 37989, 37982}, {403, 44214, 44281}, {406, 4185, 1904}, {406, 7521, 4185}, {415, 423, 422}, {419, 420, 297}, {419, 450, 421}, {419, 470, 462}, {419, 471, 463}, {427, 10301, 4}, {450, 41203, 297}, {451, 6353, 7438}, {462, 463, 460}, {470, 471, 297}, {548, 44226, 18560}, {549, 1596, 378}, {550, 1368, 16063}, {550, 11563, 44267}, {550, 16063, 7667}, {550, 44960, 4}, {597, 41585, 8541}, {631, 3089, 1593}, {632, 1595, 37119}, {851, 46554, 1375}, {851, 46555, 33305}, {852, 44888, 441}, {852, 44891, 44888}, {858, 7426, 23}, {858, 37760, 37897}, {858, 37897, 37899}, {858, 37900, 5189}, {858, 37981, 427}, {860, 37168, 1884}, {1113, 1114, 25}, {1113, 1312, 20405}, {1114, 1313, 20406}, {1304, 16080, 17986}, {1312, 1313, 427}, {1368, 10154, 22}, {1375, 33305, 851}, {1593, 3089, 1906}, {1594, 3518, 6756}, {1594, 14940, 3628}, {1598, 3526, 3541}, {1598, 3541, 1907}, {1614, 26879, 18914}, {1656, 2070, 7574}, {1656, 3517, 4}, {1656, 7574, 2072}, {1990, 6103, 16318}, {1995, 6639, 37439}, {1995, 10020, 7499}, {2045, 2046, 7395}, {2070, 5020, 37980}, {2070, 37972, 23}, {2072, 44907, 10257}, {2074, 37965, 37908}, {2211, 3231, 35325}, {2409, 7417, 25}, {2454, 2455, 40856}, {2479, 2480, 40890}, {2493, 11062, 232}, {3090, 7487, 7507}, {3147, 3542, 3}, {3147, 6353, 7493}, {3233, 16221, 16933}, {3291, 14580, 44467}, {3292, 32225, 41586}, {3515, 37196, 37460}, {3518, 14940, 1594}, {3530, 13488, 3520}, {3535, 5200, 32588}, {3542, 7493, 25}, {3542, 34351, 3575}, {3542, 35486, 4}, {3549, 6642, 7399}, {3575, 10297, 13473}, {3575, 34664, 1885}, {3575, 37454, 427}, {3628, 6756, 1594}, {3628, 23410, 5}, {3628, 34577, 7568}, {4207, 7490, 1889}, {4231, 4233, 25}, {4232, 5094, 10301}, {4232, 37934, 37904}, {4239, 30733, 25}, {4244, 7458, 25}, {5002, 5003, 16386}, {5004, 5005, 2071}, {5020, 7506, 1995}, {5064, 8889, 427}, {5095, 44102, 15471}, {5133, 7533, 3850}, {5142, 17562, 37398}, {5159, 6676, 16977}, {5159, 7426, 37899}, {5159, 16977, 10257}, {5159, 37760, 37904}, {5159, 37777, 10151}, {5159, 37911, 2}, {5189, 30745, 858}, {5640, 14389, 18583}, {5642, 41586, 3292}, {5972, 41674, 41673}, {6143, 34484, 15559}, {6240, 16868, 546}, {6240, 38322, 3575}, {6353, 7505, 1995}, {6353, 37453, 427}, {6353, 37943, 37962}, {6353, 38282, 2}, {6622, 37197, 45004}, {6623, 32216, 427}, {6639, 7506, 5}, {6640, 7517, 23335}, {6644, 10201, 15760}, {6676, 6677, 2}, {6676, 10096, 7426}, {6676, 13383, 7493}, {6676, 44232, 1995}, {6756, 45181, 10151}, {6776, 35260, 26864}, {6776, 37643, 26869}, {6995, 8889, 5064}, {7412, 7537, 37368}, {7418, 46587, 25}, {7423, 7435, 25}, {7426, 30745, 37910}, {7426, 37911, 46517}, {7426, 44214, 44210}, {7426, 44234, 7499}, {7493, 16238, 7499}, {7493, 37201, 10565}, {7519, 31133, 3627}, {7542, 10297, 10257}, {7570, 37990, 35018}, {7575, 15646, 1658}, {7575, 44234, 7542}, {7575, 44272, 24}, {7575, 44282, 5}, {7714, 35489, 37901}, {8105, 8106, 44467}, {8889, 13619, 10989}, {9707, 18912, 31804}, {9909, 21974, 2}, {9909, 30771, 1370}, {10020, 10096, 7575}, {10020, 13383, 34351}, {10020, 23410, 7568}, {10020, 44232, 5}, {10024, 45735, 31833}, {10096, 16238, 11799}, {10096, 34577, 44264}, {10096, 44452, 37971}, {10096, 44900, 44452}, {10128, 11548, 37990}, {10151, 10297, 23047}, {10192, 13567, 184}, {10295, 37934, 37931}, {10416, 15398, 16092}, {10594, 37119, 1595}, {11548, 35018, 7570}, {11563, 16532, 15646}, {11799, 44214, 3}, {11799, 44281, 1885}, {12362, 44277, 7488}, {13160, 44802, 9825}, {13371, 37440, 7553}, {13383, 16238, 3}, {13394, 37648, 182}, {13884, 13937, 6}, {14070, 18531, 44239}, {14807, 14808, 44442}, {14813, 14814, 11585}, {15078, 44440, 44241}, {15122, 44452, 140}, {15154, 15155, 44454}, {15646, 16532, 16531}, {15646, 44267, 550}, {15750, 37197, 20}, {15760, 44211, 6644}, {15765, 18585, 44218}, {16051, 37197, 427}, {16066, 17555, 37226}, {16080, 17986, 12079}, {16238, 16618, 140}, {16238, 34351, 7542}, {16272, 16304, 16322}, {16272, 16305, 16304}, {16272, 16323, 16309}, {16303, 16312, 16326}, {16303, 16321, 16312}, {16303, 16331, 16327}, {16306, 16308, 16303}, {16306, 16316, 16327}, {16306, 16321, 16313}, {16306, 16335, 16331}, {16308, 16313, 16327}, {16308, 16321, 16331}, {16308, 16335, 16316}, {16309, 16323, 16305}, {16313, 16314, 16303}, {16313, 16316, 16331}, {16313, 16331, 16312}, {16314, 16316, 16308}, {16314, 16321, 16327}, {16315, 16329, 16303}, {16316, 16325, 16312}, {16317, 16318, 44467}, {16323, 16332, 16304}, {16325, 16331, 16313}, {16387, 44452, 7499}, {16619, 25338, 37971}, {16619, 44452, 15122}, {16868, 44879, 6240}, {17506, 35491, 33923}, {18325, 37928, 37900}, {18533, 26255, 25}, {18560, 21844, 548}, {18560, 44958, 44226}, {18570, 44270, 46030}, {18571, 22249, 18579}, {18580, 31861, 44218}, {18580, 44275, 31861}, {20063, 37909, 23}, {20405, 20406, 428}, {20850, 34609, 7500}, {21284, 37920, 186}, {21841, 37934, 16619}, {21844, 44958, 18560}, {23410, 31830, 13163}, {23712, 23713, 1990}, {23712, 23715, 23713}, {23713, 23714, 23712}, {23714, 23715, 1990}, {24007, 24008, 6103}, {25338, 44234, 140}, {26864, 26869, 6776}, {26882, 26917, 34224}, {30739, 44210, 3}, {30745, 37910, 46517}, {31510, 36166, 25}, {31681, 31682, 5066}, {31726, 37955, 44246}, {32225, 44102, 12828}, {32460, 37974, 37975}, {32461, 37975, 37974}, {34477, 46030, 18570}, {34577, 44232, 23410}, {35259, 37638, 1352}, {35260, 37643, 6776}, {35471, 35488, 3627}, {35472, 35481, 8703}, {35479, 35488, 35471}, {35490, 35503, 15704}, {35491, 44959, 4}, {36168, 46619, 25}, {37165, 46549, 46552}, {37453, 37907, 37969}, {37453, 37981, 37911}, {37760, 37911, 37899}, {37777, 37920, 37969}, {37777, 37977, 21284}, {37897, 37911, 858}, {37899, 37904, 23}, {37899, 37981, 13473}, {37900, 37910, 37899}, {37904, 37931, 37969}, {37904, 46517, 37899}, {37907, 37962, 6353}, {37910, 37911, 30745}, {37917, 37954, 186}, {37928, 37973, 23}, {37933, 37943, 235}, {37934, 37942, 37984}, {37935, 37943, 10151}, {37935, 37962, 37904}, {37935, 37984, 37934}, {37937, 44893, 403}, {37942, 37977, 46517}, {37943, 37977, 37981}, {37951, 37980, 25}, {37951, 44911, 10151}, {37962, 37977, 23}, {42789, 42790, 7464}, {42807, 42808, 7526}, {44145, 46106, 2970}, {44234, 44264, 7568}, {44272, 44282, 403}, {44886, 44889, 852}, {44886, 44890, 237}, {44887, 44892, 44889}, {44888, 44892, 852}, {44889, 44890, 44886}, {44891, 44892, 441}, {45994, 45995, 3575}, {46484, 46548, 46553}, {46554, 46555, 851}, {46586, 46588, 25}, {46589, 46592, 25}


X(469) = EULER LINE INTERCEPT OF LINE X(92)X(264)

Trilinears       u csc 2A : v csc 2B : w csc 2C, where
                        u : v : w = X(386); e.g., u(a,b,c) = a(b2 + c2 + bc + ca + ab)

Barycentrics  u sec A : v sec B : w sec C

As a point on the Euler line, X(469) has Shinagawa coefficients (F,E + F + $bc$).

X(469) lies on these lines: 2,3   92,264   226,273

X(469) is the {X(2),X(4)}-harmonic conjugate of X(27). For a list of other harmonic conjugates of X(469), click Tables at the top of this page.

X(469) = inverse-in-orthocentroidal-circle of X(27)


X(470) = X(15)-CROSS CONJUGATE OF X(298)

Trilinears       sin(A + π/3) csc 2A : sin(B + π/3) csc 2B : sin(C + π/3) csc 2C
Barycentrics  sin(A + π/3) sec A : sin(B + π/3) sec B : sin(C + π/3) sec 2C
Barycentrics    tan A + sqrt(3) : :

As a point on the Euler line, X(470) has Shinagawa coefficients (31/2F, S).

X(470) lies on these lines: 2,3   18,275   264,301   298,340   302,317   343,634   394,633

X(470) = isogonal conjugate of X(36296)
X(470) = complement of X(19772)
X(470) = cevapoint of X(i) and X(j) for these {i,j}: {15, 8739}, {6111, 23714}, {6137, 30465}
X(470) = crosspoint of X(2) and X(19774)
X(470) = crosssum of X(6) and X(11243)
X(470) = trilinear pole of line X(6110)X(6782)
X(470) = orthocentroidal-circle-inverse of X(471)
X(470) = polar-circle-inverse of X(32461)
X(470) = X(15)-cross conjugate of X(298)
X(470) = X(4)-Hirst inverse of X(471)
X(470) = X(63)-isoconjugate of X(3457)
X(470) = pole wrt polar circle of trilinear polar of X(13) (line X(395)X(523))
X(470) = polar conjugate of X(13)
X(470) = {X(2),X(4)}-harmonic conjugate of X(471)


X(471) = X(16)-CROSS CONJUGATE OF X(299)

Trilinears       sin(A - π/3) csc 2A : sin(B - π/3) csc 2B : sin(C - π/3) csc 2C
Barycentrics  sin(A - π/3) sec A : sin(B - π/3) sec B : sin(C - π/3) sec 2C
Barycentrics    tan A - sqrt(3) : :

As a point on the Euler line, X(471) has Shinagawa coefficients (31/2F, -S).

X(471) lies on these lines: 2,3   17,275   264,300   299,340   303,317   343,633   394,634

X(471) = isogonal conjugate of X(36297)
X(471) = complement of X(19773)
X(471) = cevapoint of X(i) and X(j) for these {i,j}: {16, 8740}, {6110, 23715}, {6138, 30468}
X(471) = crosspoint of X(2) and X(19775)
X(471) = crosssum of X(6) and X(11244)
X(471) = trilinear pole of line X(6111)X(6783)
X(471) = inverse-in-orthocentroidal-circle of X(470)
X(471) = X(16)-cross conjugate of X(299)
X(471) = X(4)-Hirst inverse of X(470)
X(471) = pole wrt polar circle of trilinear polar of X(14) (line X(396)X(523))
X(471) = polar conjugate of X(14)
X(471) = polar-circle inverse of X(32460)
X(471) = X(63)-iaoconjugate of X(3458)
X(471) = {X(2),X(4)}-harmonic conjugate of X(470)


X(472) = X(62)-CROSS CONJUGATE OF X(303)

Trilinears       cos(A + π/3) csc 2A : cos(B + π/3) csc 2B : cos(C + π/3) csc 2C
Barycentrics  cos(A + π/3) sec A : cos(B + π/3) sec B : cos(C + π/3) sec 2C
Barycentrics    sqrt(3) tan A - 1 : :

As a point on the Euler line, X(472) has Shinagawa coefficients (F, -31/2S).

X(472) lies on these lines: 2,3   13,275   53,395   264,298   299,317   343,621   394,622

X(472) = isogonal conjugate of X(32586)
X(472) = orthocentroidal-circle-inverse of X(473)
X(472) = anticomplement of X(466)
X(472) = X(62)-cross conjugate of X(303)
X(472) = polar conjugate of X(18)
X(472) = {X(2),X(4)}-harmonic conjugate of X(473)


X(473) = X(61)-CROSS CONJUGATE OF X(302)

Trilinears       cos(A - π/3) csc 2A : cos(B - π/3) csc 2B : cos(C - π/3) csc 2C
Barycentrics  cos(A - π/3) sec A : cos(B - π/3) sec B : cos(C - π/3) sec 2C
Barycentrics    sqrt(3) tan A + 1 : :

As a point on the Euler line, X(473) has Shinagawa coefficients (F, 31/2S).

X(473) lies on these lines: 2,3   14,275   53,396   264,299   298,317   343,622   394,621

X(473) = isogonal conjugate of X(32585)
X(473) = orthocentroidal-circle-inverse of X(472)
X(473) = anticomplement of X(465)
X(473) = X(61)-cross conjugate of X(302)
X(473) = polar conjugate of X(17)
X(473) = {X(2),X(4)}-harmonic conjugate of X(472)


X(474) = EULER LINE INTERCEPT OF LINE X(10)X(56)

Trilinears       cos A - (a + b + c)/a : cos B - (a + b + c)/b : cos C - (a + b + c)/c
Trilinears       bcS - raSA : caS - rbSB : abS - rcSC)
Barycentrics  a cos A - (a + b + c) : b cos B - (a + b + c) : c cos C - (a + b + c)

As a point on the Euler line, X(474) has Shinagawa coefficients (abc$a$ - S2, S2).

X(474) lies on these lines: 2,3   8,999   10,56   35,1001   36,958   40,392   46,960   57,72   65,997   78,942   142,954   171,978   183,274   244,976   283,582   386,940   579,965   986,1054

X(474) = complement of X(2478)
X(474) = anticomplement of X(17527)


X(475) = EULER LINE INTERCEPT OF LINE X(10)X(34)

Trilinears       sec A - (a + b + c)/a : sec B - (a + b + c)/b : sec C - (a + b + c)/c
Barycentrics  a sec A - (a + b + c) : b sec B - (a + b + c) : c sec C - (a + b + c)

As a point on the Euler line, X(475) has Shinagawa coefficients ($a$F,-abc).

X(475) lies on these lines: 2,3   8,1063   10,34   264,274   318,1068

X(475) = inverse-in-orthocentroidal-circle of X(406)
X(475) = polar conjugate of isogonal conjugate of X(36743)


X(476) = TIXIER POINT

Trilinears    1/[(1 + 2 cos 2A) sin(B - C)] : :
Barycentrics    1/{(b^2 - c^2)[(a^2 - b^2 - c^2)^2 - b^2c^2]} : :

The reflection of X(110) in the Euler line; X(476) is on the circumcircle. (Michel Tixier, 5/9/98). Also, X(476) is the center of the polar conic of X(30) with respect to the Neuberg cubic; this conic is a rectangular hyperbola passing through the incenter, the excenters, and X(30). (Peter Yff, 5/23/99)

Let La, Lb, Lc be the lines through A, B, C, respectively parallel to the Euler line. Let Ma, Mb, Mc be the reflections of BC, CA, AB in La, Lb, Lc, respectively. Let A' = Mb∩Mc, and define B' and C' cyclically. The triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. The centroid of A'B'C' is X(476); see Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, November 18, 2015)

Let La, Lb, Lc be the lines through A, B, C, respectively parallel to the orthic axis. Let Ma, Mb, Mc be the reflections of BC, CA, AB in La, Lb, Lc, respectively. Let A' = Mb∩Mc, and define B' and C' cyclically. The triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. The centroid of A'B'C' is X(476); see Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, November 18, 2015)

X(476) is the center of the polar conic of X(30) with respect to the Neuberg cubic, which passes through the following points: incenter, excenters, X(30), X(5), X(523), and the excentral-isogonal conjugates of X(5663) and X(7724). (Randy Hutson, November 18, 2015)

If you have The Geometer's Sketchpad, you can view X(476).

X(476) lies on the Hutson-Parry circle and these lines: {2, 842}, {3, 477}, {4, 16221}, {5, 14979}, {13, 2379}, {14, 2378}, {20, 2693}, {21, 2687}, {22, 2697}, {23, 94}, {24, 15112}, {30, 74}, {99, 850}, {100, 4036}, {101, 4024}, {102, 7424}, {103, 5196}, {104, 1325}, {105, 7469}, {106, 7478}, {107, 7480}, {109, 13486}, {110, 523}, {111, 230}, {112, 2501}, {125, 17511}, {143, 15907}, {146, 1553}, {183, 2857}, {186, 1300}, {250, 933}, {328, 2373}, {376, 841}, {403, 1299}, {468, 3563}, {511, 9161}, {512, 9160}, {524, 9184}, {526, 16170}, {549, 11749}, {655, 15439}, {691, 4226}, {729, 11060}, {755, 14902}, {759, 2166}, {804, 20404}, {843, 6792}, {858, 1297}, {915, 2074}, {917, 2073}, {930, 23181}, {935, 4230}, {953, 3109}, {1141, 2070}, {1290, 3658}, {1292, 7475}, {1294, 2071}, {1296, 7472}, {1298, 19167}, {1302, 1632}, {1304, 4240}, {1316, 2698}, {1379, 13722}, {1380, 13636}, {1494, 9141}, {1637, 23588}, {1648, 14846}, {1789, 2716}, {1995, 2453}, {2367, 20573}, {2395, 2715}, {2407, 10420}, {2409, 10423}, {2410, 9060}, {2452, 11422}, {2688, 4184}, {2689, 7450}, {2690, 4243}, {2691, 4236}, {2694, 16049}, {2695, 4225}, {2696, 11634}, {2706, 2979}, {2752, 4228}, {2758, 7419}, {2766, 4246}, {2799, 17708}, {2858, 23342}, {2966, 9979}, {3060, 16978}, {3153, 18401}, {3154, 15059}, {3260, 15295}, {3448, 6070}, {4235, 10098}, {4238, 10101}, {4608, 6578}, {5468, 10425}, {5663, 16169}, {5897, 16386}, {5899, 13597}, {5994, 20579}, {5995, 20578}, {6032, 9831}, {6325, 11594}, {6644, 15111}, {6757, 11101}, {6795, 15080}, {7575, 13530}, {8599, 11636}, {8705, 11593}, {9033, 15395}, {9070, 15455}, {9202, 14185}, {9203, 14187}, {10296, 18300}, {10412, 15329}, {10989, 14388}, {12077, 23357}, {12113, 18318}, {14934, 15035}, {15061, 16340}, {15168, 18359}, {15646, 20480}, {16089, 18817}, {18323, 18576}, {18403, 22751}.

X(476) = reflection of X(i) in X(j) for these (i,j): (146,1553), (477,3)
X(476) = isogonal conjugate of X(526)
X(476) = isotomic conjugate of X(3268)
X(476) = anticomplement of X(3258)
X(476) = cevapoint of X(30) and X(523)
X(476) = trilinear pole of Fermat axis
X(476) = Λ(trilinear polar of X(i)) for these i: 15, 16, 186
X(476) = Ψ(X(15), X(2))
X(476) = X(1577)-isoconjugate of X(50)
X(476) = Ψ(X(54), X(5))
X(476) = Thomson-isogonal conjugate of X(5663)
X(476) = Lucas-isogonal conjugate of X(5663)
X(476) = intersection of antipedal lines of X(74) and X(110)
X(476) = polar conjugate of isogonal conjugate of X(32662)


X(477) = TIXIER ANTIPODE

Trilinears    1/[4 cos A + cos(A - 2B) + cos(A - 2C) - 3 cos(B - C)] : :
Barycentrics    (a^8 + a^6*b^2 - 4*a^4*b^4 + a^2*b^6 + b^8 - 3*a^6*c^2 + 2*a^4*b^2*c^2 + 2*a^2*b^4*c^2 - 3*b^6*c^2 + 3*a^4*c^4 - 2*a^2*b^2*c^4 + 3*b^4*c^4 - a^2*c^6 - b^2*c^6)*(a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6 + a^6*c^2 + 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 - b^6*c^2 - 4*a^4*c^4 + 2*a^2*b^2*c^4 + 3*b^4*c^4 + a^2*c^6 - 3*b^2*c^6 + c^8)
Barycentrics    1/(a^6 (b^2 + c^2) - a^4 (3 b^4 - 2 b^2 c^2 + 3 c^4) + a^2 (3 b^6 - 2 b^4 c^2 - 2 b^2 c^4 + 3 c^6) - (b^2 - c^2)^2 (b^4 + 3 b^2 c^2 + c^4)) : :

X(477) lies on these lines: {2, 9060}, {3, 476}, {4, 1304}, {5, 16166}, {20, 10420}, {22, 16167}, {23, 1302}, {24, 22239}, {30, 110}, {50, 112}, {74, 523}, {99, 3260}, {107, 186}, {109, 15228}, {111, 5915}, {125, 5627}, {140, 18319}, {265, 14851}, {376, 691}, {378, 935}, {403, 1301}, {468, 9064}, {511, 9160}, {512, 9161}, {526, 16169}, {550, 1291}, {631, 22104}, {759, 23226}, {842, 2411}, {925, 2071}, {930, 18859}, {933, 13619}, {1138, 12244}, {1141, 23286}, {1290, 3651}, {1300, 15470}, {1499, 9184}, {2222, 22342}, {2452, 2713}, {2549, 2715}, {2687, 7429}, {2688, 7440}, {2689, 7421}, {2690, 7430}, {2695, 7454}, {2752, 7425}, {2758, 7444}, {2766, 7414}, {2770, 7418}, {2782, 20404}, {3154, 14644}, {3233, 15034}, {5473, 9203}, {5474, 9202}, {5663, 14480}, {5994, 15743}, {5995, 11586}, {6080, 12290}, {6236, 11594}, {7417, 10102}, {7471, 15035}, {9781, 12052}, {10423, 18533}, {10990, 14536}, {12030, 14127}, {12084, 15112}, {13398, 16386}, {14094, 14611}, {15396, 17511}

X(477) = reflection of X(476) in X(3)
X(477) = isogonal conjugate of X(5663)
X(477) = isotomic conjugate of X(35520)
X(477) = complement of X(34193)
X(477) = trilinear pole, wrt circumcevian triangle of X(30), of line X(3)X(2453)
X(477) = reflection of X(74) in the Euler line
X(477) = Λ(X(3), X(74))
X(477) = Λ(X(40), X(2940))
X(477) = Thomson-isogonal conjugate of X(526)
X(477) = Lucas-isogonal conjugate of X(526)
X(477) = Cundy-Parry Phi transform of X(14254)
X(477) = Cundy-Parry Psi transform of X(14385)
X(477) = trilinear pole of line X(6)X(1637)
X(477) = polar conjugate of isogonal conjugate of X(32663)
X(477) = reflection of X(32111) in the orthic axis
X(477) = trilinear pole, wrt circumorthic triangle, of line X(25)X(98)


X(478) = CENTER OF YIU CONIC

Trilinears   a[a4 - 2abc(b + c - a) - (b2 - c2)2]/(b + c - a) : :
Barycentrics    a^2*(a + b - c)*(a - b + c)*(a^4 - b^4 + 2*a^2*b*c - 2*a*b^2*c - 2*a*b*c^2 + 2*b^2*c^2 - c^4)

Center of the Yiu conic, which passes through the points outside the circumcircle at which the excircles of ABC are tangent to the sidelines of ABC. See Paul Yiu's

The Clawson point and excircles.

Let Ea be the ellipse through A having foci B and C, and define Eb and Ec cyclically. The minor vertices, A1, A2; B1, B2; C1, C2 are the touchpoints of the excircle with sidelines external to the circumcircle; i.e., they lie on the Yiu conic, centered at X(478). See Figure. If ABC is a right triangle, then the Yiu conic is a pair of lines meeting in X(478). Figure. (Dan Reznik, December 13, 2021)

X(478) lies on these lines: {2, 8048}, {6, 19}, {9, 1038}, {12, 20029}, {37, 1455}, {56, 5019}, {69, 651}, {73, 2268}, {109, 573}, {154, 3195}, {198, 577}, {219, 4559}, {222, 226}, {223, 10319}, {388, 2298}, {572, 10571}, {603, 1400}, {604, 1457}, {1035, 18591}, {1122, 1407}, {1211, 5783}, {1413, 1903}, {1470, 2277}, {1766, 21147}, {2092, 11509}, {2122, 5750}, {2183, 2199}, {3142, 3330}, {3157, 10441}, {3596, 6648}, {3713, 9370}, {8231, 13388}, {8750, 18621}, {11496, 14749}.

X(478) = isogonal conjugate of X(34277)
X(478) = complement of X(8048)
X(478) = crosspoint of X(2) and X(3436)
X(478) = crosssum of X(i) and X(j) for these (i,j): {6, 3435}, {650, 2968}
X(478) = crossdifference of every pair of points on line X(521)X(14312)
. X(478) = center of the perspeconic of every pair of these triangles: ABC, 2nd extouch, 3rd extouch
X(478) = crosssum of circumcircle intercepts of excircles radical circle
X(478) = trilinear product X(i)*X(j) for these {i,j}: {6, 21147}, {7, 205}, {34, 22132}, {48, 14257}, {56, 1766}, {57, 197}, {63, 17408}, {109, 6588}, {604, 3436}, {1397, 20928}, {1400, 16049}, {1408, 21074}, {1415, 21186}


X(479) = X(269)-CROSS CONJUGATE OF X(279)

Trilinears    (tan A/2 sec A/2)2 : :
Barycentrics    tan3(A/2) : :
Barycentrics    1/(b + c - a)3 : :

Let A' be the point in which the incircle is tangent to a circle that passes through vertices B and C, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(479). For an excircle version, see X(5423).

Clark Kimberling and Peter Yff, Problem 10678, American Mathematical Monthly 105 (1998) 666.

If you have The Geometer's Sketchpad, you can view X(479).

X(479) lies on these lines: {2,5574}, {7,354}, {8,7182}, {55,3160}, {57,279}, {165,1323}, {269,614}, {348,5273}, {658,5435}, {934,1617}, {1014,5324}, {1119,1851}, {1407,1462}, {1996,5226}, {3188,6060}, {4554,8055}

X(479) = isogonal conjugate of X(480)
X(479) = isotomic conjugate of X(5423)
X(479) = X(269)-cross conjugate of X(279)
X(479) = anticomplement of X(5574)
X(479) = crosssum of X(3022) and X(4105)
X(479) = cevapoint of X(i) and X(j) for these {i,j}: {269,738}, {3271,3669}
X(479) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1088,7056,7), (3160,3599,55)
X(479) = X(i)-cross conguate of X(j) for these (i,j): (269,279), (3271,3669), (5573,2), (7195,7)
X(479) = polar conjugate of isotomic conjugate of X(30682)
X(479) = X(i)-isoconjugate of X(j) for these {i,j}: {1,480}, {2,6602}, {6,728}, {8,1253}, {9,220}, {31,5423}, {33,1260}, {41,346}, {55,200}, {78,7071}, {100,4105}, {101,4130}, {210,2328}, {212,7046}, {219,7079}, {281,1802}, {282,7368}, {284,4515}, {341,2175}, {607,3692}, {643,4524}, {644,657}, {663,4578}, {692,4163}, {756,6061}, {765,3022}, {1098,7064}, {1110,4081}, {1252,3119}, {1265,2212}, {1334,2287}, {2194,4082}, {2310,6065}, {2318,4183}, {2324,7367}, {2332,3694}, {3063,6558}, {3699,8641}, {3709,7259}, {3900,3939}, {4012,7084}, {4171,5546}, {6605,8012}


X(480) = X(200)-CEVA CONJUGATE OF X(220)

Trilinears     (cot A/2 cos A/2)2 : :
Trilinears     a(b + c - a)3 : b(c + a - b)3 : c(a + b - c)3
Barycentrics   (sin A)(cot A/2 cos A/2)2 : :

X(480) = radical center of the three circles used to construct X(57). (Peter Yff, 5/6/98)

X(480) lies on these lines: 8,344   9,55   10,954   56,78   100,144

X(480) = isogonal conjugate of X(479)
X(480) = X(200)-Ceva conjugate of X(220)
X(480) = crosssum of X(269) and X(738)
X(480) = trilinear square of X(6726)


X(481) = 1st EPPSTEIN POINT

Trilinears    1 - 2 sec A/2 cos B/2 cos C/2 : :
Trilinears    1 - 4(area)/[a(b + c - a)] : :      (E. Brisse, 3/20/01)
X(481) = s*X(1) - (r + 4R)*X(7)
X(481) = [A]/Ra + [B]/Rb + [C]/Rc - X(1)/r, where Ra, Rb, Rc = radii of Soddy circles
X(481) = [A]/Ra + [B]/Rb + [C]/Rc + X(175)/Rs', where Ra, Rb, Rc = radii of Soddy circles, and Rs' = radius of outer Soddy circle

Let S be the inner Soddy circle and SA, SB, SC the Soddy circles tangent to S. Let Ia = S∩SA, Ea = SB∩SC, and determine Ib, Ic, Eb, Ec cyclically. Then X(481) is the point of concurrence of lines IA-to-EA, IB-to-EB, IC-to-EC.

David Eppstein, "Tangent spheres and triangle centers,"American Mathematical Monthly, 108 (2001) 63-66.

Let S' be the outer Soddy center, X(175). X(481) is the point of concurrence of the Soddy lines of BS'C, CS'A, and AS'B. (Randy Hutson, September 14, 2016)

Let La be the line tangent to the outer Soddy circle at the touchpoint with the A-Soddy circle. Define Lb and Lc cyclically. Let A' = Lb∩Lc and define B' and C'cyclically. Triangle A'B'C', here introduced as the outer Soddy tangential triangle, is perspective to ABC, and the perspector is X(481). (Randy Hutson, September 14, 2016)

X(481) lies on these lines: 1,7   174,1127   226,485

X(481) = X(79)-Ceva conjugate of X(482)
X(481) = Kosnita(X(175),X(1)) point; see X(54)
X(481) = Kosnita(X(175),X(7)) point
X(481) = Kosnita(X(176),X(175)) point
X(481) = {X(7),X(176)}-harmonic conjugate of X(1373)
X(481) = 2nd-outer-Soddy-isogonal conjugate of X(32058)


X(482) = 2nd EPPSTEIN POINT

Trilinears    1 + 2 sec A/2 cos B/2 cos C/2 : :
Trilinears    1 + 4(area)/[a(b + c - a)] : :       (E. Brisse, 3/20/01)
Barycentrics    a + (a + b + c) tan(A/2) : :
X(482) = s*X(1) + (r + 4R)*X(7)
X(482) = [A]/Ra + [B]/Rb + [C]/Rc + X(1)/r, where Ra, Rb, Rc = radii of Soddy circles
X(482) = [A]/Ra + [B]/Rb + [C]/Rc + X(176)/Rs, where Ra, Rb, Rc = radii of Soddy circles, and Rs = radius of inner Soddy circle

Let S' be the outer Soddy circle and SA, SB, SC the Soddy circles tangent to S. Let JA = S'∩SA, EA = SB∩SC, and determine JB, JC, EB, EC cyclically. Then X(482) is the point of concurrence of lines JA-to-EA, JB-to-EB, JC-to-EC.

David Eppstein, "Tangent spheres and triangle centers," American Mathematical Monthly, 108 (2001) 63-66.

Let S be the inner Soddy center, X(176). X(482) is the point of concurrence of the Soddy lines of BSC, CSA, and ASB. (Randy Hutson, September 14, 2016)

Let Ia, Ib, Ic be the centers of the Elkies companion incircles. Let A' be the trilinear product Ib*Ic, and define B' and C'cyclically. The lines AA', BB', CC' concur in X(482). The lines IaA', IbB', IcC' concur in X(176). (Randy Hutson, September 14, 2016)

Let La be the line tangent to the inner Soddy circle at the touchpoint with the A-Soddy circle. Define Lb and Lc cyclically. Let A' = Lb∩Lc and define B' and C' cyclically. Triangle A'B'C', here introduced as the inner Soddy tangential triangle, is perspective to ABC, and the perspector is X(482). (Randy Hutson, September 14, 2016)

X(482) lies on these lines: 1,7   226,486

X(482) = X(79)-Ceva conjugate of X(481)
X(482) = trilinear product of centers of Elkies companion incircles
X(482) = Kosnita(X(176),X(7)) point
X(482) = Kosnita(X(176),X(176)) point
X(482) = {X(7),X(175)}-harmonic conjugate of X(1374)
X(482) = 2nd-inner-Soddy-isogonal conjugate of X(32057)
X(482) = perspector of ABC and the intangents triangle of the three Elkies companion incircles; see X(176)


X(483) = RADICAL CENTER OF AJIMA-MALFATTI CIRCLES

Trilinears    sec2A/4 : sec2B/4 : sec2C/4
Trilinears    1/(1 + cos A/2) : 1/(1 + cos B/2) : 1/(1 + cos C/2)
Barycentrics   sin A sec2A/4 : sin B sec2B/4 : sin C sec2C/4

The Ajima-Malfatti circles are described at X(179). (Peter Yff, 6/1/98)

Let (Ia) be the incircle of BCI, where I=X(1), and define (Ib) and (Ic) cyclically. Let Ea be the touchpoint of (Ia) and BC, and define Eb and Ec cyclically. Let Sa be the insimilicenter of (Ib) and (Ic), and define Sb and Sc cyclically. The lines EaSa, EbSb, EcSc concur in X(483). Let La be the internal tangent, other than AI, of (Ib) and (Ic). Define Lb and Lc cyclically. Then La, Lb, Lc concur in X(483). (Randy Hutson, January 29, 2018)

If you have The Geometer's Sketchpad, you can view X(483).

X(483) lies on these lines: 8,178   173,180   174,175

X(483) = X(372)-of-BCI-triangle
X(483) = BCI-isogonal conjugate of X(1)


X(484) = 1st EVANS PERSPECTOR

Trilinears       1 + 2(cos A - cos B - cos C) : 1 + 2(cos B - cos C - cos A) : 1 + 2(cos C - cos A - cos B)
Trilinears    a^3 + a^2 (b + c) - a (b^2 + b c + c^2) - (b - c)^2 (b + c) : :
Barycentrics  a[1 + 2(cos A - cos B - cos C)] : b[1 + 2(cos B - cos C - cos A)] : c[1 + 2(cos C - cos A - cos B)]

X(484) is the perspector of the excentral triangle and the triangle A'B'C', where A' is the reflection of vertex A in sideline BC and B', C' are determined cyclically. (Lawrence Evans, 10/22/98)

X(484) lies on the Neuberg cubic and these lines: 1,3   4,3483   10,191   12,79   13,1277   14,1276   30,80   63,535   74,3465   100,758   499,962   759,901   1046,1048   1138,3464   3466,3484

X(484) = midpoint of X(36) and X(3245)
X(484) = reflection of X(i) in X(j) for these (i,j): (1,36), (36,1155)
X(484) = isogonal conjugate of X(3065)
X(484) = inverse-in-circumcircle of X(35)
X(484) = inverse-in-Bevan-circle of X(1)
X(484) = Conway-circle-inverse of X(35631)
X(484) = X(80)-Ceva conjugate of X(1)
X(484) = crossdifference of every pair of points on line X(650)X(1100)
X(484) = {X(1)X(46)}-harmonic conjugate of X(3336)
X(484) = inverse-in-incircle of X(5045)
X(484) = perspector of ABC and the reflection of the incentral triangle in the antiorthic axis (the reflection of the cevian triangle of X(1) in the trilinear polar of X(1))
X(484) = reflection of X(36) in the antiorthic axis
X(484) = X(30)-Ceva conjugate of X(3465)
X(484) = X(186)-of-excentral-triangle
X(484) = {X(1),X(3)}-harmonic conjugate of X(37616)
X(484) = endo-homothetic center of Ehrmann vertex-triangle and dual of orthic triangle; the homothetic center of is X(3153)

leftri

Centers X(485)-X(495)

rightri
Centers X(485)-X(495), and also X(371) and X(372), are Vierkanen in een driehoek: triangle centers associated with squares.

X(485) = VECTEN POINT

Trilinears    sec(A - π/4) : sec(B - π/4) : sec(C -π/4)
Trilinears    1/(sin A + cos A) : 1/(sin B + cos B) : 1/(sin C + cos C)
Trilinears    sin A + cos(B - C) : sin B + cos(C - A) : sin C + cos(A - B)
Barycentrics  sin A sec(A - π/4) : sin B sec(B - π/4) : sin C sec(C - π/4)
Barycentrics    1/(SA + S) : :
X(485) = X[7981] + 2 X[9907], X[7981] - 4 X[12269], X[9907] + 2 X[12269], 3 X[2] - 4 X[6118], 3 X[2] + X[12222], 2 X[2] + X[22485], X[488] - 4 X[6118], 2 X[488] + 3 X[22485], 2 X[641] + X[12222], 4 X[641] + 3 X[22485], 4 X[6118] + X[12222], 8 X[6118] + 3 X[22485], 2 X[6311] - 3 X[9768], 2 X[12222] - 3 X[22485], 3 X[35822] - X[35832], 3 X[3] + X[22810], X[12124] + 2 X[12602], 3 X[12124] + 2 X[22810], X[12124] - 4 X[49104], 3 X[12602] - X[22810], X[12602] + 2 X[49104], 2 X[13882] + X[22646], X[22810] + 6 X[49104], X[32495] + 2 X[32497], X[4] + 2 X[48735], 2 X[6250] + X[12257], 2 X[13879] + X[32499], X[35821] - 3 X[35831], 4 X[5] - X[6278], 4 X[5] + X[6279], 2 X[5] + X[49318], X[6278] + 2 X[49318], X[6279] + 2 X[6289], 2 X[7584] - 3 X[49221], 4 X[13881] - X[22591], 2 X[13881] + X[22592], 5 X[19102] - 4 X[19116], 3 X[19103] - 4 X[19117], X[22591] + 2 X[22592], X[49552] + 2 X[49622], 2 X[13876] + X[22630], 2 X[13875] + X[22632], X[22628] + 2 X[22925], X[22626] + 2 X[22880], 2 X[1131] + X[45076], 2 X[1151] + 3 X[1327], X[1151] - 3 X[13846], and many others

Erect a square outwardly from each side of triangle ABC. Let A'B'C' be the triangle formed by the respective centers of the squares. The lines AA', BB', CC' concur in X(485). For details, visit Floor van Lamoen's site, Vierkanten in een driehoek: 1. Omgeschreven vierkanten (van Lamoen, 4/26/98) and his article "Friendship Among Triangle Centers," Forum Geometricorum, 1 (2001) 1-6. See also Paul Yiu's papers, "Squares Erected on the Sides of a Triangle", and "On the Squares Erected Externally on the Sides of a Triangle".

If you have The Geometer's Sketchpad, you can view Vecten Point.

X(485) lies on the Kiepert circumhyperbola, the cubics K006, K070b, K120, K122, K168, K250, K336, K424a, K690, K906, K1192, K1195, K1197, K1202, K1203, K1225, the curve Q115, and thewe lines: {1, 7981}, {2, 372}, {3, 590}, {4, 371}, {5, 6}, {8, 35641}, {10, 1686}, {11, 1124}, {12, 1335}, {13, 2043}, {14, 2044}, {15, 2041}, {16, 2042}, {17, 2045}, {18, 2046}, {20, 1131}, {22, 35776}, {25, 9922}, {30, 1151}, {32, 9987}, {39, 22717}, {40, 13893}, {46, 3377}, {55, 9646}, {56, 9661}, {61, 3367}, {62, 3392}, {64, 22838}, {69, 639}, {76, 491}, {83, 7388}, {95, 19200}, {96, 26920}, {98, 6811}, {100, 35772}, {110, 49222}, {113, 46688}, {115, 1504}, {119, 19048}, {140, 1152}, {141, 11313}, {145, 35810}, {146, 35826}, {147, 35824}, {148, 35878}, {149, 35882}, {153, 35856}, {182, 12230}, {194, 35866}, {226, 481}, {230, 6423}, {235, 3092}, {262, 3102}, {265, 49268}, {275, 1586}, {355, 7969}, {376, 14241}, {381, 1328}, {382, 6221}, {387, 36691}, {388, 35768}, {394, 1591}, {395, 18586}, {396, 18587}, {402, 12800}, {427, 3093}, {459, 3535}, {487, 2996}, {489, 671}, {490, 2459}, {492, 7752}, {493, 11209}, {494, 13005}, {495, 3298}, {496, 3297}, {497, 35802}, {498, 5414}, {499, 6502}, {511, 8992}, {515, 8983}, {516, 13912}, {517, 13911}, {524, 9974}, {542, 13662}, {546, 3592}, {547, 6432}, {548, 6411}, {549, 6410}, {550, 6409}, {607, 16033}, {608, 16027}, {615, 1656}, {616, 35753}, {617, 35850}, {627, 35848}, {628, 35846}, {631, 3316}, {632, 6426}, {637, 32489}, {642, 5491}, {847, 41516}, {925, 13520}, {944, 13902}, {946, 13883}, {952, 44635}, {962, 35610}, {1038, 19474}, {1040, 12911}, {1132, 1588}, {1270, 35794}, {1271, 35792}, {1322, 5962}, {1329, 1377}, {1346, 2466}, {1347, 2465}, {1348, 1668}, {1349, 1669}, {1368, 1578}, {1378, 2886}, {1478, 2067}, {1479, 2066}, {1482, 49232}, {1503, 13910}, {1505, 1506}, {1579, 6823}, {1585, 2052}, {1592, 10601}, {1614, 9677}, {1657, 6449}, {1659, 8953}, {1666, 2040}, {1667, 2039}, {1670, 2009}, {1671, 2010}, {1676, 1688}, {1677, 1687}, {1685, 2051}, {1689, 5404}, {1690, 5403}, {1698, 1703}, {1699, 1702}, {1737, 2362}, {1916, 33341}, {1991, 42023}, {1993, 15234}, {2048, 13332}, {2351, 3155}, {2353, 45429}, {2460, 14238}, {2549, 48773}, {2560, 2567}, {2561, 2566}, {2777, 8994}, {2781, 32303}, {2782, 32470}, {2794, 8980}, {2800, 8988}, {2829, 13913}, {2888, 12965}, {2896, 35782}, {2975, 35784}, {3054, 8376}, {3069, 3090}, {3085, 35809}, {3086, 31408}, {3089, 35765}, {3095, 49252}, {3146, 6453}, {3299, 7741}, {3301, 7951}, {3371, 3374}, {3372, 3373}, {3385, 3387}, {3386, 3388}, {3424, 5871}, {3434, 35796}, {3436, 35798}, {3448, 12375}, {3522, 43407}, {3523, 3590}, {3524, 42637}, {3525, 6454}, {3526, 6398}, {3528, 51910}, {3529, 42638}, {3530, 6412}, {3533, 43411}, {3534, 6455}, {3541, 11474}, {3542, 5413}, {3543, 43408}, {3545, 7582}, {3547, 11514}, {3549, 10898}, {3575, 13884}, {3589, 11314}, {3591, 5056}, {3594, 3628}, {3614, 19027}, {3616, 35762}, {3618, 39875}, {3627, 6425}, {3634, 13975}, {3648, 35854}, {3652, 49242}, {3815, 6421}, {3817, 49548}, {3830, 41945}, {3832, 23259}, {3839, 23263}, {3843, 6199}, {3845, 43563}, {3850, 6431}, {3851, 6417}, {3853, 6437}, {3854, 43433}, {3855, 23273}, {3858, 6470}, {3859, 6441}, {3917, 21654}, {4187, 31473}, {4240, 35790}, {5020, 8277}, {5050, 49229}, {5054, 6450}, {5055, 6418}, {5058, 5475}, {5059, 6484}, {5062, 7746}, {5066, 43341}, {5067, 32786}, {5070, 6395}, {5071, 13939}, {5072, 6427}, {5073, 6407}, {5076, 6447}, {5079, 6428}, {5217, 31499}, {5254, 6422}, {5286, 31403}, {5318, 42189}, {5321, 42190}, {5392, 11091}, {5409, 6504}, {5410, 7507}, {5417, 13428}, {5422, 15233}, {5462, 12240}, {5466, 14333}, {5476, 44501}, {5480, 36655}, {5552, 45642}, {5587, 18991}, {5591, 7376}, {5597, 12486}, {5598, 12487}, {5601, 35778}, {5602, 35780}, {5603, 19066}, {5613, 49210}, {5617, 49208}, {5657, 35611}, {5663, 13915}, {5691, 9583}, {5790, 49233}, {5818, 19065}, {5840, 13922}, {5870, 8975}, {5878, 49250}, {5886, 7968}, {5901, 44636}, {5921, 39893}, {5972, 10820}, {6033, 49212}, {6146, 19355}, {6193, 35836}, {6194, 35838}, {6201, 7000}, {6212, 7347}, {6222, 13749}, {6223, 35844}, {6224, 35852}, {6225, 35864}, {6251, 22617}, {6256, 45652}, {6259, 49234}, {6265, 49240}, {6284, 13901}, {6287, 49254}, {6288, 49256}, {6321, 49266}, {6408, 15694}, {6424, 7745}, {6430, 11539}, {6438, 16239}, {6445, 17800}, {6446, 43881}, {6451, 15696}, {6456, 15720}, {6463, 35806}, {6468, 43337}, {6471, 15699}, {6472, 43795}, {6476, 9693}, {6478, 50690}, {6480, 33703}, {6481, 43315}, {6485, 15702}, {6486, 11001}, {6497, 15693}, {6500, 19709}, {6501, 45385}, {6519, 43789}, {6568, 19056}, {6569, 35946}, {6639, 18459}, {6643, 11513}, {6667, 13977}, {6721, 13989}, {6722, 13967}, {6723, 13969}, {6748, 26868}, {6776, 14229}, {6809, 40448}, {6810, 13599}, {6812, 13380}, {6814, 45300}, {7173, 19029}, {7354, 18965}, {7375, 18841}, {7386, 12321}, {7387, 9683}, {7392, 8855}, {7395, 19006}, {7401, 10961}, {7484, 12170}, {7486, 13941}, {7505, 10881}, {7529, 44599}, {7612, 9758}, {7618, 49787}, {7694, 13748}, {7697, 49230}, {7728, 49216}, {7736, 45513}, {7737, 12963}, {7747, 9675}, {7756, 9674}, {7765, 31465}, {7787, 35766}, {7795, 45473}, {7988, 19003}, {7989, 19004}, {7998, 12275}, {7999, 12286}, {8200, 44600}, {8207, 44602}, {8220, 44627}, {8221, 44629}, {8227, 18992}, {8591, 35698}, {8681, 9823}, {8703, 43211}, {8724, 49214}, {8781, 13926}, {8901, 16029}, {8909, 12231}, {8940, 8946}, {8941, 8947}, {8991, 15311}, {8993, 29012}, {8995, 18400}, {8997, 9738}, {8998, 10819}, {9542, 43507}, {9612, 51841}, {9614, 31432}, {9616, 41869}, {9647, 12943}, {9660, 12953}, {9669, 31474}, {9676, 34148}, {9690, 49134}, {9691, 15684}, {9692, 50692}, {9733, 49028}, {9833, 10533}, {9834, 13890}, {9835, 13891}, {9838, 13899}, {9839, 13900}, {9873, 13892}, {9874, 35862}, {9956, 13973}, {9970, 49264}, {9975, 45860}, {9996, 44604}, {10008, 42060}, {10104, 44587}, {10124, 42569}, {10137, 35400}, {10175, 13936}, {10201, 11266}, {10303, 43511}, {10319, 12663}, {10356, 19012}, {10358, 18994}, {10527, 45640}, {10528, 35816}, {10529, 35818}, {10531, 45643}, {10532, 45641}, {10590, 35801}, {10591, 35803}, {10594, 35777}, {10595, 35811}, {10596, 35817}, {10597, 35819}, {10598, 35797}, {10599, 35799}, {10653, 15765}, {10654, 18585}, {10738, 48714}, {10742, 48700}, {10749, 49270}, {10796, 44586}, {10895, 18996}, {10896, 19038}, {10897, 18531}, {10942, 44643}, {10943, 44645}, {11061, 35876}, {11178, 44474}, {11265, 18569}, {11292, 12323}, {11293, 45509}, {11315, 45871}, {11316, 15294}, {11417, 37444}, {11462, 25739}, {11484, 13943}, {11488, 42173}, {11489, 42171}, {11499, 44590}, {11500, 13887}, {11511, 12598}, {11515, 12982}, {11516, 12983}, {11548, 34516}, {11824, 21737}, {11836, 40556}, {12047, 16232}, {12103, 42568}, {12110, 13885}, {12113, 13894}, {12114, 13895}, {12115, 13906}, {12116, 13907}, {12239, 13754}, {12322, 32419}, {12383, 35834}, {12384, 35828}, {12699, 49226}, {12849, 35870}, {12856, 49248}, {12900, 13990}, {12918, 49218}, {12919, 49258}, {12964, 14216}, {13219, 35880}, {13333, 43531}, {13403, 43863}, {13429, 51833}, {13644, 14230}, {13663, 37809}, {13669, 35306}, {13674, 13848}, {13678, 35872}, {13692, 49260}, {13712, 15293}, {13798, 35874}, {13812, 49262}, {13835, 41491}, {13916, 41023}, {13917, 41022}, {13921, 14244}, {13979, 15088}, {13986, 32396}, {13993, 35018}, {14061, 19055}, {14228, 14242}, {14231, 45463}, {14234, 45406}, {14269, 43504}, {14568, 45420}, {14639, 19109}, {14643, 19052}, {14644, 19111}, {14651, 35825}, {14813, 42149}, {14814, 42152}, {14853, 35841}, {14912, 39894}, {15022, 42604}, {15059, 19059}, {15061, 49217}, {15235, 17825}, {15236, 17811}, {15561, 49267}, {15682, 42413}, {15687, 52047}, {15688, 43209}, {15691, 51850}, {15692, 43256}, {15701, 42418}, {15709, 43505}, {15883, 38425}, {16267, 36468}, {16268, 36450}, {16626, 49238}, {16627, 49236}, {16772, 42197}, {16773, 42198}, {16964, 51728}, {17538, 42414}, {18404, 18457}, {18440, 49228}, {18553, 44482}, {18926, 45607}, {18939, 44518}, {19050, 26470}, {19058, 23234}, {19074, 36765}, {19081, 31272}, {19087, 40686}, {19126, 19144}, {19130, 45543}, {19423, 19491}, {19710, 43434}, {21616, 30556}, {22235, 42233}, {22237, 42231}, {22238, 35738}, {22616, 48778}, {22647, 35860}, {22758, 44606}, {22793, 31439}, {22876, 44667}, {22921, 44666}, {22955, 49246}, {23515, 46689}, {24245, 34853}, {24842, 29243}, {25555, 44657}, {26361, 49024}, {26386, 44582}, {26394, 45357}, {26410, 44584}, {26418, 45359}, {26446, 49227}, {26466, 45597}, {26467, 45595}, {26468, 44594}, {26469, 44596}, {26494, 45601}, {26503, 45599}, {26516, 35945}, {26617, 33456}, {26620, 33365}, {30398, 30428}, {30427, 32734}, {31475, 37720}, {31981, 49352}, {32354, 35858}, {32379, 49244}, {32491, 45472}, {32492, 45863}, {32503, 32509}, {33346, 33348}, {33354, 33438}, {33357, 33436}, {33358, 33442}, {33360, 33440}, {33366, 33368}, {33370, 33372}, {34507, 44502}, {35731, 36437}, {36445, 36446}, {36463, 36464}, {36473, 36553}, {36474, 36549}, {36492, 36526}, {36495, 36585}, {36496, 36581}, {36658, 36990}, {36714, 45440}, {37640, 51853}, {37641, 51852}, {38224, 49213}, {38235, 40663}, {38752, 48715}, {40330, 45577}, {41950, 41964}, {41961, 49133}, {41965, 43785}, {42085, 42200}, {42086, 42202}, {42125, 42187}, {42128, 42188}, {42159, 42281}, {42162, 42280}, {42163, 42246}, {42166, 42247}, {42168, 42684}, {42170, 42685}, {42184, 42813}, {42186, 42814}, {42193, 42817}, {42194, 42818}, {42236, 51727}, {42523, 46936}, {43257, 50687}, {43413, 43794}, {43449, 45435}, {43521, 46333}, {43816, 43867}, {43883, 50688}, {44394, 45489}, {44595, 45515}, {45101, 45544}, {45107, 45441}, {45365, 48454}, {45368, 48455}, {45498, 49038}, {45605, 48469}, {45650, 48482}, {48477, 49027}

X(485) = midpoint of X(i) and X(j) for these {i,j}: {1, 9907}, {3, 12602}, {4, 12257}, {20, 12297}, {486, 22592}, {487, 2996}, {488, 12222}, {1151, 23251}, {3070, 32497}, {6250, 48735}, {6278, 6279}, {6289, 49318}, {13873, 50720}, {22644, 42260}, {26617, 33456}, {49157, 49158}
X(485) = reflection of X(i) in X(j) for these {i,j}: {1, 12269}, {3, 49104}, {4, 6250}, {486, 13881}, {488, 641}, {641, 6118}, {1151, 8981}, {6278, 6289}, {6279, 49318}, {6289, 5}, {6304, 33446}, {6305, 33444}, {6337, 642}, {7981, 1}, {8981, 13925}, {10819, 8998}, {12124, 3}, {12257, 48735}, {12788, 10}, {12800, 402}, {13134, 10068}, {13135, 10084}, {13912, 49618}, {19145, 13910}, {22502, 50720}, {22591, 486}, {22627, 33450}, {22629, 33448}, {22644, 23251}, {32495, 3070}, {36656, 45861}, {42009, 639}, {42260, 1151}, {44647, 7583}, {48660, 22625}, {49790, 590}
X(485) = isogonal conjugate of X(371)
X(485) = isotomic conjugate of X(492)
X(485) = complement of X(488)
X(485) = anticomplement of X(641)
X(485) = polar conjugate of X(1585)
X(485) = complement of the isogonal conjugate of X(8948)
X(485) = complement of the isotomic conjugate of X(24244)
X(485) = isogonal conjugate of the anticomplement of X(639)
X(485) = isogonal conjugate of the complement of X(637)
X(485) = isotomic conjugate of the anticomplement of X(590)
X(485) = isotomic conjugate of the isogonal conjugate of X(8577)
X(485) = isogonal conjugate of the isotomic conjugate of X(34391)
X(485) = isotomic conjugate of the polar conjugate of X(41515)
X(485) = polar conjugate of the isotomic conjugate of X(11090)
X(485) = polar conjugate of the isogonal conjugate of X(6413)
X(485) = psi-transform of X(13521)
X(485) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 24246}, {493, 18589}, {1973, 13882}, {8948, 10}, {19218, 141}, {24244, 2887}, {26454, 1214}
X(485) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 24246}, {847, 486}, {16032, 6413}, {34391, 11090}, {41515, 8944}
X(485) = X(i)-cross conjugate of X(j) for these (i,j): {3, 486}, {590, 2}, {3070, 4}, {5418, 10194}, {6413, 11090}, {6560, 1328}, {8577, 41515}, {8976, 10195}, {13665, 1327}, {26920, 1586}, {26951, 11091}, {32497, 5490}, {32568, 492}, {42259, 1132}, {42261, 43571}, {43879, 3316}, {45384, 43568}
X(485) = X(i)-isoconjugate of X(j) for these (i,j): {1, 371}, {19, 5408}, {31, 492}, {47, 486}, {48, 1585}, {63, 5413}, {92, 8911}, {372, 3378}, {560, 45805}, {605, 13457}, {1748, 6414}, {2180, 16037}, {8576, 44179}
X(485) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 492}, {2, 24246}, {3, 371}, {6, 5408}, {485, 488}, {486, 34853}, {1249, 1585}, {1599, 10962}, {1993, 10960}, {3162, 5413}, {5408, 9723}, {6374, 45805}, {8576, 37864}, {8577, 44196}, {8911, 22391}, {13428, 24245}, {13882, 42009}, {33364, 39387}
X(485) = cevapoint of X(i) and X(j) for these (i,j): {3, 10665}, {6, 3155}, {590, 8035}, {3385, 3386}, {6413, 8577}
X(485) = crosspoint of X(2) and X(24244)
X(485) = crosssum of X(i) and X(j) for these (i,j): {3, 8909}, {6, 10132}, {3311, 8826}
X(485) = trilinear pole of line {523, 17431}
X(485) = internal center of similitude of nine-point circle and 2nd Lemoine circle
X(485) = pole wrt polar circle of trilinear polar of X(1585)
X(485) = X(48)-isoconjugate (polar conjugate) of X(1585)
X(485) = trilinear product of vertices of outer Vecten triangle
X(485) = X(4)-of-outer-Vecten-triangle
X(485) = X(4)-of-X(2)-quadsquares-triangle
X(485) = perspector of ABC and inner-squares triangle
X(485) = perspector of 3rd tri-squares triangle and 3rd tri-squares central triangle
X(485) = Kosnita(X(485),X(485)) point
X(485) = 3rd-tri-squares-isogonal conjugate of X(32497)
X(485) = 4th-anti-tri-squares-isogonal conjugate of X(32495)
X(485) = {X(5),X(6)}-harmonic conjugate of X(486)
X(485) = outer-Vecten-isogonal conjugate of X(641)
X(485) = barycentric product X(i)*X(j) for these {i,j}: {4, 11090}, {5, 16032}, {6, 34391}, {68, 1586}, {69, 41515}, {76, 8577}, {85, 13455}, {264, 6413}, {372, 5392}, {486, 13439}, {491, 2165}, {847, 5409}, {850, 39383}, {2996, 8944}, {5412, 20563}, {11091, 13440}, {13441, 41516}, {18819, 45472}, {21463, 42023}, {24244, 24246}, {34392, 44192}
X(485) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 492}, {3, 5408}, {4, 1585}, {6, 371}, {25, 5413}, {68, 11091}, {76, 45805}, {96, 16037}, {184, 8911}, {371, 1599}, {372, 1993}, {486, 13428}, {491, 7763}, {590, 641}, {1123, 13457}, {1321, 3535}, {1504, 32568}, {1586, 317}, {2165, 486}, {2351, 6414}, {3068, 39387}, {3070, 8968}, {3155, 10962}, {5392, 34392}, {5409, 9723}, {5412, 24}, {6413, 3}, {6414, 10666}, {8035, 590}, {8576, 44193}, {8577, 6}, {8944, 193}, {10665, 5409}, {10880, 15207}, {11090, 69}, {11091, 13430}, {11473, 15214}, {13439, 491}, {13440, 1586}, {13455, 9}, {14593, 41516}, {16032, 95}, {21463, 45420}, {24246, 488}, {26454, 8950}, {26920, 1147}, {32734, 39384}, {34391, 76}, {35764, 15188}, {39383, 110}, {41515, 4}, {41516, 13429}, {44192, 372}, {45472, 42009}
X(485) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 49157, 13135}, {1, 49158, 13134}, {2, 372, 5420}, {2, 488, 641}, {2, 1587, 372}, {2, 12222, 488}, {2, 13439, 11090}, {2, 43133, 45508}, {3, 590, 5418}, {3, 3070, 6560}, {3, 6560, 42261}, {3, 8276, 9682}, {3, 8976, 590}, {3, 13665, 3070}, {3, 13882, 49790}, {3, 13889, 8276}, {3, 45384, 8976}, {4, 371, 6561}, {4, 3068, 371}, {4, 6459, 35821}, {4, 6561, 22615}, {4, 6564, 42269}, {4, 13440, 41515}, {4, 13886, 3068}, {4, 31412, 6564}, {5, 6, 486}, {5, 486, 42274}, {5, 1352, 10515}, {5, 5875, 6214}, {5, 6215, 10516}, {5, 6290, 10514}, {5, 7583, 6}, {5, 7584, 42262}, {5, 18538, 42265}, {5, 19116, 18762}, {5, 19117, 7584}, {5, 42265, 42277}, {5, 49318, 6278}, {5, 49356, 1352}, {6, 13881, 49221}, {6, 18538, 42277}, {6, 42262, 7584}, {6, 42265, 5}, {6, 42277, 42274}, {6, 44647, 19103}, {6, 49221, 19102}, {11, 19028, 1124}, {12, 19030, 1335}, {15, 3391, 2041}, {16, 3366, 2042}, {20, 1131, 23249}, {20, 8972, 9540}, {20, 9540, 6200}, {20, 23249, 35820}, {20, 35812, 9680}, {20, 35820, 42276}, and many others


X(486) = INNER VECTEN POINT

Trilinears    sec(A + π/4) : sec(B + π/4) : sec(C + π/4)
Trilinears    1/(sin A - cos A) : 1/(sin B - cos B) : 1/(sin C - cos C) : :
Trilinears       sin A - cos(B - C) : sin B - cos(C - A) : sin C - cos(A - B) (Peter J. C. Moses, 8/22/03)
Barycentrics    sin A sec(A + π/4) : sin B sec(B + π/4) : sin C sec(C + π/4)
Barycentrics    1/(SA - S) : :
X(486) = X[7980] + 2 X[9906], X[7980] - 4 X[12268], X[9906] + 2 X[12268], 3 X[2] - 4 X[6119], 3 X[2] + X[12221], 2 X[2] + X[22484], X[487] - 4 X[6119], 2 X[487] + 3 X[22484], 2 X[642] + X[12221], 4 X[642] + 3 X[22484], 4 X[6119] + X[12221], 8 X[6119] + 3 X[22484], 2 X[6315] - 3 X[9767], 2 X[12221] - 3 X[22484], 3 X[35823] - X[35833], 3 X[3] + X[22809], X[12123] + 2 X[12601], 3 X[12123] + 2 X[22809], X[12123] - 4 X[49103], 3 X[12601] - X[22809], X[12601] + 2 X[49103], 2 X[13934] + X[22617], X[22809] + 6 X[49103], X[32492] + 2 X[32494], X[4] + 2 X[48734], 2 X[6251] + X[12256], 2 X[13933] + X[32498], X[35820] - 3 X[35830], 4 X[5] + X[6280], 4 X[5] - X[6281], 2 X[5] + X[49317], X[6280] + 2 X[6290], X[6281] + 2 X[49317], 2 X[7583] - 3 X[49220], 2 X[13881] + X[22591], 4 X[13881] - X[22592], 3 X[19104] - 4 X[19116], 5 X[19105] - 4 X[19117], 2 X[22591] + X[22592], X[49551] + 2 X[49623], 2 X[13929] + X[22601], 2 X[13928] + X[22603], X[22599] + 2 X[22926], X[22597] + 2 X[22881], 2 X[1132] + X[45077], 2 X[1152] + 3 X[1328], X[1152] - 3 X[13847], and many others

X(486) is a perspector of triangles associated with squares that circumscribe ABC. For details and references, see X(485). (Floor van Lamoen, 4/26/98)

If you have The Geometer's Sketchpad, you can view Inner Vecten Point.

X(486) lies on the Kiepert circumhyperbola, the cubics K006, K070a, K120, K122, K168, K250, K336, K424b, K690, K906, K1192, K1195, K1197, K1202, K1203, K1225, the curve Q115, and these lines: {1, 7980}, {2, 371}, {3, 615}, {4, 372}, {5, 6}, {8, 35642}, {10, 1685}, {11, 1335}, {12, 1124}, {13, 2044}, {14, 2043}, {15, 2042}, {16, 2041}, {17, 2046}, {18, 2045}, {20, 1132}, {22, 35777}, {25, 9921}, {30, 1152}, {32, 9986}, {39, 22719}, {40, 13947}, {46, 3378}, {55, 12343}, {56, 13955}, {61, 3366}, {62, 3391}, {64, 22839}, {69, 640}, {76, 492}, {83, 7389}, {95, 19199}, {96, 8911}, {98, 6813}, {100, 35773}, {110, 49223}, {113, 46689}, {115, 1505}, {119, 19047}, {140, 1151}, {141, 11314}, {145, 35811}, {146, 35827}, {147, 35825}, {148, 35879}, {149, 35883}, {153, 35857}, {182, 12229}, {194, 35867}, {226, 482}, {230, 6424}, {235, 3093}, {262, 3103}, {265, 49269}, {275, 1585}, {355, 7968}, {376, 14226}, {381, 1327}, {382, 6398}, {387, 36690}, {388, 35769}, {394, 1592}, {395, 18587}, {396, 18586}, {402, 12799}, {427, 3092}, {442, 31473}, {459, 3536}, {488, 2996}, {489, 2460}, {490, 671}, {491, 7752}, {493, 13002}, {494, 11210}, {495, 3297}, {496, 3298}, {497, 35803}, {498, 2066}, {499, 2067}, {511, 12237}, {515, 13971}, {516, 13975}, {517, 13973}, {524, 9975}, {542, 13782}, {546, 3594}, {547, 6431}, {548, 6412}, {549, 6409}, {550, 6410}, {590, 1656}, {591, 42024}, {607, 16027}, {608, 16033}, {616, 35754}, {617, 35851}, {627, 35847}, {628, 35849}, {631, 3317}, {632, 6425}, {638, 32488}, {641, 5490}, {847, 41515}, {925, 13521}, {944, 13959}, {946, 13936}, {952, 44636}, {962, 35611}, {1038, 19473}, {1040, 9631}, {1131, 1587}, {1270, 35793}, {1271, 35795}, {1321, 5962}, {1329, 1378}, {1346, 2465}, {1347, 2466}, {1348, 1669}, {1349, 1668}, {1368, 1579}, {1377, 2886}, {1478, 6502}, {1479, 5414}, {1482, 49233}, {1503, 13972}, {1504, 1506}, {1578, 6823}, {1586, 2052}, {1591, 10601}, {1657, 6450}, {1659, 8978}, {1666, 2039}, {1667, 2040}, {1670, 2010}, {1671, 2009}, {1676, 1687}, {1677, 1688}, {1686, 2048}, {1689, 5403}, {1690, 5404}, {1698, 1702}, {1699, 1703}, {1737, 16232}, {1916, 33340}, {1993, 15233}, {2047, 13332}, {2351, 3156}, {2353, 45428}, {2362, 12047}, {2459, 14234}, {2549, 48772}, {2560, 2566}, {2561, 2567}, {2777, 13969}, {2781, 32304}, {2782, 32471}, {2794, 13967}, {2800, 13976}, {2829, 13977}, {2888, 12971}, {2896, 35783}, {2975, 35785}, {3035, 9679}, {3054, 8375}, {3068, 3090}, {3085, 35808}, {3086, 35768}, {3089, 35764}, {3095, 49253}, {3146, 6454}, {3299, 7951}, {3301, 7741}, {3371, 3373}, {3372, 3374}, {3385, 3388}, {3386, 3387}, {3424, 5870}, {3434, 35797}, {3436, 35799}, {3448, 12376}, {3522, 43408}, {3523, 3591}, {3524, 42638}, {3525, 6453}, {3526, 6221}, {3528, 51911}, {3529, 42637}, {3530, 6411}, {3533, 43412}, {3534, 6456}, {3541, 11473}, {3542, 5412}, {3543, 43407}, {3545, 7581}, {3547, 11513}, {3549, 10897}, {3575, 13937}, {3589, 11313}, {3590, 5056}, {3592, 3628}, {3614, 19028}, {3616, 35763}, {3618, 39876}, {3624, 9583}, {3627, 6426}, {3634, 13912}, {3648, 35855}, {3652, 49243}, {3815, 6422}, {3817, 49547}, {3830, 41946}, {3832, 23249}, {3839, 23253}, {3843, 6395}, {3845, 43562}, {3850, 6432}, {3851, 6418}, {3853, 6438}, {3854, 43432}, {3855, 23267}, {3858, 6471}, {3859, 6442}, {3917, 21653}, {4240, 35791}, {4999, 9678}, {5012, 9677}, {5020, 8276}, {5050, 49228}, {5054, 6449}, {5055, 6417}, {5058, 7746}, {5059, 6485}, {5062, 5475}, {5066, 43340}, {5067, 32785}, {5070, 6199}, {5071, 13886}, {5072, 6428}, {5073, 6408}, {5076, 6448}, {5079, 6427}, {5204, 9647}, {5217, 9660}, {5254, 6421}, {5286, 45513}, {5318, 42187}, {5321, 42188}, {5392, 11090}, {5408, 6504}, {5411, 7507}, {5419, 13439}, {5422, 15234}, {5462, 12239}, {5466, 14334}, {5476, 44502}, {5480, 36656}, {5552, 45643}, {5587, 18992}, {5590, 7375}, {5597, 12484}, {5598, 12485}, {5601, 35781}, {5602, 35779}, {5603, 19065}, {5613, 49211}, {5617, 49209}, {5657, 35610}, {5663, 13979}, {5691, 13942}, {5705, 31438}, {5790, 49232}, {5818, 19066}, {5840, 13991}, {5871, 13949}, {5878, 49251}, {5886, 7969}, {5901, 44635}, {5921, 39894}, {5972, 10819}, {6033, 49213}, {6146, 19356}, {6193, 35837}, {6194, 35839}, {6202, 7374}, {6213, 7348}, {6223, 35845}, {6224, 35853}, {6225, 35865}, {6250, 22646}, {6256, 45653}, {6259, 49235}, {6265, 49241}, {6284, 13958}, {6287, 49255}, {6288, 49257}, {6321, 49267}, {6399, 13748}, {6407, 15694}, {6423, 7745}, {6429, 11539}, {6436, 31414}, {6437, 16239}, {6445, 43882}, {6446, 17800}, {6452, 15696}, {6455, 15720}, {6462, 35807}, {6469, 43336}, {6470, 15699}, {6473, 43796}, {6476, 9692}, {6479, 50690}, {6480, 43314}, {6481, 33703}, {6484, 15702}, {6487, 11001}, {6496, 15693}, {6500, 45384}, {6501, 19709}, {6522, 43790}, {6568, 35947}, {6569, 19055}, {6639, 18457}, {6642, 9682}, {6643, 11514}, {6667, 13913}, {6721, 8997}, {6722, 8980}, {6723, 8994}, {6776, 14244}, {6809, 8954}, {6810, 40448}, {6812, 45300}, {6814, 13380}, {7173, 19030}, {7354, 18966}, {7376, 18841}, {7386, 12320}, {7392, 8854}, {7395, 19005}, {7401, 10963}, {7484, 12169}, {7486, 8972}, {7505, 10880}, {7529, 44598}, {7603, 31481}, {7612, 9757}, {7618, 49786}, {7694, 13749}, {7697, 49231}, {7728, 49217}, {7736, 45512}, {7737, 12968}, {7749, 9675}, {7787, 35767}, {7795, 45472}, {7988, 19004}, {7989, 19003}, {7998, 12274}, {7999, 12285}, {8200, 44601}, {8207, 44603}, {8220, 44628}, {8221, 44630}, {8227, 18991}, {8591, 35699}, {8681, 9824}, {8703, 43212}, {8724, 49215}, {8781, 13873}, {8901, 16034}, {8909, 9820}, {8944, 8948}, {8945, 8949}, {8953, 30324}, {8968, 11427}, {8970, 14826}, {8995, 32396}, {8998, 12900}, {9612, 51842}, {9616, 31423}, {9632, 37696}, {9646, 19038}, {9661, 18996}, {9691, 43523}, {9732, 49029}, {9739, 12124}, {9833, 10534}, {9834, 13944}, {9835, 13945}, {9838, 13956}, {9839, 13957}, {9873, 13946}, {9874, 35863}, {9956, 13911}, {9970, 49265}, {9974, 45861}, {9996, 44605}, {10008, 42009}, {10104, 44586}, {10124, 42568}, {10138, 35400}, {10175, 13883}, {10201, 11265}, {10303, 43512}, {10319, 12662}, {10356, 19011}, {10358, 18993}, {10527, 45641}, {10528, 35817}, {10529, 35819}, {10531, 45642}, {10532, 45640}, {10590, 31408}, {10591, 35802}, {10594, 35776}, {10595, 35810}, {10596, 35816}, {10597, 35818}, {10598, 35796}, {10599, 35798}, {10653, 18585}, {10654, 15765}, {10738, 48715}, {10742, 48701}, {10749, 49271}, {10796, 44587}, {10820, 13990}, {10895, 18995}, {10896, 19037}, {10898, 18531}, {10942, 44644}, {10943, 44646}, {11061, 35877}, {11178, 44473}, {11231, 31439}, {11266, 18569}, {11291, 12322}, {11294, 45508}, {11315, 15293}, {11316, 45872}, {11418, 37444}, {11463, 25739}, {11484, 13889}, {11488, 42174}, {11489, 42172}, {11499, 44591}, {11500, 13940}, {11511, 12597}, {11515, 12980}, {11516, 12981}, {11548, 34515}, {11835, 40556}, {12103, 42569}, {12110, 13938}, {12113, 13948}, {12114, 13952}, {12115, 13964}, {12116, 13965}, {12232, 44665}, {12240, 13754}, {12323, 32421}, {12383, 35835}, {12384, 35829}, {12699, 49227}, {12849, 35871}, {12856, 49249}, {12918, 49219}, {12919, 49259}, {12970, 14216}, {13219, 35881}, {13333, 13478}, {13390, 40149}, {13403, 43864}, {13440, 51833}, {13678, 35873}, {13692, 49261}, {13712, 41490}, {13763, 14233}, {13783, 37809}, {13789, 35305}, {13794, 13988}, {13798, 35875}, {13812, 49263}, {13835, 15294}, {13880, 14229}, {13915, 15088}, {13925, 35018}, {13980, 15311}, {13981, 41023}, {13982, 41022}, {13984, 29012}, {13986, 18400}, {14061, 19056}, {14227, 14243}, {14238, 45407}, {14245, 45462}, {14269, 43503}, {14568, 45421}, {14639, 19108}, {14643, 19051}, {14644, 19110}, {14651, 35824}, {14813, 42152}, {14814, 42149}, {14853, 35840}, {14912, 39893}, {15022, 42605}, {15059, 19060}, {15061, 49216}, {15235, 17811}, {15236, 17825}, {15561, 49266}, {15682, 42414}, {15684, 43209}, {15687, 52048}, {15688, 43210}, {15691, 51849}, {15692, 43257}, {15701, 42417}, {15709, 43506}, {15884, 38426}, {16267, 36449}, {16268, 36467}, {16626, 49239}, {16627, 49237}, {16772, 42195}, {16773, 42196}, {17538, 42413}, {18404, 18459}, {18440, 49229}, {18553, 44481}, {18927, 45608}, {18940, 44518}, {19049, 26470}, {19057, 23234}, {19073, 36765}, {19082, 31272}, {19088, 40686}, {19126, 19143}, {19130, 45542}, {19422, 19490}, {19710, 43435}, {21616, 30557}, {21736, 45102}, {21737, 45552}, {22235, 42234}, {22236, 35738}, {22237, 42232}, {22645, 48779}, {22647, 35861}, {22758, 44607}, {22877, 44667}, {22922, 44666}, {22955, 49247}, {23515, 46688}, {24246, 34853}, {24843, 29243}, {25555, 44656}, {25639, 31484}, {26362, 49025}, {26363, 31453}, {26386, 44583}, {26394, 45360}, {26410, 44585}, {26418, 45358}, {26446, 49226}, {26466, 45596}, {26467, 45598}, {26468, 44595}, {26469, 44597}, {26494, 45600}, {26503, 45602}, {26521, 35944}, {26618, 33457}, {26619, 33364}, {30399, 30427}, {30428, 32734}, {31403, 31404}, {31413, 31418}, {31427, 31428}, {31432, 31434}, {31437, 31441}, {31459, 31460}, {31464, 31466}, {31471, 31476}, {31474, 31479}, {31482, 31488}, {31485, 31493}, {31981, 49351}, {32354, 35859}, {32379, 49245}, {32490, 45473}, {32495, 45862}, {32502, 32508}, {32807, 39387}, {33347, 33349}, {33355, 33437}, {33356, 33439}, {33359, 33441}, {33361, 33443}, {33367, 33369}, {33371, 33373}, {34507, 44501}, {36445, 36447}, {36463, 36465}, {36473, 36552}, {36474, 36550}, {36491, 36526}, {36495, 36584}, {36496, 36582}, {36657, 36990}, {36709, 45441}, {37640, 51855}, {37641, 51854}, {38224, 49212}, {38752, 48714}, {40330, 45576}, {41949, 41963}, {41962, 49133}, {41966, 43786}, {42085, 42199}, {42086, 42201}, {42125, 42189}, {42128, 42190}, {42159, 42280}, {42162, 42281}, {42163, 42248}, {42166, 42249}, {42167, 42684}, {42169, 42685}, {42183, 42813}, {42185, 42814}, {42191, 42817}, {42192, 42818}, {42522, 46936}, {42992, 50245}, {43256, 50687}, {43414, 43793}, {43415, 49134}, {43449, 45434}, {43508, 43524}, {43522, 46333}, {43816, 43868}, {43884, 50688}, {44392, 45488}, {44596, 45514}, {45106, 45440}, {45366, 48454}, {45367, 48455}, {45499, 49039}, {45606, 48468}, {45651, 48482}, {48476, 49026}

X(486) = midpoint of X(i) and X(j) for these {i,j}: {1, 9906}, {3, 12601}, {4, 12256}, {20, 12296}, {485, 22591}, {487, 12221}, {488, 2996}, {1152, 23261}, {3071, 32494}, {6251, 48734}, {6280, 6281}, {6290, 49317}, {13926, 50719}, {22615, 42261}, {26618, 33457}, {49155, 49156}
X(486) = reflection of X(i) in X(j) for these {i,j}: {1, 12268}, {3, 49103}, {4, 6251}, {485, 13881}, {487, 642}, {642, 6119}, {1152, 13966}, {6280, 49317}, {6281, 6290}, {6290, 5}, {6300, 33447}, {6301, 33445}, {6337, 641}, {7980, 1}, {10820, 13990}, {12123, 3}, {12256, 48734}, {12787, 10}, {12799, 402}, {13132, 10067}, {13133, 10083}, {13966, 13993}, {13975, 49619}, {19146, 13972}, {22501, 50719}, {22592, 485}, {22598, 33451}, {22600, 33449}, {22615, 23261}, {32492, 3071}, {36655, 45860}, {42060, 640}, {42261, 1152}, {44648, 7584}, {48659, 22596}, {49791, 615}
X(486) = isogonal conjugate of X(372)
X(486) = isotomic conjugate of X(491)
X(486) = complement of X(487)
X(486) = anticomplement of X(642)
X(486) = polar conjugate of X(1586)
X(486) = complement of the isogonal conjugate of X(8946)
X(486) = complement of the isotomic conjugate of X(24243)
X(486) = isogonal conjugate of the anticomplement of X(640)
X(486) = isogonal conjugate of the complement of X(638)
X(486) = isotomic conjugate of the anticomplement of X(615)
X(486) = isotomic conjugate of the isogonal conjugate of X(8576)
X(486) = isogonal conjugate of the isotomic conjugate of X(34392)
X(486) = isotomic conjugate of the polar conjugate of X(41516)
X(486) = polar conjugate of the isotomic conjugate of X(11091)
X(486) = polar conjugate of the isogonal conjugate of X(6414)
X(486) = psi-transform of X(13520)
X(486) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 24245}, {494, 18589}, {1973, 13934}, {8946, 10}, {19217, 141}, {24243, 2887}, {26461, 1214}
X(486) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 24245}, {847, 485}, {16037, 6414}, {34392, 11091}, {41516, 8940}
X(486) = X(i)-cross conjugate of X(j) for these (i,j): {3, 485}, {615, 2}, {3071, 4}, {5420, 10195}, {6414, 11091}, {6561, 1327}, {8576, 41516}, {8911, 1585}, {13785, 1328}, {13951, 10194}, {26950, 11090}, {32494, 5491}, {32575, 491}, {42258, 1131}, {42260, 43570}, {43880, 3317}, {45385, 43569}
X(486) = X(i)-isoconjugate of X(j) for these (i,j): {1, 372}, {19, 5409}, {31, 491}, {47, 485}, {48, 1586}, {63, 5412}, {92, 26920}, {371, 3377}, {560, 45806}, {604, 13461}, {1748, 6413}, {2180, 16032}, {8577, 44179}
X(486) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 491}, {2, 24245}, {3, 372}, {6, 5409}, {485, 34853}, {486, 487}, {1249, 1586}, {1600, 10960}, {1993, 10962}, {3161, 13461}, {3162, 5412}, {5409, 9723}, {6374, 45806}, {8576, 44199}, {8577, 37864}, {13439, 24246}, {13934, 42060}, {22391, 26920}, {33365, 39388}
X(486) = cevapoint of X(i) and X(j) for these (i,j): {3, 10666}, {6, 3156}, {615, 8036}, {3371, 3372}, {6414, 8576}
X(486) = crosspoint of X(2) and X(24243)
X(486) = crosssum of X(6) and X(10133)
X(486) = trilinear pole of line {523, 17432}
X(486) = external center of similitude of nine-point circle and 2nd Lemoine circle
X(486) = pole wrt polar circle of trilinear polar of X(1586)
X(486) = X(48)-isoconjugate (polar conjugate) of X(1586)
X(486) = Kosnita(X(486),X(486)) point
X(486) = trilinear product of vertices of inner Vecten triangle
X(486) = X(4) of inner Vecten triangle
X(486) = perspector of ABC and outer-squares triangle
X(486) = perspector of 4th tri-squares triangle and 4th tri-squares central triangle
X(486) = 4th-tri-squares-isogonal conjugate of X(32494)
X(486) = 3rd-anti-tri-squares-isogonal conjugate of X(32492)
X(486) = {X(5),X(6)}-harmonic conjugate of X(485)
X(486) = inner-Vecten-isogonal conjugate of X(642)
X(486) = barycentric product X(i)*X(j) for these {i,j}: {4, 11091}, {5, 16037}, {6, 34392}, {68, 1585}, {69, 41516}, {76, 8576}, {264, 6414}, {371, 5392}, {485, 13428}, {492, 2165}, {847, 5408}, {850, 39384}, {2052, 26922}, {2996, 8940}, {5413, 20563}, {11090, 13429}, {13430, 41515}, {18820, 45473}, {21464, 42024}, {24243, 24245}, {34391, 44193}
X(486) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 491}, {3, 5409}, {4, 1586}, {6, 372}, {8, 13461}, {25, 5412}, {68, 11090}, {76, 45806}, {96, 16032}, {184, 26920}, {371, 1993}, {372, 1600}, {485, 13439}, {492, 7763}, {615, 642}, {1322, 3536}, {1505, 32575}, {1585, 317}, {2165, 485}, {2351, 6413}, {3069, 39388}, {3155, 26875}, {3156, 10960}, {5392, 34391}, {5408, 9723}, {5413, 24}, {6413, 10665}, {6414, 3}, {8036, 615}, {8576, 6}, {8577, 44192}, {8911, 1147}, {8940, 193}, {10666, 5408}, {10881, 15208}, {11090, 13441}, {11091, 69}, {11474, 15217}, {13428, 492}, {13429, 1585}, {14593, 41515}, {16037, 95}, {21464, 45421}, {24245, 487}, {26922, 394}, {32734, 39383}, {34392, 76}, {35765, 15187}, {39384, 110}, {41515, 13440}, {41516, 4}, {44193, 371}, {45473, 42060}
X(486) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 49155, 13133}, {1, 49156, 13132}, {2, 371, 5418}, {2, 487, 642}, {2, 1588, 371}, {2, 12221, 487}, {2, 13428, 11091}, {2, 43134, 45509}, {3, 615, 5420}, {3, 3071, 6561}, {3, 6561, 42260}, {3, 13785, 3071}, {3, 13934, 49791}, {3, 13943, 8277}, {3, 13951, 615}, {3, 45385, 13951}, {4, 372, 6560}, {4, 3069, 372}, {4, 6460, 35820}, {4, 6560, 22644}, {4, 6565, 42268}, {4, 13429, 41516}, {4, 13939, 3069}, {4, 42561, 6565}, {5, 6, 485}, {5, 485, 42277}, {5, 1352, 10514}, {5, 5874, 6215}, {5, 6214, 10516}, {5, 6289, 10515}, {5, 7583, 42265}, {5, 7584, 6}, {5, 18762, 42262}, {5, 19116, 7583}, {5, 19117, 18538}, {5, 42262, 42274}, {5, 49317, 6281}, {5, 49355, 1352}, {6, 13881, 49220}, {6, 18762, 42274}, {6, 31411, 19103}, {6, 42262, 5}, {6, 42265, 7583}, {6, 42274, 42277}, {6, 44648, 19104}, {6, 49220, 19105}, {11, 19027, 1335}, {12, 19029, 1124}, {15, 3392, 2042}, {16, 3367, 2041}, {20, 1132, 23259}, {20, 13935, 6396}, {20, 13941, 13935}, {20, 23259, 35821}, {20, 35821, 42275}, {68, 2165, 485}, {69, 5491, 42060}, {140, 42215, 1151}, {155, 9722, 485}, {262, 19089, 3103}, {371, 10577, 2}, {371, 35823, 1588}, {372, 6565, 4}, {372, 13939, 43431}, {372, 35787, 35820}, {372, 35820, 6460}, {372, 42268, 22644}, {372, 42561, 42268}, {381, 3070, 42269}, {381, 3312, 3070}, {381, 48659, 22596}, {382, 6398, 42259}, {382, 13961, 6398}, {382, 42259, 42276}, {485, 42274, 5}, {546, 42216, 23251}, {550, 35256, 6410}, {590, 42583, 1656}, {615, 3071, 3}, {615, 13785, 6561}, {615, 32494, 13934}, {615, 43880, 13951}, {631, 3317, 32786}, {631, 6459, 6200}, {631, 23273, 6459}, {642, 6119, 2}, {944, 13959, 35762}, {946, 13936, 35774}, {1132, 13935, 35821}, {1132, 13941, 20}, {1132, 35813, 42275}, {1151, 8252, 140}, {1152, 13847, 13966}, {1322, 41516, 4}, {1328, 22615, 23261}, {1328, 42261, 22615}, {1352, 3767, 485}, {1352, 49355, 6278}, {1478, 13962, 6502}, {1479, 13963, 5414}, {1504, 1506, 31463}, {1587, 3091, 6564}, {1587, 7586, 6420}, {1588, 10577, 5418}, {1588, 12221, 35833}, {1656, 3311, 590}, {1656, 18510, 3311}, and many others


X(487) = ANTICOMPLEMENT OF X(486)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A)(1 - csc A sin B sin C)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
Barycentrics   (b2 + c2 - a2)(a2 - 2σ) : (c2 + a2 - b2)(b2 - 2σ) : (a2 + b2 - c2)(c2 - 2σ)      (M. Iliev, 5/13/07)

X(487) is a perspector of triangles associated with squares that circumscribe ABC. (Floor van Lamoen, 4/29/98)

Let OA be the circle centered at the A-vertex of the inner Vecten triangle and passing through A; define OB and OC cyclically. X(487) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

X(487) lies on these lines: 2,371   3,69   4,489   20,638   193,372   376,490   492,631

X(487) = reflection of X(486) in X(642)
X(487) = anticomplement of X(486)
X(487) = anticomplementary conjugate of X(638)
X(487) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,488), (489,20), (491,2)
X(487) = isogonal conjugate of X(8946)
X(487) = X(20)-of-inner-Vecten-triangle
X(487) = orthologic center of outer Vecten to inner Vecten triangles


X(488) = ANTICOMPLEMENT OF X(485)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A)(1 + csc A sin B sin C)
Barycentrics   (b2 + c2 - a2)(a2 + 2σ) : (c2 + a2 - b2)(b2 + 2σ) : :

X(488) is a perspector of triangles associated with squares that circumscribe ABC. For details, see Vierkanten in een driehoek: 2. Meer punten uit omgeschreven vierkanten (Floor van Lamoen, 4/29/98)

Let OA be the circle centered at the A-vertex of the outer Vecten triangle and passing through A; define OB and OC cyclically. X(488) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

X(488) lies on these lines: 2,372   3,69   4,490   193,371   376,489   491,631   591,3071

X(488) = reflection of X(485) in X(641)
X(488) = isogonal conjugate of X(8948)
X(488) = anticomplement of X(485)
X(488) = anticomplementary conjugate of X(637)
X(488) = X(i)-Ceva conjugate of X(j) , for these (i,j): (4,487), (490,20), (492,2)
X(488) = X(20)-of-outer-Vecten-triangle
X(488) = orthologic center of inner Vecten to outer Vecten triangles


X(489) = CEVAPOINT OF X(20) AND X(487)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A)(1 - csc A sin B sin C) - cos B cos C
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(489) is a perspector of triangles associated with squares that circumscribe ABC. For details and reference, see X(488). (Floor van Lamoen, 4/29/98)

X(489) lies on these lines: 3,492   4,487   20,64   30,638   176,664   376,488   485,671

X(489) = anticomplement of X(3071)
X(489) = cevapoint of X(20) and X(487)
X(489) = crosspoint of X(20) and X(487) wrt both the excentral and anticomplementary triangles


X(490) = CEVAPOINT OF X(20) AND X(488)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A)(1 + csc A sin B sin C) - cos B cos C
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(490) is a perspector of triangles associated with squares that circumscribe ABC. For details and reference, see X(488). (Floor van Lamoen, 4/29/98)

X(490) lies on these lines: 3,491   4,488   20,64   30,637   175,664   376,487   486,671

X(490) = anticomplement of X(3070)
X(490) = cevapoint of X(20) and X(488)
X(490) = crosspoint of X(20) and X(488) wrt both the excentral and anticomplementary triangles


X(491) = CEVAPOINT OF X(2) AND X(487)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A)(1 - csc A sin B sin C) + cos B cos C
Trilinears        sin(A - π/4)csc2A : sin(B - π/4)csc2B : sin(C - π/4)csc2C     (M. Iliev, 4/12/2007)
Trilinears        (1 - cot A) csc A : (1 - cot B) csc B : (1 - cot C) csc C     (M. Iliev, 4/12/2007)
Trilinears        (S - SA)/a : (S - SB)/b : (S - SC)/c     (C. Lozada, 8/07/2013)

Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
Barycentrics   b2 + c2 - a2 - 4σ : c2 + a2 - b2 - 4σ : a2 + b2 - c2 - 4σ      (M. Iliev, 5/13/07)

X(491) is a pole associated with squares that circumscribe ABC. For details and reference, see X(488). (Floor van Lamoen, 4/26/98)

X(491) lies on these lines: 2,6   3,490   4,487   5,637   76,485   315,371   372,642   488,631

X(491) = isotomic conjugate of X(486)
X(491) = anticomplement of X(615)
X(491) = X(264)-Ceva conjugate of X(492)
X(491) = cevapoint of X(2) and X(487)
X(491) = crosspoint of X(2) and X(487) wrt both the excentral and anticomplementary triangles
X(491) = {X(2),X(69)}-harmonic conjugate of X(492)
X(491) = homothetic center of ABC and unary cofactor triangle of 2nd Kenmotu diagonals triangle


X(492) = CEVAPOINT OF X(2) AND X(488)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A)(1 + csc A sin B sin C) + cos B cos C
Trilinears        sin(A + π/4)csc2A : sin(B + π/4)csc2B : sin(C + π/4)csc2C     (M. Iliev, 4/12/2007)
Trilinears        (1 + cot A) csc A : (1 + cot B) csc B : (1 + cot C) csc C     (M. Iliev, 4/12/2007)
Trilinears        (S + SA)/a : (S + SB)/b : (S + SC)/c     (C. Lozada, 8/07/2013)

Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
Barycentrics   b2 + c2 - a2 + 4σ : c2 + a2 - b2 + 4σ : a2 + b2 - c2 + 4σ      (M. Iliev, 5/13/07)

X(492) is a pole associated with squares that circumscribe ABC. For details and reference, see X(488). (Floor van Lamoen, 4/27/98)

X(492) lies on these lines: 2,6   3,489   4,488   5,638   76,486   315,372   371,641   487,631

X(492) = isotomic conjugate of X(485)
X(492) = anticomplement of X(590)
X(492) = X(264)-Ceva conjugate of X(491)
X(492) = cevapoint of X(2) and X(488)
X(492) = crosspoint of X(2) and X(488) wrt both the excentral and anticomplementary triangles
X(492) = {X(2),X(69)}-harmonic conjugate of X(491)
X(492) = homothetic center of ABC and unary cofactor triangle of 1st Kenmotu diagonals triangle


X(493) = 1st VAN LAMOEN HOMOTHETIC CENTER

Trilinears   1/(sin A + sin B sin C) : :
Barycentrics   (sin A)/(sin A + sin B sin C) : :
Barycentrics    a^2/(a^2 + S) : :

X(493) is a homothetic center of triangles associated with squares that circumscribe ABC. For details, see Vierkanten in een driehoek: 4. Ingeschreven vierkanten (Floor van Lamoen, 4/27/98)

X(493) is the homothetic center of triangle ABC and the Lucas homothetic triangle; see X(371). Writing t for the ratio L:W at X(371), let LA be the line through the intersections, other than A, of the A-Lucas(t) circle and sides CA and AB. Define LB and LC cyclically. Let A' = LB∩LC, and define B' and C' cyclically. Then the triangle A'B'C', here introduced as the Lucas(t) homothetic triangle, is homothetic to triangle ABC. If t = 1, the center of homothety is X(493); for t = -1, it is X(494); for t = 2, it is X(588); and for t = -2, it is X(589). (Randy Hutson, February 9, 2013)

Let A″ be the intersection of line BC and the common tangent to the B- and C-Lucas circles (at their touchpoint). Define B″ and C″ cyclically. Then A″B″C″ is the cevian triangle of X(493). Also, X(493) is the point of intersection of the tangents at X(371) and X(485) to the orthocubic K006. (Randy Hutson, July 23, 2015)

X(493) lies on these lines: 25,371   39,494   394,1504

X(493) = isogonal conjugate of X(3068)
X(493) = X(394)-cross conjugate of X(494)
X(493) = perspector of ABC and unary cofactor triangle of outer Vecten triangle
X(493) = perspector of ABC and unary cofactor triangle of Lucas(-1) antipodal triangle


X(494) = 2nd VAN LAMOEN HOMOTHETIC CENTER

Trilinears   1/(sin A - sin B sin C) : :
Barycentrics   (sin A)(sin A - sin B sin C) : :
Barycentrics    a^2/(a^2 - S) : :

X(494) is a homothetic center of triangles associated with squares that circumscribe ABC. For details and reference, see X(493). (Floor van Lamoen, 4/27/98)

X(494) is the homothetic center of triangle ABC and the Lucas(-1) homothetic triangle; see X(371).

Let A″ be the intersection of line BC and the common tangent to the B- and C-Lucas(-1) circles (at their touchpoint). Define B″ and C″ cyclically. Then A″B″C″ is the cevian triangle of X(494). Also, X(494) is the point of intersection of the tangents at X(372) and X(486) to the orthocubic K006. (Randy Hutson, July 23, 2015)

X(494) lies on these lines: 25,372   39,493   394,1505

X(494) = isogonal conjugate of X(3069)
X(494) = X(394)-cross conjugate of X(493)
X(494) = perspector of ABC and unary cofactor triangle of inner Vecten triangle
X(494) = perspector of ABC and unary cofactor triangle of Lucas antipodal triangle


X(495) = JOHNSON MIDPOINT

Trilinears    2 + cos(B - C) : 2 + cos(C - A) : 2 + cos(A - B)
Barycentrics    (sin A)[2 + cos(B - C)] : (sin B)[2 + cos(C - A)] : (sin C)[2 + cos(A - B)]
X(495) = 2(R/r)*X(1) + 3X(2) - X(3)

X(495) is the midpoint of segments C1-to-P1, C2-to-P2, C3-to-P3 in the Johnson four-circle configuration.

Roger A. Johnson, Advanced Euclidean Geometry, Dover, New York, 1960, page 75.

Peter Yff, "Three concurrent congruent circles 'inscribed' in a triangle," manuscript, 1998; X(495) is the point R on page 5. (See also X(496)-X(499) and X(1478), X(1479).)

If you have The Geometer's Sketchpad, you can view Johnson-Yff Circles Internal and Johnson-Yff Circles External.

X(495) lies on these lines: 1,5   2,956   3,388   4,390   8,442   10,141   30,55   35,550   36,549   56,140   202,395   203,396   226,517   381,497   392,908   429,1068   529,993   612,1060

X(495) = complement of X(956)
X(495) = X(427)-of-Fuhrmann-triangle


X(496) = {X(1),X(5)}-HARMONIC CONJUGATE OF X(495)

Trilinears    2 - cos(B - C) : 2 - cos(C - A) : 2 - cos(A - B)
Barycentrics   (sin A)[2 - cos(B - C)] : (sin B)[2 - cos(C - A)] : (sin C)[2 - cos(A - B)]
X(496) = 2(R/r)*X(1) - 3X(2) + X(3)

Peter Yff, "Three concurrent congruent circles 'inscribed' in a triangle," manuscript, 1998; X(496) is the point R' on page 5.

X(496) lies on these lines: 1,5   2,1058   3,497   4,999   30,56   35,549   36,550   55,140   149,404   202,397   203,398   381,388   390,631   613,1069   614,1062   942,946

X(496) = X(235)-of-Fuhrmann-triangle
X(496) = center of inverse-in-incircle-of-nine-point-circle
X(496) = Ursa-major-to-Ursa-minor similarity image of X(5)


X(497) = CROSSPOINT OF GERGONNE POINT AND NAGEL POINT

Trilinears    1 - cos B cos C : 1 - cos C cos A : 1 - cos A cos B
Barycentrics    (sin A)(1 - cos B cos C) : (sin B)(1 - cos C cos A) : (sin C)(1 - cos A cos B)
Barycentrics    (SW - bc)(s - a) : :
Barycentrics    (a - b - c)(a^2 + b^2 + c^2 - 2bc) : :
X(497) = 2(R/r)*X(1) - 3X(2) + 2X(3)

Let A'B'C' be the extangents-to-intangents similarity image of ABC. A'B'C' is homothetic to ABC at X(55) and to the medial triangle at X(497). (Randy Hutson, March 21, 2019)

X(497) lies on these lines: 1,4   2,11   3,496   7,354   8,210   20,56   29,1036   30,999   35,499   36,376   57,516   65,938   69,350   80,1000   212,238   329,518   381,495   452,958   614,1040   1057,1065

X(497) = isogonal conjugate of X(1037)
X(497) = isotomic conjugate of X(8817)
X(497) = anticomplement of X(1376)
X(497) = crosspoint of X(i) and X(j) for these (i,j): (7,8), (29,314)
X(497) = crosssum of X(i) and X(j) for these (i,j): (55,56), (73,1402)
X(497) = crossdifference of every pair of points on line X(652)X(665)
X(497) = homothetic center of Mandart-incircle triangle and anticomplementary triangle
X(497) = inverse-in-Feuerbach-hyperbola of X(2550)
X(497) = homothetic center of anticomplementary triangle and cross-triangle of ABC and 2nd Johnson-Yff triangle
X(497) = homothetic center of 2nd Johnson-Yff triangle triangle and cross-triangle of ABC and 2nd Johnson-Yff triangle
X(497) = {X(1),X(4)}-harmonic conjugate of X(388)


X(498) = YFF CONCURRENT CONGRUENT CIRCLES POINT

Trilinears    1 + 2 sin B sin C : 1 + 2 sin C sin A : 1 + 2 sin A sin B
Trilinears    bc + 2R^2 : :
Barycentrics    (sin A)(1 + 2 sin B sin C) : (sin B)(1 + 2 sin C sin A) : (sin C)(1 + 2 sin A sin B)
X(498) = R*X(1) + r*X(3) + 2r*X(5)

X(498) and X(499) are harmonic conjugate points with respect to X(1) and X(2), in analogy with such pairs with respect to X(1), X(4) and with respect to X(1), X(5).

Peter Yff, "Three concurrent congruent circles 'inscribed' in a triangle," manuscript, 1998; X(498) is the point S on page 6.

X(498) lies on these lines: 1,2   3,12   4,35   5,55   9,920   36,388   37,91   46,226   47,171   56,140   141,611   191,329   255,750   345,1089

X(498) = homothetic center of anti-Euler triangle and (cross-triangle of ABC and 2nd isogonal triangle of X(1))


X(499) = {X(1),X(2)}-HARMONIC CONJUGATE OF X(498)

Trilinears    1 - 2 sin B sin C : 1 - 2 sin C sin A : 1 - 2 sin A sin B
Trilinears    bc - 2R^2 : :
Barycentrics    (sin A)(1 - 2 sin B sin C) : (sin B)(1 - 2 sin C sin A) : (sin C)(1 - 2 sin A sin B)
X(499) = R*X(1) - r*X(3) - 2r*X(5)

X(499) is the harmonic conjugate of X(498) with respect to X(1) and X(2).

Peter Yff, "Three concurrent congruent circles 'inscribed' in a triangle," manuscript, 1998; X(498) is the point S' on page 6.

X(499) lies on these lines: 1,2   3,11   4,36   5,56   12,999   17,202   18,203   35,497   46,946   47,238   55,140   57,920   80,944   141,613   255,748   348,1111   484,962


X(500) = ORTHOCENTER OF THE INCENTRAL TRIANGLE

Trilinears    a(b2 +c2 - a2 + bc)[2abc + (b + c)(a2 - (b - c)2)] : :

X(500) lies on these lines: 1,30   3,6   651,943

X(500) = inverse-in-Brocard-circle of X(582)
X(500) = crosspoint of X(1) and X(35)
X(500) = crosssum of X(1) and X(79)
X(500) = X(1)-Ceva conjugate of X(942)


X(501) = MIQUEL ASSOCIATE OF INCENTER

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where
                        f(a,b,c) = a[a3 - b3 - c3 - bc(a + b + c) + ab(a - b) + ac(a - c)]/(b + c)

Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

Miquel's theorem states that if A', B', C' are points (other than A, B, C) on sidelines BC, CA, AB, respectively, then the circles AB'C', BC'A', CA'B' meet at a point. Suppose P is a point and A' = P∩BC, B' = P∩CA, C' = P∩AB; the point in which the three circles is the Miquel associate of P. (Paul Yiu, 7/6/99)

X(501) lies on these lines: 1,229   10,662   21,214   35,110   36,58   215,1364   284,942   572,992   595,1326   759,1385

X(501) = isogonal conjugate of X(502)


X(502) = ISOGONAL CONJUGATE OF X(501)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b + c)/[a3 - b3 - c3 - bc(a + b + c) + ab(a - b) + ac(a - c)]
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

Let A'B'C' be the incentral triangle. Let BCA'' be the triangle similar to A'B'C' such that the segment AA'' crosses the line BC. Define B'' and C'' cyclically. The lines AA'', BB'', CC'' concur in X(502). (Randy Hutson, 9/23/2011)

X(502) lies on these lines: 10,191   12,14873   42,14874   58,21043   313,17791   594,3678   1089,3969

X(502) = isogonal conjugate of X(501)
X(502) = X(1)-cross conjugate of X(10)

leftri

Isoscelizer points: X(503)-X(510)

rightri
Centers X(503)-X(510), and also centers X(173) = Congruent Isoscelizers Point, X(174) = Yff Center of Congruence, and X(258) = Congruent Incircles Isoscelizer Point, together with X(351)-X(364), and also X(61140)-X(61142), are associated with isoscelizers.

Definition: a line LA perpendicular to the internal bisector line of A is an A-isoscelizer. Suppose X is a point not on a sideline of ABC, and let

L(A,X) = the A-isoscelizer passing through X;
E(A,X) = L(A,X)∩AC;
F(A,X) = L(A,X)∩AB;
T(A,X) = the triangle wth vertices A, E(A,X), F(A,X);
H(A,X) = A-altitude of T(A,X);
D(A,X) = distance between E(A,X) and F(A,X);
X(A) = distance between E(A,X) and F(A,X).

Define L(B,X), E(B,X), . . . , X(B) and L(C,X), E(C,X), . . . ,X(C) cyclically.

Each of the points X(503) to X(510) is defined by Peter Yff as the point X of concurrence of isoscelers satisfying certain conditions.

Geometer's Sketchpad sketches for centers X(503)-X(510) were contributed by Peter Moses, May 7, 2005.

The following notes were contributed by Peter Moses, January 13, 2024:

H(A,X)D(A,X) = 2*area T(A,X).

H(A,X) (-Sin[A/2] + Sin[B/2] + Sin[C/2])
     = H(B,X) (Sin[A/2] - Sin[B/2] + Sin[C/2])
     = H(C,X) (Sin[A/2] + Sin[B/2] - Sin[C/2])
     = (2*R*S)/(-S + 4*R*(sa*Sin[A/2] + sb*Sin[B/2] + sc*Sin[C/2]))
     = (2*R*s)/(-s + 2*R*(Cos[A/2] + Cos[B/2] + Cos[C/2])),
     with solution X = X(503)

X(164) = solution, X, of H(A,X) = H(B,X) = H(C,X)
X(503) = solution, X, of a H(A,X) = b H(B,X) = c H(C,X)
X(504) = solution, X, of H(A,X)/a = H(B,X)/b = H(C,X)/c
X(845) = solution, X, of (-a + b + c) H(A,X) = (a - b + c) H(B,X) = (a + b - c) H(C,X)
X(173) = solution, X, of H(A,X)/(-a + b + c) = H(B,X)/(a - b + c) = H(C,X)/(a + b - c)

X(173) = solution, X, of D(A,X) = D(B,X) = D(C,X)
X(361) = solution, X, of a D(A,X) = b D(B,X) = c D(C,X)
X(362) = solution, X, of D(A,X)/a = D(B,X)/b = D(C,X)/c
X(164) = solution, X, of (-a + b + c) D(A,X) = (a - b + c) D(B,X) = (a + b - c) D(C,X)
X(61140) = solution, X, of D(A,X) / (-a + b + c) = D(B,X) / (a - b + c) = D(C,X) / (a + b - c)

X(364) = solution, X, of T(A,X) = T(B,X) = T(C,X)
X(40375) = solution, X, of a^2 T(A,X) = b^2 T(B,X) = c^2 T(C,X)
X(510) = solution, X, of T(A,X) / a^2 = T(B,X) / b^2 = T(C,X) / c^2
X(61141) = solution, X, of (-a + b + c)^2 T(A,X) = (a - b + c)^2 T(B,X) = (a + b - c)^2 T(C,X)
X(61142) = solution, X, of T(A,X) / (-a + b + c)^2 = T(B,X) / (a - b + c)^2 = T(C,X) / (a + b - c)^2


X(503) = 1st ISOSCELIZER POINT

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sec B/2 + sec C/2 - sec A/2
Barycentrics  af(A,B,C) : bf(B,C,A) : cf(C,A,B)

The isoscelizer equations aH(A,X) = bH(B,X) = cH(C,X) have solution X = X(503). (Peter Yff, 4/9/99)

If you have The Geometer's Sketchpad, you can view X(503).

X(503) lies on these lines: 1,167   164,361   173,844

X(503) = X(259)-Ceva conjugate of X(1)
X(503) = X(92)-of-excentral-triangle
X(503) = excentral-isogonal conjugate of X(504)
X(503) = ABC-to-excentral trilinear image of X(2)


X(504) = 2nd ISOSCELIZER POINT

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = b sin B/2 + c sin C/2 - a sin A/2
Barycentrics  af(A,B,C) : bf(B,C,A) : cf(C,A,B)

The isoscelizer equations [H(A,X)]/a = [H(B,X)]/b = [H(C,X)]/c have solution X = X(504). (Peter Yff, 4/9/99)

If you have The Geometer's Sketchpad, you can view X(504).

X(504) lies on this line: 164,173  

X(504) = X(48)-of-excentral-triangle
X(504) = excentral-isogonal conjugate of X(503)
X(504) = ABC-to-excentral trilinear image of X(6)


X(505) = 3rd ISOSCELIZER POINT

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/(sin B/2 + sin C/2 - sin A/2)
Barycentrics  af(A,B,C) : bf(B,C,A) : cf(C,A,B)

X(505) is the perspector of ABC and the excentral triangle of the excentral triangle of ABC. (Randy Hutson, 9/23/2011)

If you have The Geometer's Sketchpad, you can view X(505).

X(505) lies on the cubics K451 and K926 and these lines: {40, 164}, {165, 10233}, {168, 60555}, {173, 1130}, {258, 1128}, {266, 2956}, {845, 12518}, {1488, 8242}, {2089, 16664}, {3645, 32556}, {6212, 7010}, {6213, 7001}, {7597, 8111}, {7991, 10234}, {8078, 10215}, {10231, 24242}

X(505) = isogonal conjugate of X(164)
X(505) = isogonal conjugate of the anticomplement of X(21633)
X(505) = isogonal conjugate of the complement of X(9807)
X(505) = X(266)-cross conjugate of X(1)
X(505) = X(46)-of-excentral-triangle
X(505) = X(i)-isoconjugate of X(j) for these (i,j): {1, 164}, {6, 16017}, {259, 15495}, {266, 60598}
X(505) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 164}, {9, 16017}
X(505) = barycentric product X(i)*X(j) for these {i,j}: {75, 60555}, {188, 16664}
X(505) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 16017}, {6, 164}, {259, 60598}, {266, 15495}, {16664, 4146}, {60555, 1}, {61072, 21618}
X(505) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {40, 188, 164}, {20183, 52797, 164}


X(506) = 4th ISOSCELIZER POINT

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A/2)-2/3
Barycentrics  af(A,B,C) : bf(B,C,A) : cf(C,A,B)

The isoscelizer equations

X(A)[area of T(A,X)] = X(B)[area of T(B,X)] = X(C)[area of T(C,X)]

have solution X = X(506). (Peter Yff, 4/9/99)

If you have The Geometer's Sketchpad, you can view X(506).

X(506) = trilinear cube root of X(7)


X(507) = 5th ISOSCELIZER POINT

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A/2)-1/2
Barycentrics  af(A,B,C) : bf(B,C,A) : cf(C,A,B)

The isoscelizer equations

[area of T(A,X)][X(A)]2 = [area of T(B,X)][X(B)]2 = [area of T(C,X)][X(C)]2

have solution X = X(507). (Peter Yff, 4/9/99)

If you have The Geometer's Sketchpad, you can view X(507).

X(507) = trilinear square root of X(174)


X(508) = 6th ISOSCELIZER POINT

Trilinears       a-1/2sec(A/2) : b-1/2sec(B/2) : c-1/2sec(C/2)
Barycentrics  a1/2sec(A/2) : b1/2sec(B/2) : c1/2sec(C/2)

The isoscelizer equations

a[area of T(A,X)] = b[area of T(B,X)] = c[area of T(C,X)]

have solution X = X(508). (Peter Yff, 4/9/99)

If you have The Geometer's Sketchpad, you can view X(508).

X(508) = trilinear product X(366)*X(174)
X(508) = trilinear square root of X(85)
X(508) = barycentric square root of X(7)


X(509) = 7th ISOSCELIZER POINT

Trilinears    (tan A/2)1/2 : :
Trilinears    1/(semi-minor axis of A-Soddy ellipse) : :

The isoscelizer equations

[area of T(A,X)]/a = [area of T(B,X)]/b = [area of T(C,X)]/c

have solution X = X(509). (Peter Yff, 4/9/99)

If you have The Geometer's Sketchpad, you can view X(509).

X(509) = trilinear square root of X(57)


X(510) = 8th ISOSCELIZER POINT

Trilinears    b3/2 + c3/2 - a3/2 : :

The isoscelizer equations

[area of T(A,X)]/a2 = [area of T(B,X)]/b2 = [area of T(C,X)]/c2

have solution X = X(510). (Peter Yff, 4/9/99)

If you have The Geometer's Sketchpad, you can view X(510).

leftri

Centers on the line at infinity: X(511)-X(526)

rightri

In barycentric coordinates, the line at infinity, L, is the set of points X = x : y : z satisfying x + y + z = 0. The isogonal conjugate of L, given by a^2 y z + b^2 z x + c^2 x y = 0, is the circumcircle (e.g., X(98)-X(112)).

The collection of collinearities reported below for each of the centers on L comprises a family of parallel lines.

Barycentrics x: y : z (and trilinears) for X(30), X(511)-X(526), and many other triangle centers on L, are polynomials in (a,b,c). For example, barycentrics for X(511) are p(a,b,c) : p(b,c,a) : p(c,a,b), where

p(a,b,c) = a2(a2b2 + a2c2 - b4 - c4).

Such polynomials comprise a family that can be partitioned into two subfamilies: even and odd. Each subfamily can be generated as linear combination of polynomials in certain bases. For example, a basis for the even polynomials of degrees 1 to 4 follows:

degree 1: 2a - b - c
degree 2: 2a2 - b2 - c2, 2bc - ca - ab
degree 3: 2a3 - b3 - c3, a2b + a2c - b2c - b2c, a2b + a2c - b2c - b2c
degree 4: 2a4 - b4 - c4, a3b + a3c - b3c - b3c, a3b + a3c - b3c - b3c, 2b2c2 - a2b2 - a2c2, 2a2bc - ab2c - abc2

A basis for the odd polynomials of degrees 1 to 2 follows:

degree 1: b - c
degree 2: (b - c)(a + b + c), (b - c)a

See Clark Kimberling, "A combinatorial classification of triangle centers on the line at infinity," Journal of Integer Sequences 22 (2019) Article 19.5.4.


X(511) = ISOGONAL CONJUGATE OF X(98)

Trilinears    cos(A + ω) : cos(B + ω) : cos(C + ω)
Trilinears    sin A - sin(A + 2ω) : sin B - sin(B + 2ω) : sin C - sin(C + 2ω)
Trilinears    cos A + cos(A + 2ω) : cos B + cos(B + 2ω) : cos C + cos(C + 2ω)
Trilinears    a(a2b2 + a2c2 - b4 - c4) : :      (M. Iliev, 5/13/07)
Trilinears    cos A - sin A tan ω : :
Trilinears    a2cos B cos C - bc cos2A : : (R. Hutson, 1/29,15)
Trilinears    cos A + cos 2A cos(B - C) : :
Trilinears    a - 2R cos A cot ω : :
Barycentrics   sin A cos(A + ω) : sin B cos(B + ω) : sin C cos(C + ω)
Barycentrics    (cot A + cot B + cot C) tan A - (tan A + tan B + tan C) cot A : :

X(511) = X(3) - X(6)

As the isogonal conjugate of a point on the circumcircle, X(511) lies on the line at infinity.

Let L denote the line having trilinears of X(511) as coefficients. Then L is the line passing through X(6) perpendicular to the Euler line.

X(511) is the perspector of triangle ABC and the tangential triangle of the hyperbola {A, B, C, X(2), and X(110)}}.

X(511) lies on these (parallel) lines: 1,256   2,51   3,6   4,69   5,141   20,185   22,184   23,110   24,1092   25,394   26,206   30,512   35,2330   36,1428   40,1045   49,2937   54,1176   55,611   56,613   66,68   67,265   74,691   83,3399   98,385   99,2698   100,2699   101,2700   102,2701   103,2702   104,2703   105,2704   106,2705   107,450   108,2707   109,2708   111,352   112,2710   114,325   125,858   140,143 nbsp; 154,3167   155,159   165,3097   171,181   186,249   195,2916   199,1790   230,2023   232,2211   238,3271   242,1944   283,3145   287,401   291,3510   295,3509   298,1080   299,383   343,427   355,3416   376,1992   381,599   399,2930   403,1568   468,1112   549,597   550,1353   631,3567   694,3229   843,1296   852,2972   982,1401   1113,2105   1114,2104   1194,3051   1196,1613   1292,2711   1293,2712   1294,2713   1295,2714   1297,2715   1364,1936   1370,1899   1385,1386   1437,2915   1482,3242   1757,3507   1818,2183   1976,2065   2070,3447   2095,2097   2323,3220   2653,2670   3100,3270   3124,3291

X(511) = isogonal conjugate of X(98)
X(511) = isotomic conjugate of X(290)
X(511) = anticomplementary conjugate of X(147)
X(511) = complementary conjugate of X(114)
X(511) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,11672), (4,114), (290,2), (297,232)
X(511) = cevapoint of X(385) and X(401)
X(511) = X(i)-cross conjugate of X(j) for these (i,j): (4,114), (290,2), (297,232)
X(511) = crosspoint of X(i) and X(j) for these (i,j): (2,290), (297,325)
X(511) = crosssum of X(i) and X(j) for these (i,j): (2,385), (6,237), (11,659), (523,868)
X(511) = crossdifference of every pair of points on line X(6)X(523)
X(511) = orthopoint of X(512)
X(511) = X(3)-Hirst inverse of X(6)
X(511) = X(i)-line conjugate of X(j) for these (i,j): (3,6), (30,523)
X(511) = intercept of Brocard axis and the line at infinity (trilinear polars of X(110) and X(2))
X(511) = trilinear pole of line X(684)X(2491)
X(511) = radical center of Lucas(-2 tan ω) circles
X(511) = {X(3),X(6)}-harmonic conjugate of X(182)
X(511) = X(511)-of-2nd-Brocard-triangle
X(511) = X(542)-of-orthocentroidal-triangle
X(511) = X(542)-of-X(4)-Brocard triangle
X(511) = X(i)-isoconjugate of X(j) for these (i,j): (6,1821), (63,6531), (92,248)
X(511) = X(512)-of-3rd-Parry-triangle
X(511) = ideal point of PU(i) for these i: 29, 145
X(511) = bicentric sum of PU(145)
X(511) = X(542)-of-4th-anti-Brocard-triangle
X(511) = Thomson-isogonal conjugate of X(99)
X(511) = Lucas-isogonal conjugate of X(99)
X(511) = perspector of 2nd Neuberg triangle and cross-triangle of 1st and 2nd Neuberg triangles
X(511) = Cundy-Parry Phi transform of X(32)
X(511) = Cundy-Parry Psi transform of X(76)
X(511) = perspector of ABC and side-triangle of pedal triangles of PU(1)
X(511) = perspector of ABC and side-triangle of reflection triangles of PU(1)
X(511) = polar conjugate of isotomic conjugate of X(36212)


X(512) = ISOGONAL CONJUGATE OF X(99)

Trilinears    a(b2 - c2) : :
Trilinears    sin A (cos 2B - cos 2C) : :
Barycentrics    a2(b2 - c2) : :
Barycentrics    SB^2 - SC^2 : :
X(512) = P(1) - U(1)

X(512) is the point in which the line of the 1st and 2nd Brocard points meets the line at infinity.

Let A'B'C' be the 1st Brocard triangle. Let La be the reflection of B'C' in the internal angle bisector of vertex angle A, and define Lb and Lc cyclically. Then the lines La, Lb, Lc are parallel, and they concur in X(512). (Randy Hutson, September 5, 2015)

X(512) lies on these (parallel) lines: 1,875   2,3111   4,879   6,2444   25,2433   30,511   32,878   39,881   51,1640   64,2435   74,842   98,2698   99,805   100,2703   101,2702   102,2708   103,2700   104,2699   105,2711   106,2712   107,2713   108,2714   109,2701   110,249   111,843   112,2715   115,2679   187,237   263,2395   316,850   460,2501   650,2499   660,1016   670,886   764,2650   884,2440   1292,2704   1293,2705   1294,2706   1295,2707   1296,2709   1297,2710   1326,2605   1491,1734   1500,2084   1570,2451   1577,2533   1691,2483   1692,3251   1968,2909   2021,2491   2024,2507   2030,2492   2031,2510   2032,2508   2142,2143   2254,2530   2378,2379   2643,3271

X(512) = isogonal conjugate of X(99)
X(512) = isotomic conjugate of X(670)
X(512) = anticomplementary conjugate of X(148)
X(512) = complementary conjugate of X(115)
X(512) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,1084), (4,115), (66,125), (99,39), (110,6), (112,32), 1018,1500), (1306,1504), (1307,1505)
X(512) = crosspoint of X(i) and X(j) for these (i,j): (4,112), (6,110), (83,99)
X(512) = crosssum of X(i) and X(j) for these (i,j): (1,1019), (2,523), (3,525), (6,669), (39,512), (100,190), (311, 850), (514,1125), (643,662)
X(512) = orthopoint of X(511)
X(512) = vertex conjugate of X(15) and X(16)
X(512) = crossdifference of every pair of points on line X(2)X(6)
X(512) = X(112)-line conjugate of X(30)
X(512) = perspector of vertex-triangle and side triangle of circumcevian triangles of X(3) and X(6)
X(512) = bicentric difference of PU(1)
X(512) = bicentric difference of PU(2)
X(512) = ideal point of PU(1)
X(512) = ideal point of PU(2)
X(512) = ideal point of PU(26)
X(512) = vertex conjugate of PU(2)
X(512) = vertex conjugate of X(6) and X(187)
X(512) = trilinear pole of line X(351)X(865) (line is tangent to Steiner inellipse at X(1084))
X(512) = perspector of hyperbola {A,B,C,X(2),X(6)} (circumconic centered at X(1084))
X(512) = intercept of Lemoine axis and the line at infinity (trilinear polars of X(6) and X(2))
X(512) = center of circumconic that is locus of trilinear poles of lines passing through X(1084)
X(512) = perspector of ABC and the dual of the 1st Brocard triangle
X(512) = X(512)-of-2nd-Brocard triangle
X(512) = X(690)-of-orthocentroidal-triangle
X(512) = X(690)-of X(4)-Brocard-triangle
X(512) = orthic isogonal conjugate of X(115)
X(512) = incentral isogonal conjugate of X(115)
X(512) = incentral isotomic conjugate of X(2643)
X(512) = X(6)-isoconjugate of X(799)
X(512) = X(92)-isoconjugate of X(4558)
X(512) = exsimilicenter of antipedal circles of PU(1)
X(512) = harmonic center of antipedal circles of PU(1)
X(512) = trilinear pole of PU(105)
X(512) = bicentric difference of PU(191)
X(512) = ideal point of PU(191)
X(512) = perspector of ABC and unary cofactor triangle of Steiner triangle
X(512) = X(690) of 4th anti-Brocard triangle
X(512) = Thomson-isogonal conjugate of X(98)
X(512) = Lucas-isogonal conjugate of X(98)
X(512) = Cundy-Parry Psi transform of X(14265)
X(512) = barycentric product X(4)*X(647)
X(512) = barycentric product of Jerabek hyperbola intercepts of orthic axis
X(512) = barycentric product of Kiepert hyperbola intercepts of Lemoine axis
X(512) = perspector of circlecevian triangles of PU(1)
X(512) = orthic-isotomic conjugate of X(34980)


X(513) = ISOGONAL CONJUGATE OF X(100)

Trilinears    b - c : c - a : a - b
Trilinears    r - R cos A : :
Trilinears    d(a,b,c) : : , where d(a,b,c) = directed distance from A to the Nagel line
Barycentrics  ab - ac : bc - ba : ca - cb

As the isogonal conjugate of a point on the circumcircle, X(513) lies on the line at infinity.

X(513) lies on these (parallel) lines: 1,764   3,3657   6,1024   7,885   9,3126   11,3025   30,511   36,238   37,876   44,649   59,651   74,2687   98,2699   99,2703   100,765   101,1308   102,2716   103,2717   104,953   105,840   106,2718   107,2719   108,2720   109,2222   110,1290   111,2721   112,2711   190,660   269,2424   320,350   484,1734   663,855   668,889   676,2488   884,3423   927,1275   957,2401   1037,1486   1052,1054   1086,3271   1292,2742   1293,2743   1294,2744   1295,2745   1296,2746   1297,2747   1361,3319   1362,3322   1364,3326   1430,2201   1835,1874   1960,3246   2473,2487   2490,2505   2500,2532   2517,2533   2529,3239   3022,3328   3123,3248

X(513) = isogonal conjugate of X(100)
X(513) = isotomic conjugate of X(668)
X(513) = anticomplementary conjugate of X(149)
X(513) = complementary conjugate of X(11)
X(513) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,244), (4,11), (100,1), (101,354), (108,56), (109,65), (190,37), (38275,23505)
X(513) = X(244)-cross conjugate of X(1)
X(513) = crosspoint of X(i) and X(j) for these (i,j): (1,100), (4,108), (58,109), (86,190)
X(513) = crosssum of X(i) and X(j) for these (i,j): (1,513), (3,521), (6,667), (10,522), (42,649), (55,650), (142,514), (442,523), (692,906), (900,1145)
X(513) = crossdifference of every pair of points on line X(1)X(6)
X(513) = orthopoint of X(517)
X(513) = X(i)-line conjugate of X(j) for these (i,j): (30,518), (36,238)
X(513) = X(526)-of-Fuhrmann triangle
X(513) = barycentric product of PU(27)
X(513) = trilinear pole of PU(i) for these i: 27, 34
X(513) = center of circumconic that is locus of trilinear poles of lines passing through X(1015)
X(513) = X(2)-Ceva conjugate of X(1015)
X(513) = crossdifference of PU(28)
X(513) = ideal point of PU(i), for these i: 3, 6, 24, 31, 33, 41, 46, 50, 52, 53, 54, 55, 56, 57, 58, 74, 76, 78, 96, 111, 124
X(513) = bicentric difference of PU(i) for these i: 6, 31, 33, 41, 46, 50, 52, 53, 54, 55, 56, 96, 111
X(513) = trilinear product of PU(34)
X(513) = trilinear square root of X(244)
X(513) = perspector of circumconic centered at X(1015) (hyperbola {A,B,C,X(1),X(2)}
X(513) = intercept of antiorthic axis and the line at infinity (trilinear polars of X(1) and X(2))
X(513) = excentral isogonal conjugate of X(1768)
X(513) = intouch isogonal conjugate of X(11)
X(513) = orthic isogonal conjugate of X(11)
X(513) = X(6)-isoconjugate of X(190)
X(513) = X(1)-vertex conjugate of X(36)
X(513) = barycentric cube root of X(8027)
X(513) = trilinear pole of line X(244)X(665)
X(513) = Thomson-isogonal conjugate of X(104)
X(513) = Lucas-isogonal conjugate of X(104)
X(513) = Cundy-Parry Psi transform of X(14266)
X(513) = Cundy-Parry Phi transform of X(39173)
X(513) = polar conjugate of X(6335)
X(513) = pole wrt polar circle of trilinear polar of X(6335) (line X(4)X(8))
X(513) = perspector of ABC and side-triangle of Gemini triangles 5 and 6


X(514) = ISOGONAL CONJUGATE OF X(101)

Trilinears    (b - c)/a : :
Trilinears    |AP(1)| - |AU(1)| : :
Trilinears    directed distance from A to line X(1)X(6) : :
Barycentrics    b - c : :
Barycentrics    cot B/2 - cot C/2 : :

Let A7B7C7 and A8B8C8 be Gemini triangles 7 and 8, resp. Let LA be the tangent at A to conic {A,B7,C7,B8,C8}}, and define LB and LC cyclically. Let A' = LB∩LC, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(514). (Randy Hutson, January 15, 2019)

X(514) is one of two centers of inverse similitude (X(516) is the other) of the 1st and 2nd Dao equilateral triangles. (Randy Hutson, July 11, 2019)

X(514) lies on these (parallel) lines: 1,663   2,1022   10,764   11,3328   30,511   57,2401   74,2688   85,2140   98,2700   99,2702   100,1308   101,664   102,2723   103,2724   104,2717   105,2725   106,2726   107,2727   108,2728   109,929   110,2690   111,2729   116,1146   189,2399   190,1016   239,649   241,650   242,1459   330,3249   651,655   653,1461   659,667   661,693   1024,2402   1111,2170   1292,2736   1293,2737   1294,2738   1295,2739   1296,2740   1297,2741   1317,3322   1358,3323   1729,3188   1734,2254   1768,2958   1921,3261   2487,2516   4521,4885

X(514) = isogonal conjugate of X(101)
X(514) = isotomic conjugate of X(190)
X(514) = anticomplementary conjugate of X(150)
X(514) = complementary conjugate of X(116)
X(514) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,1086), (4,116), (7,11), (75,244), (100,142), (190,2)
X(514) = X(i)-cross conjugate of X(j) for these (i,j): (11,7), (244,75)
X(514) = crosspoint of X(2) and X(190)
X(514) = crosssum of X(i) and X(j) for these (i,j): (6,649), (37,650), (41,663), (48,652), (55,657), (213,667), (354,513), (1459,1473)
X(514) = crossdifference of every pair of points on line X(6)X(31)
X(514) = orthopoint of X(516)
X(514) = excentral-isogonal conjugate of X(39156)
X(514) = X(513)-Hirst inverse of X(812)
X(514) = trilinear pole of the line X(11)X(244) (which is the Feuerbach tangent line)
X(514) = pole wrt polar circle of line X(4)X(9)
X(514) = X(48)-isoconjugate (polar conjugate) of X(1897)
X(514) = X(6)-isoconjugate of X(100)
X(514) = intercept of Gergonne line and the line at infinity (trilinear polars of X(7) and X(2))
X(514) = X(690) of Fuhrmann triangle
X(514) = bicentric difference of PU(i) for these i: 10, 24, 47, 51
X(514) = ideal point of PU(i) for these i: 10, 47, 51
X(514) = trilinear product of PU(27)
X(514) = trilinear pole of PU(i) for these i: 121, 123
X(514) = perspector of circumconic centered at X(1086) (hyperbola {A,B,C,X(2),X(7)}})
X(514) = center of circumconic that is locus of trilinear poles of lines passing through X(1086)
X(514) = anticomplementary isotomic conjugate of X(514)
X(514) = Thomson-isogonal conjugate of X(103)
X(514) = Lucas-isogonal conjugate of X(103)
X(514) = barycentric square root of X(1086)
X(514) = X(63)-isoconjugate of X(8750)
X(514) = X(92)-isoconjugate of X(32656)
X(514) = perspector of ABC and vertex-triangle of Gemini triangles 7 and 8


X(515) = ISOGONAL CONJUGATE OF X(102)

Trilinears    (b + c)sec A - b sec B - c sec C :
Trilinears    r - 2 R cos B cos C : :
X(515) = X(1) - X(4)

Barycentrics    2a^4 - a^3(b + c) - a^2(b - c)^2 + a(b - c)^2(b + c) - (b^2 - c^2)^2 : :

X(515) is the perspector of triangle ABC and the tangential triangles of the conic that passes through the points A, B, C, X(2), and X(8).

As the isogonal conjugate of a point on the circumcircle, X(515) lies on the line at infinity.

X(515) lies on these (parallel) lines: 1,4   3,10   5,1125   8,20   9,3427   11,1319   12,2646   29,947   30,511   36,80   48,1826   55,1012   56,1210   58,3072   65,1071   71,1765   74,2689   78,3436   79,1389   98,2701   99,2708   100,2077   101,2723   102,1309   103,929   105,2730   106,2731   108,2733   109,2734   110,2695   111,2735   117,1455   119,214   145,962   153,908   165,376   200,3421   281,610   284,1065   381,551   382,1482   411,2975   484,1768   595,3073   602,1724   603,1771   631,1698   910,1146   936,2551   938,3333   956,3419   997,3452   1000,3062   1006,1746   1292,2751   1293,2757   1294,2762   1295,2765   1296,2768   1317,1537   1323,1565   1350,3416   1387,1538   1420,8086   1498,3173   1766,2321   1829,3575   1836,2099   1839,1963   1885,1902   2093,2096   2183,2250   3241,3543

X(515) = isogonal conjugate of X(102)
X(515) = isotomic conjugate of X(34393)
X(515) = anticomplementary conjugate of X(151)
X(515) = complementary conjugate of X(117)
X(515) = X(2)-Ceva conjugate of X(23986)
X(515) = X(4)-Ceva conjugate of X(117)
X(515) = crossdifference of every pair of points on line X(6)X(652)
X(515) = orthopoint of X(522)
X(515) = intersection of trilinear polars of X(2) and X(8)
X(515) = Thomson-isogonal conjugate of X(109)
X(515) = Lucas-isogonal conjugate of X(109)
X(515) = Cundy-Parry Phi transform of X(10570)
X(515) = Cundy-Parry Psi transform of X(10571)
X(515) = X(2)-isoconjugate of X(32677)


X(516) = ISOGONAL CONJUGATE OF X(103)

Trilinears    bc[(a - b) cot C + (a - c) cot B] : :
Trilinears    a^2 - b^2 cos C - c^2 cos B : :
Barycentrics    a(a^2 - b^2 cos C - c^2 cos B) : :
Barycentrics    2 a^3 - a^2 (b + c) - (b - c)^2 (b + c) : :
X(516) = X(1) - X(7)

X(516) is the perspector of triangle ABC and the tangential triangles of the conic that passes through the points A, B, C, X(2), and X(7).

X(516) is one of two centers of inverse similitude (X(514) is the other) of the 1st and 2nd Dao equilateral triangles. (Randy Hutson, July 11, 2019)

As the isogonal conjugate of a point on the circumcircle, X(516) lies on the line at infinity.

X(516) lies on these (parallel) lines: 1,7   2,165   3,142   4,9   8,144   11,1155   27,2328   30,511   31,1754   35,411   46,1210   55,226   57,497   63,1709   65,950   72,3059   74,2690   79,2346   80,655   98,2702   99,2700   100,908   101,2724   102,929   103,927   104,1308   105,2736   106,2737   107,2738   108,2739   109,1936   110,2688   111,2740   112,2741   118,910   146,2948   149,1768   152,1282   200,329   214,1537   238,673   255,1777   354,553   355,382   376,551   388,1697   412,1838   484,1737   550,1385   580,3073   902,3011   938,3339   944,3243   972,1543   993,1012   1058,3333   1076,1771   1086,1279   1158,3358   1284,2223   1292,2725   1293,2726   1294,2727   1295,2728   1296,2729   1317,3328   1376,3452   1389,3255   1482,1657   1490,3174   1519,2077   1538,3035   1571,2548   1572,2549   1587,1702   1588,1703   1633,3220   1698,3091   1700,2546   1701,2547   1704,2542   1705,2543   1736,2310   1829,1885   1852,1888   2017,2544   2018,2545   2321,3416   2947,3190   3021,3323   3340,3586

X(516) = isogonal conjugate of X(103)
X(516) = isotomic conjugate of X(18025)
X(516) = anticomplementary conjugate of X(152)
X(516) = complementary conjugate of X(118)
X(516) = X(2)-Ceva conjugate of X(23972)
X(516) = X(4)-Ceva conjugate of X(118)
X(516) = crosssum of X(i) and X(j) for these (i,j): (3,916), (55,672)
X(516) = crossdifference of every pair of points on line X(6)X(657)
X(516) = orthopoint of X(514)
X(516) = intercept of the Soddy line and the line at infinity (trilinear polars of X(7) and X(2))
X(516) = X(542)-of-Fuhrmann-triangle
X(516) = exsimilicenter of Bevan and anticomplementary circles
X(516) = harmonic center of Bevan and anticomplementary circles
X(516) = Thomson-isogonal conjugate of X(101)
X(516) = Lucas-isogonal conjugate of X(101)
X(516) = Cundy-Parry Phi transform of X(14377)
X(516) = Cundy-Parry Psi transform of X(3730)
X(516) = X(92)-isoconjugate of X(32657)


X(517) = ISOGONAL CONJUGATE OF X(104)

Trilinears    -1 + cos B + cos C : -1 + cos C + cos A : -1 + cos A + cos B
Trilinears    (b + c)cos A - b cos B - c cos C : :
Trilinears    b^3 + c^3 - (a^2 + bc)(b + c) + 2abc : :
Barycentrics   (sin A)(-1 + cos B + cos C) : (sin B)(-1 + cos C + cos A) : (sin C)(-1 + cos A + cos B)
Barycentrics    (a - c) cos B + (a - b) cos C : :
X(517) = X(1) - X(3)

As the isogonal conjugate of a point on the circumcircle, X(517) lies on the line at infinity.

X(517) lies on these (parallel) lines: 1,3   2,392   4,8   5,10   6,998   7,1000   9,374   11,1737   19,219   20,145   21,1389   30,511   33,1905   34,1753   37,573   42,1064   44,1168   52,1858   59,1870   63,956   71,1243   74,1290   78,945   88,1318   98,2703   99,2699   100,953   101,910   102,1807   103,1308   104,901   105,2742   106,2743   107,2744   108,2745   109,1455   110,1325   111,2746   112,2747   119,908   140,1125   151,1535   169,220   182,1386   201,2599   210,381   218,2082   221,3157   226,495   238,1052   244,1149   347,1439   376,2094   389,950   390,3488   399,2948   496,1210   500,2650   549,551   550,1483   572,1100   579,1108   580,595   582,602   601,1468   672,2170   840,1292   906,1951   936,1706   938,1058   958,3560   990,1350   997,1376   1006,1621   1042,1066   1046,2943   1051,2944   1068,1426   1113,2103   1114,2102   1124,2362   1148,1895   1293,2718   1294,2719   1295,2720   1296,2721   1297,2722   1817,3025   1352,3416   1361,1785   1362,3328   1364,3319   1391,1443   1411,2361   1437,3193   1451,1497   1457,1465   1478,1836   1479,1837   1490,2136   1656,1698   1702,3311   1703,3312   1788,3086   1830,1877   1838,1888   2171,2269   2182,2323   2270,2324   2329,3496   3022,3322   3061,3501   3085,3485   3125,3290   3190,3198   3197,3211   3474,3476

X(517) = isogonal conjugate of X(104)
X(517) = isotomic conjugate of X(18816)
X(517) = anticomplementary conjugate of X(153)
X(517) = complementary conjugate of X(119)
X(517) = X(2)-Ceva conjugate of X(23980)
X(517) = X(4)-Ceva conjugate of X(119)
X(517) = crosspoint of X(i) and X(j) for these (i,j): (1,80), (7,88)
X(517) = crosssum of X(i) and X(j) for these (i,j): (1,36), (3,912), (44,55), (56,1455)
X(517) = crossdifference of every pair of points on line X(6)X(650)
X(517) = orthopoint of X(513)
X(517) = trilinear pole of line X(1769)X(3310)
X(517) = Thomson-isogonal conjugate of X(100)
X(517) = Lucas-isogonal conjugate of X(100)
X(517) = Cundy-Parry Phi transform of X(56)
X(517) = Cundy-Parry Psi transform of X(8)
X(517) = X(30)-of-excentral-triangle


X(518) = ISOGONAL CONJUGATE OF X(105)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ab + ac - b2 - c2
Trilinears       as - Sω : bs - Sω : cs - Sω
Barycentrics    a*(a*b - b^2 + a*c - c^2) : :

As the isogonal conjugate of a point on the circumcircle, X(518) lies on the line at infinity.

Let Ia be the reflection of X(1) in the perpendicular bisector of BC, and define Ib and Ic cyclically; then X(518) = X(528) of IaIbIc. (Randy Hutson, December 26, 2015)

X(518) lies on these (parallel) lines: {1, 6}, {2, 210}, {3, 3433}, {4, 6601}, {5, 10916}, {7, 8}, {10, 141}, {11, 908}, {12, 6067}, {19, 5781}, {20, 3189}, {21, 2346}, {25, 41611}, {28, 56137}, {30, 511}, {31, 3744}, {34, 9370}, {35, 3916}, {36, 3446}, {38, 42}, {39, 3774}, {40, 1071}, {43, 982}, {46, 5687}, {51, 41581}, {55, 63}, {56, 78}, {57, 200}, {58, 5266}, {59, 765}, {66, 28787}, {67, 13211}, {74, 2691}, {80, 3254}, {81, 1390}, {84, 6769}, {86, 3786}, {88, 52925}, {89, 56151}, {92, 1859}, {98, 2704}, {99, 2711}, {100, 840}, {101, 2725}, {102, 2730}, {103, 2736}, {104, 2742}, {105, 1280}, {106, 2748}, {107, 2749}, {108, 2750}, {109, 2751}, {110, 2752}, {111, 2753}, {112, 2754}, {113, 58680}, {114, 58681}, {115, 35084}, {116, 58612}, {117, 58685}, {118, 58686}, {119, 58613}, {120, 5519}, {125, 32238}, {140, 13373}, {144, 145}, {149, 5057}, {159, 9798}, {165, 3158}, {171, 3961}, {181, 3687}, {182, 1385}, {184, 41605}, {188, 11032}, {190, 3685}, {191, 3746}, {197, 37581}, {206, 42463}, {209, 306}, {214, 5126}, {222, 8270}, {226, 2886}, {227, 37591}, {228, 16678}, {236, 8083}, {239, 335}, {240, 14571}, {241, 1458}, {243, 1897}, {244, 899}, {265, 10100}, {287, 57862}, {291, 1575}, {294, 9453}, {296, 14198}, {312, 3967}, {313, 44153}, {318, 1887}, {321, 3706}, {323, 65524}, {329, 497}, {333, 3757}, {341, 20923}, {344, 27549}, {350, 17794}, {355, 1352}, {374, 37654}, {375, 5943}, {376, 63432}, {385, 5985}, {386, 4719}, {404, 4420}, {442, 13407}, {468, 47477}, {474, 3338}, {484, 4880}, {496, 21616}, {500, 48922}, {551, 597}, {553, 49732}, {575, 15178}, {576, 5694}, {579, 3694}, {583, 1009}, {584, 13723}, {596, 40504}, {599, 3679}, {612, 940}, {614, 4383}, {620, 58590}, {643, 2651}, {644, 10699}, {648, 53204}, {650, 5098}, {651, 1456}, {664, 14189}, {666, 53214}, {668, 1921}, {670, 53222}, {672, 3693}, {677, 1814}, {692, 7193}, {693, 50765}, {694, 54980}, {748, 62869}, {750, 37520}, {756, 3720}, {759, 65885}, {846, 3750}, {869, 2274}, {872, 1193}, {894, 5263}, {895, 7984}, {896, 902}, {901, 53608}, {903, 53226}, {910, 1282}, {920, 11508}, {936, 3333}, {938, 2551}, {944, 5759}, {946, 3813}, {950, 12527}, {959, 1219}, {961, 1257}, {962, 6764}, {965, 54385}, {966, 39581}, {976, 1468}, {978, 3976}, {986, 4646}, {988, 4255}, {991, 64739}, {993, 5138}, {994, 56125}, {995, 19267}, {997, 999}, {1000, 34919}, {1012, 37569}, {1015, 21830}, {1026, 3675}, {1038, 34046}, {1040, 7074}, {1046, 5255}, {1052, 47623}, {1054, 5524}, {1056, 60987}, {1066, 3682}, {1072, 5721}, {1086, 1738}, {1120, 9432}, {1125, 3589}, {1145, 10427}, {1146, 18413}, {1149, 22220}, {1150, 26227}, {1156, 1320}, {1158, 8668}, {1159, 15346}, {1201, 21330}, {1210, 1329}, {1211, 40952}, {1214, 3190}, {1215, 3741}, {1222, 1431}, {1259, 37579}, {1260, 1617}, {1278, 3621}, {1283, 34997}, {1284, 21320}, {1292, 28914}, {1293, 65878}, {1317, 6068}, {1326, 16702}, {1330, 5015}, {1331, 2361}, {1351, 1482}, {1353, 1483}, {1362, 3323}, {1387, 18254}, {1388, 56387}, {1393, 56198}, {1400, 3965}, {1407, 60786}, {1411, 60049}, {1418, 4334}, {1423, 4073}, {1454, 11501}, {1465, 4551}, {1473, 37577}, {1475, 33299}, {1478, 3419}, {1479, 58798}, {1486, 24320}, {1490, 64077}, {1495, 41607}, {1512, 37725}, {1537, 12665}, {1538, 21635}, {1621, 3219}, {1638, 30704}, {1639, 30700}, {1647, 61176}, {1691, 11364}, {1697, 4326}, {1698, 3697}, {1699, 5927}, {1706, 3339}, {1707, 3052}, {1722, 17054}, {1727, 65144}, {1737, 5123}, {1739, 31855}, {1741, 55111}, {1754, 56178}, {1755, 20785}, {1768, 5537}, {1788, 7080}, {1824, 1889}, {1829, 1843}, {1834, 10381}, {1836, 3434}, {1837, 3436}, {1861, 1876}, {1875, 5081}, {1888, 5174}, {1898, 11415}, {1902, 12294}, {1936, 51361}, {1961, 4038}, {1962, 3989}, {1974, 11363}, {1992, 3241}, {1993, 41740}, {1998, 54408}, {1999, 21334}, {2076, 3099}, {2078, 37736}, {2093, 2097}, {2098, 11682}, {2099, 3872}, {2100, 15162}, {2101, 15163}, {2102, 2104}, {2103, 2105}, {2113, 3797}, {2136, 2951}, {2170, 20593}, {2177, 4414}, {2194, 40571}, {2223, 3286}, {2225, 46148}, {2238, 3290}, {2260, 3949}, {2262, 5819}, {2263, 6180}, {2264, 5279}, {2280, 5282}, {2283, 36819}, {2285, 3713}, {2292, 2667}, {2294, 15984}, {2308, 4722}, {2318, 25091}, {2319, 52211}, {2320, 55920}, {2321, 32118}, {2330, 2646}, {2331, 23052}, {2334, 56328}, {2352, 3998}, {2446, 24646}, {2447, 24647}, {2475, 5178}, {2476, 61013}, {2481, 33675}, {2611, 21326}, {2643, 20360}, {2650, 3728}, {2669, 18827}, {2887, 29673}, {2895, 4914}, {2900, 7580}, {2901, 24068}, {2930, 2948}, {2968, 51368}, {2979, 23155}, {2991, 15382}, {2999, 3677}, {3006, 3936}, {3008, 40538}, {3011, 17724}, {3021, 44045}, {3030, 5212}, {3035, 3660}, {3036, 12736}, {3037, 65523}, {3038, 51615}, {3039, 59814}, {3040, 12016}, {3041, 11028}, {3042, 59816}, {3058, 17781}, {3060, 41717}, {3062, 3680}, {3085, 26066}, {3086, 25681}, {3094, 9941}, {3097, 3795}, {3098, 3579}, {3110, 6629}, {3119, 52888}, {3120, 32856}, {3121, 8620}, {3126, 52228}, {3149, 12704}, {3175, 32915}, {3185, 20760}, {3187, 3891}, {3198, 24310}, {3208, 21872}, {3210, 20012}, {3214, 24443}, {3216, 3953}, {3220, 20468}, {3226, 33680}, {3240, 4003}, {3244, 3629}, {3245, 5528}, {3248, 23579}, {3252, 22116}, {3255, 5559}, {3263, 18157}, {3271, 49771}, {3293, 3670}, {3295, 3927}, {3296, 17582}, {3303, 3951}, {3304, 3984}, {3305, 3715}, {3306, 3711}, {3313, 18732}, {3315, 7292}, {3340, 4853}, {3358, 12114}, {3361, 5438}, {3421, 18391}, {3428, 18446}, {3445, 56630}, {3452, 3816}, {3474, 9965}, {3476, 12848}, {3485, 8232}, {3487, 19843}, {3501, 4515}, {3507, 18208}, {3550, 4650}, {3556, 12410}, {3576, 5085}, {3580, 15904}, {3600, 9850}, {3601, 62824}, {3616, 3618}, {3617, 3620}, {3619, 3983}, {3622, 27268}, {3623, 3890}, {3624, 4533}, {3625, 3630}, {3626, 3631}, {3628, 58561}, {3632, 3901}, {3633, 4718}, {3634, 4015}, {3635, 3884}, {3636, 3988}, {3644, 3885}, {3647, 63271}, {3649, 21926}, {3653, 38064}, {3654, 44284}, {3655, 11179}, {3656, 20423}, {3661, 31317}, {3662, 4429}, {3663, 3755}, {3664, 64007}, {3671, 18251}, {3686, 17049}, {3688, 3879}, {3691, 21808}, {3699, 5205}, {3701, 18137}, {3702, 4043}, {3705, 4417}, {3712, 3977}, {3714, 4385}, {3717, 3912}, {3721, 3780}, {3729, 3886}, {3730, 3991}, {3736, 16696}, {3756, 5121}, {3758, 36534}, {3769, 20359}, {3771, 4438}, {3772, 33137}, {3773, 17229}, {3781, 4851}, {3782, 3914}, {3784, 4952}, {3790, 17233}, {3792, 4553}, {3794, 41629}, {3799, 17310}, {3814, 59419}, {3817, 3829}, {3818, 16616}, {3821, 4085}, {3824, 3841}, {3828, 3833}, {3840, 4090}, {3846, 29655}, {3871, 7676}, {3875, 12721}, {3883, 4416}, {3896, 17147}, {3898, 4525}, {3899, 15534}, {3903, 41532}, {3909, 63071}, {3917, 61678}, {3918, 4691}, {3919, 4669}, {3921, 19875}, {3922, 4678}, {3923, 12722}, {3925, 5249}, {3929, 4428}, {3934, 58584}, {3939, 13329}, {3941, 37507}, {3943, 49763}, {3944, 33101}, {3945, 39587}, {3946, 4353}, {3952, 4009}, {3955, 20986}, {3966, 5739}, {3968, 4745}, {3971, 4891}, {3974, 34255}, {3981, 21876}, {3992, 49999}, {3994, 49982}, {3996, 32932}, {4000, 4310}, {4001, 4030}, {4004, 4668}, {4006, 16549}, {4026, 4357}, {4028, 4884}, {4032, 6737}, {4042, 5271}, {4062, 32848}, {4071, 4119}, {4075, 64536}, {4078, 17243}, {4088, 53554}, {4096, 59517}, {4104, 5743}, {4111, 4733}, {4113, 4359}, {4124, 53358}, {4133, 21848}, {4141, 4933}, {4189, 62827}, {4219, 56316}, {4252, 37552}, {4253, 25066}, {4256, 37599}, {4257, 37589}, {4267, 10461}, {4292, 63146}, {4295, 5082}, {4296, 34028}, {4297, 12437}, {4298, 6743}, {4301, 9856}, {4307, 4644}, {4308, 60941}, {4327, 5228}, {4335, 37598}, {4347, 64055}, {4349, 4667}, {4361, 21867}, {4362, 32853}, {4363, 50314}, {4364, 50290}, {4384, 64560}, {4387, 56082}, {4388, 4514}, {4395, 50013}, {4402, 15590}, {4407, 4708}, {4415, 24210}, {4418, 32940}, {4419, 64168}, {4422, 49768}, {4424, 64175}, {4439, 6541}, {4440, 62392}, {4446, 21857}, {4468, 65697}, {4485, 42711}, {4517, 17316}, {4518, 37686}, {4519, 4671}, {4523, 4852}, {4524, 17069}, {4532, 51006}, {4539, 25055}, {4541, 4987}, {4547, 19878}, {4557, 20470}, {4576, 16741}, 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{50801, 50958}, {50802, 50959}, {50803, 50960}, {50804, 50961}, {50805, 50962}, {50806, 50963}, {50807, 50964}, {50808, 50965}, {50809, 50966}, {50810, 50967}, {50812, 50968}, {50813, 50969}, {50814, 50970}, {50815, 50971}, {50816, 50972}, {50817, 50973}, {50818, 50974}, {50819, 50975}, {50820, 50976}, {50821, 50977}, {50822, 51184}, {50823, 50978}, {50824, 50979}, {50825, 50980}, {50826, 50981}, {50827, 50982}, {50828, 50983}, {50829, 50984}, {50830, 50985}, {50831, 50986}, {50832, 50987}, {50833, 50988}, {50839, 51001}, {50840, 51092}, {50841, 51158}, {50843, 51008}, {50844, 51199}, {50847, 51202}, {50848, 51011}, {50849, 51012}, {50850, 51205}, {50851, 51014}, {50852, 51015}, {50853, 51016}, {50854, 51017}, {50856, 51018}, {50857, 51019}, {50862, 51022}, {50863, 51216}, {50864, 51023}, {50865, 51024}, {50866, 51167}, {50867, 51217}, {50868, 51025}, {50869, 51026}, {50871, 51027}, {50872, 51028}, {50873, 51029}, {50874, 51164}, {51037, 51188}, {51059, 51096}, {51067, 51142}, {51069, 51143}, {51073, 51128}, {51074, 51129}, {51075, 51130}, {51076, 51131}, {51077, 51132}, {51078, 51133}, {51079, 51134}, {51080, 51135}, {51082, 51136}, {51084, 51137}, {51085, 51138}, {51086, 51139}, {51087, 51140}, {51088, 51141}, {51097, 51149}, {51104, 51153}, {51105, 51185}, {51108, 51154}, {51109, 51156}, {51114, 51159}, {51115, 51160}, {51118, 51163}, {51119, 51165}, {51120, 51166}, {51193, 63022}, {51206, 51689}, {51207, 51691}, {51223, 59760}, {51416, 64115}, {51514, 51515}, {51687, 59681}, {51694, 51730}, {51697, 51731}, {51700, 51732}, {51701, 51733}, {51702, 51734}, {51703, 51735}, {51704, 51736}, {51705, 51737}, {51707, 51739}, {51711, 51740}, {51712, 51741}, {51713, 51742}, {51718, 51744}, {51719, 51745}, {51722, 51746}, {51768, 63210}, {51837, 56147}, {52151, 52662}, {52362, 64339}, {52510, 52511}, {52805, 60902}, {52853, 52856}, {52901, 53114}, {53093, 64953}, {53298, 61428}, {53648, 56653}, {54159, 63992}, {54203, 61146}, {54289, 64349}, {54290, 61763}, {54392, 60958}, {54398, 62864}, {54400, 60689}, {55082, 60733}, {55399, 61397}, {55400, 61398}, {55582, 64202}, {55591, 63468}, {55614, 63469}, {55668, 58219}, {55673, 58221}, {55697, 58230}, {55699, 64954}, {55704, 58232}, {55711, 64952}, {55921, 56114}, {56028, 56203}, {56090, 56263}, {56101, 56262}, {56714, 56719}, {57785, 57815}, {58380, 58384}, {58385, 58386}, {58398, 58399}, {58404, 58569}, {58421, 58604}, {58440, 58614}, {58443, 59726}, {58444, 58616}, {58449, 58619}, {58453, 58625}, {58463, 58626}, {58469, 58471}, {58474, 58532}, {58490, 58491}, {58534, 58535}, {59385, 59387}, {59512, 59554}, {59516, 59615}, {59544, 59580}, {59547, 59583}, {59565, 64545}, {59646, 64543}, {59712, 61174}, {61014, 63987}, {61155, 62838}, {61157, 64343}, {61509, 61510}, {61523, 61526}, {61533, 61539}, {61534, 61551}, {61535, 61628}, {61596, 61597}, {62214, 63515}, {62777, 62800}, {63037, 63511}, {63254, 63270}, {63275, 64201}, {63333, 63384}, {63354, 63359}, {63360, 63381}, {63643, 64109}, {63688, 63698}, {63968, 64121}, {64184, 64429}, {64189, 66002}, {64204, 64698}, {64284, 64294}, {64313, 64332}, {64325, 64335}, {64526, 64539}, {64535, 64542}, {64567, 64577}, {64661, 64692}, {66007, 66008}, {66058, 66062}

X(518) = isogonal conjugate of X(105)
X(518) = isotomic conjugate of X(2481)
X(518) = polar conjugate of X(54235)
X(518) = anticomplement of the isogonal conjugate of X(105)
X(518) = isogonal conjugate of the anticomplement of X(120)
X(518) = isogonal conjugate of the complement of X(20344)
X(518) = isotomic conjugate of the anticomplement of X(6184)
X(518) = isotomic conjugate of the complement of X(39350)
X(518) = isotomic conjugate of the isogonal conjugate of X(2223)
X(518) = isogonal conjugate of the isotomic conjugate of X(3263)
X(518) = isotomic conjugate of the polar conjugate of X(5089)
X(518) = isogonal conjugate of the polar conjugate of X(46108)
X(518) = polar conjugate of the isotomic conjugate of X(25083)
X(518) = polar conjugate of the isogonal conjugate of X(20752)
X(518) = Thomson-isogonal conjugate of X(1292)
X(518) = X(i)-line-conjugate of X(j) for these (i,j): {{1, 6}, {30, 513}}
X(518) = complementary conjugate of X(120)
X(518) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,6184), (4,120), (335,37)
X(518) = crosspoint of X(1) and X(291)
X(518) = crosssum of X(i) and X(j) for these (i,j): (1,238), (56,1456)
X(518) = crossdifference of every pair of points on line X(6)X(513)
X(518) = X(i)-Hirst inverse of X(j) for these (i,j): (1,9), (6,1083)
X(518) = X(i)-line conjugate of X(j) for these (i,j): (1,6), (30,513)
X(518) = isotomic conjugate of X(2481)
X(518) = X(2781) of Fuhrmann triangle
X(518) = ideal point of PU(28)
X(518) = crossdifference of PU(i) for i in (46, 54)
X(518) = X(6)-isoconjugate of X(673)
X(518) = trilinear pole of line X(665)X(1642)
X(518) = perspector of conic {A,B,C,X(2),X(100),PU(112)}
X(518) = trilinear square root of X(4712)
X(518) = Cundy-Parry Psi transform of X(14268)
X(518) = Thomson-isogonal conjugate of X(1292)
X(518) = X(63)-isoconjugate of X(8751)
X(518) = X(92)-isoconjugate of X(32658)
X(518) = Lucas-isogonal conjugate of X(1292)
X(518) = polar conjugate of isotomic conjugate of X(25083)
X(518) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {{1, 6, 1386}, {1, 9, 1001}, {1, 37, 15569}, {1, 44, 3246}, {1, 72, 960}, {1, 238, 1279}, {1, 392, 10179}, {1, 405, 51715}, {1, 960, 58679}, {1, 984, 37}, {1, 1001, 42819}, {1, 1743, 7290}, {1, 1757, 238}, {1, 3242, 49465}, {1, 3243, 42871}, {1, 3555, 34791}, {1, 3640, 45713}, {1, 3641, 45714}, {1, 3751, 6}, {1, 3973, 60846}, {1, 4649, 1100}, {1, 5220, 15254}, {1, 5223, 9}, {1, 5234, 5436}, {1, 5247, 1104}, {1, 5525, 45765}, {1, 5588, 18992}, {1, 5589, 18991}, {1, 5692, 392}, {1, 5728, 5572}, {1, 5904, 72}, {1, 6762, 12513}, {1, 11523, 12635}, {1, 12513, 11260}, {1, 15298, 954}, {1, 15299, 42884}, {1, 16475, 38315}, {1, 16496, 3242}, {1, 18412, 5728}, {1, 18991, 45398}, {1, 18992, 45399}, {1, 19003, 11370}, {1, 19004, 11371}, {1, 34791, 58609}, {1, 35026, 5701}, {1, 41229, 405}, {1, 42871, 15570}, {1, 45426, 7968}, {1, 45427, 7969}, {1, 49448, 984}, {1, 49490, 49478}, {1, 49498, 49490}, {1, 49503, 49515}, {1, 49675, 4864}, {1, 49712, 44}, {1, 54386, 1191}, {1, 57279, 958}, {1, 60846, 35227}, {1, 64070, 4663}, {2, 210, 3740}, {2, 354, 3742}, {2, 3681, 210}, {2, 3740, 58451}, {2, 3742, 3848}, {2, 3873, 354}, {2, 4430, 3873}, {2, 4661, 3681}, {2, 5686, 38057}, {2, 11038, 38053}, {2, 47358, 51003}, {2, 50075, 51034}, {2, 50999, 47358}, {2, 59405, 38186}, {2, 59406, 38047}, {2, 63961, 61686}, {3, 3811, 56176}, {3, 12675, 58567}, {3, 63976, 58637}, {4, 39898, 64085}, {6, 45, 36404}, {6, 405, 51743}, {6, 3242, 1}, {6, 3751, 4663}, {6, 12594, 611}, {6, 12595, 613}, {6, 16496, 49465}, {6, 16777, 16972}, {6, 35963, 5701}, {6, 38315, 16475}, {6, 49509, 37}, {6, 49706, 238}, {6, 50995, 9}, {6, 64070, 3751}, {7, 8, 2550}, {7, 69, 47595}, {7, 2550, 5880}, {7, 3059, 15587}, {7, 7672, 65}, {7, 34784, 3059}, {7, 41228, 5784}, {7, 59413, 59412}, {8, 65, 5836}, {8, 69, 3416}, {8, 75, 3696}, {8, 388, 5794}, {8, 1469, 17792}, {8, 3868, 65}, {8, 4645, 32850}, {8, 8581, 15587}, {8, 24349, 75}, {8, 32049, 32537}, {8, 41228, 3059}, {8, 49499, 49483}, {8, 49698, 49694}, {8, 49707, 49698}, {8, 49714, 49702}, {8, 54383, 3779}, {8, 59412, 59413}, {9, 1001, 15254}, {9, 3243, 1}, {9, 5220, 15481}, {9, 5223, 5220}, {9, 5572, 58608}, {9, 15185, 5572}, {9, 15298, 15296}, {9, 15299, 15297}, {9, 42871, 42819}, {9, 51058, 37}, {9, 51194, 6}, {10, 141, 3844}, {10, 142, 3826}, {10, 942, 3812}, {10, 3775, 17239}, {10, 3836, 3823}, {10, 3874, 942}, {10, 5542, 142}, {10, 21620, 25466}, {10, 24325, 3739}, {10, 34790, 4662}, {10, 38054, 38204}, {10, 49479, 24325}, {10, 49504, 49510}, {10, 49505, 49511}, {10, 49510, 49457}, {10, 49511, 141}, {10, 49529, 49524}, {10, 49535, 49479}, {10, 49536, 49529}, {10, 49676, 3836}, {10, 49697, 49693}, {10, 49713, 49701}, {10, 51706, 8728}, {11, 908, 5087}, {11, 51463, 26015}, {31, 3938, 3744}, {31, 32912, 4641}, {35, 6763, 3916}, {36, 15015, 35271}, {37, 16666, 36409}, {37, 49478, 1}, {37, 49515, 984}, {37, 64546, 58620}, {38, 42, 3666}, {38, 46904, 46901}, {40, 1071, 9943}, {40, 5732, 11495}, {40, 6765, 3913}, {42, 22275, 22325}, {42, 46901, 46904}, {43, 982, 3752}, {43, 62865, 982}, {44, 1279, 238}, {44, 4864, 1279}, {44, 6603, 52985}, {55, 63, 4640}, {55, 41711, 3870}, {56, 78, 59691}, {56, 41712, 1445}, {57, 200, 1376}, {57, 17625, 63994}, {57, 46917, 64112}, {63, 3870, 55}, {63, 16465, 10391}, {65, 1469, 24471}, {65, 3059, 2550}, {65, 5784, 5880}, {65, 5836, 10107}, {65, 8581, 7}, {65, 41228, 15587}, {69, 3779, 17792}, {69, 3868, 24476}, {69, 54383, 4259}, {72, 392, 5692}, {72, 405, 45120}, {72, 3555, 1}, {72, 5728, 9}, {72, 14054, 44547}, {72, 15185, 1001}, {72, 34791, 58679}, {72, 66009, 15254}, {75, 20448, 20435}, {75, 24349, 49483}, {75, 49450, 8}, {75, 49499, 24349}, {78, 62874, 56}, {81, 3920, 3745}, {84, 6769, 64074}, {100, 3218, 1155}, {100, 3935, 3689}, {100, 62235, 3218}, {100, 62236, 3935}, {110, 41742, 32126}, {141, 49481, 3739}, {141, 49524, 10}, {141, 51150, 142}, {141, 51738, 8728}, {142, 5542, 25557}, {142, 24393, 10}, {142, 40659, 58634}, {144, 145, 390}, {144, 192, 51052}, {144, 193, 51190}, {144, 390, 5698}, {144, 3869, 64723}, {144, 30628, 14100}, {145, 192, 49470}, {145, 193, 51192}, {145, 3869, 3057}, {145, 20072, 49709}, {145, 31302, 192}, {145, 49447, 49462}, {145, 49470, 49475}, {145, 49501, 49523}, {145, 49704, 49695}, {145, 49709, 49699}, {145, 51192, 49681}, {149, 17484, 5057}, {165, 3158, 4421}, {165, 10167, 10178}, {192, 193, 49496}, {192, 31302, 49447}, {192, 49447, 49523}, {192, 49470, 49462}, {192, 49501, 49513}, {200, 62823, 57}, and many others


X(519) = ISOGONAL CONJUGATE OF X(106)

Trilinears    (2a - b - c)/a : (2b - c - a)/b : (2c - a - b)/c
Trilinears    1 - 2 csc A/2 sin B/2 sin C/2 : :
Trilinears    3 r - 2 R sin B sin C : :
Barycentrics   2a - b - c : 2b - c - a : 2c - a - b
X(519) = X(1) - X(2)

As the isogonal conjugate of a point on the circumcircle, X(519) lies on the line at infinity.

Let A'B'C' be the incentral triangle. Let A″ be the reflection of A in A', and define B″ and C″ cyclically. Let A* be the cevapoint of B″ and C″, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(519). (Randy Hutson, December 26, 2015)

Let A'B'C' be triangle T(-2,1). Let A″ be the trilinear product B'*C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(519). The lines A'A″, B'B″, C'C″ concur in X(8). (Randy Hutson, December 26, 2015)

X(519) lies on these (parallel) lines: 1,2   6,996   9,1000   30,511   35,2975   36,100   37,1573   40,376   44,2325   55,956   57,3476   58,1043   63,1727   65,553   72,950   74,2692   80,908   98,2705   99,2712   101,2726   102,2731   103,2737   104,2077   105,2748   106,1120   107,2755   108,2756   109,2757   110,2758   111,2759   112,2760   121,3544   188,1128   210,392   214,1145   226,2099   238,765   244,1739   291,3227   320,668   346,1743   350,668   355,381   388,3340   405,3303   428,1829   447,648   474,3304   484,3218   495,2886   496,1329   497,3421   549,1385   573,3169   594,1100   595,2985   597,1386   599,3242   664,1323   666,1121   672,1018   751,984   958,3295   962,3543   966,3247   999,1376   1015,1575   1056,2550   1058,2551   1107,1500   1126,1220   1150,2177   1377,3297   1378,3298   1387,3036   1420,1788   1449,2345   1478,3434   1479,3436   1697,1776   1706,3333   1785,1897   1834,3454   1837,2098   1861,1870   1862,1878   2093,2094   2654,3191   3158,3524

X(519) = isogonal conjugate of X(106)
X(519) = isotomic conjugate of X(903)
X(519) = complementary conjugate of X(121)
X(519) = anticomplementary conjugate of X(21290)
X(519) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,4370). (4,121), (80,10)
X(519) = crosssum of X(i) and X(j) for these (i,j): (6,902), (56,1457)
X(519) = crossdifference of every pair of points on line X(6)X(649)
X(519) = X(i)-Hirst inverse of X(j) for these (i,j): (513, 537), (514,545)
X(519) = X(2777)-of-Fuhrmann-triangle
X(519) = bicentric sum of PU(24)
X(519) = intercept of Nagel line and the line at infinity
X(519) = trilinear pole of line X(900)X(1635)
X(519) = X(6)-isoconjugate of X(88)
X(519) = trilinear square root of X(4738)
X(519) = barycentric cube root of X(8028)
X(519) = Cundy-Parry Psi transform of X(14261)
X(519) = Thomson-isogonal conjugate of X(1293)
X(519) = Lucas-isogonal conjugate of X(1293)
X(519) = polar conjugate of X(6336)
X(519) = pole wrt polar circle of trilinear polar of X(6336) (line X(4)X(2457))
X(519) = X(19)-isoconjugate of X(1797)
X(519) = X(63)-isoconjugate of X(8752)
X(519) = X(92)-isoconjugate of X(32659)
X(519) = barycentric square root of X(4370)
X(519) = homothetic center of Gemini triangle 28 and cross-triangle of Gemini triangles 20 and 28


X(520) = ISOGONAL CONJUGATE OF X(107)

Trilinears    (cos A)(sin 2B - sin 2C) : (cos B)(sin 2C - sin 2A) : (cos C)(sin 2A - sin 2B)
Trilinears    (cos A)(tan B - tan C) : :
Barycentrics    (sin 2A)(sin 2B - sin 2C) : (sin 2B)(sin 2C - sin 2A) : (sin 2C)(sin 2A - sin 2B)
Barycentrics    sec^2 B - sec^2 C : :
Barycentrics    tan^2 B - tan^2 C : :
Barycentrics    (cos^2 A)(b^2 - c^2) : :
Barycentrics    csc 2B - csc 2C : :

As the isogonal conjugate of a point on the circumcircle, X(520) lies on the line at infinity.

X(520) lies on these (parallel) lines: 6,2435   30,511   69,879   74,2693   98,2706   99,2713   100,2719   101,2727   102,2732   103,2738   104,2744   105,2749   106,2755   108,2761   109,2762   110,250   111,2763   112,2764   340,850   647,652   1364,2632   2451,2489   3265,4131

X(520) = isogonal conjugate of X(107)
X(520) = complementary conjugate of X(122)
X(520) = anticomplementary conjugate of X(34186)
X(520) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,35071), (4,122), (68,125), (110,3)
X(520) = crosspoint of X(3) and X(110)
X(520) = crosssum of X(i) and X(j) for these (i,j): (4,523), (51,647), (512,800)
X(520) = crossdifference of every pair of points on line X(4)X(6)
X(520) = isotomic conjugate of X(6528)
X(520) = perspector of hyperbola {A,B,C,X(2),X(3)}}
X(520) = intersection of trilinear polars of X(2) and X(3)
X(520) = X(6)-isoconjugate of X(823)
X(520) = X(92)-isoconjugate of X(112)
X(520) = trilinear pole of line X(1636)X(2972)
X(520) = Thomson-isogonal conjugate of X(1294)
X(520) = Lucas-isogonal conjugate of X(1294)
X(520) = polar conjugate of X(15352)
X(520) = perspector of circumconic centered at X(35071)
X(520) = crosssum of X(24007) and X(24008) (the Kiepert hyperbola intercepts of the orthic axis)
X(520) = pole wrt polar circle of trilinear polar of X(15352) (line X(4)X(51))
X(520) = Cundy-Parry Phi transform of X(39174)


X(521) = ISOGONAL CONJUGATE OF X(108)

Trilinears    (sec B - sec C)(csc A) : (sec C - sec A)(csc B) : (sec A - sec B)(csc C)
Trilinears    (b - c)(b + c - a)(b^2 + c^2 - a^2) : :
Trilinears    directed distance from A to line X(1)X(4) : :
Barycentrics    sec B - sec C : sec C - sec A : sec A - sec B
Barycentrics    (cos A)(cos B - cos C) : :

As the isogonal conjugate of a point on the circumcircle, X(521) lies on the line at infinity.

X(521) lies on these (parallel) lines: 6,2509   30,511   59,100   74,2694   98,2707   99,2714   101,2728   102,2733   103,2739   104,2745   109,2765   110,2766   111,2767   650,1021   651,677   656,810   1364,2968   4025,4131

X(521) = isogonal conjugate of X(108)
X(521) = isotomic conjugate of X(18026)
X(521) = complementary conjugate of X(123)
X(521) = anticomplementary conjugate of X(34188)
X(521) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,35072), (4,123), (100,3)
X(521) = perspector of hyperbola {A,B,C,X(2),X(21)}}
X(521) = crosspoint of X(8) and X(100)
X(521) = crosssum of X(i) and X(j) for these (i,j): (33,650), (56,513), (429,523), (663,1400)
X(521) = crossdifference of every pair of points on line X(6)X(19)
X(521) = polar conjugate of isogonal conjugate of X(36054)
X(521) = X(6)-isoconjugate of X(653)
X(521) = X(92)-isoconjugate of X(1415)
X(521) = Thomson-isogonal conjugate of X(1295)
X(521) = Lucas-isogonal conjugate of X(1295)
X(521) = Cundy-Parry Phi transform of X(39175)


X(522) = ISOGONAL CONJUGATE OF X(109)

Trilinears    (cos B - cos C)(csc A) : (cos C - cos A)(csc B) : (cos A - cos B)(csc C)
Trilinears    d(a,b,c) : : , where d(a,b,c) = directed distance from A to line X(1)X(3)
Barycentrics    cos B - cos C : cos C - cos A : cos A - cos B
Barycentrics    (b - c)(b + c - a) : :
Barycentrics    tan B/2 - tan C/2 : :

As the isogonal conjugate of a point on the circumcircle, X(522) lies on the line at infinity.

X(522) lies on these (parallel) lines: 1,1459   7,2400   9,657   11,3326   30,511   74,2695   75,3261   100,655   101,929   102,2734   103,2723   104,2716   105,2751   106,2757   107,2762   108,2765   109,1309   110,2689   111,2768   112,2769   124,2968   142,3126   190,666   240,656   243,652   649,3509   650,1639   663,1944   664,1275   693,1266   1026,2397   1027,2402   1090,2310   1292,2730   1293,2731   1294,2732   1295,2733   1296,2735   1317,3319   2490,2496   2526,3004   3063,3287

X(522) = isogonal conjugate of X(109)
X(522) = isotomic conjugate of X(664)
X(522) = complementary conjugate of X(124)
X(522) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,124), (8,11), (100,10), (190,9)
X(522) = X(11)-cross conjugate of X(8)
X(522) = crosspoint of X(i) and X(j) for these (i,j): (21,100), (75,190)
X(522) = crosssum of X(i) and X(j) for these (i,j): (6,663), (31,649), (55,652), (65,513), (603,1459), (692,1415)
X(522) = crossdifference of every pair of points on line X(6)X(41)
X(522) = orthopoint of X(515)
X(522) = X(i)-Hirst inverse of X(j) for these (i,j): (514,918), (519,528)
X(522) = isotomic conjugate of X(664)
X(522) = bicentric difference of PU(i) for these i: 20, 129
X(522) = ideal point of PU(i) for these i: 20, 129
X(522) = perspector of circumconic centered at X(1146)
X(522) = center of circumconic that is locus of trilinear poles of lines passing through X(1146)
X(522) = X(2)-Ceva conjugate of X(1146)
X(522) = trilinear pole of line X(11)X(1146)
X(522) = pole wrt polar circle of trilinear polar of X(653)
X(522) = X(48)-isoconjugate (polar conjugate) of X(653)
X(522) = X(6)-isoconjugate of X(651)
X(522) = extouch isogonal conjugate of X(1364)
X(522) = Thomson-isogonal conjugate of X(102)
X(522) = Lucas-isogonal conjugate of X(102)
X(522) = barycentric square root of X(1146)
X(522) = X(92)-isoconjugate of X(32660)


X(523) = ISOGONAL CONJUGATE OF X(110)

Trilinears     sin(B -C) : sin(C - A) : sin(A - B)
Barycentrics    b2 - c2 : c2 - a2 : a2 - b2
Barycentrics    cot B cos 2C - cot C cos 2B : :
Barycentrics    SB - SC : :
Barycentrics    cot B - cot C : :
Barycentrics    cos 2B - cos 2C : :

As the isogonal conjugate of a point on the circumcircle, X(523) lies on the line at infinity.

Let A'B'C' be the 1st Brocard triangle. Let A″B″C″ be the 2nd Brocard triangle. Let A* be the trilinear pole of line A'A″, and define B* and C* cyclically. The lines AA*, BB*, CC* are parallel and meet the line at infinity at X(523). (Randy Hutson, December 26, 2015)

Let A'B'C' be the Feuerbach triangle. Let La be the tangent to conic {A,B,C,B',C'}} at A, and define Lb and Lc cyclically. Let A″ = Lb∩Lc, B″ = Lc∩La, C″ = La∩Lb. The lines A'A″, B'B″, C'C″ concur in X(523); see also X(1109).

Let L be the line X(115)X(125); then X(523) = trilinear pole of L. The line L is also PU(40), the tangent to Steiner inellipse at X(115), the Lemoine axis of the 4th Brocard triangle, the Fermat axis of the 2nd Parry triangle, and the isogonal conjugate of hyperbola {A,B,C,PU(2)}) (Randy Hutson, December 26, 2015)

X(523) lies on the Kiepert parabola and these (parallel) lines: 1,2605   2,1649   4,1552   6,879   11,1090   12,2599   23,385   30,511   59,655   66,2435   74,477   75,876   98,842   99,691   100,1290   101,2690   102,2695   103,2688   104,2687   105,2752   106,2758   107,1304   108,2766   109,2689   110,476   111,2770   112,935   125,2677   140,1116   141,882   160,3164   230,231   250,648   253,2419   325,684   396,3272   656,2457   827,1287   878,3425   885,2346   930,1291   1086,2643   1101,2612   1222,2403   1292,2691   1293,2692   1294,2693   1295,2694   1296,2696   1297,2697   2525,2526   2594,2616

X(523) = isogonal conjugate of X(110)
X(523) = isotomic conjugate of X(99)
X(523) = complementary conjugate of X(125)
X(523) = anticomplementary conjugate of X(3448)
X(523) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,11), (2,115), (4,125), (99,2), (100,442), (107,4), (108,429), (110,5), (112,427), (254,136), (264,338), (476,30), (685,1503), (1113,1312), (1114,1313)
X(523) = cevapoint of X(2) and X(148)
X(523) = cevapoint of polar circle intercepts of Euler line
X(523) = X(i)-cross conjugate of X(j) for these (i,j): (115,2), (125,4)
X(523) = crosspoint of X(i) and X(j) for these (i,j): (2,99), (4,107), (54,110), (112,251)
X(523) = crossdifference of every pair of points on line X(3)X(6)
X(523) = orthopoint of X(30)
X(523) = X(30)-line conjugate of X(511)
X(523) = barycentric product of X(3413) and X(3414)
X(523) = crosssum of X(i) and X(j) for these (i,j): (3,520), (5,523), (6,512), (101,692), (141,525), (184,647), (215,654), (513,942), (521,960), (924,1147)
X(523) = orthojoin of X(115)
X(523) = X(i)-Hirst inverse of X(j) for these (i,j): (6,1316), (30, 542), (512,804)
X(523) = bicentric difference of PU(i) for these i: 4, 5, 11, 43, 45, 61, 132, 173
X(523) = ideal point of PU(i) for these i: 4, 5, 11, 43, 45, 61, 132
X(523) = crossdifference of PU(29)
X(523) = trilinear pole of PU(40); see note above
X(523) = barycentric product of PU(40)
X(523) = trilinear product of PU(71)
X(523) = perspector of the Kiepert hyperbola
X(523) = intersection of the trilinear polars of any 2 points on the Kiepert hyperbola
X(523) = intercept of orthic axis and the line at infinity (the trilinear polars of X(4) and X(2))
X(523) = center of parabola {A,B,C,X(476),X(523)}}, which is the locus of trilinear poles of lines passing through X(115)
X(523) = perspector of ABC and the side-triangle of the medial and orthic triangles
X(523) = perspector of ABC and the vertex-triangle of the tangential triangles of the medial and Feuerbach triangles
X(523) = X(526)-of-orthocentroidal-triangle
X(523) = X(526)-of-X(4)-Brocard-triangle
X(523) = X(30)-of-1st-Parry-triangle
X(523) = X(30)-of-2nd-Parry-triangle
X(523) = orthic isogonal conjugate of X(125)
X(523) = incentral isogonal conjugate of X(3024)
X(523) = pole wrt polar circle of the Euler line
X(523) = X(48)-isoconjugate (polar conjugate) of X(648)
X(523) = X(6)-isoconjugate of X(662)
X(523) = X(1101)-isoconjugate of X(523)
X(523) = X(2)-vertex conjugate of X(23)
X(523) = X(3)-vertex conjugate of X(30)
X(523) = X(4)-vertex conjugate of X(186)
X(523) = barycentric cube root of X(8029)
X(523) = perspector of side- and vertex-triangles of circumanticevian triangles of X(2) and X(4)
X(523) = barycentric square root of X(115)
X(523) = Cundy-Parry Psi transform of X(14264)
X(523) = Thomson isogonal conjugate of X(74)
X(523) = Lucas isogonal conjugate of X(74)
X(523) = trilinear square root of X(1109)
X(523) = barycentric product X(10)*X(514)
X(523) = trilinear product of Kiepert hyperbola intercepts of antiorthic axis


X(524) = ISOGONAL CONJUGATE OF X(111)

Trilinears    (2a2 - b2 - c2)/a : (2b2 - c2 - a2)/b : :
Barycentrics    2 cot A - cot B - cot C : :
X(524) = X(2) - X(6)

As the isogonal conjugate of a point on the circumcircle, X(524) lies on the line at infinity.

Let A'B'C' be the 2nd Brocard triangle. Let A″ be the trilinear pole of line B'C', and define B″, C″ cyclically. The lines AA″, BB″, CC″ are parallel, and meet the line at infinity at X(524). (Randy Hutson, July 20, 2016)

X(524) lies on these (parallel) lines: 2,6   5,576   23,2930   30,511   53,317   67,858   74,2696   76,598   98,2709   99,843   100,2721   101,2729   102,2735   103,2740   104,2746   105,2753   106,2759   107,2763   108,2767   109,2768   110,2770   140,575   182,549   187,2482   237,1634   239,320   249,1691   287,1494   297,340   316,671   319,594   332,2305   338,3260   376,1350   381,1351   397,633   398,634   428,1843   441,3284   468,2192   487,1152   488,1151   551,1386   620,2030   637,3070   638,3071   694,3228   1030,1444   1084,3229   1146,1944   1238,2965   1330,1834   1901,2893   1989,2987   2094,2097   3056,3058   3241,3242

X(524) = isogonal conjugate of X(111)
X(524) = isotomic conjugate of X(671)
X(524) = complementary conjugate of X(126)
X(524) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,126), (67,141)
X(524) = X(187)-cross conjugate of X(468)
X(524) = crosssum of X(6) and X(187)
X(524) = crossdifference of every pair of points on line X(6)X(512)
X(524) = X(i)-line conjugate of X(j) for these (i,j): (4,126), (67,141)
X(524) = intercept of line X(2)X(6) and the line at infinity (trilinear polars of X(99) and X(2))
X(524) = perspector of circumconic centered at X(2482)
X(524) = center of circumconic that is locus of trilinear poles of lines passing through X(2482)
X(524) = X(2)-Ceva conjugate of X(2482)
X(524) = X(2854)-of-4th-Brocard-triangle
X(524) = X(2854)-of-orthocentroidal-triangle
X(524) = X(2854)-of-X(4)-Brocard-triangle
X(524) = crosspoint of X(6) and X(2930) wrt both the excentral and tangential triangles
X(524) = X(6)-isoconjugate of X(897)
X(524) = polar conjugate of X(17983)
X(524) = X(63)-isoconjugate of X(8753)
X(524) = barycentric cube root of X(8030)
X(524) = perspector of ABC and unary cofactor triangle of 2nd Ehrmann triangle
X(524) = X(8705)-of-circumsymmedial-triangle
X(524) = Cundy-Parry Phi transform of X(13608)
X(524) = Cundy-Parry Psi transform of X(14262)
X(524) = Thomson-isogonal conjugate of X(1296)
X(524) = Lucas-isogonal conjugate of X(1296)
X(524) = trilinear pole of line X(351)X(690) (line is perspectrix of ABC and 2nd Brocard triangle, and the tangent to the Steiner inellipse at X(2482))


X(525) = ISOGONAL CONJUGATE OF X(112)

Trilinears    (b cos B - c cos C)/a : (c cos C - a cos A)/b : (a cos A - b cos B)/c
Trilinears    cot A sin(B - C) : :
Barycentrics    b cos B - c cos C : c cos C - a cos A : a cos A - b cos B
Barycentrics    sin 2B - sin 2C : :
Barycentrics    tan B - tan C : :
Barycentrics    tan B cos 2B - tan C cos 2C : :
Barycentrics    (b^2 - c^2)(b^2 + c^2 - a^2) : :

X(525) is the barycentric multiplier for the Jerabek hyperbola. (The barycentric product of X(525) and the circumcircle is the Jerabek hyperbola.) (Randy Hutson, August 19, 2019)

As the isogonal conjugate of a point on the circumcircle, X(525) lies on the line at infinity.

X(525) lies on these (parallel) lines: 2,1640   3,878   4,2435   30,511   74,2697   98,2710   99,249   100,2722   103,2741   104,2747   105,2754   106,2760   107,2764   109,2769   110,935   112,2867   127,1562   297,850   323,401   339,3269   441,647   669,2528   1073,2416   1636,3268   1975,2422   2474,2514   2485,2506   2513,2531   2632,2968

X(525) = isogonal conjugate of X(112)
X(525) = isotomic conjugate of X(648)
X(525) = complementary conjugate of X(127)
X(525) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,15526), (4,127), (69,125), (76,339), (99,3), (110,141), (190,440), (253,122)
X(525) = X(i)-cross conjugate of X(j) for these (i,j): (115,68), (122,253), (125,69)
X(525) = crosspoint of X(76) and X(99)
X(525) = crosssum of X(i) and X(j) for these (i,j): (6,647), (32,512), (427,523)
X(525) = crossdifference of every pair of points on line X(6)X(25)
X(525) = perspector of hyperbola {A,B,C,X(2),X(69)}}
X(525) = intersection of trilinear polars of X(2) and X(69)
X(525) = trilinear pole of line X(122)X(125)
X(525) = pole wrt polar circle of trilinear polar of X(107) (line X(4)X(6))
X(525) = X(48)-isoconjugate (polar conjugate) of X(107)
X(525) = X(6)-isoconjugate of X(162)
X(525) = X(92)-isoconjugate of X(1576)
X(525) = intersection of tangents to Steiner inellipse at X(2454) and X(2455)
X(525) = crosspoint wrt medial triangle of X(2454) and X(2455)
X(525) = Thomson-isogonal conjugate of X(1297)
X(525) = Lucas-isogonal conjugate of X(1297)
X(525) = crosssum of Jerabek hyperbola intercepts of Lemoine axis
X(525) = barycentric product of Jerabek hyperbola intercepts of de Longchamps line


X(526) = ISOGONAL CONJUGATE OF X(476)

Trilinears    (1 + 2 cos 2A)sin(B - C) : :
Trilinears    a(b^2 - c^2)[(b^2 + c^2 - a^2)^2 - b^2c^2] : :
As the isogonal conjugate of a point on the circumcircle, X(526) lies on the line at infinity.

X(526) lies on these (parallel) lines: 6,2492   30,511   67,879   110,351   125,3134   686,2433   895,2987   1177,2435   1769,2650   2611,3024

X(526) = isogonal conjugate of X(476)
X(526) = isotomic conjugate of X(35139)
X(526) = complementary conjugate of X(3258)
X(526) = X(2)-Ceva conjugate of X(18334)
X(526) = X(110)-Ceva conjugate of X(1511)
X(526) = crosspoint of X(74) and X(110)
X(526) = crosssum of X(30) and X(523)
X(526) = crossdifference of every pair of points on line X(6)X(13)
X(526) = X(i)-isoconjugate of X(j) for these (i,j): (49,2166), (265,2964)
X(526) = perspector of hyperbola {A,B,C,X(2),X(15),X(16)}}
X(526) = intersection of trilinear polars of X(2), X(15), and X(16)
X(526) = trilinear pole of line of: X(2088), tripolar centroid of X(15), tripolar centroid of X(16)
X(526) = X(523)-of-orthocentroidal-triangle
X(526) = X(523)-of-X(4)-Brocard-triangle
X(526) = X(6088)-of-circumsymmedial-triangle
X(526) = Thomson-isogonal conjugate of X(477)
X(526) = Lucas-isogonal conjugate of X(477)
X(526) = X(6)-isoconjugate of X(32680)
X(526) = X(92)-isoconjugate of X(32662)
X(526) = intersection of tangents to Walsmith rectangular hyperbola at PU(4)

leftri

Centers X(527)-X(565)

rightri

Centers X(527)-X(565) were added to ETC on the first day of 2001.

X(527) = DIRECTION OF VECTOR AX + BX + CX, where X = X(7)

Trilinears    (2ax - by - cz)/a, where x = x(a,b,c) = 1/[a(b + c - a)], y = x(b,c,a), z = x(c,a,b)
Barycentrics    2a^2 - a(b + c) - (b - c)^2 : :

X(527) lies on the line at infinity.

X(527) lies on these (parallel) lines: 2,7   30,511   44,1086   69,2321   72,1242   190,320   200,3474   239,1266   269,2324   347,1419   376,2096   381,2095   390,3241   551,993   599,2097   651,2323   666,673   896,3011   1156,3254   1478,2093   1738,1757   2340,3000   2346,3255   2551,3339   2951,3174

X(527) = isogonal conjugate of X(2291)
X(527) = isotomic conjugate of X(1121)
X(527) = X(2)-Ceva conjugate of X(35110)
X(527) = crosssum of X(6) and X(1055)
X(527) = crossdifference of every pair of points on line X(6)X(663)
X(527) = perspector of circumconic centered at X(35110)


X(528) = DIRECTION OF VECTOR AX + BX + CX, where X = X(11)

Trilinears    (2ax - by - cz)/a, where x = x(A,B,C) = 1 - cos(B-C), y = x(B,C,A), z = x(C,A,B)

Let Ia be the reflection of X(1) in the perpendicular bisector of BC, and define Ib and Ic cyclically. X(528) = X(518) of IaIbIc. (Randy Hutson, January 29, 2018)

X(528) lies on the line at infinity.

X(528) lies on these (parallel) lines: 1,1086   2,11   7,664   8,190   9,80   30,511   104,376   119,381   142,214   153,3543   377,3303   428,1824   549,1484   962,3189   1279,1738   1329,1479   1537,3174   1699,3158   1750,2900   1770,3555   2094,3474   3008,3246   3032,3034

X(528) = isogonal conjugate of X(840)
X(528) = X(2)-Ceva conjugate of X(35113)
X(528) = crossdifference of every pair of points on line X(6)X(665)
X(528) = X(519)-Hirst inverse of X(522)
X(528) = X(1503)-of-Fuhrmann-triangle
X(528) = perspector of circumconic centered at X(35113)


X(529) = DIRECTION OF VECTOR AX + BX + CX, where X = X(12)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where
                        f(a,b,c) = (2ax - by - cz)/a, x = x(A,B,C) = 1 + cos(B-C), y = x(B,C,A), z = x(C,A,B)

Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(529) lies on the line at infinity.

X(529) lies on these (parallel) lines: 2,12   8,3474   30,511   36,3035   46,1706   329,3476   428,1828   484,1145   495,993   908,1319   956,1478   1001,1056   1146,3509   1376,3421   2098,3058   2478,3304   3036,3218

X(529) = isogonal conjugate of X(38882)


X(530) = DIRECTION OF VECTOR AX + BX + CX, where X = X(13)

Trilinears    (2ax - by - cz)/a : : , where x = x(A,B,C) = csc(A + π/3), y = x(B,C,A), z = x(C,A,B)

Let A' be the nine-point center of BCX(14), and define B' and C' cyclically. A'B'C' is an equilateral triangle concyclic with X(2) and X(14) (inscribed in the 2nd Hutson circle). The isogonal (and isotomic) conjugate of X(2) wrt A'B'C' is X(530). (Randy Hutson, February 10, 2016)

X(530) lies on the line at infinity.

X(530) lies on these (parallel) lines: 2,13   14,671   30,511   99,299   115,395   148,3181   187,396   298,316   619,2482

X(530) = isogonal conjugate of X(2378)


X(531) = DIRECTION OF VECTOR AX + BX + CX, where X = X(14)

Trilinears    (2ax - by - cz)/a : : where x = x(A,B,C) = csc(A - π/3), y = x(B,C,A), z = x(C,A,B)

Let A' be the nine-point center of BCX(13), and define B' and C' cyclically. A'B'C' is an equilateral triangle concyclic with X(2) and X(13) (inscribed in the 1st Hutson circle). The isogonal (and isotomic) conjugate of X(2) wrt A'B'C' is X(531). (Randy Hutson, February 10, 2016)

X(531) lies on the line at infinity.

X(531) lies on these (parallel) lines: 2,14   13,671   30,511   99,298   115,396   148,3180   187,395   299,316   618,2482

X(531) = isogonal conjugate of X(2379)


X(532) = DIRECTION OF VECTOR AX + BX + CX, where X = X(17)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where
                        f(a,b,c) = (2ax - by - cz)/a, x = x(A,B,C) = csc(A + π/6), y = x(B,C,A), z = x(C,A,B)

Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(532) lies on the line at infinity.

X(532) lies on these (parallel) lines: 2,17   13,298   14,622   15,616   16,299   30,511   395,624   396,618   397,635

X(532) = isogonal conjugate of X(2380)


X(533) = DIRECTION OF VECTOR AX + BX + CX, where X = X(18)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where
                        f(a,b,c) = (2ax - by - cz)/a, x = x(A,B,C) = csc(A - π/6), y = x(B,C,A), z = x(C,A,B)

Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(533) lies on the line at infinity.

X(533) lies on these (parallel) lines: 2,18    13,621   14,299   15,298  16,617    30,511   395,619   396,623   398,636

X(533) = isogonal conjugate of X(2381)


X(534) = DIRECTION OF VECTOR AX + BX + CX, where X = X(19)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where
                        f(a,b,c) = (2ax - by - cz)/a, x = x(A,B,C) = tan A, y = x(B,C,A), z = x(C,A,B)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(534) lies on these (parallel) lines: 2,19   30,511   553,1407   1441,1839

X(534) = isogonal conjugate of X(38883)


X(535) = DIRECTION OF VECTOR AX + BX + CX, where X = X(36)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where
                        f(a,b,c) = (2ax - by - cz)/a, x = x(A,B,C) = 1 - 2 cos A, y = x(B,C,A), z = x(C,A,B)

Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(535) lies on the line at infinity.

X(535) lies on these (parallel) lines: 2,36   8,3245   10,1155   30,511   63,484   80,3218   214,908   226,551   376,2077   388,2078   428,1878   903,1168


X(536) =  DIRECTION OF VECTOR AX + BX + CX, where X = X(37)

Trilinears    (2ax - by - cz)/a, where x = x(a,b,c) = b + c, y = x(b,c,a), z = x(c,a,b)
Barycentrics    2bc - ca - ab : :

X(536) lies on these (parallel) lines: 2,37   30,511   42,2230   44,190   141,2321   335,903   889,3227   894,1100   1086,1266   2228,3123   2234,3009   2325,3008

X(536) = isogonal conjugate of X(739)
X(536) = isotomic conjugate of X(3227)
X(536) = X(2)-Ceva conjugate of X(13466)
X(536) = crossdifference of every pair of points on line X(6)X(667)
X(536) = barycentric cube root of X(8031)


X(537) = DIRECTION OF VECTOR AX + BX + CX, where X = X(38)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where
                        f(a,b,c) = (2ax - by - cz)/a, x = x(a,b,c) = b2 + c2, y = x(b,c,a), z = x(c,a,b)

Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(537) lies on the line at infinity.

X(537) lies on these (parallel) lines: 1,190   2,38   10,1086   30,511   37,551   75,668   192,3241

X(537) = isogonal conjugate of X(2382)
X(537) = perspector of circumconic centered at X(35123)
X(537) = X(2)-Ceva conjugate of X(35123)
X(537) = X(513)-Hirst inverse of X(519)


X(538) = DIRECTION OF VECTOR AX + BX + CX, where X = X(39)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where
                        f(a,b,c) = (2ax - by - cz)/a, x = x(a,b,c) = a(b2 + c2), y = x(b,c,a), z = x(c,a,b)

Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(538) lies on the line at infinity.

X(538) lies on these (parallel) lines: 2,39   30,511   32,1003   69,2549   75,1573   99,187   115,325   148,316   183,574   230,620   262,3545   350,1015   381,3095   591,3102   599,3094   671,1916   886,3228   1316,3292   1500,1909   1569,2021   1991,3103

X(538) = isogonal conjugate of X(729)
X(538) = isotomic conjugate of X(3228)
X(538) = X(2)-Ceva conjugate of X(35073)
X(538) = crossdifference of every pair of points on line X(6)X(669)
X(538) = perspector of circumconic centered at X(35073)


X(539) = DIRECTION OF VECTOR AX + BX + CX, where X = X(54)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where
                        f(a,b,c) = (2ax - by - cz)/a, x = x(A,B,C) = 1/cos(B-C), y = x(B,C,A), z = x(C,A,B)
Barycentrics    (a^2 - b^2 - c^2) (2 a^8 - 4 a^6 (b^2 + c^2) + 3 a^4 (b^4 + c^4) - 2 a^2 (b^2 - c^2)^2 (b^2 + c^2) + (b^2 - c^2)^4) : :

X(539) lies on the line at infinity.

X(539) lies on these (parallel) lines: 2,54   3,3519   5,1493   30,511   113,2914   155,195   265,1568   2072,3292

X(539) = isogonal conjugate of X(2383)


X(540) = DIRECTION OF VECTOR AX + BX + CX, where X = X(58)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where
                        f(a,b,c) = (2ax - by - cz)/a, x = x(a,b,c) = a/(b + c), y = x(b,c,a), z = x(c,a,b)

Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(540) lies on the line at infinity.

X(540) lies on these (parallel) lines: 2,58   30,511   340,447   376,3430


X(541) = DIRECTION OF VECTOR AX + BX + CX, where X = X(74)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where
                        f(a,b,c) = (2ax - by - cz)/a, x = x(A,B,C) = 1/(cos A - 2 cos B cos C), y = x(B,C,A), z = x(C,A,B)

Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(541) lies on the line at infinity.

X(541) lies on these (parallel) lines: 2,74   30,511   110,376   125,381   265,3426   394,399   3028,3058   3448,3543

X(541) = isogonal conjugate of X(841)


X(542) =  DIRECTION OF VECTOR AX + BX + CX, where X = X(98)

Trilinears    (2ax - by - cz)/a : : , where x = x(A,B,C) = sec(A + ω), y = x(B,C,A), z = x(C,A,B)
Barycentrics    2a^6 - 2a^4(b^2 + c^2) + a^2(b^4 + c^4) - (b^2 - c^2)^2(b^2 + c^2) : :
Barycentrics    2 a sec(A + ω) - b sec(B + ω) - c sec(C + ω) : :
Barycentrics    (3 SA - 2 SW) S^2 + 3 SW SB SC : :
X(542) = X(6) - X(13)

X(542) lies on these (parallel) lines: 2,98   3,67   4,576   5,575   6,13   30,511   68,1177   69,74   141,549   146,148   159,2931   161,1619   230,2030   428,1112   858,3292   1350,3534   1365,2606   1550,1551   1569,3094   1648,2502   1843,1986   1853,3167   3023,3028   3024,3027   3043,3044

X(542) = isogonal conjugate of X(842)
X(542) = isotomic conjugate of X(5641)
X(542) = crossdifference of every pair of points on line X(6)X(526)
X(542) = orthopoint of X(690)
X(542) = X(2)-Ceva conjugate of X(23967)
X(542) = X(30)-Hirst inverse of X(523)
X(542) = X(30)-of-1st-Brocard triangle
X(542) = X(511)-of-4th-Brocard-triangle
X(542) = X(511)-of-orthocentroidal-triangle
X(542) = X(511)-of-X(4)-Brocard-triangle
X(542) = X(524)-of-McCay-triangle
X(542) = X(524)-of-anti-McCay-triangle
X(542) = intercept of Fermat axis and the line at infinity
X(542) = perspector of hyperbola {A,B,C,X(2),X(476)}}
X(542) = Cundy-Parry Phi transform of X(14357)
X(542) = Cundy-Parry Psi transform of X(14246)
X(542) = 1st-Brocard-isogonal-conjugate of X(6795)
X(542) = homothetic center of 2nd Ehrmann triangle and Ehrmann vertex-triangle


X(543) = DIRECTION OF VECTOR AX + BX + CX, where X = X(99)

Trilinears    (2ax - by - cz)/a, x = x(a,b,c) = bc/(b2 - c2), where x : y : z = X(99)
Barycentrics    2 a^4 - b^4 - c^4 - 2 a^2 b^2 - 2 a^2 c^2 + 4 b^2 c^2 : :

X(543) lies on the line at infinity.

X(543) lies on these (parallel) lines: 2,99   22,3455   25,2936   30,511   98,376   114,381   147,3543   549,1153   598,1569   626,1975   1641,2502   2421,3016   3023,3058   3027,3325   3044,3048

X(543) = isogonal conjugate of X(843)
X(543) = X(2)-Ceva conjugate of X(35087)
X(543) = X(524)-of-1st-Brocard-triangle
X(543) = 1st-Brocard-isogonal conjugate of X(5108)
X(543) = 1st-Brocard-isotomic conjugate of X(599)
X(543) = crossdifference of every pair of points on line X(6)X(351)
X(543) = X(30)-of-anti-McCay-triangle
X(543) = SS(a → a^2) of X(545) (barycentric substitution)
X(543) = perspector of circumconic centered at X(35087)


X(544) = DIRECTION OF VECTOR AX + BX + CX, where X = X(101)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where
                         f(a,b,c) = (2ax - by - cz)/a, x = x(a,b,c) = a/(b - c), y = x(b,c,a), z = x(c,a,b)

X(544) lies on these (parallel) lines: 2,101   30,511   63,1018   103,376   118,381   152,3543   3022,3058

X(544) = isogonal conjugate of X(38884)


X(545) = DIRECTION OF VECTOR AX + BX + CX, where X = X(190)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where
                        f(a,b,c) = (2ax - by - cz)/a, x = x(a,b,c) = bc/(b - c), y = x(b,c,a), z = x(c,a,b)

Barycentrics    2 a^2 - b^2 - c^2 - 2 a b - 2 a c + 4 b c : :

X(545) lies on the line at infinity.

X(545) lies on these (parallel) lines: 2,45   30,511   44,1266   63,2161   321,1227

X(545) = isogonal conjugate of X(2384)
X(545) = isotomic conjugate of X(35168)
X(545) = X(2)-Ceva conjugate of X(35121)
X(545) = X(514)-Hirst inverse of X(519)
X(545) = perspector of circumconic centered at X(35121)


X(546) = MIDPOINT OF X(4) AND X(5)

Trilinears    3 cos(B - C) - 2 cos A : :
Trilinears    cos A + 6 cos B cos C : cos B + 6 cos C cos A : cos C + 6 cos A cos B
Barycentrics    2 a^4 + a^2 (b^2 + c^2) - 3 (b^2 - c^2)^2 : :
X(546) = X(3) - 3*X(5) = X(4) + X(5) = X(3) + 3*X(4)

As a point on the Euler line, X(546) has Shinagawa coefficients (1,5).

Let A'B'C', A″B″C″ be the outer and inner Vecten triangles, resp. Let (Oa) be the circle inscribed in square A'BA″C, and define (Ob) and (Oc) cyclically. X(546) is the radical center of circles (Oa), (Ob), (Oc). (Randy Hutson, December 2, 2017)

Let MA denote the point in which the A-median meets side BC. On 11/05/03, Andrew Crane noted that X(546) is the radical center of circles (A), (B), (C), where (A) denotes the circle centered at A and passing through MA, and (B) and (C) are defined cyclically.

X(546) lies on these lines: 2,3   13,398   14,397   113,137   156,578   946,952

X(546) = midpoint of X(i) and X(j) for these (i,j): (4,5), (382,550)
X(546) = reflection of X(i) in X(j) for these (i,j): (140,5), (548,140)
X(546) = complement of X(550)
X(546) = anticomplement of X(3530)
X(546) = circumcircle-inverse of X(37922)
X(546) = orthocentroidal-circle-inverse of X(382)
X(546) = X(1385)-of-orthic-triangle if ABC is acute
X(546) = X(5)-of-Euler-triangle
X(546) = {X(3),X(5)}-harmonic conjugate of X(3628)
X(546) = X(5)-of-Ehrmann-mid-triangle
X(546) = Ehrmann-side-to-orthic similarity image of X(5)
X(546) = Johnson-to-Ehrmann-mid similarity image of X(5)
X(546) = homothetic center of Ehrmann mid-triangle and Euler triangle
X(546) = radical center of de Longchamps circles of ABC and 1st and 2nd Ehrmann inscribed triangles


X(547) = MIDPOINT OF X(2) AND X(5)

Trilinears    5 cos(B - C) + 2 cosA : :
Trilinears    7 cos A + 10 cos B cos C : 7 cos B + 10 cos C cos A : 7 cos C + 10 cos A cos B
X(547) = X(3) + 3*X(4)

As a point on the Euler line, X(547) has Shinagawa coefficients (7,3).

X(547) lies on the Euler line. (Antreas Hatzipolakis, 1/20/00, Hyacinthos #201)

X(547) lies on these lines: 2,3   551,952

X(547) = midpoint of X(i) and X(j) for these (i,j): (2,5), (381,549)
X(547) = reflection of X(140) in X(2)
X(547) = complement of X(549)
X(547) = circumcircle-inverse of X(37923)
X(547) = center of the Vu pedal-centroidal circle of X(5)
X(547) = {X(35231),X(35232)}-harmonic conjugate of X(37952)
X(547) = pole of Fermat axis wrt conic {X(5),X(13),X(14),X(15),X(16)}}
X(547) = {X(3),X(4)}-harmonic conjugate of X(3627)


X(548) = MIDPOINT OF X(5) AND X(20)

Trilinears    - cos(B - C) + 6 cos A : :
Trilinears    5 cos A - 2 cos B cos C : 5 cos B - 2 cos C cos A : 5 cos C - 2 cos A cos B
Trilinears    2 sec A - 5 sec B sec C : 2 sec B - 5 sec C sec A : 2 sec C - 5 sec A sec B
X(548) = 3 X(2) - 7 X(3) = 3X(3) - X(5)

As a point on the Euler line, X(548) has Shinagawa coefficients (5,-7).

X(548) lies on the Euler line. (Antreas Hatzipolakis, 1/20/00, Hyacinthos #201)

X(548) = midpoint of X(i) and X(j) for these (i,j): (3,550), (5,20)
X(548) = reflection of X(i) in X(j) for these (i,j): (140,3), (546,140)
X(548) = {X(2),X(3)}-harmonic conjugate of X(15712)
X(548) = {X(3),X(5)}-harmonic conjugate of X(3530)
X(548) = radical center of circles centered at vertices of ABC with diameter equal to opposite side


X(549) = MIDPOINT OF X(2) AND X(3)

Trilinears    cos(B - C) + 4 cosA : :
Trilinears    5 cos A + 2 cos B cos C : 5 cos B + 2 cos C cos A : 5 cos C + 2 cos A cos B

Trilinears    3 cos A + 2 sin B sin C : 3 cos B + 2 sin C sin A : 3 cos C + 2 sin A sin B
Trilinears    2 sec A + 5 sec B sec C : 2 sec B + 5 sec C sec A : 2 sec C + 5 sec A sec B
Barycentrics    4a^4 - 5a^2(b^2 + c^2) + (b^2 - c^2)^2 : :
Barycentrics    2 S^2 + 3 SA (SB + SC) : :
X(549) = X[1] - 3 X[3653], 3 X[3653] + X[3654], 5 X[2] - X[4], 7 X[2] + X[20], 3 X[2] + X[376], 11 X[2] - X[382], 7 X[2] - 2 X[546], 5 X[2] + 2 X[548], 4 X[2] + X[550], X[2] - 5 X[631], 4 X[2] - 5 X[632], 7 X[2] - 5 X[1656], 13 X[2] + X[1657], 11 X[2] - 7 X[3090], 13 X[2] - 5 X[3091], 17 X[2] - X[3146], 11 X[2] + 5 X[3522], X[2] + 7 X[3523], X[2] + 3 X[3524], 7 X[2] - 11 X[3525], 5 X[2] - 7 X[3526], 13 X[2] + 7 X[3528], 19 X[2] + X[3529], X[2] + 4 X[3530], 13 X[2] - 17 X[3533], 5 X[2] + X[3534], 9 X[2] - X[3543], 37 X[2] - 17 X[3544], 7 X[2] - 3 X[3545], 8 X[2] - X[3627], 5 X[2] - 4 X[3628], 7 X[2] - X[3830], 23 X[2] - 7 X[3832], 11 X[2] - 3 X[3839], 19 X[2] - 5 X[3843], 4 X[2] - X[3845], 11 X[2] - 4 X[3850], 17 X[2] - 7 X[3851], 13 X[2] - 2 X[3853], 49 X[2] - 17 X[3854], 31 X[2] - 11 X[3855], 25 X[2] - 8 X[3856], 20 X[2] - 7 X[3857], 16 X[2] - 5 X[3858], 29 X[2] - 10 X[3859], 13 X[2] - 4 X[3860], 17 X[2] - 4 X[3861], X[2] - 3 X[5054], 5 X[2] - 3 X[5055], 19 X[2] - 11 X[5056], 31 X[2] + X[5059], 5 X[2] - 2 X[5066], 17 X[2] - 13 X[5067], 29 X[2] - 13 X[5068], 13 X[2] - 11 X[5070], 9 X[2] - 5 X[5071], 25 X[2] - 11 X[5072], 23 X[2] - X[5073], 31 X[2] - 5 X[5076], 23 X[2] - 13 X[5079], 25 X[2] - 17 X[7486], 2 X[2] + X[8703], 7 X[2] - 4 X[10109], 3 X[2] - 4 X[10124], 7 X[2] + 13 X[10299], 5 X[2] - 13 X[10303], 5 X[2] + 3 X[10304], 11 X[2] + X[11001], 2 X[2] - 3 X[11539], 5 X[2] - 8 X[11540], 53 X[2] - X[11541], 9 X[2] - 4 X[11737], X[2] - 4 X[11812], X[2] + 2 X[12100], 11 X[2] - 2 X[12101], 23 X[2] - 4 X[12102], 11 X[2] + 2 X[12103], X[2] - 8 X[12108], 19 X[2] - 8 X[12811], 17 X[2] - 10 X[12812], 9 X[2] + 5 X[14093], 13 X[2] - 3 X[14269], 2 X[2] - 7 X[14869], 5 X[2] - 12 X[14890], 3 X[2] + 4 X[14891], 13 X[2] - 6 X[14892], 9 X[2] - 2 X[14893], 35 X[2] - 19 X[15022], 25 X[2] - X[15640], 9 X[2] + X[15681], 13 X[2] - X[15682], 15 X[2] + X[15683], 15 X[2] - X[15684], 17 X[2] + X[15685], 6 X[2] + X[15686], 6 X[2] - X[15687], 7 X[2] + 3 X[15688], 11 X[2] + 3 X[15689], 7 X[2] + 2 X[15690], 9 X[2] + 2 X[15691], 3 X[2] + 5 X[15692], X[2] + 5 X[15693], 3 X[2] - 5 X[15694], 13 X[2] + 5 X[15695], 17 X[2] + 5 X[15696], 19 X[2] + 5 X[15697], 5 X[2] + 7 X[15698], 4 X[2] - 3 X[15699], 3 X[2] + 7 X[15700], X[2] - 7 X[15701], 3 X[2] - 7 X[15702], 9 X[2] - 7 X[15703], 10 X[2] + X[15704], 7 X[2] + 9 X[15705], 5 X[2] + 9 X[15706], X[2] + 9 X[15707], X[2] - 9 X[15708], 5 X[2] - 9 X[15709], 11 X[2] + 9 X[15710], 4 X[2] + 5 X[15711], 2 X[2] + 5 X[15712], 2 X[2] - 5 X[15713], 6 X[2] + 5 X[15714], 9 X[2] + 11 X[15715], 7 X[2] + 11 X[15716], 5 X[2] + 11 X[15717], 3 X[2] + 11 X[15718], X[2] + 11 X[15719], X[2] - 11 X[15720], 3 X[2] - 11 X[15721], X[2] + 17 X[15722], 9 X[2] - 11 X[15723], 5 X[2] + 4 X[15759], 7 X[2] - 8 X[16239], 2 X[2] + 3 X[17504], 23 X[2] + 5 X[17538], 37 X[2] - 5 X[17578], 25 X[2] + X[17800], 7 X[2] + 5 X[19708], 11 X[2] - 5 X[19709], 8 X[2] + X[19710], 2 X[2] + 7 X[19711], 19 X[2] + 13 X[21734], 17 X[2] + 11 X[21735], 10 X[2] - 3 X[23046], 10 X[2] - X[33699], 29 X[2] - X[33703], 7 X[2] + 4 X[33923], 3 X[2] + 2 X[34200], 13 X[2] - 8 X[35018], 37 X[2] - 55 X[35381], 123 X[2] - 65 X[35382], 143 X[2] - 5 X[35384], 39 X[2] - X[35400], 69 X[2] - 11 X[35401], 87 X[2] - 13 X[35402], 27 X[2] - 5 X[35403], 12 X[2] - X[35404], 289 X[2] - 11 X[35405], 347 X[2] - 13 X[35406], 139 X[2] - 5 X[35407], 69 X[2] - 2 X[35408], 71 X[2] - 3 X[35409], 327 X[2] + 11 X[35410], 381 X[2] + 13 X[35411], 333 X[2] + 13 X[35412], 165 X[2] + 8 X[35413], 171 X[2] + 5 X[35414], 521 X[2] - 55 X[35416], 321 X[2] - 88 X[35417], 79 X[2] + 33 X[35418], 799 X[2] - 65 X[35419], 73 X[2] + 26 X[35421], 39 X[2] - 5 X[35434], 345 X[2] + 13 X[35435], 8 X[2] - 3 X[38071], 17 X[2] - 3 X[38335], 17 X[2] - 5 X[41099], 19 X[2] - 7 X[41106], 23 X[2] + 8 X[41981], 11 X[2] + 6 X[41982], X[2] + 6 X[41983], 11 X[2] - 12 X[41984], 13 X[2] - 12 X[41985], 47 X[2] - 24 X[41986], 23 X[2] - 6 X[41987], 39 X[2] - 8 X[41988], 43 X[2] - 20 X[41989], 34 X[2] - 13 X[41990], 34 X[2] - 11 X[41991], 22 X[2] - 23 X[41992], 13 X[2] + 4 X[44245], X[2] + 8 X[44580], 4 X[2] + 7 X[44682], 12 X[2] + X[44903], 25 X[2] - 14 X[44904], 19 X[2] - 12 X[45757], 17 X[2] - 24 X[45758], 4 X[2] + 3 X[45759], 13 X[2] - 20 X[45760], 15 X[2] + 28 X[45761], 57 X[2] - 14 X[45762], 11 X[2] - 13 X[46219], 13 X[2] + 8 X[46332], 25 X[2] + 3 X[46333], 8 X[2] + 5 X[46853] (and many more)

As a point on the Euler line, X(549) has Shinagawa coefficients (5,-3).

Let Pa be the parabola with focus A and directrix BC. Let La be the polar of X(3) wrt Pa. Define Lb, Lc cyclically. Let A' = Lb∩Lc, B' = Lc∩La, C' = La∩Lb. X(141) of triangle A'B'C' is X(549). X(549) is also the centroid of the six points of tangency of lines from X(3) to Pa, Pb, and Pc. (Randy Hutson, July 20, 2016)

In the construction of the Dao 6-point circle (see X(5569)), X(549) is the centroid of AbAcBcBaCaCb. (Randy Hutson, July 20, 2016)

X(549) lies on these lines: 2,3   15,395   16,396   35,496   36,495   141,542   182,524   230,574   302,617   303,616   511,597   517,551   543,1153

X(549) lies on the cubic K1224 and these lines: {1, 3653}, {2, 3}, {6, 21843}, {7, 38065}, {8, 38066}, {9, 38067}, {10, 13624}, {11, 5010}, {12, 7280}, {13, 10646}, {14, 10645}, {15, 395}, {16, 396}, {17, 5351}, {18, 5352}, {32, 9300}, {35, 496}, {36, 495}, {39, 5306}, {40, 3656}, {46, 37737}, {49, 43572}, {51, 13363}, {52, 10627}, {54, 34483}, {55, 10072}, {56, 10056}, {57, 5719}, {61, 16773}, {62, 16772}, {69, 12017}, {74, 5655}, {76, 32516}, {83, 42788}, {95, 1494}, {98, 8724}, {99, 11632}, {103, 38774}, {104, 38762}, {109, 38786}, {110, 11694}, {113, 14677}, {114, 9167}, {115, 3054}, {119, 44847}, {125, 34153}, {126, 10163}, {128, 38429}, {141, 542}, {142, 38080}, {143, 10625}, {148, 17006}, {156, 10984}, {165, 5886}, {182, 524}, {183, 6390}, {184, 40111}, {185, 11591}, {187, 3815}, {216, 3163}, {226, 5122}, {230, 574}, {252, 25042}, {265, 15051}, {302, 617}, {303, 616}, {316, 37647}, {323, 15087}, {325, 7771}, {343, 37513}, {355, 7987}, {371, 13966}, {372, 8981}, {373, 13364}, {386, 5453}, {389, 5447}, {394, 44683}, {397, 5237}, {398, 5238}, {399, 13392}, {476, 11749}, {484, 5444}, {485, 6410}, {486, 6409}, {487, 32810}, {488, 32811}, {498, 5204}, {499, 5217}, {500, 3216}, {511, 597}, {515, 3828}, {516, 11230}, {517, 551}, {519, 1385}, {523, 9175}, {525, 45321}, {527, 31657}, {528, 1484}, {529, 31659}, {530, 6771}, {531, 6774}, {532, 33458}, {533, 33459}, {535, 23961}, {538, 13334}, {539, 10610}, {541, 5972}, {543, 1153}, {544, 6712}, {553, 6147}, {566, 14836}, {567, 43574}, {568, 2979}, {569, 43652}, {572, 17330}, {575, 8584}, {582, 37522}, {590, 6396}, {591, 43119}, {598, 11669}, {599, 3564}, {615, 6200}, {626, 34510}, {627, 33612}, {628, 33613}, {641, 13821}, {642, 13701}, {671, 21166}, {754, 13335}, {804, 9126}, {912, 11227}, {930, 12026}, {946, 19883}, {952, 3576}, {960, 40296}, {970, 5482}, {971, 38113}, {993, 3035}, {999, 5218}, {1001, 35238}, {1040, 37729}, {1071, 31835}, {1078, 3933}, {1092, 13336}, {1125, 3579}, {1141, 6592}, {1147, 37515}, {1151, 5420}, {1152, 5418}, {1154, 3917}, {1155, 4870}, {1157, 14141}, {1209, 13561}, {1216, 6102}, {1263, 34837}, {1296, 38806}, {1327, 42261}, {1328, 42260}, {1329, 5267}, {1340, 39023}, {1341, 39022}, {1350, 18583}, {1352, 13692}, {1384, 7736}, {1387, 5119}, {1482, 38314}, {1499, 11183}, {1503, 11178}, {1506, 14537}, {1539, 12900}, {1587, 6450}, {1588, 6449}, {1614, 43607}, {1621, 35000}, {1698, 18357}, {1699, 28178}, {1737, 37600}, {1991, 43118}, {1992, 5050}, {1994, 15037}, {2079, 9722}, {2080, 12150}, {2482, 2782}, {2548, 5023}, {2549, 37637}, {2771, 10176}, {2780, 11176}, {2794, 22247}, {2797, 45682}, {2808, 38772}, {2818, 38784}, {2826, 45314}, {3017, 4256}, {3053, 31401}, {3055, 5475}, {3060, 13340}, {3068, 6398}, {3069, 6221}, {3070, 41952}, {3071, 41951}, {3086, 10385}, {3095, 33706}, {3098, 3589}, {3104, 36364}, {3105, 36365}, {3241, 5657}, {3295, 7288}, {3304, 31452}, {3306, 37584}, {3311, 13935}, {3312, 9540}, {3314, 34623}, {3329, 9301}, {3336, 16139}, {3357, 16252}, {3359, 11729}, {3398, 41624}, {3411, 42504}, {3412, 42505}, {3413, 47369}, {3414, 47370}, {3419, 9945}, {3431, 44751}, {3448, 15040}, {3487, 37545}, {3567, 14449}, {3580, 14805}, {3581, 14389}, {3587, 5437}, {3592, 9680}, {3601, 12433}, {3612, 24914}, {3614, 10483}, {3616, 12702}, {3617, 18526}, {3618, 14848}, {3619, 18440}, {3620, 39899}, {3622, 8148}, {3624, 12699}, {3625, 32900}, {3631, 7908}, {3634, 18480}, {3636, 11278}, {3642, 44383}, {3643, 44382}, {3649, 37524}, {3734, 32459}, {3763, 18358}, {3767, 15815}, {3788, 7865}, {3793, 7774}, {3816, 6681}, {3817, 28146}, {3818, 34573}, {3819, 13754}, {3829, 33862}, {3847, 20107}, {3849, 7619}, {3877, 34123}, {3878, 13145}, {3911, 15935}, {3916, 17781}, {3927, 27383}, {3928, 26921}, {3929, 24467}, {3940, 5744}, {3947, 31776}, {3972, 34733}, {4257, 37662}, {4271, 10035}, {4293, 31479}, {4297, 9956}, {4421, 10267}, {4428, 11248}, {4550, 46817}, {4654, 11374}, {4669, 5882}, {4677, 30389}, {4688, 29010}, {4745, 31666}, {4846, 43713}, {4930, 34744}, {4999, 25440}, {5007, 9606}, {5012, 13339}, {5013, 5305}, {5024, 7735}, {5044, 5325}, {5126, 31397}, {5131, 11246}, {5171, 32134}, {5188, 11272}, {5206, 7745}, {5210, 7737}, {5215, 14693}, {5221, 16137}, {5226, 18541}, {5248, 6691}, {5251, 34697}, {5254, 7749}, {5259, 34630}, {5260, 26321}, {5265, 7373}, {5281, 6767}, {5303, 27529}, {5318, 33417}, {5319, 22332}, {5321, 33416}, {5326, 7951}, {5334, 42628}, {5335, 42627}, {5339, 41120}, {5340, 41119}, {5343, 42589}, {5344, 42588}, {5349, 42597}, {5350, 42596}, {5435, 15934}, {5438, 5791}, {5439, 37585}, {5440, 13151}, {5445, 10950}, {5446, 11695}, {5461, 23698}, {5462, 10263}, {5463, 21156}, {5464, 21157}, {5469, 22847}, {5470, 22893}, {5473, 20252}, {5474, 20253}, {5480, 14810}, {5481, 34897}, {5504, 13622}, {5506, 7701}, {5550, 6361}, {5562, 13630}, {5585, 31415}, {5587, 19876}, {5603, 28212}, {5609, 20417}, {5640, 13451}, {5642, 5650}, {5646, 11472}, {5648, 11579}, {5651, 35266}, {5652, 21733}, {5656, 35450}, {5703, 5708}, {5720, 10857}, {5722, 30282}, {5731, 5790}, {5762, 6173}, {5763, 37623}, {5771, 18443}, {5805, 38093}, {5840, 34126}, {5841, 38114}, {5843, 6172}, {5860, 26348}, {5861, 26341}, {5874, 45553}, {5875, 45552}, {5876, 11793}, {5878, 8567}, {5885, 31806}, {5890, 7998}, {5907, 13491}, {5926, 25423}, {6000, 10170}, {6030, 16658}, {6033, 7831}, {6054, 9751}, {6150, 14140}, {6194, 32447}, {6199, 7586}, {6241, 43903}, {6243, 15043}, {6245, 40262}, {6246, 38104}, {6247, 10282}, {6284, 10593}, {6321, 9166}, {6329, 37517}, {6337, 32836}, {6395, 7585}, {6411, 6561}, {6412, 6560}, {6419, 41963}, {6420, 31454}, {6431, 42568}, {6432, 42569}, {6445, 13941}, {6446, 8972}, {6451, 9541}, {6452, 13665}, {6453, 35813}, {6454, 35812}, {6455, 6459}, {6456, 6460}, {6496, 42638}, {6497, 42637}, {6500, 42522}, {6501, 42523}, {6502, 31499}, {6564, 32789}, {6565, 32790}, {6662, 41481}, {6666, 38082}, {6667, 22938}, {6668, 38085}, {6671, 36755}, {6672, 36756}, {6683, 14881}, {6689, 20424}, {6696, 6759}, {6700, 31445}, {6710, 38601}, {6711, 38607}, {6714, 38619}, {6715, 38620}, {6716, 38621}, {6717, 38622}, {6718, 38600}, {6719, 38623}, {6720, 9530}, {6721, 22505}, {6722, 22515}, {6723, 10113}, {6776, 21356}, {6781, 7603}, {7171, 7308}, {7294, 7741}, {7354, 10592}, {7581, 13903}, {7582, 13961}, {7610, 7618}, {7688, 34618}, {7689, 9820}, {7691, 8254}, {7697, 11149}, {7746, 11648}, {7748, 18362}, {7750, 7769}, {7752, 11057}, {7755, 9607}, {7756, 39563}, {7757, 11171}, {7761, 44377}, {7762, 10357}, {7763, 7767}, {7772, 31457}, {7775, 47101}, {7776, 32829}, {7777, 34734}, {7783, 19570}, {7786, 9821}, {7789, 7815}, {7792, 35002}, {7793, 7837}, {7801, 10104}, {7804, 15491}, {7818, 10256}, {7835, 34624}, {7840, 21445}, {7844, 44381}, {7846, 35248}, {7853, 31274}, {7857, 7884}, {7930, 9873}, {7931, 9862}, {7967, 31145}, {7988, 28182}, {7991, 31425}, {7999, 10574}, {8143, 27784}, {8182, 11184}, {8227, 16192}, {8273, 11499}, {8550, 20190}, {8552, 44818}, {8591, 14651}, {8596, 38635}, {8667, 34511}, {8717, 15448}, {8718, 43614}, {8722, 10796}, {8725, 31268}, {8726, 37700}, {8760, 44567}, {9127, 10166}, {9140, 15035}, {9143, 32609}, {9159, 15111}, {9172, 33962}, {9177, 46998}, {9517, 45319}, {9521, 45318}, {9543, 43387}, {9655, 10588}, {9668, 10589}, {9681, 43794}, {9698, 35007}, {9703, 11003}, {9709, 30478}, {9734, 44401}, {9746, 28915}, {9778, 28216}, {9780, 18525}, {9826, 37511}, {9829, 31961}, {9833, 40686}, {9880, 14971}, {9940, 24475}, {9955, 19862}, {10022, 29069}, {10031, 19914}, {10039, 37605}, {10053, 12351}, {10069, 12350}, {10095, 45186}, {10171, 28150}, {10172, 28164}, {10175, 28160}, {10197, 26286}, {10199, 26285}, {10202, 24473}, {10222, 31447}, {10247, 34631}, {10269, 11194}, {10523, 14792}, {10541, 15533}, {10564, 32225}, {10575, 45959}, {10576, 42259}, {10577, 42258}, {10601, 37483}, {10606, 32620}, {10653, 11481}, {10654, 11480}, {10706, 14643}, {10707, 34474}, {10708, 38690}, {10709, 38691}, {10710, 38692}, {10711, 38693}, {10712, 38694}, {10713, 38695}, {10714, 23239}, {10715, 38696}, {10716, 38697}, {10717, 14666}, {10718, 38699}, {10902, 10943}, {10942, 34606}, {10979, 42459}, {10990, 38795}, {10991, 38751}, {10992, 38740}, {11006, 18332}, {11064, 33533}, {11123, 16220}, {11160, 14912}, {11180, 25406}, {11204, 15311}, {11207, 45620}, {11208, 45621}, {11235, 26492}, {11236, 26487}, {11239, 16203}, {11240, 16202}, {11249, 40726}, {11362, 15178}, {11381, 45958}, {11412, 37481}, {11430, 13567}, {11438, 23292}, {11444, 31834}, {11459, 20791}, {11485, 37641}, {11486, 37640}, {11488, 42115}, {11489, 42116}, {11614, 39601}, {11623, 36521}, {11645, 24206}, {11693, 16003}, {11801, 12121}, {11804, 12893}, {12024, 44665}, {12045, 13570}, {12152, 45623}, {12153, 45624}, {12161, 37514}, {12162, 14128}, {12228, 19129}, {12243, 13188}, {12245, 37624}, {12295, 15088}, {12307, 22051}, {12322, 13798}, {12323, 13678}, {12324, 14530}, {12355, 13172}, {12506, 47074}, {12512, 22793}, {12816, 42431}, {12817, 42432}, {13108, 32522}, {13157, 14379}, {13226, 18446}, {13289, 32600}, {13292, 37476}, {13329, 17392}, {13338, 30537}, {13352, 41588}, {13353, 34148}, {13367, 31804}, {13372, 14072}, {13399, 44110}, {13416, 14708}, {13419, 17712}, {13434, 13482}, {13857, 32110}, {13886, 43511}, {13939, 43512}, {13971, 31439}, {14061, 38730}, {14131, 34466}, {14216, 17821}, {14385, 40630}, {14417, 44202}, {14561, 31884}, {14639, 38731}, {14641, 44870}, {14644, 38723}, {14711, 32523}, {14712, 17005}, {14855, 15030}, {14915, 15082}, {14988, 31165}, {15012, 15606}, {15018, 15038}, {15020, 23236}, {15033, 15360}, {15034, 15057}, {15042, 15081}, {15056, 18439}, {15066, 18445}, {15068, 17811}, {15072, 18435}, {15092, 39809}, {15305, 33879}, {15345, 16768}, {15462, 34319}, {15484, 15655}, {15489, 37536}, {15740, 44763}, {15801, 43600}, {15805, 37498}, {15806, 43601}, {16092, 46634}, {16163, 20304}, {16266, 36752}, {16317, 20481}, {16336, 35729}, {16654, 44082}, {16760, 38611}, {16808, 42145}, {16809, 42144}, {16964, 41122}, {16965, 41121}, {16966, 36969}, {16967, 36970}, {16984, 43453}, {17004, 47286}, {17008, 31859}, {17074, 23071}, {17398, 37508}, {17414, 32232}, {17595, 39544}, {17702, 34128}, {17757, 34698}, {17825, 44413}, {18016, 35720}, {18243, 22936}, {18285, 31378}, {18319, 22104}, {18388, 35254}, {18390, 47296}, {18391, 37606}, {18400, 23332}, {18483, 19878}, {18492, 19872}, {18546, 34504}, {18581, 42122}, {18582, 42123}, {18913, 19347}, {18914, 19357}, {18925, 26944}, {20112, 32479}, {20194, 38010}, {20299, 34782}, {20376, 23358}, {20379, 30714}, {20398, 36523}, {20575, 30269}, {20576, 30270}, {20583, 39561}, {21160, 31158}, {21164, 31142}, {21165, 31164}, {21309, 37665}, {21975, 35888}, {22102, 38617}, {22112, 32269}, {22236, 42149}, {22238, 42152}, {22253, 37667}, {22268, 35887}, {22660, 43839}, {22677, 41146}, {22799, 38759}, {22837, 32157}, {22937, 33668}, {23041, 31166}, {23267, 43382}, {23273, 43383}, {23300, 35228}, {23320, 23333}, {24239, 37589}, {24299, 33595}, {24390, 34745}, {24813, 41138}, {24827, 40480}, {24953, 34746}, {25524, 35239}, {25561, 29012}, {25565, 29317}, {26398, 45696}, {26422, 45697}, {26498, 45699}, {26507, 45698}, {28236, 38176}, {29181, 38136}, {29331, 41140}, {29959, 30532}, {30209, 44560}, {30308, 34595}, {30392, 34747}, {30435, 31400}, {31235, 38761}, {31253, 33697}, {31267, 34778}, {31379, 38609}, {31508, 37704}, {31662, 34641}, {31670, 38072}, {31671, 38073}, {31672, 38075}, {31673, 38076}, {31729, 32156}, {31744, 31762}, {31788, 31838}, {31945, 47050}, {32006, 32839}, {32063, 35260}, {32358, 35602}, {32419, 43120}, {32421, 43121}, {32767, 41362}, {32808, 45508}, {32809, 45509}, {32817, 32869}, {32904, 35721}, {33521, 38775}, {33537, 33540}, {33549, 46025}, {33604, 42926}, {33605, 42927}, {33899, 37837}, {34209, 47084}, {34224, 43608}, {34229, 46951}, {34312, 38700}, {34339, 44663}, {34486, 34699}, {34584, 36518}, {34841, 38608}, {35268, 35283}, {35731, 42563}, {35770, 43888}, {35771, 43887}, {35820, 42582}, {35821, 42583}, {36362, 41021}, {36363, 41020}, {36383, 36767}, {36634, 37699}, {36770, 41042}, {36836, 40694}, {36843, 40693}, {37472, 43651}, {37478, 37649}, {37506, 43653}, {37510, 46922}, {37525, 37728}, {37633, 45923}, {37698, 42043}, {37809, 42849}, {38031, 47357}, {38139, 38318}, {38141, 38319}, {38716, 38796}, {39242, 44569}, {39524, 42295}, {40329, 42329}, {41112, 42151}, {41113, 42150}, {41133, 43461}, {41313, 46475}, {41961, 41968}, {41962, 41967}, {41973, 42503}, {41974, 42502}, {41979, 42648}, {41980, 42647}, {42085, 42143}, {42086, 42146}, {42090, 42095}, {42091, 42098}, {42093, 42585}, {42094, 42584}, {42096, 42111}, {42097, 42114}, {42099, 42107}, {42100, 42110}, {42101, 42914}, {42102, 42915}, {42108, 42918}, {42109, 42919}, {42112, 42475}, {42113, 42474}, {42119, 42129}, {42120, 42132}, {42125, 42688}, {42128, 42689}, {42130, 42139}, {42131, 42142}, {42157, 42163}, {42158, 42166}, {42159, 42591}, {42162, 42590}, {42164, 42434}, {42165, 42433}, {42263, 42274}, {42264, 42277}, {42266, 42270}, {42267, 42273}, {42275, 42601}, {42276, 42600}, {42417, 42525}, {42418, 42524}, {42429, 42594}, {42430, 42595}, {42506, 42793}, {42507, 42794}, {42520, 42635}, {42521, 42636}, {42532, 42977}, {42533, 42976}, {42629, 42693}, {42630, 42692}, {42682, 42906}, {42683, 42907}, {42777, 43199}, {42778, 43200}, {42783, 43628}, {42784, 43629}, {42797, 43773}, {42798, 43774}, {42815, 42968}, {42816, 42969}, {42817, 43463}, {42818, 43464}, {42890, 43547}, {42891, 43546}, {42892, 43232}, {42893, 43233}, {42928, 43548}, {42929, 43549}, {42950, 42984}, {42951, 42985}, {42952, 43485}, {42953, 43486}, {42956, 43005}, {42957, 43004}, {42996, 43372}, {42997, 43373}, {43006, 43020}, {43007, 43021}, {43014, 43308}, {43015, 43309}, {43105, 43645}, {43106, 43646}, {43242, 43554}, {43243, 43555}, {43248, 43310}, {43249, 43311}, {43394, 43573}, {43489, 43499}, {43490, 43500}, {43575, 43836}, {43604, 44516}, {43619, 44541}, {43620, 44526}, {44204, 44564}, {44555, 45969}, {44813, 44820}, {45331, 46127}, {45410, 45420}, {45411, 45421}, {46980, 46987}, {46981, 46986}, {46983, 46990}, {46984, 46989}

X(549) = midpoint of X(i) and X(j) for these (i,j): {1, 3654}, {2, 3}, {4, 3534}, {5, 8703}, {20, 3830}, {40, 3656}, {74, 5655}, {98, 8724}, {99, 11632}, {110, 20126}, {140, 12100}, {165, 5886}, {376, 381}, {382, 11001}, {405, 44284}, {427, 44261}, {442, 44255}, {546, 15690}, {547, 34200}, {548, 5066}, {550, 3845}, {568, 2979}, {599, 11179}, {631, 15693}, {632, 15711}, {858, 44265}, {1350, 20423}, {1352, 43273}, {1368, 44273}, {1656, 19708}, {1657, 15682}, {2072, 44280}, {2482, 6055}, {3060, 13340}, {3091, 15695}, {3095, 33706}, {3098, 5476}, {3104, 36364}, {3105, 36365}, {3146, 15685}, {3241, 34718}, {3431, 44751}, {3522, 19709}, {3523, 15701}, {3524, 5054}, {3525, 15716}, {3526, 15698}, {3530, 11812}, {3543, 15681}, {3545, 15688}, {3576, 26446}, {3627, 19710}, {3628, 15759}, {3655, 3679}, {3819, 16836}, {3839, 15689}, {3843, 15697}, {3860, 44245}, {3917, 9730}, {4421, 45700}, {4669, 5882}, {4677, 37727}, {4930, 34744}, {5050, 10519}, {5055, 10304}, {5071, 14093}, {5188, 44422}, {5473, 25154}, {5474, 25164}, {5569, 7622}, {5648, 11579}, {5652, 21733}, {5656, 35450}, {5657, 10246}, {5731, 5790}, {5890, 23039}, {6054, 14830}, {6175, 28460}, {6194, 32447}, {6321, 12117}, {6863, 19704}, {6865, 19706}, {6958, 19705}, {6996, 19703}, {7502, 44287}, {7610, 7618}, {7775, 47101}, {7998, 40280}, {8182, 11184}, {8550, 22165}, {8598, 15980}, {8667, 34511}, {9126, 16235}, {9880, 38738}, {10031, 19914}, {10109, 33923}, {10124, 14891}, {10156, 33575}, {10164, 10165}, {10182, 10193}, {10192, 23328}, {10297, 47031}, {10564, 32225}, {10625, 21969}, {10717, 14666}, {11006, 18332}, {11050, 20128}, {11112, 28459}, {11113, 28458}, {11123, 16220}, {11171, 22712}, {11194, 45701}, {11202, 23329}, {11231, 17502}, {11250, 44278}, {11295, 44461}, {11296, 44465}, {11539, 17504}, {11585, 44268}, {11623, 36521}, {12101, 12103}, {12108, 44580}, {12243, 13188}, {12355, 13172}, {12506, 47074}, {13083, 13084}, {13626, 35232}, {13627, 35231}, {13632, 13634}, {13633, 13635}, {13857, 32110}, {14070, 44441}, {14417, 44202}, {14561, 31884}, {14639, 38731}, {14643, 15055}, {14644, 38723}, {14855, 15030}, {14869, 19711}, {14893, 15691}, {15035, 15061}, {15072, 18435}, {15122, 18579}, {15360, 37477}, {15561, 34473}, {15640, 17800}, {15644, 21849}, {15678, 47032}, {15683, 15684}, {15686, 15687}, {15692, 15694}, {15696, 41099}, {15699, 45759}, {15700, 15702}, {15704, 33699}, {15706, 15709}, {15707, 15708}, {15712, 15713}, {15715, 15723}, {15718, 15721}, {15719, 15720}, {15760, 44285}, {15764, 15765}, {15819, 21163}, {16092, 46634}, {17525, 37401}, {18281, 18324}, {18546, 34504}, {18570, 44262}, {19548, 19707}, {21153, 38122}, {21154, 38760}, {21166, 38224}, {28452, 37428}, {31145, 34748}, {31694, 44250}, {34152, 44282}, {34726, 44442}, {35018, 46332}, {35404, 44903}, {35955, 37348}, {36362, 41021}, {36363, 41020}, {36439, 36457}, {37506, 43653}, {37922, 44450}, {37950, 44266}, {38692, 38764}, {38693, 38752}, {38697, 38776}, {38716, 38796}, {38727, 38793}, {38737, 38748}, {41490, 41491}, {41982, 47478}, {44210, 44218}, {44882, 47354}, {46264, 47353}, {46980, 46987}, {46981, 46986}, {46983, 46990}, {46984, 46989}, {47088, 47089}, {47097, 47333}
X(549) = reflection of X(i) in X(j) for these (i,j): {2, 140}, {3, 12100}, {4, 5066}, {5, 2}, {20, 15690}, {51, 13363}, {110, 11694}, {140, 11812}, {376, 34200}, {381, 547}, {382, 12101}, {428, 23410}, {546, 10109}, {547, 10124}, {548, 15759}, {550, 8703}, {597, 10168}, {632, 15713}, {3524, 41983}, {3530, 44580}, {3534, 548}, {3543, 14893}, {3545, 47599}, {3627, 3845}, {3628, 11540}, {3656, 5901}, {3830, 546}, {3839, 47478}, {3845, 5}, {3853, 3860}, {3860, 35018}, {5055, 47598}, {5066, 3628}, {5476, 3589}, {5655, 10272}, {5946, 5892}, {7575, 18579}, {8584, 575}, {8703, 3}, {9771, 7619}, {9909, 33591}, {10109, 16239}, {10154, 34351}, {10182, 46265}, {10192, 10182}, {10263, 21849}, {10283, 38028}, {11001, 12103}, {11178, 20582}, {11539, 5054}, {11563, 44282}, {11812, 12108}, {12040, 7622}, {12100, 3530}, {12101, 3850}, {13468, 34506}, {13490, 10127}, {14269, 14892}, {14869, 15701}, {14892, 41985}, {14893, 11737}, {15060, 10170}, {15067, 3819}, {15597, 1153}, {15681, 15691}, {15682, 3853}, {15686, 376}, {15687, 381}, {15689, 41982}, {15690, 33923}, {15699, 11539}, {15704, 3534}, {15711, 15712}, {15712, 15693}, {15713, 631}, {15714, 15692}, {16160, 44257}, {16509, 15597}, {17504, 3524}, {17525, 12104}, {19710, 550}, {19711, 3523}, {20423, 18583}, {21849, 5462}, {21850, 5476}, {21969, 143}, {22165, 40107}, {23039, 44324}, {23046, 5055}, {23328, 10193}, {25154, 20252}, {25164, 20253}, {31649, 15673}, {31650, 28465}, {33699, 4}, {34200, 14891}, {34331, 34478}, {34351, 34477}, {34510, 40344}, {35404, 15687}, {36523, 20398}, {37967, 37904}, {38028, 10165}, {38034, 11230}, {38042, 11231}, {38071, 15699}, {38111, 38122}, {38112, 26446}, {38136, 38317}, {38137, 38171}, {38138, 38042}, {38139, 38318}, {38140, 10172}, {38141, 38319}, {38229, 34127}, {39884, 47354}, {41099, 12812}, {41990, 5067}, {44204, 44564}, {44213, 15330}, {44245, 46332}, {44257, 6675}, {44262, 6676}, {44265, 18571}, {44266, 468}, {44267, 47332}, {44268, 43615}, {44270, 16238}, {44275, 6677}, {44278, 10020}, {44280, 37968}, {44282, 44452}, {44422, 11272}, {44682, 19711}, {44903, 15686}, {45759, 17504}, {46853, 15711}, {47313, 12105}, {47353, 18358}, {47354, 24206}, {47478, 41984}, {47598, 14890}
X(549) = isogonal conjugate of X(14483)
X(549) = complement of X(381)
X(549) = anticomplement of X(547)
X(549) = complement of the isogonal conjugate of X(3431)
X(549) = isotomic conjugate of the isogonal conjugate of X(44109)
X(549) = isogonal conjugate of the isotomic conjugate of X(44148)
X(549) = isotomic conjugate of the polar conjugate of X(6749)
X(549) = Thomson isogonal conjugate of X(5888)
X(549) = psi-transform of X(9143)
X(549) = X(i)-complementary conjugate of X(j) for these (i,j): {3431, 10}, {43530, 20305}
X(549) = X(18317)-Ceva conjugate of X(30)
X(549) = X(44109)-cross conjugate of X(6749)
X(549) = X(1)-isoconjugate of X(14483)
X(549) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 14483}, {381, 549}
X(549) = crosssum of X(6) and X(34417)
X(549) = circumcircle-inverse of X(37924)
X(549) = orthocentroidal circle inverse of X(5055)
X(549) = orthoptic circle of the Steiner inellipe inverse of X(10989)
X(549) = first Droz-Farney circle inverse of X(30745)
X(549) = ninepoint circle of medial triangle inverse of X(15122)
X(549) = center of the Vu pedal-centroidal circle of X(20)
X(549) = X(3)-of-X(2)-Brocard-triangle
X(549) = intersection of tangents to Evans conic at X(13) and X(14)
X(549) = inverse-in-circle-O(PU(5)) of X(140)
X(549) = center of inverse-in-{circumcircle, nine-point circle}-inverter of de Longchamps line
X(549) = trisector nearest X(3) of segment X(3)X(5)
X(549) = X(115)-of-McCay-triangle
X(549) = Euler line intercept, other than X(5), of conic {X(5),X(13),X(14),X(15),X(16)}}
X(549) = QA-P39 (Midpoint of QA-P12 and QA-P20) of quadrangle ABCX(2) (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/encyclopedia-of-quadri-figures/quadrangle-objects/artikelen-qa/204-qa-p39.html)
X(549) = barycentric product X(i)*X(j) for these {i,j}: {6, 44148}, {69, 6749}, {76, 44109}
X(549) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 14483}, {6749, 4}, {44109, 6}, {44148, 76}
X(549) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 5055}, {2, 5, 15699}, {2, 20, 3545}, {2, 140, 11539}, {2, 376, 381}, {2, 381, 547}, {2, 382, 47478}, {2, 548, 23046}, {2, 550, 38071}, {2, 631, 5054}, {2, 1656, 47599}, {2, 1657, 14892}, {2, 3522, 3839}, {2, 3523, 3524}, {2, 3524, 3}, {2, 3526, 47598}, {2, 3528, 14269}, {2, 3530, 17504}, {2, 3534, 5066}, {2, 3543, 5071}, {2, 3545, 1656}, {2, 3830, 10109}, {2, 3839, 3090}, {2, 3843, 45757}, ..., {3, 4, 548}, {3, 5, 550}, {3, 20, 33923}, {3, 24, 7525}, {3, 140, 5}, {3, 376, 34200}, {3, 381, 376}, {3, 382, 3522}, {3, 547, 15686}, {3, 550, 46853}, {3, 631, 140}, {3, 632, 3627}, {3, 1006, 7508}, {3, 1656, 20}, {3, 1657, 3528}, {3, 2070, 6636}, {3, 2072, 44249}, {3, 3090, 12103}, {3, 3091, 44245}, {3, 3523, 3530}, {3, 3524, 12100}, {3, 3525, 546}, {3, 3526, 4}, {3, 3530, 15712}, {3, 3533, 3853}, {3, 3534, 10304}, {3, 3545, 15690}, {3, 3548, 12362}, {3, 3549, 31829}, {3, 3628, 15704}, {3, 3830, 15688}, {3, 3832, 41981}, {3, 3851, 15696}, {3, 5020, 35243}, {3, 5054, 2}, {3, 5055, 3534}, {3, 5070, 1657}, {3, 5071, 15691}, {3, 5079, 17538}, {3, 6640, 12605}, {3, 6644, 7502}, {3, 6863, 31775}, {3, 6883, 6914}, {3, 6958, 31789}, {3, 6989, 37281}, {3, 7393, 7526}, {3, 7395, 12084}, {3, 7483, 37356}, {3, 7484, 7514}, {3, 7489, 6909}, ... , {4, 5, 3857}, {4, 376, 15683}, {4, 548, 15704}, {4, 631, 10303}, {4, 3524, 15698}, {4, 3526, 3628}, {4, 3628, 5}, {4, 5054, 11540}, {4, 5055, 5066}, {4, 5066, 23046}, {4, 5072, 3856}, {4, 7486, 5072}, {4, 10303, 3526}, {4, 10304, 3534}, {4, 15683, 15684}, {4, 15698, 10304}, {4, 15706, 15759}, {4, 15709, 2}, {4, 15717, 3}, {4, 23046, 3845}, {4, 35472, 37931}, {4, 37453, 37942}, {4, 37942, 44957}, {4, 45761, 15714}, {4, 46333, 15640}, {5, 140, 632}, {5, 376, 35404}, {5, 550, 3627}, {5, 1368, 37938}, {5, 3524, 15711}, {5, 3530, 44682}, {5, 3627, 3858}, {5, 3845, 38071}, {5, 11539, 2}, {5, 12100, 45759}, {5, 14869, 140}, {5, 15686, 15687}, {5, 15687, 381}, {5, 15704, 4}, {5, 15712, 3}, {5, 15713, 11539}, {5, 15714, 15686}, {5, 17504, 8703}, {5, 19711, 17504}, {5, 23046, 5066}, {5, 23335, 33332}, {5, 31651, 44286}, {5, 33699, 23046}, {5, 34200, 44903}, {5, 34477, 16532}, {5, 41991, 3851}, {5, 43893, 46030}, {5, 44222, 5499}, {5, 44682, 46853}, {5, 45759, 19710}, {6, 42912, 42633}, {6, 42913, 42634}, {10, 13624, 34773}, {10, 34773, 37705}, {13, 10646, 42943}, {13, 42943, 42118}, {13, 42955, 43544}, {14, 10645, 42942}, {14, 42942, 42117}, {14, 42954, 43545}, {15, 16242, 395}, {16, 16241, 396}, {17, 5351, 42148}, {18, 5352, 42147}, {20, 1656, 546}, {20, 3524, 15716}, {20, 3525, 1656}, {20, 3545, 3830}, {20, 10299, 3}, {20, 15022, 4}, {20, 15688, 15690}, {20, 15705, 19708}, {20, 16239, 5}, {20, 19708, 15688}, {21, 13747, 17527}, {22, 12106, 37936}, {24, 1595, 7715}, {24, 14787, 23410}, {25, 35486, 37935}, {35, 496, 10386}, {35, 3582, 3058}, {35, 5433, 496}, {36, 3584, 5434}, {36, 5432, 495}, {40, 25055, 3656}, {55, 10072, 15170}, {61, 16963, 43229}, {62, 16962, 43228}, {74, 38794, 10272}, {98, 41134, 8724}, {140, 382, 41992}, {140, 546, 16239}, {140, 547, 10124}, {140, 548, 3628}, {140, 631, 14869}, {140, 3523, 15712}, {140, 3524, 8703}, {140, 3530, 3}, {140, 3628, 3526}, {140, 3850, 46219}, {140, 5054, 15713}, {140, 5066, 47598}, {140, 6676, 44452}, {140, 8703, 15699}, {140, 10124, 15694}, {140, 10212, 14118}, {140, 11276, 37251}, {140, 11540, 15709}, {140, 11737, 15723}, {140, 11812, 5054}, {140, 12101, 41984}, {140, 12108, 631}, {140, 14891, 381}, {140, 15331, 7399}, {140, 15690, 47599}, {140, 15692, 15687}, {140, 15693, 17504}, {140, 15698, 23046}, {140, 15700, 15686}, {140, 15706, 33699}, {140, 15707, 19711}, {140, 15711, 38071}, {140, 15712, 550}, {140, 15717, 15704}, {140, 15718, 15714}, {140, 15759, 5055}, {140, 16197, 16238}, {140, 16239, 3525}, {140, 16976, 18570}, {140, 17504, 3845}, {140, 19711, 45759}, {140, 33923, 1656}, {140, 34004, 10125}, {140, 34200, 547}, {140, 35018, 3533}, {140, 37298, 31650}, {140, 37968, 46029}, {140, 41983, 12100}, {140, 44245, 5070}, {140, 44580, 3524}, {140, 44682, 3627}, {140, 45761, 15684}, {140, 47598, 11540}, {186, 427, 37458}, {186, 15246, 3}, {186, 37118, 427}, {187, 3815, 18907}, {230, 574, 15048}, {371, 13966, 19116}, {372, 8981, 19117}, {376, 547, 15687}, {376, 631, 15702}, {376, 3091, 35400}, {376, 3523, 15718}, {376, 3524, 15692}, {376, 3543, 15681}, {376, 5054, 10124}, {376, 5071, 3543}, {376, 10124, 5}, {376, 14891, 15714}, {376, 15078, 47333}, {376, 15681, 15691}, {376, 15683, 3534}, {376, 15686, 550}, {376, 15687, 44903}, {376, 15692, 3}, {376, 15694, 547}, {376, 15700, 14891}, {376, 15702, 2}, {376, 15703, 14893}, {376, 15718, 12100}, {376, 15721, 15694}, {376, 15723, 11737}, {376, 33246, 37461}, {376, 34200, 8703}, {376, 35404, 19710}, {376, 44214, 44213}, {378, 468, 1596}, {381, 547, 5}, {381, 1657, 35434}, {381, 3524, 14891}, {381, 3534, 15684}, {381, 3543, 14893}, {381, 5054, 15694}, {381, 5071, 11737}, {381, 8703, 44903}, {381, 10201, 47334}, {381, 12100, 15714}, {381, 14093, 15681}, {381, 14891, 8703}, {381, 15681, 3543}, {381, 15682, 41988}, {381, 15684, 4}, {381, 15686, 35404}, {381, 15687, 3845}, {381, 15692, 34200}, {381, 15693, 15700}, {381, 15694, 2}, {381, 15700, 3}, {381, 15701, 15721}, {381, 15702, 10124}, {381, 15703, 5071}, {381, 15714, 550}, {381, 15718, 15692}, {381, 15721, 140}, {381, 15723, 15703}, {381, 31152, 31181}, {381, 34006, 37901}, {381, 34200, 15686}, {381, 35434, 14269}, {381, 44211, 44212}, {381, 44214, 44211}, {381, 44903, 3627}, {382, 3090, 3850}, {382, 3522, 12103}, {382, 3839, 12101}, {382, 15689, 11001}, {382, 19709, 3839}, {382, 46219, 3090}, {389, 5447, 6101}, {395, 16242, 42121}, {396, 16241, 42124}, {404, 7483, 8728}, {404, 37291, 7483}, {442, 4188, 17563}, {465, 466, 26906}, {474, 6910, 6675}, {484, 5444, 15950}, {498, 5204, 18990}, {499, 5217, 15171}, {546, 1656, 5}, {546, 3530, 10299}, {546, 3628, 15022}, {546, 10109, 3545}, {546, 15705, 8703}, {546, 16239, 1656}, {546, 33923, 20}, {546, 47599, 10109}, {547, 3530, 15692}, {547, 8703, 35404}, {547, 10124, 2}, {547, 11737, 5071}, {547, 11812, 15702}, {547, 12100, 34200}, {547, 14891, 376}, {547, 14893, 11737}, {547, 15330, 44211}, {547, 15683, 23046}, {547, 15686, 3845}, {547, 15691, 14893}, {547, 15692, 8703}, {547, 15700, 15714}, {547, 15702, 11539}, {547, 15714, 44903}, {547, 15718, 17504}, {547, 15759, 15683}, {547, 35404, 38071}, {547, 41981, 35401}, {547, 44213, 44212}, {547, 44580, 15718}, {547, 45761, 10304}, {547, 46332, 35400}, {548, 3526, 5}, {548, 3530, 15717}, {548, 3628, 4}, {548, 3856, 17800}, {548, 5055, 33699}, {548, 10304, 8703}, {548, 11540, 5055}, {548, 11812, 15709}, {548, 14890, 2}, {548, 15684, 15686}, {548, 15704, 550}, {548, 15759, 10304}, {548, 47598, 5066}, {550, 632, 5}, {550, 15699, 3845}, {550, 15711, 45759}, {550, 16532, 10154}, {550, 35404, 44903}, {550, 44682, 3}, {550, 44903, 15686}, {550, 45759, 8703}, {590, 6396, 42216}, {590, 41946, 35822}, {597, 10168, 38110}, {599, 5085, 11179}, {615, 6200, 42215}, {615, 41945, 35823}, {618, 619, 141}, {618, 5092, 47611}, {619, 5092, 47610}, {631, 3523, 3}, {631, 3524, 2}, {631, 3530, 5}, {631, 3534, 14890}, {631, 5054, 11812}, {631, 7485, 10257}, {631, 10299, 3525}, {631, 12100, 11539}, {631, 15692, 15694}, {631, 15698, 15709}, {631, 15700, 10124}, {631, 15702, 15721}, {631, 15706, 11540}, {631, 15707, 12100}, {631, 15708, 15701}, {631, 15712, 632}, {631, 15717, 3526}, {631, 15718, 547}, {631, 15719, 3524}, {631, 15720, 12108}, {631, 15722, 41983}, {631, 19711, 15699}, {631, 41983, 8703}, {631, 44580, 17504}, {632, 3845, 15699}, {632, 3857, 3628}, {632, 8703, 38071}, {632, 15699, 2}, {632, 15712, 46853}, {632, 17504, 19710}, {632, 35404, 547}, {632, 44682, 550}, {632, 45759, 3845}, {632, 46853, 3858}, {858, 10298, 44239}, {858, 37347, 39504}, {993, 3035, 3820}, {1006, 17549, 28443}, {1078, 7799, 37671}, {1092, 13336, 32046}, {1113, 1114, 37924}, {1125, 3579, 22791}, {1141, 6592, 14073}, and many more


X(550) = MIDPOINT OF X(3) AND X(20)

Trilinears    - cos(B - C) + 4 cos A : :
Trilinears    2 sec A - 3 sec B sec C : 2 sec B - 3 sec C sec A : 2 sec C - 3 sec A sec B
Trilinears    6 cos A + cos B cos C - 3 sin B sin C : :
Trilinears    cos A - 4 cos B cos C + 2 sin B sin C : :
Trilinears    6 cos B cos C - 2 sin B sin C - cos(B - C) : :
Trilinears    5 cos A + cos(B - C) - cos B cos C - 3 sin B sin C : :
Barycentrics    4 a^4 - 3 a^2 (b^2 + c^2) - (b^2 - c^2)^2 : :
X(550) = 3*X(3) - X(4)
As a point on the Euler line, X(550) has Shinagawa coefficients (3,-5).

Let OA be the circle centered at the A-vertex of the anticomplementary triangle and passing through A; define OB and OC cyclically. X(550) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

Let OA be the circle centered at the A-vertex of the anti-Euler triangle and passing through A; define OB and OC cyclically. X(550) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

X(550) lies on these lines: 2,3   15,397   16,398   35,495   36,496   40,952   74,930   156,1092   165,355

X(550) = midpoint of X(3) and X(20)
X(550) = reflection of X(i) in X(j) for these (i,j): (3,548), (4,140), (5,3), (382,546)
X(550) = isogonal conjugate of X(16835)
X(550) = trisector nearest X(20) of segment X(5)X(20)
X(550) = complement of X(382)
X(550) = anticomplement of X(546)
X(550) = X(5)-of-circumcevian-triangle-of-X(30)
X(550) = X(143)-of-hexyl-triangle
X(550) = intersection of tangents to Evans conic at X(17) and X(18)
X(550) = inverse-in-orthocentroidal-circle of X(3851)
X(550) = radical center of reflections of nine-point circle in A, B, C
X(550) = antipedal isogonal conjugate of X(5)
X(550) = radical trace of circumcircle and Trinh circle
X(550) = {X(2),X(3529)}-harmonic conjugate of X(382)
X(550) = {X(3),X(4)}-harmonic conjugate of X(140)
X(550) = {X(2),X(3)}-harmonic conjugate of X(3530)
X(550) = {X(2),X(20)}-harmonic conjugate of X(3529)


X(551) = MIDPOINT OF X(1) AND X(2)

Trilinears    (4a + b + c)/a : (4b + c + a)/b : (4c + a + b)/c
Trilinears    3 r + 2 R sin B sin C :
Barycentrics  4a + b + c : 4b + c + a : 4c + a + b
X(551) = X(1) + X(2) = 2 X(8) - 5 X(10) = 5 X[1] + X[8], 2 X[1] + X[10], 7 X[1] - X[145], X[1] + 2 X[1125], 7 X[1] + 5 X[1698], 3 X[1] - X[3241], 4 X[1] - X[3244], X[1] + 5 X[3616], 13 X[1] + 5 X[3617], 17 X[1] + X[3621], X[1] - 7 X[3622], 11 X[1] - 5 X[3623], 5 X[1] + 7 X[3624], 8 X[1] + X[3625], 7 X[1] + 2 X[3626], 11 X[1] + X[3632], 13 X[1] - X[3633], 3166], and many others

{1, 2}, {3, 3653}, {4, 9624}, {5, 5882}, {6, 3986}, {7, 13462}, {9, 18490}, {11, 3822}, {12, 3825}, {13, 50849}, {14, 50852}, {15, 50854}, {16, 50857}, {20, 11522}, {21, 5557}, {30, 946}, {35, 5253}, {36, 1621}, {37, 537}, {38, 4694}, {40, 3524}, {44, 61302}, {55, 4342}, {56, 553}, {58, 42028}, {63, 51816}, {65, 3884}, {69, 4909}, {72, 3881}, {74, 50878}, {75, 4717}, {79, 15678}, {80, 10031}, {81, 5315}, {85, 25723}, {86, 99}, {88, 24857}, {98, 50881}, {100, 36006}, {101, 16503}, {102, 50901}, {103, 50905}, {104, 3255}, {105, 8691}, {109, 50918}, {110, 50921}, {111, 50926}, {113, 50876}, {114, 50879}, {115, 50884}, {116, 50895}, {118, 50902}, {119, 50906}, {124, 47115}, {125, 50919}, {140, 10222}, {141, 28538}, {142, 214}, {144, 50837}, {147, 50882}, {148, 50887}, {149, 50892}, {153, 50909}, {165, 15692}, {171, 40091}, {190, 31332}, {191, 15672}, {226, 535}, {238, 46922}, {244, 4424}, {321, 4975}, {348, 58816}, {354, 392}, {355, 5055}, {366, 59438}, {376, 516}, {381, 515}, {382, 50806}, {390, 53054}, {404, 3746}, {405, 3304}, {428, 11363}, {442, 24387}, {468, 47593}, {474, 3303}, {484, 27003}, {495, 3814}, {496, 3829}, {497, 13384}, {511, 22475}, {512, 45657}, {514, 4448}, {517, 549}, {518, 597}, {522, 36220}, {524, 1386}, {527, 993}, {529, 21616}, {530, 11705}, {531, 11706}, {532, 11739}, {533, 11740}, {534, 51687}, {536, 3993}, {538, 12263}, {539, 12259}, {540, 13745}, {541, 11709}, {542, 11710}, {543, 11599}, {544, 11712}, {545, 4432}, {546, 51074}, {547, 952}, {548, 31666}, {550, 28202}, {591, 45398}, {594, 46845}, {595, 37607}, {596, 56221}, {599, 5847}, {618, 50847}, {619, 50850}, {631, 7982}, {632, 50823}, {650, 50761}, {663, 23789}, {671, 38220}, {678, 24183}, {726, 4664}, {730, 9466}, {740, 4688}, {750, 37610}, {752, 1279}, {754, 12264}, {759, 34053}, {891, 45314}, {894, 17487}, {900, 14422}, {908, 34690}, {940, 16351}, {942, 3878}, {944, 3545}, {950, 11238}, {953, 2690}, {954, 60972}, {956, 4423}, {958, 7373}, {960, 3874}, {962, 7987}, {964, 48865}, {966, 4856}, {984, 49535}, {986, 24167}, {996, 36915}, {1000, 31190}, {1010, 19796}, {1012, 43177}, {1056, 26105}, {1058, 34701}, {1064, 32486}, {1086, 35124}, {1100, 5257}, {1104, 48823}, {1108, 25081}, {1145, 52638}, {1213, 50082}, {1224, 56037}, {1255, 32923}, {1266, 41847}, {1317, 6702}, {1320, 13602}, {1323, 17079}, {1376, 6767}, {1388, 3947}, {1390, 34892}, {1420, 3485}, {1421, 54292}, {1449, 37654}, {1475, 3294}, {1482, 3654}, {1483, 9956}, {1500, 16604}, {1573, 52708}, {1616, 5711}, {1651, 11831}, {1656, 37727}, {1699, 3543}, {1724, 19738}, {1738, 49720}, {1770, 37299}, {1836, 57006}, {1908, 2666}, {1909, 18145}, {1962, 24165}, {1982, 52954}, {1991, 45399}, {1992, 16475}, {2049, 48862}, {2051, 6176}, {2087, 21332}, {2094, 4512}, {2098, 58405}, {2099, 3911}, {2238, 16971}, {2275, 25092}, {2292, 3953}, {2303, 16488}, {2320, 5561}, {2321, 3723}, {2325, 16672}, {2329, 9327}, {2345, 4072}, {2475, 4857}, {2476, 37720}, {2550, 38093}, {2646, 3058}, {2759, 9097}, {2784, 6054}, {2785, 21181}, {2787, 45342}, {2800, 10202}, {2801, 10177}, {2802, 3753}, {2975, 5259}, {3007, 36024}, {3035, 50841}, {3057, 3754}, {3090, 5881}, {3091, 50803}, {3109, 62500}, {3120, 50172}, {3146, 50869}, {3158, 25419}, {3159, 6534}, {3175, 51672}, {3227, 30571}, {3230, 3997}, {3242, 47352}, {3246, 4364}, {3247, 4098}, {3248, 25422}, {3251, 36848}, {3295, 4421}, {3306, 5119}, {3333, 3928}, {3337, 56288}, {3338, 5250}, {3339, 4323}, {3340, 7288}, {3416, 21358}, {3428, 52769}, {3445, 4052}, {3448, 50922}, {3452, 5719}, {3475, 31142}, {3476, 5219}, {3486, 50443}, {3487, 5436}, {3488, 25525}, {3522, 9589}, {3523, 5734}, {3525, 50817}, {3526, 38066}, {3528, 50812}, {3529, 50819}, {3530, 51084}, {3534, 12699}, {3555, 3678}, {3579, 12100}, {3589, 9041}, {3601, 10385}, {3612, 10624}, {3615, 50148}, {3618, 16496}, {3619, 50786}, {3620, 50788}, {3627, 58232}, {3628, 31399}, {3646, 6762}, {3647, 15673}, {3649, 17525}, {3664, 17274}, {3667, 59829}, {3670, 42040}, {3685, 50119}, {3686, 16884}, {3696, 50778}, {3699, 31514}, {3702, 4980}, {3707, 4407}, {3731, 15828}, {3739, 4709}, {3743, 37592}, {3748, 5440}, {3750, 4256}, {3751, 50999}, {3752, 4868}, {3755, 49473}, {3758, 4759}, {3763, 49681}, {3772, 56226}, {3775, 4725}, {3812, 9957}, {3813, 8728}, {3821, 17399}, {3823, 49691}, {3830, 18481}, {3832, 50868}, {3834, 49700}, {3836, 48821}, {3838, 7743}, {3839, 5691}, {3841, 24390}, {3842, 49478}, {3843, 50807}, {3844, 50949}, {3845, 9955}, {3847, 10592}, {3848, 3880}, {3850, 51078}, {3851, 50799}, {3868, 50190}, {3869, 18398}, {3873, 4525}, {3877, 4744}, {3879, 17271}, {3883, 17297}, {3887, 45328}, {3889, 4537}, {3890, 5903}, {3893, 4002}, {3897, 13407}, {3900, 44561}, {3902, 4714}, {3907, 45324}, {3913, 16408}, {3915, 37522}, {3918, 10914}, {3923, 4353}, {3929, 17561}, {3931, 52541}, {3934, 24656}, {3940, 6666}, {3943, 39260}, {3945, 16487}, {3946, 4780}, {3950, 5750}, {3956, 61686}, {3960, 23795}, {3970, 39244}, {3971, 31161}, {3984, 31259}, {4004, 52793}, {4021, 10436}, {4029, 4908}, {4031, 39782}, {4054, 19740}, {4058, 17303}, {4065, 42051}, {4066, 4968}, {4078, 41310}, {4085, 17245}, {4090, 42056}, {4125, 4358}, {4129, 48328}, {4133, 17133}, {4135, 32771}, {4138, 26128}, {4151, 45671}, {4160, 45315}, {4187, 15888}, {4190, 4309}, {4193, 37719}, {4257, 8616}, {4292, 37618}, {4304, 5249}, {4305, 9614}, {4308, 5290}, {4310, 35578}, {4311, 11114}, {4312, 59375}, {4313, 18220}, {4317, 6872}, {4321, 60967}, {4325, 15680}, {4330, 37256}, {4345, 5281}, {4356, 17301}, {4357, 17361}, {4359, 4742}, {4360, 50099}, {4361, 28640}, {4363, 4758}, {4368, 50184}, {4385, 20942}, {4389, 4896}, {4413, 8162}, {4425, 31177}, {4441, 52716}, {4464, 30598}, {4472, 28309}, {4479, 20888}, {4530, 21929}, {4568, 49528}, {4640, 4973}, {4648, 4660}, {4656, 11346}, {4657, 17313}, {4658, 11110}, {4672, 49742}, {4673, 28612}, {4675, 24692}, {4676, 49748}, {4684, 17346}, {4687, 49490}, {4690, 25358}, {4698, 49457}, {4699, 49469}, {4700, 62212}, {4702, 17395}, {4704, 49532}, {4732, 31238}, {4737, 30829}, {4740, 28522}, {4751, 49459}, {4762, 36219}, {4767, 24858}, {4781, 42026}, {4785, 36217}, {4798, 17318}, {4807, 48347}, {4844, 47779}, {4848, 5433}, {4852, 6707}, {4864, 31289}, {4867, 54357}, {4873, 26039}, {4875, 25086}, {4885, 48285}, {4906, 6703}, {4922, 59737}, {4930, 5289}, {4937, 31035}, {4945, 24709}, {4948, 48291}, {4956, 33155}, {4966, 50125}, {4967, 17393}, {4969, 52706}, {4974, 49731}, {4999, 5837}, {5010, 61155}, {5030, 60711}, {5032, 50952}, {5044, 34791}, {5046, 5270}, {5047, 5258}, {5048, 5432}, {5056, 37714}, {5059, 51119}, {5064, 49542}, {5066, 18480}, {5067, 61288}, {5070, 50804}, {5071, 5587}, {5072, 50800}, {5079, 50797}, {5091, 16384}, {5126, 25557}, {5136, 23710}, {5159, 47491}, {5180, 11552}, {5184, 26613}, {5204, 19704}, {5217, 19705}, {5218, 7962}, {5223, 61023}, {5224, 50132}, {5233, 21088}, {5242, 7026}, {5243, 7043}, {5247, 51595}, {5251, 5284}, {5260, 5288}, {5261, 6049}, {5263, 37756}, {5273, 30350}, {5276, 16784}, {5296, 16667}, {5302, 14150}, {5316, 34689}, {5333, 32943}, {5425, 59491}, {5426, 15677}, {5429, 17254}, {5435, 18421}, {5437, 31393}, {5441, 15679}, {5443, 37375}, {5450, 16203}, {5499, 35597}, {5657, 15702}, {5690, 11539}, {5698, 43180}, {5712, 16485}, {5717, 48801}, {5718, 21251}, {5722, 58463}, {5727, 10589}, {5749, 16673}, {5790, 10172}, {5794, 50740}, {5795, 25681}, {5818, 61296}, {5836, 31792}, {5844, 10124}, {5846, 20582}, {5850, 6172}, {5853, 35272}, {5860, 11371}, {5861, 11370}, {5880, 34626}, {5884, 13373}, {5887, 12005}, {5988, 8592}, {6000, 10181}, {6051, 42055}, {6161, 23814}, {6175, 10707}, {6245, 37615}, {6265, 51755}, {6361, 19708}, {6366, 44566}, {6381, 18146}, {6536, 50215}, {6541, 17359}, {6583, 8261}, {6626, 33770}, {6629, 16481}, {6667, 12735}, {6668, 38105}, {6675, 24391}, {6683, 24739}, {6687, 49701}, {6690, 15325}, {6701, 10543}, {6706, 59610}, {6713, 25485}, {6796, 16202}, {6856, 37723}, {6857, 11518}, {6881, 37726}, {6905, 34486}, {6912, 41561}, {6913, 59687}, {6921, 31452}, {6933, 37721}, {7278, 26563}, {7485, 37546}, {7677, 60932}, {7739, 9619}, {7811, 11368}, {7865, 49561}, {7968, 8983}, {7969, 13971}, {7972, 31272}, {7974, 22490}, {7975, 22489}, {7983, 9881}, {7988, 59387}, {7989, 61924}, {8025, 52680}, {8125, 30411}, {8126, 30423}, {8148, 15701}, {8167, 9708}, {8168, 61158}, {8236, 38052}, {8273, 12511}, {8543, 60952}, {8679, 15049}, {8703, 13624}, {8714, 44550}, {9028, 25362}, {9053, 38191}, {9166, 9884}, {9269, 45666}, {9310, 16783}, {9331, 17756}, {9507, 35103}, {9530, 11718}, {9569, 13731}, {9588, 10303}, {9611, 45281}, {9626, 37939}, {9670, 50239}, {9711, 51559}, {9776, 30282}, {9778, 58221}, {9779, 61985}, {9812, 15683}, {9848, 17646}, {9856, 58567}, {9864, 23234}, {9875, 41135}, {9900, 59379}, {9901, 59378}, {9909, 11365}, {9940, 45776}, {10109, 18357}, {10168, 38118}, {10178, 33574}, {10247, 15694}, {10248, 62032}, {10265, 19907}, {10299, 50809}, {10440, 45955}, {10481, 17078}, {10572, 17577}, {10584, 37708}, {10585, 37711}, {10588, 37709}, {10695, 28346}, {10966, 54430}, {11001, 41869}, {11012, 21161}, {11036, 28610}, {11049, 11900}, {11050, 16212}, {11108, 12513}, {11179, 38029}, {11207, 11366}, {11208, 11367}, {11224, 15721}, {11235, 11373}, {11236, 11374}, {11249, 28466}, {11278, 11812}, {11359, 48824}, {11364, 12150}, {11372, 43176}, {11377, 12152}, {11378, 12153}, {11523, 16845}, {11529, 34744}, {11531, 15708}, {11551, 37587}, {11570, 58625}, {11684, 15675}, {11700, 11734}, {11713, 11727}, {11714, 11728}, {11715, 11729}, {11717, 11731}, {11719, 11733}, {11737, 28224}, {12101, 33697}, {12108, 31447}, {12114, 54227}, {12245, 15709}, {12268, 32419}, {12269, 32421}, {12523, 58707}, {12559, 15829}, {12607, 17527}, {12608, 24927}, {12617, 21740}, {12640, 33895}, {12645, 61284}, {12653, 50845}, {12675, 31803}, {12680, 31871}, {12702, 15693}, {12812, 61255}, {13174, 52695}, {13374, 31786}, {13634, 48932}, {13688, 49786}, {13725, 48834}, {13751, 45288}, {13808, 49787}, {13846, 13883}, {13847, 13936}, {13902, 18992}, {13912, 35642}, {13959, 18991}, {13975, 35641}, {14005, 28618}, {14061, 50885}, {14077, 44567}, {14093, 28232}, {14151, 61015}, {14210, 26234}, {14474, 29350}, {14563, 17051}, {14636, 37620}, {14839, 28600}, {14869, 50825}, {14891, 28212}, {14893, 28186}, {15059, 50920}, {15254, 49737}, {15485, 17333}, {15570, 24393}, {15621, 19261}, {15671, 34195}, {15674, 16126}, {15681, 28150}, {15684, 28172}, {15686, 28146}, {15687, 28160}, {15689, 48661}, {15691, 28178}, {15698, 35242}, {15705, 16192}, {15712, 50833}, {15723, 59503}, {16052, 17056}, {16236, 31188}, {16393, 32577}, {16394, 48811}, {16402, 37540}, {16474, 32911}, {16478, 48839}, {16486, 37674}, {16489, 37633}, {16490, 37680}, {16494, 34230}, {16497, 37632}, {16498, 18134}, {16552, 17474}, {16589, 17448}, {16705, 17179}, {16711, 17175}, {16785, 33854}, {16801, 20132}, {16969, 17750}, {16980, 58474}, {17063, 24168}, {17067, 24693}, {17184, 30562}, {17248, 50074}, {17259, 49497}, {17264, 24295}, {17277, 49685}, {17300, 50304}, {17305, 49709}, {17317, 33076}, {17323, 60980}, {17342, 17381}, {17353, 31333}, {17390, 25498}, {17396, 24199}, {17450, 42285}, {17469, 41820}, {17502, 28174}, {17504, 31663}, {17538, 50820}, {17558, 54422}, {17575, 21031}, {17578, 50870}, {17599, 59692}, {17600, 59628}, {17606, 37734}, {17614, 33595}, {17678, 19786}, {17706, 26066}, {17718, 31141}, {17720, 27752}, {17722, 27759}, {17725, 27777}, {17757, 44847}, {17758, 48844}, {17766, 31151}, {17793, 33908}, {18140, 25303}, {18156, 33945}, {18230, 50835}, {18254, 46681}, {18447, 34823}, {18455, 34822}, {18492, 41106}, {18525, 19709}, {18526, 61261}, {19065, 49619}, {19066, 49618}, {19251, 23383}, {19290, 19765}, {19323, 37580}, {19325, 37576}, {19336, 24177}, {19701, 48863}, {19706, 34707}, {19785, 41930}, {19886, 19992}, {19933, 20000}, {19958, 27918}, {20060, 26127}, {20131, 53602}, {20195, 51102}, {20292, 36005}, {20323, 21077}, {20517, 52596}, {21075, 34749}, {21356, 50950}, {21627, 56176}, {21633, 55172}, {21735, 50813}, {21747, 26860}, {21839, 59218}, {21849, 58469}, {21879, 55343}, {21969, 31757}, {22712, 49631}, {22765, 28443}, {23156, 42450}, {23493, 52573}, {23536, 36250}, {23537, 48816}, {23708, 31266}, {23808, 53535}, {23887, 45341}, {23888, 26275}, {24046, 37598}, {24159, 51668}, {24210, 48841}, {24231, 50128}, {24299, 28452}, {24349, 51035}, {24386, 44669}, {24392, 50727}, {24408, 43282}, {24474, 28465}, {24475, 58619}, {24508, 35030}, {24552, 50102}, {24703, 34740}, {24841, 41138}, {25079, 50078}, {25351, 53534}, {25378, 26738}, {25416, 31235}, {25496, 48826}, {25526, 51669}, {25539, 49506}, {25569, 48167}, {25716, 52422}, {25996, 55134}, {26098, 48799}, {26109, 33106}, {26150, 50289}, {26365, 45696}, {26366, 45697}, {26367, 45699}, {26368, 45698}, {27268, 49448}, {27269, 31999}, {27318, 32095}, {27784, 42054}, {27804, 46895}, {27811, 35263}, {28154, 44903}, {28168, 35404}, {28182, 62139}, {28190, 62015}, {28216, 62089}, {28294, 45677}, {28358, 50620}, {28389, 50626}, {28542, 49727}, {28554, 49483}, {28629, 34607}, {28633, 62681}, {28840, 36218}, {28850, 35110}, {28877, 61673}, {29066, 45320}, {29219, 31163}, {29298, 45332}, {29311, 39550}, {30315, 46936}, {30329, 58564}, {30340, 60905}, {30893, 35548}, {31134, 50321}, {31140, 44217}, {31149, 47841}, {31150, 50760}, {31156, 31164}, {31160, 37701}, {31171, 32944}, {31172, 33065}, {31179, 33122}, {31209, 50767}, {31243, 49699}, {31252, 49695}, {31273, 50897}, {31331, 31349}, {31425, 61814}, {31479, 34717}, {31582, 49616}, {31583, 49614}, {31659, 61534}, {31806, 31838}, {32007, 41807}, {32183, 58444}, {32558, 37718}, {32900, 37705}, {32922, 50100}, {33104, 48836}, {33124, 38456}, {33144, 48833}, {33159, 49527}, {33597, 34746}, {33934, 49780}, {33940, 41875}, {33942, 39731}, {34232, 59999}, {34522, 40869}, {34611, 37571}, {34620, 57282}, {34657, 52397}, {34700, 37739}, {34712, 44442}, {34719, 49719}, {34790, 58609}, {34824, 62682}, {34831, 59647}, {34832, 38986}, {35018, 61249}, {35119, 35121}, {35148, 35168}, {35170, 35180}, {35262, 59337}, {35631, 62189}, {35652, 48820}, {35762, 35822}, {35763, 35823}, {36004, 37616}, {36011, 53302}, {36770, 50848}, {36775, 49603}, {36872, 60725}, {36911, 54389}, {37038, 48868}, {37150, 48846}, {37286, 41341}, {37593, 50083}, {37612, 40256}, {37696, 58403}, {37697, 58402}, {37712, 61912}, {37831, 49594}, {37834, 49595}, {37911, 47492}, {38035, 54131}, {38042, 61283}, {38081, 50831}, {38087, 47355}, {38092, 50839}, {38099, 50846}, {38112, 61869}, {38138, 61916}, {38150, 54051}, {38176, 61880}, {38221, 51224}, {38295, 54396}, {38330, 60172}, {39579, 40985}, {40270, 57284}, {40328, 49470}, {40341, 50791}, {40374, 59460}, {40459, 45665}, {40688, 54387}, {40718, 43266}, {41136, 50247}, {41313, 50313}, {41490, 45500}, {41491, 45501}, {41529, 62441}, {41813, 41823}, {41815, 42045}, {41816, 51597}, {42029, 51605}, {43151, 43166}, {43573, 43822}, {44212, 44662}, {44659, 45879}, {44660, 45880}, {45751, 59207}, {45757, 61246}, {46903, 46911}, {46909, 53034}, {47097, 47472}, {47274, 50145}, {47319, 58568}, {47478, 61259}, {47598, 61597}, {47599, 61281}, {47729, 50764}, {48294, 50337}, {48310, 49524}, {48406, 58156}, {48813, 48827}, {48814, 48825}, {48817, 48818}, {48829, 50130}, {48837, 50428}, {48842, 50427}, {48870, 50430}, {49135, 50873}, {49226, 52045}, {49227, 52046}, {49445, 51056}, {49456, 49726}, {49474, 51054}, {49484, 49733}, {49491, 49508}, {49549, 49563}, {49679, 50789}, {49717, 50259}, {49732, 59691}, {49739, 50103}, {49749, 50179}, {50046, 51593}, {50048, 50072}, {50049, 50071}, {50061, 50066}, {50076, 50308}, {50085, 50312}, {50120, 50281}, {50166, 50233}, {50167, 50231}, {50221, 50266}, {50235, 50265}, {50260, 50261}, {50395, 61031}, {50688, 50866}, {50691, 50874}, {50693, 51081}, {50822, 61853}, {50826, 61837}, {50830, 55859}, {50840, 60957}, {50863, 61982}, {52653, 59372}, {54447, 59388}, {54448, 61264}, {54553, 60116}, {55173, 58440}, {55174, 58709}, {55175, 58711}, {56985, 60267}, {58217, 62054}, {58219, 62057}, {58224, 62071}, {58231, 62051}, {58233, 61974}, {58234, 61949}, {58235, 61935}, {58237, 61845}, {58245, 61820}, {58386, 61661}, {58813, 60942}, {61244, 61908}, {61245, 61909}, {61248, 61911}, {61250, 61913}, {61252, 61914}, {61256, 61915}, {61262, 61922}, {61263, 61925}, {61265, 61928}, {61266, 61931}, {61270, 61942}, {61271, 61944}, {61285, 61888}, {61290, 61894}, {61292, 61896}, {61295, 61898}, {61297, 61900}, {61614, 61839}
midpoint of X(i) and X(j) for these {i,j}: {1, 2}, {3, 3656}, {4, 50811}, {5, 50824}, {6, 47358}, {7, 50836}, {8, 51093}, {9, 51099}, {10, 51071}, {11, 50843}, {12, 51112}, {13, 50849}, {14, 50852}, {15, 50854}, {16, 50857}, {20, 50865}, {74, 50878}, {79, 15678}, {80, 10031}, {98, 50881}, {99, 50886}, {100, 50891}, {101, 50898}, {102, 50901}, {103, 50905}, {104, 50908}, {105, 50913}, {106, 50915}, {109, 50918}, {110, 50921}, {111, 50926}, {145, 4677}, {354, 392}, {376, 31162}, {381, 3655}, {428, 34634}, {468, 47593}, {599, 47356}, {946, 51705}, {984, 51055}, {1125, 51103}, {1385, 51709}, {1386, 51003}, {1482, 3654}, {1699, 5731}, {1836, 57006}, {3058, 11112}, {3241, 3679}, {3242, 47359}, {3244, 4669}, {3251, 36848}, {3416, 51000}, {3534, 12699}, {3543, 34628}, {3576, 5603}, {3616, 51105}, {3617, 51097}, {3622, 51110}, {3623, 51066}, {3625, 51096}, {3626, 51091}, {3634, 51107}, {3635, 4745}, {3636, 51108}, {3649, 17525}, {3696, 50778}, {3742, 10179}, {3751, 50999}, {3753, 5919}, {3830, 18481}, {3845, 34773}, {3873, 5692}, {3877, 5902}, {3892, 10176}, {3898, 5883}, {3993, 51060}, {4301, 50808}, {4421, 34640}, {4448, 14421}, {4654, 11111}, {4664, 31178}, {4668, 51092}, {4678, 51094}, {4691, 51095}, {4767, 24858}, {4795, 24441}, {4948, 48291}, {5434, 11113}, {5441, 15679}, {5493, 51120}, {5587, 7967}, {5657, 16200}, {5698, 60963}, {5790, 61287}, {5881, 50818}, {5882, 50796}, {5886, 10246}, {6173, 47357}, {7811, 34645}, {7972, 50890}, {7982, 50810}, {7983, 9881}, {7991, 50872}, {8236, 38052}, {8703, 22791}, {9269, 45666}, {9884, 13178}, {9909, 34643}, {10222, 50821}, {10247, 26446}, {10283, 38028}, {11001, 41869}, {11194, 34647}, {11224, 59417}, {11354, 48819}, {11359, 48824}, {11362, 51077}, {11711, 12258}, {12150, 34636}, {13464, 50828}, {13745, 37631}, {15569, 51061}, {15673, 16137}, {15808, 51106}, {16394, 50068}, {16834, 50316}, {17274, 50303}, {17301, 48805}, {17378, 50296}, {17392, 49740}, {19862, 51104}, {24325, 50111}, {24349, 51035}, {24473, 31165}, {25055, 38314}, {25416, 50842}, {27804, 46895}, {28609, 34610}, {29574, 50305}, {30331, 51100}, {31145, 34747}, {31150, 50760}, {33337, 50889}, {34657, 52397}, {34712, 44442}, {34719, 49719}, {36440, 36458}, {37727, 50798}, {38042, 61283}, {38053, 38316}, {42045, 49723}, {42051, 50122}, {45316, 45667}, {47097, 47472}, {47274, 50145}, {47729, 50764}, {48813, 48827}, {48814, 48825}, {48817, 48818}, {48829, 50130}, {48830, 48854}, {49445, 51056}, {49470, 50086}, {49471, 50096}, {49474, 51054}, {49478, 51034}, {49479, 50777}, {49490, 50075}, {49511, 51005}, {49524, 50998}, {49529, 51089}, {49544, 49550}, {49566, 49579}, {49568, 49583}, {49679, 50789}, {49681, 50783}, {49684, 50781}, {49688, 50790}, {49717, 50259}, {49735, 49744}, {49739, 50169}, {49746, 50301}, {49749, 50179}, {50048, 50072}, {50049, 50071}, {50056, 50070}, {50059, 50069}, {50061, 50066}, {50063, 50064}, {50092, 50294}, {50101, 50126}, {50117, 51059}, {50166, 50233}, {50167, 50231}, {50173, 50225}, {50174, 50264}, {50221, 50266}, {50235, 50265}, {50260, 50261}, {50285, 50300}, {50949, 51147}, {50950, 51192}, {51004, 51196}, {52653, 59372}, {58560, 58679}, {58609, 58629}, {59388, 61291}, {60905, 60971}
reflection of X(i) in X(j) for these {i,j}: {1, 51103}, {2, 1125}, {3, 50828}, {4, 50802}, {7, 51098}, {8, 4745}, {10, 2}, {20, 50815}, {69, 50787}, {100, 50844}, {144, 50837}, {145, 51091}, {147, 50882}, {148, 50887}, {149, 50892}, {153, 50909}, {942, 58560}, {946, 51709}, {1125, 51108}, {1386, 51006}, {3146, 50869}, {3244, 51071}, {3448, 50922}, {3579, 12100}, {3625, 4669}, {3626, 51069}, {3635, 51107}, {3636, 41150}, {3647, 15673}, {3654, 6684}, {3656, 13464}, {3679, 3828}, {3743, 58381}, {3753, 3833}, {3817, 5886}, {3830, 18483}, {3845, 9955}, {3892, 5049}, {3898, 10179}, {3919, 5883}, {3993, 50111}, {4134, 10176}, {4297, 51705}, {4301, 3656}, {4525, 5692}, {4669, 10}, {4677, 3626}, {4701, 51070}, {4709, 50096}, {4744, 5902}, {4745, 3634}, {5066, 61272}, {5493, 50808}, {5587, 10171}, {5657, 58441}, {5790, 10172}, {5881, 50801}, {5882, 50824}, {5883, 3742}, {7991, 50814}, {8703, 13624}, {9881, 51578}, {10031, 33812}, {10164, 10165}, {10165, 38028}, {10175, 11230}, {10178, 33574}, {11362, 50821}, {11599, 12258}, {12258, 11725}, {15808, 51110}, {18357, 10109}, {18480, 5066}, {19862, 51109}, {19883, 25055}, {21849, 58469}, {21969, 31757}, {24325, 51061}, {31673, 3845}, {31730, 8703}, {33337, 50843}, {33697, 12101}, {34633, 428}, {34635, 12150}, {34637, 5434}, {34638, 376}, {34639, 4421}, {34641, 3679}, {34642, 9909}, {34644, 7811}, {34646, 11194}, {34648, 381}, {34649, 3058}, {34790, 58629}, {38054, 38053}, {38127, 11231}, {38140, 61269}, {38155, 10175}, {38201, 38204}, {43174, 50829}, {44566, 45318}, {47495, 51725}, {49504, 50075}, {49505, 47358}, {49510, 51034}, {49511, 51003}, {49520, 50777}, {49535, 51055}, {49536, 47359}, {49543, 49477}, {49580, 49591}, {49584, 49590}, {49630, 3821}, {50053, 50059}, {50062, 50063}, {50091, 17382}, {50094, 4755}, {50096, 3739}, {50111, 15569}, {50117, 51060}, {50299, 49738}, {50309, 49731}, {50777, 37}, {50781, 141}, {50796, 5}, {50808, 3}, {50810, 43174}, {50821, 140}, {50824, 15178}, {50834, 9}, {50841, 3035}, {50847, 618}, {50850, 619}, {50862, 4}, {50876, 113}, {50879, 114}, {50884, 115}, {50889, 11}, {50895, 116}, {50902, 118}, {50906, 119}, {50919, 125}, {50949, 3844}, {51004, 49511}, {51005, 1386}, {51034, 3842}, {51059, 3993}, {51060, 24325}, {51066, 31253}, {51067, 1698}, {51069, 19878}, {51071, 1}, {51077, 10222}, {51082, 5882}, {51087, 61286}, {51089, 49465}, {51093, 3635}, {51096, 3244}, {51100, 142}, {51101, 42871}, {51103, 3636}, {51106, 3622}, {51109, 3616}, {51113, 4999}, {51120, 4301}, {51196, 51005}, {51197, 51196}, {51693, 47495}, {51705, 1385}, {51709, 5901}, {59408, 38049}, {59419, 32557}, {60172, 38330}, {60963, 43180}, {61524, 11812}
isogonal conjugate of X(41434)
isotomic conjugate of X(55955)
complement of X(3679)
anticomplement of X(3828)
incircle-inverse of X(53614)
orthoptic-circle-of-the-Steiner-inellipse-inverse of X(50533)
complement of the isogonal conjugate of X(2163)
complement of the isotomic conjugate of X(39704)
isotomic conjugate of the isogonal conjugate of X(21747)
polar conjugate of the isogonal conjugate of X(22357)
Thomson-isogonal conjugate of X(37508)
medial-isogonal conjugate of X(21251)
X(28180)-anticomplementary conjugate of X(513)
X(i)-complementary conjugate of X(j) for these (i,j): {1, 21251}, {31, 16590}, {56, 17057}, {89, 141}, {513, 15614}, {667, 61073}, {692, 52593}, {2163, 10}, {2320, 1329}, {2364, 3452}, {4588, 513}, {4597, 21260}, {4604, 3835}, {5385, 27076}, {5549, 20317}, {9456, 27751}, {20569, 626}, {28607, 2}, {28658, 1211}, {30588, 21245}, {30608, 21244}, {34073, 514}, {34819, 30563}, {39704, 2887}, {52620, 21252}, {53114, 3454}, {55246, 125}, {55979, 18589}
X(i)-Ceva conjugate of X(j) for these (i,j): {2, 16590}, {4597, 514}, {4767, 6006}, {4781, 28209}, {24589, 3707}, {24858, 519}, {26860, 16666}
X(i)-cross conjugate of X(j) for these (i,j): {16666, 4031}, {21806, 16666}, {28209, 4781}
X(i)-isoconjugate of X(j) for these (i,j): {1, 41434}, {6, 40434}, {31, 55955}, {56, 56115}, {58, 56134}, {513, 28210}, {667, 58128}, {1333, 27797}
X(i)-Dao conjugate of X(j) for these (i,j): {1, 56115}, {2, 55955}, {3, 41434}, {9, 40434}, {10, 56134}, {37, 27797}, {551, 3679}, {3707, 59779}, {6631, 58128}, {16590, 2}, {39026, 28210}, {51570, 1}
cevapoint of X(21747) and X(22357)
crosspoint of X(2) and X(39704)
crosssum of X(i) and X(j) for these (i,j): {6, 2177}, {1015, 58175}
trilinear pole of line {14435, 28209}
crossdifference of every pair of points on line {649, 4491}
barycentric product X(i)*X(j) for these {i,j}: {1, 24589}, {7, 3707}, {8, 4031}, {10, 26860}, {57, 3902}, {75, 16666}, {76, 21747}, {81, 4714}, {89, 4793}, {190, 28209}, {264, 22357}, {274, 21806}, {514, 4781}, {519, 42026}, {1978, 58139}, {3616, 58859}, {3676, 30727}, {3699, 30722}, {4407, 14621}, {4555, 14435}, {16590, 39704}, {30608, 39782}
barycentric quotient X(i)/X(j) for these {i,j}: {1, 40434}, {2, 55955}, {6, 41434}, {9, 56115}, {10, 27797}, {37, 56134}, {101, 28210}, {190, 58128}, {3707, 8}, {3902, 312}, {4031, 7}, {4407, 3661}, {4714, 321}, {4781, 190}, {4793, 4671}, {14435, 900}, {16590, 3679}, {16666, 1}, {21747, 6}, {21754, 2177}, {21806, 37}, {22357, 3}, {24589, 75}, {26860, 86}, {28209, 514}, {30722, 3676}, {30727, 3699}, {39782, 5219}, {42026, 903}, {58139, 649}, {58859, 5936}
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 8, 3635}, {1, 10, 3244}, {1, 975, 30145}, {1, 1125, 10}, {1, 1193, 59301}, {1, 1201, 50604}, {1, 1698, 145}, {1, 3086, 6738}, {1, 3616, 1125}, {1, 3622, 3636}, {1, 3624, 8}, {1, 3632, 3623}, {1, 3633, 20057}, {1, 3679, 3241}, {1, 5313, 17018}, {1, 5529, 3979}, {1, 5550, 3626}, and many others


X(552) = POINT MAIA I

Trilinears    1/[a(b + c - a)(b + c)2] : 1/[b(c +a - b)(c + a)2] : 1/[c(a + b - c)(a + b)2]
Barycentrics    1/[(b + c - a)(b + c)2] : 1/[(c + a - b)(c + a)2] : 1/[(a + b - c)(a + b)2]
Barycentrics    (a + b)^2*(a + b - c)*(a + c)^2*(a - b + c) : :

X(552) lies on theselines: {57, 4573}, {86, 40998}, {261, 873}, {553, 1412}, {1088, 4616}, {4633, 30568}, {4637, 33765}, {6628, 7341}

X(552) = isogonal conjugate of X(7064)
X(552) = isotomic conjugate of X(6057)
X(552) = isotomic conjugate of the isogonal conjugate of X(7341)
X(552) = X(i)-cross conjugate of X(j) for these (i,j): {757, 1509}, {1358, 17096}, {7203, 4573}, {16714, 274}
X(552) = X(i)-isoconjugate of X(j) for these (i,j): {1, 7064}, {8, 872}, {9, 1500}, {12, 1253}, {31, 6057}, {33, 3690}, {37, 1334}, {41, 594}, {42, 210}, {55, 756}, {181, 200}, {201, 7071}, {212, 7140}, {213, 2321}, {220, 2171}, {284, 762}, {312, 7109}, {480, 1254}, {512, 4069}, {607, 3949}, {644, 4079}, {657, 21859}, {663, 40521}, {798, 30730}, {1018, 3709}, {1089, 2175}, {1109, 6066}, {1110, 4092}, {1400, 4515}, {1402, 4082}, {1802, 8736}, {1824, 2318}, {1918, 3701}, {2194, 6535}, {2197, 7079}, {2205, 30713}, {2212, 3695}, {2333, 3694}, {2643, 6065}, {3063, 4103}, {3208, 6378}, {3939, 4705}, {4041, 4557}, {4095, 40729}, {4171, 4559}, {4524, 4551}, {6354, 6602}, {6358, 14827}, {7035, 7063}, {9447, 28654}, {20665, 43265}, {21816, 33635}
X(552) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 6057}, {3, 7064}, {12, 17113}, {181, 6609}, {210, 40592}, {223, 756}, {478, 1500}, {514, 4092}, {594, 3160}, {762, 40590}, {1089, 40593}, {1214, 6535}, {1334, 40589}, {2321, 6626}, {3700, 40620}, {3701, 34021}, {4024, 40615}, {4069, 39054}, {4082, 40605}, {4103, 10001}, {4515, 40582}, {4705, 40617}, {4778, 31890}, {7140, 40837}, {21673, 38930}, {30730, 31998}
X(552) = cevapoint of X(i) and X(j) for these (i,j): {6, 16691}, {1014, 1434}, {1358, 17096}
X(552) = trilinear pole of line {17096, 30724}
X(552) = barycentric product X(i)*X(j) for these {i,j}: {7, 1509}, {57, 873}, {76, 7341}, {85, 757}, {86, 1434}, {99, 17096}, {226, 6628}, {261, 279}, {274, 1014}, {310, 1412}, {479, 7058}, {593, 6063}, {763, 1441}, {799, 7203}, {849, 20567}, {1019, 4625}, {1086, 7340}, {1088, 2185}, {1098, 23062}, {1357, 34537}, {1358, 4590}, {1407, 18021}, {1408, 6385}, {1414, 7199}, {1502, 7342}, {3669, 4623}, {3676, 4610}, {3737, 4635}, {4560, 4616}, {4573, 7192}, {4620, 17205}, {4631, 43932}, {4637, 18155}, {7055, 36419}, {7056, 46103}, {7196, 7303}, {7248, 7307}
X(552) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 6057}, {6, 7064}, {7, 594}, {21, 4515}, {56, 1500}, {57, 756}, {58, 1334}, {60, 220}, {65, 762}, {77, 3949}, {81, 210}, {85, 1089}, {86, 2321}, {99, 30730}, {222, 3690}, {226, 6535}, {249, 6065}, {261, 346}, {269, 2171}, {270, 7079}, {274, 3701}, {278, 7140}, {279, 12}, {310, 30713}, {331, 7141}, {333, 4082}, {348, 3695}, {479, 6354}, {553, 8013}, {593, 55}, {604, 872}, {651, 40521}, {662, 4069}, {664, 4103}, {738, 1254}, {757, 9}, {763, 21}, {849, 41}, {873, 312}, {934, 21859}, {1014, 37}, {1019, 4041}, {1086, 4092}, {1088, 6358}, {1098, 728}, {1119, 8736}, {1357, 3124}, {1358, 115}, {1396, 1824}, {1397, 7109}, {1407, 181}, {1408, 213}, {1412, 42}, {1414, 1018}, {1434, 10}, {1443, 4053}, {1444, 3694}, {1447, 4037}, {1509, 8}, {1790, 2318}, {1977, 7063}, {2150, 1253}, {2185, 200}, {2189, 7071}, {3669, 4705}, {3674, 20653}, {3676, 4024}, {3733, 3709}, {3737, 4171}, {3946, 21673}, {4556, 3939}, {4565, 4557}, {4573, 3952}, {4590, 4076}, {4610, 3699}, {4612, 4578}, {4616, 4552}, {4623, 646}, {4625, 4033}, {4626, 4605}, {4637, 4551}, {6063, 28654}, {6354, 6058}, {6628, 333}, {7053, 2197}, {7054, 480}, {7056, 26942}, {7058, 5423}, {7153, 7148}, {7175, 21803}, {7176, 21021}, {7177, 201}, {7185, 16886}, {7192, 3700}, {7199, 4086}, {7203, 661}, {7252, 4524}, {7304, 27538}, {7340, 1016}, {7341, 6}, {7342, 32}, {8025, 4046}, {16705, 3704}, {16726, 4516}, {16947, 1918}, {17074, 14973}, {17095, 7206}, {17096, 523}, {17103, 4095}, {17205, 21044}, {17206, 3710}, {17212, 4140}, {17474, 21704}, {18164, 21039}, {18166, 4111}, {18191, 36197}, {18600, 21031}, {23357, 6066}, {24002, 4036}, {24471, 21810}, {26856, 4081}, {30097, 21713}, {30576, 3689}, {30581, 3683}, {30593, 3686}, {30682, 6356}, {30724, 6367}, {32636, 21816}, {33295, 3985}, {33947, 4136}, {36419, 1857}, {36420, 6059}, {38859, 20616}, {40153, 40966}, {41777, 7237}, {42028, 4061}, {43924, 4079}, {46103, 7046}


X(553) = POINT MAIA II

Trilinears    bc(2a + b + c)/(b + c - a) : ca(2b + c + a)/(c + a - b) : ab(2c + a + b)/(a + b - c)
Barycentrics   (2a + b + c)/(b + c - a) : (2b + c + a)/(c + a - b) : (2c + a + b)/(a + b - c)
X(553) = 5 X[7] + X[41572], 4 X[942] - X[950], 2 X[942] + X[4292], 5 X[942] - 2 X[12433], X[942] + 2 X[24470], X[950] + 2 X[4292], 5 X[950] - 8 X[12433], X[950] + 8 X[24470], 5 X[4292] + 4 X[12433], X[4292] - 4 X[24470], 2 X[10122] + X[10123], X[12433] + 5 X[24470], X[65] + 2 X[4298], 2 X[65] + X[10106], 5 X[65] + X[10944], 4 X[4298] - X[10106], 10 X[4298] - X[10944], 5 X[5434] - X[10944], 5 X[10106] - 2 X[10944], X[72] - 4 X[12436], 2 X[210] - 3 X[46916], 3 X[354] - X[3058], X[3058] + 3 X[11246], X[5083] + 2 X[24465], 2 X[3649] + X[41551], X[1770] + 5 X[18398], X[3057] - 4 X[12577], 4 X[3812] - X[12527], X[3962] - 4 X[12447], 4 X[5045] - X[10624], 5 X[5439] - 2 X[12572], X[6284] - 4 X[6744], 2 X[6738] + X[7354], 3 X[10202] - X[28459], X[10572] - 4 X[17706], 5 X[11025] - 2 X[15006], 2 X[12575] - 5 X[17609], X[15556] - 4 X[37544], X[18990] + 2 X[31794], 2 X[31776] + X[37730]

X(553) lies on the cubic K637 and these lines: {1, 376}, {2, 7}, {4, 30304}, {6, 24177}, {10, 5221}, {11, 13159}, {12, 3828}, {20, 11518}, {30, 942}, {36, 11551}, {46, 10056}, {55, 5542}, {56, 551}, {65, 519}, {72, 12436}, {79, 10308}, {81, 3946}, {85, 42030}, {109, 9108}, {165, 3475}, {171, 24231}, {181, 1463}, {210, 5850}, {222, 5228}, {223, 7271}, {241, 4955}, {244, 41011}, {269, 45126}, {278, 39980}, {279, 39948}, {320, 3687}, {333, 24199}, {345, 17298}, {354, 516}, {377, 24391}, {381, 1210}, {388, 3339}, {390, 44841}, {428, 1876}, {481, 13389}, {482, 13388}, {497, 4312}, {498, 38068}, {515, 5902}, {524, 24471}, {528, 5083}, {534, 1407}, {535, 18838}, {536, 4032}, {540, 35650}, {542, 24235}, {545, 35652}, {547, 34753}, {549, 6147}, {552, 1412}, {554, 37772}, {559, 3639}, {664, 41823}, {752, 42053}, {903, 6648}, {938, 3543}, {940, 3663}, {946, 1709}, {962, 9841}, {999, 3656}, {1014, 40592}, {1046, 24178}, {1056, 2093}, {1081, 37773}, {1082, 3638}, {1086, 3163}, {1125, 3647}, {1155, 4995}, {1214, 1418}, {1266, 1999}, {1357, 2796}, {1358, 1366}, {1373, 1659}, {1374, 13390}, {1420, 38314}, {1427, 39974}, {1432, 43263}, {1443, 17011}, {1454, 10197}, {1465, 26740}, {1466, 11517}, {1467, 11111}, {1469, 4685}, {1617, 4428}, {1697, 11037}, {1738, 32913}, {1750, 36996}, {1770, 18398}, {1776, 3337}, {1788, 5290}, {1836, 4860}, {1837, 34648}, {1892, 5064}, {2099, 4315}, {2308, 4989}, {2325, 32933}, {2646, 12563}, {2982, 34578}, {2999, 4644}, {3008, 4641}, {3017, 23537}, {3057, 12577}, {3086, 38021}, {3146, 37723}, {3175, 17132}, {3210, 3879}, {3212, 29617}, {3241, 3340}, {3243, 17784}, {3303, 5493}, {3304, 4301}, {3333, 4295}, {3336, 3584}, {3361, 3485}, {3476, 18421}, {3486, 34628}, {3487, 3524}, {3488, 11001}, {3534, 4304}, {3545, 9612}, {3578, 3686}, {3586, 15682}, {3601, 10304}, {3654, 31397}, {3655, 4311}, {3660, 28534}, {3664, 3666}, {3665, 43054}, {3670, 5717}, {3676, 4785}, {3677, 4307}, {3729, 18141}, {3740, 5852}, {3742, 17768}, {3745, 4353}, {3746, 5557}, {3752, 17365}, {3782, 4887}, {3812, 12527}, {3817, 17728}, {3830, 5722}, {3834, 44416}, {3839, 9581}, {3873, 5853}, {3912, 32007}, {3947, 24914}, {3951, 37462}, {3962, 12447}, {3977, 18139}, {4021, 37595}, {4034, 41915}, {4035, 17740}, {4052, 40151}, {4059, 10521}, {4077, 31148}, {4102, 4431}, {4190, 11520}, {4257, 26728}, {4293, 11529}, {4310, 5269}, {4314, 34638}, {4317, 5882}, {4321, 34607}, {4327, 8270}, {4334, 42042}, {4349, 17599}, {4383, 24175}, {4413, 21060}, {4416, 19804}, {4419, 17022}, {4421, 37541}, {4512, 38053}, {4640, 25557}, {4656, 17276}, {4659, 34255}, {4664, 7201}, {4666, 44447}, {4667, 5256}, {4669, 5252}, {4684, 32932}, {4745, 40663}, {4847, 5880}, {4888, 5712}, {4891, 28530}, {4896, 17595}, {4967, 37653}, {4980, 6358}, {4982, 45222}, {5045, 10624}, {5049, 28174}, {5054, 11374}, {5071, 5714}, {5121, 33096}, {5122, 5719}, {5223, 26040}, {5270, 6901}, {5272, 24695}, {5284, 10032}, {5393, 21171}, {5425, 21578}, {5439, 12572}, {5563, 6906}, {5603, 14646}, {5698, 10582}, {5703, 15692}, {5743, 17345}, {5758, 37526}, {5759, 10857}, {5762, 11227}, {5843, 10157}, {5847, 24165}, {5919, 28228}, {6175, 6734}, {6284, 6744}, {6604, 17294}, {6703, 17235}, {6705, 26877}, {6738, 7354}, {6897, 11362}, {6904, 11523}, {7004, 40960}, {7056, 42309}, {7146, 29574}, {7175, 46922}, {7176, 29584}, {7195, 16833}, {7228, 44417}, {7247, 19797}, {7248, 10473}, {7321, 14829}, {7580, 43177}, {7702, 41565}, {7964, 43151}, {8025, 17190}, {8581, 41539}, {8703, 24929}, {8808, 10400}, {9026, 22278}, {9580, 10580}, {9613, 34627}, {9778, 10389}, {9955, 11544}, {10164, 17718}, {10165, 21165}, {10178, 38454}, {10202, 28459}, {10396, 11023}, {10569, 17642}, {10572, 17706}, {10578, 30340}, {10588, 19876}, {10592, 38083}, {10895, 38076}, {11025, 15006}, {11113, 37566}, {11375, 19883}, {11552, 30384}, {11679, 42697}, {11684, 24564}, {12351, 21636}, {12512, 37080}, {12560, 47357}, {12575, 17609}, {12625, 37435}, {12701, 21625}, {12709, 31165}, {13151, 44255}, {13624, 16137}, {14450, 41012}, {15171, 28202}, {15326, 44840}, {15556, 37544}, {15888, 43174}, {15935, 19710}, {15936, 18650}, {15956, 18655}, {16009, 16116}, {16466, 24171}, {16579, 18607}, {16834, 17079}, {16888, 17382}, {17050, 18206}, {17067, 26723}, {17074, 22464}, {17092, 28606}, {17094, 45669}, {17205, 40153}, {17364, 17490}, {17378, 41777}, {17549, 37583}, {17561, 31424}, {17757, 44848}, {18193, 26098}, {18201, 24239}, {18389, 28452}, {18625, 33150}, {18990, 28204}, {19541, 41561}, {19708, 30282}, {19723, 24796}, {19819, 36595}, {20070, 37556}, {20292, 26015}, {20367, 22097}, {21255, 32777}, {23681, 37642}, {24208, 43682}, {24210, 32857}, {24216, 33106}, {24328, 37269}, {24474, 28458}, {24588, 41140}, {24692, 29655}, {27790, 43283}, {28208, 31776}, {28301, 42044}, {28557, 32915}, {29594, 30617}, {31145, 37709}, {31995, 37655}, {34641, 41687}, {34647, 40726}, {34937, 37522}, {35242, 41870}, {36913, 40617}, {37584, 44284}, {37756, 41629}, {38055, 41166}, {39126, 42029}, {39542, 44675}, {39704, 44733}, {39782, 41150}, {41003, 41311}, {41801, 42045}, {43573, 43855}, {45638, 45639}

X(553) = midpoint of X(i) and X(j) for these {i,j}: {65, 5434}, {354, 11246}, {11112, 24473}, {24474, 28458}
X(553) = reflection of X(i) in X(j) for these {i,j}: {5434, 4298}, {10106, 5434}, {10624, 15170}, {15170, 5045}, {40998, 3742}
X(553) = isogonal conjugate of X(33635)
X(553) = isotomic conjugate of X(4102)
X(553) = complement of X(17781)
X(553) = X(10308)-complementary conjugate of X(141)
X(553) = X(i)-Ceva conjugate of X(j) for these (i,j): {7, 3649}, {4552, 3669}, {4573, 3676}, {38340, 514}
X(553) = X(i)-cross conjugate of X(j) for these (i,j): {1100, 1125}, {34502, 7}
X(553) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 4102}, {3, 33635}, {8, 1213}, {9, 3647}, {9, 32635}, {223, 1255}, {478, 1126}, {522, 35076}, {553, 17781}, {1100, 4420}, {1125, 2321}, {1214, 6539}, {1268, 3160}, {3120, 3700}, {3634, 4060}, {3649, 24048}, {4560, 16726}, {4608, 40615}, {4886, 41809}, {6540, 10001}, {31010, 40622}, {32018, 40593}
X(553) = cevapoint of X(i) and X(j) for these (i,j): {1100, 32636}, {1125, 4856}
X(553) = crosspoint of X(i) and X(j) for these (i,j): {7, 1434}, {554, 1081}
X(553) = crosssum of X(i) and X(j) for these (i,j): {55, 1334}, {1250, 10638}
X(553) = trilinear pole of line {4977, 5298}
X(553) = X(51)-of-intouch-triangle
X(553) = excentral-to-intouch similarity image of X(2)
X(553) = X(i)-isoconjugate of X(j) for these (i,j): {1, 33635}, {6, 32635}, {8, 28615}, {9, 1126}, {31, 4102}, {33, 1796}, {41, 1268}, {55, 1255}, {210, 1171}, {650, 8701}, {663, 37212}, {1334, 40438}, {2150, 6538}, {2175, 32018}, {2194, 6539}, {3063, 6540}, {3709, 4596}, {4041, 4629}
X(553) = barycentric product X(i)*X(j) for these {i,j}: {7, 1125}, {12, 30593}, {56, 1269}, {57, 4359}, {65, 16709}, {75, 32636}, {85, 1100}, {86, 3649}, {190, 30724}, {226, 8025}, {269, 3702}, {273, 3916}, {278, 4001}, {279, 3686}, {307, 31900}, {331, 22054}, {348, 1839}, {552, 8013}, {651, 4978}, {658, 4976}, {664, 4977}, {903, 5298}, {934, 4985}, {1014, 4647}, {1088, 3683}, {1213, 1434}, {1230, 1412}, {1414, 30591}, {2308, 6063}, {2355, 7182}, {3261, 36075}, {3676, 4427}, {4115, 17096}, {4554, 4979}, {4573, 4988}, {4625, 4983}, {4626, 4990}, {4697, 7249}, {4856, 27818}, {4870, 39704}, {4973, 18815}, {4974, 7233}, {6358, 30581}, {17190, 43682}, {24002, 35342}
X(553) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 32635}, {2, 4102}, {6, 33635}, {7, 1268}, {12, 6538}, {56, 1126}, {57, 1255}, {85, 32018}, {109, 8701}, {222, 1796}, {226, 6539}, {604, 28615}, {651, 37212}, {664, 6540}, {1014, 40438}, {1100, 9}, {1125, 8}, {1213, 2321}, {1230, 30713}, {1269, 3596}, {1412, 1171}, {1414, 4596}, {1434, 32014}, {1839, 281}, {1962, 210}, {2308, 55}, {2355, 33}, {3578, 42033}, {3647, 4420}, {3649, 10}, {3676, 4608}, {3683, 200}, {3686, 346}, {3702, 341}, {3775, 3790}, {3911, 31011}, {3916, 78}, {3958, 3694}, {4001, 345}, {4046, 4082}, {4115, 30730}, {4359, 312}, {4410, 4494}, {4427, 3699}, {4565, 4629}, {4573, 4632}, {4647, 3701}, {4654, 43260}, {4697, 7081}, {4856, 3161}, {4870, 3679}, {4966, 3717}, {4969, 2325}, {4970, 27538}, {4973, 4511}, {4974, 3685}, {4975, 4723}, {4976, 3239}, {4977, 522}, {4978, 4391}, {4979, 650}, {4983, 4041}, {4984, 1639}, {4985, 4397}, {4988, 3700}, {4989, 390}, {4990, 4163}, {4992, 4147}, {5298, 519}, {6533, 3702}, {7178, 31010}, {7181, 31013}, {8013, 6057}, {8025, 333}, {8040, 4046}, {16709, 314}, {20970, 1334}, {22054, 219}, {22080, 2318}, {23201, 212}, {30581, 2185}, {30591, 4086}, {30592, 14430}, {30593, 261}, {30724, 514}, {30729, 6558}, {31900, 29}, {32636, 1}, {34502, 3634}, {35327, 3939}, {35342, 644}, {36075, 101}, {41014, 3710}, {41547, 34772}, {41820, 4886}
X(553) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 7, 4654}, {2, 63, 5325}, {2, 2094, 3928}, {2, 4654, 226}, {2, 28610, 3929}, {7, 57, 226}, {7, 226, 3982}, {7, 4031, 3911}, {7, 21454, 57}, {57, 226, 3911}, {57, 4114, 3982}, {57, 4654, 2}, {57, 5219, 5435}, {57, 21454, 4031}, {65, 4298, 10106}, {142, 5325, 2}, {226, 4031, 57}, {226, 4114, 7}, {329, 5437, 5316}, {388, 3339, 4848}, {908, 27003, 6692}, {942, 4292, 950}, {942, 13369, 10122}, {942, 24470, 4292}, {1443, 17011, 47057}, {1836, 4860, 11019}, {3218, 5249, 5745}, {3218, 26842, 5249}, {3296, 6361, 1}, {3306, 5905, 3452}, {3333, 4295, 12053}, {3336, 13407, 6684}, {3339, 4355, 388}, {3649, 5298, 4870}, {3649, 32636, 1125}, {3649, 41549, 41546}, {3782, 37520, 39595}, {3911, 3982, 226}, {3916, 32636, 41547}, {3928, 6173, 2}, {4001, 4359, 3686}, {4031, 4114, 226}, {4114, 21454, 3911}, {4190, 11520, 12437}, {4312, 10980, 497}, {4641, 40688, 3008}, {4870, 5298, 1125}, {4870, 32636, 5298}, {4887, 39595, 3782}, {5221, 10404, 10}, {5228, 10481, 43035}, {6147, 37582, 13411}, {9776, 9965, 9}, {9778, 11038, 10389}, {10136, 10481, 30623}, {11019, 30424, 1836}, {17276, 37674, 4656}, {17483, 27003, 908}, {18201, 33097, 24239}


X(554) = (X(1),X(13))-ANSWER TO QUESTION A

Trilinears        sec(A/2) csc(A/2 + π/3) : sec(B/2) csc(B/2 + π/3) : sec(C/2) csc(C/2 + π/3)
Barycentrics   sin A sec(A/2) csc(A/2 + π/3) : sin B sec(B/2) csc(B/2 + π/3) : sin C sec(C/2) csc(C/2 + π/3)
Barycentrics   (a + b - c)*(a - b + c)*(a + 2*b + 2*c) + 2*Sqrt[3]*a*S : :

Suppose that X and Y are triangle centers. Let

YA = (Y of the triangle XBC),
YB = (Y of the triangle XCA),
YC = (Y of the triangle XAB).

Let A' = (XYA intersect BC), and define B' and C' cyclically. In

Clark Kimberling, "Major Centers of Triangles," Amer. Math. Monthly 104 (1997) 431-438,

Question A is this: for what choices of X and Y do the lines AA', BB', CC' concur? A solution (X,Y) will here be called the (X,Y)-answer to Question A. X(554) is the (X(1),X(13))-answer to Question A. (In the reference, see (9) on page 435, with Y = X(13).)

X(554) lies on the circumconic {A,B,C,X(2),X(7)}}, the cubics K134 and K419b, and these lines: {1, 30}, {2, 33654}, {7, 1082}, {13, 43682}, {14, 226}, {55, 10652}, {57, 41225}, {75, 299}, {396, 1652}, {497, 30344}, {553, 37772}, {675, 36073}, {1086, 11073}, {1365, 18975}, {3475, 37830}, {3639, 3982}, {5239, 5249}, {5240, 5905}, {6186, 10648}, {7043, 11092}, {11705, 26700}, {30327, 37641}

X(554) = isogonal conjugate of X(10638)
X(554) = isotomic conjugate of X(40714)
X(554) = X(i)-cross conjugate of X(j) for these (i,j): {553, 1081}, {30382, 85}
X(554) = cevapoint of X(i) and X(j) for these (i,j): {1, 1652}, {33653, 33654}
X(554) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 40714}, {3, 10638}, {223, 559}, {7043, 40579}
X(554) = X(i)-isoconjugate of X(j) for these (i,j): {1, 10638}, {16, 7126}, {31, 40714}, {35, 1251}, {55, 559}, {1250, 42677}, {2152, 7043}, {5239, 42624}, {5357, 19551}, {7005, 33653}, {7127, 7150}, {35057, 36072}
X(554) = barycentric product X(i)*X(j) for these {i,j}: {75, 33654}, {85, 33653}, {301, 7051}, {1082, 30690}, {2307, 20565}, {3261, 36073}
X(554) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 40714}, {6, 10638}, {14, 7043}, {57, 559}, {1082, 3219}, {2154, 7126}, {2160, 1251}, {2306, 42677}, {2307, 35}, {5239, 44689}, {7051, 16}, {7052, 7150}, {11073, 19551}, {19373, 5357}, {33653, 9}, {33654, 1}, {33655, 46073}, {36073, 101}, {39152, 5240}, {40713, 42033}, {41225, 5239}
X(554) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 4654, 1081}, {226, 3638, 559}, {3649, 37631, 1081}, {3782, 5434, 1081}


X(555) = (X(1),X(158))-ANSWER TO QUESTION A

Trilinears    sec3(A/2) : sec3(B/2) : sec3(C/2)
Trilinears    |AJa| : :, where Ja = A-mixtilinear excenter
Barycentrics    sin A sec3(A/2) : sin B sec3(B/2) : sin C sec3(C/2)
Barycentrics    (-a + b - c)*(a + b - c)*Sqrt[b*(a + b - c)*c*(a - b + c)] : :

X(555) lies on these lines: {7, 177}, {85, 2090}, {188, 4146}, {234, 1088}, {279, 16015}, {10481, 10489}

X(555) = isotomic conjugate of X(6731)
X(555) = isotomic conjugate of the isogonal conjugate of X(7370)
X(555) = X(i)-cross conjugate of X(j) for these (i,j): {174, 4146}, {234, 7}
X(555) = cevapoint of X(i) and X(j) for these (i,j): {7, 18886}, {174, 7371}, {514, 10491}
X(555) = barycentric product X(i)*X(j) for these {i,j}: {7, 4146}, {75, 7371}, {76, 7370}, {85, 174}, {188, 1088}, {266, 6063}, {279, 556}, {479, 7027}, {4569, 6728}, {6729, 46406}, {6730, 36838}, {6731, 23062}
X(555) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 6731}, {9, 6726}, {9, 15495}, {174, 17113}, {188, 3160}, {200, 236}, {223, 259}, {556, 40593}, {845, 7371}, {1086, 6730}, {1214, 6725}, {3082, 5431}, {6376, 7027}, {6728, 40615}, {6729, 40617}
X(555) = X(i)-isoconjugate of X(j) for these (i,j): {6, 6726}, {31, 6731}, {32, 7027}, {41, 188}, {55, 259}, {174, 1253}, {220, 266}, {480, 7370}, {556, 2175}, {657, 6733}, {692, 6730}, {1334, 6727}, {2194, 6725}, {2332, 7591}, {3939, 6729}, {4146, 14827}, {6602, 7371}
X(555) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 6726}, {2, 6731}, {7, 188}, {57, 259}, {75, 7027}, {85, 556}, {174, 9}, {188, 200}, {226, 6725}, {234, 16016}, {259, 220}, {266, 55}, {269, 266}, {279, 174}, {479, 7371}, {508, 4182}, {509, 4166}, {514, 6730}, {556, 346}, {738, 7370}, {934, 6733}, {1014, 6727}, {1088, 4146}, {1439, 7591}, {2091, 15997}, {3668, 6724}, {3669, 6729}, {3676, 6728}, {4146, 8}, {5451, 34912}, {6724, 210}, {6725, 4515}, {6726, 480}, {6727, 2328}, {6728, 3900}, {6729, 657}, {6730, 4130}, {6731, 728}, {6733, 3939}, {7027, 5423}, {7370, 6}, {7371, 1}, {7591, 2318}, {10490, 16012}, {14596, 7707}, {18886, 236}, {21456, 7028}
X(555) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {10489, 10491, 10481}, {18886, 21456, 279}


X(556) = (X(1),X(75))-ANSWER TO QUESTION A

Trilinears       csc A csc A/2 : csc B csc B/2 : csc C csc C/2
Barycentrics  csc A/2 : csc B/2 : csc C/2

X(556) lies on these lines: 8,177   75,234   312,2090

X(556) = isotomic conjugate of X(174)


X(557) = (X(1),X(9))-ANSWER TO QUESTION A

Trilinears    sec A/2 cot A/4 : sec B/2 cot B/4 : sec C/2 cot C/4
Trilinears    csc A (1 + cos A/2) : :
Barycentrics    cos2A/4 : cos2B/4 : cos2C/4

X(557) lies on these lines: 2,178   1274,1488


X(558) = (X(1),X(57))-ANSWER TO QUESTION A

Trilinears    sec A/2 tan A/4 : sec B/2 tan B/4 : sec C/2 tan C/4
Trilinears    csc A (1 - cos A/2) : :
Barycentrics  sin2(A/4) : sin2(B/4) : sin2(C/4)

X(558) lies on this line: 2,178

X(558) = X(2)-Ceva conjugate of X(39122)


X(559) = (X(1),X(15))-ANSWER TO QUESTION A

Trilinears       (sec A/2) sin(A/2 + π/3) : (sec B/2) sin(B/2 + π/3) : (sec C/2) sin(C/2 + π/3)
Barycentrics   (sin A/2) sin(A/2 + π/3) : (sin B/2) sin(B/2 + π/3) : (sin C/2) sin(C/2 + π/3)
Barycentrics   a*((a + b - c)*(a - b + c) + 2*Sqrt[3]*S) : :

X(559) lies on the cubics K134 and K341b and these lines: {1, 3}, {2, 5239}, {6, 7088}, {7, 1081}, {14, 226}, {16, 16577}, {63, 5240}, {81, 7052}, {222, 7059}, {299, 319}, {465, 17043}, {497, 37830}, {553, 3639}, {651, 7126}, {1100, 1652}, {1255, 33654}, {1277, 21475}, {1442, 10638}, {1653, 16777}, {1836, 10652}, {1962, 10647}, {2003, 5357}, {2306, 25417}, {3219, 11126}, {3474, 37833}, {3475, 30344}, {4336, 30301}, {7005, 42677}, {7051, 17011}, {7127, 37787}, {7202, 42624}, {9778, 30338}, {10391, 10650}, {10580, 30339}, {10648, 17017}, {10651, 11246}, {14100, 30357}, {16038, 17718}, {17778, 37794}, {19373, 28606}

X(559) = isogonal conjugate of X(33653)
X(559) = X(7344)-complementary conjugate of X(141)
X(559) = X(1255)-Ceva conjugate of X(1082)
X(559) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 33653}, {223, 554}, {478, 33654}, {5239, 40581}
X(559) = X(i)-isoconjugate of X(j) for these (i,j): {1, 33653}, {9, 33654}, {14, 7127}, {55, 554}, {79, 1250}, {522, 36073}, {1082, 7073}, {1251, 42680}, {2154, 5239}, {2307, 7110}, {5240, 11073}, {6186, 40713}, {7126, 41225}, {11086, 36933}, {19551, 39152}
X(559) = barycentric product X(i)*X(j) for these {i,j}: {57, 40714}, {85, 10638}, {299, 7052}, {319, 2306}, {1081, 3219}, {1251, 17095}, {18160, 36072}, {33654, 46176}
X(559) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 33653}, {16, 5239}, {56, 33654}, {57, 554}, {1081, 30690}, {1251, 7110}, {1399, 2307}, {1415, 36073}, {2003, 1082}, {2152, 7127}, {2174, 1250}, {2306, 79}, {2307, 42680}, {3219, 40713}, {5357, 5240}, {7043, 44691}, {7051, 41225}, {7052, 14}, {7150, 7043}, {10638, 9}, {19373, 39152}, {39150, 36932}, {40714, 312}, {42624, 7126}, {46073, 7026}
X(559) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 57, 1082}, {1, 37773, 37772}, {56, 20182, 1082}, {57, 1082, 37772}, {65, 37595, 1082}, {226, 3638, 554}, {241, 3748, 1082}, {940, 2099, 1082}, {1082, 37773, 57}, {1214, 24929, 1082}, {1319, 3666, 1082}, {1429, 17598, 1082}, {7146, 17716, 1082}, {13388, 13389, 37773}, {15934, 37543, 1082}


X(560) = 4th POWER POINT

Trilinears    a4 : b4 : c4
Trilinears    a2(SA - Sω) : :
Trilinears    (cot B + cot C)^2 : :
Trilinears    (SB + SC)^2 : :
Trilinears    d(a,b,c) : : , where d(a,b,c) = (distance from A to de Longchamps line)2
Barycentrics  a5 : b5 : c5

X(560) is the vertex conjugate of the foci of the inellipse that is the trilinear square of the Lemoine axis. The center of the inellipse is X(16584) and its Brianchon point (perspector) is X(31). (Randy Hutson, October 15, 2018)

X(560) lies on these lines: 1,82   31,48   41,872   42,584   100,697   101,713   110,715   717,825   719,827

X(560) = isogonal conjugate of X(561)
X(560) = isotomic conjugate of X(1928)
X(560) = complement of X(21275)
X(560) = anticomplement of X(21235)
X(560) = anticomplementary conjugate of anticomplement of X(38827)
X(560) = crosssum of X(75) and X(304)
X(560) = antigonal conjugate of X(37843)
X(560) = barycentric product of PU(9)
X(560) = X(92)-isoconjugate of X(304)


X(561) = ISOGONAL CONJUGATE OF 4th POWER POINT

Trilinears    a - 4 : b - 4 : c - 4
Trilinears    [sec^2(A/2) + csc^2(A/2)]^2 : :
Trilinears    d(a,b,c) : : , where d(a,b,c) = (distance from A to Lemoine axis)2 : :
Barycentrics   a -3 : b -3 : c-3

X(561) is the vertex conjugate of the foci of the inellipse that is the trilinear square of the de Longchamps line (with center X(21235) and perspector X(1928)). (Randy Hutson, October 15, 2018)

Let A'B'C' be the obverse triangle of X(1). Let A″B″C″ be the N-obverse triangle of X(1). Let A* be the trilinear product A'*A″, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(561). (Randy Hutson, October 15, 2018)

X(561) lies on these lines: 1,718   2,716   6,720   31,722   32,724   38,75   63,799   76,321   92,304   313,696   305,1441

X(561) = isogonal conjugate of X(560)
X(561) = isotomic conjugate of X(31)
X(561) = complement of X(17486)
X(561) = anticomplement of X(16584)
X(561) = anticomplementary conjugate of anticomplement of X(38810)
X(561) = antigonal conjugate of X(37844)
X(561) = cevapoint of X(75) and X(304)
X(561) = X(313)-cross conjugate of X(76)

X(561) = trilinear square of X(75)
X(561) = trilinear product of PU(11)
X(561) = pole wrt polar circle of trilinear polar of X(1973)
X(561) = X(48)-isoconjugate (polar conjugate) of X(1973)
X(561) = trilinear product of vertices of Gemini triangle 21
X(561) = trilinear product of vertices of Gemini triangle 22
X(561) = perspector of ABC and cross-triangle of Gemini triangles 31 and 32
X(561) = perspector of ABC and cross-triangle of ABC and Gemini triangle 31
X(561) = perspector of ABC and cross-triangle of ABC and Gemini triangle 32
X(561) = barycentric product of vertices of Gemini triangle 31
X(561) = barycentric product of vertices of Gemini triangle 32

X(562) = TRILINEAR QUOTIENT X(2)*X(50)/X(49)

Trilinears       csc A tan 3A : csc B tan 3B : csc C tan 3C
Barycentrics  tan 3A : tan 3B : tan 3C
= (3S2 - S2A)(3S2 - S2B)(3S2 - S2C)*X(3) - 16SASBSC(5SASBSC + 3S2SωS)*X(3519) (Peter Moses, January 29, 2015)

X(562) lies on these lines: {4,93}, {252,6143}

Clark Kimberling and Cyril Parry, "Products, Square Roots, and Layers in Triangle Geometry," Mathematics and Informatics Quarterly 10 (2000) 9-22; on page 21, section 8.

X(562) = polar conjugate of X(30529)


X(563) = TRILINEAR PRODUCT X(47)*X(48)

Trilinears    sin 4A : sin 4B : sin 4C
Trilinears    tan 2B + tan 2C : :
Barycentrics    sin A sin 4A : :

X(563) lies on these lines: 1,1820   19,163   48,255

Clark Kimberling and Cyril Parry, "Products, Square Roots, and Layers in Triangle Geometry," Mathematics and Informatics Quarterly 10 (2000) 9-22; on page 21, section 8.

X(563) = X(91)-isoconjugate of X(92)

X(564) = INTERSECTION OF LINES X(1)X(1048) AND X(47,91)

Trilinears    cos(2B - 2C) : cos(2C - 2A) : cos(2A - 2B)
Trilinears    b^2 c^2 (a^4 (b^4 + c^4) - 2 a^2 (b^2 - c^2)^2 (b^2 + c^2) + (b^2 - c^2)^4) : :
Barycentrics    sin A cos(2B - 2C) : sin B cos(2C - 2A) : sin C cos(2A - 2B)

X(564) lies on these lines: 1,1048   47,91

Clark Kimberling and Cyril Parry, "Products, Square Roots, and Layers in Triangle Geometry," Mathematics and Informatics Quarterly 10 (2000) 9-22; on page 21, section 8.

X(564) = {X(91),X(92)}-harmonic conjugate of X(47)


X(565) = INTERSECTION OF LINES X(49)X(93) AND X(143,324)

Trilinears    cos(3B - 3C) : cos(3C - 3A) : cos(3A - 3B)
Barycentrics   sin A cos(3B - 3C) : sin B cos(3C - 3A) : sin C cos(3A - 3B)

X(565) lies on these lines: 49,93   143,324

Clark Kimberling and Cyril Parry, "Products, Square Roots, and Layers in Triangle Geometry," Mathematics and Informatics Quarterly 10 (2000) 9-22; on page 21, section 8.

leftri

Centers X(566)-X(584)

rightri

Centers X(566)-X(584) lie on the Brocard axis, L(3,6). Each is the center X of a circle that meets the sides of triangle ABC with three equal angles at X.

Let AB, AC, BC, BA, CA, CB denote the meeting-points; e.g., AB and CB are on side CA. The equal angles are given by

D = angle(ABXAC) = angle(BCXBA) = angle(CAXCB)

Then trilinears for X are given by

X = sin A + cot D/2 cos A : sin B + cot D/2 cos B : sin C + cot D/2 cos C.

Definitions: Y is the orthogonal of X if D(X) + D(Y) = π/2;

Y is the harmonic of X if X and Y are harmonic conjugates with respect to X(3) and X(6);

Y is the orthoharmonic if Y is the harmonic of the orthogonal of X. The centers in this section were contributed by Edward Brisse, December, 2000.


X(566) = HARMONIC OF X(50)

Trilinears    sin A + cos A cot D/2 : : , where cot D/2 = 4*area/(a2 + b2 + c2 - 6R2), where R = abc/(4*area)
Trilinears    a[16(b2 + c22 - 3a2b2c2]      (M.Iliev, 5/13/07)
Barycentrics    a^2 (a^4 (b^2 + c^2) - a^2 (2 b^4 + b^2 c^2 + 2 c^4) + (b^2 - c^2)^2 (b^2 + c^2)) : :
Barycentrics    (SB + SC) (3 R^2 - SA - SW) : :

X(566) lies on these lines: {2, 94}, {3, 6}, {4, 9221}, {5, 9220}, {45, 13006}, {53, 1594}, {67, 43718}, {140, 16310}, {141, 18375}, {160, 9973}, {184, 7669}, {230, 7495}, {231, 7749}, {232, 5094}, {233, 1879}, {237, 9971}, {248, 19151}, {393, 37119}, {549, 14836}, {599, 36212}, {631, 46262}, {858, 3815}, {940, 21478}, {1180, 5306}, {1576, 37457}, {1656, 9222}, {1990, 37118}, {2165, 2963}, {2453, 47213}, {2548, 14791}, {2549, 44468}, {3054, 39576}, {3055, 15355}, {3087, 35471}, {3148, 18374}, {3291, 37637}, {5012, 14060}, {5475, 7574}, {6240, 6748}, {6749, 10295}, {7507, 14576}, {7579, 7603}, {7668, 39906}, {7736, 16063}, {7763, 35549}, {8253, 8963}, {8962, 13846}, {9019, 37184}, {10311, 21284}, {11079, 14385}, {11672, 17416}, {13160, 18353}, {13371, 42459}, {14096, 18371}, {15122, 16303}, {16776, 37465}, {17277, 22377}, {17392, 26636}, {18281, 19656}, {18570, 38872}, {19127, 37183}, {23195, 34751}, {26216, 38323}, {30537, 34288}, {34828, 40681}, {35473, 39176}, {35503, 40065}, {36990, 44437}, {37196, 44519}, {40138, 47228}, {41237, 44388}

X(566) = isogonal conjugate of X(7578)
X(566) = complement of X(44135)
X(566) = Brocard-circle-inverse of X(50)
X(566) = Schoutte-circle-inverse of X(14805)
X(566) = complement of the isotomic conjugate of X(3431)
X(566) = isogonal conjugate of the polar conjugate of X(7577)
X(566) = polar conjugate of the isotomic conjugate of X(23039)
X(566) = X(3431)-complementary conjugate of X(2887)
X(566) = X(36829)-Ceva conjugate of X(18117)
X(566) = X(18117)-cross conjugate of X(36829)
X(566) = crosspoint of X(2) and X(3431)
X(566) = crosssum of X(i) and X(j) for these (i,j): {2, 11004}, {6, 381}
X(566) = crossdifference of every pair of points on line {523, 5926}
X(566) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 7578}, {566, 44135}
X(566) = X(1)-isoconjugate of X(7578)
X(566) = barycentric product X(i)*X(j) for these {i,j}: {3, 7577}, {4, 23039}, {99, 18117}, {523, 36829}, {5562, 19177}
X(566) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 7578}, {7577, 264}, {18117, 523}, {19177, 8795}, {23039, 69}, {36829, 99}
X(566) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 50}, {6, 39, 13337}, {6, 570, 13351}, {6, 8553, 2965}, {6, 11063, 32}, {6, 15109, 577}, {6, 18573, 3003}, {6, 18578, 568}, {6, 33886, 5007}, {6, 36751, 8553}, {15, 16, 14805}, {39, 216, 3003}, {39, 3003, 6}, {50, 41335, 6}, {160, 23635, 9973}, {216, 570, 6}, {216, 3003, 18573}, {371, 372, 567}, {570, 3003, 39}, {570, 18573, 13337}, {574, 5158, 5063}, {577, 14806, 15109}, {800, 5421, 6}, {1340, 1341, 2088}, {5063, 5158, 6}, {7772, 33872, 6}, {13337, 13351, 39}, {30260, 30261, 40280}, {40695, 40696, 2}


X(567) = ORTHOGONAL OF X(50)

Trilinears    cos(B - C) + 2 cos(C - A)cos(A - B) : :
Trilinears    sin A + cos A cot D/2 : : , where cot D/2 = (a2 + b2 + c2 - 6R2)/(4*area), where R = abc/(4*area)

X(567) lies on the curve Q166 and these lines: {2, 22115}, {3, 6}, {4, 7578}, {5, 49}, {13, 11134}, {14, 11137}, {22, 39522}, {30, 5012}, {51, 2070}, {74, 45956}, {115, 9604}, {125, 43573}, {140, 3580}, {143, 7488}, {156, 3091}, {184, 381}, {185, 10620}, {186, 5946}, {195, 5562}, {206, 23049}, {215, 7951}, {323, 7550}, {373, 32609}, {382, 11424}, {399, 15030}, {498, 9666}, {499, 9653}, {546, 1614}, {547, 40111}, {549, 43574}, {550, 43576}, {597, 15462}, {631, 37644}, {1092, 3526}, {1147, 1656}, {1154, 1994}, {1173, 12107}, {1176, 21850}, {1180, 39524}, {1181, 10938}, {1199, 6102}, {1209, 10112}, {1352, 14787}, {1437, 37251}, {1493, 11591}, {1495, 7545}, {1506, 9603}, {1511, 13363}, {1657, 8717}, {1658, 3567}, {1976, 37345}, {1986, 3520}, {1993, 7514}, {2072, 23292}, {2477, 7741}, {2931, 12038}, {2937, 5446}, {3043, 20304}, {3060, 7502}, {3066, 7506}, {3090, 9545}, {3200, 37835}, {3201, 37832}, {3292, 10170}, {3518, 5944}, {3519, 12899}, {3521, 18560}, {3527, 9714}, {3545, 9544}, {3574, 31724}, {3796, 12083}, {3814, 9702}, {3843, 6759}, {3845, 14157}, {3851, 9704}, {5054, 43650}, {5055, 9306}, {5070, 16187}, {5169, 34514}, {5422, 6644}, {5462, 13367}, {5476, 18374}, {5480, 7540}, {5504, 37648}, {5544, 12309}, {5576, 6146}, {5609, 15052}, {5622, 20126}, {5640, 11464}, {5643, 15034}, {5654, 41615}, {5663, 7527}, {5876, 32136}, {5885, 43610}, {5890, 11454}, {5891, 34986}, {5921, 7404}, {6033, 39834}, {6101, 37126}, {6143, 43816}, {6193, 14786}, {6321, 39805}, {6636, 13391}, {6639, 39571}, {6723, 43817}, {6800, 7530}, {7399, 43595}, {7403, 31804}, {7503, 12161}, {7509, 16266}, {7512, 10263}, {7516, 21766}, {7517, 10982}, {7526, 7592}, {7528, 18925}, {7555, 15107}, {7556, 11002}, {7575, 15019}, {7579, 15139}, {7603, 9696}, {7728, 13198}, {7769, 10411}, {7827, 14355}, {7988, 9621}, {7989, 9622}, {8253, 9676}, {8550, 11579}, {8703, 13482}, {9677, 23261}, {9697, 39565}, {9701, 25639}, {9707, 13861}, {9777, 14070}, {9781, 37440}, {9818, 11402}, {9833, 18382}, {9920, 10282}, {10024, 12241}, {10110, 18378}, {10226, 43600}, {10254, 18390}, {10257, 45298}, {10264, 43578}, {10274, 15432}, {10574, 11250}, {10601, 47391}, {10619, 45286}, {11004, 33533}, {11422, 11459}, {11423, 12111}, {11427, 18531}, {11428, 18453}, {11429, 18455}, {11449, 15024}, {11456, 31861}, {11799, 16657}, {11808, 44515}, {11820, 47527}, {12006, 12236}, {12121, 38323}, {12162, 12308}, {12164, 34801}, {12225, 15800}, {12227, 22584}, {12229, 22809}, {12230, 22810}, {12233, 18563}, {12234, 22815}, {12295, 13403}, {12307, 14531}, {12370, 13160}, {13011, 22813}, {13012, 22814}, {13142, 34002}, {13321, 15004}, {13364, 13595}, {13366, 13754}, {13451, 37936}, {13491, 14865}, {13561, 43808}, {13564, 45186}, {13567, 45967}, {14560, 18121}, {14831, 32608}, {14845, 21308}, {14848, 19136}, {14855, 35452}, {14869, 46865}, {14912, 18917}, {14926, 41597}, {15026, 32171}, {15027, 15132}, {15035, 43584}, {15036, 43597}, {15043, 37814}, {15051, 43804}, {15053, 15646}, {15054, 43596}, {15059, 34331}, {15061, 15463}, {15062, 43602}, {15472, 20127}, {15620, 30536}, {15699, 43572}, {15720, 43652}, {15805, 35602}, {16003, 33749}, {16226, 37955}, {17809, 18451}, {17835, 43807}, {18281, 18911}, {18388, 18403}, {18434, 40276}, {18447, 19365}, {18462, 19408}, {18463, 19409}, {18494, 44077}, {18502, 40643}, {18524, 20986}, {18952, 37119}, {19127, 20423}, {19128, 37458}, {19138, 43815}, {19361, 32322}, {22529, 22808}, {23328, 43810}, {25555, 30714}, {25739, 39504}, {27866, 34153}, {32068, 44673}, {32127, 32284}, {32235, 36253}, {32245, 32272}, {32345, 41725}, {32379, 41362}, {34007, 43818}, {34396, 35934}, {35498, 43604}, {37347, 37649}, {39562, 40673}, {40114, 44275}, {40441, 45089}, {43605, 45959}, {43612, 43806}, {43831, 46686}, {43839, 46085}, {44085, 45923}

X(567) = midpoint of X(i) and X(j) for these {i,j}: {1994, 35921}, {5012, 15033}, {7527, 15032}
X(567) = reflection of X(i) in X(j) for these {i,j}: {3, 37513}, {15087, 13366}, {37347, 37649}, {41171, 5}
X(567) = isogonal conjugate of X(9221)
X(567) = Brocard-circle-inverse of X(568)
X(567) = crosssum of X(11) and X(18116)
X(567) = crossdifference of every pair of points on line {523, 2081}
X(567) = crosspoint of isogonal conjugates of PU(5)
X(567) = homothetic center of Ehrmann side-triangle and 1st anti-Conway triangle
X(567) = X(3)-Dao conjugate of X(9221)
X(567) = X(1)-isoconjugate of X(9221)
X(567) = barycentric quotient X(6)/X(9221)
X(567) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 568}, {3, 182, 13339}, {3, 568, 3581}, {3, 569, 13353}, {3, 578, 37472}, {3, 1351, 37494}, {3, 11426, 36749}, {3, 11432, 37490}, {3, 13352, 37477}, {3, 13353, 37471}, {3, 14627, 52}, {3, 15037, 9730}, {3, 36747, 37484}, {3, 36749, 6243}, {3, 36753, 37481}, {3, 37472, 37495}, {3, 37506, 14805}, {5, 49, 18350}, {5, 54, 49}, {5, 12022, 265}, {5, 44076, 6288}, {6, 14805, 3581}, {6, 37506, 3}, {51, 18475, 2070}, {52, 37505, 14627}, {54, 13434, 5}, {54, 43598, 9706}, {61, 62, 41335}, {143, 10610, 7488}, {182, 578, 13352}, {182, 13339, 37471}, {182, 13352, 3}, {184, 381, 10540}, {186, 34545, 5946}, {195, 34864, 5562}, {265, 12228, 11597}, {323, 7550, 15067}, {371, 372, 566}, {381, 18396, 18430}, {568, 13353, 19129}, {568, 14805, 3}, {568, 18438, 6243}, {569, 578, 3}, {569, 13352, 182}, {569, 37472, 37471}, {575, 9730, 15037}, {575, 11430, 9730}, {578, 13353, 37495}, {1199, 14118, 6102}, {1351, 37494, 6243}, {1993, 7514, 23039}, {2070, 15038, 51}, {3796, 44413, 12083}, {3851, 9704, 10539}, {5055, 9703, 9306}, {5085, 37483, 3}, {5462, 13367, 45735}, {5611, 5615, 41169}, {5640, 11464, 12106}, {5944, 10095, 3518}, {7503, 12161, 18436}, {7526, 7592, 34783}, {8254, 43575, 5}, {9730, 11430, 3}, {9818, 11402, 18445}, {9818, 18445, 18435}, {10564, 16836, 3}, {11425, 36752, 3}, {11426, 37506, 18438}, {11438, 39242, 3}, {12006, 43394, 22467}, {12022, 12228, 49}, {12022, 14389, 5}, {13336, 13346, 3}, {13339, 13353, 182}, {13339, 37472, 37477}, {13339, 37477, 3}, {13352, 37477, 37495}, {13353, 37472, 3}, {13353, 37477, 13339}, {14130, 43845, 185}, {15026, 32171, 44802}, {15062, 43602, 45957}, {34148, 43651, 140}, {36747, 37476, 3}, {36749, 37494, 1351}, {37471, 37495, 3}, {37472, 37477, 13352}, {37480, 44490, 37489}


X(568) = ORTHOHARMONIC OF X(50)

Trilinears    sin A + cos A cot D/2, where where cot D/2 = (6R2 - a2 - b2 - c2)/(4*area) and R = abc/(4*area)
Trilinears    a[3*R2(b2 + c2 - a2) + a2b2 + a2c2 - b4 - c4], where R = abc/(4*area)     (Wimalasiri Perera, August 29, 2011)
Trilinears    cos A - 2 cos 2A cos(B - C) : :
Barycentrics    a^2 (a^6 (b^2 + c^2) - a^4 (3 b^4 + b^2 c^2 + 3 c^4) + a^2 (3 b^6 - 2 b^4 c^2 - 2 b^2 c^4 + 3 c^6) - (b^2 - c^2)^2 (b^4 + c^4)) : :
3 X[2] - 4 X[13363], 3 X[5946] - 2 X[13363], 3 X[5946] - X[15067], 4 X[5946] - X[23039], 8 X[13363] - 3 X[23039], 4 X[15067] - 3 X[23039], X[3] + 2 X[52], X[3] - 4 X[389], 2 X[3] + X[6243], 5 X[3] - 8 X[9729], 5 X[3] - 2 X[10625], 11 X[3] - 8 X[13348], 7 X[3] - 16 X[15012], 7 X[3] - 4 X[15644], X[3] + 8 X[16625], 3 X[3] - 4 X[16836], 13 X[3] - 16 X[17704], 2 X[3] - 5 X[37481], 4 X[3] - X[37484], 2 X[3] - 3 X[40280], 4 X[6] - X[18438], X[52] + 2 X[389], 4 X[52] - X[6243], 5 X[52] + 4 X[9729], 5 X[52] + X[10625], 4 X[52] + X[13340], 11 X[52] + 4 X[13348], 7 X[52] + 8 X[15012], 7 X[52] + 2 X[15644], X[52] - 4 X[16625], 3 X[52] + 2 X[16836], 13 X[52] + 8 X[17704], 4 X[52] + 5 X[37481], 8 X[52] + X[37484], 4 X[52] + 3 X[40280], 8 X[389] + X[6243], 5 X[389] - 2 X[9729], 10 X[389] - X[10625], 8 X[389] - X[13340], 11 X[389] - 2 X[13348], 7 X[389] - 4 X[15012], 7 X[389] - X[15644], X[389] + 2 X[16625], 3 X[389] - X[16836], 13 X[389] - 4 X[17704], 8 X[389] - 5 X[37481], 16 X[389] - X[37484], 8 X[389] - 3 X[40280], 2 X[576] + X[37473], X[1351] + 2 X[19161], 2 X[3313] - 5 X[12017], 4 X[5097] - X[44439], 5 X[6243] + 16 X[9729], X[6243] + 4 X[9730], 5 X[6243] + 4 X[10625], 11 X[6243] + 16 X[13348], 7 X[6243] + 32 X[15012], 7 X[6243] + 8 X[15644], X[6243] - 16 X[16625], 3 X[6243] + 8 X[16836], 13 X[6243] + 32 X[17704], and many more

Let (A) be the circle centered at A that cuts off a segment of line BC equal to the radius of (A). Define (B) and (C) cyclically. Then X(568) is the radical center of circles (A), (B), (C). (Randy Hutson, December 2, 2017)

X(568) lies on the cubics K885 and K1139 and these lines: {2, 1154}, {3, 6}, {4, 94}, {5, 3567}, {20, 10263}, {23, 15032}, {24, 49}, {25, 10540}, {26, 6800}, {30, 3060}, {51, 381}, {54, 1658}, {68, 973}, {110, 12106}, {140, 7998}, {155, 7506}, {156, 3518}, {184, 2070}, {185, 382}, {186, 1994}, {195, 1147}, {343, 37347}, {355, 31732}, {373, 1656}, {376, 13391}, {378, 39522}, {399, 7545}, {542, 9971}, {546, 9781}, {547, 11451}, {549, 2979}, {550, 10574}, {631, 6101}, {632, 7999}, {974, 20127}, {1092, 43809}, {1181, 7517}, {1199, 7488}, {1204, 15041}, {1216, 3526}, {1263, 13505}, {1352, 16776}, {1353, 6403}, {1493, 6242}, {1503, 7540}, {1511, 11004}, {1614, 37440}, {1657, 40647}, {1843, 21852}, {1899, 31723}, {1992, 14984}, {1993, 6644}, {1995, 15068}, {2072, 13567}, {2262, 40263}, {2777, 7729}, {2781, 20126}, {2842, 31825}, {2854, 23236}, {2937, 35268}, {3090, 11591}, {3091, 5876}, {3146, 13491}, {3270, 9642}, {3292, 43586}, {3357, 17823}, {3522, 13421}, {3523, 10627}, {3525, 32142}, {3527, 34801}, {3529, 16982}, {3534, 21969}, {3545, 13364}, {3564, 11188}, {3574, 5449}, {3575, 13292}, {3627, 6241}, {3628, 11444}, {3819, 15694}, {3830, 6000}, {3832, 45959}, {3843, 10110}, {3845, 13451}, {3850, 15058}, {3851, 5907}, {3853, 12290}, {3855, 45958}, {3861, 11439}, {3917, 5054}, {5012, 7502}, {5055, 5891}, {5056, 14128}, {5067, 32205}, {5068, 18874}, {5070, 11793}, {5072, 45187}, {5073, 10575}, {5076, 11381}, {5422, 7514}, {5447, 15720}, {5448, 36518}, {5576, 12359}, {5609, 14002}, {5654, 14643}, {5655, 12824}, {5722, 18330}, {5777, 31819}, {5944, 32136}, {6033, 39835}, {6090, 6642}, {6240, 12370}, {6293, 18381}, {6321, 39806}, {6515, 18420}, {6688, 15703}, {6756, 39871}, {6759, 18378}, {6971, 39271}, {7401, 31810}, {7403, 38136}, {7528, 11411}, {7529, 12164}, {7530, 11456}, {7550, 15018}, {7553, 18914}, {7555, 15080}, {7556, 11003}, {7575, 11422}, {7576, 45968}, {7687, 22584}, {7689, 11424}, {7691, 43651}, {7720, 12803}, {7721, 12804}, {7723, 11746}, {7730, 32423}, {7731, 10264}, {7785, 15536}, {8254, 32338}, {8541, 39562}, {8550, 8705}, {8584, 44265}, {8703, 20791}, {9019, 11179}, {9544, 47485}, {9703, 34986}, {9704, 10282}, {9714, 19347}, {9777, 9818}, {9792, 19176}, {9826, 37645}, {9833, 41589}, {9969, 18440}, {9970, 37827}, {10024, 12233}, {10111, 15473}, {10124, 44299}, {10125, 22051}, {10224, 20424}, {10254, 18388}, {10255, 23515}, {10272, 12273}, {10311, 22146}, {10539, 13621}, {10594, 32139}, {10605, 44413}, {10620, 11806}, {10628, 23325}, {10653, 36978}, {10654, 36980}, {10937, 12897}, {10982, 12163}, {10984, 13564}, {11202, 37922}, {11225, 18400}, {11250, 15055}, {11264, 34799}, {11402, 14070}, {11423, 12107}, {11433, 18531}, {11435, 18453}, {11436, 18455}, {11441, 13861}, {11442, 11818}, {11455, 15687}, {11539, 44324}, {11548, 33523}, {11555, 16461}, {11556, 16462}, {11561, 12383}, {11562, 11800}, {11585, 31802}, {11624, 40693}, {11626, 40694}, {11675, 43461}, {11695, 15082}, {11745, 12134}, {11801, 12281}, {11802, 12307}, {11807, 38790}, {11819, 34224}, {12026, 13504}, {12083, 33586}, {12121, 14708}, {12188, 39846}, {12219, 20304}, {12226, 13472}, {12228, 15035}, {12235, 12429}, {12237, 22809}, {12238, 22810}, {12241, 18563}, {12242, 22815}, {12254, 32196}, {12278, 45971}, {12289, 45970}, {12324, 44544}, {12358, 37643}, {12699, 31728}, {13013, 22813}, {13014, 22814}, {13188, 39817}, {13366, 18475}, {13371, 26879}, {13403, 18562}, {13423, 36966}, {13561, 16880}, {14216, 18382}, {14269, 16194}, {14448, 36253}, {14516, 31830}, {14561, 14787}, {14790, 18916}, {14791, 18911}, {14845, 19709}, {14855, 15681}, {14865, 32138}, {15004, 15038}, {15033, 18570}, {15053, 43574}, {15061, 18281}, {15073, 47277}, {15083, 18369}, {15091, 15801}, {15317, 18532}, {15360, 44262}, {15688, 36987}, {15760, 41588}, {15800, 18569}, {16227, 44214}, {16266, 17928}, {16270, 18931}, {16618, 47582}, {16867, 38534}, {16980, 18526}, {17505, 43949}, {17800, 46850}, {17810, 18451}, {18121, 39235}, {18390, 18403}, {18404, 39571}, {18434, 21400}, {18447, 19366}, {18462, 19410}, {18463, 19411}, {18494, 47328}, {18559, 30522}, {18583, 41716}, {18952, 31815}, {19043, 19051}, {19044, 19052}, {19136, 45016}, {19362, 32322}, {19456, 44080}, {19552, 32409}, {20379, 31857}, {21661, 38594}, {21844, 43394}, {22530, 22808}, {22971, 38789}, {23049, 34146}, {23061, 43584}, {23293, 39504}, {25739, 44288}, {26876, 46025}, {26881, 37936}, {26913, 37938}, {31745, 31763}, {32062, 38335}, {32063, 41580}, {32165, 45731}, {32210, 35475}, {32246, 32272}, {32326, 34779}, {34117, 41613}, {34484, 43605}, {34545, 35921}, {34775, 34780}, {37649, 44201}, {38321, 44665}, {40266, 42450}, {41171, 41724}, {43816, 44056}, {45780, 47391}

X(568) = midpoint of X(i) and X(j) for these {i,j}: {51, 14831}, {52, 9730}, {1986, 45237}, {3060, 5890}, {5889, 11459}, {6243, 13340}, {6403, 15531}, {7576, 45968}
X(568) = reflection of X(i) in X(j) for these {i,j}: {2, 5946}, {3, 9730}, {265, 45237}, {381, 51}, {1352, 16776}, {2979, 549}, {3845, 13451}, {3917, 5892}, {5054, 16226}, {5562, 10170}, {5655, 12824}, {5891, 5943}, {9730, 389}, {10170, 5462}, {11455, 15687}, {11459, 5}, {12162, 46847}, {13340, 3}, {14643, 16222}, {15060, 13364}, {15061, 46430}, {15067, 13363}, {15072, 45956}, {15305, 3845}, {15531, 1353}, {15681, 14855}, {16776, 32191}, {18435, 381}, {18436, 11459}, {23039, 2}, {32063, 41580}, {32609, 16223}, {37484, 13340}, {41330, 15544}, {44214, 16227}, {45237, 12236}, {46847, 10110}
X(568) = anticomplement of X(15067)
X(568) = Brocard-circle-inverse of X(567)
X(568) = crosssum of X(3) and X(14852)
X(568) = Cundy-Parry Phi transform of X(50)
X(568) = Cundy-Parry Psi transform of X(94)
X(568) = homothetic center of Ehrmann side-triangle and 2nd anti-Conway triangle X(568) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 567}, {3, 52, 6243}, {3, 389, 37481}, {3, 567, 14805}, {3, 6243, 37484}, {3, 9730, 40280}, {3, 11432, 36753}, {3, 14627, 578}, {3, 15037, 182}, {3, 36747, 37495}, {3, 36749, 37472}, {3, 36752, 37471}, {3, 36753, 13353}, {3, 37489, 3581}, {3, 37493, 36749}, {3, 37496, 37480}, {4, 265, 18430}, {4, 3448, 34514}, {4, 6102, 34783}, {4, 18951, 25738}, {4, 34783, 18439}, {5, 5889, 18436}, {5, 16881, 3567}, {5, 31834, 15056}, {6, 1351, 18449}, {6, 3581, 14805}, {6, 18578, 566}, {6, 37489, 3}, {24, 12161, 49}, {25, 18445, 10540}, {52, 389, 3}, {52, 37481, 37484}, {143, 6102, 4}, {155, 7506, 18350}, {182, 37478, 3}, {185, 5446, 382}, {185, 18396, 10938}, {195, 45735, 1147}, {371, 372, 50}, {373, 5562, 10170}, {373, 10170, 1656}, {381, 13321, 51}, {389, 15644, 15012}, {389, 16625, 52}, {399, 7545, 46261}, {567, 3581, 3}, {567, 6243, 18438}, {567, 19129, 13353}, {569, 46730, 3}, {576, 11438, 13352}, {576, 44490, 6}, {1199, 7488, 32046}, {1493, 32171, 9545}, {1986, 12236, 265}, {1993, 6644, 22115}, {2070, 15087, 184}, {2979, 15045, 549}, {3557, 3558, 18114}, {3567, 5889, 5}, {3567, 11459, 5640}, {3575, 13292, 44076}, {3581, 18449, 37477}, {3853, 45957, 12290}, {3917, 5892, 5054}, {3917, 16226, 5892}, {5102, 9786, 37497}, {5102, 37497, 36747}, {5462, 5562, 1656}, {5462, 10170, 373}, {5640, 5889, 11459}, {5640, 11459, 5}, {5876, 10095, 3091}, {5890, 15072, 45956}, {5891, 5943, 5055}, {5946, 15067, 13363}, {6101, 12006, 631}, {6102, 6746, 25738}, {6243, 37481, 3}, {6243, 40280, 13340}, {7689, 11424, 14130}, {7999, 15028, 632}, {9545, 44879, 32171}, {9729, 10625, 3}, {9781, 12111, 546}, {9786, 36747, 3}, {10110, 12162, 3843}, {10115, 32352, 195}, {10224, 26917, 45622}, {10263, 13630, 20}, {10575, 13598, 5073}, {11002, 37644, 45237}, {11412, 15043, 140}, {11430, 32110, 3}, {11432, 37489, 19129}, {11438, 13352, 3}, {11444, 15024, 3628}, {11477, 37475, 37483}, {11557, 21649, 399}, {11562, 11800, 12902}, {11591, 15026, 3090}, {11806, 13417, 10620}, {11819, 43588, 34224}, {12233, 41587, 10024}, {12359, 45089, 5576}, {13321, 14831, 18435}, {13336, 46728, 3}, {13340, 37481, 40280}, {13340, 40280, 3}, {13358, 38898, 3448}, {13363, 15067, 2}, {13364, 15060, 3545}, {13382, 13598, 10575}, {17834, 36752, 3}, {18396, 40909, 382}, {18404, 39571, 43821}, {18952, 31815, 37444}, {31728, 31757, 12699}, {31732, 31760, 355}, {31830, 32358, 14516}, {34417, 46261, 7545}, {36749, 37490, 3}, {37470, 37480, 3}, {37470, 37517, 37496}, {37475, 37483, 3}, {37481, 40280, 9730}, {37486, 37514, 3}, {37490, 37493, 37472}, {40647, 45186, 1657}


X(569) = HARMONIC OF X(52)

Trilinears    sin A + cot D/2 cos A : :,
Trilinears    cot D/2 = (2e3 + e2 - e1)/[64*(area)3], where
                         e1 = a6 + b6 + c6
                         e2 = a4(b2 + c2) + b4(c2 + a2) + c4(a2 + b2)
                        e3 = a2b2c2

X(569) lies on these lines: 2,54   3,6   5,156   26,51   140,343

X(569) = inverse-in-Brocard-circle of X(52)


X(570) = ORTHOGONAL OF X(52)

Trilinears    sin A + sin 2A cos(B - C) : :     (Joe Goggins, 11/26/08)
Trilinears     (sin A)(1 - 2 sin^2 B - 2 sin^2 C) : :

X(570) lies on these lines: 2,311   3,6   53,232   115,128   140,231   157,184

X(570) = inverse-in-Brocard-circle of X(571)
X(570) = complement of X(311)
X(570) = crosspoint of X(2) and X(54)
X(570) = crosssum of X(5) and X(6)
X(570) = X(71)-of-orthic-triangle if ABC is acute
X(570) = {X(6),X(39)}-harmonic conjugate of X(5421)
X(570) = perspector of circumconic centered at X(1209)
X(570) = center of circumconic that is locus of trilinear poles of lines passing through X(1209)
X(570) = X(2)-Ceva conjugate of X(1209)
X(570) = polar conjugate of isogonal conjugate of X(23195)
X(570) = polar conjugate of isotomic conjugate of X(1216)
X(570) = X(63)-isoconjugate of X(1179)
X(570) = crossdifference of every pair of points on line X(523)X(2070) (the polar of X(5) wrt the circumcircle)


X(571) = ORTHOHARMONIC OF X(52)

Trilinears    sin A cos 2A     (M. Iliev, 4/12/07)

Let A'B'C' be the Kosnita triangle. Let A″ be the barycentric product B'*C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(571). (Randy Hutson, December 2, 2017)

X(571) lies on these lines: 3,6   4,96   66,248   112,393   160,184   206,237   230,427   608,913

X(571) = isogonal conjugate of X(5392)
X(571) = inverse-in-Brocard-circle of X(570)
X(571) = X(4)-Ceva conjugate of X(184)
X(571) = crosspoint of X(2) and X(70)
X(571) = crosssum of X(i) and X(j) for these (i,j): (6,26), (338,525)
X(571) = barycentric product of X(371) and X(372)
X(571) = X(19)-of-orthic-triangle if ABC is acute
X(571) = X(2)-isoconjugate of X(91)
X(571) = X(68)-isoconjugate of X(92)
X(571) = {X(6),X(50)}-harmonic conjugate of X(577)
X(571) = crosspoint of X(5412) and X(5413)
X(571) = Cundy-Parry Phi transform of X(52)
X(571) = Cundy-Parry Psi transform of X(96)


X(572) = ORTHOGONAL OF X(58)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos(A - U), where cot U = cot(A/2) cot(B/2) cot(C/2)     (Joe Goggins, 11/26/08)
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sin A + cot D/2 cos A,
                        cot D/2 = (a + b + c)2/(4*area)
Trilinears    s cos A + r sin A : :
Trilinears    a (a^3 - a (b^2 - b c + c^2) - b c (b + c)) : :

X(572) = s*X(3) + r*cot(ω)*X(6)

X(572) is the intersection of the Brocard axes of ABC, the Feuerbach triangle, and the Apollonius triangle. (Randy Hutson, March 21, 2019)

X(572) lies on these lines: 1,604   3,6   9,48   51,199   54,71   103,825   165,1051   169,610   184,1011   219,947   261,662   517,1100   594,952   631,966

X(572) = isogonal conjugate of X(2051)
X(572) = {X(1687),X(1688)}-harmonic conjugate of X(4279)
X(572) = inverse-in-Brocard-circle of X(573)
X(572) = crosssum of X(11) and X(661)


X(573) = ORTHOHARMONIC OF X(58)

Trilinears    cos(A + U) : : , where cot U = cot(A/2) cot(B/2) cot(C/2)      (Joe Goggins, 11/26/08)
Trilinears    (c - a)cos B sin(A - B) - (a - b)cos C sin(C - A) : :
Trilinears    a[a^2(b + c) - abc - b^3 - c^3] : :
Trilinears    s cos A - r sin A, s = semiperimeter, r = inradius
X(573) = s*X(3) - r*cot(ω)*X(6)

X(573) is the internal center of similitude of the circumcircle and Apollonius circle. The external center is X(386). (Peter J. C. Moses, 8/22/03)

Let A' be the inverse-in-excircles-radical-circle of the midpoint of BC, and define B' and C' cyclically. Triangle A'B'C' is inscribed in the Apollonius circle and homothetic to the 1st circumperp triangle at X(573). (Randy Hutson, December 2, 2017)

X(573) lies on these lines: 1,941   3,6   4,9   10,5179   20,391   36,604   37,517   43,165   51,1011   55,181   101,102   109,478   184,199   256,981   346,1018   347,1020

X(573) = reflection of X(991) in X(3)
X(573) = isogonal conjugate of X(13478)
X(573) = inverse-in-Brocard-circle of X(572)
X(573) = X(333)-Ceva conjugate of X(1)
X(573) = crosspoint of X(59) and X(190)
X(573) = crosssum of X(11) and X(649)
X(573) = crossdifference of every pair of points on line X(523)X(1459)
X(573) = X(216)-of-excentral-triangle
X(573) = perspector, wrt excentral triangle, of the excentral-hexyl ellipse
X(573) = inverse-in-excircles-radical-circle of X(5179)
X(573) = Cundy-Parry Phi transform of X(58)
X(573) = Cundy-Parry Psi transform of X(10)


X(574) = HARMONIC OF X(187)

Trilinears    sin A + cot D/2 cos A, where cot(D/2) = 12*area/(a2 + b2 + c2)
Trilinears    a(a2 - 2b2 - 2c2) : :
Trilinears    sin A + 3 cos A tan ω : :    (Peter J. C. Moses, 8/22/03)
Trilinears    2 sin(A + 2ω) - sin(A - 2ω) + sin A : :
Trilinears    3 cos A + sin A cot ω : :
Trilinears    cos(A - ω) sin 2ω - e^2 sin(A + ω) : :
Barycentrics    a^2*(a^2 - 2*b^2 - 2*c^2) : :
X(574) = 2 X[6] - 3 X[5034], X[7774] + 3 X[33008], 5 X[12017] - 3 X[35429], X[14907] - 3 X[33008], X[17131] + 2 X[31859]

Let A', B', C' be the inverse-in-Brocard-circle of A, B, C. Let A″, B″, C″ be the inverse-in-circumcircle of the vertices of 1st Brocard triangle. The lines A'A″, B'B″, C'C″ concur in X(574). (Randy Hutson, November 18, 2015)

Let X'Y'Z' be the circumcevian-inversion triangle of X(6), as defined in the preamble just before X(34864). The A-vertex of this triangle is given by

X' = (2 a^2 - b^2 - c^2) (4 a^2 + b^2 + c^2) : b^2 (-2 a^2 + b^2 - 2 c^2) (5 a^2 + 2 b^2 + 2 c^2) : c^2 (-2 a^2 - 2 b^2 + c^2) (5 a^2 + 2 b^2 + 2 c^2).

The triangles ABC and X'Y'Z' are perspective, and their perspector is X(574). (Peter Moses, December 9, 2019)

Let P1' be the tangent, other than U(1), to the Brocard circle from P(2). Let U1' be the tangent, other than P(1), to the Brocard circle from U(2). P1' and U1' are also the Schoute-circle-inverses of P(1) and U(1), resp. Then X(574) = P(1)U1'∩U(1)P1'. (Randy Hutson, January 17, 2020)

"Each one of the three circles of Apollonius of the reference triangle cuts the circumcircle in a second point. These three points form a triangle which is perspective with the base triangle through the symmedian point, the co-symmedian triangle. This triangle also possesses a Neuberg cubic, which has three points in common with the cubic of the first triangle in the circumcenter and the two isodynamic centers. Hence the remaining points of intersection must lie on a conic, which is necessarily a circle, since two of these six points are the circular points I and J." Quoted from (T. W. Moore and J. H. Neelley). The center of the circle is X(574). (Bernard Gibert, euclid 4332)

If you have GeoGebra, you can view X(574) in the context of the preceding paragraph: X(574)

X(574) lies on the cubics K284, K353, K751, K1125, and these lines: {1, 1571}, {2, 99}, {3, 6}, {4, 1506}, {5, 3055}, {10, 31456}, {11, 9664}, {12, 9651}, {20, 2548}, {22, 9699}, {23, 15302}, {24, 9700}, {25, 15433}, {30, 3815}, {35, 2241}, {36, 2242}, {37, 37599}, {40, 9619}, {42, 9346}, {51, 41275}, {55, 1015}, {56, 1500}, {69, 7810}, {74, 15920}, {76, 7781}, {83, 3552}, {98, 1569}, {100, 16975}, {101, 28563}, {106, 43077}, {109, 41160}, {110, 353}, {112, 14388}, {114, 37242}, {140, 3054}, {141, 6390}, {165, 1572}, {172, 7280}, {183, 538}, {184, 3269}, {186, 10311}, {194, 1078}, {217, 1204}, {230, 549}, {232, 378}, {237, 34417}, {248, 3431}, {262, 11170}, {274, 17684}, {315, 7764}, {316, 7775}, {325, 7761}, {352, 7998}, {376, 6781}, {381, 7603}, {382, 31467}, {384, 7782}, {385, 7757}, {395, 13084}, {396, 13083}, {404, 5283}, {474, 16589}, {493, 34994}, {494, 34995}, {498, 9597}, {499, 9598}, {515, 31398}, {517, 31443}, {548, 9606}, {550, 7745}, {597, 27088}, {598, 9855}, {599, 8575}, {625, 7841}, {626, 7763}, {631, 3767}, {647, 44814}, {691, 9831}, {729, 25424}, {754, 7774}, {755, 43357}, {805, 843}, {842, 34235}, {902, 37586}, {936, 31442}, {940, 16431}, {958, 1574}, {980, 21495}, {988, 3247}, {993, 1575}, {995, 17735}, {999, 31477}, {1003, 7804}, {1007, 32986}, {1083, 11650}, {1092, 22416}, {1100, 37589}, {1107, 25440}, {1147, 9697}, {1153, 8860}, {1180, 5354}, {1194, 7485}, {1196, 7484}, {1285, 19708}, {1296, 15921}, {1334, 9351}, {1348, 19660}, {1349, 19659}, {1352, 14981}, {1376, 1573}, {1383, 7492}, {1385, 31430}, {1420, 31426}, {1478, 31476}, {1495, 3148}, {1511, 14901}, {1593, 3199}, {1597, 33842}, {1600, 8962}, {1656, 39565}, {1657, 11742}, {1724, 37023}, {1879, 14787}, {1914, 5010}, {1968, 3520}, {1971, 11202}, {1975, 3934}, {1992, 8182}, {2023, 8179}, {2070, 44521}, {2071, 22240}, {2176, 24047}, {2177, 2223}, {2207, 3516}, {2269, 34543}, {2273, 22054}, {2291, 41157}, {2491, 44826}, {2502, 5651}, {2794, 9744}, {2896, 7796}, {2936, 9306}, {2996, 32838}, 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X(574) = midpoint of X(i) and X(j) for these {i,j}: {183, 31859}, {1296, 34241}, {7774, 14907}, {11163, 35955}, {45564, 45565}
X(574) = reflection of X(i) in X(j) for these {i,j}: {182, 39498}, {5475, 3815}, {8722, 3}, {17131, 183}, {35440, 13357}, {44773, 15810}
X(574) = isogonal conjugate of X(598)
X(574) = isotomic conjugate of X(40826)
X(574) = complement of X(11185)
X(574) = circumcircle-inverse of X(5104)
X(574) = Brocard-circle-inverse of X(187)
X(574) = 3rd-Lemoine-circle-(Ehrmann)-inverse of X(5107)
X(574) = Moses-circle-inverse of X(5107)
X(574) = Schoutte-circle-inverse of X(182)
X(574) = 2nd-Brocard-circle-inverse of X(13330)
X(574) = Ehrmann-circle-inverse of X(5107)
X(574) = complement of the isotomic conjugate of X(5486)
X(574) = isogonal conjugate of the anticomplement of X(15810)
X(574) = isogonal conjugate of the isotomic conjugate of X(599)
X(574) = isotomic conjugate of the polar conjugate of X(8541)
X(574) = isogonal conjugate of the polar conjugate of X(5094)
X(574) = Thomson-isogonal conjugate of X(22712)
X(574) = psi-transform of X(352)
X(574) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 8542}, {810, 14672}, {5486, 2887}, {30247, 21259}
X(574) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 8542}, {111, 9872}, {1296, 512}, {5094, 8541}, {9145, 17414}, {9515, 32}, {10130, 599}, {10484, 576}, {15810, 8561}, {19151, 184}, {39389, 6}, {42008, 10510}
X(574) = X(17414)-cross conjugate of X(9145)
X(574) = X(i)-isoconjugate of X(j) for these (i,j): {1, 598}, {31, 40826}, {75, 1383}, {82, 23297}, {92, 43697}, {661, 35138}, {662, 8599}, {799, 46001}, {811, 30491}, {896, 18818}, {1577, 11636}, {3112, 30489}, {10511, 16568}, {23287, 36085}
X(574) = crosspoint of X(i) and X(j) for these (i,j): {1, 7349}, {2, 5486}, {6, 21448}, {249, 12074}, {599, 5094}
X(574) = crosssum of X(i) and X(j) for these (i,j): {1, 6205}, {2, 1992}, {4, 41370}, {6, 1995}, {7, 31599}, {57, 34490}, {115, 12073}, {351, 35507}, {514, 44317}, {523, 8288}, {650, 11936}, {1383, 43697}, {20382, 23287}, {20383, 24976}, {23297, 30489}, {27085, 36415}, {32069, 42365}
X(574) = crossdifference of every pair of points on line {351, 523}
X(574) = internal center of similitude of circumcircle and Moses circle
X(574) = crossdifference of every pair of points on line X(351)X(523)
X(574) = X(6)-of-2nd-Brocard-triangle
X(574) = perspector of Lucas Brocard and Lucas(-1) Brocard triangles
X(574) = perspector of ABC and inverse-in-Brocard-circle of vertices of circumsymmedial triangle
X(574) = perspector of circumsymmedial triangle and inverse-in-Brocard-circle of A,B,C
X(574) = pole, wrt Brocard circle, of Lemoine axis
X(574) = harmonic center of Gallatly circle and Ehrmann circle
X(574) = homothetic center of circumtangential triangle and unary cofactor triangle of Stammler triangle
X(574) = perspector of ABC and unary cofactor triangle of Lemoine triangle
X(574) = X(6)-of-X(3)PU(1)
X(574) = homothetic center of Kosnita triangle and mid-triangle of inner and outer tri-equilateral triangles
X(574) = {X(15),X(16)}-harmonic conjugate of X(182)
X(574) = {X(1340),X(1341)}-harmonic conjugate of X(6)
X(574) = X(i)-isoconjugate of X(j) for these (i,j): {1, 598}, {31, 40826}, {75, 1383}, {82, 23297}, {92, 43697}, {661, 35138}, {662, 8599}, {799, 46001}, {811, 30491}, {896, 18818}, {1577, 11636}, {3112, 30489}, {10511, 16568}, {23287, 36085}
X(574) = barycentric product X(i)*X(j) for these {i,j}: {1, 36263}, {3, 5094}, {6, 599}, {32, 9464}, {39, 10130}, {67, 10510}, {69, 8541}, {74, 13857}, {99, 17414}, {106, 4141}, {110, 3906}, {111, 39785}, {187, 42008}, {249, 8288}, {512, 9146}, {513, 3908}, {523, 9145}, {524, 42007}, {690, 32583}, {1177, 19510}, {3917, 32581}, {5467, 23288}, {5486, 8542}, {9872, 34898}, {11165, 21448}, {12074, 17436}, {15810, 39389}
X(574) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 40826}, {6, 598}, {32, 1383}, {39, 23297}, {67, 10512}, {110, 35138}, {111, 18818}, {184, 43697}, {351, 23287}, {512, 8599}, {599, 76}, {669, 46001}, {1576, 11636}, {3049, 30491}, {3051, 30489}, {3455, 10511}, {3906, 850}, {3908, 668}, {4141, 3264}, {5094, 264}, {8288, 338}, {8541, 4}, {8542, 11185}, {8586, 8785}, {9145, 99}, {9146, 670}, {9464, 1502}, {9872, 11054}, {10130, 308}, {10510, 316}, {11165, 11059}, {13857, 3260}, {15810, 26235}, {17414, 523}, {19510, 1236}, {32581, 46104}, {32583, 892}, {36263, 75}, {39689, 20380}, {39785, 3266}, {42007, 671}, {42008, 18023}
X(574) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 31421, 1571}, {2, 99, 3734}, {2, 111, 8585}, {2, 671, 7617}, {2, 2549, 115}, {2, 4045, 7913}, {2, 7618, 2482}, {2, 7790, 7844}, {2, 11648, 18362}, {2, 32480, 671}, {2, 43448, 43620}, {3, 6, 187}, {3, 32, 5206}, {3, 39, 32}, {3, 187, 8588}, {3, 1384, 5210}, {3, 3053, 15513}, {3, 3095, 5171}, {3, 5013, 39}, {3, 5024, 6}, {3, 6421, 39654}, {3, 6422, 39655}, {3, 9605, 3053}, {3, 9737, 30270}, {3, 11171, 182}, {3, 11842, 38225}, {3, 13334, 37479}, {3, 14961, 577}, {3, 15655, 5585}, {3, 15815, 37512}, {3, 21309, 15655}, {3, 22121, 18472}, {3, 22332, 5007}, {3, 30435, 5023}, {3, 32447, 2080}, {3, 35002, 3098}, {3, 37512, 15515}, {4, 31401, 1506}, {4, 31415, 43457}, {6, 187, 32}, {6, 1151, 8375}, {6, 1152, 8376}, {6, 1350, 11173}, {6, 1384, 5008}, {6, 3053, 21309}, {6, 5008, 14075}, {6, 5013, 5024}, {6, 5024, 39}, {6, 5092, 5033}, {6, 5210, 1384}, {6, 5585, 3053}, {6, 6200, 9675}, {6, 6443, 6421}, {6, 6444, 6422}, {6, 8375, 5058}, {6, 8376, 5062}, {6, 8586, 576}, {6, 8589, 8588}, {6, 11173, 5052}, {6, 14806, 10979}, {6, 15602, 15515}, {6, 21309, 5007}, {6, 22246, 5041}, {6, 44453, 8586}, {15, 16, 182}, {20, 2548, 7747}, {20, 31400, 2548}, {20, 31450, 9698}, {32, 39, 7772}, {32, 8588, 187}, {32, 14075, 5008}, {32, 15515, 3}, {35, 2275, 2241}, {35, 16784, 10987}, {36, 2276, 2242}, {39, 187, 6}, {39, 5007, 9605}, {39, 5188, 46305}, {39, 5210, 14075}, {39, 8589, 187}, {39, 15513, 5007}, {39, 15515, 5206}, {39, 15602, 8589}, {39, 15815, 15515}, {39, 21163, 2021}, {39, 31652, 5013}, {39, 35007, 5041}, {39, 37512, 3}, {39, 46283, 3095}, {56, 31448, 1500}, {61, 62, 22234}, {69, 34511, 7813}, {76, 7783, 7781}, {76, 7824, 7815}, {99, 15483, 2482}, {115, 2549, 11648}, {140, 5254, 7746}, {140, 43291, 3054}, {141, 6390, 7801}, {165, 9592, 1572}, {182, 3094, 5028}, {182, 5028, 39764}, {182, 9734, 3}, {184, 3269, 39913}, {187, 5008, 1384}, {187, 8588, 5206}, {187, 8589, 3}, {187, 15513, 15655}, {187, 37512, 8589}, {194, 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5052}, {1670, 1671, 13330}, {1687, 1688, 11842}, {1689, 1690, 11171}, {1975, 3934, 17130}, {1975, 11285, 3934}, {2021, 18860, 5162}, {2022, 35424, 32}, {2028, 2029, 5107}, {2076, 13331, 5039}, {2080, 32447, 576}, {2092, 36743, 5042}, {2275, 10987, 16784}, {2548, 31400, 9698}, {2548, 31450, 31400}, {2549, 43620, 43448}, {2896, 7796, 7896}, {3053, 5007, 32}, {3053, 5013, 22332}, {3053, 5585, 15655}, {3053, 9605, 5007}, {3053, 15655, 187}, {3053, 22332, 9605}, {3054, 5254, 43291}, {3054, 43291, 7746}, {3094, 5116, 182}, {3094, 11171, 39}, {3094, 15815, 9734}, {3094, 39498, 5034}, {3095, 9301, 37517}, {3096, 7836, 7869}, {3098, 9737, 35002}, {3098, 35002, 30270}, {3314, 7799, 7908}, {3314, 7831, 7865}, {3329, 13586, 3972}, {3520, 39575, 1968}, {3524, 7735, 21843}, {3576, 9574, 9620}, {3589, 32459, 8369}, {3734, 15482, 2}, {3767, 7738, 7765}, {3785, 7758, 7826}, {3788, 6656, 7867}, {3926, 7800, 7794}, {3926, 32990, 7800}, {4255, 5021, 20970}, {4256, 5030, 6}, {4261, 5124, 5019}, {5007, 8589, 5585}, {5007, 15513, 3053}, {5013, 5116, 11171}, {5013, 9734, 5028}, {5013, 15515, 7772}, {5013, 15602, 8588}, {5013, 15815, 3}, {5013, 37512, 32}, {5013, 40349, 5065}, {5023, 5041, 32}, {5023, 30435, 35007}, {5024, 8588, 7772}, {5024, 8589, 32}, {5024, 15655, 9605}, {5024, 15815, 8589}, {5024, 37512, 8588}, {5025, 7769, 7862}, {5025, 7847, 7872}, {5034, 8722, 32}, {5038, 8586, 6}, {5038, 44453, 576}, {5041, 35007, 30435}, {5058, 39654, 32}, {5062, 39655, 32}, {5077, 11184, 31173}, {5111, 38225, 10631}, {5171, 37517, 9301}, {5206, 7772, 32}, {5461, 7619, 2}, {5585, 15513, 8588}, {5585, 15655, 15513}, {5585, 21309, 187}, {5585, 22332, 6}, {6141, 6142, 2502}, {6200, 6396, 5092}, {6292, 7863, 7795}, {6337, 7795, 7863}, {6337, 16043, 7795}, {6390, 8359, 141}, {6421, 6422, 10542}, {6421, 8375, 6}, {6421, 10541, 5062}, {6422, 8376, 6}, {6422, 10541, 5058}, {6655, 7752, 7825}, {6683, 7816, 7770}, {6781, 7753, 7737}, {7354, 31460, 9650}, {7735, 7739, 5355}, {7736, 7737, 7753}, {7739, 21843, 7735}, {7747, 9698, 2548}, {7748, 31455, 5}, {7748, 31457, 31455}, {7749, 7765, 3767}, {7757, 7771, 385}, {7760, 43459, 7793}, {7763, 7791, 626}, {7764, 7830, 315}, {7768, 7906, 7916}, {7769, 7847, 5025}, {7773, 33234, 7842}, {7774, 33008, 14907}, {7777, 7833, 316}, {7778, 11287, 7853}, {7780, 32450, 7754}, {7781, 7815, 76}, {7782, 7786, 384}, {7783, 7824, 76}, {7785, 33260, 7802}, {7789, 8362, 7822}, {7793, 33022, 43459}, {7795, 16043, 6292}, {7797, 33259, 7857}, {7799, 7831, 3314}, {7803, 16925, 6680}, {7804, 32456, 1003}, {7804, 44562, 11174}, {7810, 7813, 69}, {7814, 7910, 7885}, {7828, 7864, 7902}, {7832, 7876, 7914}, {7836, 33021, 3096}, {7851, 33233, 7886}, {7862, 7872, 5025}, {7864, 7907, 7828}, {7865, 7908, 3314}, {7870, 7937, 7931}, {7871, 7936, 7939}, {7876, 7891, 7832}, {7879, 32821, 7895}, {7888, 7935, 626}, {7904, 7906, 7768}, {7918, 7940, 7901}, {7923, 33245, 7942}, {7924, 7925, 7934}, {7928, 7947, 7922}, {8400, 26348, 8406}, {8407, 26341, 8414}, {8589, 15602, 37512}, {8589, 31652, 5024}, {9605, 15513, 32}, {9605, 15655, 21309}, {9605, 21309, 6}, {9605, 22332, 39}, {9619, 31422, 40}, {9651, 31501, 12}, {9885, 9886, 7622}, {9888, 15483, 99}, {10631, 11842, 32}, {10987, 16784, 2241}, {11063, 13337, 33872}, {12055, 35002, 39}, {13335, 43183, 32}, {13351, 15109, 571}, {14482, 15698, 46453}, {14630, 22242, 6}, {14631, 22243, 6}, {15602, 31652, 6}, {15655, 21309, 3053}, {15810, 39785, 599}, {15815, 31652, 32}, {18472, 22121, 10316}, {18860, 21163, 3}, {22246, 30435, 6}, {30435, 35007, 32}, {31401, 43619, 31415}, {31415, 43619, 4}, {31467, 44519, 39590}, {31489, 44526, 381}, {31652, 37512, 39}, {32456, 44562, 7804}, {32568, 32575, 43650}, {33215, 34511, 7810}, {39229, 39230, 6}, {41196, 41197, 5158}, {41406, 41407, 41412}, {43448, 43620, 115}


X(575) = ORTHOGONAL OF X(187)

Trilinears   sin A + cot D/2 cos A, where cot(D/2) = (a2 + b2 + c2)/(12*area)
Trilinears    3 sin A + cos A cot ω : :   (Peter J. C. Moses, 7/20/03)
Trilinears    cos A + 3 sin A tan ω : :   (Peter J. C. Moses, 8/22/03)
Trilinears    2 cos(A - 2ω) - cos(A + 2ω) + cos A : :
Barycentrics    a^2(2a^4 + b^4 + c^4 - 3a^2b^2 - 3a^2c^2 - 4b^2c^2) : :
X(575) = X(3) + 3X(6)

X(575) lies on these lines: 3,6   4,598   5,542   23,51   54,895   110,373   140,524   141,629

X(575) = midpoint of X(i) and X(j) for these (i,j): (3,576), (6,182)
X(575) = isogonal conjugate of X(7608)
X(575) = Brocard-circle-invertse of X(576)

X(575) = 2nd-Brocard-circle-inverse of X(32447)
X(575) = {X(61),X(62)}-harmonic conjugate of X(39)
X(575) = {X(1687),X(1688)}-harmonic conjugate of X(5038)
X(575) = homothetic center of Kosnita and 2nd Ehrmann triangles

X(576) = ORTHOHARMONIC OF X(187)

Trilinears    sin A + cot(D/2) cos A, where cot(D/2) = - (a2 + b2 + c2)/(12*area)
Trilinears    3 sin A - cos A cot ω : :   (Peter J. C. Moses, 7/20/03)
Trilinears    cos A - 3 sin A tan ω : :   (Peter J. C. Moses, 8/22/03)
Trilinears    2 cos(A + 2ω) - cos(A - 2ω) + cos A : :
Barycentrics    a^2(a^4 + 2b^4 + 2c^4 - 3a^2b^2 - 3a^2c^2 - 2b^2c^2) : :
X(576) = X(3) - 3X(6)

X(576) lies on these lines: 3,6   4,542   5,524   23,184   140,597   262,385

X(576) = reflection of X(i) in X(j) for these (i,j): (3,575), (182,6)
X(576) = circumcircle-inverse of X(38225)
X(576) = Brocard-circle-inverse of X(575)
X(576) = 2nd-Lemoine-circle-nverse of X(1691)
X(576) = isogonal conjugate of X(7607)
X(576) = {X(61),X(62)}-harmonic conjugate of X(32)
X(576) = pole of Lemoine axis wrt circle {X(371),X(372),PU(1),PU(39)}
X(576) = center of Ehrmann circle
X(576) = X(1) of 2nd Ehrmann triangle if ABC is acute
X(576) = Cundy-Parry Phi transform of X(187)
X(576) = Cundy-Parry Psi transform of X(671)


X(577) = HARMONIC OF X(216)

Trilinears    sin A cos2A : :
Trilinears    cos A (tan A + tan B + tan C) - sin A : :
Trilinears    cos A - sin A (cot A cot B cot C) : :
Trilinears    cos A sin 2A : :
Trilinears    cos(A - T) : :, T as at X(389)
Trilinears    sec B sec C - csc B csc C : :
Barycentrics    a^4(b^2 + c^2 - a^2)^2 : :
Barycentrics    sin^2 2A : :

X(577) lies on these lines: 2,95   3,6   20,393   22,232   30,53   48,603   69,248   112,376   141,441   160,206   172,1038   184,418   198,478   219,906   220,268   264,401   395,466   396,465

X(577) = reflection of X(36426) in X(23583)
X(577) = isogonal conjugate of X(2052)
X(577) = isotomic conjugate of X(18027)
X(577) = complement of X(317)
X(577) = X(i)-Ceva conjugate of X(j) for these (i,j): (3,184), (97,3)
X(577) = X(418)-cross conjugate of X(3)
X(577) = crosspoint of X(i) and X(j) for these (i,j): (2,68), (3,394)
X(577) = crosssum of X(i) and X(j) for these (i,j): (4,393), (6,24), (324,467)
X(577) = crossdifference of every pair of points on line X(403)X(523)
X(577) = inverse-in-Brocard-circle of X(216)
X(577) = perspector of circumconic centered at X(1147)
X(577) = center of circumconic that is locus of trilinear poles of lines passing through X(1147)
X(577) = X(2)-Ceva conjugate of X(1147)
X(577) = X(92)-isoconjugate of X(4)
X(577) = X(1577)-isoconjugate of X(107)
X(577) = {X(6),X(50)}-harmonic conjugate of X(571)
X(577) = barycentric square of X(3)
X(577) = orthic-to-ABC barycentric image of X(53)
X(577) = antipode of X(36426) in barycentric Euler inellipse
X(577) = circle-{X(371),X(372),PU(1),PU(39)}}-inverse of X(389)
X(577) = {X(371),X(372)}-harmonic conjugate of X(389)
X(577) = homothetic center of [mid-triangle of 1st & 2nd Kenmotu diagonals triangles] and [mid-triangle of orthic and dual of orthic triangles]


X(578) = ORTHOHARMONIC OF X(216)

Trilinears    sin A + cot D/2 cos A, where cot(D/2) = (a2b2c2 cos A cos B cos C)/[8*(area)3]
Trilinears    cos A + sin A (tan A + tan B + tan C) : :
Trilinears    cos A (cot A cot B cot C) + sin A : :
Barycentrics    a^2[a^8 - 3a^6(b^2 + c^2) + a^4(3b^4 + 4b^2c^2 + 3c^4) - a^2(b^2 - c^2)^2(b^2 + c^2) - 2b^2c^2(b^2 - c^2)^2] : :

Joe Goggins notes (10/1/2008), in connection with the note at X(389), that trilinears for X(578) are sin(A-T) : sin(B-T) : sin(C-T), where tan(T) = - cot A cot B cot C.

Let A'B'C' be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung). A'B'C' is homothetic to the Euler triangle at X(578). (Randy Hutson, March 14, 2018)

X(578) lies on these lines: 2,1092   3,6   4,54   24,51   49,381   156,546   185,378   436,1093

X(578) = inverse-in-Brocard-circle of X(389)

X(578) = X(12514)-of-orthic-triangle if ABC is acute

X(579) = HARMONIC OF X(284)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a2b + a2c + abc - b3 - c3)      (M.Iliev, 5/13/07)

Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(579) = s*X(3) - (r + 2R)*cot(ω)*X(6)

X(579) lies on these lines: 1,71   2,7   3,6   19,46   36,48   37,942   40,387   56,219   109,608   165,380   198,218   443,966   474,965   517,1108

X(579) = isogonal conjugate of X(1751)
X(579) = X(27)-Ceva conjugate of X(1)
X(579) = crosssum of X(11) and X(652)
X(579) = crossdifference of every pair of points on line X(523)X(663)
X(579) = inverse-in-Brocard-circle of X(284)


X(580) = ORTHOGONAL OF X(284)

Barycentrics   Sin[A] (Sin[A] + Cot[D/2] Cos[A]) : : , where
D = (a^3 - a^2*b - a*b^2 + b^3 - a^2*c - 2*a*b*c - b^2*c - a*c^2 - b*c^2 + c^3)/(4*(a + b + c)*Area);
D = 2*r*(r + 2*R) / S
D = (e1 - e2 - 2*a*b*c) / (4*Area*(a + b + c)); where
e1 = a^3 + b^3 + c^3
e2 = a^2*(b + c) + b^2*(c + a) + c^2*(a + b)

X(580) lies on these lines: 1,201   2,283   3,6   31,40   34,46   36,54   57,255   162,412   165,601   223,603   238,946   517,595

X(580) = inverse-in-Brocard-circle of X(581)
X(580) = X(270)-Ceva conjugate of X(1)
X(580) = crosspoint of X(59) and X(162)
X(580) = crosssum of X(11) and X(656)


X(581) = ORTHOHARMONIC OF X(284)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + cot D/2 cos A,
                        cot D/2 = (2abc - e1 + e2)/[4*area*(a + b + c)], where e1, e2 are as for X(579)

Barycentrics  (sin A)f(A,B,C): (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(581) lies on these lines: 1,4   3,6   35,47   40,42   81,411   84,941   222,1035   936,966   947,1036   995,1104

X(581) = inverse-in-Brocard-circle of X(580)
X(581) = crossdifference of every pair of points on line X(523)X(652)


X(582) = HARMONIC OF X(500)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + cot D/2 cos A,
                        cot D/2 = (e1 - e2 - 4abc)/[4*area*(a + b + c)], where
                        e1 = a3 + b3 + c3
                        e2 = a2(b + c) + b2(c + a) + c2(a + b)

Barycentrics  (sin A)f(A,B,C): (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(582) lies on these lines: 3,6   212,942   283,474   517,602

X(582) = inverse-in-Brocard-circle of X(500)


X(583) = ORTHOGONAL OF X(500)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a2b + a2c + 2abc - b3 - c3)      (M.Iliev, 5/13/07)

Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(583) = s*X(3) + (r - 3R)*cot(ω)*X(6)

X(583) lies on these lines: 3,6   37,38   44,992   71,1100   518,1009

X(583) = inverse-in-Brocard-circle of X(584)


X(584) = ORTHOHARMONIC OF X(500)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(ab2 + ac2 + bc2 + b2c + 2abc - a3)      (M.Iliev, 5/13/07)

Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(584) = s*X(3) + (r + 3R)*cot(ω)*X(6)

X(584) lies on these lines: 3,6   37,41   42,560   48,354

X(584) = inverse-in-Brocard-circle of X(583)


X(585) = 1st CONGRUENT SHRUNK INSQUARES POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [2*area*(1/b + 1/c - 1/a) + b + c - a]/a
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2*area*(1/b + 1/c - 1/a) + b + c - a

X(585) lies on this line: 8,192

For a discussion, see Floor van Lamoen, "Vierkanten in een driehoek: 5. Gekrompen ingeschreven vierkanten"

X(585) = {X(8),X(192)}-harmonic conjugate of X(586)

X(586) = 2nd CONGRUENT SHRUNK INSQUARES POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [2*area*(1/b + 1/c - 1/a) + a - b - c]/a
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2*area*(1/b + 1/c - 1/a) + a - b - c

X(586) lies on this line: 8,192

For a discussion, see Floor van Lamoen, "Vierkanten in een driehoek: 5. Gekrompen ingeschreven vierkanten"

X(586) = {X(8),X(192)}-harmonic conjugate of X(585)


X(587) = POINT ARCTURUS

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[2(a + b + c) + (a - b - c) tan A]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2(a + b + c) + (a - b - c) tan A

X(587) lies on this line: 2,92


X(588) = 1st KENMOTU-VAN LAMOEN POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/(a2 + 4*area(ABC))
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(588) is the perspector of triangles ABC and A'B'C', where A' is the circumcenter of X(371) and the points where the squares in the Kenmotu configuration with center X(371) meet sideline BC, and B' and C' are defined cyclically. See

Floor van Lamoen, Some concurrences from Tucker hexagons, Forum Geometricorum 2 (2002) 5-13.

X(588) is the homothetic center of triangle ABC and the Lucas(2:1) homothetic triangle; see X(371) and X(589). (Randy Hutson, 9/23/2011)

Let A'B'C' be the Lucas central triangle. Let A″ be the trilinear pole of line B'C'; define B″ and C″ cyclically. Let A* be the trilinear pole of line B″C″; define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(588). (Randy Hutson, January 29, 2015)

X(588) lies on this line: 39,589

X(588) = isogonal conjugate of X(590)
X(588) = cevapoint of X(6) and X(371)
X(588) = X(1994)-cross conjugate of X(589)


X(589) = 2nd KENMOTU-VAN LAMOEN POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/(a2 - 4*area(ABC))
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(589) is the perspector of triangles ABC and A'B'C', where A' is the circumcenter of X(372) and the points where the squares in the Kenmotu configuration with center X(372) meet sideline BC, and B' and C' are defined cyclically. See the reference at X(588).

X(588) is the homothetic center of triangle ABC and the Lucas(-2:1) homothetic triangle; see X(371) and X(588). (Randy Hutson, 9/23/2011)

Let A'B'C' be the Lucas(-1) central triangle. Let A″ be the trilinear pole of line B'C'; define B″ and C″ cyclically. Let A* be the trilinear pole of line B″C″; define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(589). (Randy Hutson, January 29, 2015)

X(589) lies on this line: 39,588

X(589) = isogonal conjugate of X(615)
X(589) = cevapoint of X(6) and X(372)
X(589) = X(1994)-cross conjugate of X(588)


X(590) = ISOGONAL CONJUGATE OF X(588)

Trilinears    (a2 + 4*area(ABC))/a : :
Trilinears    sin A + 2 sin B sin C : :
Trilinears    aR + bc : :
Barycentrics   a2 + 2S : : X(590) lies on these lines: 1,3302   2,6   3,485   4,1151   5,371   11,2066   12,2067   17,3389   32,640   39,642   53,1585   140,372   157,3155   393,3535   397,2045   398,2046   486,1656   493,2165   498,1335   499,1124   588,2963   605,748   606,750   631,1152   639,1504   1131,3522   1327,3534   1578,3546   1579,3547   1583,1609   1588,3090   1834,2047   3085,3298   3086,3297   3087,3536   3092,3542   3093,3541   3312,3526   3525,3594

X(590) = isogonal conjugate of X(588)
X(590) = complement of X(492)
X(590) = crosspoint of X(2) and X(485)
X(590) = crosssum of X(6) and X(371)
X(590) = perspector of circumconic centered at X(641)
X(590) = center of circumconic that is locus of trilinear poles of lines passing through X(641)
X(590) = X(2)-Ceva conjugate of X(641)
X(590) = homothetic center of outer Vecten and 3rd tri-squares triangles
X(590) = homothetic center of [outer Vecten of outer Vecten triangle] and pedal triangle of X(1151)


X(591) = 1st VAN LAMOEN PERPENDICULAR BISECTORS POINT

Barycentrics    2*a^2 - b^2 - c^2 - 2*S : :
X(591) = X[1991] + 2 X[5860], 3 X[1991] - 2 X[5861], 3 X[5860] + X[5861], 2 X[35684] + X[49317], 2 X[9733] + X[13748], X[9733] + 2 X[49355], X[13748] - 4 X[49355], 3 X[486] - X[42024], 2 X[45713] + X[49329], X[45713] + 2 X[49347], X[49329] - 4 X[49347], 2 X[3102] + X[49351], X[49367] + 2 X[50719], X[1160] + 2 X[48466], X[1327] - 3 X[60223], 3 X[13692] - 2 X[13710], 3 X[3524] - X[26289], 3 X[5054] - 2 X[41491], 2 X[5874] + X[61096], X[9732] - 4 X[48772], 2 X[9739] + X[49086], 2 X[14233] + X[49038]

Erect squares outwardly on the sides of triangle ABC. Two edges emanate from A; let P and Q be their endpoints. Let a' be the perpendicular bisector of PQ, and define b' and c' cyclically. Then a', b', c' concur in X(591). See also X(1991). (Floor van Lamoen, 1/4/2001, Hyacinthos #2123)

If you have GeoGebra, you can view X(591) and X(1991); the label changes with the slider position..

X(591) lies on the curve Q088 and these lines: {2, 6}, {3, 32419}, {4, 26288}, {30, 9733}, {32, 42009}, {53, 55479}, {76, 54503}, {115, 13927}, {315, 12969}, {372, 754}, {376, 12305}, {381, 6250}, {428, 45400}, {486, 42024}, {488, 3071}, {489, 6410}, {490, 23261}, {511, 49327}, {519, 45713}, {527, 60888}, {528, 48703}, {530, 49305}, {531, 49307}, {532, 49335}, {533, 49333}, {538, 3102}, {539, 49321}, {541, 49313}, {542, 49309}, {543, 49311}, {549, 43119}, {551, 45398}, {637, 1152}, {638, 42262}, {639, 3312}, {640, 13951}, {641, 3311}, {732, 49231}, {1132, 54502}, {1151, 45508}, {1160, 36733}, {1267, 17365}, {1327, 60223}, {1328, 42023}, {1350, 45510}, {1351, 45554}, {1505, 7818}, {1587, 23311}, {1588, 26619}, {1651, 45446}, {3053, 35306}, {3058, 45470}, {3095, 13692}, {3241, 45476}, {3524, 26289}, {3534, 49363}, {3564, 9757}, {3592, 39387}, {3594, 7389}, {3679, 45426}, {3734, 13763}, {3785, 19443}, {3788, 45514}, {3830, 45375}, {3845, 45438}, {3933, 44648}, {3946, 49621}, {4363, 56385}, {4399, 32794}, {4421, 45416}, {4643, 5405}, {5026, 13760}, {5054, 41491}, {5062, 41750}, {5391, 17362}, {5406, 13428}, {5420, 55040}, {5434, 45404}, {5485, 14226}, {5490, 54625}, {5874, 61096}, {6033, 49312}, {6214, 36726}, {6231, 6321}, {6278, 36714}, {6280, 6399}, {6351, 17390}, {6352, 17332}, {6390, 11157}, {6396, 47101}, {6409, 43134}, {6419, 7764}, {6420, 7759}, {6421, 11287}, {6424, 11288}, {6426, 11293}, {6463, 33272}, {6560, 44678}, {6561, 13712}, {6565, 18546}, {6748, 55474}, {6811, 11477}, {6813, 15069}, {7228, 32793}, {7618, 53130}, {7751, 11314}, {7775, 35822}, {7780, 11316}, {7811, 45434}, {7838, 45515}, {8182, 13835}, {8414, 52045}, {8716, 35949}, {8754, 32587}, {9530, 49315}, {9732, 48772}, {9739, 49086}, {9892, 49368}, {9894, 13708}, {9909, 45428}, {10056, 45490}, {10072, 45492}, {10607, 13430}, {11165, 13701}, {11194, 45436}, {11207, 45430}, {11208, 45432}, {11235, 45454}, {11236, 45456}, {11237, 45458}, {11238, 45460}, {11239, 45494}, {11240, 45496}, {11898, 45555}, {11917, 48778}, {12150, 45402}, {12152, 45467}, {12153, 45464}, {12221, 26617}, {12222, 42273}, {12322, 42259}, {12323, 42270}, {12963, 35297}, {12968, 35953}, {13678, 33457}, {13789, 49263}, {13850, 18362}, {13873, 14645}, {14233, 49038}, {17243, 30412}, {17363, 32792}, {17364, 32791}, {18586, 34509}, {18587, 34508}, {19408, 44633}, {22723, 24256}, {23251, 43133}, {26468, 45862}, {26615, 43257}, {28194, 49323}, {29617, 32802}, {32797, 49727}, {32798, 50098}, {32801, 50128}, {33210, 53483}, {33456, 42284}, {34511, 55041}, {35549, 45806}, {37350, 49262}, {38747, 45498}, {42035, 54538}, {42036, 54535}, {42215, 51123}, {42283, 58803}, {43569, 60224}, {44526, 53515}, {45345, 45696}, {45347, 45697}, {45412, 45698}, {45415, 45699}, {45422, 45700}, {45424, 45701}, {47102, 53131}, {54505, 60194}, {54506, 60300}

X(591) = midpoint of X(i) and X(j) for these {i,j}: {2, 5860}, {4, 26288}, {1160, 36733}, {49311, 49367}
X(591) = reflection of X(i) in X(j) for these {i,j}: {3, 41490}, {1991, 2}, {36733, 48466}, {49311, 50719}, {49368, 9892}
X(591) = isotomic conjugate of X(60195)
X(591) = complement of X(5861)
X(591) = complement of the isogonal conjugate of X(41445)
X(591) = complement of the isotomic conjugate of X(60208)
X(591) = isotomic conjugate of the isogonal conjugate of X(9675)
X(591) = X(i)-complementary conjugate of X(j) for these (i,j): {41445, 10}, {60208, 2887}
X(591) = X(31)-isoconjugate of X(60195)
X(591) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 60195}, {591, 5861}
X(591) = crosspoint of X(2) and X(60208)
X(591) = perspector of inner Vecten triangle and outer Vecten of outer Vecten triangle
X(591) = centroid of {Ab, Ac, Bc, Ba, Ca, Cb} used in construction of 4th Lozada circle X(591) = barycentric product X(76)*X(9675)
X(591) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 60195}, {9675, 6}
X(591) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 1992, 32787}, {2, 11160, 32811}, {2, 13757, 13847}, {2, 13759, 19053}, {2, 13847, 13783}, {2, 19053, 597}, {2, 32787, 13663}, {2, 32808, 599}, {2, 45420, 13846}, {2, 45421, 6}, {2, 62987, 45421}, {6, 492, 45472}, {6, 9766, 1991}, {69, 615, 45473}, {69, 13468, 1991}, {193, 32805, 590}, {395, 396, 62202}, {492, 5860, 9766}, {492, 13758, 44390}, {492, 45421, 2}, {492, 62987, 6}, {599, 8667, 1991}, {599, 13757, 13783}, {599, 13847, 2}, {615, 44394, 37637}, {1270, 3069, 141}, {3631, 45872, 5591}, {3830, 45375, 49361}, {5590, 7586, 3589}, {5591, 13941, 45872}, {5591, 32814, 3631}, {5858, 9761, 1991}, {5859, 9763, 1991}, {5860, 13757, 8667}, {5862, 33474, 1991}, {5863, 33475, 1991}, {6144, 8253, 62986}, {6189, 6190, 44366}, {6289, 45488, 45440}, {7585, 26340, 32455}, {7585, 26361, 45871}, {7610, 15533, 1991}, {7774, 45421, 15534}, {7778, 44390, 45472}, {8252, 40341, 491}, {8584, 9770, 1991}, {9733, 49355, 13748}, {9740, 50991, 1991}, {9771, 63064, 1991}, {11184, 15534, 1991}, {13663, 45473, 44393}, {13712, 22484, 6561}, {13757, 32808, 2}, {13846, 15534, 45420}, {13847, 15534, 22329}, {13941, 32814, 5591}, {15597, 50992, 1991}, {22165, 63029, 1991}, {26340, 26361, 7585}, {32455, 45871, 7585}, {32807, 62986, 8253}, {37640, 37641, 61322}, {39022, 39023, 44393}, {44393, 62201, 13663}, {45713, 49347, 49329}


X(592) = VAN LAMOEN CIRCUMCENTERS POINT

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/[cos A + 2 cos(B - ω) cos(C - ω)]
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

Let P be the point where the line through X(6) parallel to line CA meets line BC, and let Q be the point where the line through X(6) parallel to line AB meets line BC. Let X = X(182), and let A' be the circumcenter of the triangle PQX. Define B' and C' cyclically. The lines AA', BB', CC' concur in X(592). (Floor van Lamoen, 1/4/01)


X(593) = 1st HATZIPOLAKIS-YIU POINT

Trilinears        a/(b + c)2 : b/(c + a)2 : c/(a + b)2
Barycentrics  [a/(b + c)]2 : [b/(c + a)]2 : [c/(a + b)]2

Let O(A) be the circle tangent to line BC and to the circumcircle of triangle ABC at vertex A. Let AB and AC be where O(A) meets lines AB and AC, respectively. Let L(A) be the line joining AB and AC, and define L(B) and L(C) cyclically. Let A' be where L(B) and L(C) meet, and define B' and C' cyclically. Then triangle A'B'C' is homothetic to triangle ABC, and the center of homothety is X(593). See Antreas Hatzipolakis, Paul Yiu, Hyacinthos #2070.

X(593) lies on these lines: 2,261   31,110   36,58   81,757   115,1029   229,1104

X(593) = isogonal conjugate of X(594)
X(593) = isotomic conjugate of X(28654)
X(593) = anticomplement of complementary conjugate of X(17045)
X(593) = barycentric square of X(81)


X(594) = ISOGONAL CONJUGATE OF X(593)

Trilinears        (b + c)2/a : (c + a)2/b : (a + b)2/c
Barycentrics  (b + c)2 : (c + a)2 : (a + b)2

Let A'B'C' be the Feuerbach triangle. Let La be the tangent to the nine-point circle at A', and define Lb and Lc cyclically. La, Lb, Lc are also tangent to the Steiner inellipse. Let A″, B″, C″ be the respective touchpoints. The lines AA″, BB″, CC″ concur in X(594). (Randy Hutson, October 15, 2018)

For a construction of X(594), see Dasari Naga Vijay Krishna, "On A Simple Construction of Triangle Centers X(8), X(197), X(K) & X(594)", Scientific Inquiry and Review, Vol. 2, Issue 3, July 2018.

X(594) lies on these lines: 6,8   7,599   9,80   10,37   45,346   53,318   75,141   100,1030   210,430   220,281   313,321   319,524   519,1100   572,952   762,1089

X(594) = midpoint of X(319) and X(894)
X(594) = isogonal conjugate of X(593)
X(594) = isotomic conjugate of X(1509)
X(594) = anticomplement of X(17045)
X(594) = crosspoint of X(10) and X(321)
X(594) = crosssum of X(i) and X(j) for these (i,j): (6,595), (58,1333)
X(594) = barycentric square of X(10)
X(594) = perspector of ABC and cross-triangle of ABC and Gemini triangle 19
X(594) = barycentric product of vertices of Gemini triangle 26
X(594) = {X(6),X(8)}-harmonic conjugate of X(17362)
X(594) = {X(3661),X(3662)}-harmonic conjugate of X(17228)


X(595) = 2nd HATZIPOLAKIS-YIU POINT

Trilinears    r + R*(cos 2A - 1) : :
Trilinears    a(a2 + ab + ac - bc) : :
Trilinears    sa2 - SR : sb2 - SR sc2 - SR : :
Barycentrics    a2(a2 + ab + ac - bc) : :

Let O(A) be the circle tangent to line BC and to the circumcircle of triangle ABC at vertex A, and define O(B) and O(C) cyclically. X(595) is the radical center of O(A), O(B), O(C).

See Antreas Hatzipolakis, Paul Yiu, Hyacinthos #2070

See Antreas Hatzipolakis and Paul Yiu, Hyacinthos 2070.

X(595) lies on these lines: 1,21   3,995   10,82   32,101   35,902   40,602   46,614   55,386   56,106   110,849   171,1125   387,390   517,580

X(595) = isogonal conjugate of X(596)
X(595) = crosssum of X(244) and X(523)
X(595) = Vu tangential transform of X(1)
X(595) = {X(1),X(31)}-harmonic conjugate of X(58)


X(596) = ISOGONAL CONJUGATE OF X(595)

Trilinears        1/[a(a2 + ab + ac - bc)] : 1/[b(b2 + bc + ba - ca)] : 1/[c(c2 + ca + cb - ab)]
Barycentrics  1/(a2 + ab + ac - bc) : 1/(b2 + bc + ba - ca) : 1/(c2 + ca + cb - ab)

X(596) lies on these lines: 10,38   37,39   58,82   65,519   244,1089

X(596) = isogonal conjugate of X(595)
X(596) = anticomplement of X(4075)
X(596) = perspector of ABC and the reflection of the incentral triangle in X(1125)


X(597) = MIDPOINT OF X(2) AND X(6)

Trilinears        (4a2 + b2 + c2)/a : (4b2 + c2 + a2)/b : (4c2 + a2 + b2)/c
Barycentrics  4a2 + b2 + c2 : 4b2 + c2 + a2 : 4c2 + a2 + b2

Let OA be the circle centered at the A-vertex of the antipedal triangle of X(2) and passing through A; define OB and OC cyclically. X(597) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

X(597) lies on these lines: 2,6   5,542   30,182   39,1084   83,671   140,576   373,2854   511,549   518,551

X(597) = midpoint of X(2) and X(6)
X(597) = reflection of X(141) in X(2)
X(597) = isogonal conjugate of X(39389)
X(597) = complement of X(599)
X(597) = crosssum of X(6) and X(574)
X(597) = circumcenter of the pedal triangle of X(2)
X(597) = circumcenter of the pedal triangle of X(6)
X(597) = crosspoint of X(2) and X(598)
X(597) = centroid of pedal triangle of X(182)
X(597) = crossdifference of every pair of points on line X(512)X(5104)
X(597) = centroid of set consisting of the interiors (with or without boundaries) of the power circles
X(597) = {X(395),X(396)}-harmonic conjugate of X(3815)
X(597) = harmonic center of circles O(13,15) and O(14,16)
X(597) = center of Lemoine ellipse
X(597) = X(5)-of-anti-Artzt-triangle
X(597) = {X(2),X(1992)}-harmonic conjugate of X(599)


X(598) = ISOGONAL CONJUGATE OF X(574)

Trilinears        bc/(a2 - 2b2 - 2c2) : ca/(b2 - 2c2 - 2a2) : ab/(c2 - 2a2 - 2b2)
Barycentrics  1/(a2 - 2b2 - 2c2) : 1/(b2 - 2c2 - 2a2) : 1/(c2 - 2a2 - 2b2)

The Lemoine ellipse is the ellipse inscribed in triangle ABC having X(2) and X(6) as foci. Let A' be where this ellipse meets sideline BC, and define B' and C' cyclically. Then triangles ABC and A'B'C' are perspective, and their perspector is X(598). (Bernard Gibert, 1/5/01, Hyacinthos #2334)

X(598) lies on these lines: 2,187   4,575   6,671   30,262   76,524   98,381

X(598) = isogonal conjugate of X(574)
X(598) = isotomic conjugate of X(599)
X(598) = cyclocevian conjugate of isogonal conjugate of X(38402)
X(598) = trilinear pole of line X(351)X(523) (the Euler line of the 1st and 2nd Parry triangles)
X(598) = pole wrt polar circle of trilinear polar of X(5094)
X(598) = X(48)-isoconjugate (polar conjugate) of X(5094)
X(598) = perspector of ABC and anti-Artzt triangle
X(598) = X(262)-of-anti-Artzt-triangle


X(599) = ISOTOMIC CONJUGATE OF X(598)

Trilinears        bc(a2 - 2b2 - 2c2) : ca(b2 - 2c2 - 2a2) : ab(c2 - 2a2 - 2b2)
Barycentrics  a2 - 2b2 - 2c2 : b2 - 2c2 - 2a2 : c2 - 2a2 - 2b2

Let OA be the circle centered at the A-vertex of the Artzt triangle and passing through A; define OB and OC cyclically. X(599) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

Let OA be the circle centered at the A-vertex of the antipedal triangle of X(6) and passing through A; define OB and OC cyclically. X(599) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

X(599) lies on these lines: 2,6   3,67   7,594   8,1086   76,338   340,458   381,511

X(599) = midpoint of X(2) and X(69)
X(599) = reflection of X(i) in X(j) for these (i,j): (2,141), (6,2)
X(599) = isogonal conjugate of X(1383)
X(599) = isotomic conjugate of X(598)
X(599) = complement of X(1992)
X(599) = anticomplement of X(597)
X(599) = crosssum of X(6) and X(1384)
X(599) = X(671) of 1st Brocard triangle
X(599) = Artzt-to-McCay similarity image of X(99)
X(599) = X(3)-of-anti-Artzt-triangle
X(599) = X(6)-of-X(2)-anti-altimedial triangle
X(599) = 1st-Brocard-isogonal conjugate of X(5026)
X(599) = 1st-Brocard-isotomic conjugate of X(543)
X(599) = center of pedal circle of X(2) wrt anticomplementary triangle
X(599) = {X(2),X(1992)}-harmonic conjugate of X(597)


X(600) = 3rd HATZIPOLAKIS-YIU POINT

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(bc + 2S)[abc(b + c - a) + 2(b2 + c2 - a2)S]

Let OA be the circle with center A and radius R, the circumradius of triangle ABC. Let BA be the point where OA meets line AB farthest from B. Define CB and AC cyclically. Let CA be the point where OA meets line AC farthest from C. Define AB and BC cyclically. X(600) is the radical center of the circles ABACA, BCBAB, CACBC. If "farthest from" is changed to "nearest to" in the construction, the resulting point is X(5507). (Antreas Hatzipolakis, Paul Yiu, 1/5/01, Hyacinthos #2344, #2346-8; for a correction in coordinates, see Paul Yiu, Adgeom #202.)

If you have The Geometer's Sketchpad, you can view X(600).

X(600) lies on this line: 4640,5507.


X(601) = INTERSECTION OF LINES X(3)X(31) AND X(55)X(255)

Trilinears        sin2A + cos A : sin2B + cos B : sin2C + cos C
Barycentrics  (sin A)(sin2A + cos A) : (sin B)(sin2B + cos B) : (sin C)(sin2C + cos C)

X(601) lies on these lines: 1,104   3,31   4,171   5,750   35,47   40,58   41,906   55,255   140,748   165,580   201,920   371,606   372,605   774,1060   912,976   999,1106


X(602) = INTERSECTION OF LINES X(3)X(31) AND X(56)X(255)

Trilinears        sin2A - cos A : sin2B - cos B : sin2C - cos C
Barycentrics  (sin A)(sin2A - cos A) : (sin B)(sin2B - cos B) : (sin C)(sin2C - cos C)

X(602) lies on these lines: 1,201   3,31   4,238   5,748   36,47   40,595   56,255   140,750   171,631   371,605   372,606   517,582   774,1062


X(603) = X(58)-CEVA CONJUGATE OF X(56)

Trilinears    cos2A - cos A : :
Trilinears    (cos A) (1 - cos A) : :
Trilinears    (cos A) sin^2(A/2) : :
Trilinears    a^2 (b^2 + c^2 - a^2)/(b + c - a) : :

Barycentrics  (sin A)(cos2A - cos A) : (cos B)(cos2B - cos B) : (cos C)(cos2C - cos C)

X(603) lies on these lines: 1,104   3,73   6,1035   12,750   28,34   31,56   33,84   36,47   41,911   48,577   63,201   77,283   171,388   223,580   404,651

X(603) = isogonal conjugate of X(318)
X(603) = X(i)-Ceva conjugate of X(j) for these (i,j): (58,56), (222,48)
X(603) = X(184)-cross conjugate of X(48)
X(603) = crosspoint of X(57) and X(77)
X(603) = crosssum of X(i) and X(j) for these (i,j): (1,1158), (9,33)
X(603) = crossdifference of every pair of points on radical axis of Mandart circle and excircles radical circle
X(603) = X(4)-isoconjugate of X(8)
X(603) = X(9)-isoconjugate of X(92)


X(604) = X(56)-CEVA CONJUGATE OF X(31)

Trilinears    a(1 - cos A) : b(1 - cos B) : c(1 - cos C)
Trilinears    a2(SA - bc) : :
Trilinears    a^2/(b + c - a) : :
Trilinears    area(A'BC) : : , where A'B'C' = Fuhrmann triangle
Barycentrics    a2(1 - cos A) : :
Barycentrics    sin^2 A sin^2(A/2) : :

X(604) lies on these lines: 1,572   6,41   19,909   31,184   32,1106   36,573   57,77   65,1100   109,739   219,672   608,1042

X(604) = isogonal conjugate of X(312)
X(604) = isotomic conjugate of X(28659)
X(604) = complement of X(21286)
X(604) = anticomplement of X(21244)
X(604) = X(56)-Ceva conjugate of X(31)
X(604) = X(32)-cross conjugate of X(31)
X(604) = crosspoint of X(34) and X(57)
X(604) = crosssum of X(i) and X(j) for these (i,j): (2,329), (8,346), (9,78), (306,321)
X(604) = X(75)-isoconjugate of X(9)
X(604) = X(92)-isoconjugate of X(78)
X(604) = perspector of ABC and extraversion triangle of X(41)
X(604) = trilinear product of PU(48)
X(604) = barycentric product of PU(92)
X(604) = barycentric square of X(266)
X(604) = perspector of ABC and unary cofactor triangle of Gemini triangle 13


X(605) = INTERSECTION OF LINES X(371)X(602) AND X(372)X(601)

Trilinears    a(1 + sin A) : b(1 + sin B) : c(1 + sin c)
Trilinears    a2(S + bc) : b2(S + ca) : c2(S + ab)
Barycentrics    a2(1 + sin A) : b2(1 + sin B) : c2(1 + sin C)

X(605) lies on these lines: 6,31   371,602   372,601   590,748   615,750

X(605) = isogonal conjugate of isotomic conjugate of X(3083)
X(605) = isogonal conjugate of polar conjugate of X(6212)
X(605) = isogonal conjugate of complement of X(37881)
X(605) = isogonal conjugate of anticomplement of X(40651)


X(606) = INTERSECTION OF LINES X(371)X(601) AND X(372)X(602)

Trilinears    a(1 - sin A) : b(1 - sin B) : c(1 - sin c)
Trilinears    a2(S - bc) : b2(S - ca) : c2(S - ab)
Barycentrics    a2(1 - sin A) : b2(1 - sin B) : c2(1 - sin C)

X(606) lies on these lines: 6,31   371,601   372,602   590,750   615,748

X(606) = isogonal conjugate of isotomic conjugate of X(3084)
X(606) = isogonal conjugate of polar conjugate of X(6213)


X(607) = ISOGONAL CONJUGATE OF X(348)

Trilinears    a(1 + sec A) : b(1 + sec B) : c(1 + sec c)
Trilinears    sin A + tan A : :
Barycentrics  a2(1 + sec A) : :

X(607) lies on these lines: 1,949   4,218   6,19   8,29   9,1039   25,41   28,1002   33,210   56,911   92,239   213,1096   227,910   240,611

X(607) is the {X(6),X(19)}-harmonic conjugate of X(608). For a list of other harmonic conjugates of X(607), click Tables at the top of this page.

X(607) = isogonal conjugate of X(348)
X(607) = X(i)-Ceva conjugate of X(j) for these (i,j): (19,25), (281,55)
X(607) = X(213)-cross conjugate of X(41)
X(607) = crosspoint of X(19) and X(33)
X(607) = crosssum of X(i) and X(j) for these (i,j): (2,347), (63,77), (307,1214)

X(607) = X(92)-isoconjugate of X(1804)
X(607) = crossdifference of every pair of points on line X(4025)X(4131)

X(608) = ISOGONAL CONJUGATE OF X(345)

Trilinears        a(1 - sec A) : b(1 - sec B) : c(1 - sec c)
Trilinears    sin A - tan A : :
Barycentrics  a2(1 - sec A) : b2(1 - sec B) : c2(1 - sec C)

X(608) is the vertex conjugate of the foci of the inconic that is the polar conjugate of the isotomic conjugate of the incircle. (Randy Hutson, October 15, 2018)

X(608) lies on these lines: 6,19   7,27   9,1041   25,31   28,959   92,894   108,739   109,579   193,651   223,380   240,613   571,913   604,1042

X(608) is the {X(6),X(19)}-harmonic conjugate of X(607). For a list of other harmonic conjugates of X(608), click Tables at the top of this page.

X(608) = isogonal conjugate of X(345)
X(608) = X(i)-Ceva conjugate of X(j) for these (i,j): (34,25), (278,56)
X(608) = crosssum of X(219) and X(1259)
X(608) = X(92)-isoconjugate of X(1259)


X(609) = INTERSECTION OF LINES X(1)X(32) AND X(6)X(36)

Trilinears        area + a2 sin A : area + b2 sin B : area + c2 sin C
Barycentrics  a(area + a2 sin A) : b(area + b2 sin B) : c(area + c2 sin C)

X(609) lies on these lines: 1,32   6,36   31,101   33,112   41,58   251,614   995,1055


X(610) = X(63)-CEVA CONJUGATE OF X(1)

Trilinears    area - a2 cot A : area - b2 cot B : area - c2cot C
Trilinears    tan B + tan C - tan A : tan C + tan A - tan B : tan A + tan B - tan C    (Randy Hutson, 9/23/2011)
Trilinears    SBSC - a2SA : :      (C. Lozada, 9/07/2013)
Trilinears    3a^4 - 2a^2(b^2 + c^2) - (b^2 - c^2)^2 : :

Barycentrics   a(area - a2 cot A) : b(area - b2 cot B) : c(area - c2 cot C)

X(610) lies on these lines: 1,19   3,9   6,57   40,219   71,165   159,197   169,572   281,515   326,662

X(610) = isogonal conjugate of X(2184)
X(610) = X(63)-Ceva conjugate of X(1)
X(610) = X(204)-cross conjugate of X(1)
X(610) = X(2165)-of-excentral-triangle


X(611) = INTERSECTION OF LINES X(1)X(6) AND X(55)X(511)

Trilinears    D + sin A : D + sin B : D + sin C, where D = (a2 + b2 + c2)/(4*area)
Trilinears    a(S^2 + bc SW) : :
Barycentrics    a(D + sin A) : b(D + sin B) : c(D + sin C)

X(611) lies on these lines: 1,6   55,511   56,182   141,498   240,607   394,612


X(612) = INTERSECTION OF LINES X(1)X(2) AND X(9)X(31)

Trilinears    a2 + b2 + c2 + 2bc      (M. Iliev, 5/13/2007)
Trilinears     bc + Sω : ca + Sω : ab + Sω      (C. Lozada, 9/07/2013)
Trilinears     (1 + cos A)(1 + cos B cos C) : :
Trilinears     (1 + sec A)(1 + sec B sec C) : :

X(612) is the homothetic center of the incentral triangle and the Ayme triangle; see X(3610).

X(612) lies on these lines: 1,2   6,210   9,31   12,34   19,25   21,989   22,35   38,57   63,171   165,990   394,611   404,988   495,1060   518,940

X(612) = crossdifference of every pair of points on line X(649)X(905)
X(612) = trilinear product X(33)*X(1038)
X(612) = trilinear product X(55)*X(388)


X(613) = INTERSECTION OF LINES X(1)X(6) AND X(56)X(511)

Trilinears   W - sin A : W - sin B : W - sin C, where W = (a2 + b2 + c2)/(4*area)
Trilinears: a(S^2 - bc SW) : :
Barycentrics  a(W - sin A) : b(W - sin B) : c(W - sin C)

X(613) lies on these lines: 1,6   55,182   56,511   141,499   222,982   240,608   394,614   496,1069


X(614) = INTERSECTION OF LINES X(1)X(2) AND X(11)X(33)

Trilinears    a2 + b2 + c2 - 2bc      (M. Iliev, 5/13/07)
Trilinears    bc - Sω : ca - Sω : ab - Sω      (C. Lozada, 9/07/2013)
Trilinears    cot^2(B/2) + cot^2(C/2) : :
Trilinears    (1 - cos A)(1 - cos B cos C) : :
Trilinears    (1 - sec A)(1 - sec B sec C) : :

X(614) lies on these lines: 1,2   6,354   9,38   11,33   21,988   22,36   25,34   31,57   46,595   63,238   106,998   165,902   251,609   269,479   278,1096   305,350   394,613   496,1062   497,1040   968,1001   1616,3057

X(614) = crosspoint of X(i) and X(j) for these (i,j): (1,269), (28,86)
X(614) = crosssum of X(42) and X(72)
X(614) = trilinear product X(34)*X(1040)
X(614) = trilinear product X(56)*X(497)


X(615) = ISOGONAL CONJUGATE OF X(589)

Trilinears    (a2 - 4*area)/a : :
Trilinears    sin A - 2 sin B sin : :
Trilinears    aR- bc : :
Barycentrics    a2 - 4*area : :
Barycentrics   a2 - 2S : :

X(615) lies on these lines: 1,3300   2,6   3,486   4,1152   5,372   32,639   39,641   140,371   605,750   606,748

X(615) = isogonal conjugate of X(589)
X(615) = complement of X(491)
X(615) = crosspoint of X(2) and X(486)
X(615) = crosssum of X(6) and X(372)
X(615) = perspector of circumconic centered at X(642)
X(615) = center of circumconic that is locus of trilinear poles of lines passing through X(642)
X(615) = X(2)-Ceva conjugate of X(642)
X(615) = homothetic center of inner Vecten and 4th tri-squares triangles
X(615) = homothetic center of [outer Vecten of inner Vecten triangle] and pedal triangle of X(1152)

leftri

Centers X(616)-X(642)

rightri

Centers X(616)-X(642) were contributed by Bernard Gibert, March 2, 2001. Notation:

SA = (b2 + c2 - a2)/2       SB = (c2 + a2 - b2)/2       SC = (a2 + b2 - c2)/2


Joe Goggins (Oct. 19, 2005) found trilinears for points in this section, using the following notation. Let

F(13) = a csc(A + π/3) + b csc(B + π/3) + c csc(C + π/3)
F(14) = a csc(A - π/3) + b csc(B - π/3) + c csc(C - π/3)
These are based on trilinears for the isogonic centers, X(13) and X(14). In like manner, F(i) is defined for i = 15, 16, 17, 18, 61, 62.

Using this notation, we have, for example,
X(616) = F(13)/a - 2 csc(A + π/3) : F(13)/b - 2 csc(B + π/3) : F(13)/c - 2 csc(C + π/3)
X(617) = F(14)/a - 2 csc(A - π/3) : F(14)/b - 2 csc(B - π/3) : F(14)/c - 2 csc(C - π/3)

Trilinears of this sort are given below at X(i) for these i: 616-619, 621-624, 627-630, and 633-636.


X(616) = ANTICOMPLEMENT OF X(13)

Barycentrics    2*S^2 + (a^2 - b^2 - c^2)*(3*a^2 + Sqrt[3]*S) : :
X(616) = 3 X[2] - 4 X[618], 5 X[2] - 4 X[5459], 9 X[2] - 8 X[6669], 7 X[2] - 6 X[22489], 21 X[2] - 16 X[35019], 4 X[2] - X[35749], 2 X[2] + X[35750], X[2] + 2 X[35751], 5 X[2] - 2 X[35752], 7 X[2] - 10 X[36767], 5 X[2] - 8 X[36768], X[2] - 4 X[36769], 9 X[2] - 10 X[36770], 7 X[2] - 4 X[47865], 13 X[2] - 12 X[48311], 4 X[2] - 3 X[59378], and many others

The midpoint of X(616) and X(617) is the Steiner point, X(99).

Let OA be the circle centered at the A-vertex of the outer Napoleon triangle and passing through A; define OB and OC cyclically. X(616) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

X(616) lies on the Kiepert circumhyperbola of the anticomplementary triangle, the cubics K001, K046a, K066b, K264a, K438a, K753, K860, K867b, K1053a, the curves Q051, Q072, Q074, and these lines: {1, 8482}, {2, 13}, {3, 299}, {4, 627}, {5, 13103}, {6, 11299}, {8, 12781}, {10, 9901}, {14, 148}, {15, 532}, {18, 33412}, {20, 633}, {22, 9916}, {30, 298}, {32, 37172}, {62, 51860}, {63, 1276}, {69, 74}, {76, 54484}, {98, 60253}, {100, 12337}, {115, 11489}, {140, 59383}, {141, 11300}, {145, 7975}, {147, 5978}, {193, 51200}, {194, 3104}, {302, 381}, {303, 549}, {388, 12942}, {395, 5254}, {396, 19780}, {397, 11307}, {399, 8491}, {484, 8501}, {485, 35753}, {486, 35754}, {487, 52400}, {488, 52399}, {489, 2043}, {490, 2044}, {491, 36437}, {492, 36455}, {497, 12952}, {524, 35931}, {531, 8591}, {533, 36967}, {543, 22512}, {599, 35932}, {619, 6778}, {623, 36969}, {631, 6771}, {635, 42158}, {636, 5237}, {671, 43543}, {1138, 5675}, {1157, 8495}, {1263, 8438}, {1270, 6268}, {1271, 6270}, {1605, 34009}, {1656, 20252}, {1992, 51012}, {2076, 35917}, {2132, 8535}, {2133, 5679}, {2782, 6773}, {2896, 3642}, {2902, 11126}, {2925, 14369}, {2975, 22773}, {3065, 8434}, {3068, 49208}, {3069, 49209}, {3085, 10062}, {3086, 10078}, {3090, 16001}, {3091, 5478}, {3096, 43481}, {3146, 36961}, {3241, 50849}, {3434, 12922}, {3436, 12932}, {3441, 8530}, {3448, 36246}, {3465, 8500}, {3466, 8435}, {3481, 8497}, {3484, 8531}, {3522, 41020}, {3523, 21156}, {3524, 51388}, {3525, 20415}, {3534, 33611}, {3543, 41042}, {3545, 25154}, {3589, 59410}, {3616, 11705}, {3648, 5699}, {3679, 50847}, {3830, 33613}, {3832, 59393}, {4048, 35918}, {4240, 12793}, {5318, 44383}, {5321, 33459}, {5334, 6782}, {5460, 41135}, {5464, 52695}, {5470, 6670}, {5472, 11488}, {5552, 49144}, {5601, 12472}, {5602, 12473}, {5613, 9736}, {5624, 8445}, {5667, 8490}, {5668, 8452}, {5673, 7326}, {5681, 8431}, {5858, 33627}, {5859, 11480}, {5862, 33622}, {5981, 11177}, {5983, 21157}, {6194, 25183}, {6321, 59396}, {6462, 12990}, {6463, 12991}, {6671, 16267}, {6694, 42990}, {6772, 7739}, {6774, 14651}, {6775, 43449}, {6777, 20094}, {7164, 8508}, {7165, 8483}, {7327, 8524}, {7328, 8525}, {7329, 8507}, {7492, 13859}, {7585, 19074}, {7586, 19073}, {7684, 54138}, {7784, 11303}, {7787, 12205}, {7795, 37173}, {7799, 36755}, {7809, 11133}, {7865, 22513}, {7898, 34540}, {8173, 8446}, {8174, 8479}, {8182, 9885}, {8439, 8441}, {8444, 16883}, {8451, 8471}, {8456, 8477}, {8462, 8478}, {8486, 8514}, {8487, 8520}, {8493, 8515}, {8494, 8522}, {8703, 33609}, {8972, 13917}, {9112, 41408}, {9143, 35314}, {9166, 31695}, {9742, 9749}, {9761, 53443}, {10409, 30485}, {10527, 49143}, {10528, 13105}, {10529, 13107}, {10617, 62232}, {10645, 40901}, {10723, 41060}, {11001, 33610}, {11160, 51011}, {11289, 42148}, {11290, 22238}, {11295, 16940}, {11297, 11486}, {11301, 42974}, {11302, 42115}, {11308, 36843}, {11481, 53430}, {11645, 51016}, {12188, 47611}, {12214, 54297}, {13083, 22494}, {13172, 22507}, {13188, 36993}, {13941, 13982}, {14136, 42998}, {14144, 36776}, {14683, 37752}, {14921, 33498}, {14922, 41472}, {15300, 36331}, {15454, 19777}, {15682, 33608}, {15702, 32907}, {15928, 46471}, {16241, 34509}, {16529, 30560}, {16530, 40694}, {16644, 53452}, {16645, 22847}, {16808, 50859}, {16964, 22114}, {18581, 23005}, {19106, 21359}, {19107, 33625}, {19708, 33612}, {19779, 37848}, {20253, 38732}, {20425, 52648}, {22113, 36782}, {22493, 42430}, {22495, 45879}, {22511, 42149}, {22578, 47866}, {22580, 59373}, {22601, 32492}, {22630, 32495}, {22844, 42157}, {22846, 33415}, {23025, 31168}, {25156, 37835}, {26394, 48456}, {26418, 48457}, {26494, 49374}, {26503, 49373}, {27551, 44010}, {30472, 48656}, {31145, 50848}, {31683, 42128}, {31710, 43404}, {32037, 52204}, {33413, 47518}, {33474, 43101}, {33475, 42500}, {33560, 40334}, {33624, 42529}, {34508, 36970}, {35304, 37786}, {36324, 49807}, {36346, 49856}, {36368, 49855}, {36762, 47868}, {36763, 47855}, {36764, 47857}, {36772, 47863}, {36958, 47068}, {36995, 44461}, {37351, 42913}, {37352, 42118}, {37825, 52689}, {38412, 43540}, {40341, 42626}, {41019, 47853}, {41751, 61318}, {42035, 43541}, {42062, 43554}, {43542, 60273}, {44015, 49953}, {44250, 48876}, {45508, 48722}, {45509, 48723}, {49863, 49910}, {49879, 49922}, {50977, 51018}, {51171, 59409}, {53428, 62199}, {53458, 62198}, {54562, 60252}

X(616) = midpoint of X(i) and X(j) for these {i,j}: {5463, 35751}, {35750, 51482}
X(616) = reflection of X(i) in X(j) for these {i,j}: {1, 51114}, {2, 5463}, {4, 5617}, {6, 51159}, {8, 12781}, {13, 618}, {14, 32553}, {20, 5473}, {69, 51010}, {145, 7975}, {147, 61634}, {148, 14}, {193, 51200}, {298, 52194}, {599, 51202}, {617, 99}, {621, 298}, {622, 5979}, {628, 14145}, {1992, 51012}, {3146, 36961}, {3180, 15}, {3181, 22998}, {3241, 50849}, {3543, 41042}, {3679, 50847}, {4240, 12793}, {5318, 44383}, {5459, 36768}, {5463, 36769}, {6770, 3}, {6778, 619}, {8591, 9116}, {9901, 10}, {10653, 22687}, {10723, 41060}, {11160, 51011}, {12188, 47611}, {13103, 5}, {14683, 37752}, {20425, 52650}, {22113, 36782}, {22495, 45879}, {22578, 47866}, {31145, 50848}, {33440, 6302}, {33441, 6306}, {35749, 51482}, {35752, 5459}, {36969, 623}, {37786, 35304}, {43540, 38412}, {48656, 51872}, {49034, 49305}, {49035, 49306}, {51482, 2}, {51484, 35931}, {51485, 8595}, {54138, 7684}
X(616) = isogonal conjugate of X(3440)
X(616) = isotomic conjugate of X(19776)
X(616) = anticomplement of X(13)
X(616) = circumcircle-inverse of X(14368)
X(616) = orthoptic-circle-of-the-Steiner-circumellipse-inverse of X(5979)
X(616) = circumcircle-of-inner-Napoleon-triangle-inverse of X(618)
X(616) = circumcircle-of-outer-Napoleon-triangle-inverse of X(624)
X(616) = antigonal image of X(39132)
X(616) = anticomplement of the isogonal conjugate of X(15)
X(616) = anticomplement of the isotomic conjugate of X(298)
X(616) = isotomic conjugate of the anticomplement of X(40578)
X(616) = anticomplementary isogonal conjugate of X(621)
X(616) = psi-transform of X(619)
X(616) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 621}, {15, 8}, {31, 3180}, {48, 19772}, {163, 23870}, {298, 6327}, {470, 21270}, {1094, 616}, {1101, 35314}, {2148, 11127}, {2151, 2}, {2153, 16770}, {2154, 37779}, {2159, 11078}, {2307, 36929}, {3384, 634}, {6137, 21221}, {6149, 617}, {8739, 5905}, {17402, 7192}, {23870, 21294}, {34394, 192}, {35198, 628}, {36072, 54027}, {39152, 52367}, {44688, 3436}, {44718, 4329}, {46077, 5080}, {46112, 6360}, {51806, 622}
X(616) = X(i)-Ceva conjugate of X(j) for these (i,j): {30, 617}, {298, 2}, {621, 628}, {52194, 627}, {54556, 621}
X(616) = X(i)-cross conjugate of X(j) for these (i,j): {74, 8535}, {15768, 40578}, {40578, 2}
X(616) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3440}, {31, 19776}, {2151, 40158}
X(616) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 19776}, {3, 3440}, {15, 46060}, {618, 34296}, {40578, 40158}
X(616) = crosspoint of X(99) and X(57579)
X(616) = crossdifference of every pair of points on line {6137, 14398}
X(616) = barycentric product X(i)*X(j) for these {i,j}: {75, 19298}, {298, 40578}, {14922, 39132}, {15768, 40707}
X(616) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 19776}, {6, 3440}, {13, 40158}, {396, 34296}, {15768, 396}, {19298, 1}, {40578, 13}, {40580, 46060}
X(616) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 35750, 35749}, {2, 35751, 35750}, {2, 51482, 59378}, {3, 634, 628}, {5, 13103, 59394}, {13, 618, 2}, {13, 5463, 618}, {13, 22489, 35019}, {13, 36770, 6669}, {16, 3643, 2}, {69, 376, 617}, {74, 6148, 617}, {141, 42943, 11300}, {618, 6669, 36770}, {624, 16242, 2}, {1272, 12383, 617}, {1494, 16163, 617}, {2992, 14368, 628}, {3098, 7811, 617}, {5463, 6779, 22687}, {5463, 22489, 36767}, {5463, 35750, 59378}, {5478, 36765, 3091}, {5979, 5980, 2}, {6115, 23006, 5335}, {6302, 6306, 5463}, {6669, 36770, 2}, {6772, 9115, 37641}, {7811, 11128, 69}, {11006, 45772, 617}, {11078, 11131, 2}, {12942, 18974, 388}, {12952, 13076, 497}, {13084, 50858, 2}, {14907, 54173, 617}, {16530, 46855, 40694}, {20425, 52650, 59397}, {32833, 46264, 617}, {33440, 33441, 2}, {35019, 47865, 13}, {35749, 59378, 51482}, {35751, 36769, 2}, {35752, 36768, 2}, {36766, 47859, 18582}, {36767, 47865, 2}, {40334, 42973, 33560}, {40709, 41887, 2}, {47363, 47364, 624}, {51898, 51899, 617}


X(617) = ANTICOMPLEMENT OF X(14)

Barycentrics    2*S^2 + (a^2 - b^2 - c^2)*(3*a^2 - Sqrt[3]*S) : :
X(617) = 3 X[2] - 4 X[619], 5 X[2] - 4 X[5460], 9 X[2] - 8 X[6670], 7 X[2] - 6 X[22490], 21 X[2] - 16 X[35020], 4 X[2] - X[36327], X[2] + 2 X[36329], 5 X[2] - 2 X[36330], 2 X[2] + X[36331], 7 X[2] - 4 X[47866], X[2] - 4 X[47867], 13 X[2] - 12 X[48312], 4 X[2] - 3 X[59379], 5 X[14] - 6 X[5460], X[14] - 3 X[5464], 3 X[14] - 4 X[6670], and many others

Let OA be the circle centered at the A-vertex of the inner Napoleon triangle and passing through A; define OB and OC cyclically. X(617) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

X(617) lies on the Kiepert circumhyperbola of the anticomplementary triangle, the cubics K001, K046b, K066a, K264b, K438b, K753, K860, K867a, K1053b, the curves Q051, Q072, Q074, and these lines: {1, 8481}, {2, 14}, {3, 298}, {4, 628}, {5, 13102}, {6, 11300}, {8, 12780}, {10, 9900}, {13, 148}, {16, 533}, {17, 33413}, {20, 634}, {22, 9915}, {30, 299}, {32, 37173}, {61, 51860}, {63, 1277}, {69, 74}, {76, 54485}, {98, 60252}, {100, 12336}, {115, 11488}, {140, 59384}, {141, 11299}, {145, 7974}, {147, 5979}, {193, 51203}, {194, 3105}, {302, 549}, {303, 381}, {388, 12941}, {395, 19781}, {396, 5254}, {398, 11308}, {399, 8492}, {484, 8502}, {485, 35850}, {486, 35851}, {487, 52399}, {488, 52400}, {489, 2044}, {490, 2043}, {491, 36455}, {492, 36437}, {497, 12951}, {524, 35932}, {530, 8591}, {532, 36968}, {543, 22513}, {599, 35931}, {618, 6777}, {624, 36970}, {631, 6774}, {635, 5238}, {636, 42157}, {671, 43542}, {1138, 5674}, {1157, 8496}, {1263, 8437}, {1270, 6269}, {1271, 6271}, {1606, 34008}, {1656, 20253}, {1992, 51015}, {2076, 35918}, {2132, 8536}, {2133, 5678}, {2782, 6770}, {2896, 3643}, {2903, 11127}, {2926, 14368}, {2975, 22774}, {3065, 8433}, {3068, 49210}, {3069, 49211}, {3085, 10061}, {3086, 10077}, {3090, 16002}, {3091, 5479}, {3096, 43482}, {3146, 36962}, {3241, 50852}, {3434, 12921}, {3436, 12931}, {3440, 8528}, {3448, 36247}, {3465, 8499}, {3466, 8436}, {3481, 8498}, {3484, 8529}, {3522, 41021}, {3523, 21157}, {3524, 51387}, {3525, 20416}, {3534, 33610}, {3543, 41043}, {3545, 25164}, {3564, 44250}, {3616, 11706}, {3648, 5700}, {3679, 50850}, {3830, 33612}, {3832, 59395}, {4048, 35917}, {4240, 12792}, {5318, 33458}, {5321, 44382}, {5335, 6783}, {5459, 41135}, {5463, 52695}, {5469, 6669}, {5471, 11489}, {5552, 49146}, {5601, 12470}, {5602, 12471}, {5617, 9735}, {5623, 8455}, {5667, 8489}, {5669, 8462}, {5672, 7325}, {5682, 8431}, {5858, 11481}, {5859, 33626}, {5863, 33624}, {5980, 11177}, {5982, 21156}, {6194, 25187}, {6321, 59394}, {6462, 12988}, {6463, 12989}, {6672, 16268}, {6695, 42991}, {6771, 14651}, {6772, 43449}, {6775, 7739}, {6778, 20094}, {7164, 8509}, {7165, 8484}, {7327, 8523}, {7328, 8526}, {7329, 8506}, {7492, 13858}, {7585, 19076}, {7586, 19075}, {7685, 54139}, {7784, 11304}, {7787, 12204}, {7795, 37172}, {7799, 36756}, {7809, 11132}, {7865, 22512}, {7898, 34541}, {8172, 8456}, {8175, 8471}, {8182, 9886}, {8439, 8442}, {8446, 8469}, {8452, 8470}, {8454, 16882}, {8461, 8479}, {8486, 8513}, {8487, 8519}, {8493, 8516}, {8494, 8521}, {8703, 33608}, {8972, 13916}, {9113, 41409}, {9143, 35315}, {9166, 31696}, {9742, 9750}, {9763, 53431}, {10410, 30486}, {10527, 49145}, {10528, 13104}, {10529, 13106}, {10616, 62233}, {10646, 40900}, {10723, 41061}, {11001, 33611}, {11160, 51014}, {11289, 22236}, {11290, 42147}, {11296, 16941}, {11298, 11485}, {11301, 42116}, {11302, 42975}, {11307, 36836}, {11480, 53442}, {11645, 51018}, {12188, 47610}, {12213, 54298}, {13084, 22493}, {13172, 22509}, {13188, 36995}, {13941, 13981}, {14137, 42999}, {14145, 61634}, {14683, 37753}, {14921, 41473}, {14922, 33500}, {15300, 35750}, {15454, 19776}, {15682, 33609}, {15702, 32909}, {15928, 46470}, {16242, 34508}, {16529, 40693}, {16530, 30559}, {16644, 22893}, {16645, 53463}, {16809, 50860}, {16965, 22113}, {18582, 23004}, {19106, 33623}, {19107, 21360}, {19708, 33613}, {19778, 37850}, {20252, 38732}, {20426, 44223}, {22114, 42149}, {22494, 42429}, {22496, 45880}, {22510, 42152}, {22577, 47865}, {22579, 59373}, {22603, 32492}, {22632, 32495}, {22845, 42158}, {22891, 33414}, {23019, 31168}, {25166, 37832}, {26394, 48458}, {26418, 48459}, {26494, 49376}, {26503, 49375}, {27550, 44010}, {30471, 48655}, {31145, 50851}, {31684, 42125}, {31709, 43403}, {32036, 52203}, {33352, 35731}, {33412, 47520}, {33474, 42501}, {33475, 43104}, {33561, 40335}, {33622, 42528}, {34509, 36969}, {35303, 37785}, {36326, 49808}, {36352, 49857}, {36366, 49858}, {36959, 47066}, {36993, 44465}, {37351, 42117}, {37352, 42912}, {37824, 52688}, {40341, 42625}, {41753, 61317}, {42036, 43540}, {42063, 43555}, {43541, 56056}, {43543, 60272}, {44016, 49952}, {45508, 48724}, {45509, 48725}, {49864, 49909}, {49880, 49921}, {50977, 51016}, {53440, 62200}, {53469, 62197}, {54561, 60253}

X(617) = midpoint of X(i) and X(j) for these {i,j}: {5464, 36329}, {36331, 51483}
X(617) = reflection of X(i) in X(j) for these {i,j}: {1, 51115}, {2, 5464}, {4, 5613}, {6, 51160}, {8, 12780}, {13, 32552}, {14, 619}, {20, 5474}, {69, 51013}, {145, 7974}, {147, 36776}, {148, 13}, {193, 51203}, {299, 52193}, {599, 51205}, {616, 99}, {621, 5978}, {622, 299}, {627, 14144}, {1992, 51015}, {3146, 36962}, {3180, 22997}, {3181, 16}, {3241, 50852}, {3543, 41043}, {3679, 50850}, {4240, 12792}, {5321, 44382}, {5464, 47867}, {6773, 3}, {6777, 618}, {8591, 9114}, {9900, 10}, {10654, 22689}, {10723, 41061}, {11160, 51014}, {12188, 47610}, {13102, 5}, {14683, 37753}, {20426, 44223}, {22496, 45880}, {22577, 47865}, {31145, 50851}, {33442, 6303}, {33443, 6307}, {36327, 51483}, {36330, 5460}, {36970, 624}, {37785, 35303}, {48655, 51872}, {49036, 49307}, {49037, 49308}, {51483, 2}, {51484, 8594}, {51485, 35932}, {54139, 7685}
X(617) = isogonal conjugate of X(3441)
X(617) = isotomic conjugate of X(19777)
X(617) = anticomplement of X(14)
X(617) = circumcircle-inverse of X(14369)
X(617) = orthoptic-circle-of-the-Steiner-circumellipse-inverse of X(5978)
X(617) = circumcircle-of-inner-Napoleon-triangle-inverse of X(623)
X(617) = circumcircle of outer Napoleon triangle-inverse of X(619)
X(617) = antigonal image of X(39133)
X(617) = anticomplement of the isogonal conjugate of X(16)
X(617) = anticomplement of the isotomic conjugate of X(299)
X(617) = isotomic conjugate of the anticomplement of X(40579)
X(617) = anticomplementary isogonal conjugate of X(622)
X(617) = psi-transform of X(618)
X(617) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 622}, {16, 8}, {31, 3181}, {48, 19773}, {163, 23871}, {299, 6327}, {471, 21270}, {1095, 617}, {1101, 35315}, {2148, 11126}, {2152, 2}, {2153, 37779}, {2154, 16771}, {2159, 11092}, {3375, 633}, {6138, 21221}, {6149, 616}, {8740, 5905}, {17403, 7192}, {23871, 21294}, {34395, 192}, {35199, 627}, {36073, 54025}, {39153, 52367}, {44689, 3436}, {44719, 4329}, {46073, 5080}, {46113, 6360}, {51805, 621}
X(617) = X(i)-Ceva conjugate of X(j) for these (i,j): {30, 616}, {299, 2}, {622, 627}, {52193, 628}, {54557, 622}
X(617) = X(i)-cross conjugate of X(j) for these (i,j): {74, 8536}, {15769, 40579}, {40579, 2}
X(617) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3441}, {31, 19777}, {2152, 40159}
X(617) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 19777}, {3, 3441}, {16, 46061}, {619, 34295}, {40579, 40159}
X(617) = crosspoint of X(99) and X(57580)
X(617) = crossdifference of every pair of points on line {6138, 14398}
X(617) = barycentric product X(i)*X(j) for these {i,j}: {75, 19299}, {298, 41889}, {299, 40579}, {14921, 39133}, {15769, 40706}
X(617) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 19777}, {6, 3441}, {14, 40159}, {395, 34295}, {15769, 395}, {19299, 1}, {40579, 14}, {40581, 46061}, {41889, 13}
X(617) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 36329, 36331}, {2, 36331, 36327}, {2, 51483, 59379}, {3, 633, 627}, {5, 13102, 59396}, {14, 619, 2}, {14, 5464, 619}, {14, 22490, 35020}, {15, 3642, 2}, {69, 376, 616}, {74, 6148, 616}, {141, 42942, 11299}, {623, 16241, 2}, {1272, 12383, 616}, {1494, 16163, 616}, {2993, 14369, 627}, {3098, 7811, 616}, {5464, 6780, 22689}, {5464, 36331, 59379}, {5978, 5981, 2}, {6114, 23013, 5334}, {6303, 6307, 5464}, {6775, 9117, 37640}, {7811, 11129, 69}, {11006, 45772, 616}, {11092, 11130, 2}, {12941, 18975, 388}, {12951, 13075, 497}, {13083, 50855, 2}, {14907, 54173, 616}, {16529, 46854, 40693}, {20426, 44223, 59398}, {32833, 46264, 616}, {33442, 33443, 2}, {35020, 47866, 14}, {36327, 59379, 51483}, {36329, 47867, 2}, {40335, 42972, 33561}, {40710, 41888, 2}, {47361, 47362, 623}, {47860, 60069, 18581}, {51898, 51899, 616}


X(618) = COMPLEMENT OF X(13)

Barycentrics    Sqrt[3]*(b^2 + c^2)*S + 5*a^2*SA + 2*SB*SC : :
X(618) = 3 X[2] + X[616], 5 X[2] - 3 X[22489], 9 X[2] - 4 X[35019], 13 X[2] - X[35749], 11 X[2] + X[35750], 5 X[2] + X[35751], 7 X[2] - X[35752], X[2] + 5 X[36767], X[2] + 2 X[36768], 2 X[2] + X[36769], 3 X[2] - 5 X[36770], 4 X[2] - X[47865], 4 X[2] - 3 X[48311], 5 X[2] - X[51482], 7 X[2] - 3 X[59378], 2 X[13] - 3 X[5459], X[13] + 3 X[5463], and many others

X(618) lies on the Kiepert circumhyperbola of the medial triangle, the cubics K046a, K066a, K261b, K438a, K859b, K900, K1297, the curves Q004, Q123, and these lines: {1, 12781}, {2, 13}, {3, 635}, {4, 5473}, {5, 629}, {6, 11301}, {8, 7975}, {10, 51114}, {11, 13076}, {12, 18974}, {14, 99}, {15, 298}, {17, 33387}, {18, 11121}, {20, 36961}, {30, 623}, {39, 395}, {55, 12952}, {56, 12942}, {61, 627}, {62, 6694}, {69, 51200}, {76, 47067}, {83, 12205}, {98, 21157}, {114, 41023}, {115, 6670}, {128, 46652}, {140, 630}, {141, 542}, {148, 5469}, {299, 1078}, {376, 38412}, {396, 532}, {398, 33464}, {427, 12142}, {465, 6509}, {470, 31687}, {471, 35714}, {498, 10062}, {499, 10078}, {511, 33480}, {512, 33490}, {524, 42912}, {531, 2482}, {543, 5460}, {551, 50847}, {590, 13917}, {597, 42634}, {599, 13083}, {615, 13982}, {617, 6777}, {621, 21359}, {631, 6770}, {632, 20415}, {633, 5238}, {639, 34552}, {640, 34551}, {641, 18585}, {642, 15765}, {671, 9116}, {958, 12932}, {1080, 14538}, {1125, 11705}, {1153, 33475}, {1376, 12337}, {1649, 16234}, {1650, 12793}, {1656, 13103}, {1698, 9901}, {1992, 51011}, {2782, 6774}, {2902, 44718}, {2925, 3130}, {3068, 19074}, {3069, 19073}, {3090, 59394}, {3091, 59393}, {3096, 9982}, {3106, 10335}, {3180, 16962}, {3241, 50848}, {3412, 22844}, {3448, 37752}, {3479, 15802}, {3523, 41020}, {3526, 59383}, {3534, 33619}, {3589, 42913}, {3618, 59409}, {3628, 16001}, {3629, 42633}, {3666, 36669}, {3679, 50849}, {3763, 11302}, {3830, 33621}, {5055, 25154}, {5182, 51203}, {5237, 11289}, {5318, 33560}, {5334, 36772}, {5461, 48312}, {5464, 41134}, {5470, 14061}, {5472, 23302}, {5474, 21166}, {5479, 23698}, {5490, 49374}, {5491, 49373}, {5552, 13105}, {5590, 6268}, {5591, 6270}, {5599, 12472}, {5600, 12473}, {5613, 15561}, {5615, 33391}, {5664, 11618}, {5745, 49571}, {5858, 11485}, {5943, 53048}, {5976, 6109}, {5978, 8290}, {5983, 22510}, {6105, 11092}, {6110, 46060}, {6117, 11094}, {6292, 37341}, {6295, 51013}, {6301, 13929}, {6305, 13876}, {6321, 59402}, {6337, 10654}, {6626, 21898}, {6673, 10611}, {6674, 22846}, {6695, 16773}, {6772, 11297}, {6773, 36776}, {7496, 13859}, {7684, 52266}, {7710, 9749}, {7749, 53452}, {7831, 10646}, {7853, 37352}, {7859, 43484}, {7911, 11303}, {8222, 12990}, {8223, 12991}, {8591, 59379}, {8724, 22689}, {9112, 11488}, {9114, 51483}, {9761, 9885}, {10061, 10089}, {10077, 10086}, {10124, 32907}, {10187, 33415}, {10527, 13107}, {10576, 35753}, {10577, 35754}, {10723, 59395}, {11001, 33616}, {11119, 11581}, {11120, 46076}, {11160, 51201}, {11290, 22511}, {11295, 16942}, {11296, 42625}, {11298, 11481}, {11304, 31710}, {11305, 42155}, {11309, 42156}, {11311, 36843}, {11312, 43239}, {11813, 33396}, {12204, 39652}, {13075, 15452}, {13172, 59396}, {13188, 59384}, {13350, 25560}, {13821, 15764}, {14540, 52688}, {14905, 33273}, {14921, 38993}, {14971, 31695}, {14972, 36209}, {15300, 31696}, {15349, 49610}, {15682, 33614}, {15693, 33618}, {15701, 33620}, {15719, 33617}, {15768, 34296}, {15810, 35303}, {16256, 35314}, {16336, 33497}, {16644, 34509}, {16967, 23005}, {19075, 19109}, {19076, 19108}, {19776, 45778}, {19924, 51161}, {20377, 42581}, {20416, 51524}, {20425, 59403}, {21360, 25235}, {22114, 36781}, {22493, 51484}, {22494, 43199}, {22496, 42799}, {22507, 38750}, {22577, 41135}, {22580, 47352}, {22691, 25184}, {22737, 33385}, {22797, 61575}, {22911, 33383}, {22997, 33377}, {23234, 41043}, {24206, 44223}, {26359, 48456}, {26360, 48457}, {26361, 49034}, {26362, 49035}, {26363, 49143}, {26364, 49144}, {27550, 62651}, {31274, 42501}, {31693, 42941}, {31694, 43101}, {33393, 51855}, {33395, 51853}, {33404, 33412}, {33410, 33418}, {33411, 41040}, {33458, 42124}, {33501, 59710}, {33530, 37847}, {33615, 36318}, {33622, 49819}, {33626, 49868}, {34229, 60253}, {34541, 50860}, {34754, 40900}, {34834, 41888}, {35022, 42117}, {35697, 42129}, {35931, 50855}, {36329, 42795}, {36366, 49913}, {36386, 49953}, {36519, 41061}, {36521, 47866}, {36763, 40693}, {36764, 37640}, {36780, 44460}, {36969, 40334}, {37170, 42086}, {37171, 42910}, {37177, 42149}, {37464, 41045}, {37512, 53463}, {38738, 41060}, {39150, 51583}, {39378, 46059}, {40580, 47119}, {41000, 61371}, {41035, 41071}, {42035, 43543}, {42062, 43548}, {42100, 49901}, {42162, 47518}, {42496, 48313}, {43102, 51278}, {44667, 62653}, {45472, 49305}, {45473, 49306}, {47066, 51753}, {47355, 59410}, {47361, 62561}, {47362, 62560}, {49210, 49267}, {49211, 49266}, {49803, 49829}, {49852, 49878}, {49897, 49919}, {49941, 49959}, {54140, 59397}

X(618) = midpoint of X(i) and X(j) for these {i,j}: {1, 12781}, {2, 5463}, {3, 5617}, {4, 5473}, {6, 51010}, {8, 7975}, {10, 51114}, {13, 616}, {14, 99}, {15, 298}, {16, 5979}, {18, 14145}, {20, 36961}, {69, 51200}, {98, 61634}, {141, 51159}, {299, 22998}, {376, 41042}, {396, 52194}, {551, 50847}, {597, 51202}, {599, 51012}, {617, 6777}, {619, 32553}, {621, 36967}, {622, 6779}, {627, 36782}, {671, 9116}, {1080, 14538}, {1650, 12793}, {1992, 51011}, {3241, 50848}, {3448, 37752}, {3643, 22687}, {3679, 50849}, {5459, 36769}, {6298, 6582}, {6302, 6306}, {6773, 36776}, {8595, 50858}, {9114, 51483}, {9115, 51388}, {9761, 9885}, {11160, 51201}, {12337, 12922}, {12932, 22773}, {15300, 31696}, {16256, 35314}, {19776, 45778}, {22114, 36781}, {22493, 51484}, {35751, 51482}, {35931, 50855}, {36780, 44460}, {38738, 41060}, {47611, 51872}
reflection of X(i) in X(j) for these {i,j}: {13, 6669}, {115, 6670}, {396, 6671}, {619, 620}, {623, 44383}, {5318, 33560}, {5459, 2}, {5463, 36768}, {5478, 5}, {6108, 6672}, {6771, 140}, {7684, 52266}, {10611, 6673}, {11705, 1125}, {14136, 6694}, {16001, 20252}, {20252, 3628}, {22797, 61575}, {22846, 6674}, {31710, 33561}, {32552, 619}, {36769, 5463}, {47865, 5459}, {53048, 5943}
X(618) = isogonal conjugate of X(16459)
X(618) = isotomic conjugate of X(11119)
X(618) = complement of X(13)
X(618) = anticomplement of X(6669)
X(618) = orthoptic-circle-of-the-Steiner-inellipse-inverse of X(5979)
X(618) = circumcircle-of-inner-Napoleon-triangle-inverse of X(616)
X(618) = circumcircle-of-outer-Napoleon-triangle-inverse of X(622)
X(618) = complement of the isogonal conjugate of X(15)
X(618) = complement of the isotomic conjugate of X(298)
X(618) = medial-isogonal conjugate of X(623)
X(618) = psi-transform of X(617)
X(618) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 623}, {15, 10}, {31, 396}, {163, 23870}, {298, 2887}, {470, 20305}, {1094, 618}, {2151, 2}, {2152, 41888}, {2154, 3580}, {2624, 43962}, {3384, 636}, {6137, 8287}, {6149, 619}, {8739, 226}, {17402, 4369}, {23870, 21253}, {32679, 46651}, {34394, 37}, {35198, 630}, {35199, 33526}, {36072, 54027}, {39152, 25639}, {44688, 1329}, {44700, 20308}, {44718, 18589}, {46077, 3814}, {46112, 1214}, {51806, 624}, {60010, 34846}
X(618) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 396}, {14, 533}, {99, 23870}, {302, 33529}
X(618) = X(i)-isoconjugate of X(j) for these (i,j): {1, 16459}, {19, 47481}, {31, 11119}, {2153, 2981}, {2380, 51805}
X(618) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 11119}, {3, 16459}, {6, 47481}, {15, 39262}, {396, 2}, {618, 13}, {11127, 38403}, {14921, 299}, {30465, 523}, {30471, 40707}, {33526, 16}, {40580, 2981}, {43961, 62631}
X(618) = cevapoint of X(396) and X(15768)
X(618) = crosspoint of X(2) and X(298)
X(618) = crosssum of X(6) and X(3457)
X(618) = crossdifference of every pair of points on line {6137, 11081}
X(618) = barycentric product X(i)*X(j) for these {i,j}: {14, 14922}, {15, 41000}, {99, 35443}, {298, 396}, {301, 19294}, {470, 52194}, {532, 11092}, {7799, 61371}, {8014, 11129}, {11131, 43085}, {23870, 35314}, {56514, 59198}
X(618) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 11119}, {3, 47481}, {6, 16459}, {15, 2981}, {298, 40707}, {396, 13}, {463, 8737}, {470, 38428}, {532, 11078}, {3458, 11084}, {6671, 8838}, {8014, 11080}, {8603, 34321}, {8739, 51446}, {9115, 52039}, {11086, 2380}, {11092, 11117}, {11131, 38403}, {14446, 23283}, {14922, 299}, {15768, 40578}, {15802, 51276}, {16256, 36316}, {17402, 10409}, {19294, 16}, {23870, 62631}, {34296, 40158}, {35314, 23895}, {35329, 5995}, {35443, 523}, {36304, 11139}, {40580, 39262}, {41000, 300}, {41620, 21466}, {52194, 40709}, {52867, 34325}, {58802, 35443}, {59209, 10217}, {61371, 1989}
X(618) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 13, 6669}, {2, 616, 13}, {2, 622, 37832}, {2, 3643, 624}, {2, 5459, 48311}, {2, 5979, 6115}, {2, 5980, 6108}, {2, 8595, 40671}, {2, 11131, 41887}, {2, 13084, 45880}, {2, 16242, 6672}, {2, 33440, 6302}, {2, 33441, 6306}, {2, 36767, 36768}, {2, 36768, 36769}, {2, 36769, 47865}, {2, 51482, 22489}, {13, 5463, 616}, {13, 6669, 5459}, {13, 36770, 2}, {14, 50859, 302}, {15, 6782, 47863}, {16, 36766, 6115}, {62, 11307, 6694}, {140, 636, 630}, {141, 549, 619}, {298, 30471, 11129}, {302, 11299, 14}, {396, 9115, 41620}, {549, 47611, 5092}, {590, 49208, 13917}, {615, 49209, 13982}, {616, 36770, 6669}, {620, 32553, 32552}, {631, 6770, 21156}, {1656, 13103, 59401}, {5092, 7880, 619}, {5463, 16242, 5980}, {5463, 22489, 35751}, {5463, 36766, 5979}, {5463, 36770, 13}, {5473, 36765, 4}, {6298, 13084, 6582}, {9116, 22490, 671}, {11304, 37835, 33561}, {16530, 36782, 61}, {16644, 41745, 47857}, {18582, 23006, 47861}, {21157, 61634, 98}, {21359, 36967, 621}, {22489, 35751, 51482}, {23303, 37351, 6670}, {33440, 33441, 5463}, {41631, 41633, 41620}, {41887, 46833, 2}, {47088, 47089, 619}, {47363, 47364, 622}, {47865, 48311, 5459}, {51483, 52695, 9114}


X(619) = COMPLEMENT OF X(14)

Barycentrics    Sqrt[3]*(b^2 + c^2)*S - 5*a^2*SA - 2*SB*SC : :
X(619) = 3 X[2] + X[617], 5 X[2] - 3 X[22490], 9 X[2] - 4 X[35020], 13 X[2] - X[36327], 5 X[2] + X[36329], 7 X[2] - X[36330], 11 X[2] + X[36331], 4 X[2] - X[47866], 2 X[2] + X[47867], 4 X[2] - 3 X[48312], 5 X[2] - X[51483], 7 X[2] - 3 X[59379], 2 X[14] - 3 X[5460], X[14] + 3 X[5464], 5 X[14] - 9 X[22490], and many others

X(619) lies on the Kiepert circumhyperbola of the medial triangle, the cubics K046b, K066b, K261a, K438b, K859a, K900, K1297, the curves Q004, Q123, anjd these lines: {1, 12780}, {2, 14}, {3, 636}, {4, 5474}, {5, 630}, {6, 11302}, {8, 7974}, {10, 51115}, {11, 13075}, {12, 18975}, {13, 99}, {16, 299}, {17, 11122}, {18, 33386}, {20, 36962}, {30, 624}, {39, 396}, {55, 12951}, {56, 12941}, {61, 6695}, {62, 628}, {69, 51203}, {76, 47069}, {83, 12204}, {98, 21156}, {114, 41022}, {115, 6669}, {128, 46653}, {140, 629}, {141, 542}, {148, 5470}, {298, 1078}, {376, 41043}, {383, 14539}, {395, 533}, {397, 33465}, {427, 12141}, {466, 6509}, {470, 35715}, {471, 31688}, {498, 10061}, {499, 10077}, {511, 33481}, {512, 33491}, {524, 42913}, {530, 2482}, {543, 5459}, {551, 50850}, {590, 13916}, {597, 42633}, {599, 13084}, {615, 13981}, {616, 6778}, {622, 21360}, {631, 6773}, {632, 20416}, {634, 5237}, {639, 34551}, {640, 34552}, {641, 15765}, {642, 18585}, {671, 9114}, {958, 12931}, {1125, 11706}, {1153, 33474}, {1376, 12336}, {1649, 16233}, {1650, 12792}, {1656, 13102}, {1698, 9900}, {1992, 51014}, {2782, 6771}, {2903, 44719}, {2926, 3129}, {3068, 19076}, {3069, 19075}, {3090, 59396}, {3091, 59395}, {3096, 9981}, {3107, 10335}, {3181, 16963}, {3241, 50851}, {3411, 22845}, {3448, 37753}, {3480, 15778}, {3523, 41021}, {3526, 59384}, {3534, 33618}, {3589, 42912}, {3628, 16002}, {3629, 42634}, {3666, 36668}, {3679, 50852}, {3763, 11301}, {3830, 33620}, {5055, 25164}, {5182, 51200}, {5238, 11290}, {5321, 33561}, {5461, 48311}, {5463, 41134}, {5469, 14061}, {5471, 23303}, {5473, 21166}, {5478, 23698}, {5490, 49376}, {5491, 49375}, {5552, 13104}, {5590, 6269}, {5591, 6271}, {5599, 12470}, {5600, 12471}, {5611, 33390}, {5617, 15561}, {5664, 11617}, {5745, 49572}, {5859, 11486}, {5943, 53049}, {5976, 6108}, {5979, 8290}, {5982, 22511}, {6104, 11078}, {6111, 46061}, {6116, 11093}, {6292, 37340}, {6300, 13928}, {6304, 13875}, {6321, 59401}, {6337, 10653}, {6582, 51010}, {6626, 21869}, {6673, 22891}, {6674, 10612}, {6694, 16772}, {6770, 61634}, {6775, 11298}, {6777, 36770}, {7496, 13858}, {7685, 52263}, {7710, 9750}, {7749, 53463}, {7831, 10645}, {7853, 37351}, {7859, 43483}, {7911, 11304}, {8222, 12988}, {8223, 12989}, {8591, 59378}, {8724, 22687}, {9113, 11489}, {9116, 51482}, {9763, 9886}, {10062, 10089}, {10078, 10086}, {10124, 32909}, {10188, 33414}, {10527, 13106}, {10576, 35850}, {10577, 35851}, {10723, 59393}, {10754, 59409}, {11001, 33617}, {11119, 46072}, {11120, 11582}, {11160, 51204}, {11289, 22510}, {11295, 42626}, {11296, 16943}, {11297, 11480}, {11303, 31709}, {11306, 42154}, {11310, 42153}, {11311, 43238}, {11312, 36836}, {11813, 33397}, {12205, 39652}, {13076, 15452}, {13172, 59394}, {13188, 59383}, {13349, 25559}, {13701, 15764}, {14541, 52689}, {14904, 33273}, {14922, 38994}, {14971, 31696}, {14972, 36208}, {15300, 31695}, {15349, 49611}, {15682, 33615}, {15693, 33619}, {15701, 33621}, {15719, 33616}, {15769, 34295}, {15810, 35304}, {16255, 35315}, {16336, 33496}, {16645, 34508}, {16966, 23004}, {19073, 19109}, {19074, 19108}, {19777, 45779}, {19924, 51162}, {20378, 42580}, {20415, 51524}, {20426, 59404}, {21359, 25236}, {22493, 43200}, {22494, 51485}, {22495, 42800}, {22509, 38750}, {22578, 41135}, {22579, 47352}, {22692, 25188}, {22736, 33384}, {22796, 61575}, {22866, 33382}, {22998, 33376}, {23234, 41042}, {24206, 52650}, {26359, 48458}, {26360, 48459}, {26361, 49036}, {26362, 49037}, {26363, 49145}, {26364, 49146}, {27551, 62651}, {31274, 42500}, {31693, 43104}, {31694, 42940}, {33392, 51852}, {33394, 51854}, {33405, 33413}, {33410, 41041}, {33411, 33419}, {33459, 42121}, {33529, 37849}, {33614, 36320}, {33624, 49816}, {33627, 49865}, {34229, 60252}, {34540, 50859}, {34755, 40901}, {34834, 41887}, {35022, 42118}, {35693, 42132}, {35738, 35759}, {35751, 42796}, {35932, 50858}, {36368, 49912}, {36388, 49952}, {36519, 41060}, {36521, 47865}, {36970, 40335}, {37170, 42911}, {37171, 42085}, {37178, 42152}, {37463, 41044}, {37512, 53452}, {38738, 41061}, {39151, 51583}, {39377, 46058}, {40581, 47118}, {41001, 61370}, {41034, 41070}, {42036, 43542}, {42063, 43549}, {42099, 49902}, {42159, 47520}, {42497, 48314}, {43103, 51279}, {44250, 44667}, {44666, 62654}, {45472, 49307}, {45473, 49308}, {46709, 61719}, {47068, 51754}, {47363, 62561}, {47364, 62560}, {49208, 49267}, {49209, 49266}, {49802, 49828}, {49851, 49877}, {49898, 49920}, {49942, 49960}, {54141, 59398}

X(619) = midpoint of X(i) and X(j) for these {i,j}: {1, 12780}, {2, 5464}, {3, 5613}, {4, 5474}, {6, 51013}, {8, 7974}, {10, 51115}, {13, 99}, {14, 617}, {15, 5978}, {16, 299}, {17, 14144}, {20, 36962}, {69, 51203}, {98, 36776}, {141, 51160}, {298, 22997}, {376, 41043}, {383, 14539}, {395, 52193}, {551, 50850}, {597, 51205}, {599, 51015}, {616, 6778}, {618, 32552}, {621, 6780}, {622, 36968}, {671, 9114}, {1650, 12792}, {1992, 51014}, {3241, 50851}, {3448, 37753}, {3642, 22689}, {3679, 50852}, {5460, 47867}, {6295, 6299}, {6303, 6307}, {6770, 61634}, {8594, 50855}, {9116, 51482}, {9117, 51387}, {9763, 9886}, {11160, 51204}, {12336, 12921}, {12931, 22774}, {15300, 31695}, {16255, 35315}, {19777, 45779}, {22494, 51485}, {35932, 50858}, {36329, 51483}, {38738, 41061}, {47610, 51872}
X(619) = reflection of X(i) in X(j) for these {i,j}: {14, 6670}, {115, 6669}, {395, 6672}, {618, 620}, {624, 44382}, {5321, 33561}, {5460, 2}, {5479, 5}, {6109, 6671}, {6774, 140}, {7685, 52263}, {10612, 6674}, {11706, 1125}, {14137, 6695}, {16002, 20253}, {20253, 3628}, {22796, 61575}, {22891, 6673}, {31709, 33560}, {32553, 618}, {47866, 5460}, {47867, 5464}, {53049, 5943}
X(619) = isogonal conjugate of X(16460)
X(619) = isotomic conjugate of X(11120)
X(619) = complement of X(14)
X(619) = anticomplement of X(6670)
X(619) = orthoptic-circle-of-the-Steiner-inellipse-inverse of X(5978)
X(619) = circumcircle-of-inner-Napoleon-triangle-inverse of X(621)
X(619) = circumcircle-of-outer-Napoleon-triangle-inverse of X(617)
X(619) = complement of the isogonal conjugate of X(16)
X(619) = complement of the isotomic conjugate of X(299)
X(619) = medial-isogonal conjugate of X(624)
X(619) = psi-transform of X(616)
X(619) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 624}, {16, 10}, {31, 395}, {163, 23871}, {299, 2887}, {471, 20305}, {1095, 619}, {2151, 41887}, {2152, 2}, {2153, 3580}, {2154, 62690}, {2624, 43961}, {3375, 635}, {6138, 8287}, {6149, 618}, {8740, 226}, {17403, 4369}, {23871, 21253}, {32679, 46650}, {34395, 37}, {35198, 33527}, {35199, 629}, {36073, 54025}, {39153, 25639}, {44689, 1329}, {44701, 20308}, {44719, 18589}, {46073, 3814}, {46113, 1214}, {51805, 623}, {60009, 34846}
X(619) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 395}, {13, 532}, {99, 23871}, {303, 33530}
X(619) = X(i)-isoconjugate of X(j) for these (i,j): {1, 16460}, {19, 47482}, {31, 11120}, {2154, 6151}, {2381, 51806}
X(619) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 11120}, {3, 16460}, {6, 47482}, {16, 39261}, {395, 2}, {619, 14}, {11126, 38404}, {14922, 298}, {30468, 523}, {30472, 40706}, {33527, 15}, {40581, 6151}, {43962, 62632}
X(619) = cevapoint of X(395) and X(15769)
X(619) = crosspoint of X(2) and X(299)
X(619) = crosssum of X(6) and X(3458)
X(619) = crossdifference of every pair of points on line {6138, 11086}
X(619) = barycentric product X(i)*X(j) for these {i,j}: {13, 14921}, {16, 41001}, {99, 35444}, {299, 395}, {300, 19295}, {471, 52193}, {533, 11078}, {7799, 61370}, {8015, 11128}, {11130, 43086}, {23871, 35315}, {56515, 59199}
X(619) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 11120}, {3, 47482}, {6, 16460}, {16, 6151}, {299, 40706}, {395, 14}, {462, 8738}, {471, 38427}, {533, 11092}, {3457, 11089}, {6672, 8836}, {8015, 11085}, {8604, 34322}, {8740, 51447}, {9117, 52040}, {11078, 11118}, {11081, 2381}, {11130, 38404}, {14447, 23284}, {14921, 298}, {15769, 40579}, {15778, 51269}, {16255, 36317}, {17403, 10410}, {19295, 15}, {23871, 62632}, {34295, 40159}, {35315, 23896}, {35330, 5994}, {35444, 523}, {36305, 11138}, {40581, 39261}, {41001, 301}, {41621, 21467}, {41889, 39133}, {52193, 40710}, {52868, 34326}, {58801, 35444}, {59210, 10218}, {61370, 1989}
X(619) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 14, 6670}, {2, 617, 14}, {2, 621, 37835}, {2, 3642, 623}, {2, 5460, 48312}, {2, 5978, 6114}, {2, 5981, 6109}, {2, 8594, 40672}, {2, 11092, 62690}, {2, 11130, 41888}, {2, 13083, 45879}, {2, 16241, 6671}, {2, 33442, 6303}, {2, 33443, 6307}, {2, 47867, 47866}, {2, 51483, 22490}, {13, 50860, 303}, {14, 5464, 617}, {14, 6670, 5460}, {15, 60069, 6114}, {16, 6783, 47864}, {61, 11308, 6695}, {140, 635, 629}, {141, 549, 618}, {299, 30472, 11128}, {303, 11300, 13}, {395, 9117, 41621}, {549, 47610, 5092}, {590, 49210, 13916}, {615, 49211, 13981}, {620, 32552, 32553}, {631, 6773, 21157}, {1656, 13102, 59402}, {5092, 7880, 618}, {5464, 16241, 5981}, {5464, 22490, 36329}, {5464, 60069, 5978}, {6299, 13083, 6295}, {9114, 22489, 671}, {11303, 37832, 33560}, {16645, 41746, 47858}, {18581, 23013, 47862}, {21156, 36776, 98}, {21360, 36968, 622}, {22490, 36329, 51483}, {23302, 37352, 6669}, {33442, 33443, 5464}, {41641, 41643, 41621}, {41888, 46834, 2}, {47088, 47089, 618}, {47361, 47362, 621}, {47866, 48312, 5460}, {51482, 52695, 9116}


X(620) = MIDPOINT OF X(618) AND X(619)

Trilinears    [4SAa2 - (b4 + c4)]/a : :
X(620) = 3*X(2) + X(99)

Let S = X(99). Let A' be the centroid of the triangle BCS, and define B' and C' cyclically. Let D' be the centroid of ABC. The four centroids form a quadrilateral homothetic to the quadrilateral ABCS. The center of homothety is X(620), which is the centroid of ABCS. (Randy Hutson, 9/23/2011)

Let A'B'C' and A″B″C″ be the (equilateral) antipedal triangles of X(13) and X(14), resp. Let A* be the midpoint of A' and A″, and define B* and C* cyclically. The triangle A*B*C* is homothetic to ABC, and the center of homothety is X(620). (Randy Hutson, February 10, 2016)

X(620) = center of the hyperbola H that is the locus of perspectors of circumconics centered at a point on line X(2)X(6), which is the locus of the X(2)-Ceva conjugate of P as P moves on line X(2)X(6). Also, H is the Kiepert hyperbola of the medial triangle; H is tangent to line X(2)X(6) at X(2), and H passes X(3), X(39) and X(114). H is also the bicevian conic of X(2) and X(99) (Randy Hutson, February 10, 2016)

X(620) is the center of the inellipse that is the barycentric square of line X(2)X(6). The Brianchon point (perspector) of the inellipse is X(4590). (Randy Hutson, October 15, 2018)

X(620) lies on these lines: 2,99   3,114   30,625   98,631   141,542   187,325   230,538

X(620) = midpoint of X(i) and X(j) for these (i,j): (3,114), (99,115), (187,325), (618,619)
X(620) = complement of X(115)
X(620) = X(187)-of-X(2)-Brocard-triangle
X(620) = X(230)-of-1st-Brocard-triangle
X(620) = isotomic conjugate of isogonal conjugate of X(20976)
X(620) = polar conjugate of isogonal conjugate of X(22085)
X(620) = centroid of X(2)X(3)X(114)X(2482)
X(620) = Kosnita(X(99),X(2)) point
X(620) = crosssum of intersections of 1st and 2nd Lemoine circles (and the circle {X(1687),X(1688),PU(1),PU(2)}})
X(620) = QA-P29 center (Complement of QA-P2 wrt the Diagonal triangle) of quadrangle ABCX(2) (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/57-qa-p29.html)


X(621) = ANTICOMPLEMENT OF X(15)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [sqr(3) SBSC + 2SA area]/a

Trilinears        F(15)/a - 2 sin(A + π/3) : F(15)/b - 2 sin(B + π/3) : F(15)/c - 2 sin(C + π/3)

Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(621) lies on these lines: 2,14   3,302   4,69   5,303   13,533   20,627   30,298   183,383   265,300   299,381   325,1080   343,472   394,473

X(621) = reflection of X(i) in X(j) for these (i,j): (15,623), (616,298), (622,316)
X(621) = isogonal conjugate of X(3438)
X(621) = isotomic conjugate of X(2992)
X(621) = anticomplement of X(15)
X(621) = anticomplementary conjugate of X(616)
X(621) = X(300)-Ceva conjugate of X(2)

X(621) = crosspoint of X(616) and X(628) wrt both the excentral and anticomplementary triangles

X(622) = ANTICOMPLEMENT OF X(16)

Barycentrics   Sqrt[3]*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) + 2*(a^2 - b^2 - c^2)*S : :
X(622) = 3 X[2] - 4 X[624], 9 X[2] - 8 X[6672], 9 X[2] - 10 X[40335], 5 X[2] - 4 X[45880], 13 X[2] - 12 X[48314], 3 X[16] - 4 X[6672], 3 X[16] - 5 X[40335], 5 X[16] - 6 X[45880], 13 X[16] - 18 X[48314], X[16] - 3 X[50858], 4 X[16] - 3 X[51485], 3 X[624] - 2 X[6672], 6 X[624] - 5 X[40335], 5 X[624] - 3 X[45880], 13 X[624] - 9 X[48314], 2 X[624] - 3 X[50858], 8 X[624] - 3 X[51485], and many others

X(622) lies on the curves K060, K066a, K342b, K1053b, K1132a, Q072, Q114, and these lines: {2, 13}, {3, 303}, {4, 69}, {5, 302}, {6, 11304}, {8, 44660}, {10, 51690}, {14, 532}, {15, 9989}, {17, 6671}, {20, 628}, {30, 299}, {61, 20088}, {62, 7797}, {68, 41897}, {80, 36928}, {86, 37145}, {99, 5613}, {141, 5318}, {145, 51691}, {148, 23005}, {183, 1080}, {187, 11488}, {193, 5334}, {194, 23000}, {265, 301}, {298, 381}, {325, 383}, {343, 473}, {376, 36756}, {394, 472}, {396, 11299}, {397, 3589}, {398, 3629}, {403, 47576}, {462, 47582}, {470, 37638}, {471, 11064}, {487, 42282}, {488, 35732}, {489, 2041}, {490, 2042}, {491, 2043}, {492, 2044}, {524, 5321}, {531, 19107}, {533, 36970}, {590, 10668}, {599, 42094}, {615, 10672}, {619, 21360}, {623, 16808}, {625, 42142}, {629, 42581}, {630, 5351}, {631, 13349}, {635, 42813}, {636, 16965}, {648, 36303}, {754, 22512}, {1078, 52688}, {1141, 10410}, {1154, 52221}, {1337, 46667}, {1350, 41039}, {1656, 61514}, {1992, 44497}, {2046, 32807}, {2076, 53429}, {2080, 59397}, {2924, 34009}, {2992, 4846}, {3090, 59404}, {3091, 7685}, {3104, 7785}, {3105, 6655}, {3106, 63018}, {3130, 14369}, {3146, 36994}, {3180, 10654}, {3241, 50857}, {3412, 33465}, {3523, 21159}, {3525, 21402}, {3530, 33405}, {3545, 63106}, {3616, 11708}, {3620, 42134}, {3630, 42101}, {3631, 42102}, {3642, 36969}, {3734, 22513}, {3763, 5340}, {3832, 41037}, {3849, 42119}, {4425, 51749}, {5224, 37144}, {5309, 37171}, {5339, 6144}, {5460, 16961}, {5464, 42100}, {5468, 57597}, {5479, 54141}, {5627, 39132}, {5640, 58478}, {5859, 42154}, {5864, 7773}, {5869, 48905}, {5978, 41043}, {5981, 41023}, {5983, 33389}, {6114, 22608}, {6390, 41034}, {6670, 16963}, {6673, 33387}, {6694, 42992}, {6770, 44465}, {6773, 12188}, {6778, 22689}, {7615, 42139}, {7752, 11133}, {7758, 22913}, {7759, 33466}, {7763, 52689}, {7766, 43454}, {7787, 36760}, {7793, 54116}, {7802, 47068}, {7809, 11128}, {7822, 37178}, {7834, 10614}, {7938, 42162}, {8703, 33610}, {8836, 11126}, {9301, 37333}, {9761, 42095}, {9763, 11480}, {10109, 33608}, {10218, 23896}, {10616, 16644}, {11002, 16633}, {11092, 15442}, {11094, 64251}, {11178, 25154}, {11295, 11485}, {11296, 42127}, {11297, 42974}, {11300, 42155}, {11305, 42128}, {11306, 11486}, {11307, 42156}, {11308, 42148}, {11489, 43620}, {11542, 37340}, {11543, 31694}, {11581, 43085}, {11582, 11600}, {11606, 43539}, {14206, 44070}, {14213, 17405}, {14368, 35470}, {14458, 60253}, {15031, 16626}, {15069, 41038}, {16001, 24206}, {16268, 33561}, {16530, 22891}, {16635, 31706}, {16809, 34508}, {17035, 44711}, {17907, 36302}, {18581, 62983}, {19773, 19775}, {19924, 51013}, {20080, 42133}, {20081, 25199}, {20426, 44362}, {22114, 42159}, {22491, 42103}, {22493, 43227}, {22510, 63047}, {22576, 51011}, {22649, 48796}, {22696, 25203}, {22850, 25156}, {22893, 62233}, {25157, 63044}, {25235, 32553}, {31275, 42494}, {31693, 42138}, {32815, 44463}, {33351, 42236}, {33352, 42238}, {33359, 33395}, {33360, 33393}, {33404, 61914}, {33413, 42166}, {33458, 42942}, {33459, 33622}, {33500, 36514}, {33609, 62168}, {33611, 62160}, {33612, 62154}, {33623, 42941}, {33627, 43541}, {34316, 37901}, {34390, 34514}, {34541, 42086}, {35303, 42123}, {35304, 42124}, {35314, 44466}, {35317, 52449}, {35918, 53430}, {35932, 42088}, {36327, 42136}, {36331, 42108}, {36388, 42430}, {36758, 42998}, {36958, 43459}, {37173, 42120}, {37242, 44464}, {37341, 42118}, {37352, 43416}, {37463, 37688}, {39261, 46757}, {39555, 42152}, {40341, 42093}, {40706, 54485}, {40853, 44712}, {40901, 42085}, {41016, 64093}, {41061, 50567}, {41407, 63032}, {42528, 50860}, {42943, 44382}, {42999, 51170}, {43542, 62877}, {44459, 64018}, {44487, 63722}, {44777, 48665}, {51021, 55587}, {52399, 58804}, {52400, 58803}, {53435, 53440}, {61561, 62654}

X(622) = reflection of X(i) in X(j) for these {i,j}: {2, 50858}, {4, 20429}, {6, 51162}, {8, 50856}, {16, 624}, {20, 14539}, {69, 51018}, {99, 51388}, {145, 51691}, {148, 23005}, {193, 51207}, {616, 5979}, {617, 299}, {621, 316}, {1337, 46667}, {1992, 51019}, {3146, 36994}, {3181, 14}, {3241, 50857}, {5615, 5}, {6779, 618}, {14712, 15}, {25235, 32553}, {36514, 33500}, {36968, 619}, {36995, 3}, {37785, 31694}, {37901, 34316}, {42943, 44382}, {51485, 2}, {51690, 10}, {54141, 5479}, {63722, 44487}
X(622) = isogonal conjugate of X(3439)
X(622) = isotomic conjugate of X(2993)
X(622) = anticomplement of X(16)
X(622) = orthoptic-circle-of-the-Steiner-circumellipse-inverse of X(5980)
X(622) = circumcircle-of-anticomplementary-triangle-inverse of X(621)
X(622) = circumcircle-of-outer-Napoleon-triangle-inverse of X(618)
X(622) = antigonal image of X(1337)
X(622) = symgonal image of X(33500)
X(622) = anticomplement of the isogonal conjugate of X(14)
X(622) = anticomplement of the isotomic conjugate of X(301)
X(622) = isotomic conjugate of the anticomplement of X(40581)
X(622) = isotomic conjugate of the isogonal conjugate of X(3130)
X(622) = anticomplementary isogonal conjugate of X(617)
X(622) = psi-transform of X(623)
X(622) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 617}, {14, 8}, {301, 6327}, {2151, 18301}, {2154, 2}, {2166, 621}, {3376, 627}, {3458, 192}, {5994, 4560}, {8738, 5905}, {20579, 21221}, {23896, 7192}, {32678, 23870}, {36297, 6360}, {36309, 7253}, {36310, 17491}, {39152, 6224}, {40710, 4329}, {44691, 3436}, {46077, 3648}, {51268, 21271}, {51806, 616}, {54024, 54014}
X(622) = X(i)-Ceva conjugate of X(j) for these (i,j): {265, 621}, {301, 2}, {52220, 37779}, {54557, 617}
X(622) = X(40581)-cross conjugate of X(2)
X(622) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3439}, {31, 2993}, {1973, 64245}, {2154, 40157}, {2190, 51243}, {6149, 14373}
X(622) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 2993}, {3, 3439}, {5, 51243}, {14, 46059}, {618, 3479}, {619, 38932}, {6337, 64245}, {14993, 14373}, {40581, 40157}, {46667, 512}
X(622) = cevapoint of X(i) and X(j) for these (i,j): {532, 33500}, {617, 627}, {3181, 19773}
X(622) = crosspoint of X(i) and X(j) for these (i,j): {18020, 23896}, {35139, 57579}
X(622) = crosssum of X(6138) and X(20975)
X(622) = crossdifference of every pair of points on line {3049, 6137}
X(622) = barycentric product X(i)*X(j) for these {i,j}: {76, 3130}, {94, 14369}, {299, 51277}, {301, 40581}, {328, 64251}, {395, 46757}, {1337, 41000}, {11094, 40709}, {39261, 41001}
X(622) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 2993}, {6, 3439}, {16, 40157}, {69, 64245}, {216, 51243}, {395, 38932}, {396, 3479}, {1337, 2981}, {1989, 14373}, {3130, 6}, {3181, 58916}, {11094, 470}, {14369, 323}, {39261, 6151}, {40579, 46059}, {40581, 16}, {46757, 40706}, {51277, 14}, {64251, 186}
X(622) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 69, 621}, {4, 634, 633}, {5, 5615, 59398}, {5, 52194, 302}, {13, 3643, 2}, {15, 34509, 62984}, {16, 624, 2}, {16, 33517, 5335}, {16, 40335, 6672}, {16, 50858, 624}, {69, 621, 633}, {76, 3818, 621}, {141, 5318, 11303}, {302, 52194, 627}, {311, 41171, 621}, {315, 31670, 621}, {340, 1531, 621}, {618, 37832, 2}, {621, 634, 69}, {624, 6672, 40335}, {637, 638, 634}, {1352, 11185, 621}, {3181, 11122, 19570}, {3818, 51018, 5207}, {5207, 43453, 621}, {5562, 32002, 621}, {6672, 40335, 2}, {7768, 48895, 621}, {7850, 48901, 621}, {16771, 19779, 37779}, {20429, 51018, 316}, {21360, 36968, 619}, {31694, 37785, 59379}, {47363, 47364, 618}


X(623) = COMPLEMENT OF X(15)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where
                        f(a,b,c) = [2(b2 + c2) area + sqr(3) (SAa2 + 2SBSC)]/a

Trilinears        F(15)/a - sin(A + π/3) : F(15)/b - sin(B + π/3) : F(15)/c - sin(C + π/3)

Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(623) lies on these lines: 2,14   3,629   5,141   13,298   16,302   17,633   18,83   30,618   396,533

X(623) = midpoint of X(i) and X(j) for these (i,j): (13,298), (15,621), (16,316)
X(623) = reflection of X(624) in X(625)
X(623) = inverse-in-nine-point-circle of X(624)
X(623) = complement of X(15)
X(623) = complementary conjugate of X(618)
X(623) = crosspoint of X(2) and X(300)
X(623) = intersection of diagonals of trapezoid PU(5)PU(11); i.e., the lines P(5)P(11) and U(5)U(11))


X(624) = COMPLEMENT OF X(16)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where
                        f(a,b,c) = [2(b2 + c2) area - sqr(3)(a2SA + 2SBSC)]/a

Trilinears        F(16)/a - sin(A - π/3) : F(16)/b - sin(B - π/3) : F(16)/c - sin(C - π/3)

Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(624) lies on these lines: 2,13   3,630   5,141   14,299   15,303   17,83   30,619   395,532

X(624) = midpoint of X(i) and X(j) for these (i,j): (14,299), (15,316), (16,622)
X(624) = reflection of X(623) in X(625)
X(624) = inverse-in-nine-point-circle of X(623)
X(624) = complement of X(16)
X(624) = complementary conjugate of X(619)
X(624) = crosspoint of X(2) and X(301)
X(624) = intersection of extended legs P(5)U(11) and U(5)P(11) of trapezoid PU(5)PU(11)


X(625) = MIDPOINT OF X(623) AND X(624)

Trilinears    [2(b4 + c4 - b2c2) - a2(b2 + c2)]/a : :

A construction of X(625) is given at Hyacinthos 27519. (Antreas Hatzipolakis and Peter Moses, April 16, 2018)

X(625) lies on these lines: 2,187   5,141   30,620   115,325   126,858   230,754

X(625) = midpoint of X(i) and X(j) for these (i,j): (115,325), (187,316), (623,624)
X(625) = isotomic conjugate of isogonal conjugate of X(20977)
X(625) = polar conjugate of isogonal conjugate of X(22087)
X(625) = inverse-in-nine-point-circle of X(141)
X(625) = complement of X(187)
X(625) = complementary conjugate of X(2482)


X(626) = COMPLEMENT OF X(32)

Trilinears    (b4 + c4)/a : (c4 + a4)/b: (a4 + b4)/c
Barycentrics    b4 + c4 : c4 + a4 : a4 + b4
X(626) = 3 X[2] + X[315], 3 X[2] - 5 X[7867], 9 X[2] - X[20065], X[32] + 3 X[7818], X[32] - 5 X[7867], 3 X[32] - X[20065], X[315] + 2 X[6680], X[315] - 3 X[7818], X[315] + 5 X[7867], 3 X[315] + X[20065], 2 X[6680] + 3 X[7818], 2 X[6680] - 5 X[7867], 6 X[6680] - X[20065], 3 X[7818] + 5 X[7867], 9 X[7818] + X[20065], and many others

X(626) is the center of the inellipse that is the barycentric square of the de Longchamps line. The Brianchon point (perspector) of the inellipse is X(1502). (Randy Hutson, October 15, 2018)

X(626) lies on these lines: {1, 4769}, {2, 32}, {3, 114}, {4, 3734}, {5, 141}, {6, 7759}, {10, 760}, {11, 6029}, {13, 11305}, {14, 11306}, {20, 36997}, {30, 7789}, {37, 746}, {39, 325}, {41, 4805}, {58, 56764}, {69, 3767}, {76, 115}, {99, 6655}, {116, 20255}, {125, 13518}, {126, 46657}, {140, 13335}, {147, 12203}, {172, 30103}, {183, 6722}, {187, 7750}, {193, 5319}, {194, 7765}, {206, 39466}, {213, 4766}, {230, 7767}, {264, 27371}, {274, 33841}, {297, 3199}, {304, 34542}, {305, 47846}, {316, 384}, {371, 11314}, {372, 11313}, {385, 7755}, {491, 1504}, {492, 1505}, {524, 5305}, {538, 3933}, {542, 14880}, {543, 1975}, {548, 32459}, {549, 34510}, {550, 32456}, {574, 7763}, {575, 44380}, {577, 28405}, {599, 5461}, {631, 36998}, {637, 5591}, {638, 5590}, {641, 43121}, {642, 43120}, {710, 8265}, {712, 4136}, {724, 16584}, {732, 51848}, {742, 4153}, {744, 25346}, {766, 2887}, {805, 37841}, {858, 30749}, {980, 33736}, {1007, 15482}, {1015, 26561}, {1086, 24166}, {1196, 45201}, {1197, 29972}, {1285, 32952}, {1329, 20540}, {1352, 13355}, {1368, 2386}, {1384, 63938}, {1500, 26590}, {1502, 40359}, {1513, 5188}, {1573, 26558}, {1574, 26582}, {1656, 15271}, {1759, 4799}, {1914, 30104}, {1930, 16886}, {2023, 32189}, {2031, 3054}, {2387, 3491}, {2450, 3917}, {2458, 5207}, {2482, 7782}, {2549, 3926}, {2782, 54222}, {2882, 34517}, {2967, 39604}, {2996, 32836}, {3053, 32954}, {3061, 36230}, {3090, 9753}, {3094, 8149}, {3102, 51401}, {3103, 51395}, {3104, 51387}, {3105, 51388}, {3117, 33734}, {3118, 4121}, {3266, 31107}, {3329, 7858}, {3425, 7509}, {3493, 39092}, {3545, 14492}, {3552, 6781}, {3589, 8364}, {3618, 33221}, {3619, 31415}, {3620, 32828}, {3628, 20576}, {3630, 63926}, {3631, 43291}, {3642, 22796}, {3643, 22797}, {3721, 17211}, {3739, 3841}, {3763, 5017}, {3793, 63930}, {3815, 6683}, {3819, 14725}, {3825, 20530}, {3849, 8369}, {3852, 6697}, {3972, 7823}, {4048, 29012}, {4056, 4376}, {4071, 24211}, {4118, 4178}, {4159, 46546}, {4372, 4680}, {4381, 4837}, {4713, 24045}, {5007, 7762}, {5013, 11287}, {5023, 11288}, {5041, 41624}, {5051, 25499}, {5067, 55732}, {5074, 59512}, {5077, 34504}, {5099, 9152}, {5133, 8891}, {5171, 37466}, {5206, 14907}, {5224, 6537}, {5283, 17550}, {5286, 7758}, {5304, 33182}, {5306, 63939}, {5309, 7754}, {5346, 14614}, {5355, 7760}, {5368, 7766}, {5449, 33548}, {5475, 7770}, {5976, 46283}, {5978, 11304}, {5979, 11303}, {6036, 10104}, {6177, 47370}, {6178, 47369}, {6179, 7806}, {6248, 15980}, {6337, 32986}, {6389, 6643}, {6390, 8357}, {6661, 14537}, {7467, 23208}, {7470, 43460}, {7486, 7616}, {7495, 15822}, {7603, 31239}, {7617, 21356}, {7619, 15810}, {7622, 33215}, {7735, 14023}, {7736, 32823}, {7737, 14001}, {7738, 32818}, {7739, 33223}, {7745, 7804}, {7757, 7864}, {7769, 7824}, {7771, 7904}, {7772, 7774}, {7777, 7786}, {7783, 7799}, {7827, 7839}, {7837, 7884}, {7875, 7878}, {8024, 63797}, {8176, 21358}, {8178, 43449}, {8266, 11360}, {8355, 50991}, {8356, 37512}, {8359, 22110}, {8365, 63945}, {8367, 20582}, {8368, 63941}, {8370, 31173}, {8588, 32964}, {8667, 33240}, {8728, 36812}, {9167, 33274}, {9466, 33228}, {9478, 44772}, {9605, 9766}, {9737, 37242}, {9744, 37479}, {9770, 33230}, {9863, 10991}, {10150, 15597}, {10159, 33020}, {10282, 59706}, {10297, 47567}, {10356, 58851}, {10418, 16055}, {10979, 42406}, {11007, 11052}, {11057, 33246}, {11060, 15066}, {11184, 31467}, {11185, 14063}, {11257, 14981}, {11285, 31455}, {11286, 63956}, {11574, 18129}, {11623, 34507}, {11648, 32833}, {11750, 54076}, {11824, 36656}, {11825, 36655}, {12042, 32151}, {12162, 54074}, {12605, 54075}, {13325, 14501}, {13326, 14502}, {13330, 51396}, {14035, 62203}, {14039, 44678}, {14041, 17128}, {14046, 14568}, {14047, 16984}, {14061, 32027}, {14376, 18531}, {14561, 35389}, {14712, 33225}, {14810, 59695}, {14827, 27516}, {14929, 33186}, {14946, 42371}, {14962, 40951}, {14994, 53475}, {15031, 32993}, {15048, 32450}, {15357, 38523}, {15480, 63929}, {15513, 35297}, {15515, 32965}, {15526, 41009}, {15533, 63953}, {15589, 33199}, {15819, 35430}, {16041, 18546}, {16044, 43457}, {16052, 48844}, {16061, 30761}, {16197, 61611}, {16275, 16951}, {16589, 37664}, {16600, 25345}, {16895, 60855}, {16921, 16986}, {16974, 25598}, {16990, 32832}, {16991, 33836}, {17008, 33248}, {17030, 31488}, {17047, 21235}, {17050, 21241}, {17131, 33283}, {17137, 31023}, {17669, 18140}, {17811, 52251}, {18134, 46828}, {18375, 44558}, {18907, 33185}, {20542, 20550}, {20547, 20549}, {20819, 46508}, {20947, 30173}, {21264, 25639}, {21495, 30760}, {21536, 59563}, {21843, 32970}, {22566, 42787}, {22660, 59556}, {22712, 36519}, {22736, 33410}, {22737, 33411}, {23115, 37073}, {23660, 29990}, {24598, 63794}, {24733, 40035}, {24784, 30753}, {26019, 30819}, {26145, 56983}, {26257, 37804}, {27020, 31476}, {27259, 62420}, {30122, 41269}, {30149, 33931}, {30179, 33941}, {30435, 63932}, {30777, 37803}, {30837, 41239}, {30945, 52257}, {31090, 33835}, {31275, 33249}, {31276, 32966}, {31400, 33202}, {31406, 44562}, {31417, 63121}, {31848, 33330}, {31859, 32821}, {32457, 63923}, {32815, 32982}, {32819, 33229}, {32827, 32971}, {32829, 32990}, {32830, 33200}, {32831, 33025}, {32838, 32988}, {32955, 62992}, {32960, 62993}, {32963, 53127}, {32969, 34229}, {32973, 64018}, {32978, 34803}, {32981, 43618}, {33191, 47102}, {33195, 46453}, {33197, 37809}, {33213, 63940}, {33234, 35022}, {33238, 43619}, {33241, 63933}, {33259, 43459}, {33285, 63955}, {33292, 52713}, {33348, 33349}, {33380, 49106}, {33381, 49105}, {33466, 53428}, {33467, 53440}, {33482, 53463}, {33483, 53452}, {34505, 36523}, {34573, 41413}, {35002, 37243}, {35374, 60702}, {35431, 38317}, {35432, 53484}, {35524, 52568}, {36212, 41237}, {36213, 40643}, {36477, 48835}, {36663, 48863}, {37350, 47617}, {37636, 59197}, {37988, 41262}, {39142, 61138}, {39750, 58445}, {40050, 59560}, {40341, 63934}, {40553, 52533}, {40706, 54115}, {40707, 54116}, {42826, 58450}, {42912, 44382}, {42913, 44383}, {43456, 63722}, {43527, 62891}, {44334, 53415}, {44453, 50567}, {45284, 51429}, {47005, 48913}, {53419, 63922}, {54724, 60187}, {54841, 60128}, {55085, 63018}, {59530, 61749}, {60099, 60633}, {60278, 62890}, {63534, 64093}

X(626) = midpoint of X(i) and X(j) for these {i,j}: {1, 4769}, {2, 7818}, {3, 54393}, {4, 30270}, {20, 36997}, {32, 315}, {69, 5028}, {76, 32452}, {115, 32458}, {194, 37004}, {316, 5162}, {639, 640}, {805, 37841}, {1352, 13355}, {1975, 7748}, {2458, 5207}, {3933, 5254}, {4136, 4920}, {4381, 4837}, {5309, 7788}, {6248, 54187}, {7754, 7855}, {7801, 7841}, {7805, 7882}, {7816, 7842}, {7861, 7895}, {10297, 47567}, {11648, 32833}, {14946, 42371}
X(626) = reflection of X(i) in X(j) for these {i,j}: {32, 6680}, {3933, 7895}, {5254, 7861}, {7805, 5305}, {7816, 7789}, {7817, 8360}, {13335, 140}, {13357, 6683}, {18806, 3934}, {20576, 3628}, {39750, 58445}, {42826, 58450}
X(626) = isogonal conjugate of X(38826)
X(626) = isotomic conjugate of X(40416)
X(626) = complement of X(32)
X(626) = anticomplement of X(6680)
X(626) = nine-point-circle-inverse of X(5031)
X(626) = complement of the isogonal conjugate of X(76)
X(626) = complement of the isotomic conjugate of X(1502)
X(626) = isotomic conjugate of the isogonal conjugate of X(20859)
X(626) = isogonal conjugate of the isotomic conjugate of X(44166)
X(626) = polar conjugate of the isotomic conjugate of X(4121)
X(626) = polar conjugate of the isogonal conjugate of X(20819)
X(626) = medial-isogonal conjugate of X(39)
X(626) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 39}, {2, 37}, {4, 16583}, {6, 16584}, {7, 3752}, {8, 1212}, {9, 16588}, {10, 16589}, {19, 1196}, {27, 40941}, {31, 8265}, {37, 21838}, {57, 17053}, {58, 52535}, {63, 216}, {69, 1214}, {75, 2}, {76, 10}, {80, 49758}, {82, 1194}, {83, 16600}, {85, 1}, {86, 3666}, {88, 8610}, {91, 7746}, {92, 6}, {95, 16577}, {99, 14838}, {100, 6586}, {101, 52589}, {141, 16587}, {145, 63622}, {158, 3767}, {162, 2485}, {163, 52590}, {189, 1108}, {190, 650}, {192, 63481}, {196, 20312}, {226, 2092}, {249, 23993}, {253, 1427}, {257, 1107}, {261, 16579}, {264, 226}, {273, 3772}, {274, 1125}, {278, 20227}, {279, 52541}, {280, 46830}, {281, 20310}, {286, 40940}, {290, 16609}, {304, 3}, {305, 18589}, {306, 18591}, {307, 18592}, {308, 1215}, {309, 57}, {310, 3739}, {312, 9}, {313, 1211}, {314, 5745}, {315, 16582}, {318, 46835}, {319, 16585}, {320, 16586}, {321, 1213}, {322, 223}, {325, 16591}, {326, 6509}, {327, 16603}, {329, 40943}, {330, 16604}, {331, 1210}, {333, 40937}, {334, 3912}, {335, 1575}, {336, 441}, {338, 24040}, {341, 6554}, {348, 17102}, {349, 442}, {350, 17755}, {394, 828}, {513, 6377}, {514, 1015}, {523, 16592}, {525, 16573}, {556, 16016}, {560, 40377}, {561, 141}, {644, 52594}, {646, 4521}, {648, 16612}, {651, 6589}, {653, 6588}, {658, 6129}, {661, 1084}, {662, 647}, {664, 905}, {668, 514}, {670, 4369}, {671, 16611}, {673, 3290}, {689, 8060}, {693, 1086}, {765, 23988}, {789, 824}, {799, 523}, {811, 525}, {823, 6587}, {824, 53823}, {850, 8287}, {870, 17023}, {871, 21264}, {873, 17045}, {874, 27929}, {889, 4763}, {897, 3291}, {903, 16610}, {908, 23980}, {934, 52595}, {1016, 24036}, {1018, 52592}, {1088, 4000}, {1089, 6537}, {1101, 23584}, {1109, 23991}, {1111, 6547}, {1121, 43065}, {1218, 43223}, {1221, 6685}, {1229, 52818}, {1231, 18641}, {1240, 44417}, {1267, 40651}, {1268, 44307}, {1275, 24025}, {1441, 17056}, {1446, 1834}, {1494, 18593}, {1502, 2887}, {1577, 115}, {1581, 3229}, {1748, 40939}, {1760, 40938}, {1821, 230}, {1847, 17054}, {1895, 46829}, {1896, 52530}, {1897, 2509}, {1909, 59509}, {1916, 18904}, {1917, 40376}, {1920, 51575}, {1921, 17793}, {1925, 37890}, {1926, 39080}, {1928, 626}, {1930, 6292}, {1934, 325}, {1959, 11672}, {1965, 37891}, {1966, 5976}, {1969, 5}, {1978, 513}, {2051, 21796}, {2052, 24005}, {2084, 55050}, {2167, 570}, {2184, 800}, {2186, 3117}, {2349, 3003}, {2481, 3008}, {2580, 8105}, {2581, 8106}, {2582, 15166}, {2583, 15167}, {2995, 37646}, {2996, 16605}, {2998, 16606}, {3112, 3589}, {3113, 7792}, {3223, 6375}, {3239, 35508}, {3250, 55049}, {3257, 3310}, {3261, 11}, {3262, 52659}, {3263, 16593}, {3264, 16594}, {3265, 16595}, {3266, 16597}, {3267, 34846}, {3403, 15819}, {3596, 3452}, {3669, 16614}, {3701, 38930}, {3762, 35092}, {3766, 35119}, {3835, 40610}, {3904, 35128}, {3912, 6184}, {3936, 35069}, {3948, 35068}, {3978, 19563}, {4033, 661}, {4036, 6627}, {4043, 62646}, {4077, 17058}, {4110, 63483}, {4146, 16015}, {4358, 4370}, {4373, 16602}, {4391, 1146}, {4397, 13609}, {4417, 40590}, {4444, 39786}, {4462, 40621}, {4509, 15611}, {4554, 522}, {4555, 3960}, {4562, 665}, {4564, 13006}, {4569, 7658}, {4572, 4885}, {4583, 812}, {4590, 16598}, {4592, 52584}, {4593, 826}, {4598, 21348}, {4602, 512}, {4608, 16726}, {4609, 42327}, {4610, 31947}, {4615, 59837}, {4620, 34977}, {4623, 21196}, {4625, 17069}, {4632, 8043}, {4634, 45674}, {4639, 9508}, {4647, 51586}, {4671, 16590}, {4728, 39011}, {4789, 35135}, {4791, 61073}, {4823, 53167}, {4828, 55045}, {4858, 46101}, {4978, 35076}, {4998, 16578}, {5209, 62609}, {6063, 142}, {6331, 8062}, {6332, 35072}, {6335, 3239}, {6374, 63618}, {6381, 13466}, {6382, 34832}, {6383, 3840}, {6384, 75}, {6385, 3741}, {6386, 3835}, {6540, 48003}, {6558, 59979}, {6590, 55046}, {7003, 20311}, {7017, 20262}, {7018, 4357}, {7033, 17353}, {7035, 4422}, {7045, 23585}, {7178, 16613}, {7182, 17073}, {7199, 244}, {7209, 17063}, {7249, 28358}, {8024, 21249}, {8026, 63491}, {8061, 35971}, {8773, 36212}, {9229, 18905}, {9230, 19564}, {9239, 6656}, {9285, 45210}, {9311, 2275}, {10159, 28594}, {10405, 40133}, {13149, 21172}, {14206, 3163}, {14207, 35133}, {14208, 15526}, {14210, 2482}, {14213, 233}, {14349, 39016}, {14615, 36908}, {14616, 35466}, {14829, 56325}, {15413, 2968}, {15416, 40616}, {15455, 3700}, {16284, 3160}, {16709, 41820}, {17149, 6374}, {17206, 37565}, {17233, 40606}, {17234, 40599}, {17758, 1500}, {17762, 6626}, {17786, 52657}, {17788, 40597}, {17789, 19557}, {17791, 40612}, {18018, 16580}, {18019, 16581}, {18020, 16599}, {18021, 21233}, {18022, 20305}, {18023, 4892}, {18025, 241}, {18026, 14837}, {18027, 63840}, {18031, 518}, {18032, 239}, {18040, 40585}, {18051, 34452}, {18064, 6665}, {18070, 3124}, {18077, 55043}, {18133, 40603}, {18134, 51574}, {18135, 59577}, {18137, 40586}, {18138, 40600}, {18140, 4075}, {18145, 52872}, {18147, 62564}, {18149, 9296}, {18151, 5375}, {18152, 40607}, {18155, 4858}, {18156, 6337}, {18157, 8299}, {18158, 55065}, {18159, 6631}, {18160, 6741}, {18297, 40378}, {18298, 274}, {18359, 44}, {18669, 61067}, {18738, 62570}, {18743, 3161}, {18749, 62605}, {18750, 1249}, {18816, 3911}, {18830, 31286}, {18832, 76}, {18833, 3934}, {18834, 83}, {18891, 20333}, {18895, 3836}, {19567, 18277}, {19611, 46831}, {19804, 62648}, {20336, 440}, {20444, 32664}, {20445, 62554}, {20446, 9470}, {20448, 39046}, {20450, 39042}, {20563, 18588}, {20564, 18590}, {20565, 5249}, {20566, 908}, {20567, 2886}, {20568, 519}, {20569, 551}, {20570, 63}, {20571, 343}, {20573, 63803}, {20641, 206}, {20643, 39029}, {20884, 1560}, {20914, 36103}, {20916, 40583}, {20917, 19584}, {20920, 40584}, {20924, 214}, {20925, 40587}, {20926, 36033}, {20927, 5452}, {20928, 478}, {20929, 40589}, {20930, 6505}, {20932, 40592}, {20934, 41884}, {20935, 40593}, {20937, 40594}, {20939, 31998}, {20940, 39026}, {20941, 36830}, {20943, 40598}, {20944, 6593}, {20945, 32746}, {20946, 24771}, {20947, 6651}, {20948, 125}, {20949, 8054}, {20950, 40623}, {20951, 39054}, {20952, 55053}, {20953, 38996}, {20954, 40619}, {20955, 62650}, {21580, 6615}, {21582, 3162}, {21593, 22391}, {21596, 40611}, {21600, 40595}, {21605, 45036}, {21606, 38979}, {21609, 6600}, {21610, 55066}, {21611, 38991}, {21615, 3789}, {21739, 56531}, {23062, 5573}, {23999, 23583}, {24001, 14401}, {24002, 3756}, {24004, 6544}, {24006, 6388}, {24011, 23587}, {24018, 35071}, {24019, 52588}, {24021, 23591}, {24032, 23982}, {24037, 620}, {24039, 1649}, {24041, 34990}, {24524, 41771}, {26734, 61661}, {27424, 3061}, {27475, 2276}, {27496, 63623}, {27801, 3454}, {27805, 3709}, {27808, 4129}, {28659, 1329}, {28660, 960}, {30545, 41886}, {30565, 35125}, {30566, 35129}, {30596, 62586}, {30598, 28606}, {30635, 17237}, {30636, 17231}, {30663, 59454}, {30690, 1100}, {30693, 5574}, {30701, 25066}, {30710, 5750}, {30805, 55044}, {30806, 35110}, {30807, 23972}, {30829, 36911}, {30939, 51583}, {30963, 27481}, {31002, 536}, {31008, 59565}, {31359, 5283}, {31618, 13405}, {31623, 40942}, {31625, 24003}, {31627, 63625}, {31630, 28593}, {31643, 39595}, {31997, 41849}, {32008, 16601}, {32009, 25092}, {32014, 3743}, {32017, 17355}, {32018, 3634}, {32020, 726}, {32021, 4021}, {32023, 3663}, {32679, 18334}, {32680, 1637}, {33672, 40837}, {33677, 36905}, {33778, 52042}, {33780, 6552}, {33787, 6338}, {33805, 30}, {33806, 40368}, {33807, 40369}, {33808, 6503}, {33935, 52782}, {33939, 3647}, {34085, 676}, {34234, 8609}, {34258, 5257}, {34384, 21231}, {34393, 1465}, {34403, 52389}, {34404, 281}, {34523, 59579}, {34537, 21254}, {34538, 24017}, {35058, 39798}, {35145, 3002}, {35162, 49760}, {35164, 5723}, {35171, 1638}, {35174, 10015}, {35175, 43055}, {35181, 47784}, {35517, 39063}, {35518, 16596}, {35519, 26932}, {35538, 20343}, {35544, 46842}, {36036, 2799}, {36085, 2492}, {36100, 8607}, {36101, 8608}, {36102, 3018}, {36104, 2508}, {36105, 44817}, {36588, 31197}, {36796, 40869}, {36800, 59734}, {36803, 3716}, {36804, 1639}, {36805, 2325}, {36807, 3693}, {36838, 52596}, {36907, 53387}, {37130, 3011}, {37133, 4874}, {37134, 2491}, {37214, 8758}, {37218, 6590}, {37220, 468}, {38810, 6679}, {38847, 6680}, {39126, 63621}, {39467, 62615}, {39699, 39982}, {39700, 46838}, {39704, 4850}, {39705, 39974}, {39718, 37128}, {39727, 251}, {39733, 25}, {39735, 42}, {39749, 44798}, {39798, 21827}, {39968, 28592}, {39994, 3943}, {39995, 62571}, {40004, 3720}, {40005, 17758}, {40008, 13476}, {40009, 36907}, {40010, 321}, {40011, 72}, {40012, 2321}, {40013, 594}, {40014, 8}, {40015, 19}, {40016, 21238}, {40017, 740}, {40020, 40515}, {40023, 1698}, {40024, 3842}, {40025, 43}, {40026, 145}, {40027, 192}, {40028, 4384}, {40029, 3679}, {40030, 3696}, {40031, 3993}, {40033, 17289}, {40037, 82}, {40038, 16706}, {40039, 4358}, {40040, 4080}, {40041, 39697}, {40044, 3920}, {40071, 21530}, {40072, 21246}, {40073, 21247}, {40162, 21257}, {40216, 17245}, {40339, 40327}, {40362, 21235}, {40363, 21244}, {40364, 1368}, {40412, 25080}, {40419, 25065}, {40421, 16607}, {40424, 25091}, {40440, 23292}, {40495, 116}, {40701, 20264}, {40702, 7952}, {40703, 15595}, {40704, 50441}, {40716, 3218}, {40827, 49598}, {40834, 59720}, {40845, 238}, {41283, 17046}, {41683, 2229}, {42034, 62608}, {42311, 52542}, {42709, 62652}, {42716, 57046}, {44129, 942}, {44130, 6708}, {44150, 35075}, {44169, 20542}, {44172, 20541}, {44173, 21253}, {44179, 52032}, {44186, 7}, {44187, 3846}, {44188, 81}, {44190, 946}, {44327, 57055}, {44720, 63620}, {44733, 2277}, {46110, 6506}, {46136, 43048}, {46137, 43035}, {46234, 113}, {46238, 114}, {46244, 427}, {46254, 5972}, {46273, 511}, {46277, 524}, {46281, 24256}, {46289, 52536}, {46404, 521}, {46405, 3738}, {46406, 3900}, {46738, 15487}, {46740, 614}, {46744, 13389}, {46745, 13388}, {46746, 60249}, {46750, 3219}, {48070, 14936}, {48131, 39015}, {49780, 35124}, {51560, 918}, {51863, 34021}, {51865, 86}, {52043, 20532}, {52049, 39028}, {52137, 51578}, {52138, 10335}, {52151, 62557}, {52351, 63849}, {52379, 4999}, {52381, 63848}, {52406, 42018}, {52414, 63846}, {52611, 788}, {52612, 52601}, {52619, 17761}, {52621, 4904}, {52622, 5514}, {52935, 52597}, {53210, 43063}, {53647, 3669}, {53648, 54249}, {53653, 43049}, {53658, 47965}, {54121, 37662}, {54240, 52587}, {54957, 48404}, {54967, 11068}, {54986, 7180}, {54987, 3676}, {55106, 37543}, {55213, 17066}, {55215, 924}, {55231, 21187}, {55239, 3005}, {55553, 63843}, {55927, 9465}, {55945, 31198}, {55975, 24631}, {55983, 5222}, {55984, 34522}, {56026, 25067}, {56034, 1180}, {56074, 3672}, {56081, 25082}, {56127, 46196}, {56169, 4755}, {56212, 4687}, {56241, 25666}, {56249, 62588}, {56252, 4391}, {56596, 278}, {56784, 51582}, {56944, 46837}, {57244, 40624}, {57538, 24030}, {57581, 24009}, {57642, 56}, {57716, 9722}, {57722, 56926}, {57725, 1573}, {57773, 55}, {57774, 1088}, {57777, 12610}, {57780, 20208}, {57781, 23304}, {57785, 3946}, {57787, 16608}, {57788, 37691}, {57791, 4847}, {57792, 11019}, {57793, 20205}, {57796, 34830}, {57801, 44360}, {57806, 13567}, {57815, 6666}, {57821, 443}, {57824, 31993}, {57826, 4646}, {57830, 4415}, {57837, 6354}, {57877, 4656}, {57898, 5449}, {57904, 34825}, {57905, 12}, {57906, 65}, {57910, 26942}, {57913, 3678}, {57914, 3841}, {57918, 34822}, {57919, 34823}, {57920, 25760}, {57921, 4}, {57923, 4657}, {57925, 17279}, {57947, 17243}, {57948, 4364}, {57950, 10196}, {57955, 53415}, {57968, 30476}, {57973, 520}, {57975, 3004}, {57980, 8680}, {57995, 3834}, {57996, 516}, {57998, 394}, {57999, 625}, {58012, 3931}, {58013, 15668}, {58019, 312}, {58025, 63800}, {58026, 5718}, {58027, 30818}, {58028, 17720}, {58029, 5316}, {58817, 17071}, {58860, 53543}, {59255, 29571}, {59259, 5542}, {59437, 59440}, {59452, 59446}, {59459, 59462}, {59518, 59676}, {59759, 25078}, {60197, 4205}, {60235, 25081}, {60236, 20691}, {60244, 21024}, {60257, 21857}, {60261, 21892}, {60678, 24603}, {60706, 31336}, {61413, 59573}, {62234, 52882}, {62249, 20527}, {62250, 20334}, {62251, 20543}, {62255, 61085}, {62272, 1209}, {62273, 34836}, {62276, 140}, {62277, 46832}, {62415, 61065}, {62418, 15449}, {62465, 29017}, {62469, 62423}, {62528, 4859}, {62619, 1646}, {62884, 3986}, {62907, 4868}, {62914, 21049}, {62946, 3755}, {63225, 27751}, {63227, 45226}, {63759, 3580}, {63827, 39013}, {63895, 5988}, {63902, 49563}, {63906, 25096}, {64194, 23986}, {64211, 46836}
X(626) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 8265}, {805, 2799}, {1502, 8039}, {4630, 23881}, {16890, 20859}, {16891, 4118}, {20627, 4178}, {42371, 826}, {44166, 4121}, {55034, 523}
X(626) = X(i)-cross conjugate of X(j) for these (i,j): {4118, 7217}, {16893, 44166}, {16894, 20627}, {20819, 4121}
X(626) = X(i)-isoconjugate of X(j) for these (i,j): {1, 38826}, {31, 40416}, {32, 38847}, {75, 44167}, {560, 38830}, {711, 51904}, {1917, 44165}, {1923, 3115}, {2084, 33515}, {8061, 58114}
X(626) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 40416}, {3, 38826}, {206, 44167}, {626, 32}, {6374, 38830}, {6376, 38847}, {8265, 2}, {62452, 33515}
X(626) = cevapoint of X(20819) and X(20859)
X(626) = crosspoint of X(2) and X(1502)
X(626) = crosssum of X(i) and X(j) for these (i,j): {6, 1501}, {51318, 56915}
X(626) = crossdifference of every pair of points on line {3005, 3050}
X(626) = barycentric product X(i)*X(j) for these {i,j}: {1, 20627}, {4, 4121}, {6, 44166}, {7, 4178}, {8, 7217}, {10, 16891}, {32, 8039}, {75, 4118}, {76, 20859}, {83, 16893}, {86, 16894}, {141, 16890}, {190, 21110}, {264, 20819}, {321, 18167}, {561, 2085}, {710, 40847}, {1502, 8265}, {1799, 46508}, {3118, 40016}, {4173, 18022}, {8023, 40359}, {16717, 27801}, {23209, 44161}, {35530, 51982}, {40050, 62546}, {40362, 44164}
X(626) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 40416}, {6, 38826}, {32, 44167}, {75, 38847}, {76, 38830}, {308, 3115}, {710, 16985}, {827, 58114}, {1502, 44165}, {2085, 31}, {3118, 3051}, {4118, 1}, {4121, 69}, {4173, 184}, {4178, 8}, {4577, 33515}, {7217, 7}, {8023, 9233}, {8039, 1502}, {8265, 32}, {16717, 1333}, {16890, 83}, {16891, 86}, {16893, 141}, {16894, 10}, {18167, 81}, {20627, 75}, {20819, 3}, {20859, 6}, {21110, 514}, {23209, 14575}, {33786, 38838}, {40073, 38842}, {40362, 44163}, {40847, 57937}, {44164, 1501}, {44166, 76}, {46508, 427}, {51982, 711}, {59204, 60694}, {62546, 1974}
X(626) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 32, 6680}, {2, 83, 7889}, {2, 315, 32}, {2, 1078, 7749}, {2, 1369, 1627}, {2, 2548, 7808}, {2, 2896, 1078}, {2, 3096, 6292}, {2, 7752, 1506}, {2, 7785, 83}, {2, 7787, 7846}, {2, 7793, 7857}, {2, 7800, 7815}, {2, 7808, 6704}, {2, 7809, 7753}, {2, 7810, 34506}, {2, 7883, 7810}, {2, 7900, 7787}, {2, 7912, 7752}, {2, 7929, 7793}, {2, 7938, 3096}, {2, 20088, 10583}, {2, 26099, 25497}, {2, 26763, 27031}, {2, 26811, 26970}, {2, 32816, 2548}, {3, 3788, 620}, {3, 7761, 7830}, {3, 7778, 3788}, {3, 7784, 7761}, {4, 7795, 3734}, {5, 141, 3934}, {5, 14881, 19130}, {6, 7759, 7838}, {6, 7776, 7759}, {6, 7834, 7829}, {6, 7866, 7834}, {32, 7818, 315}, {32, 7867, 2}, {39, 325, 7764}, {39, 6656, 4045}, {39, 7821, 325}, {39, 7853, 6656}, {69, 3767, 7751}, {69, 14064, 3767}, {76, 115, 63924}, {76, 3314, 7794}, {76, 5025, 115}, {76, 7922, 3314}, {76, 7934, 5025}, {83, 3096, 10292}, {83, 7785, 7753}, {83, 7809, 7785}, {83, 7944, 2}, {99, 6655, 7756}, {99, 7836, 7863}, {99, 7909, 7836}, {99, 7911, 6655}, {115, 7794, 76}, {141, 5031, 24206}, {141, 5103, 24256}, {183, 7879, 7854}, {183, 7887, 7746}, {187, 7873, 7750}, {187, 7874, 7807}, {194, 7790, 7765}, {194, 7796, 7813}, {194, 7897, 7796}, {194, 7933, 7790}, {230, 7767, 7780}, {230, 8361, 7886}, {315, 7867, 6680}, {316, 384, 7747}, {316, 7832, 384}, {316, 7931, 7820}, {325, 6656, 39}, {325, 7853, 4045}, {384, 7832, 7820}, {384, 7885, 316}, {384, 7931, 7832}, {385, 7768, 7826}, {385, 7826, 63927}, {385, 7828, 7755}, {385, 7901, 7828}, {385, 7939, 7768}, {550, 59545, 32456}, {574, 7888, 7763}, {574, 7935, 7791}, {620, 7830, 3}, {623, 624, 19130}, {625, 3934, 5}, {625, 7849, 3934}, {635, 636, 40107}, {1007, 16043, 31401}, {1078, 2896, 7810}, {1078, 7749, 34506}, {1078, 7883, 2896}, {1078, 7899, 2}, {1078, 10350, 32}, {1506, 6292, 2}, {1975, 7841, 7748}, {1975, 7881, 7801}, {2039, 2040, 5031}, {2548, 7914, 6704}, {2548, 32816, 7775}, {2549, 3926, 7781}, {2549, 7908, 14148}, {2549, 32974, 7872}, {2887, 17046, 21240}, {2896, 7899, 7749}, {3096, 7752, 2}, {3096, 7912, 1506}, {3266, 31107, 59768}, {3314, 5025, 76}, {3314, 7934, 115}, {3329, 7941, 7858}, {3329, 7948, 7859}, {3552, 7802, 6781}, {3552, 7898, 7802}, {3552, 7945, 7835}, {3620, 32972, 32828}, {3734, 7825, 4}, {3734, 7869, 7795}, {3767, 14064, 7844}, {3788, 7761, 3}, {3788, 7784, 7830}, {3815, 8362, 6683}, {3926, 7781, 14148}, {3926, 32974, 2549}, {3933, 33184, 5254}, {3934, 7849, 141}, {3972, 7860, 7823}, {3972, 7930, 7892}, {4045, 7764, 39}, {4118, 16894, 4178}, {4680, 20267, 4372}, {4766, 24995, 213}, {5007, 7845, 7762}, {5007, 7852, 7792}, {5025, 7794, 63924}, {5025, 7922, 7794}, {5103, 24256, 19130}, {5254, 33184, 7861}, {5286, 7758, 7798}, {5286, 37668, 7758}, {5309, 7855, 7754}, {5355, 7890, 7760}, {5403, 5404, 24256}, {5475, 7822, 7770}, {6179, 7850, 7893}, {6179, 7942, 7806}, {6390, 8357, 63548}, {6655, 7836, 99}, {6655, 7909, 7863}, {6656, 7821, 7764}, {7603, 31239, 32992}, {7737, 32006, 63931}, {7738, 32818, 34511}, {7745, 7819, 7804}, {7746, 7854, 183}, {7746, 7887, 6722}, {7747, 7820, 384}, {7748, 7801, 1975}, {7749, 7810, 1078}, {7750, 7807, 187}, {7751, 7844, 3767}, {7751, 7896, 69}, {7752, 7938, 6292}, {7753, 7889, 83}, {7754, 7788, 7855}, {7754, 7851, 5309}, {7754, 33219, 7851}, {7755, 7768, 63927}, {7755, 7826, 385}, {7756, 7863, 99}, {7757, 7871, 7906}, {7757, 7918, 7864}, {7758, 33180, 7902}, {7758, 37668, 7916}, {7759, 7834, 6}, {7759, 7866, 7829}, {7760, 7779, 7890}, {7760, 7797, 5355}, {7760, 7917, 7779}, {7760, 7919, 7797}, {7761, 7778, 620}, {7762, 7792, 5007}, {7762, 8363, 7792}, {7763, 7791, 574}, {7765, 7813, 194}, {7766, 7856, 5368}, {7766, 7932, 7856}, {7766, 7946, 7877}, {7767, 8361, 230}, {7768, 7828, 385}, {7768, 7901, 7755}, {7769, 7831, 7824}, {7770, 7773, 5475}, {7770, 7868, 7822}, {7771, 7936, 7904}, {7771, 7940, 7907}, {7772, 7903, 7774}, {7772, 7913, 7803}, {7773, 7868, 7770}, {7774, 7803, 7772}, {7775, 7808, 2548}, {7775, 7914, 7808}, {7776, 7834, 7838}, {7776, 7866, 6}, {7777, 7786, 9698}, {7777, 7876, 7786}, {7778, 7784, 3}, {7779, 7797, 7760}, {7779, 7919, 5355}, {7780, 7848, 7767}, {7780, 7886, 230}, {7781, 7872, 2549}, {7781, 7908, 3926}, {7782, 7870, 7891}, {7782, 7891, 2482}, {7782, 7910, 7833}, {7783, 7924, 7847}, {7783, 7947, 7799}, {7785, 7944, 7889}, {7786, 7814, 7777}, {7786, 7937, 7876}, {7787, 7900, 7812}, {7788, 7851, 7754}, {7788, 33219, 5309}, {7790, 7796, 194}, {7790, 7897, 7813}, {7792, 8363, 7852}, {7793, 7929, 7811}, {7796, 7933, 7765}, {7797, 7917, 7890}, {7798, 7902, 5286}, {7798, 7916, 7758}, {7799, 7847, 7783}, {7802, 7835, 3552}, {7804, 7843, 7745}, {7804, 7915, 7819}, {7805, 7817, 5305}, {7806, 7893, 6179}, {7806, 14065, 7942}, {7808, 7914, 2}, {7809, 7944, 83}, {7811, 7857, 7793}, {7812, 7846, 7787}, {7814, 7876, 9698}, {7814, 7937, 7786}, {7815, 7862, 2}, {7815, 7865, 7800}, {7816, 7880, 7789}, {7817, 7882, 7805}, {7818, 7867, 32}, {7821, 7853, 39}, {7823, 7892, 3972}, {7824, 7925, 7769}, {7824, 7928, 7831}, {7825, 7869, 3734}, {7827, 7905, 7839}, {7828, 7939, 7826}, {7829, 7838, 6}, {7832, 7885, 7747}, {7833, 7870, 2482}, {7833, 7891, 7782}, {7835, 7898, 6781}, {7836, 7911, 7756}, {7837, 7920, 7894}, {7839, 7840, 7905}, {7839, 7923, 7827}, {7840, 7923, 7839}, {7841, 7881, 1975}, {7842, 7880, 7816}, {7843, 7915, 7804}, {7844, 7896, 7751}, {7845, 7852, 5007}, {7848, 7886, 7780}, {7850, 7942, 6179}, {7856, 7877, 7766}, {7858, 7859, 3329}, {7860, 7930, 3972}, {7862, 7865, 7815}, {7864, 7906, 7757}, {7870, 7910, 7782}, {7871, 7918, 7757}, {7872, 7908, 7781}, {7873, 7874, 187}, {7875, 7921, 7878}, {7877, 7932, 5368}, {7878, 7926, 7921}, {7878, 7943, 7875}, {7879, 7887, 183}, {7883, 7899, 1078}, {7884, 7894, 7920}, {7884, 7949, 7894}, {7885, 7931, 384}, {7888, 7935, 574}, {7893, 14065, 7806}, {7894, 7949, 7837}, {7897, 7933, 194}, {7898, 7945, 3552}, {7901, 7939, 385}, {7902, 7916, 7798}, {7903, 7913, 7772}, {7904, 7907, 7771}, {7907, 7940, 31274}, {7909, 7911, 99}, {7912, 7938, 2}, {7917, 7919, 7760}, {7922, 7934, 76}, {7924, 7947, 7783}, {7925, 7928, 7824}, {7926, 7943, 7878}, {7932, 7946, 7766}, {7936, 7940, 7771}, {7941, 7948, 3329}, {9466, 39565, 59635}, {10583, 20088, 12150}, {14001, 32006, 7737}, {14907, 16925, 5206}, {15810, 41133, 7619}, {16043, 31401, 15482}, {16893, 20859, 4121}, {16990, 32961, 32832}, {17211, 63817, 3721}, {24889, 24942, 2}, {27101, 27153, 2}, {32818, 33190, 7738}, {32823, 32956, 7736}, {32828, 32972, 43620}, {32830, 33200, 43448}, {32833, 33251, 11648}, {33180, 37668, 5286}, {33202, 63098, 31400}, {33228, 59635, 39565}, {33292, 52713, 63533}, {43449, 46236, 8178}


X(627) = ANTICOMPLEMENT OF X(17)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [SBSC + 2SA(a2 + sqr(3) area)]/a

Trilinears        F(17)/a - 2 sec(A - π/3) : F(17)/b - 2 sec(B - π/3) : F(17)/c - 2 sec(C - π/3)

Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

Erect equilateral triangles outwards on the sides of triangle ABC; the circumcenter of the apices is X(627). (Peter Moses, 7/16,2003)

Let A' be the isogonal conjugate of the A-vertex of the inner Napoleon triangle, and define B' and C' cyclically. Triangle A'B'C' is perspective to the inner Napoleon triangle at X(5), and to the outer Napoleon triangle at X(627). (Randy Hutson, June 7, 2019)

Let OA be the circle centered at the A-vertex of the outer Fermat triangle and passing through A; define OB and OC cyclically. X(627) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

X(627) lies on the Napoleon cubic and these lines: 2,17   3,298   4,616   5,302   16,635   20,621   54,69   61,618   140,299

X(627) = reflection of X(17) in X(629)
X(627) = isogonal conjugate of X(3489)
X(627) = anticomplement of X(17)
X(627) = anticomplementary conjugate of X(633)
X(627) = X(i)-Ceva conjugate of X(j) for these (i,j): (5,628), (302,2)
X(627) = {X(54),X(1273)}-harmonic conjugate of X(628)
X(627) = {X(69),X(631)}-harmonic conjugate of X(628)


X(628) = ANTICOMPLEMENT OF X(18)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [SBSC + 2SA(a2 - sqr(3) area)]/a

Trilinears        F(18)/a + 2 sec(A + π/3) : F(18)/b + 2 sec(B + π/3) : F(18)/c + 2 sec(C + π/3)

Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

Erect equilateral triangles inwards on the sides of triangle ABC; the circumcenter of the apices is X(628). (Peter Moses, 7/16,2003)

Let A' be the isogonal conjugate of the A-vertex of the outer Napoleon triangle, and define B' and C' cyclically. Triangle A'B'C' is perspective to the outer Napoleon triangle at X(5), and to the inner Napoleon triangle at X(628). (Randy Hutson, June 7, 2019)

Let OA be the circle centered at the A-vertex of the inner Fermat triangle and passing through A; define OB and OC cyclically. X(628) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

X(628) lies on the Napoleon cubic and these lines: 2,18   3,299   4,617   5,303   15,636   20,622   54,69   62,619   140,298

X(628) = reflection of X(18) in X(630)
X(628) = isogonal conjugate of X(3490)
X(628) = anticomplement of X(18)
X(628) = anticomplementary conjugate of X(634)
X(628) = X(i)-Ceva conjugate of X(j) for these (i,j): (5,627), (303,2)
X(628) = {X(54),X(1273)}-harmonic conjugate of X(627)
X(628) = {X(69),X(631)}-harmonic conjugate of X(627)


X(629) = COMPLEMENT OF X(17)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where
                        f(a,b,c) = [6SBSC + 7SAa2 + 2 sqr(3) (b2 + c2) area]/a

Trilinears        F(17)/a - sec(A - π/3) : F(17)/b - sec(B - π/3) : F(17)/c - sec(C - π/3)

Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(629) lies on these lines: 2,17   3,623   5,618   61,302   140,619   141,575

X(629) = midpoint of X(17) and X(627)
X(629) = complement of X(17)
X(629) = complementary conjugate of X(635)
X(629) = crosspoint of X(2) and X(302)


X(630) = COMPLEMENT OF X(18)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where
                        f(a,b,c) = [6SBSC + 7SAa2 - 2 sqr(3) (b2 + c2) area]/a

Trilinears        F(18)/a + sec(A + π/3) : F(18)/b + sec(B + π/3) : F(18)/c + sec(C + π/3)

Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(630) lies on these lines: 2,18   3,624   5,619   62,303   140,618   141,575

X(630) = midpoint of X(18) and X(628)
X(630) = complement of X(18)
X(630) = anticomplementary conjugate of X(636)
X(630) = crosspoint of X(2) and X(303)


X(631) = 3/5*OG

Trilinears    2 cos A + cos B cos C : 2 cos B + cos C cos A : 2 cos C + cos A cos B
Trilinears    cos A + sin B sin C : cos B + sin C sin A : cos C + sin A sin B : :
Trilinears    sec A + 2 sec B sec C : sec B + 2 sec C sec A : sec C + 2 sec A sec B
Trilinears    a/(b sec B + c sec C) + 2 cos A : :
Barycentrics  (sin A)(2 cos A + cos B cos C) : (sin B)(2 cos B + cos C cos A) : (sin C)(2 cos C + cos A cos B)
Barycentrics   3a^4 - 4a^2(b^2 + c^2) + (b^2 - c^2)^2 : :
Barycentrics    2 SA a^2 + SB SC : :
Barycentrics    2 S^2 - SB SC : :
Barycentrics    cot B cot C - 2 : :

As a point on the Euler line, X(631) has Shinagawa coefficients (2,-1).

Let AeBeCe and AiBiCi be the Ae and Ai triangles (aka K798e and K798i triangles). X(631) is the radical center of the circumcircles of triangles AAeAi, BBeBi, CCeCi. (Randy Hutson, June 7, 2019)

X(631) lies on these lines: {1,1000}, {2,3}, {6,9540}, {7,11374}, {8,1385}, {9,6700}, {10,944}, {11,4294}, {12,4293}, {13,5351}, {14,5352}, {15,11489}, {16,11488}, {17,5237}, {18,5238}, {32,7736}, {35,497}, {36,388}, {39,5319}, {40,1125}, {46,3485}, {49,11003}, {51,11695}, {52,2979}, {54,69}, {55,1058}, {56,1056}, {57,3487}, {61,3411}, {62,3412}, {64,5646}, {68,3431}, {72,5744}, {74,5972}, {76,6337}, {83,5171}, {84,5658}, {98,620}, {99,6036}, {100,5082}, {101,6712}, {102,6718}, {103,6710}, {104,958}, {109,6711}, {110,6699}, {113,12244}, {114,3096}, {115,13172}, {119,12248}, {122,5667}, {125,12383}, {127,13200}, {132,12253}, {141,5085}, {142,5735}, {143,11592}, {145,5690}, {146,12041}, {147,7945}, {154,6247}, {165,946}, {171,602}, {174,8127}, {183,3926}, {184,9705}, {185,5650}, {187,2548}, {193,5050}, {194,7616}, {196,8762}, {210,12675}, {212,3075}, {214,12247}, {216,1075}, {230,5013}, {238,601}, {262,5188}, {315,1007}, {323,12161}, {325,3785}, {329,3916}, {355,5731}, {371,3069}, {372,3068}, {373,10110}, {386,9568}, {387,4255}, {389,3917}, {390,496}, {394,7592}, {485,3316}, {486,3317}, {487,492}, {488,491}, {495,3600}, {511,3567}, {515,1698}, {516,8227}, {517,3616}, {551,7982}, {568,6101}, {569,6515}, {572,966}, {573,9569}, {574,3767}, {575,1992}, {576,10168}, {577,10312}, {578,11433}, {580,5712}, {581,3216}, {590,1152}, {597,11477}, {599,8550}, {603,3074}, {615,1151}, {616,6771}, {617,6774}, {618,6770}, {619,6773}, {621,13350}, {622,13349}, {629,3642}, {630,3643}, {640,8982}, {641,5590}, {642,5591}, {671,10992}, {748,3073}, {750,3072}, {908,4652}, {912,3876}, {936,5745}, {942,5435}, {952,3617}, {954,8732}, {956,7080}, {962,3579}, {978,1064}, {993,2551}, {999,5265}, {1001,6691}, {1038,1870}, {1040,6198}, {1071,5044}, {1131,6452}, {1132,6451}, {1141,13372}, {1147,5012}, {1153,5485}, {1155,4295}, {1192,11821}, {1199,1993}, {1209,12254}, {1210,3488}, {1216,5889}, {1285,3053}, {1292,6714}, {1293,6715}, {1294,6716}, {1295,6717}, {1296,6719}, {1297,6720}, {1350,3589}, {1352,3619}, {1376,4999}, {1478,4325}, {1479,4330}, {1482,3622}, {1483,3621}, {1490,6705}, {1498,6696}, {1503,3763}, {1506,5206}, {1511,3448}, {1512,8582}, {1519,4512}, {1614,9306}, {1621,10596}, {1703,8983}, {1737,3486}, {1837,4305}, {1899,13367}, {1986,13416}, {2077,5248}, {2080,7787}, {2096,3452}, {2482,11623}, {2549,7746}, {2550,10902}, {2794,7867}, {2883,10606}, {2888,10610}, {2975,3421}, {3055,5210}, {3060,5462}, {3070,6410}, {3071,6409}, {3095,6194}, {3098,7846}, {3100,9644}, {3183,3462}, {3189,10916}, {3241,3653}, {3279,3608}, {3295,5281}, {3296,3338}, {3303,4995}, {3304,5298}, {3305,7330}, {3306,5709}, {3311,7586}, {3312,7585}, {3333,13405}, {3335,3609}, {3357,6225}, {3359,5250}, {3398,7774}, {3428,6690}, {3430,6693}, {3474,4338}, {3476,10039}, {3564,3620}, {3582,10385}, {3614,12943}, {3623,5844}, {3632,13607}, {3634,4297}, {3646,12705}, {3654,10222}, {3679,5882}, {3746,10072}, {3788,7800}, {3796,9707}, {3813,4421}, {3817,12512}, {3819,5562}, {3868,10202}, {3871,10529}, {3873,13373}, {3897,5554}, {3933,9755}, {3934,11257}, {4045,9754}, {4256,5292}, {4292,5219}, {4299,5229}, {4302,5225}, {4304,9581}, {4308,5126}, {4311,9578}, {4313,5704}, {4340,5718}, {4413,8273}, {4423,11496}, {4648,5733}, {4678,12645}, {4846,11270}, {4855,6734}, {4996,10522}, {5023,7745}, {5024,5305}, {5045,10578}, {5096,5800}, {5102,6329}, {5122,5226}, {5158,5702}, {5180,10225}, {5181,5622}, {5251,5450}, {5253,10597}, {5259,8166}, {5303,11681}, {5304,9605}, {5316,6260}, {5326,7354}, {5334,11480}, {5335,11481}, {5395,10155}, {5436,9843}, {5442,5902}, {5444,5903}, {5445,10573}, {5446,5640}, {5449,12118}, {5461,12117}, {5473,6669}, {5474,6670}, {5482,5752}, {5563,10056}, {5569,7759}, {5599,11843}, {5600,11844}, {5651,6759}, {5654,5888}, {5691,10175}, {5692,5884}, {5693,10176}, {5708,5719}, {5720,10884}, {5732,5817}, {5758,9776}, {5768,5791}, {5777,10167}, {5891,12111}, {5893,5925}, {5901,12702}, {5904,12005}, {5907,6241}, {5943,9781}, {5946,6243}, {6054,9167}, {6118,12124}, {6119,12123}, {6193,12359}, {6197,10319}, {6221,7584}, {6224,12619}, {6239,12360}, {6242,12363}, {6282,6692}, {6284,7294}, {6292,7710}, {6398,7583}, {6400,12361}, {6403,11574}, {6445,9543}, {6450,8976}, {6454,8960}, {6560,10576}, {6561,10577}, {6680,9753}, {6685,9548}, {6688,13598}, {6689,7691}, {6702,12119}, {6704,12122}, {6722,7872}, {6761,12096}, {6769,10582}, {7173,12953}, {7317,12735}, {7610,9741}, {7619,7775}, {7622,7751}, {7694,7853}, {7722,12358}, {7739,7755}, {7747,8588}, {7748,8589}, {7758,7780}, {7782,11185}, {7790,9734}, {7803,7857}, {7808,8722}, {7810,7888}, {7822,8721}, {7830,7862}, {7831,7940}, {7835,12203}, {7836,10104}, {7870,11179}, {7904,7925}, {7905,11008}, {7910,13449}, {7914,9873}, {7931,9863}, {7957,13374}, {7989,10172}, {7991,13464}, {8071,10321}, {8125,8130}, {8126,8129}, {8148,10283}, {8151,9168}, {8222,11846}, {8223,11847}, {8254,12307}, {8416,8975}, {8537,11511}, {8553,9608}, {9655,10592}, {9668,10593}, {9740,12040}, {9778,11230}, {9785,11373}, {9812,9955}, {9820,12163}, {9821,10583}, {9829,12506}, {9833,11202}, {9932,12318}, {10163,12505}, {10170,12162}, {10198,10532}, {10263,11002}, {10272,10620}, {10282,11206}, {10320,10629}, {10470,10479}, {10525,10584}, {10526,10585}, {10574,11444}, {10586,10679}, {10587,10680}, {10598,11826}, {10599,11827}, {10632,11515}, {10633,11516}, {10706,10990}, {10707,10993}, {10880,11513}, {10881,11514}, {11194,12607}, {11425,13567}, {11442,11449}, {11456,13394}, {11557,13201}, {11806,12273}, {12026,13512}, {12249,12864}, {12255,13089}, {12319,12893}, {12320,12972}, {12321,12973}, {12322,12974}, {12323,12975}, {12528,13369}, {13025,13049}, {13026,13050}, {13027,13035}, {13028,13036}, {13203,13289}, {13352,13434}

X(631) is the {X(2),X(3)}-harmonic conjugate of X(4). For a list of other harmonic conjugates of X(631), click Tables at the top of this page.

X(631) = reflection of X(632) in X(140)
X(631) = isogonal conjugate of X(3527)
X(631) = isotomic conjugate of X(8797)
X(631) = complement of X(3091)
X(631) = anticomplement of X(1656)
X(631) = circumcircle-inverse of X(37925)
X(631) = orthocentroidal-circle-inverse of X(3090)
X(631) = insimilicenter of circumcircle and 1st Steiner circle; the exsimilicenter is X(20)
X(631) = pole wrt polar circle of trilinear polar of X(8796)
X(631) = X(48)-isoconjugate (polar conjugate)-of-X(8796)
X(631) = {X(3),X(4)}-harmonic conjugate of X(376)
X(631) = insimilicenter of circumcircles of Euler and anti-Euler triangles; the exsimilicenter is X(4)
X(631) = homothetic center of circumorthic triangle and mid-triangle of orthic and dual of orthic triangles
X(631) = perspector of ABC and cross-triangle of ABC and anti-Euler triangle
X(631) = orthocenter of cross-triangle of Euler and anti-Euler triangles
X(631) = homothetic center of X(4)-altimedial and X(2)-anti-altimedial triangles
X(631) = Euler line intercept, other than X(20), of conic {X(13),X(14),X(17),X(18),X(20)}}
X(631) = pole of Brocard axis wrt conic {X(2),X(15),X(16),X(17),X(18)}}


X(632) = 9/10*OG

Trilinears    (6SBSC + 7SAa2)/a
Trilinears    7 cos A + 6 cos B cos C : :
Barycentrics    cot B cot C - 7 : :

As a point on the Euler line, X(632) has Shinagawa coefficients (7,-1).

X(632) lies on these lines: 2,3   141,575

X(632) is the {X(2),X(140)}-harmonic conjugate of X(5). For a list of other harmonic conjugates of X(632), click Tables at the top of this page.

X(632) = reflection of X(631) in X(140)
X(632) = complement of X(1656)


X(633) = ANTICOMPLEMENT OF X(61)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (SBSC + 2 sqr(3) SA area)/a

Trilinears        F(61)/a - 2 cos(A - π/3) : F(61)/b - 2 cos(B - π/3) : F(61)/c - 2 cos(C - π/3)

Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(633) lies on these lines: 2,18   3,298   4,69   5,299   14,636   17,623   20,616   140,302   141,398   343,471   394,470   397,524

X(633) = isogonal conjugate of X(3442)
X(633) = anticomplement of X(61)
X(633) = anticomplementary conjugate of X(627)


X(634) = ANTICOMPLEMENT OF X(62)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (SBSC - 2 sqr(3) SA area)/a

Trilinears        F(62)/a + 2 cos(A + π/3) : F(62)/b + 2 cos(B + π/3) : F(62)/c + 2 cos(C + π/3)

Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(634) lies on these lines: 2,17   3,299   4,69   5,298   13,635   18,624   20,617   140,303   141,397   343,470   394,471   398,524

X(634) = reflection of X(i) in X(j) for these (i,j): (62,636), (61,635)
X(634) = isogonal conjugate of X(3443)
X(634) = anticomplement of X(62)
X(634) = anticomplementary conjugate of X(628)


X(635) = COMPLEMENT OF X(61)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [2SBSC + SAa2 + 2 sqr(3) (b2 + c2) area]/a

Trilinears        F(61)/a - cos(A - π/3) : F(61)/b - cos(B - π/3) : F(61)/c - cos(C - π/3)

Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(635) lies on these lines: 2,18   3,618   5,141   13,634   16,627   17,299   62,298   140,619   397,532

X(635) = midpoint of X(61) and X(633)
X(635) = complement of X(61)
X(635) = complementary conjugate of X(629)


X(636) = COMPLEMENT OF X(62)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [2SBSC + SAa2 - 2 sqr(3) (b2 + c2) area]/a

Trilinears        F(62)/a + cos(A + π/3) : F(62)/b + cos(B + π/3) : F(62)/c + cos(C + π/3)

Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(636) lies on these lines: 2,17   3,619   5,141   14,633   15,628   18,298   61,299   140,618   398,533

X(636) = midpoint of X(62) and X(634)
X(636) = complement of X(62)
X(636) = complementary conjugate of X(630)


X(637) = ANTICOMPLEMENT OF X(371)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (SBSC + 2SA area)/a
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(637) lies on these lines: 2,371   3,489   4,69   5,491   20,488   30,490

X(637) = reflection of X(i) in X(j) for these (i,j): (371,639), (638,315)
X(637) = anticomplement of X(371)
X(637) = anticomplementary conjugate of X(488)


X(638) = ANTICOMPLEMENT OF X(372)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (SBSC - 2SA area)/a
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(638) lies on these lines: 2,372   3,490   4,69   5,492   20,487   30,489

X(638) = reflection of X(i) in X(j) for these (i,j): (372,640), (637,315)
X(638) = anticomplement of X(372)
X(638) = anticomplementary conjugate of X(487)


X(639) = COMPLEMENT OF X(371)

Trilinears    [2SBSC + SAa2 + 2(b2 + c2) area]/a : :
Barycentrics    2 + sin 2B + sin 2C - cos 2B - cos 2C< : :
Barycentrics    (a2 + 4 area(ABC))(b2 + c2) - (b2 - c2)2
Barycentrics    a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4 + 2*(b^2 + c^2)*S : :
X(639) = 3 X[2] + X[637], 9 X[2] - X[43134], 3 X[371] - X[43134], 3 X[637] + X[43134], 3 X[597] - 2 X[44482], 5 X[631] - X[26441], 5 X[1656] - X[45489], 3 X[35949] - X[42266]

If you have GeoGebra, you can view X(639)

X(639) lies on thesse lines: {2, 371}, {3, 641}, {4, 5590}, {5, 141}, {6, 11313}, {10, 31557}, {32, 615}, {39, 51401}, {69, 485}, {115, 53479}, {140, 43120}, {182, 49355}, {298, 33350}, {299, 33352}, {315, 372}, {325, 3103}, {343, 1591}, {376, 13701}, {394, 8968}, {488, 6560}, {489, 6200}, {490, 35820}, {491, 5417}, {524, 7583}, {590, 1504}, {591, 3312}, {597, 19116}, {599, 42265}, {618, 34552}, {619, 34551}, {620, 43144}, {630, 35746}, {631, 26361}, {638, 6564}, {754, 13966}, {1078, 60194}, {1151, 11315}, {1152, 41490}, {1160, 15835}, {1270, 1587}, {1350, 36656}, {1352, 48467}, {1503, 6214}, {1506, 13983}, {1585, 35765}, {1599, 9683}, {1656, 45473}, {1991, 8976}, {2459, 7750}, {2460, 7807}, {3069, 7375}, {3070, 7818}, {3071, 7822}, {3090, 5591}, {3098, 45542}, {3102, 6656}, {3316, 60223}, {3525, 33365}, {3589, 7584}, {3593, 13935}, {3619, 42274}, {3620, 42277}, {3629, 19117}, {3631, 18538}, {3642, 15765}, {3643, 18585}, {3763, 11314}, {3767, 45576}, {3788, 9738}, {3836, 31558}, {4138, 31591}, {5028, 49220}, {5050, 49317}, {5054, 13821}, {5062, 44392}, {5067, 26362}, {5420, 11291}, {5860, 7581}, {5861, 13886}, {5874, 8550}, {6036, 49104}, {6228, 6250}, {6278, 6776}, {6280, 14912}, {6292, 53503}, {6302, 42241}, {6303, 42239}, {6304, 42257}, {6305, 42256}, {6306, 33394}, {6307, 33393}, {6393, 13877}, {6396, 11293}, {6420, 62987}, {6421, 35684}, {6561, 11292}, {6565, 7388}, {6566, 7873}, {6567, 7874}, {6680, 45872}, {6811, 11824}, {7376, 42561}, {7749, 50374}, {7761, 9739}, {7763, 45564}, {7778, 9732}, {7784, 9733}, {7791, 45565}, {7795, 37343}, {7800, 37342}, {7821, 51395}, {7830, 43141}, {7864, 22613}, {7867, 32490}, {7886, 8981}, {7915, 42215}, {8252, 11316}, {8253, 12962}, {8948, 32588}, {8960, 62986}, {9675, 32790}, {10194, 60205}, {10514, 40330}, {10516, 36655}, {11090, 55501}, {11178, 45543}, {11294, 35821}, {11898, 49318}, {12323, 42269}, {13335, 49103}, {13567, 15236}, {13663, 13903}, {13757, 35814}, {13880, 62202}, {13930, 59635}, {15048, 53483}, {15234, 37636}, {17811, 55887}, {18762, 34573}, {18840, 36664}, {20065, 35813}, {21737, 22716}, {22627, 42248}, {22629, 42249}, {22644, 58803}, {22883, 33426}, {22928, 33425}, {29181, 36658}, {30270, 36709}, {30435, 45487}, {31555, 45305}, {32786, 41410}, {32789, 62241}, {32807, 39388}, {32808, 35822}, {32811, 42602}, {33392, 53464}, {33395, 53453}, {33448, 53432}, {33450, 53444}, {34091, 54628}, {34391, 57904}, {34507, 44486}, {35770, 45421}, {35949, 42266}, {42260, 58804}, {42261, 55041}, {43118, 48734}, {43119, 49086}, {43121, 44390}, {43142, 48773}, {44380, 44475}, {45406, 45553}, {45444, 48746}, {45500, 49347}, {45510, 45552}, {48769, 53475}, {49039, 53033}, {53015, 61096}

X(639) = midpoint of X(i) and X(j) for these {i,j}: {4, 11825}, {69, 35840}, {315, 372}, {371, 637}, {485, 42009}, {490, 35820}, {32808, 35822}, {34507, 44486}
X(639) = reflection of X(i) in X(j) for these {i,j}: {640, 626}, {8550, 44484}, {43120, 140}
X(639) = complement of X(371)
X(639) = anticomplement of X(64691)
X(639) = nine-point-circle-inverse of X(32432)
X(639) = complement of the isogonal conjugate of X(485)
X(639) = complement of the isotomic conjugate of X(34391)
X(639) = medial-isogonal conjugate of X(641)
X(639) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 641}, {19, 8968}, {91, 640}, {485, 10}, {1820, 55890}, {3377, 642}, {6413, 1214}, {8577, 37}, {11090, 18589}, {13455, 9}, {16032, 21231}, {34391, 2887}, {36145, 54028}, {39383, 14838}, {41515, 226}, {54031, 4369}, {58825, 16592}
X(639) = X(i)-Ceva conjugate of X(j) for these (i,j): {1306, 54029}, {46134, 54028}
X(639) = X(639)-Dao conjugate of X(371)
X(639) = crosspoint of X(2) and X(34391)
X(639) = crosssum of X(10132) and X(26920)
X(639) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 487, 5418}, {2, 637, 371}, {2, 10577, 6119}, {3, 6289, 48466}, {3, 45472, 641}, {5, 141, 640}, {141, 23311, 5}, {487, 5418, 41491}, {489, 39387, 6200}, {492, 7389, 372}, {590, 32491, 6118}, {623, 635, 640}, {624, 636, 640}, {625, 40107, 640}, {638, 32489, 6564}, {2039, 2040, 32432}, {3454, 24220, 640}, {3763, 42262, 11314}, {3934, 24206, 640}, {5031, 49111, 640}, {6306, 33394, 53456}, {6307, 33393, 53467}, {7849, 19130, 640}, {11291, 32805, 5420}, {11292, 12322, 6561}, {11293, 45508, 6396}, {15067, 34827, 640}, {21245, 37536, 640}, {44392, 53487, 5062}


X(640) = COMPLEMENT OF X(372)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [2SBSC + SAa2 - 2(b2 + c2) area]/a
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(640) lies on these lines: 2,372   3,642   5,141   69,486   315,371

X(640) = midpoint of X(i) and X(j) for these (i,j): (315,371), (372,638)
X(640) = reflection of X(639) in X(626)
X(640) = complement of X(372)
X(640) = complementary conjugate of X(642)


X(641) = COMPLEMENT OF X(485)

Trilinears    [2SBSC + 3SAa2 + 2(b2 + c2) area]/a : :
Barycentrics  g(a,b,c) : 2*a^4 - 3*a^2*b^2 + b^4 - 3*a^2*c^2 - 2*b^2*c^2 + c^4 - 2*(b^2 + c^2)*S : :
X(641) = 3 X[2] + X[488], 9 X[2] - X[12222], 7 X[2] - X[22485], 3 X[485] - X[12222], 7 X[485] - 3 X[22485], X[485] + 3 X[55041], X[488] + 2 X[6118], 3 X[488] + X[12222], 7 X[488] + 3 X[22485], X[488] - 3 X[55041], 6 X[6118] - X[12222], 14 X[6118] - 3 X[22485], 2 X[6118] + 3 X[55041], 3 X[6304] + X[22627], 3 X[6304] - X[22629], and many others

Erect squares outwards on the sides of triangle ABC; the circumcenter of the centers of the squares is X(641). (Peter Moses, 7/16,2003)

X(641) lies on the Kiepert circumhyperbola of the medial triangle, the cubic K801, and these lines: {1, 12788}, {2, 372}, {3, 639}, {4, 12124}, {5, 6250}, {6, 11315}, {8, 7981}, {9, 60890}, {10, 45715}, {11, 13082}, {12, 18988}, {30, 7690}, {39, 615}, {55, 12959}, {56, 12949}, {69, 5418}, {83, 12211}, {99, 60194}, {113, 48730}, {114, 48467}, {115, 48784}, {119, 48686}, {125, 48786}, {127, 48788}, {132, 48732}, {136, 32500}, {140, 141}, {214, 48754}, {230, 45576}, {302, 42564}, {303, 42562}, {343, 55865}, {371, 492}, {373, 21654}, {427, 12148}, {486, 5490}, {487, 8913}, {490, 6564}, {498, 10068}, {499, 10084}, {524, 8981}, {549, 13821}, {574, 53480}, {590, 5062}, {591, 3311}, {597, 44474}, {618, 18585}, {619, 15765}, {620, 43120}, {623, 14814}, {624, 14813}, {626, 43121}, {629, 48752}, {630, 48750}, {631, 5590}, {636, 35744}, {637, 6200}, {946, 48740}, {958, 12939}, {1125, 12269}, {1151, 32419}, {1152, 11313}, {1209, 48774}, {1270, 9540}, {1352, 48742}, {1376, 12344}, {1504, 44392}, {1511, 48736}, {1583, 6503}, {1588, 3593}, {1599, 55532}, {1650, 12800}, {1656, 12602}, {1698, 9907}, {2482, 3071}, {2883, 48766}, {2996, 10194}, {3068, 13771}, {3069, 19102}, {3070, 6566}, {3090, 12510}, {3091, 12297}, {3096, 9987}, {3102, 5976}, {3317, 42023}, {3366, 38400}, {3367, 36779}, {3525, 5591}, {3526, 45410}, {3533, 26362}, {3589, 13966}, {3628, 23312}, {3647, 48756}, {3666, 5405}, {3763, 11316}, {3815, 18993}, {5020, 12979}, {5055, 22810}, {5058, 8997}, {5235, 64389}, {5491, 45516}, {5552, 13134}, {5599, 12486}, {5600, 12487}, {5745, 31540}, {5943, 12238}, {6119, 8252}, {6251, 23698}, {6260, 48748}, {6292, 48770}, {6396, 7389}, {6419, 62987}, {6453, 43134}, {6509, 55885}, {6560, 32499}, {6561, 51579}, {6565, 11294}, {6593, 48782}, {6626, 21909}, {6642, 12973}, {6811, 11825}, {7376, 13834}, {7388, 10577}, {7392, 12321}, {7582, 45078}, {7583, 45871}, {7710, 9757}, {7746, 53479}, {7759, 43124}, {7761, 12975}, {7769, 12218}, {7778, 43118}, {7834, 45872}, {7853, 15886}, {7866, 15884}, {7914, 35256}, {8222, 13004}, {8223, 13005}, {8225, 8299}, {8290, 33341}, {8786, 22563}, {8966, 19408}, {9306, 12230}, {9605, 45487}, {9737, 45545}, {9738, 44390}, {9744, 45552}, {9813, 12598}, {9816, 12663}, {9817, 12911}, {9818, 12985}, {10527, 13135}, {10601, 19462}, {10643, 12982}, {10644, 12983}, {10961, 12961}, {10963, 12967}, {11147, 12322}, {11165, 13951}, {11284, 12170}, {11451, 12275}, {11465, 12286}, {11479, 12304}, {11484, 12312}, {11614, 32789}, {12305, 36656}, {12323, 22646}, {12864, 48762}, {13089, 48776}, {13449, 32435}, {13651, 32785}, {13712, 22644}, {13758, 45513}, {13783, 51588}, {13875, 22848}, {13876, 22892}, {13878, 22724}, {13989, 49220}, {14229, 22718}, {14645, 44501}, {15819, 48744}, {15835, 26348}, {16441, 40592}, {17056, 63302}, {17825, 17842}, {18420, 22818}, {18586, 22635}, {18587, 22634}, {18928, 18938}, {19137, 19144}, {19146, 32954}, {19188, 19200}, {19372, 19474}, {19449, 19491}, {21616, 31591}, {22966, 48760}, {26359, 45349}, {26360, 45351}, {26363, 45526}, {26364, 45528}, {26468, 35945}, {26615, 42413}, {26619, 42561}, {30471, 35849}, {30472, 35847}, {32391, 48758}, {32489, 35820}, {32490, 32790}, {32503, 32513}, {32811, 43254}, {33346, 33370}, {33348, 33372}, {33350, 33368}, {33351, 62600}, {33352, 33366}, {33353, 62601}, {33358, 33440}, {33360, 33442}, {33425, 33426}, {34551, 44382}, {34552, 44383}, {35771, 45421}, {35812, 62986}, {35815, 45420}, {35821, 35949}, {37340, 53468}, {37341, 53457}, {40107, 44475}, {42258, 51581}, {42269, 58803}, {43291, 53482}, {43559, 60207}, {43880, 51587}, {44380, 44486}, {48769, 51373}, {48785, 62348}, {52032, 56506}, {52193, 53467}, {52194, 53456}, {56502, 62589}

X(641) =midpoint of X(i) and X(j) for these {i,j}: {1, 12788}, {2, 55041}, {3, 6289}, {4, 12124}, {8, 7981}, {99, 60270}, {371, 42009}, {485, 488}, {486, 6337}, {487, 22591}, {490, 32495}, {492, 49790}, {1650, 12800}, {3102, 13877}, {6228, 22623}, {6278, 12257}, {6304, 6305}, {6311, 6312}, {7690, 45542}, {12305, 36656}, {12322, 42260}, {12344, 12929}, {12939, 22624}, {22627, 22629}
X(641) =reflection of X(i) in X(j) for these {i,j}: {485, 6118}, {6250, 5}, {12269, 1125}, {13879, 8180}, {13881, 6119}, {48735, 49104}, {49104, 140}
X(641) =complement of X(485)
X(641) =anticomplement of X(6118)
X(641) =complement of the isogonal conjugate of X(371)
X(641) =complement of the isotomic conjugate of X(492)
X(641) =isotomic conjugate of the polar conjugate of X(44637)
X(641) =medial-isogonal conjugate of X(639)
X(641) =X(i)-complementary conjugate of X(j) for these (i,j): {1, 639}, {31, 590}, {47, 642}, {48, 55885}, {163, 54029}, {371, 10}, {486, 34825}, {492, 2887}, {1585, 20305}, {1973, 49221}, {3378, 640}, {5408, 18589}, {5413, 226}, {6414, 18588}, {8911, 1214}, {32676, 14334}, {41516, 63843}, {45805, 21235}, {54029, 21253}, {55398, 141}
X(641) =X(i)-Ceva conjugate of X(j) for these (i,j): {2, 590}, {99, 54029}
X(641) =X(i)-Dao conjugate of X(j) for these (i,j): {590, 2}, {641, 485}, {10962, 588}
X(641) =crosspoint of X(2) and X(492)
X(641) =crosssum of X(6) and X(8577)
X(641) =barycentric product X(i)*X(j) for these {i,j}: {69, 44637}, {492, 590}, {5062, 45805}
X(641) =barycentric quotient X(i)/X(j) for these {i,j}: {371, 588}, {492, 60194}, {590, 485}, {5062, 8577}, {8911, 8825}, {44637, 4}, {44647, 61390}, {52287, 41515}
X(641) ={X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 45546, 48746}, {2, 485, 6118}, {2, 488, 485}, {2, 638, 10576}, {2, 5408, 8968}, {2, 45508, 372}, {3, 45472, 639}, {3, 45554, 48466}, {4, 45522, 45498}, {5, 9739, 45544}, {39, 615, 45577}, {140, 141, 642}, {140, 48772, 182}, {182, 3788, 642}, {182, 48772, 48734}, {372, 45508, 41490}, {485, 55041, 488}, {492, 39387, 371}, {590, 5062, 45574}, {590, 44647, 13879}, {615, 32497, 49221}, {615, 49221, 13880}, {631, 45510, 45553}, {1125, 48764, 45500}, {6304, 22627, 22629}, {6305, 22629, 22627}, {6311, 13088, 6312}, {6337, 32805, 5490}, {7388, 32807, 10577}, {8252, 11314, 6119}, {10576, 35832, 485}, {11292, 32805, 486}, {26361, 33364, 4}, {33444, 33446, 2}, {33444, 33450, 485}, {33446, 33448, 485}, {33448, 33450, 488}
X(641) = X(3)-of-outer-Vecten-triangle
X(641) = outer-Vecten-isogonal conjugate of X(485)
X(641) = perspector of circumconic centered at X(590)
X(641) = center of circumconic that is locus of trilinear poles of lines passing through X(590)
X(641) = perspector of medial triangle and outer Vecten of outer Vecten triangle (note: the medial triangle is the inner Vecten of outer Vecten triangle.)


X(642) = COMPLEMENT OF X(486)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [2SBSC + 3SAa2 - 2(b2 + c2) area]/a
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

Erect squares inwards on the sides of triangle ABC; the circumcenter of the centers of the squares is X(642). (Peter Moses, 7/16,2003)

X(642) lies on these lines: 2,371   3,640   140,141   372,491

X(642) = midpoint of X(486) and X(487)
X(642) = complement of X(486)
X(642) = complementary conjugate of X(640)
X(642) = crosspoint of X(2) and X(491)
X(642) = X(3)-of-inner-Vecten-triangle
X(642) = inner-Vecten-isogonal conjugate of X(486)
X(642) = perspector of circumconic centered at X(615)
X(642) = center of circumconic that is locus of trilinear poles of lines passing through X(615)
X(642) = X(2)-Ceva conjugate of X(615)
X(642) = perspector of medial triangle and outer Vecten of inner Vecten triangle (note: the medial triangle is the inner Vecten of inner Vecten triangle.)


X(643) = TRILINEAR MULTIPLIER FOR KIEPERT PARABOLA

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)/(b2 - c2)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(643) satisfies the equation X*(incircle) = Kiepert parabola, where * denotes trilinear multiplication, defined by
(u : v : w) * (x : y : z) = ux : vy : wz.

X(643) lies on these lines: 8,1098   99,109   100,110   101,931   162,190   163,1018   212,312   283,1043

X(643) = isogonal conjugate of X(4017)
X(643) = isotomic conjugate of X(4077)
X(643) = trilinear pole of line X(9)X(21)


X(644) = TRILINEAR MULTIPLIER FOR YFF PARABOLA

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)/(b - c)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(644) satisfies the equation X*(incircle) = Yff parabola, where * denotes trilinear multiplication, defined by
(u : v : w) * (x : y : z) = ux : vy : wz.

X(644) is the perspector of the anticevian triangle of X(100) and the unary cofactor triangle of the intangents triangle. (Randy Hutson, January 15, 2019)

X(644) lies on these lines: 8,220   78,728  100,101   105,1083   145,218   190,651  219,346   645,646   666,668   813,932   934,1025

X(644) = reflection of X(i) in X(j) for these (i,j): (105,1083), (1280,1)
X(644) = isogonal conjugate of X(3669)
X(644) = isotomic conjugate of X(24002)
X(644) = anticomplement of X(4904)
X(644) = X(190)-Ceva conjugate of X(100)
X(644) = crosssum of X(764) and X(1015)
X(644) = crossdifference of every pair of points on line X(244)X(1357)
X(644) = trilinear pole of line X(9)X(55) (tangent to Feuerbach hyperbola at X(9))
X(644) = X(650)-cross conjugate of X(9)
X(644) = eigencenter of Caelum triangle
X(644) = eigencenter of 5th mixtilinear triangle
X(644) = perspector of unary cofactor triangles of 3rd and 5th mixtilinear triangles
X(644) = perspector of 5th mixtilinear triangle and unary cofactor triangle of 3rd mixtilinear triangle
X(644) = intersection, other than vertices of Gemini triangle 29, of {ABC, Gemini 29}-circumconic and {Gemini 29, Gemini 30}-circumconic


X(645) = BARYCENTRIC MULTIPLIER FOR KIEPERT PARABOLA

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)/[a(b2 - c2)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (b + c - a)/(b2 - c2)

X(645) satisfies the equation X*(incircle) = Kiepert parabola, where * denotes barycentric multiplication, defined by
(u : v : w) * (x : y : z) = ux : vy : wz (barycentric coordinates; see note at X(2)).

X(645) lies on these lines: 9,261   99,101   100,931   294,314   644,646   648,668   651,799   666,670

X(645) = anticomplement of X(17058)
X(645) = trilinear pole of line X(8)X(21)
X(645) = perspector of anticevian triangle of X(99) and unary cofactor triangle of intangents triangle


X(646) = X(650)-CROSS CONJUGATE OF X(8)

Trilinears    (b + c - a)/[a2(b - c)] : :
Barycentrics    (b + c - a)/[a(b - c)] : :

X(646) lies on these lines: 190,668   644,645

X(646) = isotomic conjugate of X(3669)
X(646) = trilinear pole of line X(8)X(210) (the tangent to Feuerbach hyperbola at X(8))
X(646) = X(650)-cross conjugate of X(8)
X(646) = X(522)-cross conjugate of X(314)
X(646) = PU(4)-harmonic conjugate of polar conjugate of X(35158)


X(647) = CROSSDIFFERENCE OF X(2) AND X(3)

Trilinears    a(b2 - c2)(b2 + c2 - a2) : :
Trilinears    sin 2A sin(B - C) : :
Trilinears    (sin A)(tan B - tan C) : :
Trilinears    cos(A - 2B) - cos(A - 2C) : :
Barycentrics    a^2*(b^2 - c^2)*SA : :
X(647) = 9 X[2] - 5 X[31072], 6 X[2] - 5 X[31277], 3 X[2] + X[31296], 3 X[2] + 2 X[41300], 3 X[351] - X[669], 3 X[351] + X[3005], 6 X[351] - X[3804], 3 X[351] - 2 X[8651], 9 X[351] - X[8664], 9 X[351] + X[8665], 2 X[669] - 3 X[8644], 3 X[669] - X[8664], 3 X[669] + X[8665], X[669] + 3 X[17414], 3 X[850] - 5 X[31072], 2 X[850] - 3 X[31174], 2 X[850] - 5 X[31277], X[850] + 3 X[36900], X[850] + 2 X[41300], X[850] - 6 X[44560], 3 X[1636] + X[17434], 3 X[1636] - X[32320], 3 X[1637] - 2 X[2501], 3 X[1637] - 4 X[6587], 3 X[1637] - X[12077], 9 X[1649] - X[2528], 3 X[2485] - 2 X[2492], 3 X[2489] - 4 X[2492], X[2501] - 3 X[9209], X[2525] - 3 X[14417], 2 X[3005] + X[3804], 2 X[3005] + 3 X[8644], X[3005] + 2 X[8651], 3 X[3005] + X[8664], 3 X[3005] - X[8665], X[3005] - 3 X[17414], 3 X[3050] + X[39232], 2 X[3265] - 3 X[14417], X[3288] + 3 X[9210], X[3569] - 3 X[9210], X[3804] - 3 X[8644], X[3804] - 4 X[8651], 3 X[3804] - 2 X[8664], 3 X[3804] + 2 X[8665], X[3804] + 6 X[17414], X[3806] + 9 X[9125], 3 X[5996] - X[44445], 2 X[6587] - 3 X[9209], 4 X[6587] - X[12077], 3 X[8644] - 4 X[8651], 9 X[8644] - 2 X[8664], 9 X[8644] + 2 X[8665], X[8644] + 2 X[17414], 6 X[8651] - X[8664], 6 X[8651] + X[8665], 2 X[8651] + 3 X[17414], X[8664] + 9 X[17414], X[8665] - 9 X[17414], 3 X[9147] + X[44445], 3 X[9148] - 5 X[31279], 6 X[9209] - X[12077], X[9409] + 3 X[15451], X[9409] - 3 X[39201], 2 X[9409] - 3 X[42658], 3 X[9979] + X[41298], 3 X[11176] - 2 X[44451], 2 X[15451] + X[42658], 6 X[30476] - 5 X[31072], 4 X[30476] - 3 X[31174], 4 X[30476] - 5 X[31277], 2 X[30476] + X[31296], 2 X[30476] + 3 X[36900], X[30476] - 3 X[44560], 10 X[31072] - 9 X[31174], 2 X[31072] - 3 X[31277], 5 X[31072] + 3 X[31296], 5 X[31072] + 9 X[36900], 5 X[31072] + 6 X[41300], 5 X[31072] - 18 X[44560], 3 X[31174] - 5 X[31277], 3 X[31174] + 2 X[31296], X[31174] + 2 X[36900], 3 X[31174] + 4 X[41300], X[31174] - 4 X[44560], 5 X[31277] + 2 X[31296], 5 X[31277] + 6 X[36900], 5 X[31277] + 4 X[41300], 5 X[31277] - 12 X[44560], X[31296] - 3 X[36900], X[31296] + 6 X[44560], 3 X[36900] - 2 X[41300], X[36900] + 2 X[44560], X[41300] + 3 X[44560]

X(647) is the point whose trilinears are coefficients for the Euler line.
X(647) = radical center of the circumcircle, nine-point circle, and Brocard circle (Wilson Stothers, 3/13/2003)
X(647) is the perspector of triangle ABC and the tangential triangle of the Jerabek hyperbola. (Randy Hutson, 9/23/2011)

Let L be the line at infinity. X(647) = X(230)X(231)∩X(441)X(525); that is, the polar conjugate of isotomic conjugate of L and the isotomic conjugate of polar conjugate of L. (Randy Hutson, February 10, 2016)

The circumcircle of the anti-orthocentroidal triangle is here named the anti-orthocentroidal circle. The point X(647) is the radical center of these three circles: circumcircle, orthocentroidal circle, anti-orthocentroidal circle. (Randy Hutson, December 10, 2016)

X(647) lies on these lines: {1, 1021}, {2, 850}, {3, 10097}, {6, 2433}, {10, 21719}, {23, 13114}, {25, 34212}, {32, 44895}, {37, 3700}, {39, 1640}, {42, 4524}, {50, 654}, {75, 21437}, {99, 9091}, {110, 2715}, {111, 842}, {112, 1304}, {114, 38975}, {115, 3258}, {122, 33504}, {125, 22264}, {136, 38970}, {141, 9030}, {184, 878}, {185, 9242}, {187, 237}, {216, 14401}, {230, 231}, {248, 35909}, {352, 9138}, {441, 525}, {511, 40283}, {513, 43060}, {514, 23723}, {520, 652}, {570, 8562}, {574, 44814}, {612, 4477}, {656, 8611}, {657, 2451}, {661, 3709}, {688, 2513}, {804, 23301}, {810, 20754}, {826, 7651}, {879, 43718}, {881, 38237}, {906, 23084}, {1015, 35090}, {1084, 6791}, {1125, 19948}, {1194, 10190}, {1196, 11123}, {1214, 17094}, {1301, 32687}, {1331, 23139}, {1499, 11615}, {1555, 19912}, {1560, 42426}, {1562, 35071}, {1570, 30219}, {1577, 27731}, {1624, 35325}, {1635, 4139}, {1962, 6608}, {1995, 33752}, {2394, 43530}, {2422, 5651}, {2430, 14642}, {2436, 18320}, {2516, 4145}, {2605, 7252}, {2611, 14936}, {2780, 32231}, {2799, 6563}, {2966, 9514}, {2972, 38356}, {3117, 11183}, {3124, 38987}, {3221, 17415}, {3261, 25511}, {3267, 4580}, {3666, 17069}, {3708, 7117}, {3766, 27293}, {3768, 23751}, {4132, 4394}, {4151, 4765}, {4391, 25902}, {4458, 16612}, {4467, 16751}, {4789, 24900}, {5466, 7607}, {5642, 11672}, {5972, 23584}, {5996, 9147}, {6331, 43188}, {6368, 14345}, {7630, 30474}, {7631, 9148}, {7656, 25925}, {7746, 23105}, {8045, 14838}, {8371, 37742}, {8673, 10317}, {8704, 9185}, {9033, 44892}, {9044, 9188}, {9137, 11580}, {9168, 9465}, {9171, 22111}, {9175, 20481}, {9178, 21448}, {9216, 32583}, {9306, 22391}, {9426, 19558}, {9517, 30491}, {9979, 41298}, {10192, 40588}, {10766, 14919}, {11063, 39180}, {11124, 40973}, {11424, 14456}, {13006, 23993}, {13558, 20998}, {14165, 14618}, {14208, 24459}, {14321, 21894}, {14341, 44817}, {14376, 23107}, {14773, 23292}, {14966, 15329}, {15328, 46262}, {15411, 15420}, {15820, 18313}, {16186, 41172}, {16598, 23988}, {16599, 23585}, {18675, 40628}, {18877, 32663}, {21053, 27711}, {21225, 26114}, {21761, 23567}, {21789, 43925}, {21837, 24290}, {22055, 22375}, {22089, 22091}, {22121, 22155}, {23786, 23887}, {23963, 34947}, {25423, 45333}, {25861, 25862}, {27648, 45746}, {30230, 40115}, {32120, 35282}, {32661, 32662}, {33843, 42733}, {39013, 39021}, {39482, 39601}, {39503, 39799}, {41336, 44896}

X(647) = midpoint of X(i) and X(j) for these {i,j}: {2, 36900}, {351, 17414}, {649, 42664}, {669, 3005}, {850, 31296}, {2966, 46245}, {3267, 4580}, {3288, 3569}, {5996, 9147}, {6137, 6138}, {6563, 33294}, {8639, 42661}, {8664, 8665}, {14270, 18117}, {14618, 15412}, {15451, 39201}, {17431, 17432}, {17434, 32320}, {21731, 42660}, {30476, 41300}
X(647) = reflection of X(i) in X(j) for these {i,j}: {2, 44560}, {125, 22264}, {669, 8651}, {850, 30476}, {1495, 42654}, {1637, 9209}, {2489, 2485}, {2501, 6587}, {2525, 3265}, {3804, 669}, {6753, 16040}, {8644, 351}, {12077, 2501}, {31174, 2}, {31296, 41300}, {42658, 39201}, {45907, 2491}
X(647) = isogonal conjugate of X(648)
X(647) = isotomic conjugate of X(6331)
X(647) = complement of X(850)
X(647) = anticomplement of X(30476)
X(647) = circumcircle-inverse of X(35901)
X(647) = Parry-circle-inverse of X(3569)
X(647) = Dao-Moses-Telv-circle inverse of X(6130)
X(647) = orthoptic-circle-of-Steiner-inellipe-inverse of X(34235)
X(647) = Moses-Parry-circle-inverse of X(1637)
X(647) = Parry-isodynamic-circle inverse of X(5191)
X(647) = polar conjugate of X(6528)
X(647) = orthic-isogonal conjugate of X(34980)
X(647) = center of the Moses radical circle
X(647) = tripolar centroid for these (i,j): {41518, 41519}
X(647) = trilinear pole of line {3269, 9409}
X(647) = crossdifference of every pair of points on line {2, 3}
X(647) = perspector of the Jerabek hyperbola
X(647) = orthojoin of X(125)
X(647) = perspector of ABC and the side-triangle of the cevian triangles of X(3) and X(6)
X(647) = PU(4)-harmonic conjugate of X(232)
X(647) = bicentric difference of PU(i) for these i: 17, 145, 157
X(647) = PU(17)-harmonic conjugate of X(185)
X(647) = crossdifference of PU(30)
X(647) = barycentric product of PU(75)
X(647) = trilinear pole of PU(109) (line X(3269)X(9409))
X(647) = center of circumconic that is locus of trilinear poles of lines passing through X(125)
X(647) = intersection of trilinear polars of any 2 points on the Jerabek hyperbola
X(647) = X(187)-of-2nd-Parry-triangle
X(647) = X(187)-of-3rd-Parry-triangle
X(647) = intersection of the Lemoine axes of ABC and the 5th Euler triangle
X(647) = intersection of orthic and Lemoine axes (trilinear polars of X(4) and X(6))
X(647) = X(92)-isoconjugate of X(110)
X(647) = pole wrt polar circle of trilinear polar of X(6528)
X(647) = X(48)-isoconjugate (polar conjugate) of X(6528)
X(647) = midpoint of PU(145)
X(647) = PU(157)-harmonic conjugate of X(51)
X(647) = X(649)-of-orthic-triangle if ABC is acute
X(647) = trilinear product of Jerabek hyperbola intercepts of antiorthic axis
X(647) = excentral-to-ABC functional image of X(649)
X(647) = orthoptic-circle-of-Steiner-inellipse-inverse of X(34235)
X(647) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {9255, 13219}, {9258, 3448}, {9292, 21221}, {9307, 21294}, {43188, 6327}
X(647) = X(i)-complementary conjugate of X(j) for these (i,j): {6, 21253}, {31, 125}, {32, 8287}, {48, 127}, {58, 21252}, {99, 21235}, {101, 21245}, {110, 2887}, {112, 20305}, {162, 21243}, {163, 141}, {184, 34846}, {249, 42327}, {250, 21259}, {560, 115}, {662, 626}, {669, 24040}, {691, 21256}, {692, 3454}, {799, 40379}, {827, 21238}, {922, 5099}, {1101, 512}, {1110, 31946}, {1333, 116}, {1397, 8286}, {1408, 17059}, {1414, 17047}, {1415, 17052}, {1501, 16592}, {1576, 10}, {1755, 36471}, {1917, 1084}, {1923, 15449}, {1924, 23991}, {1933, 2679}, {2194, 124}, {2205, 6627}, {2206, 11}, {4556, 21240}, {4565, 17046}, {4567, 21262}, {4570, 21260}, {4575, 1368}, {4590, 21263}, {4602, 40380}, {4630, 1215}, {5546, 21244}, {9247, 15526}, {9417, 35088}, {14574, 37}, {14575, 16573}, {14585, 16595}, {14586, 21231}, {16947, 4904}, {23357, 4369}, {23963, 14838}, {23990, 4129}, {23995, 523}, {24041, 23301}, {32656, 21530}, {32660, 18642}, {32661, 18589}, {32676, 5}, {32678, 34827}, {32729, 4892}, {32734, 34825}, {32739, 1211}, {34072, 3934}, {36084, 21531}, {36134, 3819}, {36142, 625}, {41280, 16613}, {46289, 7668}
X(647) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 3270}, {2, 125}, {3, 20975}, {4, 34980}, {6, 3269}, {25, 38356}, {48, 7117}, {63, 3937}, {99, 6467}, {101, 44093}, {107, 51}, {108, 40952}, {110, 184}, {112, 6}, {184, 38352}, {222, 22094}, {248, 41172}, {254, 8754}, {393, 1562}, {520, 42658}, {523, 512}, {525, 520}, {648, 185}, {650, 661}, {652, 822}, {691, 21639}, {827, 21637}, {879, 39469}, {905, 656}, {906, 18591}, {907, 19459}, {933, 13366}, {935, 2393}, {1020, 1425}, {1073, 2972}, {1113, 44125}, {1114, 44126}, {1214, 18210}, {1289, 1843}, {1296, 10602}, {1301, 25}, {1304, 1495}, {1331, 22080}, {1332, 22076}, {1415, 2092}, {1459, 810}, {1576, 39}, {1942, 35236}, {2164, 20982}, {2165, 115}, {2218, 3271}, {2394, 526}, {2407, 974}, {2433, 686}, {2489, 2519}, {2623, 30451}, {2715, 8779}, {2966, 511}, {3049, 2524}, {3267, 8673}, {3459, 24862}, {3463, 41212}, {3504, 23216}, {3563, 44114}, {3903, 23526}, {3952, 43218}, {4551, 3611}, {4557, 23620}, {4558, 3}, {4559, 1409}, {4563, 3917}, {4574, 71}, {4580, 525}, {5649, 5622}, {6331, 1899}, {7612, 6784}, {8770, 3124}, {9064, 34417}, {10423, 44102}, {14376, 15526}, {14380, 9409}, {14570, 6146}, {14586, 570}, {14618, 924}, {14910, 2088}, {14919, 16186}, {14977, 9517}, {15352, 4}, {15412, 523}, {15420, 521}, {15421, 9033}, {15459, 6000}, {16806, 21647}, {16807, 21648}, {16813, 389}, {18315, 13367}, {18890, 35071}, {22239, 44084}, {23067, 228}, {23086, 22373}, {23181, 23195}, {23286, 39201}, {26705, 40954}, {30247, 40673}, {30249, 44079}, {30441, 26937}, {32640, 3003}, {32641, 2245}, {32661, 216}, {32662, 3284}, {32734, 39643}, {33640, 44110}, {34212, 3569}, {34429, 20974}, {34568, 74}, {35178, 15073}, {36067, 44113}, {39180, 15451}, {39181, 23286}, {39183, 1510}, {39290, 21650}, {39291, 287}, {39382, 8541}, {39383, 21640}, {39384, 21641}, {39417, 1974}, {40097, 44092}, {40116, 39690}, {40323, 6388}, {40347, 1648}, {40428, 15630}, {41210, 41204}, {41676, 26926}, {41677, 32377}, {41678, 1885}, {41769, 127}, {42401, 42400}, {43188, 2}, {43679, 7668}, {44060, 154}, {44769, 21663}, {46087, 34982}
X(647) = X(i)-cross conjugate of X(j) for these (i,j): {686, 14582}, {1084, 40319}, {3049, 512}, {3124, 2351}, {3269, 6}, {3708, 2197}, {9409, 14380}, {15451, 523}, {20975, 3}, {34980, 4}, {35071, 14642}, {35236, 1942}, {38974, 232}, {39201, 520}, {39469, 879}, {41172, 248}, {42293, 39201}
X(647) = X(i)-isoconjugate of X(j) for these (i,j): {1, 648}, {2, 162}, {3, 823}, {4, 662}, {6, 811}, {19, 99}, {21, 653}, {25, 799}, {27, 100}, {28, 190}, {29, 651}, {31, 6331}, {33, 4573}, {34, 645}, {38, 42396}, {47, 30450}, {48, 6528}, {57, 36797}, {58, 6335}, {63, 107}, {69, 24019}, {74, 24001}, {75, 112}, {76, 32676}, {81, 1897}, {82, 41676}, {86, 1783}, {91, 41679}, {92, 110}, {101, 286}, {108, 333}, {109, 31623}, {158, 4558}, {163, 264}, {186, 32680}, {204, 44326}, {225, 4612}, {230, 36105}, {232, 36036}, {240, 2966}, {242, 4584}, {243, 41206}, {249, 24006}, {250, 1577}, {255, 15352}, {270, 4552}, {273, 5546}, {274, 8750}, {275, 2617}, {278, 643}, {281, 1414}, {284, 18026}, {297, 36084}, {304, 32713}, {314, 32674}, {317, 36145}, {318, 4565}, {323, 36129}, {324, 36134}, {325, 36104}, {326, 6529}, {340, 32678}, {393, 4592}, {394, 36126}, {419, 37134}, {423, 37135}, {427, 4599}, {441, 36092}, {468, 36085}, {512, 46254}, {514, 5379}, {525, 24000}, {607, 4625}, {608, 7257}, {647, 23999}, {655, 17515}, {656, 23582}, {658, 4183}, {660, 31905}, {661, 18020}, {664, 1172}, {668, 1474}, {670, 1973}, {673, 4238}, {685, 1959}, {687, 1725}, {692, 44129}, {765, 17925}, {775, 41678}, {827, 20883}, {858, 36095}, {860, 37140}, {877, 1910}, {897, 4235}, {925, 1748}, {931, 5307}, {933, 14213}, {934, 2322}, {935, 16568}, {1010, 36099}, {1019, 15742}, {1043, 32714}, {1096, 4563}, {1099, 34568}, {1101, 14618}, {1113, 2581}, {1114, 2580}, {1119, 7259}, {1235, 34072}, {1289, 1760}, {1301, 18750}, {1303, 9252}, {1304, 14206}, {1332, 8747}, {1396, 3699}, {1398, 7258}, {1415, 44130}, {1435, 7256}, {1492, 31909}, {1576, 1969}, {1625, 40440}, {1633, 40411}, {1733, 32697}, {1755, 22456}, {1784, 44769}, {1812, 36127}, {1813, 1896}, {1821, 4230}, {1824, 4610}, {1839, 4596}, {1843, 4593}, {1936, 41207}, {1953, 18831}, {1957, 43188}, {1974, 4602}, {1978, 2203}, {1981, 37142}, {1995, 37217}, {2052, 4575}, {2166, 14590}, {2167, 35360}, {2173, 16077}, {2190, 14570}, {2201, 4589}, {2204, 4572}, {2216, 41677}, {2287, 36118}, {2299, 4554}, {2313, 41208}, {2326, 4566}, {2328, 13149}, {2332, 4569}, {2333, 4623}, {2349, 4240}, {2355, 4632}, {2407, 36119}, {2421, 36120}, {2489, 24037}, {2501, 24041}, {2576, 15165}, {2577, 15164}, {2586, 8116}, {2587, 8115}, {2588, 39298}, {2589, 39299}, {2631, 42308}, {2633, 15351}, {2715, 40703}, {2964, 38342}, {3112, 35325}, {3194, 44327}, {3257, 37168}, {3260, 36131}, {3580, 36114}, {3658, 37203}, {3737, 46102}, {4143, 24022}, {4206, 37215}, {4222, 37205}, {4232, 37216}, {4233, 37206}, {4241, 36101}, {4242, 24624}, {4246, 34234}, {4248, 27834}, {4249, 37130}, {4551, 46103}, {4556, 41013}, {4560, 7012}, {4561, 5317}, {4567, 7649}, {4570, 17924}, {4577, 17442}, {4591, 38462}, {4594, 7119}, {4600, 6591}, {4603, 7009}, {4606, 31903}, {4616, 7079}, {4620, 18344}, {4622, 8756}, {4627, 5342}, {4628, 16747}, {4633, 5338}, {4635, 7071}, {4636, 40149}, {4637, 7046}, {5271, 36077}, {5468, 36128}, {6516, 8748}, {7035, 43925}, {7045, 17926}, {7115, 18155}, {7128, 7253}, {7452, 36100}, {7480, 36102}, {8753, 24039}, {8767, 34211}, {8822, 40117}, {9390, 39062}, {10423, 20884}, {11107, 38340}, {13138, 41083}, {14006, 37137}, {14165, 36061}, {14208, 23964}, {14212, 33640}, {14543, 40431}, {15149, 36086}, {15411, 24033}, {16081, 23997}, {16237, 36053}, {16813, 44706}, {17438, 33513}, {17569, 37207}, {17983, 23889}, {21582, 39417}, {23090, 24032}, {23347, 33805}, {23353, 35145}, {24018, 32230}, {27369, 37204}, {30528, 36063}, {30737, 36046}, {31900, 37212}, {31902, 37211}, {31926, 37138}, {32002, 36148}, {32696, 46238}, {32715, 46234}, {34055, 46151}, {34085, 37908}, {35201, 39290}, {36034, 46106}, {36043, 44436}, {36142, 44146}, {36149, 44134}
X(647) = cevapoint of X(i) and X(j) for these (i,j): {3, 22143}, {3049, 39201}, {15451, 42293}
X(647) = crosspoint of X(i) and X(j) for these (i,j): {1, 1020}, {2, 110}, {3, 4558}, {4, 15352}, {6, 112}, {37, 4559}, {71, 4574}, {74, 34568}, {99, 40405}, {100, 40406}, {101, 2983}, {107, 275}, {108, 2982}, {109, 40407}, {288, 933}, {523, 525}, {648, 1105}, {650, 652}, {694, 39291}, {905, 1459}, {1073, 1301}, {1113, 8116}, {1114, 8115}, {1214, 23067}, {1289, 40404}, {1304, 40384}, {1331, 1796}, {1332, 1791}, {1576, 10547}, {1799, 4563}, {1813, 40442}, {5994, 41893}, {5995, 41892}, {6331, 34405}, {10419, 43755}, {10423, 41511}, {11079, 32662}, {14533, 32661}, {15412, 23286}, {39180, 39181}
X(647) = crosssum of X(i) and X(j) for these (i,j): {1, 1021}, {2, 525}, {3, 32320}, {4, 2501}, {6, 523}, {21, 23090}, {27, 17925}, {28, 43925}, {29, 17926}, {30, 14401}, {52, 6753}, {81, 4560}, {110, 112}, {185, 647}, {216, 520}, {230, 38359}, {233, 6368}, {323, 5664}, {324, 14618}, {394, 20580}, {512, 1196}, {513, 40941}, {514, 40940}, {521, 40937}, {522, 40942}, {524, 18311}, {650, 1858}, {651, 653}, {850, 1235}, {924, 40939}, {1249, 8057}, {1625, 35360}, {1636, 40948}, {1637, 13202}, {1783, 1897}, {1829, 6591}, {1839, 7649}, {1843, 2489}, {1994, 20577}, {2322, 7253}, {2409, 15639}, {2451, 9308}, {2492, 40949}, {2574, 8106}, {2575, 8105}, {2592, 2593}, {2799, 15595}, {2967, 3569}, {3049, 40947}, {3064, 40950}, {3163, 9033}, {3288, 9755}, {3574, 12077}, {3737, 40979}, {3800, 40179}, {5095, 14273}, {5895, 6587}, {7252, 40980}, {8673, 40938}, {8743, 33294}, {14091, 30211}, {14920, 44427}, {15412, 19170}, {17434, 42441}, {31296, 41334}, {35311, 35318}, {35325, 41676}, {36054, 40946}, {41392, 41512}
X(647) = trilinear pole of line {3269, 9409}
X(647) = crossdifference of every pair of points on line {2, 3}
X(647) = barycentric product X(i)*X(j) for these {i,j}: {1, 656}, {3, 523}, {4, 520}, {5, 23286}, {6, 525}, {10, 1459}, {11, 23067}, {12, 23189}, {19, 24018}, {25, 3265}, {30, 14380}, {31, 14208}, {32, 3267}, {37, 905}, {39, 4580}, {42, 4025}, {48, 1577}, {50, 14592}, {54, 6368}, {55, 17094}, {57, 8611}, {58, 4064}, {63, 661}, {64, 8057}, {65, 521}, {66, 8673}, {67, 9517}, {68, 924}, {69, 512}, {71, 514}, {72, 513}, {73, 522}, {74, 9033}, {75, 810}, {76, 3049}, {77, 4041}, {78, 4017}, {92, 822}, {93, 37084}, {95, 15451}, {97, 12077}, {98, 684}, {99, 20975}, {100, 18210}, {101, 4466}, {106, 14429}, {107, 2972}, {110, 125}, {111, 14417}, {112, 15526}, {115, 4558}, {122, 1301}, {140, 39180}, {162, 2632}, {163, 20902}, {184, 850}, {186, 43083}, {187, 14977}, {201, 3737}, {212, 4077}, {213, 15413}, {216, 15412}, {219, 7178}, {222, 3700}, {226, 652}, {228, 693}, {233, 39181}, {248, 2799}, {250, 5489}, {251, 2525}, {253, 42658}, {255, 24006}, {264, 39201}, {265, 526}, {275, 17434}, {276, 42293}, {287, 3569}, {288, 35441}, {290, 39469}, {292, 24459}, {295, 4010}, {304, 798}, {305, 669}, {306, 649}, {307, 663}, {321, 22383}, {323, 14582}, {324, 46088}, {325, 878}, {328, 14270}, {337, 4455}, {338, 32661}, {339, 1576}, {343, 2623}, {345, 7180}, {348, 3709}, {351, 30786}, {394, 2501}, {441, 34212}, {476, 16186}, {493, 17431}, {494, 17432}, {511, 879}, {518, 10099}, {524, 10097}, {542, 35909}, {577, 14618}, {594, 7254}, {599, 30491}, {603, 4086}, {648, 3269}, {650, 1214}, {662, 3708}, {667, 20336}, {686, 2986}, {690, 895}, {694, 24284}, {804, 36214}, {823, 37754}, {826, 1176}, {868, 43754}, {882, 12215}, {906, 16732}, {912, 3657}, {933, 35442}, {1018, 3942}, {1019, 3949}, {1020, 34591}, {1021, 37755}, {1073, 6587}, {1086, 4574}, {1109, 4575}, {1231, 3063}, {1245, 23874}, {1265, 7250}, {1304, 1650}, {1331, 3120}, {1332, 3125}, {1400, 6332}, {1402, 35518}, {1409, 4391}, {1410, 4397}, {1425, 7253}, {1437, 4036}, {1439, 3900}, {1441, 1946}, {1444, 4705}, {1494, 9409}, {1495, 34767}, {1500, 15419}, {1503, 2435}, {1510, 3519}, {1565, 4557}, {1636, 16080}, {1637, 14919}, {1649, 15398}, {1790, 4024}, {1794, 23752}, {1796, 4988}, {1797, 4120}, {1799, 3005}, {1813, 21044}, {1814, 24290}, {1824, 4131}, {1826, 4091}, {1919, 40071}, {1924, 40364}, {1942, 2797}, {1976, 6333}, {1989, 8552}, {2052, 32320}, {2092, 15420}, {2169, 2618}, {2197, 4560}, {2200, 3261}, {2207, 4143}, {2318, 3676}, {2333, 30805}, {2349, 2631}, {2351, 6563}, {2359, 21124}, {2373, 42665}, {2394, 3284}, {2395, 36212}, {2419, 42671}, {2422, 6393}, {2433, 11064}, {2451, 9289}, {2485, 14376}, {2489, 3926}, {2492, 34897}, {2519, 6339}, {2524, 2998}, {2533, 7015}, {2574, 2575}, {2578, 2583}, {2579, 2582}, {2584, 2589}, {2585, 2588}, {2616, 44706}, {2629, 9392}, {2643, 4592}, {2774, 38535}, {2806, 43723}, {2850, 10693}, {2966, 41172}, {3003, 15421}, {3050, 36952}, {3064, 40152}, {3122, 4561}, {3124, 4563}, {3172, 14638}, {3199, 15414}, {3221, 43714}, {3270, 4566}, {3288, 42313}, {3289, 43665}, {3292, 5466}, {3426, 9007}, {3504, 23301}, {3564, 35364}, {3566, 6391}, {3669, 3694}, {3682, 7649}, {3690, 7192}, {3692, 7216}, {3695, 3733}, {3710, 43924}, {3800, 34817}, {3906, 43697}, {3933, 18105}, {3937, 3952}, {3990, 17924}, {3998, 6591}, {4020, 18070}, {4049, 22356}, {4055, 46107}, {4079, 17206}, {4080, 22086}, {4088, 36057}, {4171, 7177}, {4516, 6516}, {4524, 7056}, {4551, 7004}, {4552, 7117}, {4556, 21046}, {4559, 26932}, {4570, 21134}, {4581, 22076}, {4608, 22080}, {4609, 23216}, {4846, 8675}, {5027, 40708}, {5392, 30451}, {5486, 30209}, {5663, 14220}, {5664, 11079}, {6000, 43701}, {6003, 43708}, {6103, 35911}, {6130, 14941}, {6137, 40709}, {6138, 40710}, {6334, 14910}, {6340, 8651}, {6354, 23090}, {6356, 21789}, {6390, 9178}, {6394, 17994}, {6528, 34980}, {6664, 22159}, {6728, 7591}, {6742, 22094}, {6757, 23226}, {7019, 7234}, {7123, 21107}, {7193, 35352}, {7252, 26942}, {7332, 23084}, {7927, 41435}, {8061, 34055}, {8562, 15392}, {8672, 34259}, {8676, 28786}, {8677, 38955}, {8749, 41077}, {8750, 17216}, {8779, 43673}, {8808, 10397}, {8901, 23181}, {9003, 34802}, {9247, 20948}, {9255, 17478}, {9293, 22143}, {9307, 22089}, {9391, 37142}, {9426, 40050}, {10412, 22115}, {10547, 23285}, {11060, 45792}, {11077, 41078}, {11794, 38352}, {13418, 30210}, {13481, 39228}, {13611, 44060}, {13754, 15328}, {14060, 36955}, {14295, 17970}, {14314, 18316}, {14401, 40384}, {14533, 18314}, {14575, 44173}, {14837, 41087}, {14908, 35522}, {15065, 22379}, {15313, 28787}, {15352, 35071}, {15377, 21196}, {15394, 44705}, {15453, 17702}, {15470, 39170}, {16230, 17974}, {16606, 25098}, {17879, 32676}, {17898, 19614}, {17932, 44114}, {17971, 18003}, {17972, 18004}, {17973, 18006}, {17975, 18013}, {17976, 18014}, {17977, 18015}, {18019, 42659}, {18877, 41079}, {19210, 23290}, {20188, 34483}, {20563, 34952}, {20578, 44718}, {20579, 44719}, {20580, 41489}, {20906, 22381}, {21051, 23086}, {21207, 32656}, {22052, 39183}, {22054, 31010}, {22085, 42345}, {22090, 42027}, {22092, 27809}, {22096, 27808}, {22154, 40085}, {22339, 42667}, {22340, 42668}, {22341, 44426}, {22352, 31065}, {22384, 43534}, {23107, 41937}, {23224, 41013}, {23616, 23964}, {23870, 36296}, {23871, 36297}, {23872, 32585}, {23873, 32586}, {23878, 43718}, {30211, 43695}, {30465, 38414}, {30468, 38413}, {32112, 35912}, {32641, 42761}, {32740, 45807}, {34568, 39008}, {35200, 36035}, {35372, 38401}, {36054, 40149}, {38279, 45689}, {38356, 44766}, {39473, 43717}, {40161, 43060}, {40715, 42662}, {40995, 46005}, {43704, 45147}, {43709, 44665}
X(647) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 811}, {2, 6331}, {3, 99}, {4, 6528}, {6, 648}, {19, 823}, {25, 107}, {31, 162}, {32, 112}, {37, 6335}, {39, 41676}, {42, 1897}, {48, 662}, {50, 14590}, {51, 35360}, {54, 18831}, {55, 36797}, {63, 799}, {65, 18026}, {68, 46134}, {69, 670}, {71, 190}, {72, 668}, {73, 664}, {74, 16077}, {77, 4625}, {78, 7257}, {98, 22456}, {110, 18020}, {112, 23582}, {115, 14618}, {125, 850}, {162, 23999}, {184, 110}, {187, 4235}, {212, 643}, {213, 1783}, {216, 14570}, {217, 1625}, {219, 645}, {222, 4573}, {228, 100}, {237, 4230}, {248, 2966}, {251, 42396}, {255, 4592}, {265, 35139}, {275, 42405}, {287, 43187}, {293, 36036}, {295, 4589}, {304, 4602}, {305, 4609}, {306, 1978}, {307, 4572}, {339, 44173}, {351, 468}, {393, 15352}, {394, 4563}, {418, 23181}, {511, 877}, {512, 4}, {513, 286}, {514, 44129}, {520, 69}, {521, 314}, {522, 44130}, {523, 264}, {525, 76}, {526, 340}, {560, 32676}, {570, 41677}, {571, 41679}, {577, 4558}, {603, 1414}, {649, 27}, {650, 31623}, {652, 333}, {656, 75}, {657, 2322}, {661, 92}, {662, 46254}, {663, 29}, {665, 15149}, {667, 28}, {669, 25}, {684, 325}, {686, 3580}, {688, 1843}, {690, 44146}, {692, 5379}, {798, 19}, {800, 41678}, {804, 17984}, {810, 1}, {822, 63}, {826, 1235}, {850, 18022}, {878, 98}, {879, 290}, {881, 17980}, {895, 892}, {905, 274}, {906, 4567}, {924, 317}, {1015, 17925}, {1042, 36118}, {1073, 44326}, {1084, 2489}, {1096, 36126}, {1173, 33513}, {1176, 4577}, {1214, 4554}, {1260, 7256}, {1298, 41208}, {1301, 44181}, {1304, 42308}, {1331, 4600}, {1332, 4601}, {1400, 653}, {1402, 108}, {1409, 651}, {1410, 934}, {1425, 4566}, {1427, 13149}, {1439, 4569}, {1444, 4623}, {1459, 86}, {1495, 4240}, {1510, 32002}, {1576, 250}, {1577, 1969}, {1636, 11064}, {1637, 46106}, {1649, 34336}, {1790, 4610}, {1796, 4632}, {1797, 4615}, {1799, 689}, {1802, 7259}, {1812, 4631}, {1813, 4620}, {1843, 46151}, {1918, 8750}, {1919, 1474}, {1924, 1973}, {1945, 41207}, {1946, 21}, {1949, 41206}, {1960, 37168}, {1973, 24019}, {1974, 32713}, {1976, 685}, {1977, 43925}, {1980, 2203}, {2081, 14918}, {2084, 17442}, {2088, 44427}, {2165, 30450}, {2173, 24001}, {2193, 4612}, {2196, 4584}, {2197, 4552}, {2200, 101}, {2207, 6529}, {2223, 4238}, {2281, 36099}, {2318, 3699}, {2351, 925}, {2353, 1289}, {2395, 16081}, {2422, 6531}, {2433, 16080}, {2435, 35140}, {2451, 9308}, {2485, 17907}, {2489, 393}, {2491, 232}, {2492, 37765}, {2501, 2052}, {2510, 14568}, {2519, 6392}, {2523, 30599}, {2524, 194}, {2525, 8024}, {2530, 16747}, {2574, 15165}, {2575, 15164}, {2578, 2581}, {2579, 2580}, {2616, 40440}, {2623, 275}, {2631, 14206}, {2632, 14208}, {2643, 24006}, {2799, 44132}, {2963, 38342}, {2966, 41174}, {2972, 3265}, {3003, 16237}, {3005, 427}, {3049, 6}, {3050, 36794}, {3051, 35325}, {3063, 1172}, {3120, 46107}, {3121, 6591}, {3122, 7649}, {3124, 2501}, {3125, 17924}, {3221, 3186}, {3250, 31909}, {3265, 305}, {3267, 1502}, {3269, 525}, {3270, 7253}, {3284, 2407}, {3288, 458}, {3289, 2421}, {3292, 5468}, {3455, 935}, {3457, 36306}, {3458, 36309}, {3504, 3222}, {3519, 46139}, {3569, 297}, {3657, 46133}, {3682, 4561}, {3690, 3952}, {3692, 7258}, {3694, 646}, {3695, 27808}, {3700, 7017}, {3708, 1577}, {3709, 281}, {3724, 4242}, {3804, 6995}, {3917, 4576}, {3937, 7192}, {3942, 7199}, {3949, 4033}, {3990, 1332}, {4010, 40717}, {4017, 273}, {4025, 310}, {4041, 318}, {4055, 1331}, {4064, 313}, {4079, 1826}, {4091, 17206}, {4120, 46109}, {4171, 7101}, {4455, 242}, {4466, 3261}, {4516, 44426}, {4524, 7046}, {4557, 15742}, {4558, 4590}, {4559, 46102}, {4563, 34537}, {4574, 1016}, {4575, 24041}, {4580, 308}, {4592, 24037}, {4705, 41013}, {4730, 38462}, {4822, 5342}, {4834, 31902}, {5027, 419}, {5029, 423}, {5040, 422}, {5075, 415}, {5113, 420}, {5191, 7473}, {5466, 46111}, {5489, 339}, {5504, 18878}, {6041, 6103}, {6130, 16089}, {6137, 470}, {6138, 471}, {6140, 37943}, {6332, 28660}, {6367, 44143}, {6368, 311}, {6377, 17921}, {6391, 35136}, {6529, 34538}, {6587, 15466}, {6753, 11547}, {7004, 18155}, {7015, 4594}, {7053, 4616}, {7099, 4637}, {7116, 4603}, {7117, 4560}, {7177, 4635}, {7178, 331}, {7180, 278}, {7216, 1847}, {7234, 7009}, {7250, 1119}, {7252, 46103}, {7254, 1509}, {7669, 30716}, {7927, 44142}, {8029, 2970}, {8034, 2969}, {8057, 14615}, {8061, 20883}, {8552, 7799}, {8611, 312}, {8618, 4249}, {8632, 31905}, {8636, 17520}, {8638, 37908}, {8639, 4185}, {8641, 4183}, {8642, 4233}, {8643, 4248}, {8644, 4232}, {8646, 4206}, {8648, 17515}, {8651, 6353}, {8653, 461}, {8663, 430}, {8664, 428}, {8665, 5064}, {8673, 315}, {8675, 44134}, {8677, 17139}, {8749, 15459}, {8779, 34211}, {8794, 42401}, {8882, 16813}, {9007, 44133}, {9033, 3260}, {9178, 17983}, {9210, 11331}, {9247, 163}, {9391, 44150}, {9407, 23347}, {9409, 30}, {9426, 1974}, {9491, 11325}, {9494, 27369}, {9517, 316}, {10097, 671}, {10099, 2481}, {10316, 4611}, {10397, 27398}, {10412, 18817}, {10547, 827}, {11079, 39290}, {12077, 324}, {12215, 880}, {13366, 35311}, {14060, 33799}, {14208, 561}, {14270, 186}, {14273, 37778}, {14380, 1494}, {14398, 1990}, {14401, 36789}, {14407, 8756}, {14417, 3266}, {14429, 3264}, {14533, 18315}, {14575, 1576}, {14582, 94}, {14585, 32661}, {14592, 20573}, {14600, 2715}, {14601, 32696}, {14618, 18027}, {14824, 5186}, {14908, 691}, {14910, 687}, {14936, 17926}, {14977, 18023}, {14984, 14221}, {15391, 39291}, {15412, 276}, {15413, 6385}, {15420, 40827}, {15421, 40832}, {15422, 8794}, {15451, 5}, {15475, 6344}, {15526, 3267}, {15905, 36841}, {16186, 3268}, {16573, 17899}, {17094, 6063}, {17414, 5094}, {17434, 343}, {17970, 805}, {17971, 17929}, {17972, 17930}, {17973, 17931}, {17974, 17932}, {17975, 17933}, {17976, 17934}, {17977, 17935}, {17978, 17936}, {17979, 17937}, {17989, 17987}, {17990, 17927}, {17992, 17985}, {17994, 6530}, {18001, 17982}, {18002, 17981}, {18105, 32085}, {18117, 7577}, {18210, 693}, {18344, 1896}, {18877, 44769}, {18890, 30441}, {19627, 14591}, {20336, 6386}, {20727, 33946}, {20775, 1634}, {20902, 20948}, {20975, 523}, {21044, 46110}, {21123, 17171}, {21134, 21207}, {21731, 403}, {21796, 17906}, {21828, 17923}, {21837, 17916}, {21906, 14273}, {22061, 18047}, {22080, 4427}, {22085, 14588}, {22086, 16704}, {22089, 1975}, {22090, 33296}, {22093, 17103}, {22094, 4467}, {22096, 3733}, {22115, 10411}, {22143, 31998}, {22159, 7760}, {22260, 8754}, {22341, 6516}, {22352, 10330}, {22363, 1633}, {22369, 4436}, {22370, 36860}, {22373, 4367}, {22381, 932}, {22383, 81}, {22384, 33295}, {22386, 16695}, {22388, 4184}, {23067, 4998}, {23090, 7058}, {23092, 7304}, {23099, 2971}, {23189, 261}, {23200, 5467}, {23216, 669}, {23220, 859}, {23224, 1444}, {23225, 3286}, {23286, 95}, {23610, 42068}, {23616, 36793}, {23620, 3732}, {23874, 44154}, {23878, 44144}, {24018, 304}, {24284, 3978}, {24290, 46108}, {24459, 1921}, {25098, 31008}, {30209, 11185}, {30451, 1993}, {30491, 598}, {32320, 394}, {32585, 32036}, {32586, 32037}, {32654, 32697}, {32656, 4570}, {32659, 4591}, {32661, 249}, {32662, 39295}, {32663, 30528}, {32676, 24000}, {32713, 32230}, {33581, 1301}, {34055, 4593}, {34212, 6330}, {34449, 39418}, {34952, 24}, {34980, 520}, {35072, 15411}, {35236, 2797}, {35364, 35142}, {35518, 40072}, {35909, 5641}, {36051, 36105}, {36054, 1812}, {36058, 4622}, {36060, 36085}, {36212, 2396}, {36214, 18829}, {36296, 23895}, {36297, 23896}, {37084, 44180}, {37754, 24018}, {38352, 31296}, {38356, 33294}, {39005, 38380}, {39013, 15423}, {39180, 40410}, {39181, 31617}, {39201, 3}, {39228, 7782}, {39469, 511}, {39665, 2479}, {39666, 2480}, {39687, 23090}, {40016, 42395}, {40319, 3565}, {40352, 1304}, {40353, 34568}, {40354, 32695}, {40373, 14574}, {40947, 1632}, {41087, 44327}, {41172, 2799}, {41221, 23290}, {41435, 35137}, {42065, 10425}, {42293, 216}, {42651, 41203}, {42653, 451}, {42658, 20}, {42659, 23}, {42660, 378}, {42661, 429}, {42662, 447}, {42663, 460}, {42664, 469}, {42665, 858}, {42666, 860}, {42667, 1113}, {42668, 1114}, {42669, 1981}, {42670, 2074}, {42671, 2409}, {42702, 42717}, {42752, 39534}, {43083, 328}, {43693, 35169}, {43697, 35138}, {43722, 33514}, {43925, 36419}, {44093, 14543}, {44112, 23353}, {44114, 16230}, {44173, 44161}, {44705, 14249}, {44729, 44721}, {46005, 18848}, {46088, 97}, {46112, 17402}, {46113, 17403}
X(647) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 850, 30476}, {2, 26080, 24960}, {2, 30476, 31277}, {2, 31296, 850}, {75, 21610, 21437}, {110, 36830, 23357}, {110, 45215, 35319}, {111, 9213, 8430}, {111, 10562, 10561}, {351, 669, 8651}, {351, 3005, 669}, {351, 42665, 42659}, {650, 6129, 6591}, {650, 6589, 3310}, {650, 21348, 6590}, {652, 1459, 22383}, {652, 22383, 22086}, {661, 21828, 7180}, {669, 8651, 8644}, {669, 8665, 8664}, {669, 17414, 3005}, {850, 30476, 31174}, {850, 36900, 31296}, {905, 2522, 2523}, {905, 25098, 4025}, {1636, 17434, 32320}, {1637, 12077, 2501}, {2501, 6587, 1637}, {2501, 6753, 2489}, {2501, 9209, 6587}, {2525, 14417, 3265}, {2623, 23286, 46088}, {2966, 18020, 9514}, {3005, 8651, 3804}, {3005, 8664, 8665}, {3005, 9420, 21646}, {3049, 32320, 30451}, {3288, 9210, 3569}, {3709, 7180, 661}, {3804, 8644, 669}, {5638, 5639, 3569}, {6041, 34291, 10567}, {6129, 6586, 33525}, {6586, 6589, 650}, {6587, 16040, 2485}, {8105, 8106, 1637}, {8644, 42658, 42659}, {9213, 10561, 10562}, {10561, 10562, 8430}, {16751, 28606, 4467}, {17414, 39201, 42665}, {24007, 24008, 6130}, {24622, 25594, 2}, {24782, 25084, 2}, {31174, 31277, 30476}, {31296, 36900, 41300}, {31296, 44560, 31277}, {36900, 44560, 31174}, {39665, 39666, 9409}, {41300, 44560, 30476}, {42667, 42668, 42659}


X(648) = TRILINEAR POLE OF EULER LINE

Trilinears    1/[a(b2 - c2)(b2 + c2 - a2)]
Trilinears    csc 2A csc(B - C) : :
Trilinears    (csc A)/(tan B - tan C) : :
Barycentrics    (a^2 - b^2)*(a^2 - c^2)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) : :
X(648) = 3 X[2] - 4 X[23583], 7 X[2] - 8 X[40477], 9 X[2] - 8 X[40484], 3 X[250] - 2 X[7473], 2 X[297] - 3 X[37765], X[340] - 4 X[1990], X[340] - 3 X[37765], X[1494] - 4 X[3163], 3 X[1494] - 4 X[15526], 3 X[1494] - 8 X[23583], 3 X[1494] - 2 X[39352], X[1494] + 2 X[39358], 7 X[1494] - 16 X[40477], 9 X[1494] - 16 X[40484], 4 X[1576] - 3 X[35278], 2 X[1632] - 3 X[35278], 4 X[1990] - 3 X[37765], 3 X[3163] - X[15526], 3 X[3163] - 2 X[23583], 6 X[3163] - X[39352], 2 X[3163] + X[39358], 7 X[3163] - 4 X[40477], 9 X[3163] - 4 X[40484], 3 X[6530] - 2 X[44228], 4 X[7473] - 3 X[30716], X[9410] - 4 X[36435], 2 X[15526] + 3 X[39358], 7 X[15526] - 12 X[40477], 3 X[15526] - 4 X[40484], X[16077] - 3 X[23582], 2 X[16077] - 3 X[39062], 4 X[23583] - X[39352], 4 X[23583] + 3 X[39358], 7 X[23583] - 6 X[40477], 3 X[23583] - 2 X[40484], X[35474] - 3 X[41204], X[39352] + 3 X[39358], 7 X[39352] - 24 X[40477], 3 X[39352] - 8 X[40484], 7 X[39358] + 8 X[40477], 9 X[39358] + 8 X[40484], 9 X[40477] - 7 X[40484], 3 X[44579] - 2 X[45312]

X(648) is constructed as the pole of the Euler line L as follows: let A″, B″, C″ be the points where L meets the sidelines BC, CA, AB of ABC. Let A', B', C' be the harmonic conjugates of A″, B″, C″ with respect to {B,C}, {C,A}, {A,B}, respectively, The lines AA', BB', CC' concur in X(648).

Let T be the anticevian triangle of X(4), and let U be the bianticevian conic of X(1) and X(4). Let TT be the tangential triangle of U with respect to T. Then ABC and TT are perspective, and X(648) is their perspector. (Randy Hutson, December 26, 2015)

X(648) lies on the Steiner circumellipse, the MacBeath circumconic, the cubics K256 and K676, and these lines: {1, 1982}, {2, 1494}, {4, 542}, {5, 42873}, {6, 264}, {19, 18827}, {20, 9530}, {24, 6179}, {25, 3228}, {27, 903}, {28, 3227}, {29, 1121}, {30, 16075}, {33, 35144}, {34, 35176}, {52, 8884}, {53, 3629}, {69, 1249}, {76, 8743}, {81, 16082}, {83, 1235}, {86, 1815}, {92, 14616}, {94, 275}, {95, 216}, {98, 2452}, {99, 112}, {100, 36077}, {107, 110}, {108, 931}, {132, 147}, {133, 6053}, {146, 10152}, {155, 1093}, {162, 190}, {183, 45141}, {185, 1105}, {186, 41626}, {193, 317}, {194, 1968}, {217, 9291}, {225, 2907}, {232, 385}, {233, 40410}, {242, 35173}, {243, 2651}, {249, 687}, {250, 523}, {253, 45245}, {273, 3759}, {276, 41334}, {281, 35141}, {283, 8764}, {286, 1172}, {297, 340}, {305, 3162}, {314, 41364}, {315, 34163}, {316, 5523}, {318, 3758}, {323, 46106}, {325, 16318}, {333, 34393}, {378, 7757}, {394, 15466}, {399, 34334}, {401, 3284}, {403, 14568}, {415, 23710}, {419, 35146}, {422, 35155}, {423, 8756}, {427, 41624}, {436, 34986}, {445, 42045}, {447, 519}, {450, 3292}, {459, 44877}, {467, 41628}, {468, 18823}, {470, 23713}, {471, 23712}, {472, 8737}, {473, 8738}, {511, 35474}, {514, 35169}, {525, 15459}, {527, 44331}, {530, 6110}, {531, 6111}, {532, 11094}, {533, 11093}, {536, 44330}, {538, 14581}, {543, 40890}, {575, 37124}, {577, 3164}, {599, 11331}, {608, 35159}, {621, 36302}, {622, 36303}, {645, 668}, {651, 823}, {653, 662}, {666, 5379}, {670, 2421}, {677, 7253}, {691, 30247}, {754, 40889}, {799, 36099}, {827, 1289}, {847, 2904}, {850, 17708}, {889, 43925}, {892, 2501}, {925, 933}, {935, 11636}, {1020, 1021}, {1033, 3964}, {1043, 44698}, {1075, 1092}, {1078, 39575}, {1179, 6152}, {1196, 40413}, {1294, 40948}, {1300, 1986}, {1302, 1304}, {1350, 15576}, {1351, 33971}, {1352, 41371}, {1441, 40582}, {1474, 3226}, {1503, 44704}, {1560, 30786}, {1568, 6761}, {1585, 45420}, {1586, 45421}, {1594, 7858}, {1625, 6528}, {1629, 3060}, {1634, 4230}, {1792, 8885}, {1799, 40938}, {1826, 2905}, {1843, 14970}, {1855, 32004}, {1861, 35163}, {1886, 17731}, {1896, 3193}, {1936, 2659}, {1947, 2003}, {1948, 2323}, {1973, 18826}, {1974, 3186}, {1975, 3172}, {1993, 2052}, {2055, 41481}, {2201, 35166}, {2203, 18825}, {2207, 7754}, {2211, 17984}, {2287, 46137}, {2326, 4360}, {2331, 27958}, {2332, 33296}, {2404, 23977}, {2451, 43187}, {2489, 18829}, {2592, 8115}, {2593, 8116}, {2633, 21187}, {2777, 30227}, {2782, 22143}, {2906, 41013}, {2914, 6344}, {2970, 19504}, {2990, 40571}, {2991, 5317}, {3003, 44375}, {3064, 35154}, {3087, 43981}, {3168, 9306}, {3187, 36419}, {3199, 7805}, {3260, 15262}, {3270, 37142}, {3289, 16089}, {3564, 6530}, {3574, 14860}, {3580, 14165}, {3618, 32000}, {3732, 24019}, {4238, 32041}, {4241, 32040}, {4552, 5546}, {4555, 17906}, {4560, 4565}, {4562, 15742}, {4567, 32698}, {4569, 4573}, {4570, 32699}, {4577, 42396}, {4586, 32676}, {4590, 32697}, {4591, 32705}, {4616, 15419}, {4664, 11107}, {5089, 35152}, {5094, 11163}, {5097, 39530}, {5467, 35178}, {5468, 35179}, {5649, 18311}, {5655, 11251}, {5664, 30528}, {5667, 16163}, {5702, 41145}, {5895, 18848}, {5921, 10002}, {5965, 39569}, {5994, 23873}, {5995, 23872}, {6353, 41360}, {6394, 9475}, {6515, 11547}, {6526, 32605}, {6527, 17037}, {6591, 35147}, {6748, 32455}, {6749, 8584}, {7258, 42384}, {7480, 14480}, {7488, 38808}, {7649, 35148}, {7669, 36176}, {7728, 18507}, {7762, 27376}, {7766, 10311}, {7768, 41366}, {7772, 37337}, {7838, 27371}, {7840, 41358}, {8057, 32230}, {8681, 34854}, {8744, 41617}, {8745, 21447}, {8755, 35149}, {8879, 34254}, {9182, 46144}, {9229, 37895}, {9307, 40947}, {9381, 40393}, {9410, 36435}, {9487, 15471}, {9514, 12077}, {9755, 40801}, {10602, 39646}, {10762, 18338}, {10979, 43980}, {11008, 32001}, {11054, 37855}, {11062, 44376}, {11185, 41370}, {11416, 14957}, {11441, 14249}, {11477, 37200}, {11511, 37190}, {12160, 41365}, {12829, 14772}, {13582, 39284}, {14024, 35167}, {14060, 34473}, {14361, 37669}, {14379, 45255}, {14401, 42308}, {14571, 19623}, {14615, 20806}, {14618, 18557}, {14627, 14978}, {14639, 41221}, {14918, 37766}, {14927, 31887}, {14941, 41219}, {15069, 15274}, {15149, 18821}, {15384, 39191}, {15395, 41512}, {15796, 42456}, {15905, 20477}, {15912, 19210}, {16175, 20410}, {16263, 40909}, {16264, 21850}, {16276, 17409}, {16704, 46136}, {16813, 18315}, {17035, 36412}, {17346, 17555}, {17378, 37448}, {17402, 32036}, {17403, 32037}, {17778, 18679}, {17911, 33770}, {17924, 35156}, {17926, 35157}, {17932, 20031}, {18019, 40583}, {18024, 40601}, {18487, 40885}, {18685, 21287}, {18687, 28754}, {18822, 31905}, {18828, 41209}, {19041, 24243}, {19042, 24244}, {20022, 21459}, {21166, 22085}, {22052, 45845}, {22456, 26714}, {23895, 36306}, {23896, 36309}, {23964, 33294}, {25054, 36849}, {25986, 37631}, {27369, 43094}, {30450, 46134}, {31150, 37966}, {31510, 44552}, {31909, 43097}, {32711, 39295}, {34545, 40684}, {34568, 39290}, {34810, 35908}, {34990, 40879}, {35168, 37168}, {35174, 46102}, {36212, 40888}, {36830, 41298}, {37644, 43462}, {37778, 37784}, {37783, 46141}, {38253, 46206}, {38342, 46139}, {40412, 40937}, {40414, 40715}, {40448, 42441}, {40996, 44334}, {41203, 41586}, {41363, 44132}, {42401, 42405}, {43093, 44129}, {44579, 45312}

X(648) = midpoint of X(i) and X(j) for these {i,j}: {2, 39358}, {193, 40867}
X(648) = reflection of X(i) in X(j) for these {i,j}: {2, 3163}, {69, 15595}, {95, 39081}, {99, 40866}, {287, 6}, {297, 1990}, {317, 36426}, {340, 297}, {401, 3284}, {1494, 2}, {1632, 1576}, {1972, 216}, {6330, 1249}, {15014, 14581}, {15526, 23583}, {30716, 250}, {39062, 23582}, {39352, 15526}, {40715, 40940}, {40885, 18487}, {40996, 44334}, {46239, 40938}
X(648) = isogonal conjugate of X(647)
X(648) = isotomic conjugate of X(525)
X(648) = complement of X(39352)
X(648) = anticomplement of X(15526)
X(648) = cevapoint of X(110) and X(112)
X(648) = X(i)-cross conjugate of X(j) for these (i,j): (6,250), (110,99), (112,107), (520,95), (523,264)
X(648) = trilinear pole of PU(30)
X(648) = polar-circle-inverse of X(16278)
X(648) = polar conjugate of X(523)
X(648) = cyclocevian conjugate of X(13573)
X(648) = crossdifference of PU(109)
X(648) = MacBeath circumconic antipode of X(287)
X(648) = pole wrt polar circle of trilinear polar of X(523) (line X(115)X(125))
X(648) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {107, 21294}, {162, 13219}, {163, 34186}, {250, 4329}, {1101, 6527}, {23582, 6327}, {23590, 5906}, {23964, 8}, {23999, 315}, {24000, 69}, {24019, 3448}, {24021, 317}, {24022, 6515}, {32230, 21270}, {32676, 39352}, {32713, 21221}, {36131, 45289}, {41937, 192}
X(648) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 39062}, {9390, 141}, {9392, 127}, {9406, 32750}, {15351, 2887}
X(648) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 39062}, {250, 36794}, {264, 30716}, {687, 14590}, {811, 36797}, {4590, 317}, {6331, 99}, {6528, 107}, {15459, 16237}, {16077, 4240}, {16813, 41679}, {18020, 4}, {18831, 110}, {22456, 4230}, {23582, 2}, {32230, 17907}, {33513, 35360}, {34538, 15466}, {42308, 30}, {42396, 112}, {42405, 6528}, {44181, 20}
X(648) = X(i)-cross conjugate of X(j) for these (i,j): {2, 23582}, {4, 18020}, {6, 250}, {19, 15742}, {20, 44181}, {30, 42308}, {69, 44183}, {92, 46102}, {110, 99}, {112, 107}, {193, 4590}, {385, 41174}, {512, 40413}, {514, 40414}, {520, 95}, {521, 40412}, {523, 264}, {525, 2}, {647, 1105}, {651, 662}, {653, 823}, {850, 83}, {1172, 5379}, {1249, 32230}, {1499, 10603}, {1503, 39297}, {1625, 110}, {1636, 1294}, {1783, 162}, {1897, 811}, {1993, 249}, {2407, 687}, {2420, 476}, {2421, 32697}, {2451, 6}, {2489, 32085}, {2501, 4}, {2799, 6330}, {3187, 1016}, {3288, 40801}, {3562, 7045}, {3569, 98}, {3580, 39295}, {3732, 799}, {3868, 4998}, {4230, 22456}, {4240, 16077}, {4391, 14534}, {4560, 31623}, {5664, 46106}, {6368, 40410}, {6587, 18848}, {6753, 8884}, {7192, 40411}, {7253, 86}, {7754, 34537}, {8057, 69}, {8115, 39299}, {8116, 39298}, {8673, 1799}, {8743, 23964}, {9033, 1494}, {9209, 18850}, {9517, 2373}, {9979, 671}, {10015, 24624}, {12077, 14860}, {14273, 17983}, {14316, 1916}, {14391, 265}, {14401, 30}, {14543, 190}, {14544, 664}, {14570, 30450}, {14618, 275}, {15639, 2409}, {16230, 35142}, {17434, 40448}, {17498, 274}, {17924, 40395}, {17925, 27}, {17926, 29}, {18311, 37765}, {18314, 40393}, {20577, 324}, {20580, 15466}, {21300, 40415}, {23090, 21}, {23616, 46115}, {23878, 42330}, {24978, 94}, {25259, 32014}, {31296, 276}, {32320, 3}, {32661, 925}, {33294, 76}, {34211, 2966}, {35311, 18831}, {35318, 35360}, {35324, 930}, {35325, 112}, {35360, 6528}, {35441, 14938}, {36828, 11636}, {39470, 37202}, {40571, 4567}, {41079, 2986}, {41676, 6331}, {41677, 38342}, {41678, 15352}, {43925, 28}, {44409, 7}, {44427, 16080}, {45292, 31621}
X(648) = cevapoint of X(i) and X(j) for these (i,j): {1, 1021}, {2, 525}, {3, 32320}, {4, 2501}, {6, 523}, {21, 23090}, {27, 17925}, {28, 43925}, {29, 17926}, {30, 14401}, {52, 6753}, {81, 4560}, {110, 112}, {185, 647}, {216, 520}, {230, 38359}, {233, 6368}, {323, 5664}, {324, 14618}, {394, 20580}, {512, 1196}, {513, 40941}, {514, 40940}, {521, 40937}, {522, 40942}, {524, 18311}, {650, 1858}, {651, 653}, {850, 1235}, {924, 40939}, {1249, 8057}, {1625, 35360}, {1636, 40948}, {1637, 13202}, {1783, 1897}, {1829, 6591}, {1839, 7649}, {1843, 2489}, {1994, 20577}, {2322, 7253}, {2409, 15639}, {2451, 9308}, {2492, 40949}, {2574, 8106}, {2575, 8105}, {2592, 2593}, {2799, 15595}, {2967, 3569}, {3049, 40947}, {3064, 40950}, {3163, 9033}, {3288, 9755}, {3574, 12077}, {3737, 40979}, {3800, 40179}, {5095, 14273}, {5895, 6587}, {7252, 40980}, {8673, 40938}, {8743, 33294}, {14091, 30211}, {14920, 44427}, {15412, 19170}, {17434, 42441}, {31296, 41334}, {35311, 35318}, {35325, 41676}, {36054, 40946}, {41392, 41512}
X(648) = crosspoint of X(i) and X(j) for these (i,j): {2, 15351}, {99, 44326}, {110, 44828}, {6331, 6528}, {18831, 42405}
X(648) = crosssum of X(i) and X(j) for these (i,j): {3, 22143}, {3049, 39201}, {15451, 42293}
X(648) = trilinear pole of line {2, 3}
X(648) = crossdifference of every pair of points on line {3269, 9409}
X(648) = Feuerbach image of X(4) if ABC is acute
X(648) = Brianchon point (perspector) of inscribed parabola with focus X(112)
X(648) = perspector of hyperbola {A,B,C,X(16077),X(18020),X(22456)}} (the circumconic through the polar conjugates of PU(40))
X(648) = X(92)-isoconjugate of X(39201)
X(648) = Steiner-circumellipse-X(1)-antipode of X(35145)
X(648) = Steiner-circumellipse-X(4)-antipode of X(671)
X(648) = Steiner-circumellipse-X(6)-antipode of X(290)
X(648) = trilinear product of circumcircle intercepts of line X(19)X(27)
X(648) = X(i)-isoconjugate of X(j) for these (i,j): {1, 647}, {2, 810}, {3, 661}, {4, 822}, {6, 656}, {10, 22383}, {19, 520}, {25, 24018}, {31, 525}, {32, 14208}, {37, 1459}, {41, 17094}, {42, 905}, {48, 523}, {56, 8611}, {63, 512}, {65, 652}, {69, 798}, {71, 513}, {72, 649}, {73, 650}, {74, 2631}, {75, 3049}, {77, 3709}, {78, 7180}, {91, 30451}, {92, 39201}, {101, 18210}, {107, 37754}, {110, 3708}, {112, 2632}, {115, 4575}, {125, 163}, {158, 32320}, {162, 3269}, {184, 1577}, {201, 7252}, {212, 7178}, {213, 4025}, {216, 2616}, {219, 4017}, {222, 4041}, {225, 36054}, {226, 1946}, {228, 514}, {244, 4574}, {255, 2501}, {265, 2624}, {293, 3569}, {295, 21832}, {304, 669}, {305, 1924}, {306, 667}, {307, 3063}, {326, 2489}, {336, 2491}, {521, 1400}, {522, 1409}, {560, 3267}, {577, 24006}, {603, 3700}, {657, 1439}, {662, 20975}, {663, 1214}, {672, 10099}, {684, 1910}, {686, 36053}, {690, 36060}, {692, 4466}, {693, 2200}, {756, 7254}, {823, 34980}, {850, 9247}, {872, 15419}, {878, 1959}, {879, 1755}, {895, 2642}, {896, 10097}, {906, 3120}, {922, 14977}, {923, 14417}, {924, 1820}, {1018, 3937}, {1019, 3690}, {1020, 3270}, {1021, 1425}, {1109, 32661}, {1176, 8061}, {1245, 2522}, {1254, 23090}, {1260, 7216}, {1331, 3125}, {1332, 3122}, {1333, 4064}, {1402, 6332}, {1410, 3239}, {1437, 4024}, {1444, 4079}, {1576, 20902}, {1636, 36119}, {1637, 35200}, {1650, 36131}, {1790, 4705}, {1796, 4983}, {1797, 4730}, {1799, 2084}, {1807, 21828}, {1813, 4516}, {1821, 39469}, {1824, 4091}, {1826, 23224}, {1911, 24459}, {1918, 15413}, {1919, 20336}, {1953, 23286}, {1964, 4580}, {1967, 24284}, {1973, 3265}, {1980, 40071}, {2088, 36061}, {2148, 6368}, {2155, 8057}, {2156, 8673}, {2157, 9517}, {2159, 9033}, {2167, 15451}, {2169, 12077}, {2170, 23067}, {2171, 23189}, {2173, 14380}, {2184, 42658}, {2190, 17434}, {2196, 4010}, {2197, 3737}, {2247, 35909}, {2249, 9391}, {2250, 8677}, {2252, 3657}, {2281, 23874}, {2312, 2435}, {2314, 43709}, {2315, 15328}, {2318, 3669}, {2333, 4131}, {2349, 9409}, {2451, 9255}, {2524, 3223}, {2525, 46289}, {2533, 7116}, {2574, 2579}, {2575, 2578}, {2582, 42667}, {2583, 42668}, {2584, 8106}, {2585, 8105}, {2618, 14533}, {2623, 44706}, {2643, 4558}, {2972, 24019}, {3005, 34055}, {3064, 22341}, {3121, 4561}, {3124, 4592}, {3292, 23894}, {3682, 6591}, {3692, 7250}, {3694, 43924}, {3725, 15420}, {3733, 3949}, {3835, 22381}, {3942, 4557}, {3990, 7649}, {4033, 22096}, {4049, 23202}, {4055, 17924}, {4088, 32658}, {4120, 36058}, {4171, 7053}, {4524, 7177}, {4551, 7117}, {4559, 7004}, {4602, 23216}, {4674, 22086}, {6129, 41087}, {6149, 14582}, {6587, 19614}, {6729, 7591}, {7084, 21107}, {7138, 17926}, {7148, 23092}, {8766, 34212}, {8818, 23226}, {9258, 22089}, {9396, 22143}, {9406, 34767}, {9426, 40364}, {9456, 14429}, {14575, 20948}, {14642, 17898}, {15352, 42080}, {15373, 21051}, {15389, 20910}, {15409, 21720}, {15526, 32676}, {16186, 32678}, {16606, 22090}, {16732, 32656}, {17438, 39180}, {18070, 20775}, {18344, 40152}, {18793, 22092}, {18877, 36035}, {21044, 36059}, {21789, 37755}, {21834, 23086}, {22373, 27805}, {23201, 31010}, {23493, 25098}, {23503, 43714}, {24290, 36057}, {30491, 36263}, {35071, 36126}, {36084, 41172}, {36516, 43963}, {40440, 42293}
X(648) = barycentric product X(i)*X(j) for these {i,j}: {1, 811}, {3, 6528}, {4, 99}, {5, 18831}, {6, 6331}, {7, 36797}, {19, 799}, {21, 18026}, {24, 46134}, {25, 670}, {27, 190}, {28, 668}, {29, 664}, {30, 16077}, {33, 4625}, {34, 7257}, {63, 823}, {69, 107}, {75, 162}, {76, 112}, {81, 6335}, {83, 41676}, {86, 1897}, {92, 662}, {94, 14590}, {95, 35360}, {98, 877}, {100, 286}, {101, 44129}, {108, 314}, {109, 44130}, {110, 264}, {140, 33513}, {141, 42396}, {158, 4592}, {163, 1969}, {186, 35139}, {216, 42405}, {232, 43187}, {240, 36036}, {242, 4589}, {249, 14618}, {250, 850}, {273, 643}, {274, 1783}, {275, 14570}, {276, 1625}, {278, 645}, {281, 4573}, {290, 4230}, {297, 2966}, {298, 36306}, {299, 36309}, {304, 24019}, {305, 32713}, {308, 35325}, {310, 8750}, {311, 933}, {315, 1289}, {316, 935}, {317, 925}, {318, 1414}, {324, 18315}, {325, 685}, {326, 36126}, {331, 5546}, {332, 36127}, {333, 653}, {340, 476}, {343, 16813}, {393, 4563}, {394, 15352}, {403, 18878}, {415, 35154}, {419, 18829}, {422, 35147}, {423, 35148}, {427, 4577}, {428, 35137}, {447, 35169}, {459, 36841}, {468, 892}, {470, 23895}, {471, 23896}, {472, 32037}, {473, 32036}, {511, 22456}, {523, 18020}, {525, 23582}, {561, 32676}, {646, 1396}, {651, 31623}, {656, 23999}, {658, 2322}, {661, 46254}, {666, 15149}, {671, 4235}, {687, 3580}, {689, 1843}, {691, 44146}, {693, 5379}, {801, 41678}, {805, 17984}, {827, 1235}, {880, 17980}, {930, 32002}, {1016, 17925}, {1043, 36118}, {1113, 15165}, {1114, 15164}, {1119, 7256}, {1172, 4554}, {1236, 10423}, {1249, 44326}, {1275, 17926}, {1288, 44128}, {1301, 14615}, {1302, 44134}, {1303, 9291}, {1304, 3260}, {1309, 17139}, {1435, 7258}, {1474, 1978}, {1494, 4240}, {1513, 41074}, {1576, 18022}, {1632, 34405}, {1634, 46104}, {1733, 36105}, {1799, 46151}, {1824, 4623}, {1826, 4610}, {1839, 4632}, {1847, 7259}, {1880, 4631}, {1896, 6516}, {1944, 41207}, {1948, 41206}, {1973, 4602}, {1974, 4609}, {1981, 35145}, {1993, 30450}, {1994, 38342}, {2052, 4558}, {2074, 35156}, {2201, 4639}, {2203, 6386}, {2287, 13149}, {2299, 4572}, {2349, 24001}, {2396, 6531}, {2407, 16080}, {2409, 35140}, {2421, 16081}, {2479, 2480}, {2481, 4238}, {2489, 34537}, {2501, 4590}, {2580, 2581}, {2592, 39298}, {2593, 39299}, {2617, 40440}, {2715, 44132}, {2986, 16237}, {3064, 4620}, {3186, 3222}, {3265, 32230}, {3267, 23964}, {3518, 46139}, {3569, 41174}, {3658, 46133}, {3732, 40411}, {3926, 6529}, {4143, 23590}, {4183, 4569}, {4226, 35142}, {4232, 35179}, {4241, 18025}, {4242, 14616}, {4246, 18816}, {4249, 43093}, {4552, 46103}, {4555, 37168}, {4560, 46102}, {4561, 8747}, {4562, 31905}, {4565, 7017}, {4567, 17924}, {4570, 46107}, {4576, 32085}, {4586, 31909}, {4591, 46109}, {4593, 17442}, {4594, 7009}, {4599, 20883}, {4600, 7649}, {4601, 6591}, {4611, 43678}, {4612, 40149}, {4614, 5342}, {4615, 8756}, {4616, 7046}, {4621, 31917}, {4622, 38462}, {4635, 7079}, {4637, 7101}, {5094, 35138}, {5117, 33514}, {5383, 17921}, {5392, 41679}, {5467, 46111}, {5468, 17983}, {5641, 7473}, {6035, 6103}, {6330, 34211}, {6344, 10411}, {6353, 35136}, {6393, 20031}, {6530, 17932}, {6540, 31900}, {6578, 44143}, {7012, 18155}, {7119, 7260}, {7192, 15742}, {7452, 34393}, {7953, 44142}, {8057, 44181}, {8754, 31614}, {8795, 23181}, {9033, 42308}, {9064, 44133}, {9308, 43188}, {10301, 42367}, {10420, 44138}, {10425, 44145}, {11064, 15459}, {11185, 30247}, {11412, 39418}, {11794, 36794}, {13485, 30716}, {14052, 33799}, {14208, 24000}, {14221, 40118}, {14543, 40414}, {14574, 44161}, {14591, 20573}, {14920, 39290}, {15351, 39062}, {15411, 23984}, {17515, 35174}, {17569, 41072}, {17708, 37765}, {17907, 44766}, {17927, 17930}, {17929, 17987}, {17931, 17985}, {17934, 17982}, {17935, 17981}, {18027, 32661}, {20884, 36095}, {24006, 24041}, {24039, 36128}, {26705, 33297}, {26714, 44144}, {27369, 42371}, {28660, 32674}, {30737, 44770}, {31617, 35318}, {31625, 43925}, {31902, 32042}, {31926, 32041}, {32038, 44734}, {32428, 41208}, {32691, 44154}, {33294, 44183}, {33640, 44136}, {34336, 34574}, {34568, 36789}, {35311, 40410}, {35314, 38428}, {35315, 38427}, {35575, 43976}, {36077, 44140}, {36084, 40703}, {36104, 46238}, {36131, 46234}, {36831, 43752}, {37908, 46135}, {37966, 46141}, {39291, 39931}, {39295, 44427}, {40393, 41677}, {41083, 44327}, {41331, 42395}, {41762, 42297}, {44769, 46106}
X(648) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 656}, {2, 525}, {3, 520}, {4, 523}, {5, 6368}, {6, 647}, {7, 17094}, {9, 8611}, {10, 4064}, {19, 661}, {20, 8057}, {21, 521}, {22, 8673}, {23, 9517}, {24, 924}, {25, 512}, {27, 514}, {28, 513}, {29, 522}, {30, 9033}, {31, 810}, {32, 3049}, {33, 4041}, {34, 4017}, {48, 822}, {49, 37084}, {51, 15451}, {53, 12077}, {54, 23286}, {58, 1459}, {59, 23067}, {60, 23189}, {63, 24018}, {69, 3265}, {74, 14380}, {75, 14208}, {76, 3267}, {81, 905}, {83, 4580}, {86, 4025}, {92, 1577}, {94, 14592}, {98, 879}, {99, 69}, {100, 72}, {101, 71}, {105, 10099}, {107, 4}, {108, 65}, {109, 73}, {110, 3}, {111, 10097}, {112, 6}, {125, 5489}, {141, 2525}, {154, 42658}, {158, 24006}, {162, 1}, {163, 48}, {184, 39201}, {186, 526}, {190, 306}, {216, 17434}, {217, 42293}, {232, 3569}, {233, 35441}, {237, 39469}, {239, 24459}, {242, 4010}, {249, 4558}, {250, 110}, {264, 850}, {265, 43083}, {270, 3737}, {273, 4077}, {274, 15413}, {275, 15412}, {278, 7178}, {281, 3700}, {284, 652}, {286, 693}, {288, 39181}, {297, 2799}, {314, 35518}, {317, 6563}, {318, 4086}, {323, 8552}, {324, 18314}, {325, 6333}, {333, 6332}, {340, 3268}, {376, 9007}, {378, 8675}, {385, 24284}, {393, 2501}, {415, 2785}, {419, 804}, {420, 9479}, {422, 2787}, {423, 2786}, {425, 2798}, {427, 826}, {428, 7927}, {430, 6367}, {441, 39473}, {450, 2797}, {458, 23878}, {461, 4843}, {468, 690}, {469, 23879}, {470, 23870}, {471, 23871}, {472, 23873}, {473, 23872}, {476, 265}, {477, 14220}, {511, 684}, {512, 20975}, {513, 18210}, {514, 4466}, {519, 14429}, {520, 2972}, {523, 125}, {524, 14417}, {525, 15526}, {526, 16186}, {571, 30451}, {577, 32320}, {593, 7254}, {607, 3709}, {608, 7180}, {643, 78}, {644, 3694}, {645, 345}, {647, 3269}, {651, 1214}, {653, 226}, {656, 2632}, {661, 3708}, {662, 63}, {664, 307}, {668, 20336}, {670, 305}, {671, 14977}, {685, 98}, {687, 2986}, {691, 895}, {692, 228}, {799, 304}, {805, 36214}, {811, 75}, {822, 37754}, {823, 92}, {827, 1176}, {842, 35909}, {850, 339}, {851, 9391}, {859, 8677}, {860, 6370}, {862, 4155}, {877, 325}, {892, 30786}, {906, 3990}, {907, 34817}, {915, 3657}, {925, 68}, {930, 3519}, {931, 34259}, {933, 54}, {934, 1439}, {935, 67}, {1010, 23874}, {1018, 3949}, {1019, 3942}, {1020, 37755}, {1021, 34591}, {1101, 4575}, {1113, 2575}, {1114, 2574}, {1172, 650}, {1173, 39180}, {1235, 23285}, {1249, 6587}, {1252, 4574}, {1286, 18124}, {1287, 18125}, {1288, 70}, {1289, 66}, {1291, 43704}, {1294, 43701}, {1297, 2435}, {1299, 43709}, {1300, 15328}, {1301, 64}, {1302, 4846}, {1304, 74}, {1305, 28786}, {1309, 38955}, {1325, 2850}, {1331, 3682}, {1332, 3998}, {1333, 22383}, {1383, 30491}, {1396, 3669}, {1398, 7250}, {1414, 77}, {1415, 1409}, {1435, 7216}, {1437, 23224}, {1444, 4131}, {1474, 649}, {1494, 34767}, {1495, 9409}, {1509, 15419}, {1576, 184}, {1577, 20902}, {1611, 2519}, {1613, 2524}, {1624, 185}, {1625, 216}, {1627, 22159}, {1632, 1899}, {1633, 17441}, {1634, 3917}, {1783, 37}, {1784, 36035}, {1790, 4091}, {1813, 40152}, {1822, 2585}, {1823, 2584}, {1824, 4705}, {1826, 4024}, {1838, 23752}, {1839, 4988}, {1843, 3005}, {1845, 42768}, {1848, 21124}, {1861, 4088}, {1877, 30572}, {1884, 6089}, {1895, 17898}, {1896, 44426}, {1897, 10}, {1957, 17478}, {1968, 2451}, {1969, 20948}, {1973, 798}, {1974, 669}, {1976, 878}, {1978, 40071}, {1981, 8680}, {1989, 14582}, {1990, 1637}, {1995, 30209}, {2052, 14618}, {2073, 2774}, {2074, 8674}, {2075, 2773}, {2173, 2631}, {2189, 7252}, {2190, 2616}, {2193, 36054}, {2194, 1946}, {2201, 21832}, {2203, 667}, {2204, 3063}, {2207, 2489}, {2211, 2491}, {2299, 663}, {2303, 2522}, {2322, 3239}, {2326, 1021}, {2332, 657}, {2333, 4079}, {2355, 4983}, {2393, 42665}, {2396, 6393}, {2407, 11064}, {2409, 1503}, {2420, 3284}, {2421, 36212}, {2445, 42671}, {2452, 22264}, {2485, 38356}, {2489, 3124}, {2501, 115}, {2576, 2579}, {2577, 2578}, {2580, 2583}, {2581, 2582}, {2586, 2589}, {2587, 2588}, {2605, 22094}, {2617, 44706}, {2633, 2629}, {2690, 38535}, {2713, 1942}, {2715, 248}, {2722, 43723}, {2766, 10693}, {2905, 21196}, {2906, 31947}, {2914, 8562}, {2966, 287}, {2967, 41167}, {2970, 23105}, {2971, 22260}, {2986, 15421}, {3003, 686}, {3050, 38352}, {3064, 21044}, {3068, 17431}, {3069, 17432}, {3120, 21134}, {3163, 14401}, {3186, 23301}, {3194, 6129}, {3222, 43714}, {3233, 16163}, {3266, 45807}, {3267, 36793}, {3284, 1636}, {3285, 22086}, {3518, 1510}, {3563, 35364}, {3565, 6391}, {3569, 41172}, {3580, 6334}, {3581, 14314}, {3658, 912}, {3699, 3710}, {3732, 18589}, {3733, 3937}, {3737, 7004}, {3867, 3806}, {3926, 4143}, {3939, 2318}, {3952, 3695}, {4000, 21107}, {4024, 21046}, {4025, 17216}, {4143, 23974}, {4183, 3900}, {4185, 8672}, {4186, 4139}, {4206, 8678}, {4221, 9051}, {4222, 4132}, {4226, 3564}, {4227, 9001}, {4230, 511}, {4232, 1499}, {4233, 3309}, {4234, 9031}, {4235, 524}, {4236, 34381}, {4237, 9028}, {4238, 518}, {4240, 30}, {4241, 516}, {4242, 758}, {4243, 916}, {4244, 3827}, {4246, 517}, {4247, 9002}, {4248, 3667}, {4249, 674}, {4250, 20718}, {4427, 41014}, {4551, 201}, {4552, 26942}, {4554, 1231}, {4556, 1790}, {4557, 3690}, {4558, 394}, {4559, 2197}, {4560, 26932}, {4563, 3926}, {4565, 222}, {4566, 6356}, {4567, 1332}, {4570, 1331}, {4573, 348}, {4575, 255}, {4576, 3933}, {4577, 1799}, {4589, 337}, {4590, 4563}, {4591, 1797}, {4592, 326}, {4594, 7019}, {4599, 34055}, {4600, 4561}, {4602, 40364}, {4609, 40050}, {4610, 17206}, {4611, 20806}, {4612, 1812}, {4616, 7056}, {4625, 7182}, {4629, 1796}, {4630, 10547}, {4636, 283}, {4637, 7177}, {5009, 22384}, {5064, 7950}, {5089, 24290}, {5094, 3906}, {5095, 1649}, {5101, 4808}, {5317, 6591}, {5338, 4822}, {5342, 4815}, {5379, 100}, {5467, 3292}, {5468, 6390}, {5502, 21663}, {5546, 219}, {5994, 36297}, {5995, 36296}, {6011, 43708}, {6103, 1640}, {6330, 43673}, {6331, 76}, {6335, 321}, {6336, 4049}, {6344, 10412}, {6353, 3566}, {6368, 35442}, {6525, 44705}, {6528, 264}, {6529, 393}, {6530, 16230}, {6531, 2395}, {6587, 1562}, {6591, 3125}, {6663, 34979}, {6733, 7591}, {6799, 42059}, {6995, 3800}, {7009, 2533}, {7012, 4551}, {7054, 23090}, {7058, 15411}, {7071, 4524}, {7076, 21831}, {7079, 4171}, {7115, 4559}, {7128, 1020}, {7192, 1565}, {7252, 7117}, {7253, 2968}, {7256, 1265}, {7257, 3718}, {7259, 3692}, {7419, 32475}, {7431, 9000}, {7435, 6001}, {7436, 8999}, {7452, 515}, {7463, 8679}, {7468, 14984}, {7471, 17702}, {7473, 542}, {7476, 2836}, {7480, 5663}, {7482, 2854}, {7649, 3120}, {7799, 45792}, {7953, 41435}, {8057, 122}, {8541, 17414}, {8737, 20578}, {8738, 20579}, {8739, 6137}, {8740, 6138}, {8743, 2485}, {8744, 2492}, {8745, 6753}, {8747, 7649}, {8748, 3064}, {8749, 2433}, {8750, 42}, {8753, 9178}, {8754, 8029}, {8756, 4120}, {8882, 2623}, {9033, 1650}, {9060, 34802}, {9064, 3426}, {9218, 22143}, {9291, 42331}, {9306, 22089}, {9308, 30476}, {9390, 9392}, {9426, 23216}, {9544, 39228}, {10015, 42761}, {10098, 5505}, {10101, 10100}, {10295, 9003}, {10301, 12073}, {10311, 3288}, {10312, 3050}, {10330, 7767}, {10420, 5504}, {10423, 1177}, {10425, 43705}, {11062, 2081}, {11064, 41077}, {11101, 30212}, {11107, 35057}, {11325, 3221}, {11413, 30211}, {11634, 8681}, {11636, 43697}, {11794, 36952}, {12092, 16867}, {12111, 40494}, {12383, 38401}, {13149, 1446}, {13397, 28787}, {13398, 15316}, {13450, 23290}, {13486, 7100}, {13621, 30210}, {13739, 6003}, {14004, 4151}, {14006, 3907}, {14013, 28846}, {14015, 9013}, {14024, 3716}, {14052, 36955}, {14129, 20577}, {14165, 44427}, {14208, 17879}, {14273, 1648}, {14401, 39008}, {14480, 15061}, {14533, 46088}, {14534, 15420}, {14543, 440}, {14544, 18641}, {14570, 343}, {14574, 14575}, {14581, 14398}, {14586, 14533}, {14587, 15958}, {14590, 323}, {14591, 50}, {14611, 6699}, {14618, 338}, {14918, 41078}, {14920, 5664}, {14953, 39470}, {14966, 3289}, {15014, 9035}, {15149, 918}, {15164, 22340}, {15165, 22339}, {15329, 13754}, {15352, 2052}, {15384, 1301}, {15411, 23983}, {15459, 16080}, {15471, 9125}, {15526, 23616}, {15639, 23976}, {15742, 3952}, {15958, 19210}, {16077, 1494}, {16080, 2394}, {16081, 43665}, {16166, 11559}, {16230, 868}, {16237, 3580}, {16757, 18187}, {16806, 32585}, {16807, 32586}, {16813, 275}, {17094, 1367}, {17104, 23226}, {17171, 16892}, {17206, 30805}, {17402, 44718}, {17403, 44719}, {17442, 8061}, {17515, 3738}, {17520, 832}, {17569, 30665}, {17708, 34897}, {17906, 4415}, {17907, 33294}, {17921, 21138}, {17923, 4707}, {17924, 16732}, {17925, 1086}, {17926, 1146}, {17927, 18004}, {17932, 6394}, {17938, 17970}, {17939, 17971}, {17940, 17972}, {17941, 12215}, {17942, 17975}, {17943, 17976}, {17944, 17977}, {17945, 17978}, {17980, 882}, {17981, 18015}, {17982, 18014}, {17983, 5466}, {17984, 14295}, {17985, 18006}, {17987, 18003}, {17994, 44114}, {18020, 99}, {18022, 44173}, {18026, 1441}, {18047, 4019}, {18155, 17880}, {18315, 97}, {18344, 4516}, {18374, 42659}, {18384, 15475}, {18808, 12079}, {18829, 40708}, {18831, 95}, {18879, 43755}, {19118, 8651}, {20031, 6531}, {20189, 34483}, {20626, 6145}, {21789, 3270}, {22239, 11744}, {22456, 290}, {23067, 7066}, {23090, 35072}, {23181, 5562}, {23189, 1364}, {23347, 1495}, {23353, 851}, {23357, 32661}, {23590, 6529}, {23710, 30574}, {23712, 9200}, {23713, 9201}, {23714, 14446}, {23715, 14447}, {23895, 40709}, {23896, 40710}, {23964, 112}, {23977, 16318}, {23999, 811}, {24000, 162}, {24001, 14206}, {24006, 1109}, {24019, 19}, {24021, 36126}, {24041, 4592}, {26283, 30213}, {26704, 15232}, {26705, 15320}, {26714, 43718}, {27369, 688}, {27644, 25098}, {30221, 38729}, {30247, 5486}, {30248, 13418}, {30249, 43695}, {30450, 5392}, {30512, 44665}, {30716, 3448}, {30733, 15313}, {31510, 2777}, {31623, 4391}, {31900, 4977}, {31901, 28195}, {31902, 4802}, {31903, 4778}, {31904, 28840}, {31905, 812}, {31907, 28859}, {31908, 28882}, {31909, 824}, {31912, 4785}, {31914, 28855}, {31916, 30519}, {31917, 3776}, {31921, 28890}, {31925, 30520}, {31926, 4762}, {32002, 41298}, {32036, 40712}, {32037, 40711}, {32230, 107}, {32320, 35071}, {32640, 18877}, {32656, 4055}, {32661, 577}, {32674, 1400}, {32676, 31}, {32687, 43717}, {32691, 1245}, {32695, 8749}, {32696, 1976}, {32697, 2987}, {32708, 14910}, {32710, 15453}, {32713, 25}, {32714, 1427}, {32715, 40352}, {32729, 14908}, {32734, 2351}, {32739, 2200}, {33294, 127}, {33513, 40410}, {33640, 11270}, {33803, 14060}, {34211, 441}, {34386, 15414}, {34397, 14270}, {34403, 14638}, {34484, 20188}, {34538, 15352}, {34568, 40384}, {34574, 15398}, {34854, 17994}, {34859, 2211}, {34983, 41212}, {35136, 6340}, {35139, 328}, {35140, 2419}, {35169, 40715}, {35265, 44810}, {35278, 6776}, {35311, 140}, {35318, 233}, {35324, 22052}, {35325, 39}, {35327, 22080}, {35342, 3958}, {35360, 5}, {35907, 6103}, {35908, 32112}, {36034, 35200}, {36036, 336}, {36049, 41087}, {36059, 22341}, {36084, 293}, {36092, 8767}, {36104, 1910}, {36105, 8773}, {36114, 36053}, {36118, 3668}, {36126, 158}, {36127, 225}, {36128, 23894}, {36129, 2166}, {36131, 2159}, {36134, 2169}, {36142, 36060}, {36145, 1820}, {36306, 13}, {36309, 14}, {36419, 17925}, {36420, 43925}, {36421, 17926}, {36793, 23107}, {36794, 31296}, {36797, 8}, {36829, 23039}, {36830, 22146}, {36831, 44715}, {36839, 10217}, {36840, 10218}, {36841, 37669}, {36898, 34290}, {37168, 900}, {37669, 20580}, {37765, 9979}, {37766, 24978}, {37908, 926}, {37937, 2781}, {37943, 45147}, {37962, 2780}, {37963, 2775}, {37966, 2771}, {38294, 45689}, {38342, 11140}, {38832, 22090}, {38861, 13198}, {38936, 15470}, {39062, 39352}, {39201, 34980}, {39284, 39183}, {39297, 2867}, {39298, 8115}, {39299, 8116}, {39352, 38240}, {39383, 6413}, {39384, 6414}, {39417, 34207}, {39534, 42759}, {40097, 43703}, {40117, 1903}, {40138, 9209}, {40596, 10117}, {40979, 40628}, {41013, 4036}, {41083, 14837}, {41174, 43187}, {41204, 6130}, {41206, 40843}, {41207, 1952}, {41321, 17747}, {41357, 16278}, {41364, 6588}, {41502, 9404}, {41512, 39170}, {41610, 24562}, {41676, 141}, {41677, 37636}, {41678, 13567}, {41679, 1993}, {41906, 28788}, {42067, 8034}, {42068, 23099}, {42308, 16077}, {42396, 83}, {42405, 276}, {43188, 9289}, {43351, 42021}, {43717, 34212}, {43754, 17974}, {43925, 1015}, {43976, 30735}, {44060, 3532}, {44077, 34952}, {44080, 42660}, {44084, 21731}, {44089, 5027}, {44090, 5113}, {44091, 8664}, {44092, 42661}, {44097, 42653}, {44099, 42663}, {44100, 8653}, {44102, 351}, {44103, 42664}, {44113, 42666}, {44123, 42667}, {44124, 42668}, {44129, 3261}, {44130, 35519}, {44134, 30474}, {44146, 35522}, {44162, 9426}, {44183, 44766}, {44326, 34403}, {44695, 14308}, {44698, 21172}, {44734, 23880}, {44766, 14376}, {44769, 14919}, {44770, 1297}, {44828, 40800}, {45215, 22416}, {45662, 39474}, {46102, 4552}, {46103, 4560}, {46106, 41079}, {46107, 21207}, {46134, 20563}, {46151, 427}, {46254, 799}
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 39352, 15526}, {6, 264, 36794}, {6, 9308, 264}, {53, 3629, 27377}, {53, 27377, 32002}, {69, 1249, 17907}, {98, 2452, 13479}, {98, 11596, 20975}, {99, 36841, 4558}, {110, 35360, 107}, {112, 4558, 41679}, {112, 41676, 99}, {162, 1897, 36797}, {185, 1941, 1105}, {193, 393, 317}, {297, 1990, 37765}, {324, 1994, 275}, {340, 37765, 297}, {1576, 1632, 35278}, {2407, 4558, 36841}, {2407, 14570, 4558}, {2407, 16237, 14590}, {2407, 41678, 41679}, {2452, 9512, 20975}, {2452, 20975, 11596}, {3163, 15526, 23583}, {3163, 39358, 1494}, {4558, 14570, 99}, {4558, 41679, 14590}, {9512, 20975, 98}, {11596, 20975, 13479}, {14570, 41676, 41677}, {14570, 41678, 16237}, {15526, 23583, 2}, {15526, 39352, 1494}, {16237, 41677, 14570}, {16813, 18831, 18315}, {17037, 36413, 6527}, {32713, 46151, 107}, {35311, 35360, 110}, {37766, 37779, 14918}, {39298, 39299, 14590}, {41194, 41195, 41254}, {41676, 41678, 14570}, {41677, 41679, 99}


X(649) = CROSSDIFFERENCE OF X(1) and X(2)

Trilinears    a(b - c) : b(c - a) : c(a - b)
Trilinears    csc^2 (B/2) - csc^2 (C/2) : :
Trilinears    cot^2 (B/2) - cot^2 (C/2) : :
Barycentrics  a2(b - c) : b2(c - a) : c2(a - b)
X(649) = 9 X[2] - 5 X[26798], 3 X[2] + X[26853], 3 X[2] - 5 X[27013], 9 X[2] - 7 X[27138], 6 X[2] - 5 X[30835], 6 X[2] - 7 X[31207], 3 X[2] - 4 X[31286], 5 X[2] - 4 X[45339], 3 X[165] - 2 X[15599], 3 X[351] - X[8663], 2 X[650] - 3 X[1635], 3 X[650] - 4 X[2516], 4 X[650] - X[4813], 4 X[650] - 3 X[4893], 2 X[650] + X[4979], 3 X[657] - 4 X[22108], X[661] - 3 X[1635], 3 X[661] - 8 X[2516], X[661] - 4 X[4394], X[661] + 2 X[4790], 2 X[661] - 3 X[4893], 3 X[663] - 4 X[1960], 3 X[663] - 2 X[4775], X[663] + 2 X[4834], 2 X[663] - 3 X[8643], 3 X[663] - 5 X[8656], 3 X[667] - 2 X[1960], 3 X[667] - X[4775], 4 X[667] - 3 X[8643], 6 X[667] - 5 X[8656], 2 X[693] - 3 X[4379], 3 X[798] - X[3768], 2 X[1019] + X[4498], 3 X[1019] + X[21385], 9 X[1635] - 8 X[2516], 3 X[1635] - 4 X[4394], 3 X[1635] + 2 X[4790], 6 X[1635] - X[4813], 3 X[1635] + X[4979], 3 X[1638] - 4 X[2487], 3 X[1638] - X[23729], 3 X[1639] - 4 X[2490], 3 X[1639] - 2 X[14321], 2 X[1960] + 3 X[4834], 8 X[1960] - 9 X[8643], 4 X[1960] - 5 X[8656], 3 X[2483] - 2 X[2515], 3 X[2484] - 4 X[2515], 4 X[2487] - X[23729], 2 X[2516] - 3 X[4394], 4 X[2516] + 3 X[4790], 16 X[2516] - 3 X[4813], 16 X[2516] - 9 X[4893], 8 X[2516] + 3 X[4979], 4 X[2527] - X[3700], 8 X[2529] - X[4838], 4 X[3239] - 3 X[4120], 4 X[3676] - 3 X[6545], 4 X[3739] - 3 X[27485], 2 X[3768] - 3 X[20979], 2 X[3776] - 3 X[4453], 4 X[3798] - 3 X[4750], 2 X[3798] - 3 X[4786], 4 X[3798] - X[16892], 6 X[3835] - 5 X[26798], 2 X[3835] + X[26853], 2 X[3835] - 5 X[27013], 6 X[3835] - 7 X[27138], 4 X[3835] - 5 X[30835], 4 X[3835] - 3 X[31147], 4 X[3835] - 7 X[31207], X[3835] - 3 X[45313], 5 X[3835] - 6 X[45339], X[4024] + 3 X[4984], 2 X[4025] - 3 X[4750], X[4025] - 3 X[4786], 3 X[4063] - X[21385], 2 X[4106] - 3 X[4728], 2 X[4106] - 5 X[24924], 3 X[4120] - 8 X[43061], 3 X[4367] - X[21343], 4 X[4369] - 3 X[4379], 2 X[4369] + X[4380], 4 X[4369] - X[4382], 3 X[4379] + 2 X[4380], 3 X[4379] - X[4382], 2 X[4380] + X[4382], 2 X[4394] + X[4790], 8 X[4394] - X[4813], 8 X[4394] - 3 X[4893], 4 X[4394] + X[4979], 3 X[4449] - 2 X[21343], 2 X[4468] - 3 X[6546], 3 X[4498] - 2 X[21385], 2 X[4500] - 3 X[4789], 2 X[4507] + 3 X[8027], 8 X[4521] - 9 X[6544], X[4724] - 4 X[4782], X[4724] + 2 X[4784], 3 X[4728] - 4 X[4885], 3 X[4728] - 5 X[24924], 3 X[4750] - X[16892], 3 X[4763] - 2 X[25666], 6 X[4763] - 5 X[31209], 4 X[4765] - X[4988], X[4765] - 4 X[14351], 4 X[4765] - 9 X[14435], 3 X[4773] - X[4976], X[4775] + 3 X[4834], 4 X[4775] - 9 X[8643], 2 X[4775] - 5 X[8656], 3 X[4776] - 4 X[25666], 3 X[4776] - 5 X[31209], 2 X[4782] + X[4784], 6 X[4786] - X[16892], 4 X[4790] + X[4813], 4 X[4790] + 3 X[4893], X[4813] - 3 X[4893], X[4813] + 2 X[4979], 2 X[4820] - 3 X[4931], X[4822] - 4 X[6050], 4 X[4834] + 3 X[8643], 6 X[4834] + 5 X[8656], 4 X[4885] - 5 X[24924], 3 X[4893] + 2 X[4979], 2 X[4932] + X[17494], 3 X[4984] + 2 X[6590], X[4988] - 16 X[14351], X[4988] - 9 X[14435], 3 X[6546] - 4 X[11068], 9 X[8643] - 10 X[8656], 3 X[8644] - 2 X[8653], 16 X[14351] - 9 X[14435], 8 X[14351] - X[45745], 9 X[14435] - 2 X[45745], 3 X[20295] - 5 X[26798], X[20295] - 5 X[27013], 3 X[20295] - 7 X[27138], 2 X[20295] - 5 X[30835], 2 X[20295] - 3 X[31147], 2 X[20295] - 7 X[31207], X[20295] - 4 X[31286], X[20295] - 6 X[45313], 5 X[20295] - 12 X[45339], 2 X[21196] - 3 X[27486], X[21196] - 3 X[45679], 4 X[21212] - 3 X[44435], 2 X[21212] - 3 X[45674], 3 X[21297] - 5 X[26985], 2 X[23813] - 3 X[45320], 4 X[25380] - 3 X[44429], 4 X[25666] - 5 X[31209], 5 X[26777] - X[31290], 5 X[26798] + 3 X[26853], X[26798] - 3 X[27013], 5 X[26798] - 7 X[27138], 2 X[26798] - 3 X[30835], 10 X[26798] - 9 X[31147], 10 X[26798] - 21 X[31207], 5 X[26798] - 12 X[31286], 5 X[26798] - 18 X[45313], 25 X[26798] - 36 X[45339], X[26853] + 5 X[27013], 3 X[26853] + 7 X[27138], 2 X[26853] + 5 X[30835], 2 X[26853] + 3 X[31147], 2 X[26853] + 7 X[31207], X[26853] + 4 X[31286], X[26853] + 6 X[45313], 5 X[26853] + 12 X[45339], 15 X[27013] - 7 X[27138], 10 X[27013] - 3 X[31147], 10 X[27013] - 7 X[31207], 5 X[27013] - 4 X[31286], 5 X[27013] - 6 X[45313], 25 X[27013] - 12 X[45339], 14 X[27138] - 15 X[30835], 14 X[27138] - 9 X[31147], 2 X[27138] - 3 X[31207], 7 X[27138] - 12 X[31286], 7 X[27138] - 18 X[45313], 35 X[27138] - 36 X[45339], 3 X[27486] - X[45746], 3 X[28398] - 2 X[29738], 3 X[30565] - X[44449], 5 X[30835] - 3 X[31147], 5 X[30835] - 7 X[31207], 5 X[30835] - 8 X[31286], 5 X[30835] - 12 X[45313], 25 X[30835] - 24 X[45339], 3 X[31147] - 7 X[31207], 3 X[31147] - 8 X[31286], X[31147] - 4 X[45313], 5 X[31147] - 8 X[45339], 3 X[31148] - 2 X[43067], 7 X[31207] - 8 X[31286], 7 X[31207] - 12 X[45313], 35 X[31207] - 24 X[45339], 2 X[31286] - 3 X[45313], 5 X[31286] - 3 X[45339], 5 X[45313] - 2 X[45339], 6 X[45679] - X[45746]

X(649) is the perspector of triangle ABC and the tangential triangle of the conic {A, B, C, X(1), X(6)}}. (Randy Hutson, 9/23/2011)

A'B'C' be the excentral triangle and H the hyperbola {A',B',C',X(1),X(9}}, so that H is the Jerabek hyperbola of A'B'C'. Let TT be the tangential triangle, wrt A'B'C', of H. Then T and TT are perspective, and their perspector is X(649). (Randy Hutson, December 26, 2015)

The circle that has center X(649) and passes through the isodynamic points, X(15) and X(16), also passes through the isodynamic points of the excentral triangle (X1276) and X(1277), as well as X(32622) and X(32623). This circle is here named the Gheorghe circle, in recognition of the work of Liliana Gheorghe. The Gheorghe circle is coaxial with the three Apollonius circles and also the Moses radical circle, which has center X(647) and passes through X(6116) and X(6117), these being the isodynamic points of the orthic triangle. The centers of the Gheorghe circle, Moses radical circle, and the three Apollonius circles all lie on the Lemoine axis, X(187)X(237); for related circles, see X(15). (Dan Reznik, April 16, 2021)

See Gheorghe Circle and Gheorghe Circle (zoomed in). See also Gheorghe Circle in motion.

The Moses radical circle is the Gheorghe circle of the orthic triangle. (Dan Reznik, April 20, 2021)

X(649) lies on the Yff parabola and these lines: {1, 3249}, {2, 3835}, {3, 14825}, {6, 2441}, {8, 25301}, {9, 4521}, {10, 21720}, {19, 3064}, {27, 46107}, {31, 884}, {32, 8578}, {40, 28292}, {42, 788}, {44, 513}, {45, 14437}, {55, 23865}, {57, 1024}, {58, 43926}, {63, 4468}, {75, 20909}, {81, 18200}, {89, 1022}, {100, 660}, {101, 901}, {103, 43079}, {106, 2384}, {109, 919}, {110, 2702}, {111, 2712}, {165, 15599}, {171, 18098}, {187, 237}, {190, 889}, {239, 514}, {244, 38346}, {522, 3509}, {523, 4773}, {604, 2423}, {614, 9315}, {662, 17929}, {693, 812}, {727, 9111}, {739, 2382}, {764, 1475}, {802, 4374}, {814, 2533}, {824, 4467}, {830, 1734}, {834, 1459}, {840, 2291}, {854, 1404}, {876, 40747}, {891, 4378}, {893, 17187}, {894, 21211}, {900, 2527}, {905, 28372}, {918, 4897}, {926, 4105}, {995, 30650}, {1015, 43922}, {1018, 4781}, {1027, 2279}, {1125, 19949}, {1146, 39006}, {1201, 9433}, {1293, 6078}, {1334, 6161}, {1415, 32675}, {1436, 2432}, {1461, 7339}, {1577, 29013}, {1638, 2487}, {1639, 2490}, {1768, 21382}, {1769, 6588}, {1797, 10756}, {1977, 6377}, {1978, 4598}, {2082, 23764}, {2160, 21131}, {2161, 14442}, {2170, 15635}, {2269, 24118}, {2284, 23832}, {2308, 8034}, {2319, 17155}, {2340, 9320}, {2473, 2521}, {2504, 7178}, {2529, 4777}, {2664, 4040}, {2666, 3720}, {2786, 21391}, {2787, 4474}, {3004, 17069}, {3051, 23575}, {3120, 43920}, {3219, 10196}, {3239, 3667}, {3246, 39310}, {3259, 46101}, {3261, 10566}, {3306, 14475}, {3309, 3803}, {3666, 14751}, {3669, 7216}, {3709, 4502}, {3726, 4132}, {3739, 27485}, {3762, 29148}, {3766, 24601}, {3776, 4453}, {3801, 29025}, {3837, 24719}, {3900, 4729}, {3904, 28468}, {3929, 45670}, {3937, 14936}, {3960, 14438}, {4010, 4874}, {4014, 38991}, {4017, 6591}, {4041, 8678}, {4057, 4079}, {4083, 4367}, {4106, 4728}, {4122, 29078}, {4129, 27045}, {4139, 4501}, {4142, 29118}, {4148, 4462}, {4191, 20674}, {4266, 14812}, {4384, 14433}, {4391, 6002}, {4414, 25800}, {4432, 40614}, {4481, 16751}, {4486, 26248}, {4491, 14407}, {4500, 4789}, {4584, 4600}, {4586, 35009}, {4588, 28875}, {4603, 4610}, {4640, 40586}, {4672, 35353}, {4730, 4814}, {4761, 29066}, {4762, 31148}, {4763, 4776}, {4774, 29236}, {4791, 29178}, {4804, 7662}, {4820, 4926}, {4822, 6050}, {4823, 29270}, {4841, 4977}, {4879, 23506}, {4940, 31287}, {4944, 4958}, {4978, 29302}, {5701, 34583}, {6003, 23691}, {6004, 24290}, {6014, 6017}, {6084, 21104}, {6085, 23650}, {6182, 6608}, {6332, 28478}, {6363, 20980}, {6372, 21763}, {6373, 20983}, {6615, 40134}, {7004, 21339}, {7045, 36146}, {7199, 29771}, {7265, 29216}, {7289, 23730}, {7649, 21190}, {8056, 23834}, {8611, 15313}, {8677, 36054}, {8697, 28891}, {9085, 29242}, {9097, 17222}, {9259, 9262}, {9294, 17759}, {9359, 9361}, {10015, 29126}, {10459, 30203}, {14349, 14838}, {14377, 23100}, {14425, 39386}, {14474, 30950}, {14991, 23657}, {15487, 28589}, {16231, 42403}, {16557, 32925}, {16560, 24129}, {16612, 23800}, {16695, 40627}, {17029, 26824}, {17072, 21053}, {17159, 21225}, {17204, 17215}, {17412, 23224}, {17439, 38242}, {18154, 20954}, {18278, 21791}, {20517, 29158}, {21003, 21005}, {21051, 25636}, {21122, 43925}, {21146, 29362}, {21204, 27003}, {21212, 31095}, {21260, 24960}, {21297, 26985}, {21302, 28470}, {21348, 21834}, {21349, 21350}, {22092, 45902}, {22094, 23647}, {22224, 26048}, {23301, 30968}, {23726, 26934}, {23794, 30024}, {23803, 26983}, {23813, 45320}, {23845, 35326}, {24462, 26249}, {24484, 46125}, {24560, 25902}, {24562, 25900}, {24592, 27855}, {24756, 30957}, {24804, 35355}, {25008, 25981}, {25009, 26017}, {25128, 31330}, {25380, 44429}, {25511, 42327}, {25604, 40474}, {25627, 26037}, {26049, 27527}, {26777, 31290}, {27009, 44312}, {27014, 28758}, {27451, 30095}, {28255, 28286}, {28840, 31150}, {28867, 30565}, {29179, 43361}, {32669, 32674}, {33570, 35270}, {35595, 45684}, {38325, 42322}, {38344, 38345}, {38348, 40789}, {38367, 40733}

X(649) = midpoint of X(i) and X(j) for these {i,j}: {650, 4790}, {659, 4784}, {661, 4979}, {667, 4834}, {693, 4380}, {1019, 4063}, {7192, 17494}, {17159, 21225}, {20295, 26853}, {21302, 31291}, {29545, 29807}
X(649) = reflection of X(i) in X(j) for these {i,j}: {2, 45313}, {650, 4394}, {659, 4782}, {661, 650}, {663, 667}, {693, 4369}, {1459, 3733}, {1491, 9508}, {2484, 2483}, {3004, 17069}, {3239, 43061}, {3250, 665}, {3835, 31286}, {4010, 4874}, {4024, 6590}, {4025, 3798}, {4040, 4401}, {4079, 6586}, {4106, 4885}, {4382, 693}, {4449, 4367}, {4468, 11068}, {4498, 4063}, {4502, 3709}, {4724, 659}, {4750, 4786}, {4775, 1960}, {4776, 4763}, {4804, 7662}, {4813, 661}, {4814, 4730}, {4893, 1635}, {4940, 31287}, {4958, 4944}, {4979, 4790}, {4988, 45745}, {7192, 4932}, {14321, 2490}, {14349, 14838}, {16892, 4025}, {17217, 21191}, {20295, 3835}, {20979, 798}, {21301, 17072}, {21834, 21348}, {23655, 7234}, {24719, 3837}, {27486, 45679}, {30835, 27013}, {31147, 2}, {38325, 42322}, {42664, 647}, {44435, 45674}, {45745, 4765}, {45746, 21196}
X(649) = reflection of X(i) in X(j) for these (i,j): (661,650), (663,667)
X(649) = isogonal conjugate of X(190)
X(649) = isotomic conjugate of X(1978)
X(649) = complement of X(20295)
X(649) = anticomplement of X(3835)
X(649) = excentral-isogonal conjugate of X(9355)
X(649) = tangential-isogonal conjugate of X(16873)
X(649) = trilinear pole of line {1015, 1960}
X(649) = Parry-circle-inverse of X(5029)
X(649) = Parry-isodynamic-circle-inverse of X(5168)
X(649) = crossdifference of every pair of points on line {1, 2}
X(649) = excentral isogonal conjugate of X(9355)
X(649) = tangential isogonal conjugate of X(16873)
X(649) = bicentric difference of PU(i) for i in (8, 48, 58, 84, 92, 98)
X(649) = PU(8)-harmonic conjugate of X(42)
X(649) = trilinear pole of PU(25) (line X(1015)X(1960))
X(649) = barycentric product of PU(34)
X(649) = PU(48)-harmonic conjugate of X(31)
X(649) = PU(58)-harmonic conjugate of X(899)
X(649) = PU(84)-harmonic conjugate of X(3720)
X(649) = PU(92)-harmonic conjugate of X(1201)
X(649) = PU(98)-harmonic conjugate of X(1149)
X(649) = intersection of antiorthic and Lemoine axes (trilinear polars of X(1) and X(6))
X(649) = pole, with respect to Bevan circle, of line X(2)X(7)
X(649) = X(6)-isoconjugate of X(668)
X(649) = X(647)-of-excentral-triangle
X(649) = perspector of the excentral Jerabek hyperbola
X(649) = complement of polar conjugate of isogonal conjugate of X(22154)
X(649) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {87, 150}, {109, 20537}, {110, 17149}, {330, 21293}, {651, 20350}, {664, 20559}, {692, 21219}, {932, 69}, {1576, 36857}, {2053, 37781}, {2162, 149}, {2319, 33650}, {4598, 6327}, {5383, 21301}, {7121, 4440}, {16606, 3448}, {18830, 315}, {21759, 148}, {22381, 39352}, {23493, 21221}, {32739, 41840}, {34071, 8}, {42027, 21294}
X(649) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 8054}, {596, 21252}, {692, 4075}, {8050, 2887}, {20615, 17059}, {34594, 3741}, {37205, 21240}, {39798, 116}, {40085, 21253}, {40148, 11}, {40519, 10}
X(649) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 3248}, {2, 8054}, {6, 1015}, {19, 2170}, {25, 20974}, {27, 3120}, {31, 38346}, {56, 3271}, {57, 244}, {58, 3122}, {64, 3270}, {84, 2310}, {87, 38986}, {99, 2309}, {100, 42}, {101, 6}, {108, 40958}, {109, 31}, {110, 2308}, {163, 2260}, {190, 1}, {267, 2643}, {513, 663}, {514, 1459}, {651, 1201}, {653, 3924}, {658, 614}, {660, 3009}, {662, 1193}, {664, 7032}, {667, 20979}, {673, 27846}, {692, 1475}, {799, 3720}, {813, 672}, {825, 21764}, {893, 3121}, {901, 902}, {932, 22343}, {934, 20978}, {1018, 1100}, {1019, 513}, {1020, 1104}, {1086, 8578}, {1292, 2293}, {1293, 55}, {1407, 3937}, {1415, 1400}, {1436, 7117}, {1461, 56}, {1576, 40955}, {1919, 23572}, {2156, 21339}, {2160, 3125}, {2161, 2087}, {2162, 1977}, {2176, 23560}, {2226, 43922}, {2248, 3124}, {2423, 21758}, {2702, 1914}, {2705, 8540}, {3224, 21762}, {3257, 1149}, {3345, 2638}, {3437, 20975}, {3444, 20982}, {3500, 3123}, {3572, 5029}, {3669, 43924}, {3733, 667}, {3903, 23532}, {4556, 58}, {4557, 20963}, {4559, 2300}, {4584, 238}, {4588, 21747}, {4591, 36}, {4598, 2}, {4603, 81}, {4604, 995}, {4607, 899}, {4610, 17187}, {4628, 5299}, {4629, 1203}, {4638, 106}, {4817, 4724}, {6010, 2269}, {6011, 14547}, {6014, 2177}, {6577, 1918}, {7087, 23646}, {7096, 17463}, {7097, 34591}, {7252, 22383}, {8050, 40148}, {8052, 5262}, {8056, 17477}, {8699, 3052}, {8709, 20464}, {8750, 16502}, {8917, 24012}, {9500, 35505}, {10566, 514}, {11051, 14936}, {13478, 11}, {13610, 4128}, {14377, 1086}, {16695, 23464}, {18830, 23493}, {21362, 20323}, {23345, 1960}, {23355, 6373}, {23892, 3768}, {28469, 3056}, {28477, 2268}, {28486, 2330}, {28847, 2280}, {29014, 71}, {29217, 5301}, {30554, 5332}, {32039, 87}, {32665, 2183}, {32674, 604}, {32675, 1404}, {32714, 1042}, {34067, 20459}, {34071, 2275}, {34080, 2347}, {34183, 38363}, {34234, 1647}, {34440, 3269}, {34445, 23456}, {35342, 4272}, {36146, 1458}, {36614, 23470}, {37133, 21352}, {37138, 869}, {37205, 1125}, {37209, 30950}, {37211, 386}, {37215, 17017}, {38828, 32577}, {39950, 16726}, {40151, 1357}, {40188, 3942}, {40519, 39}, {40737, 4117}, {40770, 21755}, {41405, 41395}, {42467, 7004}, {43077, 2276}, {43739, 11998}, {43924, 8643}, {43929, 8632}
X(649) = X(i)-cross conjugate of X(j) for these (i,j): {6, 9262}, {244, 2350}, {512, 513}, {665, 3572}, {667, 43924}, {798, 667}, {1015, 6}, {1960, 23345}, {1977, 40148}, {3063, 663}, {3121, 42}, {3122, 58}, {3125, 1400}, {3248, 1}, {3249, 3248}, {3271, 56}, {3768, 23892}, {4083, 20979}, {6373, 23355}, {6377, 2}, {7180, 6591}, {7234, 18108}, {8659, 43929}, {8661, 43922}, {14936, 25}, {16614, 39956}, {21123, 514}, {21143, 1015}, {21191, 23572}, {23456, 34445}, {23470, 30651}, {38986, 87}
X(649) = X(i)-isoconjugate of X(j) for these (i,j): {1, 190}, {2, 100}, {3, 6335}, {4, 1332}, {6, 668}, {7, 644}, {8, 651}, {9, 664}, {10, 662}, {11, 31615}, {12, 4612}, {19, 4561}, {21, 4552}, {31, 1978}, {32, 6386}, {35, 15455}, {36, 36804}, {37, 99}, {40, 44327}, {41, 4572}, {42, 799}, {43, 4598}, {44, 4555}, {45, 4597}, {55, 4554}, {56, 646}, {57, 3699}, {58, 4033}, {59, 4391}, {63, 1897}, {65, 645}, {69, 1783}, {71, 811}, {72, 648}, {74, 42716}, {75, 101}, {76, 692}, {78, 653}, {80, 4585}, {81, 3952}, {82, 4568}, {83, 4553}, {85, 3939}, {86, 1018}, {87, 4595}, {88, 17780}, {89, 4767}, {92, 1331}, {98, 42717}, {102, 42718}, {103, 42719}, {104, 2397}, {105, 42720}, {106, 24004}, {107, 3998}, {108, 345}, {109, 312}, {110, 321}, {111, 42721}, {112, 20336}, {145, 27834}, {162, 306}, {163, 313}, {171, 27805}, {181, 4631}, {192, 932}, {200, 658}, {210, 4573}, {213, 670}, {219, 18026}, {220, 4569}, {226, 643}, {228, 6331}, {238, 4562}, {239, 660}, {241, 36802}, {244, 6632}, {249, 4036}, {256, 18047}, {257, 4579}, {261, 21859}, {264, 906}, {269, 6558}, {273, 4587}, {274, 4557}, {278, 4571}, {279, 4578}, {281, 6516}, {286, 4574}, {291, 3570}, {292, 874}, {294, 883}, {304, 8750}, {314, 4559}, {318, 1813}, {322, 36049}, {329, 13138}, {333, 4551}, {335, 3573}, {341, 1461}, {344, 1292}, {346, 934}, {350, 813}, {386, 37218}, {476, 42701}, {480, 36838}, {512, 4601}, {513, 1016}, {514, 765}, {517, 13136}, {518, 666}, {519, 3257}, {521, 46102}, {522, 4564}, {523, 4567}, {524, 5380}, {525, 5379}, {536, 898}, {556, 6733}, {561, 32739}, {598, 3908}, {612, 37215}, {649, 7035}, {650, 4998}, {655, 4511}, {661, 4600}, {667, 31625}, {673, 1026}, {675, 42723}, {677, 30807}, {689, 21814}, {691, 42713}, {693, 1252}, {728, 4626}, {739, 41314}, {740, 4584}, {751, 4482}, {752, 5386}, {754, 5389}, {756, 4610}, {757, 4103}, {788, 5388}, {789, 2276}, {812, 5378}, {823, 3682}, {824, 5384}, {825, 33931}, {831, 17289}, {833, 32777}, {835, 28606}, {839, 4261}, {840, 42722}, {869, 37133}, {889, 3230}, {891, 5381}, {892, 21839}, {894, 3903}, {899, 4607}, {900, 5376}, {901, 4358}, {903, 1023}, {905, 15742}, {908, 36037}, {914, 36106}, {918, 5377}, {919, 3263}, {925, 42700}, {927, 3693}, {931, 31993}, {933, 42698}, {958, 32038}, {960, 6648}, {982, 4621}, {983, 33946}, {984, 4586}, {985, 3807}, {1001, 32041}, {1014, 30730}, {1020, 1043}, {1025, 14942}, {1042, 7258}, {1089, 4556}, {1098, 4605}, {1100, 6540}, {1110, 3261}, {1125, 37212}, {1212, 6606}, {1213, 4596}, {1214, 36797}, {1215, 4603}, {1220, 3882}, {1222, 21362}, {1255, 4427}, {1257, 14543}, {1260, 13149}, {1262, 4397}, {1265, 32714}, {1267, 6135}, {1268, 35342}, {1275, 3900}, {1278, 29227}, {1290, 32849}, {1293, 18743}, {1296, 42724}, {1301, 42699}, {1302, 42704}, {1305, 27396}, {1308, 17264}, {1310, 2345}, {1319, 4582}, {1333, 27808}, {1334, 4625}, {1376, 30610}, {1400, 7257}, {1414, 2321}, {1415, 3596}, {1427, 7256}, {1429, 36801}, {1434, 4069}, {1441, 5546}, {1473, 42384}, {1476, 25268}, {1492, 3661}, {1500, 4623}, {1509, 40521}, {1575, 8709}, {1576, 27801}, {1577, 4570}, {1633, 30701}, {1698, 37211}, {1757, 35148}, {1824, 4563}, {1826, 4592}, {1911, 27853}, {1914, 4583}, {1918, 4602}, {1921, 34067}, {1930, 4628}, {1962, 4632}, {1969, 32656}, {1983, 20566}, {1997, 30236}, {2087, 6635}, {2149, 35519}, {2162, 36863}, {2176, 18830}, {2205, 4609}, {2214, 33948}, {2222, 32851}, {2223, 36803}, {2238, 4589}, {2283, 36796}, {2284, 2481}, {2287, 4566}, {2295, 4594}, {2323, 35174}, {2339, 14594}, {2340, 34085}, {2398, 36101}, {2427, 18816}, {2702, 20947}, {2703, 17790}, {2715, 42703}, {2742, 37788}, {2743, 37758}, {2748, 37756}, {2753, 37857}, {2832, 5387}, {3006, 36087}, {3035, 31628}, {3112, 46148}, {3175, 8690}, {3198, 44326}, {3219, 6742}, {3222, 21877}, {3227, 23343}, {3239, 7045}, {3240, 37209}, {3262, 32641}, {3264, 32665}, {3290, 35574}, {3293, 37205}, {3616, 4606}, {3666, 8707}, {3667, 5382}, {3668, 7259}, {3669, 4076}, {3672, 6574}, {3679, 4604}, {3681, 43190}, {3687, 36098}, {3692, 36118}, {3701, 4565}, {3717, 36146}, {3718, 32674}, {3719, 36127}, {3739, 8708}, {3747, 4639}, {3752, 8706}, {3762, 9268}, {3783, 37207}, {3797, 30664}, {3799, 14621}, {3869, 44765}, {3870, 37206}, {3888, 17743}, {3909, 40394}, {3912, 36086}, {3935, 37143}, {3943, 4622}, {3954, 4577}, {3969, 13486}, {3990, 6528}, {3992, 4591}, {3995, 34594}, {4024, 24041}, {4037, 36066}, {4039, 37134}, {4041, 4620}, {4043, 43076}, {4062, 36085}, {4079, 24037}, {4082, 4637}, {4083, 5383}, {4115, 40438}, {4357, 36147}, {4359, 8701}, {4370, 4618}, {4384, 37138}, {4417, 36050}, {4420, 38340}, {4421, 42343}, {4436, 32009}, {4441, 8693}, {4463, 44766}, {4505, 40746}, {4512, 4624}, {4515, 4616}, {4558, 41013}, {4576, 18098}, {4588, 4671}, {4590, 4705}, {4593, 21035}, {4599, 15523}, {4613, 40773}, {4614, 5257}, {4615, 21805}, {4617, 5423}, {4619, 24026}, {4629, 4647}, {4633, 37593}, {4636, 6358}, {4638, 4738}, {4663, 35177}, {4664, 29351}, {4687, 6013}, {4699, 29199}, {4752, 39704}, {4756, 25417}, {4777, 5385}, {4781, 40434}, {4812, 29026}, {4850, 9059}, {4980, 28176}, {4997, 23703}, {5222, 37223}, {5223, 32040}, {5291, 35147}, {5293, 8052}, {5297, 37210}, {5360, 43187}, {5375, 8047}, {5391, 6136}, {5435, 31343}, {5526, 35171}, {5545, 42712}, {6011, 33116}, {6012, 17279}, {6014, 30829}, {6065, 24002}, {6079, 16610}, {6163, 6630}, {6164, 6634}, {6332, 7012}, {6376, 34071}, {6381, 34075}, {6542, 37135}, {6554, 8269}, {6559, 41353}, {6577, 18137}, {6603, 35157}, {6605, 35312}, {6614, 30693}, {6631, 9282}, {6735, 37136}, {6745, 37139}, {6790, 46119}, {7017, 36059}, {7080, 37141}, {7081, 37137}, {7115, 35518}, {7239, 40415}, {7260, 20964}, {8050, 32911}, {8056, 43290}, {8652, 28605}, {8684, 33891}, {8694, 19804}, {8699, 20942}, {9058, 17740}, {9067, 17756}, {9070, 32779}, {9265, 9296}, {9266, 9295}, {9271, 17487}, {9278, 17934}, {9361, 9362}, {11124, 31619}, {11495, 42303}, {11611, 17944}, {13396, 17281}, {13397, 17776}, {14947, 40865}, {15322, 28653}, {15624, 31624}, {16514, 41072}, {16593, 39272}, {16777, 32042}, {16785, 35181}, {17459, 35572}, {17787, 29055}, {17796, 35156}, {17863, 29163}, {17930, 20693}, {18140, 40519}, {18147, 29014}, {19604, 30720}, {19799, 32691}, {20332, 23354}, {20440, 20640}, {20453, 20696}, {20568, 23344}, {20901, 31616}, {20911, 32736}, {20940, 40150}, {21272, 23617}, {21453, 35341}, {21802, 35137}, {21833, 31614}, {21874, 35136}, {21899, 37880}, {22003, 40430}, {22456, 42702}, {23067, 31623}, {23493, 36860}, {23704, 35160}, {23832, 36805}, {23845, 32017}, {23891, 37129}, {23981, 36795}, {23990, 40495}, {24589, 28210}, {25001, 43344}, {25660, 29151}, {26700, 42033}, {26706, 28420}, {26711, 33168}, {26714, 42711}, {28148, 42029}, {28162, 42034}, {28474, 41316}, {28583, 41315}, {28847, 30758}, {30555, 31130}, {30625, 42301}, {30963, 43077}, {31633, 42552}, {32008, 35338}, {32018, 35327}, {32676, 40071}, {32718, 35543}, {32931, 35009}, {33113, 33637}, {33157, 43348}, {35280, 39749}, {35517, 36039}, {36077, 42706}, {36080, 44140}, {37204, 41267}, {38828, 44720}, {40728, 46132}, {41839, 43350}, {44426, 44717}
X(649) = cevapoint of X(i) and X(j) for these (i,j): {1, 9359}, {2, 21224}, {6, 9259}, {512, 798}, {513, 4083}, {514, 21191}, {667, 3063}, {1015, 21143}, {3121, 8027}, {3248, 3249}
X(649) = crosspoint of X(i) and X(j) for these (i,j): {1, 190}, {2, 8050}, {6, 101}, {19, 32674}, {28, 32714}, {56, 1461}, {57, 109}, {58, 4556}, {81, 100}, {87, 32039}, {99, 40409}, {106, 4638}, {108, 40397}, {110, 1171}, {274, 18830}, {513, 3669}, {514, 7649}, {651, 1476}, {662, 2363}, {799, 40439}, {893, 4603}, {901, 2226}, {1019, 3733}, {1293, 40151}, {1333, 1415}, {2162, 4598}, {2702, 9506}, {15397, 32682}
X(649) = crosssum of X(i) and X(j) for these (i,j): {1, 649}, {2, 514}, {6, 4057}, {7, 31605}, {8, 3239}, {9, 522}, {10, 4024}, {37, 513}, {57, 30719}, {63, 6332}, {71, 4064}, {75, 20954}, {100, 644}, {101, 1331}, {115, 12078}, {145, 31182}, {192, 20979}, {213, 8640}, {239, 4375}, {321, 4391}, {440, 525}, {512, 21838}, {519, 6544}, {523, 1213}, {650, 3057}, {656, 18674}, {657, 4319}, {659, 17475}, {661, 2292}, {663, 20665}, {667, 21757}, {668, 36863}, {693, 20880}, {798, 2667}, {812, 17755}, {900, 4370}, {918, 16593}, {1016, 6634}, {1018, 3952}, {1019, 8025}, {1252, 14887}, {1635, 17460}, {1960, 23552}, {2183, 23757}, {2254, 17464}, {2398, 3234}, {2786, 6651}, {3059, 4130}, {3161, 3667}, {3219, 7265}, {3294, 17494}, {3699, 30720}, {3700, 21677}, {3730, 25259}, {3768, 42083}, {3835, 33890}, {3995, 4063}, {4025, 17170}, {4079, 21035}, {4115, 4427}, {4368, 21832}, {4473, 24129}, {4498, 30568}, {4560, 17185}, {4568, 46148}, {4775, 23553}, {4777, 16590}, {4785, 27481}, {4893, 17461}, {5513, 23887}, {6006, 36911}, {6364, 40651}, {6545, 31647}, {6546, 32094}, {7192, 17175}, {8058, 38015}, {8714, 40586}, {16892, 17192}, {21362, 25268}, {24979, 24980}, {28840, 31336}
X(649) = barycentric product X(i)*X(j) for these {i,j}: {1, 513}, {3, 7649}, {4, 1459}, {6, 514}, {7, 663}, {8, 43924}, {9, 3669}, {10, 3733}, {11, 109}, {19, 905}, {21, 4017}, {25, 4025}, {27, 647}, {28, 656}, {31, 693}, {32, 3261}, {34, 521}, {37, 1019}, {38, 18108}, {39, 10566}, {41, 24002}, {42, 7192}, {43, 43931}, {44, 1022}, {48, 17924}, {54, 21102}, {55, 3676}, {56, 522}, {57, 650}, {58, 523}, {59, 21132}, {63, 6591}, {64, 21172}, {65, 3737}, {71, 17925}, {74, 11125}, {75, 667}, {76, 1919}, {77, 18344}, {78, 43923}, {79, 2605}, {81, 661}, {82, 2530}, {83, 21123}, {84, 6129}, {85, 3063}, {86, 512}, {87, 4083}, {88, 1635}, {89, 4893}, {91, 34948}, {92, 22383}, {99, 3122}, {100, 244}, {101, 1086}, {103, 676}, {104, 1769}, {105, 2254}, {106, 900}, {108, 7004}, {110, 3120}, {111, 4750}, {112, 4466}, {115, 4556}, {158, 23224}, {162, 18210}, {163, 16732}, {174, 6729}, {184, 46107}, {190, 1015}, {200, 43932}, {210, 7203}, {213, 7199}, {222, 3064}, {225, 23189}, {226, 7252}, {238, 876}, {239, 3572}, {241, 1024}, {249, 21131}, {250, 21134}, {251, 16892}, {256, 4367}, {257, 20981}, {266, 6728}, {267, 31947}, {269, 3900}, {273, 1946}, {274, 798}, {278, 652}, {279, 657}, {284, 7178}, {286, 810}, {291, 659}, {292, 812}, {306, 43925}, {310, 669}, {330, 20979}, {333, 7180}, {335, 8632}, {350, 875}, {386, 43927}, {393, 4091}, {479, 4105}, {516, 2424}, {518, 1027}, {519, 23345}, {520, 8747}, {525, 1474}, {536, 23892}, {560, 40495}, {561, 1980}, {593, 4024}, {595, 40086}, {596, 4057}, {603, 44426}, {604, 4391}, {608, 6332}, {651, 2170}, {653, 7117}, {654, 2006}, {658, 14936}, {660, 27846}, {662, 3125}, {664, 3271}, {665, 673}, {668, 3248}, {679, 3251}, {692, 1111}, {694, 4107}, {726, 23355}, {727, 3837}, {738, 4130}, {739, 4728}, {741, 4010}, {751, 4378}, {753, 4809}, {757, 4705}, {764, 765}, {788, 870}, {799, 3121}, {813, 27918}, {824, 40746}, {832, 977}, {834, 43531}, {849, 4036}, {850, 2206}, {871, 8630}, {884, 9436}, {885, 1458}, {890, 31002}, {891, 37129}, {893, 4369}, {897, 14419}, {898, 19945}, {899, 43928}, {901, 1647}, {902, 6548}, {903, 1960}, {904, 4374}, {908, 2423}, {909, 10015}, {918, 1438}, {932, 3123}, {934, 2310}, {959, 17418}, {961, 17420}, {963, 7661}, {967, 45745}, {983, 3777}, {985, 1491}, {996, 9002}, {998, 9001}, {1002, 4724}, {1014, 4041}, {1016, 21143}, {1018, 16726}, {1021, 1427}, {1026, 43921}, {1042, 7253}, {1043, 7250}, {1054, 6164}, {1088, 8641}, {1096, 4131}, {1106, 4397}, {1120, 6085}, {1126, 4977}, {1146, 1461}, {1149, 23836}, {1155, 35348}, {1156, 14413}, {1169, 21124}, {1170, 21127}, {1171, 4988}, {1173, 21103}, {1174, 21104}, {1175, 23752}, {1176, 21108}, {1177, 21109}, {1178, 2533}, {1193, 4581}, {1220, 6371}, {1222, 6363}, {1252, 6545}, {1255, 4979}, {1262, 42462}, {1279, 35355}, {1293, 3756}, {1318, 39771}, {1319, 23838}, {1323, 23351}, {1326, 18014}, {1331, 2969}, {1333, 1577}, {1334, 17096}, {1357, 3699}, {1358, 3939}, {1364, 36127}, {1365, 4636}, {1395, 35518}, {1396, 8611}, {1397, 35519}, {1400, 4560}, {1402, 18155}, {1407, 3239}, {1408, 4086}, {1411, 3738}, {1412, 3700}, {1413, 8058}, {1414, 4516}, {1415, 4858}, {1417, 4768}, {1421, 42552}, {1422, 14298}, {1431, 3907}, {1432, 3287}, {1434, 3709}, {1436, 14837}, {1437, 24006}, {1457, 43728}, {1472, 2517}, {1476, 6615}, {1486, 26721}, {1492, 4475}, {1509, 4079}, {1565, 8750}, {1576, 21207}, {1581, 4164}, {1638, 2291}, {1643, 37131}, {1646, 4607}, {1751, 43060}, {1783, 3942}, {1790, 2501}, {1795, 39534}, {1813, 8735}, {1826, 7254}, {1875, 37628}, {1876, 23696}, {1897, 3937}, {1911, 3766}, {1914, 4444}, {1924, 6385}, {1929, 9508}, {1967, 14296}, {1973, 15413}, {1977, 1978}, {2087, 3257}, {2109, 25381}, {2149, 40166}, {2156, 16757}, {2160, 14838}, {2161, 3960}, {2162, 3835}, {2163, 4777}, {2164, 21188}, {2183, 2401}, {2191, 3309}, {2194, 4077}, {2195, 43042}, {2203, 14208}, {2207, 30805}, {2208, 17896}, {2214, 14349}, {2215, 23882}, {2217, 21189}, {2218, 23800}, {2221, 6590}, {2226, 6544}, {2248, 21196}, {2258, 43067}, {2276, 4817}, {2279, 4762}, {2287, 7216}, {2296, 2978}, {2297, 8712}, {2299, 17094}, {2308, 4608}, {2311, 7212}, {2316, 30725}, {2319, 43051}, {2333, 15419}, {2334, 4778}, {2340, 43930}, {2348, 37626}, {2349, 14399}, {2350, 17494}, {2353, 21178}, {2354, 15420}, {2364, 43052}, {2382, 36848}, {2384, 14475}, {2395, 17209}, {2426, 15634}, {2432, 34050}, {2433, 18653}, {2486, 43076}, {2488, 21453}, {2489, 17206}, {2504, 9085}, {2509, 40188}, {2516, 36603}, {2526, 39958}, {2611, 13486}, {2616, 18180}, {2623, 17167}, {2718, 24457}, {2720, 35015}, {2786, 17962}, {2787, 17954}, {2832, 34893}, {2973, 32656}, {2983, 29162}, {2985, 23751}, {2998, 23572}, {3011, 35365}, {3022, 4626}, {3049, 44129}, {3119, 4617}, {3124, 4610}, {3224, 21191}, {3226, 6373}, {3227, 3768}, {3249, 31625}, {3250, 14621}, {3270, 36118}, {3285, 4049}, {3310, 34234}, {3433, 21185}, {3435, 21186}, {3437, 21187}, {3444, 21192}, {3445, 3667}, {3446, 21201}, {3447, 21203}, {3449, 21118}, {3450, 21119}, {3451, 21120}, {3453, 21121}, {3455, 21205}, {3500, 21348}, {3668, 21789}, {3675, 36086}, {3762, 9456}, {3778, 7255}, {3798, 8770}, {3801, 38813}, {3803, 23051}, {3805, 40763}, {3862, 23597}, {3880, 37627}, {3912, 43929}, {3954, 39179}, {4040, 13476}, {4062, 43926}, {4063, 39798}, {4081, 6614}, {4128, 4594}, {4132, 39949}, {4142, 34250}, {4160, 34916}, {4162, 19604}, {4163, 7023}, {4303, 14775}, {4306, 23289}, {4373, 8643}, {4379, 30650}, {4382, 30651}, {4394, 8056}, {4401, 7241}, {4440, 9262}, {4449, 9309}, {4453, 6187}, {4455, 18827}, {4458, 8852}, {4459, 29055}, {4462, 38266}, {4467, 6186}, {4481, 40747}, {4491, 39697}, {4498, 39956}, {4521, 40151}, {4534, 38828}, {4551, 18191}, {4557, 17205}, {4559, 17197}, {4561, 42067}, {4565, 21044}, {4584, 39786}, {4598, 6377}, {4600, 8034}, {4603, 16592}, {4618, 42084}, {4637, 36197}, {4638, 35092}, {4707, 34079}, {4775, 39704}, {4784, 30571}, {4786, 21448}, {4790, 25430}, {4791, 28607}, {4813, 25417}, {4823, 34819}, {4834, 30598}, {4885, 9315}, {4932, 39967}, {4943, 16079}, {4957, 34073}, {4960, 28625}, {4978, 28615}, {4983, 40438}, {5009, 35352}, {5029, 6650}, {5209, 18002}, {5317, 24018}, {5331, 8672}, {5620, 42741}, {6005, 10013}, {6006, 41436}, {6149, 43082}, {6336, 22086}, {6372, 40433}, {6381, 23349}, {6384, 8640}, {6549, 23344}, {6550, 9268}, {6551, 24188}, {6586, 14377}, {6588, 42467}, {6589, 13478}, {6610, 23893}, {6629, 9178}, {6730, 7370}, {7035, 8027}, {7077, 43041}, {7087, 20517}, {7121, 20906}, {7169, 21174}, {7234, 32010}, {7260, 21755}, {7316, 14432}, {7339, 23615}, {7658, 11051}, {8050, 8054}, {8059, 38357}, {8578, 44184}, {8648, 18815}, {8656, 36588}, {8659, 36807}, {8677, 36123}, {8690, 21963}, {8713, 10579}, {8714, 34444}, {8917, 17427}, {9217, 21200}, {9259, 42555}, {9265, 21211}, {9267, 9359}, {9269, 9325}, {9292, 17215}, {9299, 18149}, {9311, 20980}, {9361, 38238}, {9505, 38348}, {9506, 27929}, {10428, 23757}, {10490, 10495}, {10492, 18888}, {10509, 10581}, {14370, 21194}, {14554, 21786}, {15378, 21133}, {15382, 20504}, {16099, 42662}, {16606, 18197}, {16695, 42027}, {16702, 23894}, {16737, 40729}, {16887, 18105}, {17216, 32713}, {17217, 23493}, {17222, 45677}, {17435, 36146}, {17731, 18001}, {17758, 21007}, {17780, 43922}, {18018, 21122}, {18101, 46153}, {18359, 21758}, {18771, 21105}, {18772, 21106}, {18830, 38986}, {20295, 40148}, {20516, 34183}, {20518, 41528}, {20908, 34077}, {20974, 43190}, {21003, 39714}, {21110, 38826}, {21113, 42346}, {21138, 34071}, {21173, 34434}, {21175, 34436}, {21176, 34437}, {21179, 34441}, {21180, 34442}, {21183, 34446}, {21190, 34427}, {21202, 34179}, {21206, 36615}, {21208, 40519}, {21385, 39981}, {21763, 42328}, {21828, 24624}, {21832, 37128}, {22084, 26705}, {22108, 34578}, {22350, 43933}, {23100, 23990}, {23707, 30691}, {23723, 34429}, {23729, 38825}, {23807, 34248}, {23845, 40451}, {23989, 32739}, {24027, 42455}, {25426, 28840}, {26932, 32674}, {26933, 32691}, {28209, 41434}, {29198, 39972}, {29226, 36598}, {30723, 34820}, {30724, 33635}, {32039, 40610}, {32641, 42754}, {32714, 34591}, {34858, 36038}, {35014, 36110}, {36037, 42753}, {40397, 40628}, {40409, 40627}, {40738, 45882}, {41799, 45877}, {42290, 45755}
X(649) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 668}, {2, 1978}, {3, 4561}, {6, 190}, {7, 4572}, {9, 646}, {10, 27808}, {11, 35519}, {19, 6335}, {21, 7257}, {25, 1897}, {27, 6331}, {28, 811}, {31, 100}, {32, 101}, {34, 18026}, {37, 4033}, {39, 4568}, {41, 644}, {42, 3952}, {43, 36863}, {44, 24004}, {48, 1332}, {55, 3699}, {56, 664}, {57, 4554}, {58, 99}, {75, 6386}, {81, 799}, {86, 670}, {87, 18830}, {100, 7035}, {101, 1016}, {106, 4555}, {109, 4998}, {110, 4600}, {163, 4567}, {172, 18047}, {184, 1331}, {190, 31625}, {212, 4571}, {213, 1018}, {220, 6558}, {238, 874}, {239, 27853}, {244, 693}, {269, 4569}, {274, 4602}, {284, 645}, {291, 4583}, {292, 4562}, {310, 4609}, {351, 4062}, {386, 33948}, {512, 10}, {513, 75}, {514, 76}, {521, 3718}, {522, 3596}, {523, 313}, {525, 40071}, {560, 692}, {593, 4610}, {603, 6516}, {604, 651}, {608, 653}, {647, 306}, {650, 312}, {652, 345}, {654, 32851}, {656, 20336}, {657, 346}, {659, 350}, {661, 321}, {662, 4601}, {663, 8}, {665, 3912}, {667, 1}, {669, 42}, {672, 42720}, {673, 36803}, {676, 35517}, {688, 21035}, {692, 765}, {693, 561}, {727, 8709}, {738, 36838}, {739, 4607}, {741, 4589}, {757, 4623}, {764, 1111}, {788, 984}, {798, 37}, {810, 72}, {812, 1921}, {822, 3998}, {830, 33941}, {834, 5224}, {869, 3799}, {870, 46132}, {872, 40521}, {875, 291}, {876, 334}, {884, 14942}, {890, 899}, {891, 6381}, {893, 27805}, {896, 42721}, {899, 41314}, {900, 3264}, {902, 17780}, {904, 3903}, {905, 304}, {909, 13136}, {910, 42719}, {923, 5380}, {926, 3717}, {984, 4505}, {985, 789}, {1014, 4625}, {1015, 514}, {1019, 274}, {1022, 20568}, {1024, 36796}, {1027, 2481}, {1042, 4566}, {1084, 4079}, {1086, 3261}, {1106, 934}, {1111, 40495}, {1126, 6540}, {1171, 4632}, {1178, 4594}, {1201, 21272}, {1252, 6632}, {1253, 4578}, {1326, 17934}, {1333, 662}, {1334, 30730}, {1357, 3676}, {1395, 108}, {1397, 109}, {1398, 36118}, {1400, 4552}, {1402, 4551}, {1407, 658}, {1408, 1414}, {1411, 35174}, {1412, 4573}, {1415, 4564}, {1416, 927}, {1435, 13149}, {1436, 44327}, {1437, 4592}, {1438, 666}, {1458, 883}, {1459, 69}, {1460, 14594}, {1461, 1275}, {1462, 34085}, {1472, 1310}, {1474, 648}, {1491, 33931}, {1500, 4103}, {1501, 32739}, {1576, 4570}, {1577, 27801}, {1635, 4358}, {1646, 4728}, {1734, 33932}, {1755, 42717}, {1769, 3262}, {1790, 4563}, {1911, 660}, {1912, 12782}, {1914, 3570}, {1918, 4557}, {1919, 6}, {1922, 813}, {1924, 213}, {1946, 78}, {1960, 519}, {1964, 4553}, {1973, 1783}, {1974, 8750}, {1979, 9362}, {1980, 31}, {2084, 3954}, {2087, 3762}, {2149, 31615}, {2150, 4612}, {2160, 15455}, {2161, 36804}, {2162, 4598}, {2163, 4597}, {2170, 4391}, {2173, 42716}, {2175, 3939}, {2176, 4595}, {2177, 4767}, {2182, 42718}, {2183, 2397}, {2185, 4631}, {2194, 643}, {2195, 36802}, {2200, 4574}, {2203, 162}, {2206, 110}, {2208, 13138}, {2210, 3573}, {2214, 37218}, {2221, 37215}, {2223, 1026}, {2225, 42723}, {2242, 4482}, {2246, 42722}, {2251, 1023}, {2254, 3263}, {2275, 33946}, {2276, 3807}, {2279, 32041}, {2287, 7258}, {2299, 36797}, {2300, 3882}, {2308, 4427}, {2310, 4397}, {2316, 4582}, {2328, 7256}, {2347, 25268}, {2423, 34234}, {2424, 18025}, {2440, 31638}, {2441, 31227}, {2483, 17289}, {2484, 2345}, {2485, 4150}, {2488, 4847}, {2489, 1826}, {2492, 21094}, {2516, 20942}, {2520, 17860}, {2522, 19799}, {2530, 1930}, {2533, 1237}, {2605, 319}, {2624, 42701}, {2642, 42713}, {2643, 4036}, {2969, 46107}, {2978, 31330}, {3005, 15523}, {3009, 23354}, {3022, 4163}, {3049, 71}, {3051, 46148}, {3052, 43290}, {3063, 9}, {3064, 7017}, {3120, 850}, {3121, 661}, {3122, 523}, {3123, 20906}, {3124, 4024}, {3125, 1577}, {3221, 21080}, {3230, 23891}, {3248, 513}, {3249, 1015}, {3250, 3661}, {3251, 4738}, {3261, 1502}, {3271, 522}, {3287, 17787}, {3310, 908}, {3572, 335}, {3669, 85}, {3676, 6063}, {3700, 30713}, {3709, 2321}, {3716, 4087}, {3733, 86}, {3737, 314}, {3752, 21580}, {3766, 18891}, {3768, 536}, {3777, 33930}, {3803, 39731}, {3835, 6382}, {3837, 35538}, {3900, 341}, {3937, 4025}, {3939, 4076}, {3942, 15413}, {3960, 20924}, {4010, 35544}, {4014, 20907}, {4017, 1441}, {4024, 28654}, {4025, 305}, {4040, 17143}, {4041, 3701}, {4057, 4360}, {4063, 18140}, {4079, 594}, {4083, 6376}, {4091, 3926}, {4105, 5423}, {4107, 3978}, {4128, 2533}, {4130, 30693}, {4142, 18835}, {4162, 44720}, {4164, 1966}, {4258, 30728}, {4367, 1909}, {4369, 1920}, {4378, 3761}, {4391, 28659}, {4394, 18743}, {4435, 3975}, {4444, 18895}, {4453, 40075}, {4455, 740}, {4466, 3267}, {4491, 17160}, {4498, 18135}, {4516, 4086}, {4521, 44723}, {4524, 4082}, {4556, 4590}, {4560, 28660}, {4565, 4620}, {4581, 1240}, {4586, 5388}, {4610, 34537}, {4636, 6064}, {4705, 1089}, {4724, 4441}, {4728, 35543}, {4730, 3992}, {4750, 3266}, {4762, 21615}, {4770, 4125}, {4775, 3679}, {4782, 30963}, {4785, 10009}, {4786, 11059}, {4790, 19804}, {4802, 30596}, {4809, 35548}, {4813, 28605}, {4832, 5257}, {4834, 1698}, {4874, 4485}, {4879, 25280}, {4893, 4671}, {4895, 4723}, {4905, 33933}, {4977, 1269}, {4979, 4359}, {4983, 4647}, {4988, 1230}, {5027, 4039}, {5029, 6542}, {5040, 17763}, {5299, 33951}, {5317, 823}, {5532, 23104}, {6004, 33937}, {6085, 1266}, {6129, 322}, {6139, 6745}, {6161, 4986}, {6186, 6742}, {6363, 3663}, {6371, 4357}, {6372, 20888}, {6373, 726}, {6377, 3835}, {6544, 36791}, {6545, 23989}, {6551, 42372}, {6586, 17233}, {6588, 20928}, {6589, 4417}, {6591, 92}, {6615, 20895}, {6729, 556}, {7004, 35518}, {7023, 4626}, {7032, 3888}, {7077, 36801}, {7113, 4585}, {7117, 6332}, {7121, 932}, {7122, 4579}, {7178, 349}, {7180, 226}, {7192, 310}, {7199, 6385}, {7202, 18160}, {7216, 1446}, {7234, 1215}, {7250, 3668}, {7252, 333}, {7254, 17206}, {7335, 6517}, {7337, 36127}, {7366, 4617}, {7649, 264}, {8027, 244}, {8034, 3120}, {8042, 16727}, {8054, 20295}, {8574, 21092}, {8578, 150}, {8630, 869}, {8631, 17033}, {8632, 239}, {8633, 4362}, {8634, 30167}, {8635, 3920}, {8636, 976}, {8637, 386}, {8638, 2340}, {8640, 43}, {8641, 200}, {8642, 3870}, {8643, 145}, {8645, 3935}, {8646, 612}, {8648, 4511}, {8649, 6633}, {8650, 7292}, {8651, 4028}, {8653, 4061}, {8654, 3938}, {8655, 17018}, {8656, 3241}, {8657, 16834}, {8658, 41140}, {8659, 3008}, {8660, 1149}, {8661, 1647}, {8662, 2999}, {8663, 8013}, {8678, 4385}, {8735, 46110}, {8747, 6528}, {8750, 15742}, {9002, 4389}, {9008, 14620}, {9247, 906}, {9259, 6631}, {9262, 6630}, {9268, 6635}, {9297, 2228}, {9299, 9361}, {9313, 4429}, {9315, 30610}, {9359, 9296}, {9404, 42033}, {9426, 1918}, {9454, 2284}, {9456, 3257}, {9459, 23344}, {9494, 41267}, {9508, 20947}, {10566, 308}, {11125, 3260}, {14296, 1926}, {14349, 33935}, {14377, 31624}, {14399, 14206}, {14404, 3994}, {14407, 3943}, {14413, 30806}, {14419, 14210}, {14428, 4156}, {14575, 32656}, {14598, 34067}, {14621, 37133}, {14623, 9065}, {14838, 33939}, {14936, 3239}, {14991, 22011}, {15413, 40364}, {16502, 3732}, {16584, 7239}, {16612, 2064}, {16695, 33296}, {16702, 24039}, {16726, 7199}, {16732, 20948}, {16757, 20641}, {16892, 8024}, {16947, 4565}, {17186, 4611}, {17187, 4576}, {17209, 2396}, {17411, 34528}, {17424, 30827}, {17458, 30473}, {17477, 4106}, {17494, 18152}, {17924, 1969}, {17925, 44129}, {17954, 35147}, {17962, 35148}, {17990, 6541}, {18001, 11599}, {18105, 18082}, {18108, 3112}, {18155, 40072}, {18191, 18155}, {18196, 34022}, {18197, 31008}, {18200, 8033}, {18210, 14208}, {18268, 4584}, {18344, 318}, {20228, 21362}, {20229, 35341}, {20295, 40087}, {20517, 40365}, {20954, 40088}, {20970, 4115}, {20974, 25259}, {20975, 4064}, {20979, 192}, {20980, 3729}, {20981, 894}, {20982, 7265}, {20983, 32925}, {21003, 32922}, {21005, 32926}, {21007, 17277}, {21102, 311}, {21103, 1232}, {21104, 1233}, {21108, 1235}, {21109, 1236}, {21110, 44166}, {21122, 22}, {21123, 141}, {21124, 1228}, {21126, 42554}, {21127, 1229}, {21131, 338}, {21132, 34387}, {21134, 339}, {21143, 1086}, {21172, 14615}, {21178, 40073}, {21191, 6374}, {21194, 40035}, {21205, 40074}, {21207, 44173}, {21297, 40089}, {21348, 17786}, {21385, 18145}, {21747, 4781}, {21758, 3218}, {21761, 7283}, {21762, 20979}, {21763, 25264}, {21789, 1043}, {21814, 35309}, {21828, 3936}, {21832, 3948}, {21835, 21834}, {21837, 4006}, {22086, 3977}, {22096, 1459}, {22108, 17264}, {22260, 21043}, {22383, 63}, {22386, 22090}, {23189, 332}, {23220, 22350}, {23224, 326}, {23225, 1818}, {23345, 903}, {23349, 37129}, {23355, 3226}, {23472, 17350}, {23503, 21877}, {23506, 25287}, {23524, 4499}, {23572, 194}, {23649, 25272}, {23655, 32937}, {23751, 3782}, {23752, 1234}, {23807, 18837}, {23892, 3227}, {23979, 4619}, {24002, 20567}, {24533, 41318}, {25142, 8026}, {27644, 36860}, {27846, 3766}, {27929, 18035}, {28607, 4604}, {28615, 37212}, {29226, 20943}, {31947, 20932}, {32660, 44717}, {32665, 5376}, {32666, 5377}, {32674, 46102}, {32676, 5379}, {32702, 39294}, {32719, 9268}, {32735, 39293}, {32739, 1252}, {33917, 19945}, {34067, 5378}, {34069, 5384}, {34071, 5383}, {34073, 5385}, {34075, 5381}, {34080, 5382}, {34591, 15416}, {34819, 37211}, {34858, 36037}, {34916, 35181}, {34948, 44179}, {35519, 40363}, {36054, 3719}, {37128, 4639}, {37129, 889}, {38238, 18149}, {38266, 27834}, {38346, 17494}, {38367, 17475}, {38389, 17894}, {38986, 4083}, {39201, 3682}, {40148, 8050}, {40432, 7260}, {40495, 1928}, {40610, 23886}, {40627, 21024}, {40746, 4586}, {41405, 6634}, {41935, 4638}, {42067, 7649}, {42078, 15632}, {42336, 1122}, {42462, 23978}, {42649, 27529}, {42653, 21081}, {42661, 20653}, {42662, 16086}, {42753, 36038}, {43041, 18033}, {43049, 21609}, {43051, 30545}, {43060, 18134}, {43922, 6548}, {43923, 273}, {43924, 7}, {43925, 27}, {43928, 31002}, {43929, 673}, {43931, 6384}, {43932, 1088}, {45755, 28809}, {45902, 17760}, {46107, 18022}, {46288, 4628}
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3835, 30835}, {2, 20295, 3835}, {2, 26798, 27138}, {2, 26853, 20295}, {2, 27013, 31286}, {2, 31286, 31207}, {75, 20952, 20909}, {101, 41405, 1252}, {190, 9362, 7035}, {650, 652, 657}, {650, 654, 652}, {650, 661, 4893}, {650, 4394, 1635}, {650, 4979, 4813}, {650, 14300, 21127}, {661, 1635, 650}, {663, 667, 8643}, {663, 8656, 1960}, {665, 8632, 5029}, {665, 8658, 1960}, {665, 8659, 8632}, {667, 669, 8655}, {667, 1960, 8656}, {667, 3250, 5029}, {667, 4775, 1960}, {667, 8640, 669}, {667, 8641, 8642}, {667, 8646, 8635}, {669, 890, 8640}, {669, 2978, 663}, {693, 4369, 4379}, {798, 2484, 657}, {901, 1252, 41405}, {1019, 18197, 7192}, {1021, 4765, 45755}, {1635, 4790, 4813}, {1635, 4979, 661}, {1919, 21123, 1459}, {1960, 4775, 663}, {1960, 8656, 8643}, {1960, 8658, 8632}, {1960, 8659, 8658}, {1977, 6377, 8054}, {2488, 6139, 8641}, {2488, 8641, 663}, {2490, 14321, 1639}, {2590, 2591, 1635}, {3063, 20981, 23472}, {3250, 8632, 663}, {3733, 21007, 1919}, {3798, 4025, 4750}, {3835, 20295, 31147}, {3835, 27013, 31207}, {3835, 31286, 2}, {3835, 45313, 31286}, {3937, 14936, 20974}, {4025, 4786, 3798}, {4106, 4885, 4728}, {4369, 4380, 4382}, {4379, 4382, 693}, {4394, 4790, 661}, {4394, 4979, 4893}, {4468, 11068, 6546}, {4498, 45755, 21832}, {4607, 7035, 9362}, {4728, 24924, 4885}, {4750, 16892, 4025}, {4763, 25666, 31209}, {4776, 31209, 25666}, {4782, 4784, 4724}, {4813, 4893, 661}, {4817, 24623, 4379}, {5638, 5639, 5029}, {6589, 43060, 21828}, {7252, 43060, 1459}, {8641, 8642, 8645}, {14351, 45745, 14435}, {20295, 27013, 2}, {20295, 27138, 26798}, {20295, 27673, 28398}, {20295, 31286, 30835}, {20295, 45313, 31207}, {22044, 24089, 4024}, {25924, 25955, 2}, {26798, 27138, 3835}, {26853, 27013, 3835}, {26853, 31286, 31147}, {26853, 45313, 30835}, {27293, 30023, 30835}, {27486, 45746, 21196}, {27673, 29487, 30835}, {27673, 29807, 31147}, {29487, 29545, 20295}, {29545, 31286, 28398}, {30835, 31147, 3835}, {30835, 31207, 2}, {31147, 31207, 30835}, {31286, 45313, 27013}, {39665, 39666, 42662}


X(650) = CROSSDIFFERENCE OF X(1) AND X(3)

Trilinears    cos B - cos C : cos C - cos A : cos A - cos B
Trilinears    (b - c)(b + c - a) : (c - a)(c + a - b) : (a - b)(a + b - c)
Trilinears    cos(A/2) sin(B/2 - C/2) : :
Trilinears    sin^2(B/2) - sin^2(C/2) : :
Trilinears    cos^2(B/2) - cos^2(C/2) : :
Barycentrics    sin A (cos B - cos C) : sin B (cos C - cos A) : sin C (cos A - cos B)
Barycentrics    a(b - c)(b + c - a) : b(c - a)(c + a - b) : c(a - b)(a + b - c)
X(650) = 3 X[2] + X[17494], 3 X[2] + 5 X[26777], 9 X[2] - X[26824], 9 X[2] - 5 X[26985], 3 X[2] - 7 X[27115], 3 X[2] - 5 X[31209], 6 X[2] - 5 X[31250], 3 X[2] - 4 X[31287], X[649] - 3 X[1635], X[649] - 4 X[2516], 3 X[649] + X[4813], X[649] + 3 X[4893], 3 X[649] - X[4979], 2 X[659] + X[2526], X[661] + 3 X[1635], X[661] + 4 X[2516], X[661] + 2 X[4394], 2 X[661] + X[4790], 3 X[661] - X[4813], X[661] - 3 X[4893], 3 X[661] + X[4979], 3 X[663] + X[4814], 3 X[663] - X[4895], 5 X[663] - X[4959], X[693] + 5 X[26777], 3 X[693] - X[26824], 3 X[693] - 5 X[26985], X[693] - 7 X[27115], X[693] + 3 X[31150], X[693] - 5 X[31209], 2 X[693] - 5 X[31250], X[693] - 4 X[31287], X[693] - 6 X[44567], 2 X[693] - 3 X[45320], 3 X[905] - 2 X[3960], 3 X[1635] - 4 X[2516], 3 X[1635] - 2 X[4394], 6 X[1635] - X[4790], 9 X[1635] + X[4813], 9 X[1635] - X[4979], 3 X[1638] - 2 X[3676], 3 X[1638] - 4 X[7658], 3 X[1638] - X[21104], 3 X[1639] - 2 X[3239], 3 X[1639] - X[3700], 3 X[1639] - 4 X[4521], 3 X[1639] + 2 X[4765], 6 X[1639] - X[4820], 3 X[1639] + X[4976], 4 X[2490] - X[6590], 2 X[2490] - 3 X[14425], 4 X[2490] + X[45745], 8 X[2516] - X[4790], 12 X[2516] + X[4813], 4 X[2516] + 3 X[4893], 12 X[2516] - X[4979], X[3004] + 2 X[11068], 4 X[3239] - X[4820], 4 X[3239] - 3 X[4944], 2 X[3239] + X[4976], 3 X[3669] - 4 X[3960], X[3669] - 4 X[14838], X[3700] - 4 X[4521], X[3700] + 2 X[4765], 2 X[3700] - 3 X[4944], 3 X[3709] - X[4526], X[3762] + 3 X[45671], X[3960] - 3 X[14838], X[4024] - 9 X[6544], 2 X[4041] + X[4162], 3 X[4041] - X[4814], 3 X[4041] + X[4895], 5 X[4041] + X[4959], X[4105] - 3 X[14392], X[4106] - 4 X[25666], 3 X[4162] + 2 X[4814], 3 X[4162] - 2 X[4895], 5 X[4162] - 2 X[4959], 2 X[4163] - 3 X[44729], X[4369] - 3 X[4763], X[4378] - 3 X[14419], 3 X[4379] - 5 X[24924], 3 X[4379] - 7 X[31207], X[4380] + 3 X[4776], X[4380] + 2 X[4940], X[4382] - 3 X[4728], X[4382] - 5 X[30835], 4 X[4394] - X[4790], 6 X[4394] + X[4813], 2 X[4394] + 3 X[4893], 6 X[4394] - X[4979], X[4467] - 3 X[27486], X[4467] + 3 X[30565], X[4468] + 2 X[17069], X[4474] - 3 X[14430], 2 X[4521] + X[4765], 8 X[4521] - X[4820], 8 X[4521] - 3 X[4944], 4 X[4521] + X[4976], X[4560] + 2 X[20317], X[4705] + 2 X[6050], 3 X[4728] - 2 X[23813], 3 X[4728] - 5 X[30835], 7 X[4751] - 3 X[4828], 3 X[4763] - 2 X[31286], 6 X[4763] - X[43067], 4 X[4765] + X[4820], 4 X[4765] + 3 X[4944], 3 X[4776] - 2 X[4940], 3 X[4776] - X[20295], 3 X[4790] + 2 X[4813], X[4790] + 6 X[4893], 3 X[4790] - 2 X[4979], 2 X[4791] - 3 X[45664], X[4813] - 9 X[4893], 5 X[4814] + 3 X[4959], X[4820] - 3 X[4944], X[4820] + 2 X[4976], X[4841] + 8 X[31182], X[4841] + 4 X[43061], 2 X[4885] + X[17494], 2 X[4885] + 5 X[26777], 6 X[4885] - X[26824], 6 X[4885] - 5 X[26985], 2 X[4885] - 7 X[27115], 2 X[4885] + 3 X[31150], 2 X[4885] - 5 X[31209], 4 X[4885] - 5 X[31250], X[4885] - 3 X[44567], 4 X[4885] - 3 X[45320], 9 X[4893] + X[4979], 5 X[4895] - 3 X[4959], X[4932] - 3 X[45313], 3 X[4944] + 2 X[4976], 2 X[4949] + 3 X[4984], 3 X[6546] + X[16892], X[6590] - 6 X[14425], X[6608] + 3 X[14392], X[7192] - 5 X[27013], 4 X[7653] - 3 X[31148], 4 X[7658] - X[21104], X[7659] - 4 X[9508], 3 X[10196] + X[21196], 3 X[11193] - 2 X[11934], 3 X[11193] - 4 X[17115], X[13401] + 2 X[14298], X[13401] + 4 X[40137], X[14300] + 2 X[40137], 3 X[14395] - X[36054], 3 X[14404] - X[20983], 6 X[14425] + X[45745], X[17494] - 5 X[26777], 3 X[17494] + X[26824], 3 X[17494] + 5 X[26985], X[17494] + 7 X[27115], X[17494] - 3 X[31150], X[17494] + 5 X[31209], 2 X[17494] + 5 X[31250], X[17494] + 4 X[31287], X[17494] + 6 X[44567], 2 X[17494] + 3 X[45320], 2 X[21188] - 3 X[41800], X[21222] - 3 X[44550], 3 X[21297] - 7 X[27138], 2 X[23813] - 5 X[30835], 5 X[24924] - 7 X[31207], X[25259] + 3 X[27486], X[25259] - 3 X[30565], 15 X[26777] + X[26824], 3 X[26777] + X[26985], 5 X[26777] + 7 X[27115], 5 X[26777] - 3 X[31150], 2 X[26777] + X[31250], 5 X[26777] + 4 X[31287], 5 X[26777] + 6 X[44567], 10 X[26777] + 3 X[45320], X[26824] - 5 X[26985], X[26824] - 21 X[27115], X[26824] + 9 X[31150], X[26824] - 15 X[31209], 2 X[26824] - 15 X[31250], X[26824] - 12 X[31287], X[26824] - 18 X[44567], 2 X[26824] - 9 X[45320], 5 X[26985] - 21 X[27115], 5 X[26985] + 9 X[31150], X[26985] - 3 X[31209], 2 X[26985] - 3 X[31250], 5 X[26985] - 12 X[31287], 5 X[26985] - 18 X[44567], 10 X[26985] - 9 X[45320], 7 X[27115] + 3 X[31150], 7 X[27115] - 5 X[31209], 14 X[27115] - 5 X[31250], 7 X[27115] - 4 X[31287], 7 X[27115] - 6 X[44567], 14 X[27115] - 3 X[45320], 3 X[31150] + 5 X[31209], 6 X[31150] + 5 X[31250], 3 X[31150] + 4 X[31287], X[31150] + 2 X[44567], 2 X[31150] + X[45320], 5 X[31209] - 4 X[31287], 5 X[31209] - 6 X[44567], 10 X[31209] - 3 X[45320], 5 X[31250] - 8 X[31287], 5 X[31250] - 12 X[44567], 5 X[31250] - 3 X[45320], 4 X[31286] - X[43067], 2 X[31287] - 3 X[44567], 8 X[31287] - 3 X[45320], 3 X[38238] - 2 X[43931], 4 X[44567] - X[45320], X[45669] + 2 X[45670]

X(650) is the perspector of triangle ABC and the tangential triangle of the Feuerbach hyperbola. (Randy Hutson, 9/23/2011)

Let T1, T2, T3 denote the intouch, extouch,and incentral triangles. Let T4 be the side triangle of T2 and T3; T5 that of T3 and T1, and T6 that of T1 and T2. Then X(650) is the perspector of triangle ABC and T4, of ABC and T5, and of ABC and T6. (Randy Hutson, 9/23/2011)

X(650) is the point of intersection of orthic axis, antiorthic axis and Gergonne line, these being the trilinear polars of X(4), X(1) and X(7), respectively. More generally, X(650) is the intersection of the trilinear polars of every pair of points on the Feuerbach hyperbola. (Randy Hutson, December 26, 2015)

Note from Keita Miyamoto, August 4, 2024: X(650) is the center of the Stevanović circle. This circle is orthogonal to the following circles: circumcircle [1], nine-point circle [1], excircles radical circle [1], tangential circle [2], Bevan circle [1, referred to as excentral circle], orthocentroidal circle [2], Apollonius circle [1], radical circle of the mixtilinear excircles [3], Apollonius circle of the mixtilinear excircles [3], orthoptic circle of the Steiner inellipse [2], polar circle (when the triangle is obtuse) [2], inner Apollonius circle of the three circles whose diameters are the sidelengths of the triangle [3, referred to as inner Miyamoto-Moses-Apollonius circle], outer Apollonius circle of the three circles whose diameters are the sidelengths of the triangle [3, referred to as outer Miyamoto-Moses-Apollonius circle].

The circle [3] has radius (1/2)*sqrt((a*b*c*(a^5-a^4*b-a*b^4+b^5-a^4*c+a^3*b*c+a*b^3*c-b^4*c+a*b*c^3-a*c^4-b*c^4+c^5))/((a-b)^2*(a-c)^2*(b-c)^2)). (described in [2])

References:

[1] Milorad R. Stevanović. The Apollonius Circle and Related Triangle Centers. Forum Geometricorum, 3, pp.187-195, 2003.
[2] Weisstein, Eric W. "Stevanović Circle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/StevanovicCircle.html
[3] Teruyama Momiji, Stevanović Circle of a Triangle, 2024. https://youtube.com/watch?v=xKs9ZDhl2E4&si=2gQZ6RDQsa9UuEz-

X(650) lies on the GEOS circle, the cubic K925, and these lines: {1, 1643}, {2, 693}, {3, 8760}, {6, 2423}, {9, 15737}, {10, 21721}, {11, 1566}, {19, 2432}, {20, 8142}, {35, 11247}, {37, 4024}, {39, 7626}, {42, 21727}, {44, 513}, {45, 24457}, {55, 884}, {56, 9373}, {65, 1938}, {75, 21438}, {88, 37131}, {100, 919}, {101, 2222}, {105, 43079}, {108, 40116}, {110, 9090}, {111, 2752}, {112, 2766}, {115, 5520}, {123, 38972}, {141, 9015}, {226, 23806}, {230, 231}, {241, 514}, {244, 38375}, {354, 9443}, {512, 2499}, {518, 5098}, {521, 1021}, {522, 1639}, {644, 5548}, {651, 7045}, {663, 861}, {666, 4998}, {667, 4705}, {812, 3835}, {814, 21051}, {824, 3666}, {830, 3803}, {876, 23768}, {885, 5218}, {900, 14321}, {918, 4025}, {926, 2488}, {949, 23696}, {1015, 4534}, {1022, 39963}, {1027, 3126}, {1040, 41795}, {1054, 39344}, {1084, 35079}, {1086, 24198}, {1146, 7117}, {1212, 21132}, {1376, 5452}, {1415, 1783}, {1418, 23730}, {1427, 23727}, {1459, 24532}, {1577, 23882}, {1734, 3309}, {1758, 21189}, {1814, 25954}, {1919, 38469}, {1946, 2202}, {1960, 4770}, {1980, 5040}, {2170, 33646}, {2174, 21860}, {2276, 4448}, {2291, 2717}, {2457, 2529}, {2517, 26080}, {2520, 6139}, {2523, 2527}, {2532, 6371}, {2605, 17412}, {2646, 2649}, {2820, 15599}, {3035, 15914}, {3057, 9366}, {3119, 7004}, {3122, 45743}, {3210, 25271}, {3246, 39308}, {3250, 4083}, {3251, 4825}, {3261, 24622}, {3569, 32126}, {3570, 42717}, {3572, 16606}, {3667, 4773}, {3699, 36801}, {3730, 7634}, {3739, 4411}, {3746, 32195}, {3752, 6546}, {3762, 45671}, {3798, 4897}, {3837, 25126}, {3887, 4794}, {3904, 28938}, {3907, 4147}, {3910, 6332}, {3975, 4391}, {4063, 14349}, {4120, 4926}, {4129, 29013}, {4132, 42664}, {4140, 4397}, {4142, 23877}, {4160, 30234}, {4163, 44729}, {4171, 42312}, {4367, 4490}, {4370, 35129}, {4378, 14419}, {4379, 24924}, {4380, 4776}, {4382, 4728}, {4449, 17427}, {4455, 8640}, {4462, 17496}, {4467, 25259}, {4474, 14430}, {4481, 18197}, {4498, 8712}, {4500, 44307}, {4546, 8710}, {4551, 35326}, {4554, 30610}, {4562, 9362}, {4730, 4775}, {4751, 4828}, {4785, 45315}, {4791, 45664}, {4802, 4988}, {4806, 29328}, {4834, 4983}, {4838, 28165}, {4843, 4990}, {4850, 31992}, {4927, 44432}, {4931, 28205}, {4932, 28840}, {4949, 4984}, {4962, 14350}, {4995, 21795}, {5190, 13999}, {5432, 15584}, {5514, 10017}, {5540, 34460}, {6004, 8659}, {6164, 21893}, {6174, 6184}, {6367, 42653}, {6506, 8735}, {6545, 16602}, {6615, 14418}, {6666, 23810}, {7192, 16751}, {7212, 26146}, {7642, 10567}, {7650, 21960}, {7653, 31148}, {8056, 37626}, {8632, 21901}, {8642, 23687}, {9010, 14404}, {9013, 15990}, {9320, 20683}, {9355, 9357}, {9364, 20980}, {9590, 39227}, {10164, 40606}, {11502, 30706}, {11672, 35083}, {13259, 45695}, {13405, 14746}, {13609, 38357}, {14296, 20906}, {14475, 31197}, {14735, 17355}, {14749, 40942}, {14812, 16670}, {14935, 40062}, {15313, 17796}, {16592, 41180}, {16601, 21201}, {16706, 25603}, {16757, 26248}, {16975, 30583}, {17072, 29051}, {17161, 28606}, {17780, 42723}, {17924, 37799}, {17926, 44426}, {18004, 29078}, {18108, 26249}, {20316, 24718}, {20954, 27293}, {21052, 25637}, {21053, 29274}, {21105, 40133}, {21183, 44902}, {21192, 23875}, {21222, 44550}, {21260, 29070}, {21297, 27138}, {21758, 39521}, {22091, 44408}, {23100, 24774}, {23758, 28151}, {23972, 35116}, {24002, 37757}, {24098, 25067}, {24115, 25091}, {24141, 25099}, {24430, 41796}, {24484, 34583}, {24720, 25380}, {24789, 24793}, {25900, 25955}, {27009, 40619}, {27345, 28758}, {28161, 40500}, {28292, 38324}, {28798, 36796}, {28851, 45674}, {28867, 45679}, {29226, 40464}, {30023, 42327}, {30764, 30865}, {31615, 31628}, {32688, 40097}, {33573, 35015}, {34591, 38345}, {35091, 35508}, {36067, 40117}, {37998, 42322}, {40131, 42758}, {45669, 45670}

X(650) = midpoint of X(i) and X(j) for these {i,j}: {2, 31150}, {649, 661}, {657, 21127}, {659, 1491}, {663, 4041}, {667, 4705}, {693, 17494}, {1635, 4893}, {1734, 4040}, {1960, 4770}, {2254, 4724}, {2488, 4524}, {3239, 4765}, {3250, 21832}, {3251, 4825}, {3700, 4976}, {3716, 4913}, {4025, 4468}, {4063, 14349}, {4105, 6608}, {4367, 4490}, {4380, 20295}, {4391, 4560}, {4462, 17496}, {4467, 25259}, {4730, 4775}, {4813, 4979}, {4814, 4895}, {4834, 4983}, {6590, 45745}, {14298, 14300}, {17418, 17420}, {20906, 21225}, {26777, 31209}, {27486, 30565}
X(650) = reflection of X(i) in X(j) for these {i,j}: {2, 44567}, {11, 10006}, {20, 8142}, {649, 4394}, {667, 6050}, {693, 4885}, {905, 14838}, {2526, 1491}, {3239, 4521}, {3669, 905}, {3676, 7658}, {3700, 3239}, {3776, 21212}, {3803, 4401}, {3835, 25666}, {4025, 17069}, {4106, 3835}, {4162, 663}, {4369, 31286}, {4382, 23813}, {4391, 20317}, {4394, 2516}, {4411, 3739}, {4790, 649}, {4820, 3700}, {4885, 31287}, {4897, 3798}, {4927, 44432}, {4944, 1639}, {4976, 4765}, {7178, 14837}, {7655, 656}, {7662, 4874}, {11934, 17115}, {13401, 14300}, {14298, 40137}, {20295, 4940}, {21104, 3676}, {21183, 44902}, {21348, 6586}, {24720, 25380}, {31250, 31209}, {43051, 24782}, {43052, 10015}, {43061, 31182}, {43067, 4369}, {45320, 2}
X(650) = isogonal conjugate of X(651)
X(650) = isotomic conjugate of X(4554)
X(650) = complement of X(693)
X(650) = complementary conjugate of X(21252)
X(650) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {692, 41792}, {9309, 150}, {9311, 21293}, {9315, 149}, {9439, 37781}, {30610, 6327}
X(650) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 21252}, {6, 116}, {31, 11}, {32, 1086}, {37, 21253}, {41, 26932}, {42, 125}, {55, 124}, {56, 17059}, {59, 17072}, {71, 127}, {100, 2887}, {101, 141}, {109, 2886}, {110, 3741}, {112, 34830}, {163, 3739}, {184, 2968}, {190, 626}, {212, 123}, {213, 8287}, {228, 34846}, {251, 44312}, {560, 1015}, {604, 4904}, {644, 21244}, {651, 17046}, {662, 21240}, {663, 46100}, {664, 17047}, {668, 21235}, {692, 10}, {765, 21260}, {813, 20541}, {825, 21264}, {901, 21241}, {902, 3259}, {906, 18589}, {919, 20335}, {1016, 21262}, {1018, 21245}, {1110, 513}, {1252, 3835}, {1253, 5514}, {1331, 1368}, {1333, 17761}, {1397, 3756}, {1402, 8286}, {1415, 142}, {1461, 21258}, {1501, 6377}, {1576, 1125}, {1783, 20305}, {1813, 18639}, {1897, 21243}, {1918, 115}, {1919, 6547}, {1978, 40379}, {2149, 4885}, {2175, 1146}, {2177, 15614}, {2187, 7358}, {2194, 34589}, {2200, 15526}, {2205, 16592}, {2206, 244}, {2209, 5518}, {2210, 38989}, {2212, 6506}, {2284, 20540}, {2426, 118}, {3052, 5510}, {3573, 20542}, {3939, 1329}, {4055, 122}, {4557, 3454}, {4559, 17052}, {4565, 17050}, {4567, 42327}, {4570, 512}, {4588, 21242}, {4600, 23301}, {4601, 21263}, {4628, 3934}, {4630, 29654}, {5379, 21259}, {5380, 21256}, {5384, 788}, {5546, 21246}, {6066, 4521}, {7109, 6627}, {7122, 40608}, {8685, 17792}, {8750, 5}, {9454, 35094}, {9459, 35092}, {14598, 39786}, {14599, 35119}, {14827, 13609}, {17943, 20339}, {21059, 5511}, {23344, 121}, {23357, 21196}, {23979, 7658}, {23990, 514}, {23995, 31947}, {24027, 3900}, {32642, 516}, {32652, 946}, {32653, 24220}, {32656, 3}, {32660, 17073}, {32665, 3834}, {32666, 518}, {32674, 16608}, {32676, 942}, {32682, 674}, {32699, 916}, {32718, 4871}, {32719, 519}, {32736, 3831}, {32739, 2}, {34067, 3836}, {34069, 24325}, {34071, 20255}, {34073, 34824}, {34080, 21255}, {36049, 21239}, {36059, 34822}, {36086, 20544}, {41267, 15449}, {41280, 16614}, {46148, 21248}, {46288, 21208}
X(650) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 2310}, {2, 11}, {4, 3270}, {6, 11998}, {7, 3022}, {8, 3271}, {9, 2170}, {11, 11193}, {19, 38345}, {21, 4516}, {33, 38358}, {55, 38347}, {57, 7004}, {99, 11997}, {100, 55}, {101, 37}, {107, 1859}, {108, 33}, {174, 10501}, {189, 1364}, {190, 3057}, {200, 38375}, {264, 11988}, {274, 4459}, {277, 1086}, {278, 38357}, {280, 4081}, {281, 1146}, {282, 34591}, {294, 17435}, {330, 24840}, {333, 18191}, {346, 4534}, {513, 4162}, {514, 513}, {522, 3900}, {598, 11936}, {643, 3683}, {644, 9}, {645, 960}, {646, 8}, {648, 1858}, {651, 1}, {653, 65}, {655, 517}, {658, 354}, {662, 2646}, {664, 14100}, {666, 518}, {668, 3056}, {670, 42397}, {693, 11934}, {799, 21334}, {885, 926}, {905, 6129}, {929, 44670}, {979, 3248}, {1018, 2269}, {1019, 42312}, {1020, 2654}, {1021, 652}, {1024, 4435}, {1029, 3024}, {1043, 45743}, {1119, 11918}, {1222, 40528}, {1247, 2643}, {1290, 9629}, {1292, 4319}, {1783, 6}, {1897, 1864}, {2006, 35015}, {2401, 900}, {3239, 14298}, {3257, 5048}, {3362, 2638}, {3699, 210}, {3737, 663}, {3882, 8240}, {3911, 33646}, {3939, 1212}, {4069, 3691}, {4373, 36639}, {4391, 521}, {4551, 14547}, {4552, 950}, {4554, 497}, {4559, 14749}, {4560, 522}, {4573, 10391}, {4581, 512}, {4582, 3880}, {4598, 20359}, {4603, 17611}, {4606, 1697}, {4612, 21}, {4626, 10939}, {4627, 6051}, {5380, 8540}, {5546, 40937}, {6135, 7133}, {6136, 42013}, {6335, 1837}, {7040, 42069}, {7110, 21044}, {7658, 17427}, {8046, 3025}, {8051, 1357}, {8056, 244}, {8817, 14935}, {10492, 6728}, {13138, 30223}, {13149, 1836}, {13486, 7073}, {14838, 31947}, {15352, 42385}, {17924, 15313}, {17926, 3064}, {18830, 23497}, {19605, 3119}, {23838, 4895}, {24002, 3309}, {26704, 1824}, {26705, 1827}, {26706, 7071}, {27527, 25128}, {27818, 4014}, {27834, 2098}, {30610, 2}, {30730, 3686}, {31343, 200}, {31628, 100}, {32038, 3486}, {32041, 390}, {32641, 8609}, {32714, 1854}, {34085, 20358}, {34277, 2968}, {34529, 35604}, {35326, 14746}, {36049, 1108}, {36050, 31}, {36086, 41339}, {36098, 3745}, {36100, 35014}, {36796, 4124}, {36838, 7}, {36910, 4530}, {37136, 1319}, {37137, 3666}, {37139, 1155}, {37141, 56}, {37143, 18839}, {37206, 17642}, {37222, 14115}, {37887, 3120}, {38340, 942}, {39272, 28071}, {39956, 1015}, {40097, 25}, {40116, 5089}, {40117, 19}, {40400, 2087}, {41207, 243}, {41514, 3318}, {41790, 3942}, {41791, 26932}, {42343, 5274}, {42380, 42378}, {42381, 42379}, {42388, 42386}, {42389, 42387}, {42408, 14923}, {43069, 37548}, {43760, 3675}, {44178, 17463}, {44426, 18344}, {45877, 6729}
X(650) = X(i)-cross conjugate of X(j) for these (i,j): {55, 42552}, {647, 652}, {657, 3900}, {661, 3064}, {663, 513}, {926, 885}, {1566, 5089}, {1946, 521}, {2170, 9}, {2310, 1}, {3022, 7}, {3063, 18344}, {3119, 33}, {3270, 4}, {3271, 8}, {3310, 2432}, {3709, 663}, {4041, 522}, {4516, 21}, {4895, 23838}, {7117, 6}, {10581, 657}, {11124, 100}, {14936, 55}, {17435, 294}, {21044, 37}, {21127, 514}, {33525, 21789}, {36197, 7073}, {42454, 38347}
X(650) = cevapoint of X(i) and X(j) for these (i,j): {1, 9355}, {9, 4919}, {11, 42454}, {514, 21195}, {522, 4147}, {647, 661}, {657, 663}, {1946, 3063}, {3126, 42341}, {3709, 4041}, {10581, 21127}, {21044, 42462}
X(650) = crosspoint of X(i) and X(j) for these (i,j): {1, 651}, {2, 100}, {7, 36838}, {8, 646}, {9, 644}, {21, 4612}, {29, 653}, {57, 108}, {81, 13486}, {101, 284}, {107, 40395}, {112, 1169}, {190, 1222}, {274, 4594}, {280, 37141}, {281, 1783}, {282, 40117}, {333, 3699}, {514, 522}, {655, 40437}, {658, 21453}, {662, 40430}, {664, 23618}, {1018, 14624}, {1021, 17926}, {1897, 40444}, {3737, 4560}, {3939, 10482}, {4391, 44426}, {4554, 8817}, {6135, 7133}, {6136, 42013}, {6335, 34406}, {8056, 31343}, {9503, 36086}, {10492, 10495}, {13149, 34398}, {19607, 26704}, {34277, 40097}, {39272, 43760}
X(650) = crosssum of X(i) and X(j) for these (i,j): {1, 650}, {2, 17496}, {3, 36054}, {6, 513}, {9, 521}, {57, 3669}, {73, 652}, {101, 109}, {213, 7234}, {218, 43049}, {221, 14298}, {222, 905}, {226, 514}, {408, 32320}, {500, 9404}, {520, 18591}, {522, 20262}, {525, 1211}, {649, 1201}, {654, 34586}, {656, 18675}, {657, 2293}, {661, 2650}, {663, 1200}, {665, 1362}, {1019, 40153}, {1400, 43924}, {1415, 36059}, {2254, 9502}, {2850, 40584}, {3063, 7083}, {3287, 28369}, {3676, 10481}, {4551, 4559}, {6180, 20980}, {6364, 13389}, {6365, 13388}, {8677, 23980}, {15730, 43050}, {30719, 45204}, {39063, 39470}
X(650) = trilinear pole of line {926, 2170}
X(650) = crossdifference of every pair of points on line {1, 3}
X(650) = polar conjugate of X(18026)
X(650) = orthojoin of X(1521)
X(650) = trilinear pole of line X(926)X(2170) (the tangent to the incircle at X(3022))
X(650) = anticomplement of X(4885)
X(650) = perspector of the Feuerbach hyperbola
X(650) = center of circumconic that is locus of trilinear poles of lines passing through X(11)
X(650) = bicentric difference of PU(i) for i in (15, 57, 59, 60, 80, 94, 125)
X(650) = PU(15)-harmonic conjugate of X(65)
X(650) = PU(57)-harmonic conjugate of X(1155)
X(650) = PU(59)-harmonic conjugate of X(3057)
X(650) = PU(60)-harmonic conjugate of X(56)
X(650) = PU(80)-harmonic conjugate of X(2646)
X(650) = PU(94)-harmonic conjugate of X(354)
X(650) = PU(125)-harmonic conjugate of X(3)
X(650) = medial-isogonal conjugate of X(21252)
X(650) = X(6)-isoconjugate of X(664)
X(650) = crosspoint of X(1) and X(651)
X(650) = radical center of {circumcircle, nine-point circle, Bevan circle}
X(650) = radical center of {circumcircle, nine-point circle, Apollonius circle}
X(650) = bicentric difference of PU(112)
X(650) = PU(112)-harmonic conjugate of X(55)
X(650) = X(2501)-of-excentral-triangle
X(650) = PU(4)-harmonic conjugate of X(5089)
X(650) = trilinear square root of X(2310)
X(650) = trilinear product of Feuerbach hyperbola intercepts of antiorthic axis
X(650) = crosssum of Feuerbach hyperbola intercepts of antiorthic axis
X(650) = excentral-to-ABC barycentric image of X(649)
X(650) = X(i)-isoconjugate of X(j) for these (i,j): {1, 651}, {2, 109}, {3, 653}, {4, 1813}, {6, 664}, {7, 101}, {8, 1461}, {9, 934}, {10, 4565}, {11, 4619}, {12, 4556}, {19, 6516}, {21, 1020}, {27, 23067}, {31, 4554}, {32, 4572}, {34, 1332}, {35, 38340}, {36, 655}, {37, 1414}, {40, 37141}, {41, 4569}, {42, 4573}, {48, 18026}, {55, 658}, {56, 190}, {57, 100}, {58, 4552}, {59, 514}, {60, 4605}, {63, 108}, {65, 662}, {69, 32674}, {73, 648}, {75, 1415}, {77, 1783}, {78, 32714}, {81, 4551}, {83, 46153}, {85, 692}, {86, 4559}, {88, 23703}, {92, 36059}, {99, 1400}, {102, 2406}, {104, 24029}, {105, 1025}, {107, 40152}, {110, 226}, {112, 307}, {145, 38828}, {162, 1214}, {163, 1441}, {171, 37137}, {174, 6733}, {181, 4610}, {200, 4617}, {210, 4637}, {212, 13149}, {213, 4625}, {219, 36118}, {220, 4626}, {221, 44327}, {222, 1897}, {223, 13138}, {225, 4558}, {241, 36086}, {244, 31615}, {264, 32660}, {269, 644}, {273, 906}, {278, 1331}, {279, 3939}, {284, 4566}, {294, 41353}, {296, 1981}, {320, 32675}, {329, 8059}, {331, 32656}, {346, 6614}, {347, 36049}, {348, 8750}, {349, 1576}, {393, 6517}, {394, 36127}, {512, 4620}, {513, 4564}, {517, 37136}, {518, 36146}, {521, 7128}, {522, 1262}, {527, 14733}, {553, 8701}, {579, 1305}, {603, 6335}, {604, 668}, {608, 4561}, {643, 1427}, {645, 1042}, {646, 1106}, {649, 4998}, {650, 7045}, {660, 1429}, {663, 1275}, {665, 39293}, {666, 1458}, {672, 927}, {673, 2283}, {677, 43035}, {693, 2149}, {738, 4578}, {757, 21859}, {765, 3669}, {799, 1402}, {811, 1409}, {813, 1447}, {823, 22341}, {825, 7179}, {851, 41206}, {883, 1438}, {893, 6649}, {894, 29055}, {901, 3911}, {905, 7012}, {908, 2720}, {919, 9436}, {932, 1423}, {951, 14543}, {961, 3882}, {1014, 1018}, {1016, 43924}, {1026, 1462}, {1037, 3732}, {1038, 36099}, {1055, 35157}, {1110, 24002}, {1119, 4587}, {1121, 23346}, {1155, 37139}, {1156, 23890}, {1170, 35338}, {1174, 35312}, {1193, 6648}, {1231, 32676}, {1252, 3676}, {1253, 36838}, {1254, 4612}, {1268, 36075}, {1284, 4584}, {1292, 1445}, {1293, 5435}, {1308, 37787}, {1310, 2285}, {1317, 4638}, {1319, 3257}, {1334, 4616}, {1357, 6632}, {1397, 1978}, {1399, 15455}, {1403, 4598}, {1404, 4555}, {1405, 4597}, {1407, 3699}, {1408, 4033}, {1411, 4585}, {1412, 3952}, {1416, 42720}, {1417, 24004}, {1420, 27834}, {1428, 4562}, {1431, 18047}, {1432, 4579}, {1434, 4557}, {1435, 4571}, {1457, 13136}, {1459, 46102}, {1460, 37215}, {1465, 36037}, {1468, 32038}, {1469, 4586}, {1471, 32041}, {1475, 6606}, {1476, 21362}, {1492, 7146}, {1617, 37206}, {1633, 7131}, {1634, 18097}, {1708, 13397}, {1880, 4592}, {1983, 18815}, {2002, 44059}, {2003, 6742}, {2078, 37143}, {2082, 8269}, {2099, 4604}, {2221, 14594}, {2222, 3218}, {2223, 34085}, {2338, 23973}, {2347, 6613}, {2425, 34393}, {2701, 17950}, {2742, 30379}, {2743, 37789}, {3212, 34071}, {3219, 26700}, {3239, 7339}, {3262, 32669}, {3361, 4606}, {3451, 21272}, {3649, 4629}, {3662, 8685}, {3665, 4628}, {3666, 36098}, {3668, 5546}, {3671, 4627}, {3674, 32736}, {3888, 7132}, {3903, 7175}, {3912, 32735}, {3982, 28176}, {4017, 4567}, {4025, 7115}, {4031, 28210}, {4079, 7340}, {4103, 7341}, {4105, 23586}, {4114, 28214}, {4130, 24013}, {4163, 23971}, {4357, 8687}, {4391, 24027}, {4570, 7178}, {4575, 40149}, {4588, 5219}, {4591, 40663}, {4600, 7180}, {4621, 7248}, {4636, 6354}, {4654, 8652}, {5221, 37211}, {5226, 28162}, {5228, 37138}, {5249, 15439}, {5257, 5545}, {5364, 34083}, {5750, 29279}, {5905, 36082}, {6063, 32739}, {6183, 40131}, {6558, 7023}, {6577, 17077}, {6734, 32651}, {7013, 40117}, {7113, 35174}, {7649, 44717}, {8545, 14074}, {8677, 39294}, {8690, 28387}, {8693, 40719}, {8694, 21454}, {8697, 31231}, {9268, 30725}, {9316, 30610}, {9357, 9358}, {9454, 46135}, {10030, 34067}, {10436, 32693}, {10571, 44765}, {11608, 17942}, {12848, 28291}, {13486, 16577}, {14544, 40407}, {16947, 27808}, {17080, 36050}, {17484, 34921}, {17966, 35154}, {18830, 41526}, {19369, 35180}, {21105, 38809}, {21446, 35280}, {21453, 35326}, {22464, 32641}, {23113, 40446}, {23353, 40843}, {23845, 40420}, {23979, 35519}, {23981, 34234}, {24016, 40869}, {24032, 36054}, {24035, 36055}, {24471, 36147}, {27339, 28624}, {27382, 36079}, {29052, 41246}, {29363, 36538}, {30598, 36074}, {30806, 36141}, {32636, 37212}, {32643, 35516}, {32652, 40702}, {32666, 40704}, {32689, 33864}, {34055, 46152}, {34075, 43037}, {34080, 39126}, {36048, 40937}, {37607, 43069}, {40151, 43290}, {40576, 44178}
X(650) = barycentric product X(i)*X(j) for these {i,j}: {1, 522}, {3, 44426}, {4, 521}, {6, 4391}, {7, 3900}, {8, 513}, {9, 514}, {10, 3737}, {11, 100}, {19, 6332}, {21, 523}, {25, 35518}, {27, 8611}, {29, 656}, {31, 35519}, {33, 4025}, {37, 4560}, {41, 3261}, {42, 18155}, {48, 46110}, {55, 693}, {56, 4397}, {57, 3239}, {58, 4086}, {59, 42455}, {60, 4036}, {63, 3064}, {65, 7253}, {69, 18344}, {75, 663}, {76, 3063}, {78, 7649}, {79, 35057}, {80, 3738}, {81, 3700}, {84, 8058}, {85, 657}, {86, 4041}, {87, 4147}, {88, 1639}, {89, 4944}, {92, 652}, {99, 4516}, {101, 4858}, {102, 14304}, {104, 2804}, {106, 4768}, {108, 2968}, {109, 24026}, {115, 4612}, {123, 40097}, {124, 36050}, {149, 42552}, {174, 6730}, {178, 10495}, {188, 6728}, {189, 14298}, {190, 2170}, {200, 3676}, {210, 7192}, {212, 46107}, {219, 17924}, {220, 24002}, {226, 1021}, {241, 28132}, {244, 3699}, {256, 3907}, {257, 3287}, {261, 4705}, {264, 1946}, {269, 4163}, {270, 4064}, {274, 3709}, {279, 4130}, {280, 6129}, {281, 905}, {282, 14837}, {283, 24006}, {284, 1577}, {291, 3716}, {294, 918}, {312, 649}, {314, 512}, {318, 1459}, {321, 7252}, {333, 661}, {335, 4435}, {341, 43924}, {345, 6591}, {346, 3669}, {517, 43728}, {518, 885}, {519, 23838}, {520, 1896}, {525, 1172}, {527, 23893}, {556, 6729}, {607, 15413}, {608, 15416}, {643, 3120}, {644, 1086}, {645, 3125}, {646, 1015}, {647, 31623}, {651, 1146}, {653, 34591}, {654, 18359}, {658, 3119}, {659, 4518}, {660, 4124}, {662, 21044}, {664, 2310}, {665, 36796}, {666, 17435}, {667, 3596}, {668, 3271}, {669, 40072}, {679, 4543}, {692, 34387}, {751, 4474}, {764, 4076}, {765, 21132}, {798, 28660}, {810, 44130}, {812, 4876}, {824, 2344}, {850, 2194}, {875, 4087}, {876, 3685}, {884, 3263}, {891, 36798}, {897, 14432}, {900, 1320}, {903, 4895}, {926, 2481}, {934, 4081}, {941, 23880}, {952, 46041}, {960, 4581}, {983, 3810}, {985, 4522}, {1018, 17197}, {1019, 2321}, {1022, 2325}, {1024, 3912}, {1027, 3717}, {1036, 2517}, {1039, 23874}, {1043, 4017}, {1088, 4105}, {1109, 4636}, {1111, 3939}, {1126, 4985}, {1156, 6366}, {1214, 17926}, {1220, 17420}, {1222, 6615}, {1252, 40166}, {1255, 4976}, {1265, 43923}, {1293, 4939}, {1295, 14312}, {1309, 35014}, {1332, 8735}, {1334, 7199}, {1358, 4578}, {1392, 4926}, {1415, 23978}, {1432, 4529}, {1434, 4171}, {1441, 21789}, {1476, 42337}, {1635, 4997}, {1638, 41798}, {1783, 26932}, {1785, 37628}, {1807, 44428}, {1809, 39534}, {1812, 2501}, {1857, 4131}, {1861, 23696}, {1880, 15411}, {1897, 7004}, {1919, 28659}, {1980, 40363}, {2052, 36054}, {2053, 20906}, {2087, 4582}, {2090, 45877}, {2161, 3904}, {2175, 40495}, {2182, 2399}, {2184, 14331}, {2185, 4024}, {2191, 44448}, {2192, 17896}, {2193, 14618}, {2204, 3267}, {2218, 20294}, {2254, 14942}, {2287, 7178}, {2298, 3910}, {2299, 14208}, {2316, 3762}, {2319, 3835}, {2320, 4777}, {2328, 4077}, {2334, 4811}, {2335, 23882}, {2339, 6590}, {2341, 4707}, {2342, 36038}, {2346, 6362}, {2349, 14400}, {2364, 4791}, {2400, 41339}, {2509, 41791}, {2516, 38255}, {2520, 34409}, {2618, 35196}, {2648, 2785}, {2651, 18013}, {2771, 14224}, {2787, 11609}, {2798, 43746}, {2806, 43735}, {2826, 34894}, {2827, 12641}, {2969, 4571}, {2997, 8676}, {3022, 4569}, {3122, 7257}, {3124, 4631}, {3223, 25128}, {3227, 4526}, {3254, 3887}, {3257, 4530}, {3270, 18026}, {3307, 3308}, {3309, 6601}, {3310, 36795}, {3345, 14302}, {3572, 3975}, {3667, 3680}, {3675, 36802}, {3683, 4608}, {3684, 4444}, {3689, 6548}, {3694, 17925}, {3701, 3733}, {3703, 18108}, {3756, 31343}, {3766, 7077}, {3837, 8851}, {3868, 23289}, {3871, 40086}, {3876, 43927}, {3880, 23836}, {3903, 4459}, {3952, 18191}, {3960, 36910}, {4009, 43928}, {4051, 25576}, {4069, 17205}, {4082, 7203}, {4083, 7155}, {4102, 4979}, {4140, 40432}, {4162, 4373}, {4183, 17094}, {4367, 4451}, {4394, 6557}, {4458, 7281}, {4467, 7073}, {4477, 7249}, {4515, 17096}, {4521, 8056}, {4534, 27834}, {4542, 4618}, {4546, 19604}, {4553, 18101}, {4554, 14936}, {4564, 42462}, {4573, 36197}, {4594, 40608}, {4617, 23970}, {4626, 24010}, {4723, 23345}, {4762, 40779}, {4765, 25430}, {4778, 4866}, {4801, 34820}, {4813, 42030}, {4814, 39704}, {4820, 25417}, {4893, 30608}, {4900, 6006}, {4913, 30571}, {4919, 42555}, {4957, 5549}, {4959, 39707}, {4977, 32635}, {4978, 33635}, {5423, 43932}, {5514, 37141}, {5546, 16732}, {5853, 35355}, {6001, 43737}, {6003, 6598}, {6063, 8641}, {6335, 7117}, {6336, 14418}, {6364, 13454}, {6365, 13426}, {6516, 42069}, {6588, 34277}, {6605, 21104}, {6607, 42311}, {6608, 21453}, {6741, 13486}, {6745, 35348}, {7017, 22383}, {7045, 23615}, {7105, 8062}, {7106, 17899}, {7110, 14838}, {7220, 24720}, {7658, 19605}, {8047, 11193}, {8648, 20566}, {8674, 11604}, {8678, 30479}, {8702, 10266}, {8713, 42015}, {8748, 24018}, {8750, 17880}, {8817, 17115}, {9001, 30513}, {9404, 30690}, {9439, 20907}, {10309, 30201}, {10492, 16016}, {10566, 33299}, {10570, 21189}, {10581, 31618}, {11125, 44693}, {11934, 13577}, {13138, 38357}, {14077, 34919}, {14330, 21446}, {14395, 16080}, {14430, 37129}, {14589, 31611}, {15313, 43740}, {16596, 40117}, {16606, 27527}, {16726, 30730}, {17418, 31359}, {17424, 34523}, {18210, 36797}, {20317, 39956}, {20979, 27424}, {21011, 39177}, {21102, 44687}, {21120, 23617}, {21127, 32008}, {21172, 44692}, {21272, 40528}, {21302, 40505}, {21666, 36059}, {21832, 36800}, {21859, 26856}, {23090, 40149}, {23104, 24027}, {23189, 41013}, {23351, 30806}, {24031, 36127}, {26546, 40141}, {26704, 34588}, {27475, 45755}, {27538, 43931}, {27846, 36801}, {28071, 43042}, {31628, 46101}, {33573, 37139}, {34860, 42312}, {34896, 40577}, {35015, 36037}, {35508, 36838}, {36121, 39471}, {37206, 38375}, {38345, 44765}, {38358, 43190}, {40444, 40628}, {40716, 42657}
X(650) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 664}, {2, 4554}, {3, 6516}, {4, 18026}, {6, 651}, {7, 4569}, {8, 668}, {9, 190}, {11, 693}, {19, 653}, {21, 99}, {25, 108}, {29, 811}, {31, 109}, {32, 1415}, {33, 1897}, {34, 36118}, {37, 4552}, {41, 101}, {42, 4551}, {48, 1813}, {55, 100}, {56, 934}, {57, 658}, {58, 1414}, {65, 4566}, {75, 4572}, {78, 4561}, {80, 35174}, {81, 4573}, {86, 4625}, {100, 4998}, {101, 4564}, {105, 927}, {109, 7045}, {171, 6649}, {184, 36059}, {200, 3699}, {210, 3952}, {212, 1331}, {213, 4559}, {219, 1332}, {220, 644}, {228, 23067}, {244, 3676}, {255, 6517}, {261, 4623}, {269, 4626}, {278, 13149}, {279, 36838}, {281, 6335}, {282, 44327}, {283, 4592}, {284, 662}, {294, 666}, {312, 1978}, {314, 670}, {333, 799}, {346, 646}, {354, 35312}, {480, 4578}, {512, 65}, {513, 7}, {514, 85}, {518, 883}, {521, 69}, {522, 75}, {523, 1441}, {525, 1231}, {604, 1461}, {607, 1783}, {608, 32714}, {612, 14594}, {643, 4600}, {644, 1016}, {645, 4601}, {646, 31625}, {647, 1214}, {649, 57}, {651, 1275}, {652, 63}, {654, 3218}, {656, 307}, {657, 9}, {659, 1447}, {661, 226}, {662, 4620}, {663, 1}, {665, 241}, {667, 56}, {669, 1402}, {672, 1025}, {673, 34085}, {692, 59}, {693, 6063}, {728, 6558}, {764, 1358}, {788, 1469}, {798, 1400}, {810, 73}, {812, 10030}, {822, 40152}, {830, 7247}, {876, 7233}, {884, 105}, {885, 2481}, {891, 43037}, {893, 37137}, {902, 23703}, {904, 29055}, {905, 348}, {906, 44717}, {909, 37136}, {918, 40704}, {926, 518}, {941, 32038}, {1014, 4616}, {1015, 3669}, {1019, 1434}, {1021, 333}, {1024, 673}, {1036, 1310}, {1037, 8269}, {1043, 7257}, {1055, 23890}, {1086, 24002}, {1096, 36127}, {1106, 6614}, {1146, 4391}, {1156, 35157}, {1172, 648}, {1252, 31615}, {1253, 3939}, {1260, 4571}, {1261, 8706}, {1318, 4618}, {1320, 4555}, {1333, 4565}, {1334, 1018}, {1357, 43932}, {1364, 4131}, {1400, 1020}, {1407, 4617}, {1412, 4637}, {1415, 1262}, {1434, 4635}, {1436, 37141}, {1438, 36146}, {1456, 23973}, {1458, 41353}, {1459, 77}, {1476, 6613}, {1486, 40576}, {1491, 7179}, {1500, 21859}, {1577, 349}, {1635, 3911}, {1638, 37780}, {1639, 4358}, {1643, 5723}, {1734, 33298}, {1769, 22464}, {1783, 46102}, {1802, 4587}, {1812, 4563}, {1843, 46152}, {1896, 6528}, {1919, 604}, {1944, 15418}, {1946, 3}, {1960, 1319}, {1964, 46153}, {1973, 32674}, {1980, 1397}, {2053, 932}, {2082, 3732}, {2087, 30725}, {2149, 4619}, {2150, 4556}, {2160, 38340}, {2161, 655}, {2170, 514}, {2171, 4605}, {2175, 692}, {2182, 2406}, {2183, 24029}, {2185, 4610}, {2192, 13138}, {2193, 4558}, {2194, 110}, {2195, 36086}, {2202, 1981}, {2204, 112}, {2208, 8059}, {2212, 8750}, {2218, 1305}, {2223, 2283}, {2249, 41206}, {2254, 9436}, {2264, 14543}, {2269, 3882}, {2287, 645}, {2291, 37139}, {2293, 35338}, {2298, 6648}, {2299, 162}, {2310, 522}, {2311, 4584}, {2316, 3257}, {2319, 4598}, {2320, 4597}, {2321, 4033}, {2323, 4585}, {2325, 24004}, {2328, 643}, {2329, 18047}, {2330, 4579}, {2339, 37215}, {2340, 1026}, {2342, 36037}, {2344, 4586}, {2346, 6606}, {2347, 21362}, {2364, 4604}, {2423, 34051}, {2424, 43736}, {2432, 36100}, {2481, 46135}, {2484, 2285}, {2488, 354}, {2489, 1880}, {2501, 40149}, {2509, 28739}, {2520, 1836}, {2530, 3665}, {2605, 1442}, {2646, 17136}, {2648, 35154}, {2651, 17933}, {2804, 3262}, {2826, 38468}, {2968, 35518}, {2978, 10473}, {3022, 3900}, {3049, 1409}, {3056, 3888}, {3057, 21272}, {3061, 33946}, {3063, 6}, {3064, 92}, {3119, 3239}, {3120, 4077}, {3121, 7180}, {3122, 4017}, {3125, 7178}, {3158, 43290}, {3208, 4595}, {3239, 312}, {3248, 43924}, {3250, 7146}, {3251, 1317}, {3254, 35171}, {3261, 20567}, {3270, 521}, {3271, 513}, {3287, 894}, {3309, 6604}, {3310, 1465}, {3423, 6183}, {3452, 21580}, {3596, 6386}, {3667, 39126}, {3669, 279}, {3675, 43042}, {3676, 1088}, {3683, 4427}, {3684, 3570}, {3685, 874}, {3688, 4553}, {3689, 17780}, {3693, 42720}, {3699, 7035}, {3700, 321}, {3701, 27808}, {3709, 37}, {3711, 4767}, {3712, 42721}, {3715, 4756}, {3716, 350}, {3733, 1014}, {3737, 86}, {3738, 320}, {3766, 18033}, {3777, 7185}, {3790, 4505}, {3810, 33930}, {3835, 30545}, {3876, 33948}, {3900, 8}, {3904, 20924}, {3907, 1909}, {3910, 20911}, {3939, 765}, {3960, 17078}, {3975, 27853}, {4009, 41314}, {4017, 3668}, {4024, 6358}, {4025, 7182}, {4036, 34388}, {4041, 10}, {4079, 2171}, {4081, 4397}, {4083, 3212}, {4086, 313}, {4091, 7183}, {4092, 4036}, {4105, 200}, {4124, 3766}, {4130, 346}, {4131, 7055}, {4140, 3963}, {4147, 6376}, {4148, 3975}, {4155, 7235}, {4162, 145}, {4163, 341}, {4171, 2321}, {4183, 36797}, {4367, 7176}, {4369, 7196}, {4374, 7205}, {4378, 7223}, {4382, 7243}, {4390, 4482}, {4391, 76}, {4394, 5435}, {4397, 3596}, {4435, 239}, {4449, 9312}, {4455, 1284}, {4459, 4374}, {4468, 21609}, {4474, 3761}, {4477, 7081}, {4501, 4361}, {4502, 7201}, {4515, 30730}, {4516, 523}, {4517, 3799}, {4518, 4583}, {4521, 18743}, {4522, 33931}, {4524, 210}, {4526, 536}, {4528, 4723}, {4529, 17787}, {4530, 3762}, {4534, 4462}, {4543, 4738}, {4546, 44720}, {4560, 274}, {4578, 4076}, {4581, 31643}, {4612, 4590}, {4617, 23586}, {4626, 24011}, {4631, 34537}, {4636, 24041}, {4705, 12}, {4724, 40719}, {4729, 4848}, {4730, 40663}, {4765, 19804}, {4768, 3264}, {4775, 2099}, {4790, 21454}, {4813, 4654}, {4814, 3679}, {4820, 28605}, {4822, 3671}, {4827, 391}, {4834, 5221}, {4858, 3261}, {4876, 4562}, {4893, 5219}, {4895, 519}, {4919, 6631}, {4936, 30720}, {4944, 4671}, {4959, 3632}, {4976, 4359}, {4979, 553}, {4983, 3649}, {4985, 1269}, {4990, 3702}, {5040, 5061}, {5075, 1758}, {5276, 14612}, {5375, 31633}, {5532, 42455}, {5546, 4567}, {5547, 5380}, {5548, 5376}, {5549, 5385}, {6004, 30617}, {6050, 7288}, {6129, 347}, {6139, 1155}, {6182, 2550}, {6186, 26700}, {6187, 2222}, {6332, 304}, {6362, 20880}, {6363, 1122}, {6364, 13453}, {6365, 13436}, {6366, 30806}, {6371, 24471}, {6372, 4059}, {6373, 1463}, {6377, 43051}, {6589, 17080}, {6591, 278}, {6607, 3059}, {6608, 4847}, {6614, 24013}, {6615, 3663}, {6728, 4146}, {6729, 174}, {6730, 556}, {7004, 4025}, {7054, 4612}, {7058, 4631}, {7064, 40521}, {7073, 6742}, {7077, 660}, {7083, 1633}, {7110, 15455}, {7117, 905}, {7118, 36049}, {7154, 40117}, {7155, 18830}, {7178, 1446}, {7180, 1427}, {7252, 81}, {7253, 314}, {7359, 42716}, {7649, 273}, {7658, 31627}, {8012, 35341}, {8027, 1357}, {8058, 322}, {8611, 306}, {8632, 1429}, {8638, 2223}, {8640, 1403}, {8641, 55}, {8642, 1617}, {8643, 1420}, {8645, 2078}, {8646, 1460}, {8648, 36}, {8653, 37593}, {8655, 16878}, {8656, 13462}, {8676, 3868}, {8678, 388}, {8735, 17924}, {8748, 823}, {8750, 7012}, {8755, 24035}, {8851, 8709}, {9247, 32660}, {9355, 10001}, {9404, 3219}, {9447, 32739}, {10581, 1212}, {11124, 3035}, {11193, 149}, {11604, 35156}, {11609, 35147}, {11934, 3434}, {11998, 17496}, {14224, 46141}, {14298, 329}, {14300, 9776}, {14302, 33672}, {14304, 35516}, {14330, 30854}, {14331, 18750}, {14349, 33949}, {14392, 6745}, {14395, 11064}, {14399, 6357}, {14400, 14206}, {14411, 1642}, {14413, 1323}, {14418, 3977}, {14419, 7181}, {14421, 43038}, {14427, 2325}, {14430, 6381}, {14432, 14210}, {14837, 40702}, {14838, 17095}, {15997, 45876}, {16686, 40577}, {16726, 17096}, {16757, 17076}, {17115, 497}, {17197, 7199}, {17418, 10436}, {17420, 4357}, {17425, 34522}, {17427, 31527}, {17435, 918}, {17924, 331}, {17926, 31623}, {18000, 2652}, {18155, 310}, {18191, 7192}, {18210, 17094}, {18265, 34067}, {18344, 4}, {20229, 35326}, {20293, 44139}, {20317, 18135}, {20684, 7239}, {20979, 1423}, {20980, 6180}, {20981, 7175}, {21044, 1577}, {21120, 26563}, {21124, 45196}, {21127, 142}, {21132, 1111}, {21172, 33673}, {21195, 40593}, {21761, 1935}, {21789, 21}, {21795, 35310}, {21811, 22003}, {21828, 18593}, {21832, 16609}, {22108, 37787}, {22382, 7364}, {22383, 222}, {23090, 1812}, {23189, 1444}, {23224, 1804}, {23289, 2997}, {23351, 1156}, {23572, 1424}, {23615, 24026}, {23696, 31637}, {23838, 903}, {23864, 13588}, {23880, 34284}, {23893, 1121}, {24010, 4163}, {24012, 4105}, {24026, 35519}, {25128, 17149}, {25430, 4624}, {26932, 15413}, {27527, 31008}, {27538, 36863}, {27837, 27829}, {27846, 43041}, {28071, 36802}, {28132, 36796}, {28660, 4602}, {29226, 17090}, {29278, 4968}, {30584, 41318}, {31623, 6331}, {31628, 31619}, {31947, 41808}, {32635, 6540}, {32674, 7128}, {32739, 2149}, {33299, 4568}, {33525, 40937}, {33635, 37212}, {33950, 33951}, {34068, 14733}, {34387, 40495}, {34591, 6332}, {34820, 4606}, {34858, 2720}, {35015, 36038}, {35057, 319}, {35355, 35160}, {35508, 4130}, {35518, 305}, {35519, 561}, {36054, 394}, {36086, 39293}, {36127, 24032}, {36197, 3700}, {36796, 36803}, {36798, 889}, {36800, 4639}, {36910, 36804}, {37908, 4238}, {38266, 38828}, {38347, 17494}, {38357, 17896}, {38358, 25259}, {38365, 4040}, {38375, 4468}, {38991, 21390}, {39201, 22341}, {39687, 36054}, {39786, 7212}, {40072, 4609}, {40137, 18228}, {40166, 23989}, {40495, 41283}, {40608, 2533}, {40627, 45208}, {40779, 32041}, {40869, 42719}, {40972, 46148}, {41339, 2398}, {42067, 43923}, {42069, 44426}, {42084, 39771}, {42312, 3875}, {42317, 32040}, {42325, 32007}, {42337, 20895}, {42454, 40619}, {42455, 34387}, {42462, 4858}, {42552, 8047}, {42649, 3336}, {42657, 484}, {42670, 5172}, {42771, 42758}, {43035, 24015}, {43049, 17093}, {43050, 37757}, {43728, 18816}, {43921, 43930}, {43923, 1119}, {43924, 269}, {43925, 1396}, {43929, 1462}, {43932, 479}, {44426, 264}, {45755, 4384}, {45902, 28391}, {46041, 46136}, {46110, 1969}
X(650) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 693, 4885}, {2, 4885, 31250}, {2, 17494, 693}, {2, 24562, 25925}, {2, 26114, 25511}, {2, 26641, 24562}, {2, 26777, 17494}, {2, 26824, 26985}, {2, 26854, 27193}, {2, 27115, 31209}, {2, 27648, 27674}, {2, 28132, 28143}, {2, 31209, 31287}, {11, 14936, 38347}, {75, 21611, 21438}, {100, 1252, 14589}, {100, 5375, 1252}, {647, 3310, 6589}, {647, 6590, 21348}, {647, 6591, 6129}, {649, 652, 654}, {649, 657, 652}, {649, 1635, 4394}, {649, 4813, 4979}, {649, 4893, 661}, {649, 21127, 14300}, {651, 9358, 7045}, {652, 657, 9404}, {652, 14300, 4790}, {652, 21127, 13401}, {654, 9404, 652}, {654, 13401, 4790}, {657, 661, 14298}, {657, 17418, 3287}, {661, 1635, 649}, {661, 4394, 4790}, {661, 4979, 4813}, {663, 4814, 4895}, {663, 45755, 4435}, {665, 7180, 43060}, {693, 4885, 45320}, {693, 24948, 27674}, {693, 26641, 28984}, {693, 26695, 29005}, {693, 27115, 31287}, {693, 30024, 18154}, {693, 31150, 17494}, {693, 31209, 2}, {693, 31287, 31250}, {1021, 3737, 7252}, {1146, 7117, 11998}, {1638, 21104, 3676}, {1639, 3700, 3239}, {1639, 4765, 4820}, {1639, 4976, 3700}, {2516, 4394, 1635}, {2516, 4893, 4790}, {2590, 2591, 654}, {3119, 7004, 38358}, {3239, 3700, 4944}, {3239, 4521, 1639}, {3239, 4976, 4820}, {3310, 40134, 6129}, {3676, 7658, 1638}, {3676, 43049, 3669}, {3676, 43050, 43049}, {4041, 4895, 4814}, {4369, 4763, 31286}, {4379, 31207, 24924}, {4380, 4776, 20295}, {4382, 4728, 23813}, {4382, 30835, 4728}, {4394, 14298, 654}, {4394, 40137, 652}, {4467, 30565, 25259}, {4521, 4765, 3700}, {4521, 4976, 4944}, {4560, 27527, 18155}, {4776, 20295, 4940}, {4820, 4944, 3700}, {4885, 31287, 2}, {4885, 44567, 31287}, {6586, 6589, 647}, {6588, 6589, 6129}, {6591, 40134, 6588}, {6608, 14392, 4105}, {7045, 37139, 9358}, {11934, 17115, 11193}, {15280, 15283, 11}, {17494, 26114, 29771}, {17494, 26777, 31150}, {17494, 26985, 26824}, {17494, 27014, 25667}, {17494, 27115, 2}, {17494, 27139, 29739}, {17494, 27346, 18071}, {17494, 27648, 28374}, {17494, 28834, 40166}, {17494, 31209, 4885}, {17494, 31287, 45320}, {17494, 44567, 31250}, {18154, 25511, 4885}, {18154, 29771, 693}, {24900, 24948, 2}, {24900, 27648, 4885}, {24948, 31150, 28374}, {25259, 27486, 4467}, {26049, 27014, 2}, {26114, 30024, 4885}, {26777, 27115, 693}, {26824, 26985, 693}, {27014, 30061, 4885}, {27115, 31150, 4885}, {27115, 31209, 44567}, {28423, 28714, 24562}, {28984, 31287, 25925}, {29427, 29739, 693}, {31150, 31209, 693}, {31150, 44567, 45320}, {31250, 45320, 4885}, {31287, 44567, 31209}


X(651) = TRILINEAR POLE OF LINE X(1)X(3)

Trilinears    1/(cos B - cos C) : 1/(cos C - cos A) : 1/(cos A - cos B)
Trilinears    1/[(b - c)(b + c - a)] : 1/[(c - a)(c + a - b)] : 1[(a - b)(a + b - c)]
Barycentrics    (sin A)/(cos B - cos C) : (sin B)/(cos C - cos A) : (sin C)/(cos A - cos B)
Barycentrics    a/[(b - c)(b + c - a)] : b/[(c - a)(c + a - b)] : c/[(a - b)(a + b - c)]
X(651) = 3 X[2] - 4 X[36949], 4 X[692] - 3 X[35280], 2 X[1633] - 3 X[35280], 4 X[36949] - X[37781]

Let E1 and E9 be the circumellipses centered at X(1) and X(9), respectively. The line tangent to E1 at X(100) meets E9 at X(651). (Dan Reznik, July 7, 2019)

X(651) lies on the MacBeath circumconic and these lines: 1, 1156}, {2, 222}, {3, 23707}, {4, 3157}, {5, 23070}, {6, 7}, {8, 221}, {9, 77}, {10, 34043}, {12, 8614}, {20, 7078}, {21, 73}, {30, 23071}, {34, 3868}, {37, 1442}, {40, 38674}, {43, 9316}, {44, 241}, {56, 19245}, {57, 88}, {59, 513}, {63, 223}, {65, 895}, {69, 478}, {72, 4296}, {75, 28968}, {78, 1394}, {79, 35197}, {80, 6126}, {81, 226}, {85, 3758}, {92, 2988}, {99, 32038}, {100, 109}, {101, 934}, {104, 34586}, {105, 1362}, {108, 110}, {112, 13395}, {141, 28780}, {144, 219}, {145, 34040}, {153, 15501}, {155, 1068}, {169, 2002}, {172, 28391}, {175, 1335}, {176, 1124}, {190, 644}, {193, 608}, {198, 1804}, {200, 34033}, {212, 7411}, {213, 7176}, {218, 279}, {220, 3160}, {225, 3193}, {238, 1458}, {239, 40862}, {255, 411}, {269, 1445}, {273, 2989}, {278, 1993}, {281, 5942}, {287, 894}, {307, 2287}, {312, 28997}, {321, 1943}, {323, 6357}, {329, 394}, {344, 28965}, {348, 37214}, {349, 37219}, {377, 19349}, {401, 44354}, {404, 603}, {481, 3301}, {482, 3299}, {500, 943}, {514, 655}, {518, 1456}, {521, 677}, {523, 21784}, {527, 2323}, {545, 41803}, {559, 7126}, {604, 1423}, {610, 7013}, {611, 4307}, {613, 4310}, {645, 799}, {648, 823}, {650, 7045}, {653, 1783}, {658, 1638}, {660, 6163}, {662, 1414}, {666, 1275}, {668, 6648}, {850, 43189}, {851, 17975}, {857, 3330}, {896, 1758}, {899, 9364}, {905, 1262}, {906, 32651}, {908, 22128}, {912, 1870}, {919, 6183}, {920, 1079}, {927, 8693}, {932, 29055}, {940, 5226}, {961, 17946}, {971, 3100}, {978, 1106}, {1000, 1480}, {1014, 1400}, {1018, 4606}, {1026, 4578}, {1030, 7279}, {1035, 1259}, {1037, 7083}, {1038, 3876}, {1040, 11220}, {1042, 5247}, {1046, 1254}, {1066, 3073}, {1069, 38295}, {1082, 19551}, {1100, 7269}, {1103, 2956}, {1110, 4905}, {1119, 3211}, {1170, 10481}, {1172, 7282}, {1190, 3599}, {1191, 4308}, {1201, 1476}, {1203, 4298}, {1214, 2349}, {1231, 37220}, {1252, 43049}, {1253, 1742}, {1305, 36080}, {1310, 8687}, {1319, 3246}, {1323, 5526}, {1386, 8581}, {1396, 40571}, {1398, 42461}, {1402, 37132}, {1404, 1429}, {1406, 1788}, {1407, 4383}, {1413, 27383}, {1418, 16669}, {1421, 3315}, {1424, 41526}, {1425, 29958}, {1427, 4641}, {1428, 1463}, {1431, 43763}, {1440, 27508}, {1444, 40590}, {1447, 33854}, {1449, 7190}, {1455, 4511}, {1465, 3218}, {1471, 4334}, {1498, 6223}, {1616, 6049}, {1617, 17127}, {1625, 40518}, {1654, 40999}, {1718, 11570}, {1723, 4341}, {1724, 4306}, {1757, 5018}, {1762, 37755}, {1768, 24025}, {1776, 8758}, {1812, 33066}, {1817, 40152}, {1818, 23693}, {1876, 34381}, {1880, 2987}, {1897, 13138}, {1936, 2635}, {1944, 30807}, {1958, 2199}, {1992, 6604}, {1994, 17483}, {2122, 7080}, {2124, 36973}, {2149, 36087}, {2161, 7202}, {2182, 7291}, {2183, 11349}, {2222, 4588}, {2261, 7289}, {2263, 3751}, {2265, 3942}, {2283, 4557}, {2286, 17257}, {2293, 2346}, {2295, 40765}, {2298, 7261}, {2317, 18162}, {2324, 34488}, {2348, 34855}, {2427, 23113}, {2428, 35312}, {2550, 45729}, {2580, 8115}, {2581, 8116}, {2639, 32320}, {2647, 2648}, {2720, 9058}, {2808, 3270}, {2895, 26942}, {2911, 41563}, {2975, 10571}, {2982, 21907}, {2986, 40149}, {3000, 9441}, {3008, 30379}, {3013, 8287}, {3074, 4303}, {3075, 6915}, {3091, 41344}, {3149, 23072}, {3215, 35976}, {3240, 37541}, {3248, 36267}, {3257, 3669}, {3292, 41349}, {3297, 17805}, {3298, 17802}, {3468, 44706}, {3487, 36742}, {3570, 4598}, {3573, 4499}, {3580, 37799}, {3600, 16466}, {3616, 34046}, {3638, 5357}, {3639, 5353}, {3660, 7292}, {3664, 21617}, {3668, 41572}, {3674, 5280}, {3676, 5375}, {3681, 8270}, {3684, 6168}, {3713, 32099}, {3759, 39126}, {3770, 34388}, {3784, 19649}, {3869, 21147}, {3873, 34036}, {3908, 40521}, {3911, 37222}, {3928, 36636}, {3945, 8232}, {3947, 37559}, {3952, 14594}, {3955, 4220}, {4014, 5091}, {4017, 37135}, {4032, 21741}, {4192, 22161}, {4219, 9637}, {4223, 45963}, {4224, 26892}, {4228, 20122}, {4253, 38859}, {4315, 5315}, {4321, 16469}, {4327, 16475}, {4328, 16667}, {4329, 22132}, {4331, 24695}, {4347, 5904}, {4350, 16572}, {4391, 6335}, {4417, 28774}, {4513, 25718}, {4556, 37140}, {4572, 37133}, {4584, 37134}, {4604, 5549}, {4607, 4998}, {4627, 4637}, {4649, 16133}, {4654, 43758}, {4715, 36589}, {5047, 37523}, {5120, 27624}, {5125, 5906}, {5219, 26738}, {5232, 5783}, {5236, 9028}, {5257, 24557}, {5260, 37558}, {5261, 5711}, {5276, 7179}, {5278, 27339}, {5279, 30456}, {5291, 43059}, {5328, 25934}, {5452, 7056}, {5703, 36746}, {5707, 5714}, {5744, 22129}, {5758, 37498}, {5782, 29611}, {5811, 17814}, {5848, 21293}, {5889, 41227}, {5932, 27382}, {6127, 10090}, {6147, 36750}, {6198, 40263}, {6326, 11700}, {6354, 18625}, {6505, 34052}, {6510, 43044}, {6603, 43064}, {6606, 32041}, {6646, 17086}, {6733, 43192}, {6909, 22350}, {6996, 20744}, {7004, 13243}, {7071, 42460}, {7074, 9778}, {7100, 35194}, {7191, 17625}, {7339, 14298}, {7352, 37117}, {7460, 23161}, {7580, 22117}, {7673, 12652}, {7952, 11441}, {8125, 34034}, {8126, 34026}, {8271, 30628}, {8540, 20358}, {8652, 26700}, {8732, 37681}, {9310, 34497}, {9317, 21742}, {9359, 36276}, {9446, 20229}, {9776, 10601}, {9809, 38357}, {9945, 22141}, {10025, 20752}, {10755, 25048}, {11573, 37431}, {11678, 34041}, {11679, 34044}, {11680, 34029}, {11681, 34030}, {11682, 34039}, {11685, 34037}, {11686, 34038}, {11687, 34031}, {11688, 34027}, {11690, 34025}, {12709, 17016}, {12738, 33649}, {12831, 45946}, {13149, 14545}, {13257, 15252}, {13397, 15439}, {14074, 14733}, {14100, 30621}, {14315, 23346}, {14511, 39756}, {15066, 31018}, {15251, 38055}, {15253, 33148}, {15507, 36280}, {15726, 41339}, {15906, 36125}, {16577, 33761}, {16610, 37789}, {16732, 18815}, {16948, 37583}, {17075, 17347}, {17077, 17277}, {17095, 17256}, {17120, 41246}, {17134, 22134}, {17234, 28741}, {17332, 41808}, {17335, 31225}, {17346, 33298}, {17349, 27342}, {17353, 23617}, {17379, 26125}, {17615, 34049}, {17796, 43066}, {17811, 18228}, {17950, 20072}, {18134, 28776}, {18230, 25878}, {18239, 40658}, {18624, 37672}, {18629, 20806}, {18631, 22133}, {18632, 23130}, {18743, 28996}, {18750, 28950}, {18771, 43947}, {18816, 40624}, {20245, 23124}, {20277, 24430}, {20348, 23125}, {20921, 28951}, {21189, 24027}, {21367, 44708}, {21859, 36074}, {23131, 23512}, {23541, 33650}, {23585, 23980}, {23704, 27834}, {24035, 44765}, {24149, 30690}, {24237, 24618}, {24341, 28125}, {24470, 37509}, {25737, 30719}, {26001, 44356}, {26006, 40880}, {26580, 26637}, {26657, 26685}, {26668, 27509}, {26842, 34545}, {26884, 33849}, {26890, 37261}, {27540, 37669}, {27994, 34061}, {28370, 41426}, {28606, 45126}, {28979, 33116}, {28999, 42719}, {29000, 42718}, {29353, 40910}, {29497, 38869}, {30572, 36237}, {30625, 34059}, {30684, 31164}, {31019, 37695}, {31231, 37687}, {32674, 36099}, {32736, 36098}, {34045, 35614}, {35466, 37797}, {36040, 36088}, {36049, 37141}, {36067, 43347}, {36075, 37211}, {36084, 43754}, {36092, 44770}, {36905, 39047}, {37516, 41230}, {37543, 37685}, {38459, 43065}, {38948, 45929}, {40443, 40606}, {40861, 44351}, {44112, 44151}, {44350, 44352}

X(651) = reflection of X(i) in X(j) for these {i,j}: {7, 39063}, {81, 40584}, {1633, 692}, {1814, 6}, {4318, 1456}, {7291, 2182}, {22464, 43035}, {26932, 36949}, {36101, 9}, {37781, 26932}, {40577, 59}
X(651) = isogonal conjugate of X(650)
X(651) = isotomic conjugate of X(4391)
X(651) = complement of X(37781)
X(651) = cevapoint of X(101) and X(109)
X(651) = polar conjugate of X(44426)
X(651) = trilinear pole of line {1, 3}
X(651) = crossdifference of every pair of points on line {926, 2170}
X(651) = X(i)-cross conjugate of X(j) for these (i,j): (6,59), (101,100), (513,7), (514,81), (521,77)
X(651) = crosssum of X(i) and X(j) for these (i,j): (647,661), (657,663)
X(651) = MacBeath circumconic antipode of X(1814)
X(651) = trilinear pole wrt tangential triangle of line X(1)X(3)
X(651) = orthocorrespondent of X(11)
X(651) = perspector of conic {A,B,C,PU(57)}}
X(651) = eigencenter of Honsberger triangle
X(651) = X(8754)-of-excentral-triangle
X(651) = barycentric product of circumcircle intercepts of line X(7)X(8)
X(651) = cevapoint of Feuerbach hyperbola intercepts of antiorthic axis
X(651) = intersection, other than vertices of Gemini triangle 30, of {ABC, Gemini 30}-circumconic and {Gemini 29, Gemini 30}-circumconic
X(651) = trilinear product of circumcircle intercepts of line X(2)X(7)
X(651) = BSS(a^2→a) of X(112)
X(651) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 9358}, {7, 40577}, {648, 653}, {658, 934}, {662, 1813}, {664, 100}, {765, 1445}, {927, 2283}, {1016, 5435}, {1262, 17074}, {1275, 7}, {1414, 109}, {3257, 24029}, {4554, 6516}, {4564, 57}, {4569, 40576}, {4573, 664}, {4620, 1014}, {4998, 56}, {5376, 37789}, {6183, 35280}, {6648, 4552}, {7045, 1}, {7128, 17080}, {13136, 2406}, {18026, 108}, {23586, 3160}, {24011, 7676}, {24032, 411}, {31615, 4564}, {37139, 23890}, {39293, 7677}, {44327, 37141}, {46102, 2}
X(651) = X(i)-cross conjugate of X(j) for these (i,j): {1, 7045}, {6, 59}, {9, 7012}, {56, 4998}, {57, 4564}, {101, 100}, {109, 934}, {218, 1252}, {221, 7339}, {222, 1262}, {223, 7128}, {238, 39293}, {513, 7}, {514, 81}, {521, 77}, {649, 1476}, {650, 1}, {652, 21}, {654, 104}, {657, 2346}, {661, 17097}, {665, 105}, {905, 2}, {906, 13397}, {928, 43363}, {978, 7035}, {1020, 653}, {1046, 24041}, {1415, 108}, {1461, 37141}, {1742, 24011}, {1743, 765}, {1745, 24032}, {1783, 13138}, {1983, 1290}, {2254, 43736}, {2283, 927}, {2348, 5377}, {2427, 901}, {2530, 7249}, {3063, 1037}, {3173, 44717}, {3287, 2298}, {3669, 57}, {3676, 1170}, {3960, 88}, {4040, 21453}, {4369, 1258}, {4383, 1016}, {4551, 664}, {4559, 109}, {4641, 4567}, {4790, 7091}, {4905, 1088}, {5247, 4600}, {6180, 1275}, {6364, 13389}, {6365, 13388}, {7175, 4620}, {7178, 2982}, {7180, 961}, {7234, 7176}, {7252, 951}, {7460, 2864}, {7655, 8809}, {9404, 943}, {10015, 2990}, {13401, 3296}, {14298, 8}, {14300, 5558}, {14349, 44733}, {14395, 1807}, {14837, 40399}, {14838, 1255}, {16610, 5376}, {16612, 1257}, {17496, 17074}, {18599, 24000}, {20980, 6}, {21007, 41431}, {21173, 86}, {21189, 75}, {21362, 190}, {21390, 82}, {21786, 106}, {23067, 1305}, {23090, 40442}, {23187, 95}, {23511, 5382}, {23800, 273}, {23890, 37139}, {24029, 655}, {30725, 34051}, {34032, 23984}, {34033, 24013}, {34048, 46102}, {35326, 101}, {36054, 3}, {36059, 6516}, {36075, 26700}, {39470, 1814}, {40137, 7320}, {43049, 279}, {43050, 34056}, {43051, 7132}, {43060, 28}, {43924, 1014}, {44319, 42311}
X(651) = cevapoint of X(i) and X(j) for these (i,j): {1, 650}, {2, 17496}, {3, 36054}, {6, 513}, {9, 521}, {57, 3669}, {73, 652}, {101, 109}, {213, 7234}, {218, 43049}, {221, 14298}, {222, 905}, {226, 514}, {408, 32320}, {500, 9404}, {520, 18591}, {522, 20262}, {525, 1211}, {649, 1201}, {654, 34586}, {656, 18675}, {657, 2293}, {661, 2650}, {663, 1200}, {665, 1362}, {1019, 40153}, {1400, 43924}, {1415, 36059}, {2254, 9502}, {2850, 40584}, {3063, 7083}, {3287, 28369}, {3676, 10481}, {4551, 4559}, {6180, 20980}, {6364, 13389}, {6365, 13388}, {8677, 23980}, {15730, 43050}, {30719, 45204}, {39063, 39470}
X(651) = crosspoint of X(i) and X(j) for these (i,j): {1, 9357}, {190, 44327}, {648, 662}, {658, 664}, {1414, 4573}, {4554, 18026}, {4564, 31615}
X(651) = crosssum of X(i) and X(j) for these (i,j): {1, 9355}, {9, 4919}, {11, 42454}, {514, 21195}, {522, 4147}, {647, 661}, {657, 663}, {1946, 3063}, {3126, 42341}, {3709, 4041}, {10581, 21127}, {21044, 42462}
X(651) = X(i)-isoconjugate of X(j) for these (i,j): {1, 650}, {2, 663}, {3, 3064}, {4, 652}, {6, 522}, {7, 657}, {8, 649}, {9, 513}, {10, 7252}, {11, 101}, {19, 521}, {21, 661}, {25, 6332}, {28, 8611}, {29, 647}, {31, 4391}, {32, 35519}, {33, 905}, {37, 3737}, {41, 693}, {42, 4560}, {44, 23838}, {48, 44426}, {55, 514}, {56, 3239}, {57, 3900}, {58, 3700}, {59, 42462}, {60, 4024}, {63, 18344}, {64, 14331}, {65, 1021}, {73, 17926}, {74, 14400}, {75, 3063}, {78, 6591}, {79, 9404}, {80, 654}, {81, 4041}, {84, 14298}, {85, 8641}, {86, 3709}, {88, 4895}, {89, 4814}, {92, 1946}, {100, 2170}, {106, 1639}, {108, 34591}, {109, 1146}, {110, 21044}, {111, 14432}, {115, 4636}, {124, 32653}, {158, 36054}, {184, 46110}, {188, 6729}, {190, 3271}, {200, 3669}, {210, 1019}, {212, 17924}, {213, 18155}, {219, 7649}, {220, 3676}, {225, 23090}, {226, 21789}, {244, 644}, {256, 3287}, {259, 6728}, {261, 4079}, {266, 6730}, {269, 4130}, {279, 4105}, {281, 1459}, {282, 6129}, {283, 2501}, {284, 523}, {291, 4435}, {292, 3716}, {294, 2254}, {312, 667}, {314, 798}, {318, 22383}, {332, 2489}, {333, 512}, {346, 43924}, {515, 2432}, {518, 1024}, {520, 8748}, {525, 2299}, {527, 23351}, {579, 23289}, {604, 4397}, {607, 4025}, {643, 3125}, {645, 3122}, {646, 3248}, {651, 2310}, {653, 3270}, {656, 1172}, {658, 3022}, {659, 4876}, {662, 4516}, {664, 14936}, {665, 14942}, {669, 28660}, {672, 885}, {673, 926}, {676, 2338}, {692, 4858}, {728, 43932}, {739, 14430}, {749, 4501}, {810, 31623}, {812, 7077}, {813, 4124}, {822, 1896}, {875, 3975}, {876, 3684}, {884, 3912}, {893, 3907}, {900, 2316}, {901, 4530}, {909, 2804}, {918, 2195}, {934, 3119}, {941, 17418}, {1002, 45755}, {1014, 4171}, {1015, 3699}, {1018, 18191}, {1022, 3689}, {1027, 3693}, {1036, 6590}, {1039, 2522}, {1043, 7180}, {1086, 3939}, {1110, 40166}, {1121, 6139}, {1126, 4976}, {1155, 23893}, {1170, 6608}, {1174, 6362}, {1178, 4140}, {1252, 21132}, {1253, 24002}, {1262, 23615}, {1292, 38375}, {1293, 4534}, {1318, 6544}, {1320, 1635}, {1331, 8735}, {1333, 4086}, {1334, 7192}, {1357, 6558}, {1395, 15416}, {1400, 7253}, {1407, 4163}, {1414, 36197}, {1415, 24026}, {1431, 4529}, {1432, 4477}, {1434, 4524}, {1436, 8058}, {1458, 28132}, {1461, 4081}, {1491, 2344}, {1577, 2194}, {1638, 4845}, {1647, 5548}, {1751, 8676}, {1783, 7004}, {1813, 42069}, {1826, 23189}, {1857, 4091}, {1897, 7117}, {1919, 3596}, {1924, 40072}, {1960, 4997}, {1973, 35518}, {1980, 28659}, {2053, 3835}, {2090, 45878}, {2125, 17427}, {2149, 42455}, {2150, 4036}, {2160, 35057}, {2161, 3738}, {2162, 4147}, {2163, 4944}, {2175, 3261}, {2183, 43728}, {2185, 4705}, {2189, 4064}, {2192, 14837}, {2193, 24006}, {2204, 14208}, {2212, 15413}, {2226, 4543}, {2258, 23880}, {2265, 46041}, {2269, 4581}, {2287, 4017}, {2291, 6366}, {2298, 17420}, {2311, 4010}, {2318, 17925}, {2319, 4083}, {2320, 4893}, {2321, 3733}, {2325, 23345}, {2328, 7178}, {2332, 17094}, {2334, 4765}, {2339, 8678}, {2342, 10015}, {2346, 21127}, {2348, 35355}, {2364, 4777}, {2423, 6735}, {2424, 40869}, {2425, 15633}, {2484, 30479}, {2488, 32008}, {2590, 3308}, {2591, 3307}, {2605, 7110}, {2643, 4612}, {2785, 17963}, {2968, 32674}, {2969, 4587}, {3049, 44130}, {3120, 5546}, {3121, 7257}, {3208, 43931}, {3224, 25128}, {3254, 22108}, {3445, 4521}, {3451, 42337}, {3572, 3685}, {3680, 4394}, {3688, 10566}, {3692, 43923}, {3710, 43925}, {3717, 43929}, {3732, 14935}, {3768, 36798}, {3904, 6187}, {4009, 23892}, {4069, 16726}, {4076, 21143}, {4092, 4556}, {4162, 8056}, {4451, 20981}, {4455, 36800}, {4474, 30650}, {4515, 7203}, {4517, 4817}, {4518, 8632}, {4522, 40746}, {4526, 37129}, {4542, 4638}, {4546, 40151}, {4557, 17197}, {4603, 40608}, {4617, 24010}, {4619, 5532}, {4626, 35508}, {4724, 40779}, {4750, 5547}, {4768, 9456}, {4775, 30608}, {4778, 34820}, {4790, 4866}, {4825, 30607}, {4834, 42030}, {4885, 9439}, {4913, 25426}, {4919, 6164}, {4939, 34080}, {4953, 38828}, {4959, 26745}, {4977, 33635}, {4979, 32635}, {4985, 28615}, {5060, 18013}, {5075, 17947}, {5089, 23696}, {5452, 26721}, {5514, 8059}, {5540, 42552}, {6059, 30805}, {6065, 6545}, {6066, 23100}, {6169, 42341}, {6182, 39273}, {6364, 13456}, {6365, 13427}, {6373, 36799}, {6506, 36082}, {6557, 8643}, {6589, 10570}, {6603, 35348}, {6607, 10509}, {6614, 23970}, {6615, 23617}, {6740, 21828}, {7072, 21188}, {7073, 14838}, {7091, 40137}, {7106, 8062}, {7108, 21761}, {7118, 17896}, {7131, 17115}, {7152, 14302}, {7155, 20979}, {7160, 14300}, {7162, 13401}, {7255, 20684}, {7707, 10495}, {8602, 30201}, {8638, 18031}, {8640, 27424}, {8648, 18359}, {8750, 26932}, {9447, 40495}, {10397, 40836}, {10482, 21104}, {10492, 16012}, {10579, 14282}, {10581, 21453}, {11125, 15627}, {11934, 44178}, {12032, 28143}, {12077, 35196}, {14304, 32677}, {14392, 34056}, {14395, 36119}, {14399, 44693}, {14413, 41798}, {14418, 36125}, {14571, 37628}, {14733, 33573}, {15313, 39943}, {15420, 40976}, {15997, 45877}, {17435, 36086}, {18000, 40882}, {18101, 46148}, {18108, 33299}, {21172, 30457}, {21185, 40141}, {21362, 40528}, {21390, 40505}, {21666, 32660}, {21739, 42657}, {21807, 39177}, {23104, 23979}, {23493, 27527}, {24012, 36838}, {24225, 40523}, {28291, 43960}, {29226, 36630}, {30731, 43922}, {31182, 33963}, {32641, 35015}, {32739, 34387}, {35072, 36127}, {36049, 38357}, {36050, 38345}, {39924, 45902}, {39956, 42312}, {42064, 43050}
X(651) = barycentric product X(i)*X(j) for these {i,j}: {1, 664}, {3, 18026}, {4, 6516}, {6, 4554}, {7, 100}, {8, 934}, {9, 658}, {10, 1414}, {21, 4566}, {31, 4572}, {34, 4561}, {36, 35174}, {37, 4573}, {42, 4625}, {55, 4569}, {56, 668}, {57, 190}, {59, 693}, {63, 653}, {65, 99}, {69, 108}, {73, 811}, {75, 109}, {76, 1415}, {77, 1897}, {78, 36118}, {81, 4552}, {85, 101}, {86, 4551}, {92, 1813}, {105, 883}, {110, 1441}, {112, 1231}, {158, 6517}, {162, 307}, {163, 349}, {181, 4623}, {200, 4626}, {210, 4616}, {219, 13149}, {220, 36838}, {222, 6335}, {223, 44327}, {225, 4592}, {226, 662}, {241, 666}, {256, 6649}, {264, 36059}, {269, 3699}, {273, 1331}, {274, 4559}, {278, 1332}, {279, 644}, {286, 23067}, {304, 32674}, {312, 1461}, {319, 26700}, {320, 2222}, {321, 4565}, {322, 8059}, {326, 36127}, {329, 37141}, {331, 906}, {333, 1020}, {341, 6614}, {345, 32714}, {346, 4617}, {347, 13138}, {348, 1783}, {354, 6606}, {377, 13395}, {388, 1310}, {479, 4578}, {497, 8269}, {513, 4998}, {514, 4564}, {518, 927}, {522, 7045}, {527, 37139}, {552, 40521}, {553, 37212}, {604, 1978}, {643, 3668}, {645, 1427}, {646, 1407}, {648, 1214}, {650, 1275}, {655, 3218}, {660, 1447}, {661, 4620}, {670, 1402}, {672, 34085}, {673, 1025}, {692, 6063}, {738, 6558}, {765, 3676}, {789, 1469}, {799, 1400}, {813, 10030}, {823, 40152}, {831, 7247}, {894, 37137}, {898, 43037}, {903, 23703}, {905, 46102}, {908, 37136}, {919, 40704}, {932, 3212}, {940, 32038}, {1014, 3952}, {1016, 3669}, {1018, 1434}, {1042, 7257}, {1086, 31615}, {1088, 3939}, {1119, 4571}, {1121, 23890}, {1122, 8706}, {1155, 35157}, {1252, 24002}, {1262, 4391}, {1284, 4589}, {1292, 6604}, {1293, 39126}, {1305, 3868}, {1317, 4618}, {1319, 4555}, {1334, 4635}, {1397, 6386}, {1403, 18830}, {1408, 27808}, {1409, 6331}, {1412, 4033}, {1423, 4598}, {1428, 4583}, {1429, 4562}, {1432, 18047}, {1439, 36797}, {1442, 6742}, {1445, 37206}, {1446, 5546}, {1449, 4624}, {1462, 42720}, {1463, 8709}, {1465, 13136}, {1476, 21272}, {1492, 7179}, {1509, 21859}, {1633, 8817}, {1758, 35154}, {1799, 46152}, {1847, 4587}, {1880, 4563}, {1909, 29055}, {1945, 15418}, {1969, 32660}, {1981, 40843}, {2003, 15455}, {2006, 4585}, {2078, 35171}, {2099, 4597}, {2149, 3261}, {2171, 4610}, {2185, 4605}, {2223, 46135}, {2254, 39293}, {2283, 2481}, {2284, 34018}, {2285, 37215}, {2321, 4637}, {2338, 24015}, {2346, 35312}, {2397, 34051}, {2398, 43736}, {2406, 36100}, {2550, 6183}, {2652, 17933}, {2720, 3262}, {2742, 38468}, {3057, 6613}, {3112, 46153}, {3219, 38340}, {3257, 3911}, {3263, 32735}, {3451, 21580}, {3573, 7233}, {3598, 37223}, {3649, 4596}, {3666, 6648}, {3671, 4614}, {3674, 36147}, {3732, 7131}, {3779, 34083}, {3903, 7176}, {3912, 36146}, {4017, 4600}, {4025, 7012}, {4032, 4603}, {4059, 8708}, {4076, 43932}, {4077, 4570}, {4105, 24011}, {4130, 23586}, {4146, 6733}, {4163, 24013}, {4357, 36098}, {4397, 7339}, {4556, 6358}, {4558, 40149}, {4567, 7178}, {4579, 7249}, {4584, 16609}, {4586, 7146}, {4595, 7153}, {4601, 7180}, {4604, 5219}, {4606, 21454}, {4612, 6354}, {4619, 4858}, {4621, 41777}, {4622, 40663}, {4654, 37211}, {4705, 7340}, {4968, 29279}, {5061, 35147}, {5172, 35156}, {5221, 32042}, {5228, 32041}, {5252, 13396}, {5376, 30725}, {5377, 43042}, {5378, 43041}, {5379, 17094}, {5380, 7181}, {5382, 30719}, {5383, 43051}, {5385, 43052}, {5389, 7214}, {5435, 27834}, {6012, 30617}, {6135, 13453}, {6136, 13436}, {6180, 30610}, {6332, 7128}, {6528, 22341}, {6540, 32636}, {6734, 36048}, {7035, 43924}, {7115, 15413}, {7132, 33946}, {7175, 27805}, {7182, 8750}, {7235, 36066}, {7265, 35049}, {7316, 42721}, {7672, 43349}, {8047, 40577}, {8680, 41206}, {8685, 33930}, {8687, 20911}, {8707, 24471}, {9357, 10001}, {9436, 36086}, {10509, 35341}, {13486, 40999}, {13577, 40576}, {14612, 39957}, {14733, 30806}, {14942, 41353}, {17080, 44765}, {17090, 29227}, {17097, 17136}, {17791, 34921}, {17924, 44717}, {18033, 34067}, {18743, 38828}, {18816, 23981}, {19604, 43290}, {20567, 32739}, {20924, 32675}, {20930, 36082}, {21362, 40420}, {21453, 35338}, {22464, 36037}, {24027, 35519}, {24029, 34234}, {30456, 44326}, {30545, 34071}, {31618, 35326}, {32018, 36075}, {32693, 34284}, {33864, 36094}, {34855, 36802}, {35516, 36040}, {36049, 40702}, {37138, 40719}, {37143, 37787}
X(651) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 522}, {2, 4391}, {3, 521}, {4, 44426}, {6, 650}, {7, 693}, {8, 4397}, {9, 3239}, {10, 4086}, {11, 42455}, {12, 4036}, {19, 3064}, {21, 7253}, {25, 18344}, {31, 663}, {32, 3063}, {34, 7649}, {35, 35057}, {36, 3738}, {37, 3700}, {40, 8058}, {41, 657}, {42, 4041}, {43, 4147}, {44, 1639}, {45, 4944}, {48, 652}, {55, 3900}, {56, 513}, {57, 514}, {58, 3737}, {59, 100}, {63, 6332}, {65, 523}, {69, 35518}, {71, 8611}, {73, 656}, {75, 35519}, {77, 4025}, {81, 4560}, {85, 3261}, {86, 18155}, {92, 46110}, {99, 314}, {100, 8}, {101, 9}, {104, 43728}, {105, 885}, {106, 23838}, {107, 1896}, {108, 4}, {109, 1}, {110, 21}, {112, 1172}, {162, 29}, {163, 284}, {171, 3907}, {172, 3287}, {181, 4705}, {184, 1946}, {190, 312}, {198, 14298}, {200, 4163}, {201, 4064}, {213, 3709}, {220, 4130}, {221, 6129}, {222, 905}, {223, 14837}, {225, 24006}, {226, 1577}, {238, 3716}, {241, 918}, {244, 21132}, {249, 4612}, {259, 6730}, {266, 6728}, {269, 3676}, {273, 46107}, {278, 17924}, {279, 24002}, {284, 1021}, {294, 28132}, {307, 14208}, {345, 15416}, {347, 17896}, {348, 15413}, {349, 20948}, {354, 6362}, {388, 2517}, {404, 20293}, {478, 6588}, {512, 4516}, {513, 11}, {514, 4858}, {515, 14304}, {517, 2804}, {519, 4768}, {521, 2968}, {522, 24026}, {553, 4978}, {577, 36054}, {603, 1459}, {604, 649}, {608, 6591}, {610, 14331}, {643, 1043}, {644, 346}, {648, 31623}, {649, 2170}, {650, 1146}, {652, 34591}, {653, 92}, {655, 18359}, {657, 3119}, {658, 85}, {659, 4124}, {660, 4518}, {661, 21044}, {662, 333}, {663, 2310}, {664, 75}, {665, 17435}, {666, 36796}, {667, 3271}, {668, 3596}, {670, 40072}, {678, 4543}, {692, 55}, {693, 34387}, {750, 4474}, {764, 7336}, {765, 3699}, {799, 28660}, {811, 44130}, {813, 4876}, {825, 2344}, {874, 4087}, {883, 3263}, {896, 14432}, {898, 36798}, {899, 14430}, {901, 1320}, {902, 4895}, {905, 26932}, {906, 219}, {919, 294}, {927, 2481}, {932, 7155}, {934, 7}, {940, 23880}, {953, 46041}, {961, 4581}, {982, 3810}, {984, 4522}, {1014, 7192}, {1016, 646}, {1018, 2321}, {1019, 17197}, {1020, 226}, {1023, 2325}, {1025, 3912}, {1026, 3717}, {1038, 23874}, {1042, 4017}, {1086, 40166}, {1100, 4976}, {1101, 4636}, {1106, 43924}, {1110, 3939}, {1125, 4985}, {1155, 6366}, {1172, 17926}, {1193, 17420}, {1201, 6615}, {1214, 525}, {1231, 3267}, {1252, 644}, {1253, 4105}, {1275, 4554}, {1284, 4010}, {1290, 11604}, {1292, 6601}, {1293, 3680}, {1295, 43737}, {1305, 2997}, {1308, 3254}, {1310, 30479}, {1319, 900}, {1331, 78}, {1332, 345}, {1333, 7252}, {1334, 4171}, {1357, 764}, {1361, 42757}, {1362, 3126}, {1381, 3308}, {1382, 3307}, {1388, 4926}, {1393, 21102}, {1394, 21172}, {1396, 17925}, {1397, 667}, {1398, 43923}, {1399, 2605}, {1400, 661}, {1401, 2530}, {1402, 512}, {1403, 4083}, {1404, 1635}, {1405, 4893}, {1407, 3669}, {1408, 3733}, {1409, 647}, {1412, 1019}, {1414, 86}, {1415, 6}, {1416, 1027}, {1417, 23345}, {1418, 21104}, {1419, 7658}, {1420, 3667}, {1421, 21201}, {1423, 3835}, {1424, 21191}, {1427, 7178}, {1428, 659}, {1429, 812}, {1434, 7199}, {1437, 23189}, {1438, 1024}, {1439, 17094}, {1441, 850}, {1442, 4467}, {1443, 4453}, {1445, 4468}, {1447, 3766}, {1449, 4765}, {1456, 676}, {1457, 1769}, {1458, 2254}, {1459, 7004}, {1460, 8678}, {1461, 57}, {1463, 3837}, {1465, 10015}, {1468, 17418}, {1469, 1491}, {1470, 9001}, {1471, 4724}, {1475, 21127}, {1477, 35355}, {1486, 11934}, {1490, 14302}, {1576, 2194}, {1617, 3309}, {1633, 497}, {1635, 4530}, {1740, 25128}, {1743, 4521}, {1758, 2785}, {1769, 35015}, {1783, 281}, {1795, 37628}, {1804, 4131}, {1812, 15411}, {1813, 63}, {1870, 44428}, {1875, 39534}, {1880, 2501}, {1897, 318}, {1914, 4435}, {1935, 8062}, {1943, 17899}, {1946, 3270}, {1978, 28659}, {1981, 1948}, {1983, 2323}, {2003, 14838}, {2078, 3887}, {2099, 4777}, {2149, 101}, {2170, 42462}, {2171, 4024}, {2173, 14400}, {2174, 9404}, {2175, 8641}, {2177, 4814}, {2193, 23090}, {2194, 21789}, {2218, 23289}, {2222, 80}, {2223, 926}, {2241, 4501}, {2280, 45755}, {2283, 518}, {2284, 3693}, {2285, 6590}, {2286, 2522}, {2291, 23893}, {2293, 6608}, {2295, 4140}, {2310, 23615}, {2329, 4529}, {2330, 4477}, {2352, 8676}, {2425, 2182}, {2426, 41339}, {2652, 18013}, {2687, 14224}, {2701, 2648}, {2703, 11609}, {2714, 43746}, {2720, 104}, {2722, 43735}, {2742, 34894}, {2743, 12641}, {3052, 4162}, {3057, 42337}, {3063, 14936}, {3158, 4546}, {3198, 14308}, {3212, 20906}, {3218, 3904}, {3230, 4526}, {3251, 4542}, {3257, 4997}, {3284, 14395}, {3339, 28147}, {3340, 28161}, {3361, 4778}, {3446, 42552}, {3485, 7650}, {3570, 3975}, {3573, 3685}, {3598, 30804}, {3616, 4811}, {3649, 30591}, {3659, 42017}, {3660, 2826}, {3666, 3910}, {3667, 4939}, {3668, 4077}, {3669, 1086}, {3671, 4815}, {3674, 4509}, {3676, 1111}, {3683, 4990}, {3684, 4148}, {3689, 4528}, {3699, 341}, {3709, 36197}, {3733, 18191}, {3745, 29278}, {3752, 21120}, {3799, 3790}, {3868, 20294}, {3870, 44448}, {3882, 3687}, {3888, 3705}, {3900, 4081}, {3903, 4451}, {3911, 3762}, {3915, 42312}, {3939, 200}, {3952, 3701}, {4017, 3120}, {4025, 17880}, {4033, 30713}, {4069, 4082}, {4077, 21207}, {4105, 24010}, {4130, 23970}, {4162, 4953}, {4242, 5081}, {4258, 4827}, {4306, 23800}, {4334, 24720}, {4350, 31605}, {4367, 4459}, {4383, 20317}, {4391, 23978}, {4394, 4534}, {4401, 4965}, {4427, 3702}, {4436, 3706}, {4482, 4494}, {4551, 10}, {4552, 321}, {4553, 3703}, {4554, 76}, {4556, 2185}, {4557, 210}, {4558, 1812}, {4559, 37}, {4561, 3718}, {4564, 190}, {4565, 81}, {4566, 1441}, {4567, 645}, {4569, 6063}, {4570, 643}, {4571, 1265}, {4572, 561}, {4573, 274}, {4574, 3694}, {4575, 283}, {4578, 5423}, {4579, 7081}, {4584, 36800}, {4585, 32851}, {4587, 3692}, {4588, 2320}, {4590, 4631}, {4592, 332}, {4595, 4110}, {4598, 27424}, {4600, 7257}, {4604, 30608}, {4605, 6358}, {4612, 7058}, {4617, 279}, {4619, 4564}, {4620, 799}, {4623, 18021}, {4624, 40023}, {4625, 310}, {4626, 1088}, {4636, 1098}, {4637, 1434}, {4649, 4913}, {4654, 4823}, {4705, 4092}, {4752, 4873}, {4781, 3902}, {4848, 4404}, {4849, 44729}, {4905, 17059}, {4998, 668}, {5018, 4458}, {5061, 2787}, {5172, 8674}, {5193, 2827}, {5219, 4791}, {5221, 4802}, {5228, 4762}, {5376, 4582}, {5377, 36802}, {5378, 36801}, {5379, 36797}, {5435, 4462}, {5546, 2287}, {5606, 10266}, {5930, 17898}, {6001, 14312}, {6011, 6598}, {6014, 4900}, {6063, 40495}, {6065, 4578}, {6099, 45393}, {6129, 38357}, {6135, 13454}, {6136, 13426}, {6180, 4885}, {6335, 7017}, {6386, 40363}, {6516, 69}, {6517, 326}, {6558, 30693}, {6575, 42015}, {6584, 6595}, {6586, 38358}, {6589, 38345}, {6591, 8735}, {6594, 38376}, {6610, 1638}, {6614, 269}, {6648, 30710}, {6649, 1909}, {6733, 188}, {7012, 1897}, {7023, 43932}, {7045, 664}, {7083, 17115}, {7113, 654}, {7115, 1783}, {7125, 4091}, {7128, 653}, {7146, 824}, {7175, 4369}, {7176, 4374}, {7178, 16732}, {7180, 3125}, {7183, 30805}, {7201, 4500}, {7203, 17205}, {7210, 21178}, {7223, 4411}, {7225, 4382}, {7234, 40608}, {7239, 4136}, {7248, 3777}, {7335, 23224}, {7339, 934}, {7340, 4623}, {8059, 84}, {8269, 8817}, {8614, 31947}, {8641, 3022}, {8677, 35014}, {8683, 3893}, {8685, 983}, {8686, 23836}, {8687, 2298}, {8693, 40779}, {8694, 4866}, {8697, 1392}, {8701, 32635}, {8750, 33}, {9058, 30513}, {9312, 20907}, {9316, 4449}, {9358, 39351}, {9455, 8638}, {10310, 30201}, {10473, 784}, {10571, 21189}, {11011, 28183}, {13136, 36795}, {13138, 280}, {13149, 331}, {13397, 43740}, {13444, 177}, {13462, 6006}, {13486, 3615}, {13588, 21300}, {14074, 34919}, {14298, 5514}, {14544, 23661}, {14594, 4385}, {14733, 1156}, {14882, 8702}, {15385, 40097}, {15386, 36050}, {15439, 943}, {16577, 7265}, {16680, 11997}, {16686, 11193}, {16777, 4820}, {16878, 6005}, {17074, 17496}, {17082, 23807}, {17095, 18160}, {17096, 16727}, {17197, 40213}, {17496, 40624}, {17780, 4723}, {17942, 2651}, {18026, 264}, {18047, 17787}, {18097, 18070}, {18108, 18101}, {18199, 16759}, {18344, 42069}, {18421, 28169}, {18593, 4707}, {20122, 39212}, {20229, 10581}, {20615, 40086}, {20958, 11124}, {21007, 38347}, {21132, 1090}, {21147, 21186}, {21173, 34589}, {21189, 124}, {21272, 20895}, {21362, 3452}, {21454, 4801}, {21859, 594}, {22341, 520}, {22356, 14418}, {22383, 7117}, {22464, 36038}, {23067, 72}, {23144, 24562}, {23224, 1364}, {23343, 4009}, {23344, 3689}, {23346, 1155}, {23353, 243}, {23363, 21334}, {23586, 36838}, {23703, 519}, {23706, 1785}, {23832, 3880}, {23845, 3057}, {23890, 527}, {23971, 4617}, {23979, 1415}, {23981, 517}, {24002, 23989}, {24013, 4626}, {24016, 43736}, {24019, 8748}, {24026, 23104}, {24027, 109}, {24029, 908}, {24033, 36127}, {24443, 21119}, {24471, 3004}, {25577, 4051}, {26665, 29003}, {26700, 79}, {26716, 42317}, {27644, 27527}, {27834, 6557}, {29055, 256}, {30239, 10309}, {30456, 6587}, {31343, 6556}, {31615, 1016}, {32038, 34258}, {32636, 4977}, {32643, 32677}, {32651, 2982}, {32652, 2192}, {32656, 212}, {32660, 48}, {32661, 2193}, {32665, 2316}, {32666, 2195}, {32669, 909}, {32674, 19}, {32675, 2161}, {32676, 2299}, {32677, 2432}, {32691, 1039}, {32693, 941}, {32714, 278}, {32728, 34068}, {32735, 105}, {32739, 41}, {34036, 21185}, {34051, 2401}, {34067, 7077}, {34068, 23351}, {34071, 2319}, {34073, 2364}, {34074, 34820}, {34085, 18031}, {34497, 21195}, {34543, 17424}, {34855, 43042}, {34921, 3065}, {35174, 20566}, {35280, 390}, {35281, 3872}, {35307, 21011}, {35312, 20880}, {35326, 1212}, {35327, 3683}, {35338, 4847}, {35342, 3686}, {36039, 2338}, {36040, 102}, {36049, 282}, {36050, 10570}, {36054, 35072}, {36057, 23696}, {36059, 3}, {36067, 36121}, {36072, 1251}, {36073, 33653}, {36074, 16777}, {36075, 1100}, {36076, 1061}, {36079, 8809}, {36080, 2335}, {36082, 90}, {36086, 14942}, {36094, 1311}, {36098, 1220}, {36100, 2399}, {36110, 36123}, {36113, 32706}, {36118, 273}, {36127, 158}, {36134, 35196}, {36141, 2291}, {36146, 673}, {37136, 34234}, {37137, 257}, {37139, 1121}, {37141, 189}, {37211, 42030}, {37212, 4102}, {37523, 28623}, {37541, 14077}, {37543, 23882}, {37579, 15313}, {37580, 6182}, {37583, 6003}, {37593, 4843}, {37595, 26732}, {37694, 20316}, {37787, 30565}, {37800, 26546}, {38296, 2516}, {38340, 30690}, {38809, 31628}, {38828, 8056}, {39630, 15910}, {39633, 6597}, {40097, 43742}, {40117, 7003}, {40149, 14618}, {40152, 24018}, {40499, 4073}, {40521, 6057}, {40576, 3434}, {40577, 149}, {41206, 35145}, {41280, 1980}, {41349, 2798}, {41353, 9436}, {41405, 4919}, {41426, 30198}, {41526, 20979}, {41777, 3776}, {42289, 4804}, {42552, 34896}, {42757, 3326}, {43040, 20908}, {43049, 4904}, {43051, 21138}, {43052, 4957}, {43192, 178}, {43290, 44720}, {43736, 2400}, {43909, 42547}, {43923, 2969}, {43924, 244}, {43932, 1358}, {43947, 43974}, {44327, 34404}, {44426, 21666}, {44717, 1332}, {45219, 14284}, {45874, 15997}, {45875, 2090}, {46102, 6335}, {46148, 33299}, {46152, 427}, {46153, 38}
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 9355, 2310}, {2, 222, 17074}, {2, 37781, 26932}, {4, 3157, 3562}, {6, 6180, 7}, {7, 37771, 1086}, {9, 1419, 77}, {44, 241, 37787}, {44, 6610, 241}, {57, 43048, 88}, {63, 223, 17080}, {73, 1935, 21}, {101, 1461, 1813}, {109, 4551, 100}, {190, 664, 4552}, {190, 1332, 644}, {190, 4585, 1332}, {221, 9370, 8}, {222, 34048, 2}, {222, 43043, 34051}, {226, 2003, 81}, {238, 1458, 7677}, {241, 6610, 1443}, {255, 1745, 411}, {269, 1445, 17092}, {269, 1743, 1445}, {603, 37694, 404}, {650, 7045, 9358}, {662, 1414, 4565}, {692, 1633, 35280}, {948, 4644, 7}, {1020, 1461, 934}, {1020, 21362, 24029}, {1086, 5723, 37771}, {1201, 9363, 1476}, {1253, 1742, 7676}, {1400, 7175, 1014}, {1407, 4383, 5435}, {1421, 5083, 3315}, {1442, 29007, 37}, {1443, 37787, 241}, {1633, 40576, 40577}, {1783, 32714, 653}, {1936, 2635, 36002}, {2263, 3751, 7672}, {2265, 3942, 16560}, {2293, 9440, 2346}, {2310, 9355, 1156}, {2647, 2650, 17097}, {3000, 9441, 30295}, {3074, 4303, 6986}, {3157, 8757, 4}, {3888, 4579, 100}, {3939, 35338, 100}, {4014, 20958, 5091}, {4334, 16468, 1471}, {4566, 14543, 653}, {10481, 17745, 1170}, {26932, 36949, 2}, {30725, 45273, 664}, {34028, 37659, 77}, {36280, 36942, 15507}


X(652) = CROSSDIFFERENCE OF X(1) AND X(4)

Trilinears    sec B - sec C : sec C - sec A : sec A - sec B
Trilinears    a (b - c) (b + c - a) (b^2 + c^2 - a^2) : :
Barycentrics  sin A (sec B - sec C) : sin B (sec C - sec A) : sin C (sec A - sec B)
X(652) = 2 X[3064] - 3 X[14400], X[8611] - 3 X[14418], 4 X[14331] - 3 X[14400]

X(652) is the perspector of triangle ABC and the tangential triangle of the conic that passes through the points A, B, C, X(1), and X(3). (Randy Hutson, 9/23/2011)

X(652) lies on these lines: {6, 2432}, {9, 3239}, {19, 42755}, {31, 21122}, {44, 513}, {55, 4105}, {57, 7658}, {63, 4025}, {73, 9240}, {101, 2149}, {109, 7115}, {112, 2762}, {142, 25604}, {228, 22388}, {243, 522}, {333, 35519}, {514, 23726}, {520, 647}, {521, 8611}, {653, 2639}, {663, 1951}, {905, 4091}, {926, 8641}, {1769, 6591}, {2170, 35065}, {2260, 42768}, {2504, 21107}, {2523, 14395}, {2636, 9394}, {2654, 2657}, {2655, 21173}, {3101, 20298}, {3215, 14825}, {3217, 27780}, {3219, 25259}, {3835, 28834}, {3928, 44551}, {4131, 24562}, {4382, 40166}, {4498, 21120}, {4524, 6139}, {4814, 42657}, {4885, 25924}, {4976, 6362}, {5075, 23655}, {5227, 9031}, {5271, 17894}, {6332, 15411}, {6608, 11934}, {7082, 23615}, {7117, 39006}, {7180, 20980}, {9029, 23865}, {10319, 20319}, {14330, 28589}, {16612, 21189}, {17094, 23727}, {17215, 25511}, {17975, 23146}, {20316, 26080}, {20760, 23093}, {22382, 23226}, {23090, 23189}, {25955, 31287}, {26049, 26652}, {26694, 27139}, {27527, 28960}, {32674, 36040}, {34591, 38344}, {35072, 38353}, {38983, 39687}, {38991, 43960}, {43060, 43924}

X(652) = reflection of X(i) in X(j) for these {i,j}: {663, 21789}, {3064, 14331}, {23727, 17094}
X(652) = isogonal conjugate of X(653)
X(652) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,38983), (101,48), (109,55)
X(652) = crosspoint of X(i) and X(j) for these (i,j): (9,101), (109, 222)
X(652) = crosssum of X(i) and X(j) for these (i,j): (1,652), (57,514), (65,650), (281,522), (513,1108)
X(652) = crossdifference of every pair of points on line X(1)X(4)
X(652) = trilinear pole of line {2637, 2638}
X(652) = bicentric difference of PU(i) for i in (16, 76, 82, 126)
X(652) = PU(16)-harmonic conjugate of X(73)
X(652) = PU(76)-harmonic conjugate of X(2635)
X(652) = PU(82)-harmonic conjugate of X(2654)
X(652) = PU(126)-harmonic conjugate of X(4)
X(652) = polar conjugate of isotomic conjugate of isogonal conjugate of X(36127)
X(652) = trilinear product of PU(77)
X(652) = trilinear pole of PU(77)
X(652) = intersection of trilinear polars of X(1) and X(3)
X(652) = trilinear product of circumcircle intercepts of line X(2)X(92)
X(652) = X(31)-complementary conjugate of X(38983)
X(652) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 2638}, {2, 38983}, {3, 3270}, {9, 34591}, {48, 38344}, {63, 7004}, {90, 2310}, {101, 48}, {109, 55}, {162, 14547}, {163, 2269}, {219, 7117}, {222, 1364}, {268, 35072}, {522, 663}, {651, 73}, {653, 1}, {658, 20277}, {823, 2654}, {905, 1459}, {906, 71}, {1020, 2646}, {1021, 650}, {1248, 42080}, {1331, 212}, {1332, 22072}, {1751, 11}, {1783, 22063}, {1813, 3}, {2762, 10535}, {3939, 7124}, {4558, 22361}, {4561, 20753}, {4563, 22421}, {4571, 2318}, {4574, 22074}, {4587, 219}, {7094, 38345}, {8059, 2188}, {8064, 1436}, {23067, 23207}, {23090, 36054}, {23189, 1946}, {36039, 672}, {36049, 6}, {36050, 42}, {36059, 40945}, {36093, 2356}, {37203, 35015}, {39943, 2170}, {39946, 3248}, {39947, 244}, {40117, 41087}, {40518, 40944}, {41206, 1936}
X(652) = X(i)-cross conjugate of X(j) for these (i,j): {647, 650}, {822, 36054}, {1946, 1459}, {2638, 1}, {3270, 3}, {7117, 219}
X(652) = cevapoint of X(i) and X(j) for these (i,j): {1, 2636}, {647, 822}
X(652) = crosspoint of X(i) and X(j) for these (i,j): {1, 653}, {3, 1813}, {6, 32653}, {9, 101}, {21, 651}, {57, 8059}, {63, 1331}, {100, 40399}, {109, 222}, {190, 40436}, {219, 4587}, {268, 36049}, {333, 4636}, {521, 905}, {522, 6332}, {906, 2193}, {1021, 23090}, {1172, 40117}, {1812, 4571}
X(652) = crosssum of X(i) and X(j) for these (i,j): {1, 652}, {4, 3064}, {6, 39199}, {9, 8058}, {19, 7649}, {57, 514}, {65, 650}, {108, 1783}, {109, 32674}, {196, 14837}, {281, 522}, {407, 2501}, {513, 1108}, {523, 1901}, {649, 3924}, {656, 2294}, {657, 4336}, {822, 2658}, {1828, 6591}, {1880, 43923}, {3676, 7195}, {6087, 23986}, {8063, 40837}, {8755, 42755}, {14331, 44696}, {17924, 40149}
X(652) = crossdifference of every pair of points on line {1, 4}
X(652) = X(i)-isoconjugate of X(j) for these (i,j): {1, 653}, {2, 108}, {4, 651}, {6, 18026}, {7, 1783}, {8, 32714}, {9, 36118}, {19, 664}, {25, 4554}, {27, 4551}, {28, 4552}, {29, 1020}, {33, 658}, {34, 190}, {55, 13149}, {56, 6335}, {57, 1897}, {59, 17924}, {63, 36127}, {65, 648}, {73, 823}, {75, 32674}, {83, 46152}, {85, 8750}, {92, 109}, {99, 1880}, {100, 278}, {101, 273}, {102, 24035}, {107, 1214}, {110, 40149}, {112, 1441}, {158, 1813}, {162, 226}, {196, 13138}, {208, 44327}, {225, 662}, {264, 1415}, {270, 4605}, {281, 934}, {286, 4559}, {307, 24019}, {318, 1461}, {331, 692}, {342, 36049}, {347, 40117}, {349, 32676}, {388, 36099}, {393, 6516}, {513, 46102}, {514, 7012}, {521, 23984}, {522, 7128}, {607, 4569}, {608, 668}, {644, 1119}, {645, 1426}, {646, 1398}, {652, 24032}, {655, 1870}, {666, 1876}, {693, 7115}, {811, 1400}, {851, 41207}, {883, 8751}, {901, 37790}, {908, 36110}, {927, 5089}, {1016, 43923}, {1025, 36124}, {1041, 3732}, {1118, 1332}, {1172, 4566}, {1231, 32713}, {1262, 44426}, {1275, 18344}, {1290, 37799}, {1295, 2405}, {1309, 1465}, {1395, 1978}, {1396, 3952}, {1402, 6331}, {1409, 6528}, {1414, 1826}, {1427, 36797}, {1435, 3699}, {1476, 17906}, {1769, 39294}, {1785, 37136}, {1824, 4573}, {1829, 6648}, {1847, 3939}, {1848, 36098}, {1861, 36146}, {1874, 4584}, {1875, 13136}, {1877, 3257}, {1937, 1981}, {1952, 23353}, {1973, 4572}, {2006, 4242}, {2052, 36059}, {2149, 46107}, {2222, 17923}, {2333, 4625}, {2356, 34085}, {2406, 36121}, {2639, 9394}, {2766, 37798}, {2969, 31615}, {3064, 7045}, {3262, 32702}, {3669, 15742}, {4185, 32038}, {4564, 7649}, {4565, 41013}, {4617, 7046}, {4624, 5338}, {4626, 7079}, {4998, 6591}, {5236, 36086}, {5379, 7178}, {6198, 38340}, {6332, 24033}, {6336, 23703}, {6517, 6520}, {6614, 7101}, {7009, 37137}, {7071, 36838}, {7952, 37141}, {14733, 37805}, {14838, 34922}, {15352, 22341}, {16082, 23981}, {17080, 26704}, {21362, 40446}, {23706, 34234}, {23710, 37139}, {23985, 35518}, {23987, 36100}, {24027, 46110}, {24029, 36123}, {26706, 37800}, {32652, 40701}, {32667, 35516}, {32735, 46108}, {36126, 40152}, {41321, 43736}
X(652) = barycentric product X(i)*X(j) for these {i,j}: {1, 521}, {3, 522}, {6, 6332}, {8, 1459}, {9, 905}, {10, 23189}, {11, 1331}, {21, 656}, {29, 520}, {31, 35518}, {33, 4131}, {41, 15413}, {48, 4391}, {55, 4025}, {60, 4064}, {63, 650}, {69, 663}, {71, 4560}, {72, 3737}, {73, 7253}, {75, 1946}, {77, 3900}, {78, 513}, {81, 8611}, {88, 14418}, {92, 36054}, {100, 7004}, {101, 26932}, {102, 39471}, {108, 24031}, {109, 2968}, {125, 4636}, {184, 35519}, {189, 10397}, {190, 7117}, {212, 693}, {219, 514}, {222, 3239}, {226, 23090}, {228, 18155}, {244, 4571}, {255, 44426}, {268, 14837}, {271, 6129}, {281, 4091}, {283, 523}, {284, 525}, {295, 3716}, {304, 3063}, {306, 7252}, {307, 21789}, {312, 22383}, {314, 810}, {318, 23224}, {326, 18344}, {332, 512}, {333, 647}, {345, 649}, {348, 657}, {394, 3064}, {517, 37628}, {518, 23696}, {577, 46110}, {603, 4397}, {604, 15416}, {607, 30805}, {643, 18210}, {644, 3942}, {651, 34591}, {653, 35072}, {661, 1812}, {664, 3270}, {667, 3718}, {692, 17880}, {822, 31623}, {885, 1818}, {895, 14432}, {906, 4858}, {926, 31637}, {1019, 3694}, {1021, 1214}, {1024, 25083}, {1036, 23874}, {1073, 14331}, {1086, 4587}, {1146, 1813}, {1156, 14414}, {1172, 24018}, {1259, 7649}, {1260, 3676}, {1265, 43924}, {1332, 2170}, {1334, 15419}, {1364, 1897}, {1400, 15411}, {1433, 8058}, {1437, 4086}, {1444, 4041}, {1565, 3939}, {1577, 2193}, {1639, 1797}, {1769, 1809}, {1790, 3700}, {1791, 17420}, {1792, 4017}, {1795, 2804}, {1796, 4976}, {1802, 24002}, {1807, 3738}, {1808, 4010}, {2188, 17896}, {2194, 14208}, {2269, 15420}, {2289, 17924}, {2299, 3265}, {2310, 6516}, {2311, 24459}, {2318, 7192}, {2319, 25098}, {2321, 7254}, {2327, 7178}, {2328, 17094}, {2338, 39470}, {2339, 2522}, {2349, 14395}, {2359, 3910}, {2501, 6514}, {2638, 18026}, {2785, 17973}, {3049, 28660}, {3271, 4561}, {3478, 9031}, {3504, 25128}, {3669, 3692}, {3699, 3937}, {3708, 4612}, {3709, 17206}, {3710, 3733}, {3719, 6591}, {3907, 7015}, {4105, 7056}, {4130, 7177}, {4147, 23086}, {4163, 7053}, {4451, 22093}, {4466, 5546}, {4467, 8606}, {4516, 4592}, {4518, 22384}, {4557, 17219}, {4558, 21044}, {4574, 17197}, {4768, 36058}, {4997, 22086}, {5440, 23838}, {5931, 42658}, {6056, 46107}, {6368, 35196}, {6510, 23893}, {6517, 42069}, {6608, 40443}, {7016, 8062}, {7100, 35057}, {7107, 17899}, {7108, 22382}, {7155, 22090}, {7182, 8641}, {7358, 8059}, {14298, 41081}, {14304, 36055}, {14400, 14919}, {16596, 36049}, {17418, 34259}, {17926, 40152}, {21186, 39167}, {22092, 36799}, {22350, 43728}, {23614, 24032}, {23978, 32660}, {23983, 32674}, {24026, 36059}, {32653, 40626}, {32656, 34387}, {34588, 36050}, {35014, 36037}, {39201, 44130}, {40399, 40628}, {42462, 44717}
X(652) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 18026}, {3, 664}, {6, 653}, {9, 6335}, {11, 46107}, {21, 811}, {25, 36127}, {29, 6528}, {31, 108}, {32, 32674}, {41, 1783}, {48, 651}, {55, 1897}, {56, 36118}, {57, 13149}, {63, 4554}, {69, 4572}, {71, 4552}, {73, 4566}, {77, 4569}, {78, 668}, {101, 46102}, {108, 24032}, {184, 109}, {212, 100}, {219, 190}, {222, 658}, {228, 4551}, {255, 6516}, {268, 44327}, {283, 99}, {284, 648}, {332, 670}, {333, 6331}, {345, 1978}, {512, 225}, {513, 273}, {514, 331}, {520, 307}, {521, 75}, {522, 264}, {525, 349}, {577, 1813}, {603, 934}, {604, 32714}, {647, 226}, {649, 278}, {650, 92}, {654, 17923}, {656, 1441}, {657, 281}, {661, 40149}, {663, 4}, {665, 5236}, {667, 34}, {692, 7012}, {798, 1880}, {810, 65}, {822, 1214}, {884, 36124}, {905, 85}, {906, 4564}, {926, 1861}, {1021, 31623}, {1092, 6517}, {1146, 46110}, {1172, 823}, {1259, 4561}, {1260, 3699}, {1331, 4998}, {1364, 4025}, {1409, 1020}, {1415, 7128}, {1437, 1414}, {1444, 4625}, {1459, 7}, {1635, 37790}, {1639, 46109}, {1790, 4573}, {1792, 7257}, {1802, 644}, {1807, 35174}, {1808, 4589}, {1812, 799}, {1813, 1275}, {1814, 34085}, {1818, 883}, {1919, 608}, {1946, 1}, {1951, 1981}, {1960, 1877}, {1964, 46152}, {1980, 1395}, {2170, 17924}, {2175, 8750}, {2182, 24035}, {2188, 13138}, {2193, 662}, {2194, 162}, {2197, 4605}, {2200, 4559}, {2204, 24019}, {2249, 41207}, {2289, 1332}, {2299, 107}, {2310, 44426}, {2318, 3952}, {2327, 645}, {2328, 36797}, {2342, 1309}, {2347, 17906}, {2359, 6648}, {2361, 4242}, {2605, 7282}, {2636, 39060}, {2638, 521}, {2968, 35519}, {3049, 1400}, {3063, 19}, {3064, 2052}, {3239, 7017}, {3248, 43923}, {3270, 522}, {3271, 7649}, {3669, 1847}, {3692, 646}, {3694, 4033}, {3709, 1826}, {3710, 27808}, {3716, 40717}, {3718, 6386}, {3737, 286}, {3900, 318}, {3937, 3676}, {3939, 15742}, {3942, 24002}, {3955, 6649}, {4025, 6063}, {4041, 41013}, {4055, 23067}, {4064, 34388}, {4079, 8736}, {4091, 348}, {4105, 7046}, {4130, 7101}, {4131, 7182}, {4391, 1969}, {4455, 1874}, {4516, 24006}, {4546, 44721}, {4558, 4620}, {4560, 44129}, {4571, 7035}, {4587, 1016}, {4612, 46254}, {4636, 18020}, {4895, 38462}, {5075, 17985}, {6056, 1331}, {6129, 342}, {6139, 23710}, {6332, 76}, {6514, 4563}, {6518, 15418}, {7004, 693}, {7053, 4626}, {7085, 14594}, {7099, 4617}, {7116, 37137}, {7117, 514}, {7118, 40117}, {7124, 3732}, {7177, 36838}, {7252, 27}, {7253, 44130}, {7254, 1434}, {8606, 6742}, {8611, 321}, {8638, 2356}, {8641, 33}, {8648, 1870}, {8676, 5125}, {8677, 22464}, {8748, 15352}, {9247, 1415}, {10397, 329}, {10581, 1855}, {14331, 15466}, {14395, 14206}, {14400, 46106}, {14413, 38461}, {14414, 30806}, {14418, 4358}, {14432, 44146}, {14578, 37136}, {14585, 32660}, {14837, 40701}, {14936, 3064}, {15411, 28660}, {15413, 20567}, {15416, 28659}, {17880, 40495}, {17973, 35154}, {18210, 4077}, {18344, 158}, {20752, 1025}, {20753, 3888}, {20775, 46153}, {21044, 14618}, {21132, 2973}, {21761, 1940}, {21789, 29}, {22053, 35312}, {22072, 21272}, {22074, 3882}, {22079, 35338}, {22086, 3911}, {22090, 3212}, {22091, 9312}, {22092, 43040}, {22093, 7176}, {22096, 43924}, {22361, 17136}, {22368, 46177}, {22379, 1443}, {22382, 1943}, {22383, 57}, {22384, 1447}, {23090, 333}, {23189, 86}, {23202, 23703}, {23220, 1457}, {23224, 77}, {23225, 1458}, {23226, 1442}, {23614, 24031}, {23615, 21666}, {23696, 2481}, {24018, 1231}, {24031, 35518}, {24562, 21609}, {25098, 30545}, {26932, 3261}, {31637, 46135}, {32320, 40152}, {32641, 39294}, {32656, 59}, {32658, 36146}, {32660, 1262}, {32674, 23984}, {32739, 7115}, {34591, 4391}, {34858, 36110}, {35014, 36038}, {35072, 6332}, {35196, 18831}, {35518, 561}, {35519, 18022}, {36054, 63}, {36057, 927}, {36059, 7045}, {37628, 18816}, {38344, 17496}, {39201, 73}, {39471, 35516}, {40628, 17862}, {42658, 5930}, {43924, 1119}, {46110, 18027}
X(652) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {647, 22086, 22383}, {647, 22383, 1459}, {649, 657, 650}, {650, 654, 649}, {650, 4790, 14300}, {650, 9404, 657}, {650, 13401, 21127}, {653, 2639, 24032}, {654, 9404, 650}, {822, 2522, 22443}, {3064, 14331, 14400}, {4394, 40137, 650}, {4979, 21127, 13401}


X(653) = TRILINEAR POLE OF LINE X(1)X(4)

Trilinears    1/(sec B - sec C) : 1/(sec C - sec A) : 1/(sec A - sec B)
Barycentrics    (sin A)/(sec B - sec C) : (sin B)/(sec C - sec A) : (sin C)/(sec A - sec B)
Barycentrics    1/[(b - c)(b + c - a)(b^2 + c^2 - a^2)] : :
X(653) = 3 X[2] - 4 X[40535]

X(653) lies on the curve CC9 and these lines: {1, 2636}, {2, 196}, {3, 1148}, {4, 1156}, {7, 281}, {8, 44696}, {9, 342}, {19, 273}, {20, 3176}, {27, 18815}, {29, 65}, {34, 37129}, {40, 1895}, {46, 158}, {56, 23772}, {57, 92}, {63, 1767}, {77, 2331}, {78, 207}, {85, 2002}, {88, 278}, {100, 108}, {101, 4605}, {107, 109}, {112, 1305}, {190, 6335}, {208, 318}, {225, 897}, {226, 2349}, {241, 14571}, {242, 1876}, {243, 1155}, {297, 17950}, {307, 6330}, {329, 40837}, {331, 37130}, {347, 1249}, {349, 37220}, {425, 2651}, {450, 17975}, {468, 17985}, {484, 1784}, {514, 1461}, {607, 3212}, {608, 20332}, {646, 42384}, {648, 662}, {651, 1783}, {652, 2639}, {655, 17924}, {658, 13149}, {660, 43923}, {685, 36084}, {692, 23353}, {799, 4572}, {823, 15352}, {934, 2405}, {962, 40836}, {1013, 37541}, {1020, 2637}, {1025, 37206}, {1041, 40987}, {1047, 7138}, {1075, 20764}, {1118, 1788}, {1119, 8732}, {1214, 41083}, {1360, 4081}, {1400, 1821}, {1416, 21210}, {1430, 9364}, {1441, 2322}, {1447, 5089}, {1452, 37390}, {1585, 13437}, {1586, 13459}, {1708, 1748}, {1737, 7541}, {1826, 7282}, {1828, 40446}, {1844, 12432}, {1845, 12736}, {1846, 19636}, {1847, 7719}, {1857, 3474}, {1880, 7233}, {1957, 9316}, {2501, 9358}, {2656, 2658}, {3064, 37139}, {3162, 30918}, {3209, 9308}, {3257, 46102}, {3339, 39585}, {3559, 7098}, {3658, 35360}, {3729, 7101}, {3911, 17923}, {4296, 44698}, {4312, 39531}, {4554, 37215}, {4599, 42396}, {4848, 5174}, {4997, 37768}, {5236, 8756}, {5307, 43759}, {6332, 44327}, {6354, 18679}, {6360, 7011}, {7012, 7649}, {7013, 45738}, {7017, 32939}, {7049, 38284}, {7079, 8545}, {7115, 36087}, {7156, 34059}, {7551, 39542}, {8755, 22464}, {9056, 36067}, {9778, 44695}, {14257, 17555}, {14837, 23984}, {16813, 32660}, {17074, 18676}, {17077, 17913}, {18097, 37221}, {18588, 30841}, {21186, 24033}, {21617, 25993}, {23982, 23986}, {26003, 37787}, {30725, 46118}, {34393, 40626}, {35486, 38295}, {36090, 36110}, {37222, 37789}, {37797, 37799}

X(653) = reflection of X(16596) in X(40535)
X(653) = isogonal conjugate of X(652)
X(653) = isotomic conjugate of X(6332)
X(653) = anticomplement of X(16596)
X(653) = polar conjugate of X(522)
X(653) = trilinear product of PU(76)
X(653) = crossdifference of PU(77)
X(653) = trilinear pole wrt tangential triangle of line X(1)X(4)
X(653) = pole wrt polar circle of trilinear polar of X(522) (line X(11)X(1146))
X(653) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {7115, 6223}, {36049, 34188}, {40117, 33650}
X(653) = X(31)-complementary conjugate of X(39053)
X(653) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 2639}, {2, 39053}, {648, 651}, {4564, 37279}, {4998, 5125}, {7045, 412}, {13149, 36118}, {18026, 1897}, {24032, 1}, {39294, 17923}, {46102, 278}
X(653) = X(i)-cross conjugate of X(j) for these (i,j): {1, 24032}, {19, 7012}, {46, 7045}, {57, 7128}, {108, 36118}, {109, 664}, {196, 23984}, {278, 46102}, {514, 92}, {522, 7}, {579, 59}, {650, 29}, {652, 1}, {1020, 651}, {1021, 17097}, {1708, 4564}, {1721, 24011}, {1722, 7035}, {1723, 765}, {1734, 21453}, {1738, 39293}, {1781, 24000}, {1783, 1897}, {1786, 35049}, {1788, 4998}, {1826, 34922}, {2637, 23707}, {3064, 4}, {3144, 18020}, {3911, 39294}, {4142, 7249}, {6591, 40446}, {7649, 273}, {7661, 1440}, {8058, 342}, {8063, 329}, {10900, 5382}, {14331, 8}, {14400, 80}, {14837, 2}, {16612, 1220}, {17906, 6335}, {17924, 27}, {18679, 23582}, {21173, 40440}, {21180, 18815}, {21185, 1088}, {21186, 75}, {21189, 86}, {22443, 1037}, {32674, 36127}, {35349, 3699}, {42755, 22464}
X(653) = cevapoint of X(i) and X(j) for these (i,j): {1, 652}, {4, 3064}, {6, 39199}, {9, 8058}, {19, 7649}, {57, 514}, {65, 650}, {108, 1783}, {109, 32674}, {196, 14837}, {281, 522}, {407, 2501}, {513, 1108}, {523, 1901}, {649, 3924}, {656, 2294}, {657, 4336}, {822, 2658}, {1828, 6591}, {1880, 43923}, {3676, 7195}, {6087, 23986}, {8063, 40837}, {8755, 42755}, {14331, 44696}, {17924, 40149}
X(653) = crosspoint of X(i) and X(j) for these (i,j): {1, 9394}, {648, 823}, {13149, 18026}
X(653) = crosssum of X(i) and X(j) for these (i,j): {1, 2636}, {647, 822}
X(653) = trilinear pole of line {1, 4}
X(653) = crossdifference of every pair of points on line {2637, 2638}
X(653) = X(i)-isoconjugate of X(j) for these (i,j): {1, 652}, {2, 1946}, {3, 650}, {4, 36054}, {6, 521}, {8, 22383}, {9, 1459}, {11, 906}, {21, 647}, {29, 822}, {31, 6332}, {32, 35518}, {33, 4091}, {37, 23189}, {41, 4025}, {48, 522}, {55, 905}, {58, 8611}, {63, 663}, {65, 23090}, {69, 3063}, {71, 3737}, {72, 7252}, {73, 1021}, {74, 14395}, {77, 657}, {78, 649}, {84, 10397}, {100, 7117}, {101, 7004}, {106, 14418}, {108, 35072}, {109, 34591}, {184, 4391}, {210, 7254}, {212, 514}, {219, 513}, {222, 3900}, {228, 4560}, {244, 4587}, {255, 3064}, {268, 6129}, {281, 23224}, {283, 661}, {284, 656}, {295, 4435}, {314, 3049}, {332, 798}, {333, 810}, {345, 667}, {348, 8641}, {394, 18344}, {512, 1812}, {520, 1172}, {523, 2193}, {525, 2194}, {577, 44426}, {603, 3239}, {607, 4131}, {644, 3937}, {646, 22096}, {651, 3270}, {653, 2638}, {654, 1807}, {672, 23696}, {692, 26932}, {884, 25083}, {885, 20752}, {926, 1814}, {1015, 4571}, {1019, 2318}, {1024, 1818}, {1036, 2522}, {1146, 36059}, {1167, 40628}, {1214, 21789}, {1259, 6591}, {1260, 3669}, {1320, 22086}, {1331, 2170}, {1332, 3271}, {1364, 1783}, {1397, 15416}, {1402, 15411}, {1409, 7253}, {1415, 2968}, {1433, 14298}, {1437, 3700}, {1444, 3709}, {1639, 36058}, {1790, 4041}, {1792, 7180}, {1793, 21828}, {1797, 4895}, {1802, 3676}, {1803, 6608}, {1808, 21832}, {1809, 3310}, {1813, 2310}, {1896, 32320}, {1919, 3718}, {2053, 25098}, {2150, 4064}, {2175, 15413}, {2183, 37628}, {2188, 14837}, {2196, 3716}, {2200, 18155}, {2204, 3265}, {2212, 30805}, {2289, 7649}, {2291, 14414}, {2299, 24018}, {2319, 22090}, {2327, 4017}, {2359, 17420}, {2431, 6001}, {2637, 23707}, {2804, 14578}, {3287, 7015}, {3692, 43924}, {3694, 3733}, {3708, 4636}, {3907, 7116}, {3939, 3942}, {4105, 7177}, {4130, 7053}, {4147, 15373}, {4163, 7099}, {4516, 4558}, {4574, 18191}, {4575, 21044}, {4581, 22074}, {4612, 20975}, {4768, 32659}, {4858, 32656}, {4876, 22384}, {5546, 18210}, {6056, 17924}, {6510, 23351}, {6516, 14936}, {6588, 39167}, {7100, 9404}, {7105, 22382}, {7107, 8062}, {7110, 23226}, {8606, 14838}, {8851, 22092}, {9247, 35519}, {10581, 40443}, {14331, 19614}, {14400, 35200}, {14432, 36060}, {15420, 20967}, {16596, 32652}, {17880, 32739}, {17926, 22341}, {18026, 39687}, {22144, 42552}, {22356, 23838}, {22379, 36910}, {22381, 27527}, {23113, 40528}, {23146, 40505}, {23220, 36795}, {23225, 36796}, {23614, 23984}, {24026, 32660}, {24031, 32674}, {31623, 39201}, {32641, 35014}, {32653, 34588}, {32677, 39471}
X(653) = barycentric product X(i)*X(j) for these {i,j}: {1, 18026}, {4, 664}, {7, 1897}, {8, 36118}, {9, 13149}, {19, 4554}, {25, 4572}, {27, 4552}, {29, 4566}, {33, 4569}, {34, 668}, {57, 6335}, {59, 46107}, {65, 811}, {69, 36127}, {73, 6528}, {75, 108}, {76, 32674}, {85, 1783}, {92, 651}, {99, 225}, {100, 273}, {101, 331}, {107, 307}, {109, 264}, {112, 349}, {158, 6516}, {162, 1441}, {190, 278}, {196, 44327}, {226, 648}, {281, 658}, {286, 4551}, {312, 32714}, {318, 934}, {342, 13138}, {514, 46102}, {521, 24032}, {608, 1978}, {644, 1847}, {646, 1435}, {655, 17923}, {662, 40149}, {666, 5236}, {693, 7012}, {799, 1880}, {823, 1214}, {883, 36124}, {927, 1861}, {1020, 31623}, {1093, 6517}, {1118, 4561}, {1119, 3699}, {1231, 24019}, {1262, 46110}, {1275, 3064}, {1305, 5125}, {1309, 22464}, {1395, 6386}, {1396, 4033}, {1400, 6331}, {1414, 41013}, {1415, 1969}, {1426, 7257}, {1461, 7017}, {1813, 2052}, {1824, 4625}, {1826, 4573}, {1848, 6648}, {1870, 35174}, {1874, 4589}, {1877, 4555}, {1952, 1981}, {2356, 46135}, {2501, 4620}, {3112, 46152}, {3257, 37790}, {3261, 7115}, {3262, 36110}, {3668, 36797}, {3676, 15742}, {4077, 5379}, {4242, 18815}, {4391, 7128}, {4467, 34922}, {4559, 44129}, {4564, 17924}, {4605, 46103}, {4610, 8736}, {4617, 7101}, {4626, 7046}, {4998, 7649}, {5089, 34085}, {5307, 32038}, {6063, 8750}, {6332, 23984}, {6742, 7282}, {7035, 43923}, {7045, 44426}, {7079, 36838}, {8680, 41207}, {9394, 39060}, {10015, 39294}, {15352, 40152}, {16082, 24029}, {17906, 40420}, {17985, 35154}, {18027, 32660}, {18097, 41676}, {18816, 23706}, {21272, 40446}, {23710, 35157}, {23987, 34393}, {24033, 35518}, {24035, 36100}, {26705, 33298}, {28017, 42384}, {35516, 36067}, {36049, 40701}, {36146, 46108}, {37139, 37805}, {40117, 40702}, {46104, 46153}
X(653) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 521}, {2, 6332}, {4, 522}, {6, 652}, {7, 4025}, {12, 4064}, {19, 650}, {25, 663}, {27, 4560}, {28, 3737}, {29, 7253}, {31, 1946}, {33, 3900}, {34, 513}, {37, 8611}, {44, 14418}, {48, 36054}, {56, 1459}, {57, 905}, {58, 23189}, {59, 1331}, {65, 656}, {73, 520}, {75, 35518}, {77, 4131}, {85, 15413}, {92, 4391}, {99, 332}, {100, 78}, {101, 219}, {104, 37628}, {105, 23696}, {107, 29}, {108, 1}, {109, 3}, {110, 283}, {112, 284}, {158, 44426}, {162, 21}, {163, 2193}, {190, 345}, {196, 14837}, {198, 10397}, {208, 6129}, {222, 4091}, {225, 523}, {226, 525}, {242, 3716}, {250, 4636}, {264, 35519}, {273, 693}, {278, 514}, {281, 3239}, {284, 23090}, {286, 18155}, {307, 3265}, {312, 15416}, {318, 4397}, {331, 3261}, {333, 15411}, {342, 17896}, {348, 30805}, {349, 3267}, {388, 23874}, {393, 3064}, {468, 14432}, {513, 7004}, {514, 26932}, {515, 39471}, {521, 24031}, {522, 2968}, {603, 23224}, {604, 22383}, {607, 657}, {608, 649}, {643, 1792}, {644, 3692}, {648, 333}, {649, 7117}, {650, 34591}, {651, 63}, {652, 35072}, {658, 348}, {662, 1812}, {663, 3270}, {664, 69}, {668, 3718}, {692, 212}, {693, 17880}, {765, 4571}, {811, 314}, {823, 31623}, {906, 2289}, {927, 31637}, {933, 35196}, {934, 77}, {1018, 3694}, {1020, 1214}, {1025, 25083}, {1096, 18344}, {1108, 40628}, {1118, 7649}, {1119, 3676}, {1155, 14414}, {1172, 1021}, {1214, 24018}, {1249, 14331}, {1252, 4587}, {1262, 1813}, {1331, 1259}, {1332, 3719}, {1365, 21134}, {1395, 667}, {1396, 1019}, {1398, 43924}, {1399, 23226}, {1400, 647}, {1402, 810}, {1403, 22090}, {1404, 22086}, {1409, 822}, {1412, 7254}, {1414, 1444}, {1415, 48}, {1423, 25098}, {1426, 4017}, {1428, 22384}, {1434, 15419}, {1435, 3669}, {1441, 14208}, {1445, 24562}, {1457, 8677}, {1459, 1364}, {1461, 222}, {1474, 7252}, {1633, 1040}, {1769, 35014}, {1783, 9}, {1785, 2804}, {1788, 20315}, {1813, 394}, {1824, 4041}, {1826, 3700}, {1827, 6608}, {1828, 6615}, {1829, 17420}, {1839, 4976}, {1840, 4140}, {1846, 23757}, {1847, 24002}, {1848, 3910}, {1870, 3738}, {1874, 4010}, {1875, 1769}, {1876, 2254}, {1877, 900}, {1880, 661}, {1893, 4804}, {1897, 8}, {1940, 8062}, {1946, 2638}, {1947, 17899}, {1950, 22382}, {1973, 3063}, {1981, 1944}, {1990, 14400}, {2052, 46110}, {2149, 906}, {2173, 14395}, {2201, 4435}, {2212, 8641}, {2222, 1807}, {2283, 1818}, {2285, 2522}, {2299, 21789}, {2331, 14298}, {2333, 3709}, {2356, 926}, {2501, 21044}, {2637, 33572}, {2638, 23614}, {2701, 17973}, {2720, 1795}, {2969, 21132}, {3064, 1146}, {3176, 14302}, {3186, 25128}, {3476, 9031}, {3668, 17094}, {3669, 3942}, {3676, 1565}, {3699, 1265}, {3732, 27509}, {3939, 1260}, {3952, 3710}, {4017, 18210}, {4064, 7068}, {4185, 17418}, {4186, 42312}, {4242, 4511}, {4551, 72}, {4552, 306}, {4554, 304}, {4557, 2318}, {4558, 6514}, {4559, 71}, {4561, 1264}, {4564, 1332}, {4565, 1790}, {4566, 307}, {4569, 7182}, {4572, 305}, {4573, 17206}, {4605, 26942}, {4617, 7177}, {4619, 44717}, {4620, 4563}, {4626, 7056}, {4998, 4561}, {5125, 20294}, {5236, 918}, {5307, 23880}, {5342, 4811}, {5379, 643}, {5546, 2327}, {5930, 8057}, {6198, 35057}, {6331, 28660}, {6332, 23983}, {6335, 312}, {6516, 326}, {6517, 3964}, {6528, 44130}, {6529, 8748}, {6558, 30681}, {6591, 2170}, {6614, 7053}, {7009, 3907}, {7012, 100}, {7045, 6516}, {7046, 4163}, {7071, 4105}, {7079, 4130}, {7115, 101}, {7119, 3287}, {7128, 651}, {7178, 4466}, {7192, 17219}, {7282, 4467}, {7649, 11}, {7952, 8058}, {8058, 7358}, {8059, 1433}, {8687, 2359}, {8735, 42462}, {8736, 4024}, {8748, 17926}, {8750, 55}, {8751, 1024}, {8756, 1639}, {9088, 3478}, {9316, 22091}, {13138, 271}, {13149, 85}, {13486, 1789}, {14257, 21186}, {14331, 40616}, {14776, 2342}, {14837, 16596}, {15439, 1794}, {15742, 3699}, {16609, 24459}, {17094, 17216}, {17906, 3452}, {17923, 3904}, {17924, 4858}, {17925, 17197}, {17985, 2785}, {18026, 75}, {18097, 4580}, {18344, 2310}, {20613, 2509}, {21186, 123}, {21189, 34588}, {21666, 23104}, {21859, 3949}, {23067, 3682}, {23353, 1936}, {23363, 22421}, {23703, 5440}, {23706, 517}, {23710, 6366}, {23845, 22072}, {23890, 6510}, {23979, 32660}, {23981, 22350}, {23985, 32674}, {23987, 515}, {24019, 1172}, {24027, 36059}, {24032, 18026}, {24033, 108}, {26700, 7100}, {26704, 10570}, {29055, 7015}, {32652, 2188}, {32656, 6056}, {32660, 577}, {32667, 32677}, {32669, 14578}, {32674, 6}, {32676, 2194}, {32691, 1036}, {32702, 909}, {32713, 2299}, {32714, 57}, {32727, 34078}, {32735, 36057}, {34922, 6742}, {35349, 42018}, {36037, 1809}, {36040, 36055}, {36044, 1295}, {36049, 268}, {36059, 255}, {36067, 102}, {36075, 22054}, {36076, 3422}, {36082, 1069}, {36093, 26703}, {36098, 1791}, {36099, 2339}, {36106, 45393}, {36110, 104}, {36118, 7}, {36123, 43728}, {36124, 885}, {36125, 23838}, {36126, 1896}, {36127, 4}, {36140, 32726}, {36146, 1814}, {36797, 1043}, {37141, 41081}, {37790, 3762}, {38462, 4768}, {39199, 38983}, {39294, 13136}, {39534, 35015}, {40116, 2338}, {40117, 282}, {40149, 1577}, {40663, 14429}, {40983, 2488}, {40987, 17115}, {41013, 4086}, {41207, 35145}, {41321, 40869}, {42069, 23615}, {42070, 4543}, {42755, 10017}, {43035, 39470}, {43290, 44722}, {43923, 244}, {43924, 3937}, {44327, 44189}, {44426, 24026}, {44696, 21172}, {46102, 190}, {46107, 34387}, {46110, 23978}, {46152, 38}, {46153, 3917}, {46254, 4631}
X(653) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 2636, 2638}, {46, 158, 412}, {65, 1940, 29}, {652, 24032, 2639}, {1118, 1788, 5125}, {1148, 8762, 3}, {1783, 32714, 651}, {2636, 2638, 23707}, {4566, 14543, 651}, {14837, 23984, 39053}, {16596, 40535, 2}
X(653) = cevapoint of X(i) and X(j) for these (i,j): (1,652), (57,514), (65,650), (281,522), (513,1108)
X(653) = X(6)-isoconjugate of X(521)
X(653) = perspector of conic {A,B,C,PU(76)}}


X(654) = CROSSDIFFERENCE OF X(1) AND X(5)

Trilinears    cos(A - B) - cos(C - A) : :
Barycentrics    a^2 (b - c) (a - b - c) (a^2 - b^2 + b c - c^2) : :

X(654) lies on the Terzic circle and these lines: {3, 41155}, {6, 3310}, {9, 1639}, {44, 513}, {50, 647}, {55, 926}, {57, 1638}, {63, 918}, {81, 42744}, {101, 109}, {103, 2291}, {171, 24462}, {182, 41158}, {514, 23737}, {517, 41162}, {812, 40166}, {900, 2161}, {909, 2432}, {1021, 2596}, {1768, 2826}, {1946, 8676}, {2170, 4435}, {2352, 23225}, {2423, 14578}, {2594, 2598}, {2774, 41164}, {2786, 41163}, {2824, 41165}, {3196, 27780}, {3218, 4453}, {3219, 30565}, {3305, 45326}, {3306, 44902}, {3669, 4091}, {3887, 41166}, {4041, 42649}, {4063, 21120}, {4131, 28984}, {4282, 7252}, {4895, 42657}, {4984, 14400}, {5220, 30700}, {6589, 22383}, {8659, 41156}, {14418, 42552}, {17455, 21758}, {20295, 28834}, {22160, 44410}, {25924, 31250}, {32641, 32675}, {35365, 36057}, {36054, 43060}

X(654) = reflection of X(55) in X(6139)
X(654) = reflection of X(654) in the anti-orthic axis
X(654) = isogonal conjugate of X(655)
X(654) = circumcircle-inverse of X(41155)
X(654) = Brocard-circle-inverse of X(41158)
X(654) = X(31)-complementary conjugate of X(38984)
X(654) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 38984}, {54, 41218}, {74, 3270}, {101, 17455}, {110, 215}, {651, 34586}, {655, 1}, {901, 55}, {1983, 2245}, {2250, 11998}, {3065, 2310}, {3461, 41211}, {3466, 2638}, {23838, 663}, {24029, 1319}, {24624, 11}, {32641, 6}, {40215, 3025}
X(654) = X(i)-cross conjugate of X(j) for these (i,j): {2081, 2600}, {41218, 54}
X(654) = cevapoint of X(2081) and X(2624)
X(654) = crosspoint of X(i) and X(j) for these (i,j): {1, 655}, {9, 5548}, {57, 2720}, {100, 2990}, {101, 2316}, {104, 651}, {110, 24624}, {275, 1309}, {901, 40215}, {1983, 4282}, {3738, 3960}, {3904, 44428}, {4560, 43728}
X(654) = crosssum of X(i) and X(j) for these (i,j): {1, 654}, {6, 39200}, {9, 2804}, {57, 30725}, {216, 8677}, {513, 8609}, {514, 3911}, {517, 650}, {523, 2245}, {2599, 2600}, {4559, 23981}
X(654) = crossdifference of every pair of points on line {1, 5}
X(654) = perspector of hyperbola {A,B,C,X(1),X(36)}}
X(654) = bicentric difference of PU(68)
X(654) = PU(68)-harmonic conjugate of X(2594) X(654) = X(i)-isoconjugate of X(j) for these (i,j): {1, 655}, {2, 2222}, {6, 35174}, {12, 37140}, {56, 36804}, {75, 32675}, {80, 651}, {100, 2006}, {101, 18815}, {109, 18359}, {190, 1411}, {476, 16577}, {653, 1807}, {664, 2161}, {759, 4552}, {901, 14628}, {934, 36910}, {1020, 6740}, {1415, 20566}, {2341, 4566}, {2594, 32680}, {3257, 14584}, {3738, 23592}, {4551, 24624}, {4554, 6187}, {4559, 14616}, {4565, 15065}, {4573, 34857}, {6358, 36069}, {14213, 36078}, {21741, 35139}, {24029, 40437}, {26700, 41226}, {32671, 34388}, {32678, 40999}, {34242, 44765}, {35307, 39277}, {37222, 37630}
X(654) = barycentric product X(i)*X(j) for these {i,j}: {1, 3738}, {3, 44428}, {6, 3904}, {9, 3960}, {36, 522}, {54, 6369}, {55, 4453}, {60, 6370}, {75, 8648}, {214, 23838}, {261, 42666}, {284, 4707}, {312, 21758}, {318, 22379}, {320, 663}, {333, 21828}, {513, 4511}, {514, 2323}, {521, 1870}, {526, 3615}, {649, 32851}, {650, 3218}, {652, 17923}, {655, 35128}, {656, 17515}, {657, 17078}, {693, 2361}, {758, 3737}, {860, 23189}, {1021, 18593}, {1443, 3900}, {1459, 5081}, {1464, 7253}, {1577, 4282}, {1639, 40215}, {1845, 37628}, {1983, 4858}, {2167, 2600}, {2170, 4585}, {2185, 2610}, {2245, 4560}, {2364, 23884}, {3063, 20924}, {3064, 22128}, {3724, 18155}, {3936, 7252}, {3939, 4089}, {4242, 7004}, {4391, 7113}, {4768, 16944}, {21012, 39178}, {21789, 41804}, {34586, 43728}, {35011, 45950}
X(654) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 35174}, {6, 655}, {9, 36804}, {31, 2222}, {32, 32675}, {36, 664}, {320, 4572}, {513, 18815}, {522, 20566}, {526, 40999}, {649, 2006}, {650, 18359}, {657, 36910}, {663, 80}, {667, 1411}, {1443, 4569}, {1464, 4566}, {1635, 14628}, {1870, 18026}, {1946, 1807}, {1960, 14584}, {1983, 4564}, {2150, 37140}, {2245, 4552}, {2323, 190}, {2361, 100}, {2600, 14213}, {2610, 6358}, {2624, 16577}, {3025, 4453}, {3063, 2161}, {3218, 4554}, {3615, 35139}, {3724, 4551}, {3737, 14616}, {3738, 75}, {3904, 76}, {3960, 85}, {4041, 15065}, {4282, 662}, {4453, 6063}, {4511, 668}, {4707, 349}, {6369, 311}, {6370, 34388}, {7113, 651}, {7252, 24624}, {8648, 1}, {9404, 41226}, {14270, 2594}, {17515, 811}, {21758, 57}, {21789, 6740}, {21828, 226}, {22379, 77}, {32675, 23592}, {32851, 1978}, {34544, 4585}, {35128, 3904}, {41218, 6369}, {42666, 12}, {44428, 264}
X(654) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {649, 652, 650}, {650, 652, 9404}, {650, 4790, 13401}, {2590, 2591, 650}, {2610, 2624, 2245}, {3310, 22086, 6}, {4394, 14298, 650}


X(655) = TRILINEAR POLE OF LINE X(1)X(5)

Trilinears    1/[cos(A - B) - cos(C- A)] : :
Barycentrics    (a - b)*(a - c)*(a + b - c)*(a - b + c)*(a^2 - a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2) : :
X(655) = 3 X[2] - 4 X[40536]

X(655) lies on the curve CC9 and these lines: {2, 40536}, {7, 37131}, {54, 45238}, {57, 14628}, {59, 523}, {80, 516}, {88, 2006}, {100, 522}, {162, 7012}, {190, 4391}, {476, 15439}, {514, 651}, {517, 40437}, {527, 36101}, {653, 17924}, {658, 24002}, {662, 4552}, {664, 4604}, {673, 2161}, {885, 36086}, {908, 36100}, {925, 36078}, {1020, 38340}, {1025, 37143}, {1087, 3460}, {1090, 2957}, {1155, 14204}, {1411, 4318}, {1807, 3100}, {2349, 17484}, {2401, 2406}, {2595, 2597}, {3218, 18359}, {3322, 4542}, {3904, 13136}, {4581, 36098}, {6740, 37142}, {7672, 36815}, {10015, 23592}, {10030, 37130}, {11041, 34232}, {16548, 21368}, {30379, 43760}, {37214, 40704}

X(655) = isogonal conjugate of X(654)
X(655) = isotomic conjugate of X(3904)
X(655) = polar conjugate of X(44428)
X(655) = isotomic conjugate of the isogonal conjugate of X(32675)
X(655) = X(i)-cross conjugate of X(j) for these (i,j): {484, 7045}, {650, 40437}, {654, 1}, {900, 7}, {1983, 6742}, {2245, 59}, {2600, 3615}, {3465, 24032}, {10015, 2}, {16548, 7012}, {23703, 664}, {24029, 651}, {24715, 39293}, {30725, 14628}, {35466, 1016}, {40663, 4998}

X(655) = cevapoint of X(i) and X(j) for these (i,j): {1, 654}, {6, 39200}, {9, 2804}, {57, 30725}, {216, 8677}, {513, 8609}, {514, 3911}, {517, 650}, {523, 2245}, {2599, 2600}, {4559, 23981}
X(655) = crosssum of X(2081) and X(2624)
X(655) = trilinear pole of line {1, 5}
X(655) = X(i)-isoconjugate of X(j) for these (i,j): {1, 654}, {2, 8648}, {6, 3738}, {8, 21758}, {11, 1983}, {21, 21828}, {31, 3904}, {36, 650}, {41, 4453}, {48, 44428}, {54, 2600}, {55, 3960}, {60, 2610}, {281, 22379}, {320, 3063}, {513, 2323}, {514, 2361}, {522, 7113}, {523, 4282}, {647, 17515}, {649, 4511}, {652, 1870}, {657, 1443}, {663, 3218}, {667, 32851}, {758, 7252}, {1021, 1464}, {1639, 16944}, {1835, 23090}, {1946, 17923}, {2148, 6369}, {2150, 6370}, {2185, 42666}, {2194, 4707}, {2222, 35128}, {2245, 3737}, {2432, 11700}, {2624, 3615}, {3271, 4585}, {3724, 4560}, {4242, 7117}, {4895, 40215}, {5081, 22383}, {8641, 17078}, {17455, 23838}, {18344, 22128}, {18593, 21789}, {36110, 38353}
X(655) = barycentric product X(i)*X(j) for these {i,j}: {1, 35174}, {57, 36804}, {75, 2222}, {76, 32675}, {80, 664}, {100, 18815}, {109, 20566}, {190, 2006}, {311, 36078}, {476, 40999}, {651, 18359}, {658, 36910}, {668, 1411}, {1414, 15065}, {1807, 18026}, {2161, 4554}, {2594, 35139}, {3257, 14628}, {3904, 23592}, {4551, 14616}, {4552, 24624}, {4555, 14584}, {4566, 6740}, {4572, 6187}, {4585, 34535}, {4625, 34857}, {6358, 37140}, {16577, 32680}, {34388, 36069}, {35175, 37630}, {38340, 41226}
X(655) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3738}, {2, 3904}, {4, 44428}, {5, 6369}, {6, 654}, {7, 4453}, {12, 6370}, {31, 8648}, {57, 3960}, {80, 522}, {100, 4511}, {101, 2323}, {108, 1870}, {109, 36}, {162, 17515}, {163, 4282}, {181, 42666}, {190, 32851}, {226, 4707}, {476, 3615}, {603, 22379}, {604, 21758}, {651, 3218}, {653, 17923}, {654, 35128}, {658, 17078}, {664, 320}, {692, 2361}, {759, 3737}, {934, 1443}, {1020, 18593}, {1168, 23838}, {1400, 21828}, {1411, 513}, {1415, 7113}, {1807, 521}, {1813, 22128}, {1897, 5081}, {1953, 2600}, {1983, 34544}, {2006, 514}, {2149, 1983}, {2161, 650}, {2171, 2610}, {2222, 1}, {2341, 1021}, {2594, 526}, {3676, 4089}, {4551, 758}, {4552, 3936}, {4554, 20924}, {4559, 2245}, {4564, 4585}, {4566, 41804}, {4572, 40075}, {5219, 23884}, {6187, 663}, {6740, 7253}, {7012, 4242}, {14584, 900}, {14616, 18155}, {14628, 3762}, {15065, 4086}, {16577, 32679}, {18359, 4391}, {18815, 693}, {20566, 35519}, {21741, 2624}, {21859, 4053}, {23703, 214}, {23706, 1845}, {23981, 34586}, {24029, 16586}, {24624, 4560}, {27818, 27836}, {32671, 2150}, {32675, 6}, {34079, 7252}, {34242, 21189}, {34857, 4041}, {35013, 45950}, {35174, 75}, {35308, 21012}, {36061, 1789}, {36069, 60}, {36078, 54}, {36804, 312}, {36815, 3716}, {36910, 3239}, {37140, 2185}, {37630, 2802}, {39200, 38984}, {40172, 4895}, {40437, 43728}, {40999, 3268}
X(655) = {X(2595),X(2599)}-harmonic conjugate of X(3615)


X(656) = CROSSDIFFERENCE OF X(1) AND X(19)

Trilinears    tan B - tan C : tan C - tan A : tan A - tan B
Trilinears    sin 2B - sin 2C : :
Trilinears    cos A sin(B - C) : :
Trilinears    b cos(C - A) - c cos(B - A) : :
Trilinears    (b^2 - c^2)(b^2 + c^2 - a^2) : :
Trilinears    d(a,b,c) : : , where d(a,b,c) = directed distance from A to Euler line
Barycentrics    sin A (tan B - tan C) : sin B (tan C - tan A) : sin C (tan A - tan B)
X(656) = 2 X[1734] + X[1769], 3 X[2457] - 2 X[7178], 3 X[2457] - X[23752], 2 X[4036] - 3 X[21052], X[4064] - 3 X[14429], 2 X[7253] - 3 X[45686], 4 X[8062] - 3 X[45686], 3 X[11125] - 2 X[44409], 3 X[19875] - 2 X[45660], 3 X[41800] - X[44409]>

X(656) lies on these lines: {1, 7629}, {2, 7253}, {3, 23189}, {10, 4086}, {11, 38981}, {19, 8768}, {37, 4171}, {44, 513}, {48, 9253}, {63, 14209}, {65, 42768}, {69, 15419}, {71, 10099}, {75, 17893}, {100, 39026}, {101, 2722}, {108, 2762}, {109, 2766}, {115, 35091}, {122, 35580}, {125, 7004}, {162, 2633}, {201, 6368}, {240, 522}, {244, 8286}, {407, 42755}, {514, 23800}, {521, 810}, {523, 2457}, {525, 4064}, {647, 8611}, {662, 1101}, {663, 15313}, {667, 832}, {676, 6608}, {900, 6615}, {1021, 16612}, {1834, 23757}, {1924, 19559}, {1953, 9249}, {1955, 2616}, {2092, 3126}, {2310, 8287}, {2345, 4529}, {2523, 22383}, {2530, 6371}, {2605, 8674}, {2629, 9390}, {2972, 39004}, {3122, 21945}, {3139, 35015}, {3269, 16573}, {3667, 4129}, {3700, 6587}, {3709, 24290}, {3733, 5096}, {3737, 5127}, {3776, 13258}, {3837, 28116}, {3900, 6129}, {3914, 42767}, {3942, 20975}, {4010, 23301}, {4019, 4580}, {4025, 14208}, {4036, 21052}, {4122, 23954}, {4132, 4729}, {4139, 4730}, {4140, 21958}, {4151, 4815}, {4367, 38469}, {4391, 20316}, {4524, 7250}, {4575, 36061}, {4705, 8672}, {4778, 4905}, {4804, 30591}, {4811, 25009}, {4985, 8714}, {5224, 18160}, {5903, 35050}, {6332, 20315}, {6739, 24028}, {7254, 20746}, {7658, 21172}, {8057, 17094}, {9001, 43924}, {10015, 21102}, {11125, 41800}, {14077, 14353}, {14192, 37305}, {14288, 44316}, {14315, 28183}, {17496, 20293}, {18155, 24718}, {19875, 45660}, {21912, 22069}, {22084, 26932}, {22091, 23224}, {22342, 23286}, {22379, 23187}, {23738, 28195}, {24031, 34846}, {24410, 37041}, {24457, 28221}, {24893, 24921}, {25299, 27710}, {25440, 36033}, {30765, 42327}, {35069, 35116}, {35200, 36062}, {36031, 40577}, {36036, 46254}, {37154, 43728}, {40952, 42758}

X(656) = reflection of X(i) in X(j) for these {i,j}: {1459, 905}, {1769, 21189}, {2517, 17072}, {2605, 31947}, {3737, 14838}, {4086, 10}, {4391, 20316}, {4804, 30591}, {6332, 20315}, {7253, 8062}, {7649, 14837}, {11125, 41800}, {14288, 44316}, {21102, 10015}, {21172, 7658}, {23752, 7178}, {45686, 2}
X(656) = midpoint of X(i) and X(j) for these {i,j}: {650, 7655}, {1734, 21189}, {2254, 17420}, {4017, 4041}, {4524, 7250}, {17496, 20293}
X(656) = isogonal conjugate of X(162)
X(656) = isotomic conjugate of X(811)
X(656) = complement of X(7253)
X(656) = anticomplement of X(8062)
X(656) = polar conjugate of X(823)
X(656) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {2713, 14956}, {7016, 34188}, {7105, 33650}, {7106, 37781}
X(656) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 34591}, {42, 5514}, {56, 34589}, {65, 124}, {73, 123}, {108, 34831}, {109, 960}, {213, 13609}, {228, 40616}, {603, 34588}, {604, 4858}, {651, 21246}, {658, 21240}, {934, 3741}, {1020, 141}, {1042, 11}, {1106, 244}, {1254, 125}, {1262, 4369}, {1275, 42327}, {1400, 26932}, {1402, 1146}, {1407, 17761}, {1409, 16596}, {1410, 2968}, {1415, 5745}, {1425, 34846}, {1427, 116}, {1461, 3739}, {1918, 35508}, {3122, 34530}, {3668, 21252}, {4017, 46100}, {4551, 1329}, {4552, 21244}, {4559, 3452}, {4565, 21233}, {4566, 2887}, {4605, 21245}, {4617, 17050}, {6354, 21253}, {6614, 3742}, {7045, 512}, {7128, 30476}, {7138, 122}, {7143, 8286}, {23067, 34823}, {23979, 14838}, {24027, 523}, {24033, 520}, {32674, 6708}, {32714, 34830}, {36059, 34851}, {37755, 127}
X(656) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 2632}, {2, 34591}, {3, 7004}, {63, 3708}, {69, 3942}, {72, 18210}, {75, 20902}, {90, 2611}, {91, 1109}, {92, 37754}, {100, 18673}, {109, 2292}, {162, 1}, {163, 38}, {190, 18674}, {307, 4466}, {521, 520}, {522, 523}, {525, 8611}, {651, 18675}, {653, 2294}, {662, 48}, {799, 18671}, {811, 6508}, {823, 1953}, {905, 647}, {1018, 17441}, {1020, 37}, {1332, 3958}, {1577, 661}, {1751, 2170}, {4025, 525}, {4551, 73}, {4566, 37755}, {4575, 44706}, {4592, 63}, {7093, 4128}, {7094, 21340}, {8769, 2643}, {14208, 24018}, {14837, 6587}, {17072, 23301}, {23067, 201}, {23604, 3120}, {36034, 1725}, {36036, 1959}, {36037, 758}, {36048, 25080}, {36050, 2650}, {36084, 8766}, {39946, 17476}, {43724, 1364}
X(656) = X(i)-cross conjugate of X(j) for these (i,j): {1, 9392}, {125, 201}, {810, 661}, {822, 24018}, {2632, 1}, {2643, 1820}, {3269, 37755}, {3708, 63}, {16573, 2}, {18210, 72}, {20975, 3949}, {22094, 3}, {37754, 92}
X(656) = cevapoint of X(i) and X(j) for these (i,j): {1, 2629}, {3, 22156}, {810, 822}
X(656) = crosspoint of X(i) and X(j) for these (i,j): {1, 162}, {2, 4566}, {3, 23067}, {10, 4551}, {63, 4592}, {75, 662}, {100, 1257}, {521, 522}, {525, 17094}, {823, 40440}, {905, 4025}, {1020, 1439}, {1331, 1794}, {1577, 14208}, {1822, 1823}, {2169, 4575}
X(656) = crosssum of X(i) and X(j) for these (i,j): {1, 656}, {2, 17498}, {6, 21789}, {25, 43925}, {31, 661}, {58, 3737}, {81, 16751}, {108, 109}, {163, 32676}, {513, 1104}, {652, 14547}, {663, 1195}, {1021, 4183}, {1577, 20883}, {1769, 1845}, {1783, 8750}, {1838, 7649}, {2355, 6591}, {2588, 2589}, {2631, 42074}
X(656) = trilinear pole of line {2631, 2632}
X(656) = crossdifference of every pair of points on line {1, 19}
X(656) = bicentric difference of PU(i) for i in (21, 22, 74, 127)
X(656) = PU(21)-harmonic conjugate of X(1953)
X(656) = PU(22)-harmonic conjugate of X(48)
X(656) = PU(74)-harmonic conjugate of X(2173)
X(656) = trilinear product of PU(75)
X(656) = trilinear pole of PU(75) (line X(2631)X(2632))
X(656) = PU(127)-harmonic conjugate of X(19)
X(656) = perspector of hyperbola {A,B,C,X(1),X(63)}
X(656) = intersection of trilinear polars of X(1) and X(63)
X(656) = pole wrt polar circle of trilinear polar of X(823) (line X(1)X(29))
X(656) = X(48)-isoconjugate (polar conjugate) of X(823)
X(656) = X(92)-isoconjugate of X(163)
X(656) = X(6)-isoconjugate of X(648)
X(656) = center of conic {A,B,C,X(1021),X(2765)} (the isogonal conjugate of line X(108)X(109))
X(656) = trilinear square root of X(2632)
X(656) = barycentric square root of X(3269)
X(656) = circle-{X(11),X(36),X(65)}}-inverse of X(36035)
X(656) = {X(2588),X(2599)}-harmonic conjugate of X(36035)
X(656) = X(i)-isoconjugate of X(j) for these (i,j): {1, 162}, {2, 112}, {3, 107}, {4, 110}, {5, 933}, {6, 648}, {15, 36306}, {16, 36309}, {19, 662}, {20, 1301}, {21, 108}, {22, 1289}, {23, 935}, {24, 925}, {25, 99}, {26, 1288}, {27, 101}, {28, 100}, {29, 109}, {30, 1304}, {31, 811}, {32, 6331}, {33, 1414}, {34, 643}, {39, 42396}, {48, 823}, {51, 18831}, {53, 18315}, {54, 35360}, {56, 36797}, {58, 1897}, {63, 24019}, {69, 32713}, {74, 4240}, {75, 32676}, {81, 1783}, {83, 35325}, {86, 8750}, {92, 163}, {94, 14591}, {98, 4230}, {102, 7452}, {103, 4241}, {104, 4246}, {105, 4238}, {111, 4235}, {158, 4575}, {184, 6528}, {186, 476}, {190, 1474}, {216, 16813}, {217, 42405}, {225, 4636}, {230, 32697}, {232, 2966}, {237, 22456}, {240, 36084}, {249, 2501}, {250, 523}, {251, 41676}, {255, 36126}, {264, 1576}, {270, 4551}, {275, 1625}, {278, 5546}, {281, 4565}, {283, 36127}, {284, 653}, {286, 692}, {288, 35318}, {297, 2715}, {317, 32734}, {324, 14586}, {325, 32696}, {333, 32674}, {340, 14560}, {376, 9064}, {378, 1302}, {382, 33640}, {393, 4558}, {394, 6529}, {403, 10420}, {405, 36077}, {415, 2701}, {419, 805}, {422, 2703}, {423, 2702}, {425, 2714}, {427, 827}, {428, 7953}, {430, 6578}, {436, 1303}, {441, 32687}, {450, 2713}, {458, 26714}, {460, 10425}, {461, 5545}, {462, 10410}, {463, 10409}, {467, 32692}, {468, 691}, {470, 5995}, {471, 5994}, {472, 16807}, {473, 16806}, {477, 7480}, {511, 685}, {512, 18020}, {513, 5379}, {520, 32230}, {525, 23964}, {571, 30450}, {577, 15352}, {607, 4573}, {608, 645}, {644, 1396}, {647, 23582}, {651, 1172}, {656, 24000}, {658, 2332}, {664, 2299}, {668, 2203}, {670, 1974}, {675, 4249}, {687, 3003}, {689, 27369}, {759, 4242}, {798, 46254}, {799, 1973}, {810, 23999}, {813, 31905}, {825, 31909}, {833, 17520}, {842, 7473}, {856, 36068}, {857, 36071}, {858, 10423}, {859, 1309}, {860, 36069}, {862, 36066}, {877, 1976}, {892, 44102}, {901, 37168}, {907, 6995}, {915, 3658}, {917, 4243}, {919, 15149}, {927, 37908}, {930, 3518}, {931, 4185}, {934, 4183}, {1010, 32691}, {1016, 43925}, {1020, 2326}, {1021, 7128}, {1096, 4592}, {1101, 24006}, {1105, 1624}, {1113, 1114}, {1173, 35311}, {1176, 46151}, {1235, 4630}, {1252, 17925}, {1262, 17926}, {1286, 21213}, {1287, 21284}, {1290, 2074}, {1291, 37943}, {1292, 4233}, {1293, 4248}, {1295, 7435}, {1296, 4232}, {1297, 2409}, {1299, 30512}, {1300, 15329}, {1306, 5200}, {1310, 4206}, {1311, 7463}, {1325, 2766}, {1331, 8747}, {1332, 5317}, {1333, 6335}, {1370, 39417}, {1395, 7257}, {1398, 7256}, {1415, 31623}, {1435, 7259}, {1461, 2322}, {1494, 23347}, {1495, 16077}, {1503, 44770}, {1585, 39383}, {1586, 39384}, {1634, 32085}, {1725, 36114}, {1748, 36145}, {1784, 36034}, {1813, 8748}, {1817, 40117}, {1822, 2587}, {1823, 2586}, {1826, 4556}, {1839, 4629}, {1843, 4577}, {1880, 4612}, {1884, 6083}, {1896, 36059}, {1951, 41207}, {1959, 36104}, {1968, 43188}, {1981, 2249}, {1989, 14590}, {1990, 44769}, {1995, 30247}, {2052, 32661}, {2071, 22239}, {2073, 2690}, {2075, 2689}, {2159, 24001}, {2165, 41679}, {2189, 4552}, {2190, 2617}, {2194, 18026}, {2201, 4584}, {2202, 41206}, {2204, 4554}, {2207, 4563}, {2211, 43187}, {2212, 4625}, {2222, 17515}, {2287, 32714}, {2303, 36099}, {2328, 36118}, {2333, 4610}, {2355, 4596}, {2374, 11634}, {2407, 8749}, {2420, 16080}, {2421, 6531}, {2445, 35140}, {2485, 44183}, {2489, 4590}, {2491, 41174}, {2576, 2581}, {2577, 2580}, {2633, 9390}, {2687, 37966}, {2691, 37963}, {2693, 31510}, {2696, 37962}, {2697, 37937}, {2752, 7476}, {2770, 7482}, {2881, 39297}, {2965, 38342}, {2967, 41173}, {2971, 31614}, {3135, 39418}, {3146, 44060}, {3163, 34568}, {3168, 44828}, {3172, 44326}, {3194, 13138}, {3222, 11325}, {3260, 32715}, {3267, 41937}, {3284, 15459}, {3447, 30716}, {3542, 13398}, {3559, 36082}, {3563, 4226}, {3565, 6353}, {3580, 32708}, {3733, 15742}, {3737, 7012}, {4143, 23975}, {4184, 26705}, {4186, 8690}, {4221, 9107}, {4222, 34594}, {4225, 26704}, {4227, 9058}, {4228, 26706}, {4234, 9088}, {4236, 15344}, {4237, 9085}, {4244, 26703}, {4247, 9059}, {4559, 46103}, {4560, 7115}, {4567, 6591}, {4570, 7649}, {4574, 36419}, {4591, 8756}, {4599, 17442}, {4603, 7119}, {4609, 44162}, {4611, 13854}, {4614, 5338}, {4616, 7071}, {4628, 17171}, {4637, 7079}, {5064, 7954}, {5094, 11636}, {5095, 34574}, {5467, 17983}, {5468, 8753}, {5649, 6103}, {6011, 13739}, {6149, 36129}, {6198, 13486}, {6530, 43754}, {6620, 35575}, {7252, 46102}, {7419, 32704}, {7426, 10098}, {7431, 9057}, {7436, 9056}, {7450, 32706}, {7468, 40118}, {7469, 10101}, {7471, 32710}, {7472, 40119}, {7488, 20626}, {7493, 39382}, {8057, 15384}, {8105, 39298}, {8106, 39299}, {8541, 35138}, {8652, 31902}, {8693, 31926}, {8694, 31903}, {8701, 31900}, {8737, 17402}, {8738, 17403}, {8739, 23895}, {8740, 23896}, {8743, 44766}, {8744, 17708}, {8766, 36092}, {8772, 36105}, {8882, 14570}, {8884, 23181}, {9060, 10295}, {9070, 14015}, {9091, 15014}, {9409, 42308}, {10096, 13863}, {10301, 12074}, {10312, 11794}, {10411, 18384}, {10594, 43351}, {10641, 32037}, {10642, 32036}, {11064, 32695}, {11101, 30250}, {11103, 36076}, {11107, 26700}, {11413, 30249}, {12030, 37964}, {12092, 16868}, {13366, 33513}, {13397, 30733}, {13450, 15958}, {13573, 40596}, {13589, 39439}, {13619, 16166}, {13621, 30248}, {14004, 43076}, {14006, 29055}, {14013, 28847}, {14052, 33803}, {14165, 32662}, {14206, 36131}, {14574, 18022}, {14587, 23290}, {14618, 23357}, {14776, 17139}, {14910, 16237}, {14953, 40116}, {14966, 16081}, {15164, 44124}, {15165, 44123}, {15388, 33294}, {15411, 23985}, {16049, 40097}, {17562, 43356}, {17569, 30664}, {17927, 17940}, {17932, 34854}, {17938, 17984}, {17939, 17987}, {17941, 17980}, {17943, 17982}, {17944, 17981}, {18669, 36095}, {18829, 44089}, {18878, 44084}, {19118, 35136}, {20031, 36212}, {20189, 34484}, {20883, 34072}, {23090, 23984}, {23353, 37142}, {23889, 36128}, {23997, 36120}, {26283, 30251}, {28196, 31901}, {28841, 31904}, {28856, 31914}, {28883, 31908}, {28891, 31921}, {30554, 31916}, {30555, 31925}, {30737, 32649}, {31912, 43077}, {32002, 32737}, {32320, 34538}, {32640, 46106}, {32646, 44436}, {32693, 44734}, {32729, 44146}, {32738, 44134}, {32739, 44129}, {34211, 43717}, {34397, 35139}, {35137, 44091}, {35188, 37855}, {35278, 40801}, {35324, 39284}, {35329, 38428}, {35330, 38427}, {36049, 41083}, {36841, 41489}, {38340, 41502}, {38936, 41512}, {39176, 39290}, {41363, 44767}, {41678, 41890}, {42658, 44181}, {44077, 46134}
X(656) = barycentric product X(i)*X(j) for these {i,j}: {1, 525}, {3, 1577}, {4, 24018}, {6, 14208}, {7, 8611}, {9, 17094}, {10, 905}, {19, 3265}, {31, 3267}, {37, 4025}, {38, 4580}, {42, 15413}, {48, 850}, {63, 523}, {65, 6332}, {69, 661}, {71, 693}, {72, 514}, {73, 4391}, {75, 647}, {76, 810}, {77, 3700}, {78, 7178}, {81, 4064}, {82, 2525}, {88, 14429}, {92, 520}, {97, 2618}, {99, 3708}, {100, 4466}, {110, 20902}, {112, 17879}, {115, 4592}, {125, 662}, {162, 15526}, {163, 339}, {184, 20948}, {190, 18210}, {201, 4560}, {204, 14638}, {219, 4077}, {222, 4086}, {226, 521}, {228, 3261}, {255, 14618}, {264, 822}, {265, 32679}, {291, 24459}, {293, 2799}, {304, 512}, {305, 798}, {306, 513}, {307, 650}, {313, 22383}, {321, 1459}, {326, 2501}, {328, 2624}, {336, 3569}, {337, 21832}, {338, 4575}, {343, 2616}, {345, 4017}, {348, 4041}, {349, 1946}, {394, 24006}, {522, 1214}, {561, 3049}, {648, 2632}, {649, 20336}, {652, 1441}, {663, 1231}, {667, 40071}, {669, 40364}, {684, 1821}, {756, 15419}, {799, 20975}, {811, 3269}, {823, 2972}, {826, 34055}, {878, 46238}, {879, 1959}, {896, 14977}, {897, 14417}, {906, 21207}, {914, 3657}, {923, 45807}, {1018, 1565}, {1019, 3695}, {1020, 2968}, {1021, 6356}, {1042, 15416}, {1073, 17898}, {1089, 7254}, {1096, 4143}, {1109, 4558}, {1111, 4574}, {1254, 15411}, {1265, 7216}, {1331, 16732}, {1332, 3120}, {1400, 35518}, {1409, 35519}, {1439, 3239}, {1444, 4024}, {1494, 2631}, {1581, 24284}, {1725, 15421}, {1783, 17216}, {1790, 4036}, {1791, 21124}, {1796, 30591}, {1799, 8061}, {1807, 4707}, {1814, 4088}, {1820, 6563}, {1824, 30805}, {1826, 4131}, {1910, 6333}, {1924, 40050}, {1969, 39201}, {2166, 8552}, {2167, 6368}, {2169, 18314}, {2173, 34767}, {2181, 15414}, {2184, 8057}, {2197, 18155}, {2200, 40495}, {2292, 15420}, {2312, 2419}, {2318, 24002}, {2349, 9033}, {2524, 18832}, {2574, 2583}, {2575, 2582}, {2578, 22340}, {2579, 22339}, {2584, 2593}, {2585, 2592}, {2623, 18695}, {2642, 30786}, {2643, 4563}, {3125, 4561}, {3504, 20910}, {3669, 3710}, {3676, 3694}, {3682, 17924}, {3690, 7199}, {3709, 7182}, {3718, 7180}, {3737, 26942}, {3912, 10099}, {3916, 31010}, {3917, 18070}, {3937, 4033}, {3942, 3952}, {3949, 7192}, {3958, 4608}, {3990, 46107}, {3998, 7649}, {4049, 5440}, {4091, 41013}, {4171, 7056}, {4551, 26932}, {4552, 7004}, {4559, 17880}, {4566, 34591}, {4567, 21134}, {4705, 17206}, {4858, 23067}, {5486, 14209}, {6149, 14592}, {6334, 36053}, {6358, 23189}, {6390, 23894}, {6516, 21044}, {6528, 37754}, {6529, 24020}, {6587, 19611}, {7100, 7265}, {7253, 37755}, {8766, 43673}, {8767, 39473}, {9247, 44173}, {9255, 30476}, {9289, 17478}, {9390, 38240}, {9391, 35145}, {9392, 39352}, {9409, 33805}, {10097, 14210}, {14206, 14380}, {14213, 23286}, {14919, 36035}, {15412, 44706}, {15455, 22094}, {16186, 32680}, {17219, 21859}, {17434, 40440}, {17896, 41087}, {20571, 30451}, {20769, 35352}, {20879, 39180}, {22341, 46110}, {23616, 24000}, {23800, 40161}, {23994, 32661}, {24290, 31637}, {25098, 42027}, {27832, 44729}, {32676, 36793}, {35200, 41079}, {35522, 36060}, {36036, 41172}, {36037, 42761}, {36119, 41077}, {37220, 42665}, {39469, 46273}, {40152, 44426}
X(656) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 648}, {2, 811}, {3, 662}, {4, 823}, {6, 162}, {9, 36797}, {10, 6335}, {19, 107}, {25, 24019}, {30, 24001}, {31, 112}, {32, 32676}, {37, 1897}, {38, 41676}, {42, 1783}, {47, 41679}, {48, 110}, {63, 99}, {65, 653}, {69, 799}, {71, 100}, {72, 190}, {73, 651}, {75, 6331}, {77, 4573}, {78, 645}, {82, 42396}, {91, 30450}, {92, 6528}, {99, 46254}, {101, 5379}, {112, 24000}, {115, 24006}, {125, 1577}, {158, 15352}, {162, 23582}, {163, 250}, {184, 163}, {201, 4552}, {212, 5546}, {213, 8750}, {216, 2617}, {219, 643}, {222, 1414}, {226, 18026}, {228, 101}, {244, 17925}, {248, 36084}, {255, 4558}, {265, 32680}, {283, 4612}, {287, 36036}, {293, 2966}, {295, 4584}, {296, 41206}, {304, 670}, {305, 4602}, {306, 668}, {307, 4554}, {326, 4563}, {332, 4631}, {336, 43187}, {337, 4639}, {339, 20948}, {345, 7257}, {348, 4625}, {393, 36126}, {394, 4592}, {512, 19}, {513, 27}, {514, 286}, {520, 63}, {521, 333}, {522, 31623}, {523, 92}, {525, 75}, {577, 4575}, {603, 4565}, {647, 1}, {648, 23999}, {649, 28}, {650, 29}, {652, 21}, {654, 17515}, {657, 4183}, {659, 31905}, {661, 4}, {662, 18020}, {663, 1172}, {667, 1474}, {669, 1973}, {672, 4238}, {684, 1959}, {686, 1725}, {693, 44129}, {774, 41678}, {798, 25}, {810, 6}, {822, 3}, {826, 20883}, {850, 1969}, {851, 1981}, {878, 1910}, {879, 1821}, {895, 36085}, {896, 4235}, {905, 86}, {906, 4570}, {910, 4241}, {924, 1748}, {1018, 15742}, {1042, 32714}, {1096, 6529}, {1109, 14618}, {1176, 4599}, {1177, 36095}, {1214, 664}, {1231, 4572}, {1245, 36099}, {1260, 7259}, {1265, 7258}, {1331, 4567}, {1332, 4600}, {1400, 108}, {1402, 32674}, {1409, 109}, {1410, 1461}, {1425, 1020}, {1427, 36118}, {1437, 4556}, {1439, 658}, {1444, 4610}, {1459, 81}, {1491, 31909}, {1562, 17898}, {1565, 7199}, {1577, 264}, {1635, 37168}, {1637, 1784}, {1725, 16237}, {1755, 4230}, {1796, 4596}, {1797, 4622}, {1799, 4593}, {1820, 925}, {1821, 22456}, {1822, 39298}, {1823, 39299}, {1880, 36127}, {1910, 685}, {1919, 2203}, {1924, 1974}, {1937, 41207}, {1946, 284}, {1953, 35360}, {1959, 877}, {1964, 35325}, {1973, 32713}, {1976, 36104}, {1989, 36129}, {2083, 1632}, {2084, 1843}, {2148, 933}, {2153, 36306}, {2154, 36309}, {2155, 1301}, {2156, 1289}, {2157, 935}, {2158, 1288}, {2159, 1304}, {2167, 18831}, {2169, 18315}, {2173, 4240}, {2182, 7452}, {2183, 4246}, {2190, 16813}, {2193, 4636}, {2197, 4551}, {2200, 692}, {2215, 36077}, {2225, 4249}, {2245, 4242}, {2247, 7473}, {2250, 1309}, {2252, 3658}, {2253, 4243}, {2254, 15149}, {2281, 32691}, {2310, 17926}, {2312, 2409}, {2314, 30512}, {2315, 15329}, {2318, 644}, {2349, 16077}, {2351, 36145}, {2357, 40117}, {2395, 36120}, {2433, 36119}, {2451, 1957}, {2484, 4206}, {2489, 1096}, {2501, 158}, {2519, 33781}, {2522, 1010}, {2523, 25526}, {2524, 1740}, {2525, 1930}, {2530, 17171}, {2574, 2581}, {2575, 2580}, {2578, 1114}, {2579, 1113}, {2582, 15165}, {2583, 15164}, {2584, 8116}, {2585, 8115}, {2610, 860}, {2616, 275}, {2618, 324}, {2623, 2190}, {2624, 186}, {2629, 39062}, {2631, 30}, {2632, 525}, {2638, 23090}, {2642, 468}, {2643, 2501}, {2799, 40703}, {2962, 38342}, {2972, 24018}, {2987, 36105}, {3005, 17442}, {3049, 31}, {3063, 2299}, {3064, 1896}, {3120, 17924}, {3122, 6591}, {3123, 17921}, {3125, 7649}, {3248, 43925}, {3265, 304}, {3267, 561}, {3270, 1021}, {3287, 14006}, {3289, 23997}, {3292, 23889}, {3569, 240}, {3657, 37203}, {3668, 13149}, {3682, 1332}, {3690, 1018}, {3692, 7256}, {3694, 3699}, {3695, 4033}, {3700, 318}, {3708, 523}, {3709, 33}, {3710, 646}, {3737, 46103}, {3777, 31917}, {3900, 2322}, {3937, 1019}, {3942, 7192}, {3949, 3952}, {3958, 4427}, {3990, 1331}, {3998, 4561}, {4017, 278}, {4020, 1634}, {4024, 41013}, {4025, 274}, {4041, 281}, {4055, 906}, {4064, 321}, {4077, 331}, {4079, 1824}, {4086, 7017}, {4088, 46108}, {4091, 1444}, {4120, 38462}, {4131, 17206}, {4171, 7046}, {4391, 44130}, {4394, 4248}, {4435, 14024}, {4455, 2201}, {4466, 693}, {4516, 3064}, {4524, 7079}, {4551, 46102}, {4558, 24041}, {4559, 7012}, {4561, 4601}, {4563, 24037}, {4574, 765}, {4575, 249}, {4580, 3112}, {4592, 4590}, {4642, 17906}, {4705, 1826}, {4724, 31926}, {4730, 8756}, {4782, 31912}, {4784, 31904}, {4790, 31903}, {4813, 31902}, {4832, 5338}, {4841, 5342}, {4979, 31900}, {4983, 1839}, {5486, 37217}, {5489, 20902}, {6129, 41083}, {6149, 14590}, {6332, 314}, {6333, 46238}, {6368, 14213}, {6390, 24039}, {6516, 4620}, {6529, 24021}, {6587, 1895}, {6591, 8747}, {6676, 18063}, {7004, 4560}, {7015, 4603}, {7019, 7260}, {7053, 4637}, {7056, 4635}, {7117, 3737}, {7177, 4616}, {7178, 273}, {7180, 34}, {7216, 1119}, {7234, 7119}, {7250, 1435}, {7252, 270}, {7254, 757}, {7767, 18062}, {8057, 18750}, {8061, 427}, {8105, 2587}, {8106, 2586}, {8611, 8}, {8641, 2332}, {8672, 5307}, {8673, 1760}, {8766, 34211}, {9033, 14206}, {9178, 36128}, {9247, 1576}, {9255, 43188}, {9391, 8680}, {9392, 15351}, {9404, 11107}, {9406, 23347}, {9409, 2173}, {9508, 423}, {9517, 16568}, {10097, 897}, {10099, 673}, {10547, 34072}, {14208, 76}, {14209, 11185}, {14220, 36102}, {14380, 2349}, {14401, 1099}, {14417, 14210}, {14429, 4358}, {14533, 36134}, {14582, 2166}, {14908, 36142}, {14910, 36114}, {14977, 46277}, {15412, 40440}, {15413, 310}, {15419, 873}, {15451, 1953}, {15526, 14208}, {16186, 32679}, {16318, 24024}, {16562, 30716}, {16573, 8062}, {16732, 46107}, {16758, 17215}, {16892, 16747}, {17094, 85}, {17206, 4623}, {17216, 15413}, {17418, 44734}, {17434, 44706}, {17438, 35311}, {17441, 3732}, {17442, 46151}, {17478, 9308}, {17879, 3267}, {17898, 15466}, {18070, 46104}, {18187, 21178}, {18210, 514}, {18344, 8748}, {18673, 14543}, {18675, 14544}, {18877, 36034}, {19611, 44326}, {20336, 1978}, {20727, 3888}, {20902, 850}, {20948, 18022}, {20975, 661}, {21044, 44426}, {21046, 4036}, {21107, 3673}, {21134, 16732}, {21789, 2326}, {21828, 1870}, {21832, 242}, {22061, 4579}, {22073, 3909}, {22076, 3882}, {22080, 35342}, {22084, 16751}, {22089, 1958}, {22090, 27644}, {22092, 18792}, {22094, 14838}, {22143, 2644}, {22156, 39054}, {22159, 33760}, {22341, 1813}, {22373, 20981}, {22381, 34071}, {22383, 58}, {22443, 13588}, {23067, 4564}, {23090, 1098}, {23189, 2185}, {23216, 1924}, {23224, 1790}, {23226, 40214}, {23286, 2167}, {23503, 11325}, {23616, 17879}, {23620, 1633}, {23894, 17983}, {23928, 17914}, {24006, 2052}, {24018, 69}, {24019, 32230}, {24020, 4143}, {24031, 15411}, {24284, 1966}, {24290, 1861}, {24459, 350}, {25098, 33296}, {26932, 18155}, {30451, 47}, {30572, 37790}, {30574, 37805}, {32320, 255}, {32661, 1101}, {32676, 23964}, {32679, 340}, {34055, 4577}, {34212, 8767}, {34591, 7253}, {34767, 33805}, {34980, 822}, {35200, 44769}, {35518, 28660}, {36035, 46106}, {36036, 41174}, {36051, 32697}, {36053, 687}, {36054, 283}, {36058, 4591}, {36060, 691}, {36061, 39295}, {36062, 30528}, {36119, 15459}, {36126, 34538}, {36214, 37134}, {37754, 520}, {37755, 4566}, {39201, 48}, {39469, 1755}, {40071, 6386}, {40152, 6516}, {40352, 36131}, {40364, 4609}, {40440, 42405}, {41087, 13138}, {42080, 32320}, {42658, 610}, {42665, 18669}, {42667, 2576}, {42668, 2577}, {42669, 23353}, {42761, 36038}, {43717, 36092}, {43924, 1396}, {44706, 14570}, {46088, 2169}
X(656) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 7253, 8062}, {3, 23189, 23226}, {75, 20948, 17893}, {125, 22094, 7004}, {162, 2633, 24000}, {798, 8061, 661}, {1577, 17898, 24006}, {1577, 24006, 2618}, {2457, 23752, 7178}, {2588, 2589, 36035}, {2618, 36035, 24006}, {2642, 8061, 798}, {3269, 16573, 34591}, {7253, 8062, 45686}, {17898, 24006, 36035}


X(657) = CROSSDIFFERENCE OF X(1) AND X(7)

Trilinears    (1 + cos A)(cos B - cos C) : :
Trilinears    sec^2(B/2) - sec^2(C/2) : :
Trilinears    tan^2(B/2) - tan^2(C/2) : :
Trilinears    a(b - c)(b + c - a)^2 : :
X(657) = 2 X[9] + X[45755], X[649] - 4 X[22108], 2 X[4130] - 3 X[14427], X[4171] + 2 X[4827], X[4171] - 3 X[14427], 2 X[4827] + 3 X[14427], 4 X[14330] - X[42462]

X(657) lies on these lines: 9,522   44,513   59,101   663,853

X(657) = midpoint of X(4130) and X(4827)
X(657) = reflection of X(i) in X(j) for these {i,j}: {7, 21195}, {4171, 4130}, {17410, 4394}, {21127, 650}, {23748, 31605}
X(657) = isogonal conjugate of X(658)
X(657) = X(31)-complementary conjugate of X(14714)
X(657) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 24012}, {2, 14714}, {6, 3270}, {9, 2310}, {55, 3022}, {101, 55}, {190, 4319}, {220, 14936}, {282, 2638}, {650, 663}, {651, 2293}, {653, 4336}, {658, 1}, {1021, 3900}, {1783, 42}, {3732, 3938}, {3900, 4105}, {3939, 1253}, {6558, 200}, {7079, 3119}, {8750, 7071}, {21789, 8641}, {32641, 902}, {36049, 31}, {40523, 3688}, {44178, 7004}
X(657) = X(i)-cross conjugate of X(j) for these (i,j): {3022, 55}, {4524, 3900}, {8641, 663}, {10581, 650}, {14936, 220}, {24012, 1}, {35508, 7071}, {36197, 1334}
X(657) = crosspoint of X(i) and X(j) for these (i,j): {1, 658}, {6, 8750}, {9, 3939}, {55, 101}, {200, 6558}, {644, 1261}, {650, 3900}, {651, 2346}, {1021, 21789}, {1783, 4183}, {2287, 4578}
X(657) = crosssum of X(i) and X(j) for these (i,j): {1, 657}, {2, 4025}, {6, 44408}, {7, 514}, {57, 3676}, {77, 4091}, {354, 650}, {513, 40133}, {522, 23058}, {614, 649}, {651, 934}, {652, 20277}, {905, 1439}, {1020, 4566}, {1122, 3669}, {1427, 43932}, {1446, 24002}, {1638, 3321}, {7658, 9533}, {31605, 41785}, {39063, 43042}
X(657) = trilinear pole of line {3022, 14936}
X(657) = crossdifference of every pair of points on line {1, 7}
X(657) = point of concurrence of trilinear polars of vertices of the intangents triangle
X(657) = bicentric difference of PU(104)
X(657) = PU(104)-harmonic conjugate of X(2293)
X(657) = perspector of circumconic through vertices of intangents triangle
X(657) = X(i)-isoconjugate of X(j) for these (i,j): {1, 658}, {2, 934}, {3, 13149}, {6, 4569}, {7, 651}, {8, 4617}, {9, 4626}, {10, 4637}, {37, 4616}, {42, 4635}, {55, 36838}, {56, 4554}, {57, 664}, {59, 24002}, {63, 36118}, {65, 4573}, {69, 32714}, {75, 1461}, {77, 653}, {81, 4566}, {85, 109}, {86, 1020}, {99, 1427}, {100, 279}, {101, 1088}, {103, 24015}, {108, 348}, {110, 1446}, {190, 269}, {222, 18026}, {226, 1414}, {241, 927}, {273, 1813}, {278, 6516}, {312, 6614}, {331, 36059}, {347, 37141}, {479, 644}, {513, 1275}, {514, 7045}, {552, 21859}, {555, 6733}, {604, 4572}, {646, 7023}, {648, 1439}, {655, 1443}, {657, 24011}, {662, 3668}, {666, 34855}, {668, 1407}, {673, 41353}, {693, 1262}, {738, 3699}, {757, 4605}, {799, 1042}, {883, 1462}, {948, 6183}, {1014, 4552}, {1016, 43932}, {1106, 1978}, {1111, 4619}, {1119, 1332}, {1170, 35312}, {1254, 4610}, {1292, 17093}, {1308, 37757}, {1310, 7365}, {1323, 37139}, {1331, 1847}, {1358, 31615}, {1400, 4625}, {1410, 6331}, {1415, 6063}, {1418, 6606}, {1426, 4563}, {1432, 6649}, {1434, 4551}, {1435, 4561}, {1441, 4565}, {1442, 38340}, {1458, 34085}, {1633, 30705}, {1783, 7056}, {1897, 7177}, {1996, 14074}, {2222, 17078}, {2283, 34018}, {3239, 24013}, {3261, 24027}, {3361, 4624}, {3669, 4998}, {3674, 36098}, {3676, 4564}, {3752, 6613}, {3900, 23586}, {3939, 23062}, {4000, 8269}, {4017, 4620}, {4025, 7128}, {4091, 24032}, {4131, 23984}, {4320, 37215}, {4350, 37206}, {4391, 7339}, {4397, 23971}, {4586, 7204}, {4600, 7216}, {4601, 7250}, {4612, 6046}, {4631, 7143}, {5249, 36048}, {6335, 7053}, {6610, 35157}, {6648, 24471}, {7176, 37137}, {7182, 32674}, {7183, 36127}, {7196, 29055}, {8059, 40702}, {9436, 36146}, {10509, 35338}, {13138, 14256}, {14733, 37780}, {17095, 26700}, {18750, 36079}, {22464, 37136}, {23973, 36101}, {23979, 40495}, {24016, 30807}, {24033, 30805}, {32668, 35517}, {32735, 40704}, {35326, 42311}, {37138, 42309}, {37143, 38459}, {38828, 39126}, {40933, 44326}
X(657) = barycentric product X(i)*X(j) for these {i,j}: {1, 3900}, {6, 3239}, {7, 4105}, {8, 663}, {9, 650}, {10, 21789}, {11, 3939}, {21, 4041}, {31, 4397}, {33, 521}, {37, 1021}, {41, 4391}, {42, 7253}, {55, 522}, {56, 4163}, {57, 4130}, {59, 23615}, {71, 17926}, {75, 8641}, {78, 18344}, {81, 4171}, {86, 4524}, {88, 14427}, {100, 2310}, {101, 1146}, {106, 4528}, {109, 4081}, {190, 14936}, {200, 513}, {210, 3737}, {212, 44426}, {219, 3064}, {220, 514}, {244, 4578}, {256, 4477}, {259, 6730}, {281, 652}, {282, 14298}, {284, 3700}, {292, 4148}, {312, 3063}, {318, 1946}, {333, 3709}, {341, 667}, {346, 649}, {480, 3676}, {512, 1043}, {523, 2328}, {525, 2332}, {607, 6332}, {643, 4516}, {644, 2170}, {647, 2322}, {651, 3119}, {654, 36910}, {656, 4183}, {658, 35508}, {661, 2287}, {662, 36197}, {664, 3022}, {665, 6559}, {672, 28132}, {692, 24026}, {693, 1253}, {728, 3669}, {884, 3717}, {885, 2340}, {893, 4529}, {905, 7079}, {926, 14942}, {934, 24010}, {1015, 6558}, {1019, 4515}, {1024, 3693}, {1098, 4705}, {1110, 42455}, {1126, 4990}, {1156, 14392}, {1172, 8611}, {1252, 42462}, {1260, 7649}, {1261, 6615}, {1293, 4953}, {1318, 4543}, {1320, 4895}, {1331, 42069}, {1334, 4560}, {1392, 4959}, {1459, 7046}, {1461, 23970}, {1639, 2316}, {1783, 34591}, {1802, 17924}, {1826, 23090}, {1897, 3270}, {1973, 15416}, {2053, 4147}, {2175, 35519}, {2192, 8058}, {2194, 4086}, {2212, 35518}, {2254, 28071}, {2297, 40137}, {2320, 4814}, {2321, 7252}, {2327, 2501}, {2333, 15411}, {2342, 2804}, {2346, 6608}, {2364, 4944}, {2968, 8750}, {3121, 7258}, {3122, 7256}, {3125, 7259}, {3190, 23289}, {3261, 14827}, {3271, 3699}, {3445, 4546}, {3680, 4162}, {3689, 23838}, {3692, 6591}, {3716, 7077}, {3733, 4082}, {3887, 42064}, {4024, 7054}, {4025, 7071}, {4069, 18191}, {4079, 7058}, {4092, 4636}, {4435, 4876}, {4530, 5548}, {4569, 24012}, {4587, 8735}, {4765, 34820}, {4827, 25430}, {4845, 6366}, {4943, 33963}, {4976, 33635}, {5423, 43924}, {5514, 36049}, {5546, 21044}, {5547, 14432}, {6065, 21132}, {6362, 10482}, {6556, 8643}, {6602, 24002}, {6603, 23893}, {6605, 21127}, {6607, 21453}, {6726, 6728}, {6729, 6731}, {6745, 23351}, {7003, 10397}, {7037, 14302}, {7073, 35057}, {7101, 22383}, {7110, 9404}, {7367, 14837}, {10581, 32008}, {14331, 30457}, {14400, 15627}, {21666, 32656}, {23707, 30692}, {23978, 32739}, {33525, 40435}, {40779, 45755}
X(657) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 4569}, {6, 658}, {8, 4572}, {9, 4554}, {19, 13149}, {21, 4625}, {25, 36118}, {31, 934}, {32, 1461}, {33, 18026}, {41, 651}, {42, 4566}, {55, 664}, {56, 4626}, {57, 36838}, {58, 4616}, {81, 4635}, {101, 1275}, {200, 668}, {212, 6516}, {213, 1020}, {220, 190}, {284, 4573}, {294, 34085}, {341, 6386}, {346, 1978}, {480, 3699}, {512, 3668}, {513, 1088}, {521, 7182}, {522, 6063}, {604, 4617}, {607, 653}, {649, 279}, {650, 85}, {652, 348}, {654, 17078}, {661, 1446}, {663, 7}, {667, 269}, {669, 1042}, {692, 7045}, {728, 646}, {788, 7204}, {798, 1427}, {810, 1439}, {910, 24015}, {926, 9436}, {934, 24011}, {1021, 274}, {1024, 34018}, {1043, 670}, {1098, 4623}, {1146, 3261}, {1253, 100}, {1260, 4561}, {1333, 4637}, {1334, 4552}, {1397, 6614}, {1459, 7056}, {1461, 23586}, {1500, 4605}, {1802, 1332}, {1919, 1407}, {1946, 77}, {1973, 32714}, {1980, 1106}, {2170, 24002}, {2175, 109}, {2194, 1414}, {2195, 927}, {2212, 108}, {2223, 41353}, {2287, 799}, {2293, 35312}, {2310, 693}, {2322, 6331}, {2327, 4563}, {2328, 99}, {2330, 6649}, {2332, 648}, {2340, 883}, {2484, 7365}, {2488, 10481}, {2638, 4131}, {3022, 522}, {3063, 57}, {3064, 331}, {3119, 4391}, {3121, 7216}, {3239, 76}, {3248, 43932}, {3270, 4025}, {3271, 3676}, {3287, 7196}, {3669, 23062}, {3700, 349}, {3709, 226}, {3716, 18033}, {3900, 75}, {3907, 7205}, {3939, 4998}, {4041, 1441}, {4079, 6354}, {4081, 35519}, {4082, 27808}, {4105, 8}, {4130, 312}, {4148, 1921}, {4162, 39126}, {4163, 3596}, {4171, 321}, {4183, 811}, {4391, 20567}, {4397, 561}, {4435, 10030}, {4477, 1909}, {4501, 7243}, {4515, 4033}, {4516, 4077}, {4524, 10}, {4528, 3264}, {4529, 1920}, {4578, 7035}, {4636, 7340}, {4827, 19804}, {4845, 35157}, {4990, 1269}, {5546, 4620}, {6056, 6517}, {6059, 36127}, {6139, 1323}, {6558, 31625}, {6559, 36803}, {6591, 1847}, {6602, 644}, {6607, 4847}, {6608, 20880}, {6729, 555}, {7054, 4610}, {7071, 1897}, {7079, 6335}, {7084, 8269}, {7118, 37141}, {7252, 1434}, {7253, 310}, {7259, 4601}, {7367, 44327}, {8551, 35341}, {8611, 1231}, {8638, 1458}, {8641, 1}, {8642, 4350}, {8645, 38459}, {8646, 4320}, {8648, 1443}, {8653, 3671}, {9404, 17095}, {9447, 1415}, {10482, 6606}, {10581, 142}, {14298, 40702}, {14392, 30806}, {14427, 4358}, {14827, 101}, {14936, 514}, {14942, 46135}, {15416, 40364}, {17115, 3673}, {17425, 6173}, {17926, 44129}, {18344, 273}, {18889, 37139}, {21007, 33765}, {21761, 6359}, {21789, 86}, {22108, 37757}, {22383, 7177}, {23090, 17206}, {23289, 15467}, {23615, 34387}, {23990, 4619}, {24010, 4397}, {24012, 3900}, {24026, 40495}, {28132, 18031}, {30706, 3732}, {32739, 1262}, {33525, 5249}, {33581, 36079}, {34591, 15413}, {34820, 4624}, {35072, 30805}, {35508, 3239}, {35519, 41283}, {36054, 7183}, {36197, 1577}, {39687, 4091}, {42064, 35171}, {42069, 46107}, {42462, 23989}, {43924, 479}
X(657) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 6586, 1459}, {650, 652, 649}, {650, 3287, 17418}, {650, 9404, 652}, {650, 14298, 661}, {665, 20980, 43924}, {798, 2484, 649}, {3063, 3709, 663}, {3063, 33525, 17412}, {3709, 10581, 33525}, {4148, 4529, 4397}, {4171, 14427, 4130}, {4524, 8641, 4105}


X(658) = TRILINEAR POLE OF LINE X(1)X(7)

Trilinears    1/[(1 + cos A)(cos B - cos C)] : :
Trilinears    1/(sec^2(B/2) - sec^2(C/2)) : :
Trilinears    1/(tan^2(B/2) - tan^2(C/2)) : :
Barycentrics    1/((b - c) (a - b - c)^2) : :
X(658) = 3 X[2] - 4 X[40537]

X(658) lies on the curve CC9 and these lines: {1, 24012}, {2, 7056}, {7, 11}, {44, 41351}, {55, 31526}, {57, 673}, {63, 31627}, {64, 45239}, {77, 23707}, {85, 3306}, {86, 1439}, {88, 279}, {100, 664}, {108, 6183}, {109, 927}, {144, 17113}, {162, 1414}, {190, 1020}, {269, 37129}, {273, 43764}, {333, 37202}, {348, 5744}, {354, 9446}, {479, 5435}, {514, 37139}, {651, 1638}, {653, 13149}, {655, 24002}, {657, 24011}, {660, 43932}, {662, 1461}, {799, 4635}, {883, 3699}, {897, 3668}, {899, 41355}, {1042, 37132}, {1043, 40933}, {1097, 3182}, {1155, 14189}, {1262, 36087}, {1275, 3257}, {1407, 20332}, {1427, 37128}, {1434, 1446}, {1445, 23062}, {1447, 34855}, {1565, 13226}, {1821, 7196}, {1847, 37203}, {1897, 4025}, {2349, 17095}, {2898, 3474}, {2968, 18025}, {3218, 37780}, {3599, 10580}, {3732, 24015}, {3817, 10136}, {3911, 37131}, {4417, 7055}, {4551, 37138}, {4552, 4606}, {4605, 37212}, {4998, 43290}, {6172, 36888}, {6614, 36098}, {7013, 33673}, {7070, 45742}, {7182, 14829}, {7183, 40702}, {7339, 36094}, {7658, 23586}, {9057, 24016}, {9312, 17106}, {9358, 21104}, {9778, 31527}, {9812, 36620}, {13257, 28344}, {14837, 24013}, {15252, 45276}, {17277, 37214}, {17352, 30705}, {17605, 42386}, {17728, 30623}, {18240, 24203}, {23587, 23972}, {24009, 39156}, {27834, 27836}, {30988, 40593}, {32040, 44551}, {32714, 36099}, {37208, 41245}, {37650, 41356}, {38285, 43750}, {41798, 44005}

X(658) = reflection of X(13609) in X(40537)
X(658) = isogonal conjugate of X(657)
X(658) = isotomic conjugate of X(3239)
X(658) = anticomplement of X(13609)
X(658) = X(7339)-anticomplementary conjugate of X(31527)
X(658) = X(i)-Ceva conjugate of X(j) for these (i,j): {1275, 279}, {4569, 664}, {4616, 934}, {4635, 4569}, {4998, 17093}, {7045, 33765}, {23586, 7056}, {24011, 1}, {34085, 24015}, {36838, 4626}
X(658) = X(i)-cross conjugate of X(j) for these (i,j): {1, 24011}, {57, 7045}, {169, 7012}, {223, 24032}, {279, 1275}, {514, 7}, {650, 21453}, {651, 664}, {657, 1}, {905, 86}, {934, 4626}, {1020, 934}, {1445, 4564}, {1461, 36118}, {3008, 39293}, {3676, 1088}, {3732, 190}, {4025, 7056}, {4091, 77}, {4253, 59}, {4566, 4569}, {5435, 4998}, {7658, 2}, {9533, 23586}, {10015, 903}, {14282, 5558}, {14331, 34402}, {14543, 99}, {14837, 75}, {16572, 765}, {20521, 6384}, {21188, 273}, {21212, 7249}, {21390, 1476}, {23511, 7035}, {23730, 514}, {23799, 310}, {24002, 1434}, {24782, 40418}, {37681, 1016}, {39470, 18025}, {41800, 1268}
X(658) = cevapoint of X(i) and X(j) for these (i,j): {1, 657}, {2, 4025}, {6, 44408}, {7, 514}, {57, 3676}, {77, 4091}, {354, 650}, {513, 40133}, {522, 23058}, {614, 649}, {651, 934}, {652, 20277}, {905, 1439}, {1020, 4566}, {1122, 3669}, {1427, 43932}, {1446, 24002}, {1638, 3321}, {7658, 9533}, {31605, 41785}, {39063, 43042}
X(658) = crosspoint of X(i) and X(j) for these (i,j): {799, 44326}, {4569, 36838}, {4616, 4635}
X(658) = crossdifference of every pair of points on line {3022, 14936}
X(658) = trilinear pole of line {1, 7}
X(658) = trilinear pole wrt tangential triangle of line X(1)X(7)
X(658) = BSS(a^2→a) of X(107)
X(658) = X(i)-isoconjugate of X(j) for these (i,j): {1, 657}, {2, 8641}, {6, 3900}, {8, 3063}, {9, 663}, {21, 3709}, {31, 3239}, {32, 4397}, {33, 652}, {37, 21789}, {41, 522}, {42, 1021}, {55, 650}, {56, 4130}, {57, 4105}, {58, 4171}, {81, 4524}, {100, 14936}, {101, 2310}, {106, 14427}, {109, 3119}, {110, 36197}, {200, 649}, {210, 7252}, {212, 3064}, {213, 7253}, {219, 18344}, {220, 513}, {228, 17926}, {281, 1946}, {284, 4041}, {294, 926}, {341, 1919}, {346, 667}, {480, 3669}, {512, 2287}, {514, 1253}, {521, 607}, {604, 4163}, {644, 3271}, {647, 4183}, {651, 3022}, {656, 2332}, {658, 24012}, {661, 2328}, {665, 28071}, {692, 1146}, {693, 14827}, {728, 43924}, {798, 1043}, {810, 2322}, {884, 3693}, {893, 4477}, {904, 4529}, {905, 7071}, {906, 42069}, {934, 35508}, {943, 33525}, {949, 6182}, {1015, 4578}, {1024, 2340}, {1098, 4079}, {1110, 42462}, {1170, 6607}, {1174, 6608}, {1260, 6591}, {1334, 3737}, {1415, 4081}, {1459, 7079}, {1461, 24010}, {1783, 3270}, {1792, 2489}, {1802, 7649}, {1824, 23090}, {1857, 36054}, {1911, 4148}, {1974, 15416}, {2149, 23615}, {2170, 3939}, {2175, 4391}, {2192, 14298}, {2194, 3700}, {2212, 6332}, {2223, 28132}, {2291, 14392}, {2299, 8611}, {2316, 4895}, {2334, 4827}, {2346, 10581}, {2364, 4814}, {2488, 6605}, {3121, 7256}, {3122, 7259}, {3248, 6558}, {3676, 6602}, {3733, 4515}, {4435, 7077}, {4516, 5546}, {4528, 9456}, {4546, 38266}, {4631, 7063}, {4705, 7054}, {4953, 34080}, {4990, 28615}, {5514, 32652}, {6066, 40166}, {6129, 7367}, {6139, 41798}, {6366, 18889}, {6603, 23351}, {6726, 6729}, {7008, 10397}, {7046, 22383}, {7050, 40137}, {7073, 9404}, {7118, 8058}, {7123, 17115}, {7162, 17412}, {8638, 36796}, {8648, 36910}, {8750, 34591}, {9447, 35519}, {10482, 21127}, {11934, 40141}, {22108, 42064}, {23990, 42455}, {24026, 32739}, {30692, 32726}, {34526, 46006}
X(658) = barycentric product X(i)*X(j) for these {i,j}: {1, 4569}, {7, 664}, {8, 4626}, {9, 36838}, {10, 4616}, {37, 4635}, {56, 4572}, {57, 4554}, {63, 13149}, {65, 4625}, {69, 36118}, {75, 934}, {76, 1461}, {77, 18026}, {85, 651}, {86, 4566}, {99, 3668}, {100, 1088}, {108, 7182}, {109, 6063}, {190, 279}, {226, 4573}, {241, 34085}, {269, 668}, {273, 6516}, {274, 1020}, {304, 32714}, {312, 4617}, {321, 4637}, {331, 1813}, {348, 653}, {349, 4565}, {479, 3699}, {514, 1275}, {644, 23062}, {646, 738}, {655, 17078}, {662, 1446}, {670, 1042}, {693, 7045}, {789, 7204}, {799, 1427}, {811, 1439}, {927, 9436}, {1025, 34018}, {1106, 6386}, {1119, 4561}, {1254, 4623}, {1262, 3261}, {1323, 35157}, {1332, 1847}, {1407, 1978}, {1414, 1441}, {1415, 20567}, {1434, 4552}, {1443, 35174}, {1458, 46135}, {1509, 4605}, {1897, 7056}, {2481, 41353}, {3239, 23586}, {3596, 6614}, {3663, 6613}, {3673, 8269}, {3674, 6648}, {3676, 4998}, {3732, 30705}, {3900, 24011}, {4131, 24032}, {4397, 24013}, {4564, 24002}, {4601, 7216}, {4610, 6354}, {4619, 23989}, {4620, 7178}, {4624, 21454}, {4631, 7147}, {6335, 7177}, {6606, 10481}, {6649, 7249}, {7035, 43932}, {7055, 36127}, {7128, 15413}, {7196, 37137}, {7205, 29055}, {7339, 35519}, {7365, 37215}, {10004, 32040}, {14256, 44327}, {14615, 36079}, {14727, 41355}, {17093, 37206}, {17095, 38340}, {18025, 23973}, {21453, 35312}, {23984, 30805}, {24015, 36101}, {24016, 35517}, {24027, 40495}, {32041, 42309}, {35171, 38459}, {35338, 42311}, {36146, 40704}, {36908, 44326}, {37139, 37780}, {37141, 40702}, {37143, 37757}, {39293, 43042}
X(658) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3900}, {2, 3239}, {6, 657}, {7, 522}, {8, 4163}, {9, 4130}, {11, 23615}, {27, 17926}, {31, 8641}, {34, 18344}, {37, 4171}, {42, 4524}, {44, 14427}, {55, 4105}, {56, 663}, {57, 650}, {58, 21789}, {59, 3939}, {65, 4041}, {75, 4397}, {77, 521}, {81, 1021}, {85, 4391}, {86, 7253}, {99, 1043}, {100, 200}, {101, 220}, {108, 33}, {109, 55}, {110, 2328}, {112, 2332}, {145, 4546}, {162, 4183}, {171, 4477}, {174, 6730}, {190, 346}, {222, 652}, {223, 14298}, {226, 3700}, {239, 4148}, {269, 513}, {273, 44426}, {278, 3064}, {279, 514}, {304, 15416}, {331, 46110}, {347, 8058}, {348, 6332}, {354, 6608}, {479, 3676}, {513, 2310}, {514, 1146}, {519, 4528}, {522, 4081}, {553, 4976}, {603, 1946}, {604, 3063}, {614, 17115}, {644, 728}, {646, 30693}, {648, 2322}, {649, 14936}, {650, 3119}, {651, 9}, {653, 281}, {655, 36910}, {657, 35508}, {661, 36197}, {662, 2287}, {663, 3022}, {664, 8}, {666, 6559}, {668, 341}, {673, 28132}, {692, 1253}, {693, 24026}, {738, 3669}, {765, 4578}, {883, 3717}, {894, 4529}, {905, 34591}, {906, 1802}, {927, 14942}, {934, 1}, {1014, 3737}, {1016, 6558}, {1018, 4515}, {1020, 37}, {1025, 3693}, {1042, 512}, {1086, 42462}, {1088, 693}, {1106, 667}, {1111, 42455}, {1119, 7649}, {1122, 6615}, {1125, 4990}, {1155, 14392}, {1214, 8611}, {1254, 4705}, {1262, 101}, {1275, 190}, {1308, 42064}, {1317, 4543}, {1319, 4895}, {1323, 6366}, {1331, 1260}, {1332, 3692}, {1358, 21132}, {1388, 4959}, {1400, 3709}, {1407, 649}, {1410, 810}, {1412, 7252}, {1414, 21}, {1415, 41}, {1416, 884}, {1418, 21127}, {1420, 4162}, {1421, 11193}, {1427, 661}, {1429, 4435}, {1434, 4560}, {1435, 6591}, {1439, 656}, {1441, 4086}, {1442, 35057}, {1443, 3738}, {1446, 1577}, {1447, 3716}, {1449, 4827}, {1458, 926}, {1459, 3270}, {1461, 6}, {1462, 1024}, {1475, 10581}, {1633, 4319}, {1638, 33573}, {1783, 7079}, {1790, 23090}, {1813, 219}, {1847, 17924}, {1897, 7046}, {2003, 9404}, {2099, 4814}, {2260, 33525}, {2263, 6182}, {2283, 2340}, {2293, 6607}, {2635, 30692}, {2720, 2342}, {2999, 40137}, {3212, 4147}, {3239, 23970}, {3261, 23978}, {3663, 42337}, {3667, 4953}, {3668, 523}, {3669, 2170}, {3671, 4843}, {3674, 3910}, {3676, 11}, {3699, 5423}, {3732, 6554}, {3882, 3965}, {3888, 4073}, {3900, 24010}, {3911, 1639}, {3939, 480}, {3952, 4082}, {4017, 4516}, {4025, 2968}, {4032, 4140}, {4091, 35072}, {4131, 24031}, {4298, 29278}, {4306, 8676}, {4308, 8710}, {4320, 8678}, {4341, 15313}, {4350, 3309}, {4551, 210}, {4552, 2321}, {4554, 312}, {4556, 7054}, {4558, 2327}, {4559, 1334}, {4561, 1265}, {4564, 644}, {4565, 284}, {4566, 10}, {4567, 7259}, {4569, 75}, {4572, 3596}, {4573, 333}, {4592, 1792}, {4600, 7256}, {4601, 7258}, {4605, 594}, {4610, 7058}, {4616, 86}, {4617, 57}, {4619, 1252}, {4620, 645}, {4625, 314}, {4626, 7}, {4635, 274}, {4636, 6061}, {4637, 81}, {4654, 4820}, {4848, 44729}, {4998, 3699}, {5219, 4944}, {5222, 14330}, {5228, 45755}, {5435, 4521}, {5930, 14308}, {5932, 14302}, {6049, 4943}, {6063, 35519}, {6335, 7101}, {6354, 4024}, {6356, 4064}, {6357, 14400}, {6359, 8062}, {6516, 78}, {6517, 1259}, {6604, 44448}, {6613, 1222}, {6614, 56}, {6649, 7081}, {6733, 6726}, {7011, 10397}, {7023, 43924}, {7045, 100}, {7053, 1459}, {7056, 4025}, {7099, 22383}, {7125, 36054}, {7128, 1783}, {7175, 3287}, {7176, 3907}, {7177, 905}, {7178, 21044}, {7179, 4522}, {7181, 14432}, {7182, 35518}, {7185, 3810}, {7203, 18191}, {7204, 1491}, {7216, 3125}, {7223, 4474}, {7225, 4501}, {7250, 3122}, {7339, 109}, {7365, 6590}, {7370, 6729}, {7371, 6728}, {7649, 42069}, {7658, 13609}, {8059, 2192}, {8641, 24012}, {8750, 7071}, {9533, 7658}, {10481, 6362}, {13149, 92}, {13444, 16012}, {14256, 14837}, {14594, 3974}, {14733, 4845}, {14837, 5514}, {16727, 40213}, {17078, 3904}, {17082, 25128}, {17093, 4468}, {17096, 17197}, {17136, 6737}, {17206, 15411}, {18026, 318}, {18623, 14331}, {21132, 5532}, {21188, 6506}, {21272, 6736}, {21454, 4765}, {22464, 2804}, {23062, 24002}, {23067, 2318}, {23224, 2638}, {23703, 3689}, {23723, 34969}, {23788, 14010}, {23890, 6603}, {23971, 1461}, {23973, 516}, {23979, 32739}, {24002, 4858}, {24011, 4569}, {24013, 934}, {24015, 30807}, {24016, 103}, {24027, 692}, {24471, 17420}, {26700, 7073}, {30719, 4534}, {30725, 4530}, {30805, 23983}, {32651, 2259}, {32668, 911}, {32674, 607}, {32714, 19}, {32735, 2195}, {32739, 14827}, {33765, 17494}, {34036, 11934}, {34056, 23893}, {34085, 36796}, {34387, 23104}, {34496, 45743}, {34855, 2254}, {35312, 4847}, {35326, 8012}, {35338, 3059}, {35341, 45791}, {36048, 943}, {36049, 7367}, {36059, 212}, {36079, 64}, {36082, 7072}, {36086, 28071}, {36118, 4}, {36127, 1857}, {36141, 18889}, {36146, 294}, {36838, 85}, {36908, 6587}, {37139, 41798}, {37141, 282}, {37757, 30565}, {38340, 7110}, {38459, 3887}, {38859, 4040}, {39293, 36802}, {39771, 4542}, {41353, 518}, {41355, 42341}, {42309, 4762}, {43037, 14430}, {43041, 4124}, {43049, 38375}, {43290, 6555}, {43924, 3271}, {43932, 244}, {44408, 14714}, {44717, 4587}, {46107, 21666}, {46153, 3688}
X(658) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 9533, 7056}, {57, 1088, 33765}, {100, 35312, 664}, {354, 9446, 21453}, {479, 5435, 17093}, {934, 4566, 664}, {1439, 6359, 86}, {4573, 4616, 4637}, {13609, 40537, 2}


X(659) = CROSSDIFFERENCE OF X(1) AND X(39)

Trilinears    (a2 - bc)(b - c) : :
Barycentrics    a*(b - c)*(a^2 - b*c) : :
X(659) = 2 X[1] - 3 X[25569], 6 X[2] - 5 X[30795], 5 X[2] - 4 X[45340], 3 X[650] - X[2526], 3 X[667] - X[4378], 2 X[676] - 3 X[26275], X[764] - 3 X[14419], 3 X[1491] - 2 X[2526], 3 X[1635] - X[2254], 3 X[1635] - 2 X[9508], 2 X[1769] - 3 X[28396], 4 X[1960] - X[21343], 2 X[1960] + X[21385], 4 X[1960] - 3 X[25569], 2 X[3716] - 3 X[4448], 4 X[3716] - 3 X[4800], 4 X[3716] - X[4810], X[3777] - 4 X[6050], 2 X[3803] + X[4490], 4 X[3837] - 5 X[30795], X[3837] - 3 X[45314], 5 X[3837] - 6 X[45340], 2 X[3960] - 3 X[14419], X[4010] - 3 X[4448], 2 X[4010] - 3 X[4800], X[4010] + 2 X[4830], X[4088] - 3 X[6546], 3 X[4367] - 2 X[4378], X[4367] - 4 X[4401], X[4378] - 6 X[4401], 6 X[4448] - X[4810], 3 X[4448] + 2 X[4830], X[4449] - 3 X[8643], 2 X[4458] - 3 X[4809], 2 X[4498] + X[4879], X[4724] + 2 X[4782], 2 X[4724] + X[4784], 3 X[4763] - 2 X[25380], 4 X[4782] - X[4784], 3 X[4800] - X[4810], 3 X[4800] + 4 X[4830], 3 X[4809] - 4 X[13246], X[4810] + 4 X[4830], 2 X[18004] - 3 X[30565], X[21343] + 2 X[21385], X[21343] - 3 X[25569], 2 X[21385] + 3 X[25569], X[23770] - 3 X[26275], 4 X[25380] - 3 X[36848], 5 X[30795] - 12 X[45314], 25 X[30795] - 24 X[45340], 5 X[31209] - 3 X[44429], 5 X[45314] - 2 X[45340]

X(659) lies on these lines: {1, 891}, {2, 3837}, {3, 2826}, {10, 21722}, {11, 33311}, {23, 385}, {36, 6550}, {40, 2821}, {44, 513}, {56, 24097}, {75, 21439}, {88, 16505}, {100, 190}, {105, 676}, {145, 25574}, {244, 3248}, {291, 875}, {292, 665}, {512, 4040}, {514, 667}, {521, 13256}, {522, 4122}, {663, 4083}, {666, 34067}, {678, 38349}, {693, 4874}, {764, 2832}, {784, 8637}, {804, 1281}, {812, 3716}, {814, 4391}, {830, 4705}, {834, 2978}, {885, 8638}, {898, 4555}, {905, 3777}, {918, 8301}, {926, 1282}, {952, 19916}, {1001, 6009}, {1019, 6372}, {1022, 14422}, {1054, 6085}, {1280, 23834}, {1283, 6089}, {1376, 4925}, {1577, 29070}, {1734, 6004}, {1757, 6165}, {1769, 5075}, {1929, 18001}, {2517, 6133}, {2530, 14838}, {2533, 29051}, {2782, 19926}, {2785, 5592}, {2787, 3762}, {2802, 41191}, {2804, 13222}, {2808, 19921}, {2814, 38324}, {2827, 14664}, {2975, 24093}, {3004, 3733}, {3126, 19593}, {3218, 8661}, {3226, 23355}, {3667, 13252}, {3669, 23765}, {3709, 21389}, {3722, 4145}, {3737, 6371}, {3738, 13277}, {3751, 9032}, {3766, 39044}, {3803, 4490}, {3808, 4164}, {3835, 24719}, {3887, 4730}, {3900, 13165}, {4088, 6546}, {4093, 4132}, {4147, 28470}, {4155, 8298}, {4369, 21146}, {4380, 29328}, {4435, 8632}, {4449, 8643}, {4462, 29324}, {4474, 29236}, {4508, 14433}, {4560, 16695}, {4583, 8709}, {4707, 29102}, {4728, 44304}, {4750, 8650}, {4761, 29188}, {4762, 7662}, {4763, 25380}, {4765, 8662}, {4774, 29066}, {4775, 4794}, {4778, 4932}, {4785, 45673}, {4791, 29033}, {4802, 8655}, {4806, 20295}, {4824, 7234}, {4834, 6005}, {4885, 24747}, {4948, 31150}, {5027, 19580}, {5253, 24099}, {5276, 25808}, {6164, 14122}, {6363, 21173}, {6631, 9266}, {7265, 29106}, {8297, 24286}, {8424, 24141}, {8636, 29025}, {8648, 21132}, {8659, 24578}, {8660, 21222}, {8674, 38325}, {8712, 35683}, {9318, 24447}, {9451, 42341}, {10015, 29240}, {14296, 27855}, {14667, 15914}, {17930, 36066}, {17954, 18015}, {17990, 21391}, {19903, 28850}, {19915, 28915}, {20142, 28209}, {20316, 25636}, {21051, 21301}, {21053, 25638}, {21115, 28195}, {21201, 42670}, {21789, 29162}, {22379, 24126}, {23301, 26049}, {23569, 23650}, {23768, 43060}, {23815, 31288}, {23888, 30580}, {24193, 27846}, {24350, 24353}, {24488, 24499}, {24512, 25815}, {24666, 24768}, {24720, 31286}, {24721, 25381}, {25259, 29078}, {25299, 26033}, {25634, 33115}, {25686, 27293}, {26073, 26076}, {26114, 44451}, {27045, 31946}, {27167, 40086}, {27628, 28283}, {27648, 28241}, {27915, 27952}, {28373, 29274}, {28602, 31073}, {31209, 44429}, {33660, 33663}, {43051, 43924}

X(659) = midpoint of X(i) and X(j) for these {i,j}: {1, 21385}, {100, 13266}, {649, 4724}, {663, 4498}, {3716, 4830}, {4040, 4063}, {4730, 6161}
X(659) = reflection of X(i) in X(j) for these {i,j}: {1, 1960}, {2, 45314}, {3, 44805}, {649, 4782}, {667, 4401}, {693, 4874}, {764, 3960}, {876, 665}, {905, 6050}, {1022, 14422}, {1491, 650}, {2254, 9508}, {2517, 6133}, {2530, 14838}, {3777, 905}, {3801, 4142}, {4010, 3716}, {4367, 667}, {4458, 13246}, {4486, 27929}, {4728, 45666}, {4775, 4794}, {4784, 649}, {4800, 4448}, {4810, 4010}, {4879, 663}, {4948, 31150}, {20295, 4806}, {21146, 4369}, {21301, 21051}, {21343, 1}, {23765, 3669}, {23770, 676}, {23815, 31288}, {24097, 30725}, {24462, 17990}, {24719, 3835}, {24720, 31286}, {24721, 25381}, {31131, 28602}, {36848, 4763}, {38348, 8632}
X(659) = isogonal conjugate of X(660)
X(659) = isotomic conjugate of X(4583)
X(659) = complement of isotomic conjugate of isogonal conjugate of X(21003)
X(659) = X(98)-Ceva conjugate of X(11)
X(659) = crosspoint of X(100) and X(105)
X(659) = crosssum of X(i) and X(j) for these (i,j): (1,659), (9,926), (141,918), (291,876), (292,875), (513,518)
X(659) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {101, 20355}, {692, 39354}, {727, 149}, {3226, 21293}, {8709, 6327}, {8851, 33650}, {18793, 3448}, {20332, 150}, {27809, 21294}, {34077, 4440}
X(659) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 40623}, {39714, 21252}, {39979, 116}
X(659) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 40623}, {98, 11}, {99, 4366}, {100, 8299}, {190, 17475}, {238, 27846}, {242, 4124}, {513, 38348}, {660, 1}, {666, 6}, {812, 4435}, {874, 239}, {927, 6654}, {1026, 1279}, {1027, 513}, {1447, 27918}, {1929, 244}, {2665, 3248}, {3225, 21762}, {3512, 2170}, {3570, 2238}, {3572, 4367}, {3573, 238}, {7166, 3123}, {7168, 38986}, {7351, 7004}, {8709, 2}, {17930, 81}, {18793, 8054}, {37128, 1015}, {37207, 24512}, {40737, 38978}, {43747, 2310}
X(659) = X(i)-cross conjugate of X(j) for these (i,j): {4455, 8632}, {21832, 812}, {27846, 238}
X(659) = cevapoint of X(i) and X(j) for these (i,j): {665, 6373}, {4455, 21832}
X(659) = crosspoint of X(i) and X(j) for these (i,j): {1, 660}, {57, 927}, {99, 37128}, {100, 105}, {190, 3226}, {238, 3573}, {239, 874}, {251, 919}, {812, 43041}, {3570, 33295}, {3733, 23355}, {17930, 40725}
X(659) = crosssum of X(i) and X(j) for these (i,j): {1, 659}, {6, 21003}, {9, 926}, {141, 918}, {291, 876}, {292, 875}, {512, 2238}, {513, 518}, {514, 20335}, {649, 3009}, {665, 40730}, {2284, 4557}, {3570, 18047}, {3952, 23354}, {4705, 20708}
X(659) = trilinear pole of line {27846, 38989}
X(659) = crossdifference of every pair of points on line {1, 39}
X(659) = bicentric difference of PU(134)
X(659) = PU(134)-harmonic conjugate of X(8299)
X(659) = complement of polar conjugate of isogonal conjugate of X(22155)
X(659) = X(i)-isoconjugate of X(j) for these (i,j): {1, 660}, {2, 813}, {6, 4562}, {31, 4583}, {37, 4584}, {38, 36081}, {42, 4589}, {56, 36801}, {75, 34067}, {100, 291}, {101, 335}, {109, 4518}, {110, 43534}, {190, 292}, {213, 4639}, {295, 1897}, {334, 692}, {337, 8750}, {513, 5378}, {651, 4876}, {664, 7077}, {666, 3252}, {668, 1911}, {694, 18047}, {741, 3952}, {756, 36066}, {765, 876}, {805, 1215}, {869, 41072}, {875, 7035}, {919, 40217}, {932, 41531}, {982, 8684}, {984, 30664}, {1016, 3572}, {1018, 37128}, {1237, 17938}, {1252, 4444}, {1492, 3864}, {1581, 4579}, {1922, 1978}, {2196, 6335}, {2276, 37207}, {2283, 33676}, {2295, 37134}, {2311, 4552}, {3573, 30663}, {3862, 4586}, {3903, 18787}, {3939, 7233}, {4033, 18268}, {4557, 18827}, {4559, 36800}, {4570, 35352}, {4572, 18265}, {6386, 14598}, {8709, 40155}, {18099, 46161}, {18829, 20964}, {18895, 32739}, {22116, 36086}, {33888, 39420}, {34071, 40848}, {35309, 39276}, {37135, 40794}, {40093, 40519}, {40936, 41209}
X(659) = barycentric product X(i)*X(j) for these {i,j}: {1, 812}, {6, 3766}, {7, 4435}, {9, 43041}, {21, 7212}, {28, 24459}, {57, 3716}, {75, 8632}, {81, 4010}, {86, 21832}, {88, 4448}, {89, 4800}, {92, 22384}, {99, 39786}, {100, 27918}, {190, 27846}, {238, 514}, {239, 513}, {242, 905}, {244, 3570}, {256, 4107}, {257, 4164}, {269, 4148}, {274, 4455}, {291, 4375}, {292, 27855}, {350, 649}, {512, 30940}, {522, 1429}, {650, 1447}, {651, 4124}, {656, 31905}, {660, 35119}, {661, 33295}, {663, 10030}, {666, 38989}, {667, 1921}, {693, 1914}, {740, 1019}, {751, 4508}, {804, 40432}, {862, 15419}, {874, 1015}, {876, 4366}, {885, 34253}, {893, 14296}, {984, 23597}, {985, 4486}, {1022, 4432}, {1024, 39775}, {1027, 17755}, {1086, 3573}, {1284, 4560}, {1428, 4391}, {1509, 4155}, {1577, 5009}, {1635, 27922}, {1919, 18891}, {1929, 27929}, {1980, 44169}, {2201, 4025}, {2210, 3261}, {2238, 7192}, {2254, 6654}, {2401, 15507}, {2665, 27854}, {2786, 40767}, {3063, 18033}, {3248, 27853}, {3572, 39044}, {3669, 3685}, {3676, 3684}, {3733, 3948}, {3737, 16609}, {3747, 7199}, {3783, 4817}, {3808, 17743}, {3835, 34252}, {3960, 36815}, {3975, 43924}, {3985, 7203}, {4083, 39914}, {4367, 17493}, {4369, 18786}, {4433, 17096}, {4444, 8300}, {4465, 43928}, {4810, 25417}, {4830, 25430}, {6164, 27912}, {6632, 24193}, {6650, 38348}, {7193, 17924}, {7255, 18904}, {7649, 20769}, {8852, 27951}, {9508, 40725}, {14433, 37129}, {14599, 40495}, {14621, 30665}, {22383, 40717}, {24018, 34856}, {24685, 35348}
X(659) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 4562}, {2, 4583}, {6, 660}, {9, 36801}, {31, 813}, {32, 34067}, {58, 4584}, {81, 4589}, {86, 4639}, {101, 5378}, {238, 190}, {239, 668}, {242, 6335}, {244, 4444}, {251, 36081}, {261, 36806}, {350, 1978}, {513, 335}, {514, 334}, {593, 36066}, {649, 291}, {650, 4518}, {661, 43534}, {663, 4876}, {665, 22116}, {667, 292}, {693, 18895}, {740, 4033}, {788, 3862}, {804, 3963}, {812, 75}, {874, 31625}, {876, 40098}, {905, 337}, {985, 37207}, {1015, 876}, {1019, 18827}, {1024, 33676}, {1178, 37134}, {1284, 4552}, {1428, 651}, {1429, 664}, {1447, 4554}, {1580, 18047}, {1691, 4579}, {1914, 100}, {1919, 1911}, {1921, 6386}, {1977, 875}, {1980, 1922}, {2201, 1897}, {2210, 101}, {2238, 3952}, {2254, 40217}, {3063, 7077}, {3125, 35352}, {3248, 3572}, {3250, 3864}, {3261, 44172}, {3570, 7035}, {3572, 30663}, {3573, 1016}, {3669, 7233}, {3684, 3699}, {3685, 646}, {3716, 312}, {3733, 37128}, {3737, 36800}, {3747, 1018}, {3766, 76}, {3783, 3807}, {3797, 4505}, {3808, 3662}, {3948, 27808}, {4010, 321}, {4063, 40093}, {4083, 40848}, {4093, 35309}, {4107, 1909}, {4124, 4391}, {4148, 341}, {4155, 594}, {4164, 894}, {4366, 874}, {4367, 30669}, {4375, 350}, {4378, 7245}, {4432, 24004}, {4433, 30730}, {4435, 8}, {4448, 4358}, {4455, 37}, {4465, 41314}, {4475, 23596}, {4486, 33931}, {4508, 3761}, {4760, 42721}, {4800, 4671}, {4810, 28605}, {4830, 19804}, {5009, 662}, {5027, 2295}, {5029, 40794}, {7192, 40017}, {7193, 1332}, {7212, 1441}, {7255, 40834}, {8299, 42720}, {8300, 3570}, {8632, 1}, {10030, 4572}, {14296, 1920}, {14433, 6381}, {14599, 692}, {14621, 41072}, {15507, 2397}, {16514, 3799}, {17475, 23354}, {17494, 40094}, {18786, 27805}, {18892, 32739}, {20769, 4561}, {20979, 41531}, {20981, 18787}, {21385, 40095}, {21832, 10}, {22383, 295}, {22384, 63}, {23597, 870}, {24193, 6545}, {24459, 20336}, {27846, 514}, {27855, 1921}, {27918, 693}, {27929, 20947}, {30654, 40790}, {30665, 3661}, {30940, 670}, {31905, 811}, {33295, 799}, {34252, 4598}, {34253, 883}, {34856, 823}, {35119, 3766}, {36815, 36804}, {38348, 6542}, {38367, 3009}, {38989, 918}, {39044, 27853}, {39786, 523}, {39914, 18830}, {40432, 18829}, {40495, 44170}, {40746, 30664}, {40767, 35148}, {41333, 4557}, {43041, 85}
X(659) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1960, 25569}, {2, 3837, 30795}, {2, 26855, 27194}, {75, 21612, 21439}, {764, 14419, 3960}, {1635, 2246, 22108}, {1635, 2254, 9508}, {1635, 3768, 20331}, {1960, 21385, 21343}, {3716, 4010, 4800}, {4010, 4448, 3716}, {4448, 4830, 4810}, {4458, 13246, 4809}, {4724, 4782, 4784}, {4800, 4810, 4010}, {8632, 21832, 4435}, {13245, 13246, 45695}, {21343, 25569, 1}, {23770, 26275, 676}, {27015, 27075, 2}, {27294, 30025, 30865}, {31596, 31597, 45695}


X(660) = TRILINEAR POLE OF LINE X(1)X(39)

Trilinears    1/[(a2 - bc)(b - c)] : :
Barycentrics    a*(a - b)*(a - c)*(-b^2 + a*c)*(a*b - c^2) : :
X(660) = 3 X[2] - 4 X[40538]

X(660) lies on the curve CC9 and these lines: {2, 38989}, {9, 3252}, {42, 40796}, {43, 40155}, {44, 292}, {45, 24482}, {63, 39344}, {67, 37221}, {88, 291}, {100, 649}, {101, 1492}, {109, 8684}, {110, 4599}, {162, 35325}, {190, 513}, {238, 1911}, {239, 335}, {295, 4518}, {320, 334}, {337, 37214}, {512, 1016}, {651, 6163}, {653, 43923}, {658, 43932}, {662, 765}, {663, 9266}, {668, 37133}, {670, 37204}, {672, 36906}, {694, 43763}, {799, 3952}, {805, 8707}, {823, 46151}, {874, 23354}, {875, 4607}, {876, 3257}, {883, 34085}, {891, 4555}, {897, 46154}, {898, 1960}, {926, 36802}, {932, 40499}, {984, 25800}, {1026, 3572}, {1155, 14200}, {1156, 4876}, {1757, 2664}, {1821, 4645}, {2144, 27920}, {2235, 3862}, {2254, 36238}, {2284, 3573}, {2349, 46147}, {2580, 46166}, {2581, 46167}, {3570, 37207}, {3681, 24586}, {3699, 4598}, {3766, 36803}, {3903, 4613}, {4083, 6631}, {4436, 37212}, {4444, 37143}, {4551, 37137}, {4715, 7245}, {4767, 37209}, {5220, 22116}, {6005, 32094}, {6372, 32028}, {6633, 29350}, {7233, 43762}, {8701, 36066}, {9458, 37222}, {14621, 19586}, {17763, 24624}, {17944, 37140}, {18278, 43761}, {20072, 30669}, {20683, 40098}, {23691, 36100}, {27495, 40740}, {29936, 36256}, {32680, 46155}, {36085, 36827}, {37132, 46156}, {37134, 46161}, {37218, 43927}, {37220, 46165}, {40432, 40936}

X(660) = reflection of X(38989) in X(40538)
X(660) = isogonal conjugate of X(659)
X(660) = isotomic conjugate of X(3766)
X(660) = anticomplement of X(38989)
X(660) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {813, 39353}, {919, 39362}, {5377, 17794}, {5378, 20344}
X(660) = X(i)-Ceva conjugate of X(j) for these (i,j): {4584, 813}, {4589, 4562}, {5378, 291}, {36081, 34067}
X(660) = X(i)-cross conjugate of X(j) for these (i,j): {291, 5378}, {511, 59}, {659, 1}, {665, 2}, {875, 292}, {876, 291}, {926, 3252}, {1026, 100}, {1027, 1280}, {1757, 765}, {2238, 1016}, {2664, 7035}, {3509, 4564}, {3570, 3903}, {3572, 37128}, {4447, 4998}, {6165, 513}, {6211, 7012}, {6373, 6}, {17990, 37}, {18788, 7045}, {20683, 1252}, {21391, 82}, {21830, 31625}, {24462, 75}
X(660) = cevapoint of X(i) and X(j) for these (i,j): {1, 659}, {6, 21003}, {9, 926}, {141, 918}, {291, 876}, {292, 875}, {512, 2238}, {513, 518}, {514, 20335}, {649, 3009}, {665, 40730}, {2284, 4557}, {3570, 18047}, {3952, 23354}, {4705, 20708}
X(660) = crosspoint of X(i) and X(j) for these (i,j): {666, 8709}, {4584, 4589}, {37134, 41209}
X(660) = crosssum of X(i) and X(j) for these (i,j): {665, 6373}, {4455, 21832}
X(660) = trilinear pole of line {1, 39}
X(660) = crossdifference of every pair of points on line {27846, 38989}
X(660) = X(i)-isoconjugate of X(j) for these (i,j): {1, 659}, {2, 8632}, {4, 22384}, {6, 812}, {31, 3766}, {55, 43041}, {56, 3716}, {57, 4435}, {58, 4010}, {81, 21832}, {86, 4455}, {100, 27846}, {101, 27918}, {106, 4448}, {109, 4124}, {238, 513}, {239, 649}, {242, 1459}, {244, 3573}, {256, 4164}, {284, 7212}, {292, 4375}, {350, 667}, {512, 33295}, {514, 1914}, {520, 34856}, {522, 1428}, {523, 5009}, {647, 31905}, {650, 1429}, {662, 39786}, {663, 1447}, {665, 6654}, {693, 2210}, {739, 14433}, {740, 3733}, {757, 4155}, {798, 30940}, {804, 1178}, {813, 35119}, {874, 3248}, {875, 39044}, {876, 8300}, {884, 39775}, {893, 4107}, {904, 14296}, {905, 2201}, {983, 3808}, {985, 30665}, {1015, 3570}, {1019, 2238}, {1024, 34253}, {1027, 8299}, {1284, 3737}, {1407, 4148}, {1474, 24459}, {1874, 23189}, {1911, 27855}, {1919, 1921}, {1929, 38348}, {1960, 27922}, {1977, 27853}, {1980, 18891}, {2163, 4800}, {2276, 23597}, {2334, 4830}, {3063, 10030}, {3253, 6373}, {3261, 14599}, {3572, 4366}, {3669, 3684}, {3685, 43924}, {3747, 7192}, {4083, 34252}, {4367, 18786}, {4432, 23345}, {4433, 7203}, {4465, 23892}, {4486, 40746}, {4508, 30650}, {4817, 16514}, {5027, 32010}, {5029, 40725}, {6591, 20769}, {7193, 7649}, {7199, 41333}, {7252, 16609}, {9262, 27912}, {9508, 40767}, {17493, 20981}, {17755, 43929}, {17793, 23355}, {17962, 27929}, {18264, 20518}, {18892, 40495}, {20979, 39914}, {30654, 40738}, {32020, 38367}, {36086, 38989}
X(660) = barycentric product X(i)*X(j) for these {i,j}: {1, 4562}, {6, 4583}, {10, 4584}, {37, 4589}, {42, 4639}, {57, 36801}, {75, 813}, {76, 34067}, {100, 335}, {101, 334}, {141, 36081}, {181, 36806}, {190, 291}, {292, 668}, {295, 6335}, {337, 1783}, {514, 5378}, {594, 36066}, {644, 7233}, {651, 4518}, {662, 43534}, {664, 4876}, {666, 22116}, {692, 18895}, {741, 4033}, {765, 4444}, {789, 3862}, {805, 3963}, {875, 31625}, {876, 1016}, {932, 40848}, {984, 37207}, {1018, 18827}, {1025, 33676}, {1215, 37134}, {1581, 18047}, {1911, 1978}, {1916, 4579}, {1922, 6386}, {2276, 41072}, {2295, 18829}, {3570, 30663}, {3572, 7035}, {3573, 40098}, {3661, 30664}, {3662, 8684}, {3864, 4586}, {3903, 30669}, {3952, 37128}, {4551, 36800}, {4554, 7077}, {4557, 40017}, {4567, 35352}, {4598, 41531}, {5384, 23596}, {16587, 41209}, {18268, 27808}, {18787, 27805}, {32739, 44172}, {35148, 40794}, {36086, 40217}, {36803, 40730}
X(660) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 812}, {2, 3766}, {6, 659}, {9, 3716}, {31, 8632}, {37, 4010}, {42, 21832}, {44, 4448}, {45, 4800}, {48, 22384}, {55, 4435}, {57, 43041}, {65, 7212}, {72, 24459}, {99, 30940}, {100, 239}, {101, 238}, {109, 1429}, {162, 31905}, {163, 5009}, {171, 4107}, {172, 4164}, {190, 350}, {200, 4148}, {213, 4455}, {238, 4375}, {239, 27855}, {291, 514}, {292, 513}, {295, 905}, {334, 3261}, {335, 693}, {337, 15413}, {512, 39786}, {513, 27918}, {644, 3685}, {646, 4087}, {649, 27846}, {650, 4124}, {651, 1447}, {659, 35119}, {662, 33295}, {664, 10030}, {665, 38989}, {668, 1921}, {692, 1914}, {741, 1019}, {750, 4508}, {765, 3570}, {805, 40432}, {813, 1}, {875, 1015}, {876, 1086}, {894, 14296}, {899, 14433}, {906, 7193}, {932, 39914}, {984, 4486}, {985, 23597}, {1016, 874}, {1018, 740}, {1023, 4432}, {1025, 39775}, {1026, 17755}, {1252, 3573}, {1331, 20769}, {1415, 1428}, {1449, 4830}, {1500, 4155}, {1757, 27929}, {1783, 242}, {1911, 649}, {1922, 667}, {1978, 18891}, {2196, 1459}, {2275, 3808}, {2276, 30665}, {2283, 34253}, {2284, 8299}, {2295, 804}, {2311, 3737}, {2427, 15507}, {2664, 27854}, {2702, 40767}, {3252, 2254}, {3257, 27922}, {3570, 39044}, {3572, 244}, {3573, 4366}, {3699, 3975}, {3799, 3797}, {3862, 1491}, {3864, 824}, {3888, 33891}, {3903, 17493}, {3939, 3684}, {3952, 3948}, {3963, 14295}, {4033, 35544}, {4069, 3985}, {4169, 4783}, {4444, 1111}, {4482, 4495}, {4518, 4391}, {4551, 16609}, {4554, 18033}, {4557, 2238}, {4559, 1284}, {4562, 75}, {4579, 385}, {4583, 76}, {4584, 86}, {4589, 274}, {4639, 310}, {4645, 27951}, {4752, 4693}, {4876, 522}, {5378, 190}, {6163, 27912}, {6335, 40717}, {6386, 44169}, {7035, 27853}, {7077, 650}, {7233, 24002}, {7245, 4411}, {8684, 17743}, {8750, 2201}, {14598, 1919}, {16777, 4810}, {17735, 38348}, {18047, 1966}, {18265, 3063}, {18268, 3733}, {18787, 4369}, {18827, 7199}, {18895, 40495}, {18897, 1980}, {21003, 40623}, {21143, 24193}, {21801, 42767}, {21859, 7235}, {22116, 918}, {23343, 4465}, {24019, 34856}, {30663, 4444}, {30664, 14621}, {30669, 4374}, {30671, 4475}, {32739, 2210}, {34067, 6}, {34071, 34252}, {35342, 4974}, {35352, 16732}, {36066, 1509}, {36081, 83}, {36086, 6654}, {36800, 18155}, {36801, 312}, {36806, 18021}, {37128, 7192}, {37134, 32010}, {37135, 40725}, {37207, 870}, {37593, 4839}, {39026, 27943}, {40093, 20949}, {40095, 21606}, {40521, 4037}, {40730, 665}, {40794, 2786}, {40848, 20906}, {41531, 3835}, {43262, 4406}, {43534, 1577}
X(660) = {X(38989),X(40538)}-harmonic conjugate of X(2)


X(661) = CROSSDIFFERENCE OF X(1) AND X(21)

Trilinears    cot B - cot C : cot C - cot A : cot A - cot B
Trilinears    b2 - c2 : :
Trilinears    SB - SC : :
Trilinears    cos 2B - cos 2C : :
Trilinears    d(a,b,c) : : , where d(a,b,c) = directed distance from A to line X(2)X(6)
Barycentrics    sin A (cot B - cot C) : sin B (cot C - cot A) : sin C (cot A - cot B)
X(661) = X(661) = 6 X[2] - 5 X[24924], 3 X[2] - 4 X[25666], 3 X[2] + X[31290], 5 X[2] - 4 X[45663], 2 X[649] - 3 X[1635], 5 X[649] - 8 X[2516], 3 X[649] - 4 X[4394], 3 X[649] - 2 X[4790], X[649] - 3 X[4893], 4 X[650] - 3 X[1635], 5 X[650] - 4 X[2516], 3 X[650] - 2 X[4394], 3 X[650] - X[4790], 2 X[650] + X[4813], 2 X[650] - 3 X[4893], 4 X[650] - X[4979], 2 X[693] - 3 X[4728], X[693] - 3 X[4776], 15 X[1635] - 16 X[2516], 9 X[1635] - 8 X[4394], 9 X[1635] - 4 X[4790], 3 X[1635] + 2 X[4813], 3 X[1635] - X[4979], 8 X[2490] - 9 X[6544], 6 X[2516] - 5 X[4394], 12 X[2516] - 5 X[4790], 8 X[2516] + 5 X[4813], 8 X[2516] - 15 X[4893], 16 X[2516] - 5 X[4979], 2 X[2527] - 3 X[14425], 2 X[2533] - 3 X[21052], 3 X[3239] - 4 X[14350], 2 X[3700] - 3 X[4120], 4 X[3700] - X[4838], 4 X[3700] - 3 X[4931], 2 X[3700] + X[4988], 4 X[3776] - 3 X[21115], 2 X[3776] - 3 X[44435], 4 X[3835] - 3 X[4728], 2 X[3835] - 3 X[4776], X[4024] - 3 X[4120], X[4024] + 2 X[4841], 2 X[4024] - 3 X[4931], X[4024] - 4 X[14321], 3 X[4041] - 2 X[4730], 3 X[4041] - 4 X[4770], X[4041] + 2 X[4983], 2 X[4106] - 3 X[31147], 6 X[4120] - X[4838], 3 X[4120] + 2 X[4841], 3 X[4120] + X[4988], 3 X[4120] - 4 X[14321], 4 X[4369] - 5 X[24924], 4 X[4369] - 3 X[31148], 2 X[4369] + X[31290], X[4369] - 3 X[45315], 5 X[4369] - 6 X[45663], 2 X[4378] - 3 X[14413], 3 X[4379] - 4 X[4885], 3 X[4379] - 5 X[30835], 3 X[4379] - 2 X[43067], X[4380] - 3 X[31150], X[4382] - 4 X[4940], X[4382] - 3 X[31147], 4 X[4394] + 3 X[4813], 4 X[4394] - 9 X[4893], 8 X[4394] - 3 X[4979], 2 X[4411] - 3 X[27485], 3 X[4453] - 4 X[21212], 4 X[4705] - X[4729], 3 X[4705] - X[4730], 3 X[4705] - 2 X[4770], 2 X[4705] + X[4822], 3 X[4729] - 4 X[4730], 3 X[4729] - 8 X[4770], X[4729] + 2 X[4822], X[4729] + 4 X[4983], 2 X[4730] + 3 X[4822], X[4730] + 3 X[4983], 3 X[4750] - 2 X[4897], 3 X[4750] - 4 X[17069], 6 X[4763] - 5 X[27013], 6 X[4763] - 7 X[27115], 4 X[4770] + 3 X[4822], 2 X[4770] + 3 X[4983], 2 X[4790] + 3 X[4813], 2 X[4790] - 9 X[4893], 4 X[4790] - 3 X[4979], X[4804] - 4 X[4806], X[4804] + 2 X[4824], 2 X[4806] + X[4824], X[4813] + 3 X[4893], 2 X[4813] + X[4979], X[4838] + 4 X[4841], X[4838] - 3 X[4931], X[4838] + 2 X[4988], X[4838] - 8 X[14321], 4 X[4841] + 3 X[4931], X[4841] + 2 X[14321], 4 X[4885] - 5 X[30835], 6 X[4893] - X[4979], 6 X[4928] - 5 X[26985], 6 X[4928] - 7 X[27138], 3 X[4931] + 2 X[4988], 3 X[4931] - 8 X[14321], 2 X[4932] - 5 X[31209], 4 X[4940] - 3 X[31147], 3 X[4958] + 4 X[45745], X[4988] + 4 X[14321], 3 X[6545] - 2 X[21104], 3 X[6546] + X[23731], 3 X[6590] - 8 X[14350], 2 X[7192] - 5 X[24924], X[7192] - 4 X[25666], 2 X[7192] - 3 X[31148], X[7192] - 6 X[45315], 5 X[7192] - 12 X[45663], 3 X[8029] - 2 X[12072], 2 X[8689] - 3 X[45673], 4 X[21051] - 3 X[21052], 2 X[21196] + X[44449], 3 X[21297] - 5 X[26798], 3 X[21297] - X[26824], 2 X[24720] - 3 X[44429], 5 X[24924] - 8 X[25666], 5 X[24924] - 3 X[31148], 5 X[24924] + 2 X[31290], 5 X[24924] - 12 X[45315], 25 X[24924] - 24 X[45663], 8 X[25666] - 3 X[31148], 4 X[25666] + X[31290], 2 X[25666] - 3 X[45315], 5 X[25666] - 3 X[45663], 5 X[26777] - X[26853], 5 X[26798] - X[26824], 5 X[26985] - 7 X[27138], 5 X[27013] - 7 X[27115], 5 X[30835] - 2 X[43067], 3 X[31148] + 2 X[31290], X[31148] - 4 X[45315], 5 X[31148] - 8 X[45663], 7 X[31207] - 8 X[31287], 5 X[31209] - 4 X[31286], X[31290] + 6 X[45315], 5 X[31290] + 12 X[45663], 5 X[45315] - 2 X[45663]

X(661) is the perspector of triangle ABC and the tangential triangle of the conic {A, B, C, X(1), X(10)}}. (Randy Hutson, 9/23/2011)

X(661) lies on the cubic K1005 and these lines: {1, 4160}, {2, 3572}, {4, 17926}, {6, 4833}, {9, 35354}, {10, 4761}, {11, 20974}, {31, 9256}, {37, 42768}, {38, 3805}, {44, 513}, {63, 8774}, {71, 3657}, {100, 2702}, {101, 1290}, {109, 9090}, {115, 2170}, {125, 1566}, {162, 36084}, {163, 32678}, {190, 35147}, {226, 4077}, {239, 40459}, {244, 3124}, {430, 23615}, {442, 14825}, {512, 4041}, {514, 693}, {522, 4502}, {523, 3700}, {525, 21124}, {647, 3709}, {662, 2644}, {663, 810}, {665, 2530}, {669, 4455}, {756, 3005}, {764, 14434}, {784, 23657}, {786, 1926}, {788, 2978}, {799, 27805}, {812, 17494}, {824, 25259}, {830, 1580}, {832, 3063}, {875, 25836}, {876, 25823}, {900, 4976}, {905, 27674}, {918, 3004}, {923, 36150}, {926, 2499}, {940, 18199}, {1011, 23864}, {1015, 38979}, {1019, 1931}, {1021, 2651}, {1022, 40434}, {1024, 1174}, {1084, 3123}, {1211, 6546}, {1213, 2490}, {1252, 14513}, {1459, 6588}, {1638, 28902}, {1639, 3837}, {1643, 6161}, {1734, 6005}, {1769, 6589}, {1901, 42462}, {1953, 8818}, {1960, 14438}, {1962, 9279}, {1980, 8633}, {2051, 40213}, {2149, 2222}, {2159, 36151}, {2171, 12077}, {2276, 24577}, {2292, 18015}, {2294, 17422}, {2310, 20975}, {2488, 40952}, {2520, 8641}, {2527, 14425}, {2533, 20486}, {2623, 21741}, {2640, 9395}, {2650, 2653}, {2786, 4467}, {3064, 24006}, {3136, 28143}, {3139, 33573}, {3309, 45755}, {3310, 6615}, {3667, 4765}, {3676, 28878}, {3716, 23656}, {3722, 21341}, {3733, 20472}, {3737, 5053}, {3738, 23650}, {3776, 21115}, {3777, 23738}, {3801, 21125}, {3907, 24534}, {3942, 8287}, {3952, 7239}, {4025, 28846}, {4049, 30588}, {4051, 23942}, {4063, 29807}, {4071, 4086}, {4083, 4490}, {4105, 6182}, {4106, 4382}, {4132, 4826}, {4148, 21302}, {4151, 4170}, {4155, 8663}, {4367, 5029}, {4374, 24622}, {4378, 14413}, {4379, 4885}, {4380, 4785}, {4411, 27485}, {4449, 17478}, {4453, 21212}, {4521, 4778}, {4534, 36637}, {4559, 35307}, {4560, 6002}, {4562, 7035}, {4581, 26080}, {4750, 4897}, {4763, 27013}, {4773, 39386}, {4775, 4895}, {4777, 4820}, {4802, 4944}, {4808, 7927}, {4815, 22044}, {4817, 26277}, {4874, 24674}, {4926, 4949}, {4927, 21116}, {4928, 26985}, {4932, 31209}, {4948, 45676}, {4984, 28217}, {5051, 5592}, {5701, 46125}, {5949, 40475}, {6075, 46101}, {6084, 23729}, {6545, 17056}, {6586, 43060}, {6587, 7178}, {6627, 41180}, {6791, 17058}, {7004, 23647}, {7199, 18154}, {7265, 23879}, {7668, 38990}, {8029, 12072}, {8054, 40623}, {8286, 38375}, {8635, 21005}, {8645, 23865}, {8689, 45673}, {8713, 14330}, {8819, 40628}, {9258, 17871}, {9810, 18200}, {10196, 25381}, {10933, 15526}, {11124, 22080}, {14437, 24457}, {14470, 29226}, {14936, 38389}, {15523, 21962}, {16546, 17799}, {17072, 25627}, {17435, 42753}, {17452, 24117}, {17893, 20440}, {18155, 23658}, {18197, 24900}, {18210, 36197}, {18591, 42769}, {18635, 23730}, {19584, 30584}, {20483, 20659}, {20508, 28006}, {20954, 29771}, {20980, 22383}, {21054, 23941}, {21117, 21141}, {21191, 29978}, {21232, 27071}, {21260, 23818}, {21272, 26794}, {21297, 26798}, {21301, 29051}, {21339, 38358}, {21604, 33946}, {21719, 21958}, {22043, 24083}, {22260, 23928}, {22382, 23695}, {23090, 30212}, {23466, 27014}, {23764, 38930}, {23897, 24119}, {24019, 36131}, {24103, 27042}, {24130, 26772}, {24459, 27731}, {24506, 45686}, {24562, 25924}, {24719, 29362}, {26049, 27345}, {26114, 27265}, {26777, 26853}, {27081, 31992}, {27440, 27451}, {27486, 28867}, {27575, 27587}, {27648, 27673}, {28225, 43061}, {28292, 38329}, {28374, 28398}, {29029, 33299}, {30061, 30094}, {31207, 31287}, {33570, 42438}, {35068, 35123}, {35069, 35129}, {37998, 38325}, {38347, 38390}, {40134, 43924}, {40619, 44312}

X(661) = midpoint of X(i) and X(j) for these {i,j}: {649, 4813}, {2978, 20983}, {3700, 4841}, {4010, 4824}, {4024, 4988}, {4041, 4822}, {4467, 44449}, {4705, 4983}, {7192, 31290}, {17494, 20295}, {25259, 45746}
X(661) = reflection of X(i) in X(j) for these {i,j}: {2, 45315}, {649, 650}, {693, 3835}, {1019, 14838}, {1577, 4129}, {1635, 4893}, {2254, 1491}, {2484, 2509}, {2533, 21051}, {3700, 14321}, {4010, 4806}, {4024, 3700}, {4041, 4705}, {4106, 4940}, {4122, 18004}, {4369, 25666}, {4382, 4106}, {4467, 21196}, {4728, 4776}, {4729, 4041}, {4730, 4770}, {4761, 10}, {4784, 9508}, {4789, 45661}, {4790, 4394}, {4804, 4010}, {4822, 4983}, {4838, 4024}, {4895, 4775}, {4897, 17069}, {4931, 4120}, {4932, 31286}, {4948, 45676}, {4979, 649}, {4988, 4841}, {6590, 3239}, {7192, 4369}, {7199, 42327}, {16892, 3004}, {21115, 44435}, {21116, 4927}, {21146, 3837}, {21834, 4079}, {23738, 3777}, {23755, 7178}, {31148, 2}, {40471, 3005}, {43067, 4885}
X(661) = isogonal conjugate of X(662)
X(661) = isotomic conjugate of X(799)
X(661) = complement of X(7192)
X(661) = anticomplement of X(4369)
X(661) = polar conjugate of X(811)
X(661) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,244), (162,31)
X(661) = crosspoint of X(i) and X(j) for these (i,j): (92,162), (513, 514)
X(661) = crosssum of X(i) and X(j) for these (i,j): (1,661), (21,1021), (48,656), (81,1019), (100,101), (513,1100), (649,1193), (667,1197), (820,822)
X(661) = orthic-isogonal conjugate of X(2310)
X(661) = bicentric difference of PU(i) for these i: 23, 32, 35, 78, 81, 128
X(661) = PU(23)-harmonic conjugate of X(31)
X(661) = PU(32)-harmonic conjugate of X(1962)
X(661) = PU(35)-harmonic conjugate of X(38)
X(661) = barycentric product of PU(71)
X(661) = PU(78)-harmonic conjugate of X(896)
X(661) = trilinear product of PU(79)
X(661) = trilinear pole of PU(79) (line X(2642)X(2643))
X(661) = PU(81)-harmonic conjugate of X(2650)
X(661) = PU(128)-harmonic conjugate of X(63)
X(661) = perspector of the Stammler hyperbola wrt the excentral triangle
X(661) = perspector of circumconic centered at X(244), which is the hyperbola {A,B,C,X(1),X(10)}}
X(661) = center of circumconic that is locus of trilinear poles of lines passing through X(244)
X(661) = intersection of trilinear polars of X(1) and X(10)
X(661) = antigonal image of X(1338)
X(661) = pole wrt polar circle of trilinear polar of X(811) (line X(19)X(27))
X(661) = X(48)-isoconjugate (polar conjugate) of X(811)
X(661) = X(6)-isoconjugate of X(99)
X(661) = trilinear square root of X(2643)
X(661) = trilinear product X(6)*X(523)
X(661) = trilinear product X(37)*X(513)
X(661) = trilinear product X(4)*X(647)
X(661) = trilinear product of circumcircle intercepts of line X(115)X(125)
X(661) = trilinear product of Jerabek hyperbola intercepts of orthic axis
X(661) = trilinear product of Kiepert hyperbola intercepts of Lemoine axis
X(661) = barycentric product of Kiepert hyperbola intercepts of antiorthic axis
X(661) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {100, 30660}, {256, 150}, {257, 21293}, {805, 30941}, {893, 149}, {904, 4440}, {3573, 25332}, {3903, 69}, {4594, 17137}, {4603, 17135}, {7104, 9263}, {7260, 17138}, {27805, 6327}, {29055, 7}, {30670, 4441}, {32739, 30661}, {34067, 30662}, {37137, 3434}, {40729, 148}
X(661) = X(i)-complementary conjugate of X(j) for these (i,j): {6, 17761}, {10, 21252}, {31, 244}, {37, 116}, {41, 4858}, {42, 11}, {55, 34589}, {65, 17059}, {100, 3741}, {101, 3739}, {109, 3742}, {163, 17045}, {181, 8286}, {190, 21240}, {210, 124}, {212, 34588}, {213, 1086}, {228, 2968}, {594, 21253}, {644, 21246}, {651, 17050}, {692, 1125}, {756, 125}, {765, 512}, {798, 6547}, {872, 115}, {902, 34590}, {1016, 42327}, {1018, 141}, {1020, 21258}, {1110, 523}, {1252, 4369}, {1253, 34591}, {1334, 26932}, {1400, 4904}, {1402, 3756}, {1415, 3946}, {1500, 8287}, {1783, 34830}, {1918, 1015}, {2149, 17069}, {2205, 6377}, {2209, 38986}, {2318, 123}, {3690, 34846}, {3725, 15611}, {3747, 38989}, {3939, 960}, {3949, 127}, {3952, 2887}, {4033, 626}, {4041, 46100}, {4069, 1329}, {4103, 21245}, {4551, 2886}, {4552, 17046}, {4557, 10}, {4559, 142}, {4564, 17066}, {4574, 18589}, {4849, 5510}, {4878, 5511}, {5546, 21233}, {7035, 23301}, {7084, 17463}, {7109, 16592}, {8701, 27798}, {8750, 942}, {15742, 21259}, {18098, 44312}, {20964, 40608}, {21805, 3259}, {21859, 17052}, {23067, 34822}, {23344, 34587}, {23990, 14838}, {27808, 21235}, {28615, 24185}, {30730, 21244}, {31625, 21263}, {32656, 37565}, {32665, 4395}, {32739, 3666}, {34067, 740}, {35309, 21248}, {39258, 35094}, {40521, 3454}, {41333, 35119}
X(661) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 2643}, {2, 244}, {4, 2310}, {6, 2170}, {10, 3122}, {19, 3708}, {37, 3125}, {57, 2611}, {65, 4516}, {76, 3123}, {87, 4128}, {92, 1109}, {99, 23928}, {100, 1962}, {101, 2294}, {162, 31}, {163, 1953}, {190, 2292}, {226, 3120}, {253, 24010}, {304, 17876}, {502, 21043}, {512, 21834}, {513, 512}, {514, 523}, {523, 4041}, {650, 647}, {651, 2650}, {662, 1}, {664, 23668}, {668, 3728}, {799, 38}, {811, 17872}, {823, 774}, {876, 4155}, {1018, 37}, {1019, 4132}, {1020, 65}, {1022, 4145}, {1292, 42446}, {1427, 18210}, {1577, 656}, {1783, 40977}, {1826, 21044}, {1897, 40973}, {1978, 21020}, {2051, 11}, {2127, 21723}, {2184, 2632}, {2222, 3724}, {2321, 21950}, {2616, 810}, {3064, 2501}, {3223, 4117}, {3239, 6587}, {3257, 758}, {3669, 4139}, {3835, 23301}, {3882, 10459}, {3952, 756}, {4017, 4729}, {4033, 10}, {4052, 21963}, {4083, 22226}, {4103, 1213}, {4129, 31946}, {4444, 2254}, {4551, 42}, {4552, 4642}, {4557, 21808}, {4559, 2171}, {4562, 740}, {4584, 20360}, {4593, 17446}, {4599, 17469}, {4602, 75}, {4605, 1834}, {6010, 17452}, {6011, 55}, {7096, 21340}, {7178, 4017}, {7192, 40471}, {7199, 4151}, {8056, 17476}, {8818, 115}, {13478, 7004}, {14554, 1647}, {15322, 16777}, {15455, 24443}, {16606, 3121}, {17758, 1086}, {18070, 1577}, {18795, 38978}, {20570, 23672}, {20691, 22227}, {21353, 21824}, {21604, 27707}, {21859, 2092}, {23894, 2642}, {24019, 19}, {27805, 2}, {32676, 17442}, {32678, 2173}, {32680, 1725}, {35342, 3723}, {35354, 2610}, {36084, 8772}, {36085, 896}, {36086, 3747}, {36104, 2312}, {36126, 2181}, {36142, 18669}, {36145, 48}, {36146, 44661}, {36148, 17438}, {36632, 23953}, {36907, 4466}, {37134, 1959}, {37212, 3743}, {37216, 36263}, {39733, 17879}, {39735, 1111}, {39748, 3248}, {39797, 17463}, {39798, 1015}, {39957, 4475}, {39979, 27846}, {39981, 2087}, {40011, 23676}, {40515, 2486}, {40521, 16589}, {43739, 38345}
X(661) = X(i)-cross conjugate of X(j) for these (i,j): {1, 9396}, {115, 2171}, {512, 4017}, {810, 656}, {1084, 7148}, {2084, 798}, {2632, 2156}, {2642, 23894}, {2643, 1}, {3122, 10}, {3124, 756}, {3125, 37}, {3708, 19}, {3709, 4041}, {4079, 512}, {4155, 876}, {4516, 65}, {4705, 523}, {4983, 513}, {8034, 244}, {8061, 1577}, {16592, 2}, {20975, 1254}, {20982, 6}, {21051, 21834}, {36197, 1824}, {37754, 2155}
X(661) = cevapoint of X(i) and X(j) for these (i,j): {1, 2640}, {2, 21220}, {6, 21004}, {10, 21100}, {37, 21888}, {512, 3709}, {523, 21051}, {798, 810}, {1577, 20910}, {2084, 8061}, {3124, 8034}, {4024, 21720}, {4079, 4705}
X(661) = crosspoint of X(i) and X(j) for these (i,j): {1, 662}, {2, 3952}, {6, 4559}, {10, 4033}, {19, 24019}, {37, 1018}, {57, 26700}, {65, 1020}, {75, 4602}, {92, 162}, {100, 1255}, {101, 2259}, {163, 2148}, {190, 1220}, {226, 4551}, {513, 514}, {523, 7178}, {650, 3064}, {651, 17097}, {668, 1221}, {799, 3112}, {821, 823}, {1168, 3257}, {1577, 24006}, {2222, 34535}, {2576, 2577}, {2580, 2587}, {2581, 2586}, {4562, 30663}, {22116, 40526}, {32676, 46289}, {36127, 40573}
X(661) = crosssum of X(i) and X(j) for these (i,j): {1, 661}, {2, 4560}, {3, 23090}, {6, 3733}, {9, 35057}, {21, 1021}, {31, 1924}, {48, 656}, {63, 24018}, {81, 1019}, {100, 101}, {110, 5546}, {163, 4575}, {214, 1635}, {284, 3737}, {513, 1100}, {514, 5249}, {523, 5949}, {526, 35069}, {649, 1193}, {650, 2646}, {651, 1813}, {667, 1197}, {693, 17866}, {798, 1964}, {820, 822}, {1577, 14213}, {1930, 14208}, {2238, 38348}, {2578, 2585}, {2579, 2584}, {2582, 2583}, {2642, 42081}, {3687, 6332}, {3738, 34544}, {7192, 17169}, {8061, 17457}, {8300, 8632}, {14838, 40214}
X(661) = trilinear pole of line {2642, 2643}
X(661) = crossdifference of every pair of points on line {1, 21}
X(661) = X(i)-isoconjugate of X(j) for these (i,j): {1, 662}, {2, 110}, {3, 648}, {4, 4558}, {5, 18315}, {6, 99}, {7, 5546}, {8, 4565}, {9, 1414}, {10, 4556}, {13, 17402}, {14, 17403}, {15, 23895}, {16, 23896}, {19, 4592}, {21, 651}, {22, 44766}, {23, 17708}, {25, 4563}, {27, 1331}, {28, 1332}, {29, 1813}, {30, 44769}, {31, 799}, {32, 670}, {38, 4599}, {39, 4577}, {41, 4625}, {42, 4610}, {44, 4622}, {48, 811}, {49, 38342}, {50, 35139}, {54, 14570}, {55, 4573}, {56, 645}, {57, 643}, {58, 190}, {59, 4560}, {60, 4552}, {61, 32036}, {62, 32037}, {63, 162}, {64, 36841}, {65, 4612}, {66, 4611}, {68, 41679}, {69, 112}, {74, 2407}, {75, 163}, {76, 1576}, {81, 100}, {83, 1634}, {86, 101}, {92, 4575}, {95, 1625}, {97, 35360}, {98, 2421}, {107, 394}, {108, 1812}, {109, 333}, {111, 5468}, {141, 827}, {154, 44326}, {171, 4603}, {172, 4594}, {183, 26714}, {184, 6331}, {187, 892}, {193, 3565}, {200, 4637}, {213, 4623}, {216, 18831}, {220, 4616}, {222, 36797}, {226, 4636}, {230, 10425}, {232, 17932}, {237, 43187}, {238, 4584}, {248, 877}, {249, 523}, {250, 525}, {251, 4576}, {255, 823}, {261, 4559}, {264, 32661}, {265, 14590}, {269, 7259}, {274, 692}, {275, 23181}, {283, 653}, {284, 664}, {286, 906}, {287, 4230}, {290, 14966}, {297, 43754}, {298, 5995}, {299, 5994}, {302, 16806}, {303, 16807}, {304, 32676}, {310, 32739}, {311, 14586}, {314, 1415}, {323, 476}, {324, 15958}, {325, 2715}, {326, 24019}, {328, 14591}, {332, 32674}, {340, 32662}, {343, 933}, {352, 9080}, {385, 805}, {391, 5545}, {395, 10410}, {396, 10409}, {403, 43755}, {418, 42405}, {441, 44770}, {470, 38414}, {471, 38413}, {491, 39384}, {492, 39383}, {511, 2966}, {512, 4590}, {513, 4567}, {514, 4570}, {519, 4591}, {520, 23582}, {524, 691}, {526, 39295}, {538, 32717}, {542, 5649}, {560, 4602}, {571, 46134}, {574, 35138}, {576, 35178}, {577, 6528}, {593, 3952}, {595, 37205}, {597, 12074}, {598, 9145}, {599, 11636}, {604, 7257}, {644, 1014}, {646, 1408}, {647, 18020}, {649, 4600}, {658, 2328}, {661, 24041}, {663, 4620}, {666, 3286}, {667, 4601}, {668, 1333}, {669, 34537}, {671, 5467}, {677, 14953}, {685, 36212}, {687, 13754}, {689, 3051}, {694, 17941}, {729, 23342}, {741, 3570}, {757, 1018}, {758, 37140}, {759, 4585}, {763, 40521}, {765, 1019}, {785, 27164}, {798, 24037}, {801, 1624}, {810, 46254}, {813, 33295}, {822, 23999}, {825, 30966}, {842, 14999}, {843, 9182}, {849, 4033}, {850, 23357}, {859, 13136}, {874, 18268}, {880, 9468}, {886, 33875}, {895, 4235}, {896, 36085}, {897, 23889}, {901, 16704}, {902, 4615}, {905, 5379}, {907, 3618}, {919, 30941}, {923, 24039}, {925, 1993}, {930, 1994}, {931, 940}, {932, 27644}, {934, 2287}, {935, 22151}, {1016, 3733}, {1020, 1098}, {1021, 7045}, {1043, 1461}, {1092, 15352}, {1100, 4596}, {1101, 1577}, {1106, 7258}, {1110, 7199}, {1113, 8116}, {1114, 8115}, {1125, 4629}, {1147, 30450}, {1171, 4427}, {1172, 6516}, {1176, 41676}, {1178, 18047}, {1213, 6578}, {1252, 7192}, {1253, 4635}, {1262, 7253}, {1275, 21789}, {1289, 20806}, {1290, 37783}, {1291, 37779}, {1292, 41610}, {1293, 41629}, {1296, 1992}, {1297, 34211}, {1301, 37669}, {1302, 15066}, {1304, 11064}, {1306, 3068}, {1307, 3069}, {1310, 2303}, {1326, 35148}, {1379, 6189}, {1380, 6190}, {1383, 9146}, {1384, 35179}, {1396, 4571}, {1402, 4631}, {1407, 7256}, {1412, 3699}, {1434, 3939}, {1437, 6335}, {1444, 1783}, {1449, 4614}, {1474, 4561}, {1492, 40773}, {1494, 2420}, {1501, 4609}, {1502, 14574}, {1509, 4557}, {1511, 39290}, {1580, 37134}, {1613, 3222}, {1633, 40403}, {1636, 42308}, {1648, 45773}, {1649, 34539}, {1691, 18829}, {1755, 36036}, {1790, 1897}, {1792, 32714}, {1799, 35325}, {1814, 4238}, {1815, 4241}, {1817, 13138}, {1821, 23997}, {1822, 2581}, {1823, 2580}, {1914, 4589}, {1923, 37204}, {1930, 34072}, {1931, 37135}, {1936, 41206}, {1959, 36084}, {1964, 4593}, {1976, 2396}, {1978, 2206}, {1983, 14616}, {1989, 10411}, {2030, 46144}, {2063, 30249}, {2149, 18155}, {2167, 2617}, {2185, 4551}, {2193, 18026}, {2194, 4554}, {2210, 4639}, {2234, 36133}, {2238, 36066}, {2251, 4634}, {2308, 4632}, {2327, 36118}, {2360, 44327}, {2363, 3882}, {2410, 34210}, {2482, 34574}, {2502, 9170}, {2574, 39298}, {2575, 39299}, {2644, 9395}, {2696, 41617}, {2701, 40882}, {2702, 17731}, {2703, 19623}, {2705, 37792}, {2709, 22329}, {2713, 40888}, {2965, 46139}, {2981, 35314}, {2986, 15329}, {2987, 4226}, {2988, 7450}, {2989, 4243}, {2990, 3658}, {2991, 4236}, {3003, 18878}, {3053, 35136}, {3094, 33514}, {3108, 10330}, {3124, 31614}, {3164, 44828}, {3180, 36515}, {3181, 36514}, {3219, 13486}, {3225, 41337}, {3228, 5118}, {3231, 9150}, {3233, 40384}, {3260, 32640}, {3265, 23964}, {3266, 32729}, {3284, 16077}, {3285, 4555}, {3289, 22456}, {3329, 43357}, {3448, 40173}, {3564, 32697}, {3573, 37128}, {3580, 10420}, {3589, 7953}, {3616, 4627}, {3709, 7340}, {3736, 4586}, {3737, 4564}, {3763, 7954}, {3917, 42396}, {3926, 32713}, {3936, 36069}, {3964, 6529}, {3978, 17938}, {4184, 43190}, {4225, 44765}, {4240, 14919}, {4267, 6648}, {4273, 4597}, {4282, 35174}, {4383, 8690}, {4436, 40408}, {4481, 5384}, {4562, 5009}, {4566, 7054}, {4579, 40432}, {4588, 5235}, {4598, 38832}, {4604, 4653}, {4628, 16887}, {4630, 8024}, {4658, 37211}, {4833, 5385}, {4921, 8697}, {4998, 7252}, {5006, 35147}, {5007, 35137}, {5008, 42367}, {5012, 11794}, {5027, 39292}, {5060, 35154}, {5191, 6035}, {5291, 17929}, {5333, 8652}, {5380, 16702}, {5383, 16695}, {5422, 43351}, {5502, 44877}, {5504, 16237}, {5562, 16813}, {5663, 30528}, {5664, 15395}, {5970, 14607}, {6064, 7180}, {6065, 17096}, {6082, 11580}, {6083, 35466}, {6149, 32680}, {6151, 35315}, {6233, 11163}, {6370, 9273}, {6393, 32696}, {6507, 36126}, {6514, 36127}, {6515, 13398}, {6517, 8748}, {6542, 17940}, {6572, 16285}, {6577, 29767}, {6650, 17943}, {6742, 40214}, {6759, 30441}, {7121, 36860}, {7122, 7260}, {7239, 7305}, {7254, 15742}, {7341, 30730}, {7488, 16039}, {7578, 36829}, {7735, 35575}, {7763, 32734}, {7766, 25424}, {7769, 32737}, {7799, 14560}, {7840, 32694}, {8025, 8701}, {8059, 27398}, {8594, 9197}, {8595, 9196}, {8598, 9190}, {8600, 8860}, {8623, 41209}, {8673, 44183}, {8694, 42028}, {8707, 40153}, {8708, 18166}, {8750, 17206}, {8822, 36049}, {9058, 26637}, {9060, 40112}, {9063, 18899}, {9066, 9463}, {9124, 26613}, {9160, 40879}, {9181, 18823}, {9202, 37786}, {9203, 37785}, {9217, 31998}, {9218, 35511}, {9306, 43188}, {9426, 44168}, {9513, 40866}, {10302, 35357}, {11003, 36886}, {11130, 36840}, {11131, 36839}, {11591, 30527}, {11634, 41909}, {11849, 34357}, {13397, 40571}, {13485, 36830}, {13571, 13578}, {14206, 36034}, {14210, 36142}, {14213, 36134}, {14587, 18314}, {14614, 39639}, {15384, 20580}, {15455, 17104}, {15534, 33638}, {15631, 41932}, {16163, 34568}, {16705, 32736}, {16948, 27834}, {17139, 32641}, {17185, 36098}, {17198, 31616}, {17277, 43076}, {17379, 43359}, {17735, 17930}, {17790, 17939}, {17931, 17966}, {17933, 17963}, {17934, 17962}, {17935, 17961}, {17942, 17947}, {17944, 17946}, {18122, 39448}, {18157, 32666}, {18191, 31615}, {18206, 36086}, {19622, 35156}, {20185, 41628}, {20189, 34545}, {20948, 23995}, {20998, 37880}, {22052, 33513}, {23067, 46103}, {23189, 46102}, {23963, 44173}, {24000, 24018}, {24001, 35200}, {25507, 28148}, {26860, 28210}, {27867, 31296}, {27958, 29055}, {28196, 42025}, {28419, 39417}, {28724, 46151}, {30247, 41614}, {30512, 43756}, {30939, 32665}, {30940, 34067}, {31623, 36059}, {31626, 35311}, {31631, 36082}, {32014, 35327}, {32656, 44129}, {32660, 44130}, {32671, 35550}, {32692, 39113}, {32738, 32833}, {32911, 34594}, {33296, 34071}, {33803, 36953}, {33946, 38813}, {35049, 35057}, {35193, 38340}, {35278, 40802}, {35296, 44768}, {35319, 39287}, {35324, 40410}, {35329, 40707}, {35330, 40706}, {35342, 40438}, {35356, 39389}, {36145, 44179}, {36213, 39291}, {36277, 37216}, {36306, 44718}, {36309, 44719}, {36790, 41173}, {36831, 43768}, {37183, 44767}, {37215, 44119}, {37685, 43356}, {39054, 39137}, {39469, 41174}, {40441, 41677}, {41331, 42371}, {41517, 46294}, {42294, 42296}, {42295, 42297}
X(661) = barycentric product X(i)*X(j) for these {i,j}: {1, 523}, {3, 24006}, {4, 656}, {5, 2616}, {6, 1577}, {7, 4041}, {8, 4017}, {9, 7178}, {10, 513}, {11, 4551}, {12, 3737}, {19, 525}, {25, 14208}, {28, 4064}, {31, 850}, {32, 20948}, {33, 17094}, {37, 514}, {39, 18070}, {42, 693}, {44, 4049}, {48, 14618}, {54, 2618}, {55, 4077}, {56, 4086}, {57, 3700}, {58, 4036}, {63, 2501}, {64, 17898}, {65, 522}, {71, 17924}, {72, 7649}, {73, 44426}, {74, 36035}, {75, 512}, {76, 798}, {81, 4024}, {82, 826}, {83, 8061}, {85, 3709}, {86, 4705}, {87, 21051}, {88, 4120}, {89, 4931}, {91, 924}, {92, 647}, {94, 2624}, {99, 2643}, {100, 3120}, {101, 16732}, {105, 4088}, {107, 2632}, {110, 1109}, {112, 20902}, {115, 662}, {125, 162}, {148, 9396}, {158, 520}, {163, 338}, {181, 18155}, {190, 3125}, {210, 3676}, {213, 3261}, {225, 521}, {226, 650}, {228, 46107}, {238, 35352}, {240, 879}, {244, 3952}, {256, 2533}, {264, 810}, {274, 4079}, {278, 8611}, {279, 4171}, {291, 4010}, {293, 16230}, {304, 2489}, {306, 6591}, {307, 18344}, {308, 2084}, {312, 7180}, {313, 667}, {321, 649}, {330, 21834}, {334, 4455}, {335, 21832}, {336, 17994}, {339, 32676}, {341, 7250}, {346, 7216}, {349, 3063}, {351, 46277}, {393, 24018}, {502, 31947}, {524, 23894}, {526, 2166}, {560, 44173}, {561, 669}, {594, 1019}, {596, 4132}, {643, 1365}, {648, 3708}, {651, 21044}, {652, 40149}, {657, 1446}, {658, 36197}, {659, 43534}, {663, 1441}, {664, 4516}, {668, 3122}, {671, 2642}, {673, 24290}, {684, 36120}, {688, 18833}, {690, 897}, {692, 21207}, {740, 876}, {756, 7192}, {759, 6370}, {799, 3124}, {804, 1581}, {811, 20975}, {822, 2052}, {823, 3269}, {824, 40747}, {868, 36084}, {875, 35544}, {878, 40703}, {881, 1926}, {882, 1966}, {891, 41683}, {896, 5466}, {899, 35353}, {900, 4674}, {903, 4730}, {905, 1826}, {918, 18785}, {923, 35522}, {943, 23752}, {983, 3801}, {985, 4122}, {1002, 4804}, {1015, 4033}, {1018, 1086}, {1020, 1146}, {1021, 6354}, {1022, 3943}, {1027, 3932}, {1042, 4397}, {1084, 4602}, {1088, 4524}, {1089, 3733}, {1096, 3265}, {1100, 31010}, {1101, 23105}, {1111, 4557}, {1126, 30591}, {1156, 30574}, {1214, 3064}, {1221, 40627}, {1245, 2517}, {1254, 7253}, {1255, 4988}, {1268, 4983}, {1292, 21945}, {1320, 30572}, {1334, 24002}, {1358, 4069}, {1400, 4391}, {1402, 35519}, {1409, 46110}, {1414, 4092}, {1427, 3239}, {1432, 4140}, {1459, 41013}, {1491, 40718}, {1500, 7199}, {1502, 1924}, {1510, 2962}, {1576, 23994}, {1635, 4080}, {1637, 2349}, {1648, 36085}, {1725, 15328}, {1733, 35364}, {1734, 15320}, {1737, 3657}, {1755, 43665}, {1757, 18014}, {1758, 18013}, {1769, 38955}, {1783, 4466}, {1784, 14380}, {1821, 3569}, {1822, 39240}, {1823, 39241}, {1824, 4025}, {1861, 10099}, {1880, 6332}, {1897, 18210}, {1903, 14837}, {1910, 2799}, {1918, 40495}, {1919, 27801}, {1928, 9426}, {1929, 18004}, {1930, 18105}, {1934, 5027}, {1953, 15412}, {1956, 6130}, {1959, 2395}, {1962, 4608}, {1967, 14295}, {1969, 3049}, {1973, 3267}, {1978, 3121}, {1989, 32679}, {2088, 32680}, {2148, 18314}, {2153, 23870}, {2154, 23871}, {2156, 33294}, {2157, 9979}, {2159, 41079}, {2160, 7265}, {2161, 4707}, {2167, 12077}, {2169, 23290}, {2170, 4552}, {2171, 4560}, {2173, 2394}, {2184, 6587}, {2186, 23878}, {2190, 6368}, {2214, 23879}, {2238, 4444}, {2247, 14223}, {2250, 10015}, {2254, 13576}, {2292, 4581}, {2298, 21124}, {2310, 4566}, {2312, 43673}, {2321, 3669}, {2333, 15413}, {2334, 4815}, {2357, 17896}, {2401, 21801}, {2422, 46238}, {2433, 14206}, {2491, 46273}, {2509, 36907}, {2530, 18082}, {2574, 2589}, {2575, 2588}, {2578, 2593}, {2579, 2592}, {2582, 8106}, {2583, 8105}, {2605, 6757}, {2606, 7137}, {2607, 2614}, {2610, 24624}, {2611, 6742}, {2617, 8901}, {2623, 14213}, {2627, 18114}, {2631, 16080}, {2640, 9293}, {2648, 18006}, {2652, 2785}, {2786, 9278}, {2970, 4575}, {2972, 36126}, {3005, 3112}, {3221, 18832}, {3223, 23301}, {3224, 20910}, {3248, 27808}, {3293, 40086}, {3445, 4404}, {3456, 18076}, {3500, 21958}, {3565, 17876}, {3566, 8769}, {3572, 3948}, {3668, 3900}, {3701, 43924}, {3710, 43923}, {3800, 23051}, {3835, 16606}, {3837, 18793}, {3949, 17925}, {3954, 10566}, {3971, 43931}, {3992, 23345}, {3994, 43928}, {4052, 4394}, {4063, 40085}, {4082, 43932}, {4083, 42027}, {4103, 16726}, {4117, 4609}, {4129, 39798}, {4139, 34860}, {4145, 39697}, {4151, 13476}, {4155, 18827}, {4170, 7241}, {4373, 4729}, {4453, 34857}, {4475, 4613}, {4492, 4761}, {4559, 4858}, {4562, 39786}, {4567, 21131}, {4580, 17442}, {4592, 8754}, {4594, 21725}, {4599, 39691}, {4610, 21833}, {4695, 23836}, {4770, 39704}, {4782, 34475}, {4791, 28658}, {4822, 5936}, {4823, 28625}, {4824, 30571}, {4832, 40023}, {4838, 25417}, {4841, 25430}, {4876, 7212}, {4893, 30588}, {4979, 6539}, {5379, 21134}, {5489, 24000}, {5620, 8674}, {6003, 41501}, {6011, 8286}, {6057, 7203}, {6129, 39130}, {6149, 10412}, {6164, 21093}, {6358, 7252}, {6367, 40438}, {6521, 32320}, {6544, 30575}, {6548, 21805}, {6724, 6728}, {6741, 26700}, {6791, 37216}, {7018, 7234}, {7035, 8034}, {7148, 17217}, {7237, 7255}, {7260, 21823}, {8029, 24041}, {8056, 14321}, {8599, 36263}, {8672, 31359}, {8714, 40504}, {8808, 14298}, {8809, 14308}, {8811, 14302}, {8818, 14838}, {9033, 36119}, {9148, 37132}, {9178, 14210}, {9255, 16229}, {9258, 30476}, {9267, 21100}, {9292, 17893}, {9307, 17478}, {9309, 21052}, {9395, 10278}, {9402, 18298}, {9404, 43682}, {9508, 11599}, {10693, 21180}, {13212, 36117}, {13486, 21054}, {14207, 21448}, {14220, 36063}, {14398, 33805}, {14404, 31002}, {14407, 20568}, {14417, 36128}, {14429, 36125}, {14431, 37129}, {14554, 21894}, {14616, 42666}, {15232, 21189}, {15313, 23604}, {15352, 37754}, {15451, 40440}, {15455, 20982}, {15523, 18108}, {15526, 24019}, {16186, 36129}, {16592, 27805}, {16892, 18098}, {17197, 21859}, {17205, 40521}, {17415, 46281}, {17431, 19218}, {17432, 19217}, {17438, 39183}, {17467, 42345}, {17469, 31065}, {17763, 18015}, {17768, 35347}, {17879, 32713}, {17881, 32734}, {17926, 37755}, {17954, 18003}, {17955, 34763}, {17956, 18008}, {17957, 18010}, {17990, 18032}, {18001, 20947}, {18359, 21828}, {18691, 46005}, {19604, 44729}, {19611, 44705}, {20332, 21053}, {20571, 34952}, {20906, 23493}, {21186, 43703}, {21192, 21353}, {21727, 39734}, {21836, 42328}, {21867, 26721}, {21888, 42555}, {21950, 27834}, {21951, 25576}, {22260, 24037}, {23285, 46289}, {23503, 40162}, {23785, 40516}, {23800, 41506}, {23838, 40663}, {27807, 40471}, {28161, 31503}, {31946, 39748}, {34920, 38469}, {35235, 36061}, {35354, 35466}, {36036, 44114}, {36037, 42759}, {40327, 40345}, {40437, 42768}, {42653, 44188}
X(661) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 99}, {2, 799}, {3, 4592}, {4, 811}, {6, 662}, {7, 4625}, {8, 7257}, {9, 645}, {10, 668}, {11, 18155}, {19, 648}, {25, 162}, {31, 110}, {32, 163}, {33, 36797}, {37, 190}, {38, 4576}, {41, 5546}, {42, 100}, {48, 4558}, {51, 2617}, {55, 643}, {56, 1414}, {57, 4573}, {63, 4563}, {65, 664}, {71, 1332}, {72, 4561}, {73, 6516}, {75, 670}, {76, 4602}, {81, 4610}, {82, 4577}, {83, 4593}, {86, 4623}, {88, 4615}, {91, 46134}, {92, 6331}, {98, 36036}, {99, 24037}, {100, 4600}, {101, 4567}, {106, 4622}, {107, 23999}, {110, 24041}, {111, 36085}, {115, 1577}, {125, 14208}, {158, 6528}, {162, 18020}, {163, 249}, {181, 4551}, {184, 4575}, {187, 23889}, {190, 4601}, {192, 36860}, {200, 7256}, {210, 3699}, {213, 101}, {220, 7259}, {225, 18026}, {226, 4554}, {228, 1331}, {237, 23997}, {240, 877}, {244, 7192}, {251, 4599}, {256, 4594}, {257, 7260}, {269, 4616}, {279, 4635}, {284, 4612}, {291, 4589}, {292, 4584}, {293, 17932}, {308, 37204}, {313, 6386}, {321, 1978}, {333, 4631}, {335, 4639}, {338, 20948}, {346, 7258}, {351, 896}, {393, 823}, {512, 1}, {513, 86}, {514, 274}, {520, 326}, {521, 332}, {522, 314}, {523, 75}, {524, 24039}, {525, 304}, {560, 1576}, {561, 4609}, {594, 4033}, {604, 4565}, {610, 36841}, {643, 6064}, {647, 63}, {648, 46254}, {649, 81}, {650, 333}, {651, 4620}, {652, 1812}, {656, 69}, {657, 2287}, {659, 33295}, {662, 4590}, {663, 21}, {665, 18206}, {667, 58}, {669, 31}, {688, 1964}, {690, 14210}, {692, 4570}, {693, 310}, {694, 37134}, {729, 36133}, {740, 874}, {741, 36066}, {756, 3952}, {762, 4103}, {764, 17205}, {784, 10471}, {788, 3736}, {798, 6}, {799, 34537}, {804, 1966}, {810, 3}, {812, 30940}, {822, 394}, {826, 1930}, {850, 561}, {872, 4557}, {875, 741}, {876, 18827}, {878, 293}, {879, 336}, {881, 1967}, {882, 1581}, {888, 2234}, {893, 4603}, {896, 5468}, {897, 892}, {900, 30939}, {903, 4634}, {905, 17206}, {918, 18157}, {922, 5467}, {923, 691}, {924, 44179}, {1015, 1019}, {1018, 1016}, {1019, 1509}, {1020, 1275}, {1021, 7058}, {1042, 934}, {1084, 798}, {1086, 7199}, {1089, 27808}, {1096, 107}, {1109, 850}, {1126, 4596}, {1245, 1310}, {1254, 4566}, {1255, 4632}, {1333, 4556}, {1334, 644}, {1357, 7203}, {1365, 4077}, {1400, 651}, {1402, 109}, {1407, 4637}, {1409, 1813}, {1414, 7340}, {1426, 36118}, {1427, 658}, {1441, 4572}, {1459, 1444}, {1491, 30966}, {1500, 1018}, {1576, 1101}, {1577, 76}, {1580, 17941}, {1581, 18829}, {1635, 16704}, {1637, 14206}, {1649, 24038}, {1734, 33297}, {1755, 2421}, {1757, 17934}, {1758, 17933}, {1769, 17139}, {1821, 43187}, {1824, 1897}, {1826, 6335}, {1880, 653}, {1903, 44327}, {1910, 2966}, {1917, 14574}, {1918, 692}, {1919, 1333}, {1924, 32}, {1927, 17938}, {1929, 17930}, {1945, 41206}, {1946, 283}, {1953, 14570}, {1959, 2396}, {1962, 4427}, {1964, 1634}, {1966, 880}, {1967, 805}, {1973, 112}, {1974, 32676}, {1976, 36084}, {1980, 2206}, {1989, 32680}, {1990, 24001}, {2054, 37135}, {2084, 39}, {2088, 32679}, {2092, 3882}, {2148, 18315}, {2151, 17402}, {2152, 17403}, {2153, 23895}, {2154, 23896}, {2156, 44766}, {2157, 17708}, {2159, 44769}, {2166, 35139}, {2170, 4560}, {2171, 4552}, {2172, 4611}, {2173, 2407}, {2179, 1625}, {2181, 35360}, {2184, 44326}, {2190, 18831}, {2194, 4636}, {2200, 906}, {2205, 32739}, {2207, 24019}, {2234, 23342}, {2238, 3570}, {2245, 4585}, {2247, 14999}, {2250, 13136}, {2254, 30941}, {2258, 931}, {2295, 18047}, {2310, 7253}, {2312, 34211}, {2318, 4571}, {2321, 646}, {2333, 1783}, {2334, 4614}, {2356, 4238}, {2357, 13138}, {2394, 33805}, {2395, 1821}, {2422, 1910}, {2433, 2349}, {2451, 1958}, {2484, 2303}, {2485, 1760}, {2486, 20954}, {2488, 17194}, {2489, 19}, {2491, 1755}, {2492, 16568}, {2501, 92}, {2514, 17446}, {2517, 44154}, {2519, 2128}, {2520, 17188}, {2530, 16887}, {2533, 1909}, {2576, 39298}, {2577, 39299}, {2578, 8116}, {2579, 8115}, {2588, 15165}, {2589, 15164}, {2610, 3936}, {2611, 4467}, {2616, 95}, {2618, 311}, {2623, 2167}, {2624, 323}, {2631, 11064}, {2632, 3265}, {2640, 31998}, {2642, 524}, {2643, 523}, {2644, 31632}, {2648, 17931}, {2650, 17136}, {2652, 35154}, {2667, 4436}, {2787, 5209}, {2799, 46238}, {2962, 46139}, {2978, 10458}, {3004, 16739}, {3005, 38}, {3049, 48}, {3050, 18042}, {3063, 284}, {3064, 31623}, {3112, 689}, {3120, 693}, {3121, 649}, {3122, 513}, {3123, 17217}, {3125, 514}, {3221, 1740}, {3223, 3222}, {3248, 3733}, {3250, 40773}, {3261, 6385}, {3267, 40364}, {3269, 24018}, {3271, 3737}, {3287, 27958}, {3402, 26714}, {3563, 36105}, {3566, 18156}, {3569, 1959}, {3572, 37128}, {3589, 18062}, {3668, 4569}, {3669, 1434}, {3675, 23829}, {3700, 312}, {3708, 525}, {3709, 9}, {3721, 33946}, {3733, 757}, {3737, 261}, {3747, 3573}, {3773, 4505}, {3777, 33947}, {3778, 3888}, {3800, 39731}, {3801, 33930}, {3835, 31008}, {3900, 1043}, {3930, 42720}, {3942, 15419}, {3943, 24004}, {3948, 27853}, {3952, 7035}, {3954, 4568}, {3971, 36863}, {3994, 41314}, {4010, 350}, {4014, 17218}, {4017, 7}, {4024, 321}, {4027, 46295}, {4033, 31625}, {4036, 313}, {4041, 8}, {4049, 20568}, {4062, 42721}, {4064, 20336}, {4069, 4076}, {4077, 6063}, {4079, 37}, {4083, 33296}, {4086, 3596}, {4088, 3263}, {4092, 4086}, {4117, 669}, {4120, 4358}, {4122, 33931}, {4128, 4367}, {4129, 18140}, {4132, 4360}, {4139, 3875}, {4140, 17787}, {4145, 17160}, {4151, 17143}, {4155, 740}, {4171, 346}, {4367, 17103}, {4369, 8033}, {4391, 28660}, {4394, 41629}, {4415, 21580}, {4444, 40017}, {4455, 238}, {4466, 15413}, {4501, 4483}, {4515, 6558}, {4516, 522}, {4524, 200}, {4531, 40499}, {4551, 4998}, {4557, 765}, {4559, 4564}, {4602, 44168}, {4642, 21272}, {4674, 4555}, {4705, 10}, {4707, 20924}, {4729, 145}, {4730, 519}, {4750, 16741}, {4770, 3679}, {4775, 4653}, {4790, 42028}, {4804, 4441}, {4806, 30963}, {4808, 33937}, {4813, 5333}, {4814, 4720}, {4822, 3616}, {4825, 4803}, {4826, 16777}, {4832, 1449}, {4834, 4658}, {4838, 28605}, {4841, 19804}, {4843, 4673}, {4849, 43290}, {4893, 5235}, {4931, 4671}, {4977, 16709}, {4979, 8025}, {4983, 1125}, {4988, 4359}, {5027, 1580}, {5029, 1931}, {5075, 2651}, {5113, 17799}, {5466, 46277}, {5489, 17879}, {5620, 35156}, {6004, 33953}, {6041, 2247}, {6129, 8822}, {6140, 1749}, {6149, 10411}, {6186, 13486}, {6367, 4647}, {6368, 18695}, {6370, 35550}, {6372, 17175}, {6373, 18792}, {6377, 18197}, {6520, 15352}, {6524, 36126}, {6544, 16729}, {6545, 16727}, {6587, 18750}, {6591, 27}, {6615, 17183}, {6753, 1748}, {6791, 14207}, {7064, 4069}, {7170, 9425}, {7178, 85}, {7180, 57}, {7192, 873}, {7200, 16737}, {7202, 16755}, {7203, 552}, {7212, 10030}, {7216, 279}, {7234, 171}, {7250, 269}, {7252, 2185}, {7265, 33939}, {7649, 286}, {7745, 18063}, {8029, 1109}, {8034, 244}, {8061, 141}, {8105, 2581}, {8106, 2580}, {8287, 18160}, {8574, 16562}, {8611, 345}, {8639, 1468}, {8640, 38832}, {8641, 2328}, {8643, 16948}, {8644, 36277}, {8646, 44119}, {8651, 1707}, {8653, 4512}, {8655, 39673}, {8663, 1962}, {8664, 17469}, {8672, 10436}, {8678, 1010}, {8680, 15418}, {8750, 5379}, {8754, 24006}, {8769, 35136}, {8772, 4226}, {8818, 15455}, {9009, 36289}, {9178, 897}, {9247, 32661}, {9258, 43188}, {9278, 35148}, {9279, 24342}, {9395, 37880}, {9396, 35511}, {9402, 1045}, {9406, 2420}, {9417, 14966}, {9426, 560}, {9427, 1924}, {9456, 4591}, {9494, 1923}, {9508, 17731}, {9979, 20944}, {10099, 31637}, {10278, 20939}, {11060, 32678}, {11123, 20903}, {12071, 42005}, {12077, 14213}, {14061, 33809}, {14207, 11059}, {14208, 305}, {14270, 6149}, {14295, 1926}, {14298, 27398}, {14321, 18743}, {14398, 2173}, {14399, 18653}, {14404, 899}, {14407, 44}, {14419, 6629}, {14428, 2244}, {14431, 6381}, {14574, 23995}, {14618, 1969}, {14838, 34016}, {14936, 1021}, {15451, 44706}, {15475, 2166}, {16230, 40703}, {16583, 3732}, {16592, 4369}, {16600, 33951}, {16606, 4598}, {16613, 6002}, {16732, 3261}, {16892, 16703}, {17094, 7182}, {17414, 36263}, {17415, 3116}, {17442, 41676}, {17467, 14588}, {17469, 10330}, {17476, 4897}, {17478, 1975}, {17747, 42719}, {17763, 17935}, {17898, 14615}, {17924, 44129}, {17954, 17929}, {17955, 34760}, {17958, 17936}, {17959, 17937}, {17989, 17763}, {17990, 1757}, {17991, 17960}, {17992, 1758}, {17993, 17955}, {17994, 240}, {17995, 18028}, {17997, 34054}, {17999, 17959}, {18000, 2648}, {18001, 1929}, {18002, 17954}, {18004, 20947}, {18014, 18032}, {18070, 308}, {18105, 82}, {18155, 18021}, {18197, 7304}, {18210, 4025}, {18266, 17943}, {18344, 29}, {18384, 36129}, {18785, 666}, {18793, 8709}, {18833, 42371}, {19610, 9396}, {20683, 1026}, {20691, 4595}, {20902, 3267}, {20910, 6374}, {20948, 1502}, {20964, 4579}, {20966, 3909}, {20970, 35342}, {20974, 16751}, {20975, 656}, {20979, 27644}, {20982, 14838}, {20998, 2644}, {21004, 39054}, {21006, 33760}, {21035, 4553}, {21043, 4036}, {21044, 4391}, {21051, 6376}, {21055, 30473}, {21056, 22028}, {21100, 9296}, {21104, 16708}, {21108, 16747}, {21123, 16696}, {21124, 20911}, {21127, 16713}, {21131, 16732}, {21143, 16726}, {21144, 23807}, {21207, 40495}, {21448, 37216}, {21720, 40603}, {21725, 2533}, {21727, 4651}, {21731, 1725}, {21755, 20981}, {21759, 34071}, {21789, 1098}, {21795, 35341}, {21796, 21362}, {21801, 2397}, {21805, 17780}, {21806, 4781}, {21809, 25268}, {21814, 46148}, {21816, 4115}, {21824, 7265}, {21828, 3218}, {21831, 7283}, {21832, 239}, {21833, 4024}, {21834, 192}, {21835, 20979}, {21836, 25264}, {21837, 3730}, {21888, 6631}, {21893, 9362}, {21905, 17466}, {21906, 2642}, {21950, 4462}, {21958, 17786}, {21963, 4106}, {22172, 4499}, {22229, 3501}, {22260, 2643}, {22341, 6517}, {22373, 22093}, {22383, 1790}, {23105, 23994}, {23282, 42714}, {23301, 17149}, {23493, 932}, {23503, 1613}, {23610, 4117}, {23655, 13588}, {23751, 16700}, {23878, 3403}, {23879, 33935}, {23894, 671}, {23928, 21295}, {23994, 44173}, {23996, 15631}, {24006, 264}, {24018, 3926}, {24019, 23582}, {24041, 31614}, {24290, 3912}, {25430, 4633}, {28615, 4629}, {28625, 37211}, {28658, 4604}, {30574, 30806}, {30591, 1269}, {31010, 32018}, {31290, 33779}, {31296, 33764}, {31946, 18133}, {32320, 6507}, {32671, 9273}, {32676, 250}, {32678, 39295}, {32679, 7799}, {32713, 24000}, {32740, 36142}, {33294, 20641}, {34079, 37140}, {34294, 18070}, {34591, 15411}, {34952, 47}, {35347, 35141}, {35352, 334}, {35353, 31002}, {35364, 8773}, {35519, 40072}, {36035, 3260}, {36051, 10425}, {36053, 18878}, {36054, 6514}, {36119, 16077}, {36120, 22456}, {36151, 30528}, {36197, 3239}, {36263, 9146}, {36800, 36806}, {37132, 9150}, {37134, 39292}, {38237, 39337}, {38252, 3565}, {38986, 16695}, {39201, 255}, {39258, 2284}, {39786, 812}, {39798, 37205}, {40148, 34594}, {40345, 40301}, {40352, 36034}, {40354, 36131}, {40471, 17140}, {40608, 3907}, {40627, 1107}, {40718, 789}, {40747, 4586}, {40934, 1633}, {40977, 14543}, {41079, 46234}, {41221, 2618}, {41683, 889}, {42027, 18830}, {42074, 3233}, {42327, 34022}, {42653, 191}, {42661, 2292}, {42663, 8772}, {42664, 28606}, {42666, 758}, {42667, 1822}, {42668, 1823}, {42670, 5127}, {42752, 1769}, {42753, 23788}, {42759, 36038}, {43534, 4583}, {43665, 46273}, {43763, 41209}, {43924, 1014}, {44113, 4242}, {44173, 1928}, {44426, 44130}, {44445, 18064}, {44705, 1895}, {44729, 44720}, {45775, 33919}, {45801, 20941}, {45907, 19591}, {46281, 9063}, {46288, 34072}, {46289, 827}
X(661) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4369, 24924}, {2, 7192, 4369}, {2, 26545, 25008}, {2, 26775, 27114}, {2, 26822, 27167}, {2, 27469, 28372}, {2, 31290, 7192}, {115, 20982, 2170}, {647, 7180, 21828}, {649, 650, 1635}, {649, 4893, 650}, {650, 4790, 4394}, {650, 4813, 4979}, {650, 14298, 657}, {662, 2644, 24041}, {693, 3835, 4728}, {693, 4776, 3835}, {798, 8061, 656}, {1635, 4979, 649}, {2533, 21051, 21052}, {2578, 2579, 2624}, {3124, 16592, 244}, {3700, 4024, 4931}, {3700, 4988, 4838}, {3700, 8611, 4171}, {3700, 14321, 4120}, {3709, 7180, 647}, {3952, 7239, 35309}, {4024, 4120, 3700}, {4024, 42664, 21834}, {4106, 4940, 31147}, {4120, 4841, 4838}, {4120, 4988, 4024}, {4369, 4481, 28372}, {4369, 7192, 31148}, {4369, 25666, 2}, {4369, 27929, 26248}, {4369, 29717, 29402}, {4369, 45315, 25666}, {4379, 30835, 4885}, {4382, 31147, 4106}, {4394, 4790, 649}, {4444, 25666, 30765}, {4455, 7234, 669}, {4705, 4730, 4770}, {4705, 4822, 4729}, {4730, 4770, 4041}, {4806, 4824, 4804}, {4813, 4893, 649}, {4838, 4931, 4024}, {4841, 14321, 4024}, {4885, 43067, 4379}, {4897, 17069, 4750}, {4988, 14321, 4931}, {7192, 25666, 24924}, {7192, 26775, 29402}, {7192, 27647, 28372}, {16751, 24948, 14838}, {18070, 20953, 20948}, {20906, 21438, 20909}, {20949, 21611, 20952}, {24041, 36085, 2644}, {24924, 31148, 4369}, {25666, 29512, 27045}, {25666, 29717, 27114}, {25666, 31290, 31148}, {26798, 26824, 21297}, {26983, 27045, 2}, {26985, 27138, 4928}, {27013, 27115, 4763}, {27167, 29457, 24924}, {27469, 27647, 4481}, {27527, 28758, 2}, {28938, 28983, 26640}, {31290, 45315, 24924}


X(662) = TRILINEAR POLE OF LINE X(1)X(21)

Trilinears    1/(cot B - cot C) : 1/(cot C - cot A) : 1/(cot A - cot B)
Trilinears    1/(b2 - c2) : :
Trilinears    1/(cos 2B - cos 2C) : :
Trilinears    d(a,b,c) : : , where d(a,b,c) = directed distance of A to line X(115)X(125)
Barycentrics    (sin A)/(cot B - cot C) : (sin B)/(cot C - cot A) : (sin C)/(cot A - cot B)
Barycentrics    cos(B - C) - cos(C - A) cos(A - B) : :
Barycentrics    sec(B - C) - sec(C - A) sec(A - B) : :
X(662) = 6 X[2] - 5 X[31278], 3 X[2] + X[31297], 3 X[2] - 4 X[40539], 4 X[8287] - 3 X[31175], 4 X[8287] - 5 X[31278], 2 X[8287] + X[31297], 2 X[21221] - 3 X[31175], 2 X[21221] - 5 X[31278], X[21221] - 4 X[40539], 3 X[31175] - 5 X[31278], 3 X[31175] + 2 X[31297], 3 X[31175] - 8 X[40539], 5 X[31278] + 2 X[31297], 5 X[31278] - 8 X[40539], X[31297] + 4 X[40539]

Let A'B'C' be the excentral triangle and H the Stammler hyperbola. Let T be the tangential triangle, with respect to A'B'C', of H. Then T and ABC are perspective, and their perspector is X(662). (Randy Hutson, December 26, 2015)

Let A2B2C2 and A3B3C3 be the 2nd and 3rd Parry triangles. Let A' be the trilinear product A2*A3, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(662). (Randy Hutson, February 10, 2016)

Let La be the A-extraversion of line X(1)X(21), and define Lb, Lc cyclically. Let A' = Lb ∩ Lc, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(662). (Randy Hutson, January 29, 2018)

Let La be the A-extraversion of line X(7)X(21), and define Lb, Lc cyclically. Let A' = Lb ∩ Lc, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(662). (Randy Hutson, January 29, 2018)

Let La be the A-extraversion of line X(8)X(21), and define Lb, Lc cyclically. Let A' = Lb ∩ Lc, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(662). (Randy Hutson, January 29, 2018)

Let La be the A-extraversion of line X(10)X(21), and define Lb, Lc cyclically. Let A' = Lb ∩ Lc, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(662). (Randy Hutson, January 29, 2018)

Let La be the A-extraversion of line X(21)X(36), and define Lb, Lc cyclically. Let A' = Lb ∩ Lc, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(662). (Randy Hutson, January 29, 2018)

X(662) lies on the cubic K1004, the curve CC9, and these lines: {1, 897}, {2, 2185}, {3, 1098}, {6, 757}, {7, 40622}, {9, 39256}, {10, 501}, {11, 19642}, {19, 8771}, {21, 1156}, {27, 913}, {28, 14868}, {29, 1800}, {31, 36289}, {37, 34053}, {41, 3758}, {43, 6043}, {44, 1931}, {48, 75}, {58, 16477}, {60, 404}, {63, 2159}, {65, 35991}, {69, 24580}, {76, 30882}, {81, 88}, {82, 1178}, {86, 142}, {99, 101}, {100, 110}, {109, 931}, {112, 1310}, {115, 24957}, {141, 24614}, {162, 2617}, {163, 2642}, {214, 759}, {224, 13739}, {229, 34772}, {238, 1326}, {239, 7113}, {243, 425}, {249, 14838}, {250, 32673}, {261, 572}, {270, 37304}, {274, 2224}, {283, 23707}, {293, 23996}, {304, 2172}, {305, 30918}, {306, 38822}, {310, 30895}, {319, 40589}, {320, 20769}, {326, 610}, {333, 909}, {409, 2646}, {448, 1944}, {513, 17940}, {560, 1740}, {584, 17379}, {593, 5035}, {604, 3759}, {644, 4606}, {648, 653}, {649, 17929}, {651, 1414}, {655, 4552}, {656, 1101}, {658, 1461}, {660, 765}, {661, 2644}, {668, 37218}, {670, 4586}, {689, 787}, {691, 8691}, {741, 3248}, {775, 820}, {798, 37134}, {799, 3570}, {805, 30670}, {811, 823}, {827, 831}, {849, 3216}, {892, 35180}, {894, 2174}, {908, 18653}, {911, 1444}, {922, 1580}, {932, 43359}, {985, 40734}, {997, 17512}, {1014, 41610}, {1018, 4596}, {1019, 3257}, {1021, 36141}, {1043, 2360}, {1086, 24617}, {1100, 1963}, {1125, 15792}, {1155, 2651}, {1169, 39979}, {1171, 30581}, {1193, 2363}, {1306, 6136}, {1307, 6135}, {1325, 4511}, {1333, 18274}, {1375, 37796}, {1412, 41629}, {1429, 24378}, {1434, 43762}, {1442, 40582}, {1474, 14013}, {1492, 1576}, {1509, 4251}, {1577, 15455}, {1625, 23113}, {1633, 4236}, {1691, 15994}, {1780, 37288}, {1781, 18714}, {1812, 1817}, {1818, 23692}, {1914, 2106}, {1924, 24037}, {1930, 34055}, {1933, 2227}, {1959, 2173}, {1966, 9417}, {1973, 18156}, {2126, 21081}, {2134, 21879}, {2167, 14213}, {2183, 17209}, {2194, 13588}, {2244, 17799}, {2245, 19308}, {2267, 17335}, {2268, 4687}, {2304, 31997}, {2326, 37448}, {2327, 8822}, {2341, 16578}, {2407, 38340}, {2421, 4603}, {2503, 24504}, {2576, 2580}, {2577, 2581}, {2635, 23695}, {2669, 9454}, {2748, 12074}, {2893, 28755}, {2939, 18719}, {2948, 4736}, {3109, 10609}, {3110, 3271}, {3204, 17350}, {3218, 37783}, {3219, 40592}, {3571, 5147}, {3573, 4436}, {3615, 35195}, {3682, 19842}, {3684, 17731}, {3737, 4570}, {3869, 37405}, {3939, 37138}, {4033, 7257}, {4069, 35368}, {4268, 17349}, {4276, 40110}, {4287, 17259}, {4473, 31059}, {4551, 36098}, {4559, 43069}, {4562, 4577}, {4563, 37215}, {4589, 37207}, {4590, 20981}, {4593, 4602}, {4594, 17941}, {4598, 34071}, {4599, 34072}, {4600, 4607}, {4601, 36860}, {4604, 34073}, {4620, 4625}, {4664, 9310}, {4670, 40744}, {4676, 11104}, {4858, 14616}, {4921, 16723}, {5009, 18792}, {5057, 5196}, {5086, 37158}, {5224, 25447}, {5235, 43757}, {5333, 25383}, {5468, 37210}, {5539, 23648}, {5698, 35915}, {5949, 7332}, {6061, 7411}, {6224, 6740}, {6507, 18750}, {6514, 27398}, {6518, 37774}, {6578, 15322}, {6626, 17256}, {7045, 23090}, {7054, 37659}, {7058, 14829}, {7192, 37143}, {7256, 43290}, {7258, 24004}, {7321, 18162}, {7452, 36797}, {8043, 21891}, {8052, 34076}, {8286, 26141}, {8301, 25048}, {8625, 34996}, {8652, 43356}, {8684, 43357}, {8690, 8694}, {8701, 34594}, {8708, 43076}, {9070, 36069}, {10330, 42720}, {13178, 21043}, {14206, 36102}, {14210, 36150}, {14534, 14554}, {14543, 17136}, {14953, 17139}, {14999, 17069}, {15507, 37019}, {16521, 40773}, {16545, 18049}, {16546, 18715}, {16598, 21381}, {16610, 37791}, {16704, 37222}, {16751, 36087}, {17104, 25440}, {17190, 27065}, {17206, 37214}, {17227, 25940}, {17289, 41534}, {17346, 34016}, {17438, 17868}, {17745, 46194}, {18020, 22382}, {18048, 18143}, {18051, 34082}, {18157, 34081}, {18206, 37131}, {18315, 39177}, {18594, 18713}, {18595, 18716}, {18596, 18717}, {18597, 18718}, {18598, 18720}, {18599, 18721}, {18740, 36145}, {19559, 34070}, {20337, 44387}, {20932, 34066}, {20948, 46254}, {20975, 24500}, {21008, 40432}, {21252, 30995}, {21254, 24714}, {22054, 28287}, {23363, 23831}, {23582, 32670}, {23999, 36139}, {24000, 36092}, {24018, 36131}, {24267, 25688}, {24557, 25887}, {24582, 24619}, {24636, 25469}, {24723, 35916}, {27665, 28283}, {27834, 34080}, {28210, 40522}, {28753, 37382}, {28841, 36066}, {30576, 37680}, {30962, 44081}, {30966, 37208}, {31998, 35148}, {32669, 37136}, {32851, 37793}, {33116, 40605}, {33297, 37213}, {34990, 35069}, {35049, 44769}, {36034, 36083}, {36036, 36132}

X(662) = midpoint of X(21221) and X(31297)
X(662) = reflection of X(i) in X(j) for these {i,j}: {8287, 40539}, {21221, 8287}, {31175, 2}, {40438, 39042}
X(662) = isogonal conjugate of X(661)
X(662) = isotomic conjugate of X(1577)
X(662) = complement of X(21221)
X(662) = anticomplement of X(8287)
X(662) = polar conjugate of X(24006)
X(662) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1983, 14731}, {6742, 21294}, {13486, 150}, {35049, 3434}
X(662) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 39054}, {39137, 141}
X(662) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 2644}, {2, 39054}, {99, 643}, {249, 2185}, {765, 33766}, {799, 4592}, {811, 162}, {1101, 18042}, {4567, 81}, {4573, 1414}, {4590, 757}, {4593, 799}, {4596, 100}, {4599, 163}, {4600, 58}, {4610, 99}, {4612, 4558}, {4620, 86}, {18020, 1098}, {23999, 255}, {24000, 1760}, {24037, 31}, {24041, 1}, {36036, 23997}, {36066, 3573}, {36085, 23889}, {46254, 75}
X(662) = X(i)-cross conjugate of X(j) for these (i,j): {1, 24041}, {2, 4564}, {3, 7045}, {6, 765}, {31, 24037}, {48, 1101}, {58, 4600}, {81, 4567}, {100, 99}, {101, 110}, {110, 1414}, {154, 24013}, {163, 162}, {255, 23999}, {284, 4570}, {404, 4998}, {513, 40438}, {572, 59}, {573, 7012}, {610, 24000}, {644, 8690}, {649, 2363}, {650, 40430}, {651, 648}, {656, 75}, {661, 1}, {798, 82}, {822, 775}, {1018, 34594}, {1019, 81}, {1021, 21}, {1577, 2167}, {1635, 759}, {1813, 4558}, {1924, 31}, {1958, 46254}, {1983, 36037}, {2254, 18827}, {2287, 5379}, {2617, 811}, {2624, 36053}, {2642, 897}, {3216, 7035}, {3573, 36066}, {3733, 757}, {3737, 86}, {3738, 14616}, {3882, 190}, {3888, 670}, {3909, 668}, {4040, 40439}, {4251, 1252}, {4383, 5382}, {4560, 2185}, {4575, 4592}, {4579, 4577}, {4585, 3257}, {5053, 9268}, {5276, 5384}, {5546, 643}, {6003, 7}, {7437, 927}, {8632, 741}, {13329, 39293}, {14208, 34055}, {14838, 2}, {15309, 25417}, {16612, 40436}, {16751, 274}, {20981, 6}, {21383, 3903}, {22382, 3}, {23090, 1098}, {23889, 36085}, {23997, 36036}, {24018, 63}, {24948, 32009}, {25900, 7131}, {27644, 4601}, {32679, 2349}, {32911, 1016}, {33854, 5378}, {35342, 100}, {37633, 5385}, {37659, 1275}, {37680, 5376}, {40214, 249}
X(662) = antigonal image of X(1337)
X(662) = cevapoint of X(i) and X(j) for these (i,j): {1, 661}, {2, 4560}, {3, 23090}, {6, 3733}, {9, 35057}, {21, 1021}, {31, 1924}, {48, 656}, {63, 24018}, {81, 1019}, {100, 101}, {110, 5546}, {163, 4575}, {214, 1635}, {284, 3737}, {513, 1100}, {514, 5249}, {523, 5949}, {526, 35069}, {649, 1193}, {650, 2646}, {651, 1813}, {667, 1197}, {693, 17866}, {798, 1964}, {820, 822}, {1577, 14213}, {1930, 14208}, {2238, 38348}, {2578, 2585}, {2579, 2584}, {2582, 2583}, {2642, 42081}, {3687, 6332}, {3738, 34544}, {7192, 17169}, {8061, 17457}, {8300, 8632}, {14838, 40214}
X(662) = crosspoint of X(i) and X(j) for these (i,j): {1, 9395}, {99, 4573}, {799, 811}, {4593, 4599}
X(662) = crosssum of X(i) and X(j) for these (i,j): {1, 2640}, {2, 21220}, {6, 21004}, {10, 21100}, {37, 21888}, {512, 3709}, {523, 21051}, {798, 810}, {1577, 20910}, {2084, 8061}, {3124, 8034}, {4024, 21720}, {4079, 4705}
X(662) = trilinear pole of line {1, 21}
X(662) = crossdifference of every pair of points on line {2642, 2643}
X(662) = trilinear product X(2)*X(110)
X(662) = trilinear product X(81)*X(100)
X(662) = trilinear product of circumcircle intercepts of line X(2)X(6)
X(662) = trilinear product of PU(78)
X(662) = crossdifference of PU(79)
X(662) = barycentric product of PU(90)
X(662) = perspector of conic {A,B,C,PU(78)}}
X(662) = trilinear product X(6)*X(99)
X(662) = trilinear product of PU(145)
X(662) = trilinear product of Steiner circumellipse intercepts of Brocard axis
X(662) = trilinear product of MacBeath circumconic intercepts of Euler line
X(662) = X(i)-isoconjugate of X(j) for these (i,j): {1, 661}, {2, 512}, {3, 2501}, {4, 647}, {5, 2623}, {6, 523}, {7, 3709}, {8, 7180}, {9, 4017}, {10, 649}, {11, 4559}, {12, 7252}, {13, 6137}, {14, 6138}, {15, 20578}, {16, 20579}, {19, 656}, {25, 525}, {30, 2433}, {31, 1577}, {32, 850}, {34, 8611}, {37, 513}, {41, 4077}, {42, 514}, {48, 24006}, {50, 10412}, {51, 15412}, {53, 23286}, {54, 12077}, {55, 7178}, {56, 3700}, {57, 4041}, {58, 4024}, {64, 6587}, {65, 650}, {66, 2485}, {67, 2492}, {68, 6753}, {69, 2489}, {71, 7649}, {72, 6591}, {73, 3064}, {74, 1637}, {75, 798}, {76, 669}, {80, 21828}, {81, 4705}, {82, 8061}, {83, 3005}, {86, 4079}, {87, 21834}, {88, 4730}, {89, 4770}, {92, 810}, {94, 14270}, {98, 3569}, {99, 3124}, {100, 3125}, {101, 3120}, {105, 24290}, {106, 4120}, {107, 3269}, {109, 21044}, {110, 115}, {111, 690}, {112, 125}, {141, 18105}, {158, 822}, {162, 3708}, {163, 1109}, {181, 4560}, {184, 14618}, {186, 14582}, {187, 5466}, {190, 3122}, {200, 7216}, {210, 3669}, {213, 693}, {225, 652}, {226, 663}, {228, 17924}, {230, 35364}, {232, 879}, {237, 43665}, {244, 1018}, {248, 16230}, {249, 8029}, {251, 826}, {257, 7234}, {262, 3288}, {263, 23878}, {264, 3049}, {269, 4171}, {275, 15451}, {279, 4524}, {287, 17994}, {290, 2491}, {291, 21832}, {292, 4010}, {297, 878}, {308, 688}, {313, 1919}, {321, 667}, {323, 15475}, {325, 2422}, {335, 4455}, {338, 1576}, {346, 7250}, {351, 671}, {385, 882}, {393, 520}, {459, 42658}, {468, 10097}, {476, 2088}, {511, 2395}, {521, 1880}, {522, 1400}, {524, 9178}, {526, 1989}, {542, 14998}, {560, 20948}, {561, 1924}, {574, 8599}, {594, 3733}, {598, 17414}, {599, 46001}, {604, 4086}, {607, 17094}, {648, 20975}, {651, 4516}, {657, 3668}, {660, 39786}, {662, 2643}, {665, 13576}, {668, 3121}, {670, 1084}, {684, 6531}, {685, 41172}, {686, 1300}, {691, 1648}, {692, 16732}, {694, 804}, {727, 21053}, {729, 9148}, {739, 14431}, {740, 3572}, {755, 14420}, {756, 1019}, {759, 2610}, {827, 39691}, {842, 1640}, {843, 8371}, {847, 30451}, {868, 2715}, {872, 7199}, {875, 3948}, {876, 2238}, {881, 3978}, {886, 1645}, {887, 34087}, {888, 3228}, {892, 21906}, {893, 2533}, {895, 14273}, {896, 23894}, {897, 2642}, {902, 4049}, {903, 14407}, {905, 1824}, {924, 2165}, {934, 36197}, {941, 8672}, {1015, 3952}, {1016, 8034}, {1020, 2310}, {1021, 1254}, {1022, 21805}, {1027, 3930}, {1029, 42653}, {1042, 3239}, {1073, 44705}, {1086, 4557}, {1093, 32320}, {1096, 24018}, {1126, 4988}, {1141, 2081}, {1171, 6367}, {1214, 18344}, {1245, 6590}, {1255, 4983}, {1289, 38356}, {1291, 10413}, {1293, 21950}, {1296, 6791}, {1301, 1562}, {1333, 4036}, {1334, 3676}, {1357, 30730}, {1365, 5546}, {1379, 13636}, {1380, 13722}, {1383, 3906}, {1402, 4391}, {1409, 44426}, {1425, 17926}, {1427, 3900}, {1431, 4140}, {1438, 4088}, {1441, 3063}, {1446, 8641}, {1459, 1826}, {1474, 4064}, {1491, 40747}, {1494, 14398}, {1495, 2394}, {1499, 21448}, {1500, 7192}, {1501, 44173}, {1502, 9426}, {1503, 34212}, {1510, 2963}, {1625, 8901}, {1634, 34294}, {1635, 4674}, {1649, 10630}, {1650, 32695}, {1769, 2250}, {1783, 18210}, {1843, 4580}, {1903, 6129}, {1914, 35352}, {1916, 5027}, {1918, 3261}, {1946, 40149}, {1953, 2616}, {1960, 4080}, {1964, 18070}, {1973, 14208}, {1974, 3267}, {1976, 2799}, {1977, 27808}, {1980, 27801}, {1987, 6130}, {1990, 14380}, {2028, 30508}, {2029, 30509}, {2030, 34246}, {2052, 39201}, {2054, 2786}, {2084, 3112}, {2086, 18829}, {2092, 4581}, {2148, 2618}, {2155, 17898}, {2159, 36035}, {2162, 21051}, {2163, 4931}, {2166, 2624}, {2170, 4551}, {2171, 3737}, {2200, 46107}, {2205, 40495}, {2207, 3265}, {2254, 18785}, {2259, 23752}, {2279, 4804}, {2281, 2517}, {2291, 30574}, {2308, 31010}, {2316, 30572}, {2321, 43924}, {2333, 4025}, {2334, 4841}, {2350, 4151}, {2353, 33294}, {2357, 14837}, {2378, 9200}, {2379, 9201}, {2380, 14446}, {2381, 14447}, {2396, 15630}, {2420, 12079}, {2423, 17757}, {2424, 17747}, {2435, 16318}, {2436, 34209}, {2451, 9307}, {2502, 9180}, {2530, 18098}, {2574, 8106}, {2575, 8105}, {2578, 2589}, {2579, 2588}, {2592, 42667}, {2593, 42668}, {2605, 8818}, {2631, 36119}, {2632, 24019}, {2640, 9396}, {2679, 39291}, {2966, 44114}, {2969, 4574}, {2970, 32661}, {2971, 4563}, {2972, 6529}, {2986, 21731}, {2996, 8651}, {2998, 3221}, {3003, 15328}, {3018, 15453}, {3050, 3613}, {3108, 7927}, {3114, 17415}, {3224, 23301}, {3227, 14404}, {3230, 35353}, {3248, 4033}, {3250, 40718}, {3251, 30575}, {3268, 11060}, {3271, 4552}, {3284, 18808}, {3310, 38955}, {3413, 5639}, {3414, 5638}, {3426, 9209}, {3445, 14321}, {3447, 45801}, {3455, 9979}, {3457, 23870}, {3458, 23871}, {3565, 6388}, {3566, 8770}, {3657, 8609}, {3690, 17925}, {3694, 43923}, {3695, 43925}, {3747, 4444}, {3768, 41683}, {3800, 39951}, {3804, 18840}, {3835, 23493}, {3903, 16592}, {3932, 43929}, {3943, 23345}, {3954, 18108}, {3960, 34857}, {3994, 23892}, {4052, 8643}, {4057, 40085}, {4083, 16606}, {4092, 4565}, {4108, 30495}, {4117, 4602}, {4122, 40746}, {4128, 27805}, {4129, 40148}, {4132, 39798}, {4139, 39956}, {4145, 39981}, {4155, 37128}, {4169, 43922}, {4374, 40729}, {4404, 38266}, {4466, 8750}, {4515, 43932}, {4556, 21043}, {4558, 8754}, {4566, 14936}, {4570, 21131}, {4590, 22260}, {4594, 21823}, {4603, 21725}, {4608, 20970}, {4609, 9427}, {4707, 6187}, {4729, 8056}, {4775, 30588}, {4777, 28658}, {4802, 28625}, {4822, 25430}, {4824, 25426}, {4826, 30598}, {4832, 5936}, {5007, 31065}, {5029, 11599}, {5040, 11611}, {5075, 11608}, {5089, 10099}, {5094, 30491}, {5098, 43671}, {5113, 11606}, {5191, 14223}, {5291, 18015}, {5392, 34952}, {5485, 8644}, {5489, 23964}, {5503, 9135}, {5562, 15422}, {5641, 6041}, {5664, 40355}, {5967, 8430}, {5969, 14606}, {5970, 11182}, {5994, 30468}, {5995, 30465}, {6088, 34898}, {6094, 9023}, {6103, 35909}, {6140, 13582}, {6145, 16040}, {6164, 21888}, {6186, 7265}, {6328, 36830}, {6354, 21789}, {6368, 8882}, {6370, 34079}, {6371, 14624}, {6373, 27809}, {6378, 17217}, {6464, 6562}, {6542, 18001}, {6586, 15320}, {6588, 43703}, {6589, 15232}, {6650, 17990}, {6664, 21006}, {6724, 6729}, {6742, 20982}, {6748, 39180}, {7064, 17096}, {7077, 7212}, {7140, 7254}, {7148, 18197}, {7332, 21784}, {7578, 18117}, {7950, 39955}, {8057, 41489}, {8288, 11636}, {8552, 18384}, {8562, 11071}, {8574, 13485}, {8632, 43534}, {8639, 34258}, {8663, 32014}, {8664, 10159}, {8665, 43527}, {8673, 13854}, {8675, 34288}, {8704, 11166}, {8714, 40147}, {8735, 23067}, {8736, 23189}, {8749, 9033}, {8752, 14429}, {8753, 14417}, {8781, 42663}, {8791, 9517}, {8795, 42293}, {8884, 17434}, {8946, 17432}, {8948, 17431}, {9009, 9462}, {9035, 16098}, {9171, 18823}, {9188, 36882}, {9208, 43535}, {9210, 14458}, {9217, 10278}, {9258, 17478}, {9262, 21093}, {9267, 21893}, {9278, 9508}, {9279, 40776}, {9292, 30476}, {9293, 20998}, {9315, 21052}, {9409, 16080}, {9468, 14295}, {9479, 46286}, {9486, 43667}, {9491, 40162}, {9494, 40016}, {10561, 14357}, {10566, 21035}, {10579, 14324}, {11081, 23284}, {11086, 23283}, {11167, 11186}, {12073, 39389}, {13136, 42752}, {13212, 32712}, {13366, 39183}, {13450, 46088}, {13486, 21824}, {14316, 41533}, {14318, 42006}, {14428, 43098}, {14443, 34539}, {14498, 45687}, {14533, 23290}, {14534, 42661}, {14573, 15415}, {14574, 23962}, {14579, 45147}, {14581, 34767}, {14592, 34397}, {14775, 18591}, {14776, 42761}, {14977, 44102}, {14991, 27807}, {15065, 21758}, {15318, 30442}, {15352, 34980}, {15420, 44092}, {15421, 44084}, {15526, 32713}, {16081, 39469}, {16310, 43709}, {16459, 35443}, {16460, 35444}, {16726, 40521}, {17402, 30452}, {17403, 30453}, {17411, 18812}, {17735, 18014}, {17790, 18002}, {17946, 17989}, {17947, 17992}, {17949, 17997}, {17950, 18000}, {17961, 18003}, {17962, 18004}, {17963, 18006}, {17964, 34763}, {17965, 18008}, {17966, 18013}, {17980, 24284}, {18082, 21123}, {18191, 21859}, {18310, 22259}, {18315, 41221}, {18830, 21835}, {18832, 23503}, {19222, 45907}, {19610, 31998}, {20691, 43931}, {20902, 32676}, {20906, 21759}, {20910, 34248}, {20976, 42345}, {20979, 42027}, {21207, 32739}, {21353, 31947}, {21461, 23872}, {21462, 23873}, {21646, 43679}, {21727, 39950}, {21905, 44182}, {22105, 46154}, {22172, 25576}, {22383, 41013}, {23099, 34537}, {23105, 23357}, {23285, 46288}, {23287, 42007}, {23350, 34369}, {23610, 44168}, {23992, 34574}, {24007, 39665}, {24008, 39666}, {24624, 42666}, {24978, 34448}, {26958, 46005}, {27375, 31296}, {28615, 30591}, {30496, 44451}, {30735, 40799}, {31946, 39964}, {32112, 35906}, {32641, 42759}, {32662, 35235}, {32740, 35522}, {34289, 42660}, {34845, 36198}, {36126, 37754}, {36955, 39024}, {37137, 40608}, {38237, 46274}, {39022, 41880}, {39023, 41881}, {40151, 44729}, {40338, 40345}, {40352, 41079}, {40418, 40627}, {41167, 41932}, {41178, 41209}, {41506, 43060}, {42344, 45773}, {42659, 46105}, {42664, 43531}, {42671, 43673}, {44127, 46245}
X(662) = barycentric product X(i)*X(j) for these {i,j}: {1, 99}, {3, 811}, {4, 4592}, {6, 799}, {7, 643}, {8, 1414}, {9, 4573}, {19, 4563}, {21, 664}, {27, 1332}, {28, 4561}, {29, 6516}, {31, 670}, {32, 4602}, {37, 4610}, {38, 4577}, {39, 4593}, {42, 4623}, {44, 4615}, {47, 46134}, {48, 6331}, {55, 4625}, {56, 7257}, {57, 645}, {58, 668}, {59, 18155}, {63, 648}, {69, 162}, {75, 110}, {76, 163}, {77, 36797}, {81, 190}, {82, 4576}, {85, 5546}, {86, 100}, {92, 4558}, {95, 2617}, {101, 274}, {107, 326}, {108, 332}, {109, 314}, {111, 24039}, {112, 304}, {141, 4599}, {171, 4594}, {172, 7260}, {200, 4616}, {220, 4635}, {226, 4612}, {238, 4589}, {239, 4584}, {240, 17932}, {249, 1577}, {250, 14208}, {255, 6528}, {261, 4551}, {264, 4575}, {269, 7256}, {279, 7259}, {283, 18026}, {284, 4554}, {286, 1331}, {290, 23997}, {293, 877}, {305, 32676}, {310, 692}, {311, 36134}, {312, 4565}, {319, 13486}, {321, 4556}, {323, 32680}, {325, 36084}, {333, 651}, {336, 4230}, {340, 36061}, {346, 4637}, {385, 37134}, {394, 823}, {511, 36036}, {512, 24037}, {513, 4600}, {514, 4567}, {519, 4622}, {520, 23999}, {523, 24041}, {524, 36085}, {538, 36133}, {552, 4069}, {560, 4609}, {561, 1576}, {593, 4033}, {610, 44326}, {644, 1434}, {646, 1412}, {647, 46254}, {649, 4601}, {650, 4620}, {653, 1812}, {656, 18020}, {658, 2287}, {660, 33295}, {661, 4590}, {666, 18206}, {671, 23889}, {689, 1964}, {691, 14210}, {693, 4570}, {740, 36066}, {741, 874}, {757, 3952}, {763, 4103}, {765, 7192}, {785, 10471}, {789, 3736}, {798, 34537}, {805, 1966}, {813, 30940}, {827, 1930}, {849, 27808}, {850, 1101}, {873, 4557}, {880, 1967}, {892, 896}, {894, 4603}, {897, 5468}, {901, 30939}, {902, 4634}, {906, 44129}, {907, 39731}, {919, 18157}, {925, 44179}, {931, 10436}, {932, 33296}, {933, 18695}, {934, 1043}, {1010, 1310}, {1014, 3699}, {1016, 1019}, {1018, 1509}, {1020, 7058}, {1021, 1275}, {1098, 4566}, {1100, 4632}, {1102, 6529}, {1125, 4596}, {1252, 7199}, {1333, 1978}, {1400, 4631}, {1407, 7258}, {1415, 28660}, {1441, 4636}, {1444, 1897}, {1449, 4633}, {1492, 30966}, {1580, 18829}, {1581, 17941}, {1634, 3112}, {1707, 35136}, {1725, 18878}, {1733, 10425}, {1740, 3222}, {1755, 43187}, {1757, 17930}, {1758, 17931}, {1760, 44766}, {1783, 17206}, {1790, 6335}, {1792, 36118}, {1813, 31623}, {1817, 44327}, {1821, 2421}, {1822, 15165}, {1823, 15164}, {1896, 6517}, {1910, 2396}, {1914, 4639}, {1923, 42371}, {1924, 44168}, {1926, 17938}, {1928, 14574}, {1929, 17934}, {1931, 35148}, {1944, 41206}, {1958, 43188}, {1959, 2966}, {1969, 32661}, {1992, 37216}, {2162, 36860}, {2166, 10411}, {2167, 14570}, {2184, 36841}, {2185, 4552}, {2194, 4572}, {2206, 6386}, {2234, 9150}, {2236, 41209}, {2247, 6035}, {2249, 15418}, {2303, 37215}, {2327, 13149}, {2328, 4569}, {2349, 2407}, {2420, 33805}, {2580, 8116}, {2581, 8115}, {2582, 39298}, {2583, 39299}, {2640, 37880}, {2643, 31614}, {2644, 35511}, {2648, 17933}, {2651, 35154}, {2703, 5209}, {2715, 46238}, {2964, 46139}, {3051, 37204}, {3108, 18062}, {3257, 16704}, {3260, 36034}, {3265, 24000}, {3266, 36142}, {3403, 26714}, {3564, 36105}, {3565, 18156}, {3570, 37128}, {3573, 18827}, {3616, 4614}, {3732, 40403}, {3733, 7035}, {3737, 4998}, {3875, 8690}, {3882, 14534}, {3888, 40415}, {3903, 17103}, {3926, 24019}, {3936, 37140}, {3964, 36126}, {4008, 35575}, {4017, 6064}, {4025, 5379}, {4041, 7340}, {4076, 7203}, {4226, 8773}, {4238, 31637}, {4358, 4591}, {4359, 4629}, {4360, 34594}, {4427, 40438}, {4436, 40439}, {4560, 4564}, {4579, 32010}, {4583, 5009}, {4585, 24624}, {4586, 40773}, {4597, 4653}, {4598, 27644}, {4604, 5235}, {4606, 42028}, {4627, 19804}, {4628, 16703}, {4638, 16729}, {4647, 6578}, {4658, 32042}, {4673, 5545}, {5127, 35156}, {5333, 37211}, {5377, 23829}, {5380, 6629}, {5383, 18197}, {5467, 46277}, {6012, 33953}, {6149, 35139}, {6385, 32739}, {6393, 36104}, {6507, 15352}, {6518, 41207}, {6606, 17194}, {6628, 40521}, {6632, 16726}, {6648, 17185}, {7045, 7253}, {7128, 15411}, {7763, 36145}, {7769, 36148}, {7799, 32678}, {8024, 34072}, {8025, 37212}, {8701, 16709}, {8708, 17175}, {8709, 18792}, {8822, 13138}, {9066, 36289}, {9395, 31998}, {9396, 31632}, {9424, 9425}, {11794, 18042}, {13398, 33808}, {14206, 44769}, {14213, 18315}, {14919, 24001}, {14966, 46273}, {15455, 40214}, {16568, 17708}, {16705, 36147}, {16739, 32736}, {17136, 40430}, {17139, 36037}, {17143, 43076}, {17197, 31615}, {17394, 43356}, {17469, 35137}, {17731, 37135}, {17763, 17929}, {17935, 17954}, {17940, 20947}, {17943, 18032}, {18047, 40432}, {18268, 27853}, {18645, 31628}, {18830, 38832}, {18831, 44706}, {20941, 40173}, {20948, 23357}, {23181, 40440}, {23342, 37132}, {23582, 24018}, {23995, 44173}, {24038, 34574}, {27398, 37141}, {27834, 41629}, {27958, 37137}, {30941, 36086}, {31008, 34071}, {31997, 43359}, {32014, 35342}, {32640, 46234}, {32679, 39295}, {32833, 36149}, {32911, 37205}, {33951, 40398}, {34055, 41676}, {35138, 36263}, {35179, 36277}, {35550, 36069}, {36059, 44130}, {37206, 41610}, {37217, 41614}, {40703, 43754}, {41517, 46295}
X(662) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 523}, {2, 1577}, {3, 656}, {4, 24006}, {5, 2618}, {6, 661}, {7, 4077}, {8, 4086}, {9, 3700}, {10, 4036}, {19, 2501}, {20, 17898}, {21, 522}, {27, 17924}, {28, 7649}, {29, 44426}, {30, 36035}, {31, 512}, {32, 798}, {37, 4024}, {38, 826}, {39, 8061}, {41, 3709}, {42, 4705}, {43, 21051}, {44, 4120}, {45, 4931}, {47, 924}, {48, 647}, {50, 2624}, {54, 2616}, {55, 4041}, {56, 4017}, {57, 7178}, {58, 513}, {59, 4551}, {60, 3737}, {63, 525}, {69, 14208}, {72, 4064}, {75, 850}, {76, 20948}, {77, 17094}, {81, 514}, {83, 18070}, {86, 693}, {88, 4049}, {92, 14618}, {99, 75}, {100, 10}, {101, 37}, {107, 158}, {108, 225}, {109, 65}, {110, 1}, {111, 23894}, {112, 19}, {145, 4404}, {162, 4}, {163, 6}, {171, 2533}, {184, 810}, {187, 2642}, {190, 321}, {194, 20910}, {204, 44705}, {213, 4079}, {219, 8611}, {220, 4171}, {238, 4010}, {240, 16230}, {250, 162}, {255, 520}, {261, 18155}, {274, 3261}, {283, 521}, {284, 650}, {286, 46107}, {291, 35352}, {293, 879}, {304, 3267}, {310, 40495}, {314, 35519}, {323, 32679}, {326, 3265}, {332, 35518}, {333, 4391}, {394, 24018}, {476, 2166}, {501, 31947}, {512, 2643}, {513, 3120}, {514, 16732}, {518, 4088}, {520, 2632}, {523, 1109}, {525, 20902}, {560, 669}, {561, 44173}, {563, 30451}, {577, 822}, {593, 1019}, {595, 4132}, {604, 7180}, {610, 6587}, {643, 8}, {644, 2321}, {645, 312}, {646, 30713}, {647, 3708}, {648, 92}, {649, 3125}, {650, 21044}, {651, 226}, {653, 40149}, {656, 125}, {657, 36197}, {658, 1446}, {660, 43534}, {661, 115}, {663, 4516}, {664, 1441}, {667, 3122}, {668, 313}, {670, 561}, {672, 24290}, {685, 36120}, {689, 18833}, {691, 897}, {692, 42}, {693, 21207}, {741, 876}, {757, 7192}, {758, 6370}, {765, 3952}, {798, 3124}, {799, 76}, {805, 1581}, {810, 20975}, {811, 264}, {822, 3269}, {823, 2052}, {825, 40747}, {827, 82}, {849, 3733}, {850, 23994}, {859, 1769}, {874, 35544}, {877, 40703}, {880, 1926}, {892, 46277}, {896, 690}, {897, 5466}, {898, 41683}, {899, 14431}, {901, 4674}, {902, 4730}, {905, 4466}, {906, 71}, {907, 23051}, {919, 18785}, {922, 351}, {923, 9178}, {925, 91}, {930, 2962}, {931, 31359}, {932, 42027}, {933, 2190}, {934, 3668}, {942, 23752}, {982, 3801}, {984, 4122}, {1001, 4804}, {1010, 2517}, {1014, 3676}, {1016, 4033}, {1018, 594}, {1019, 1086}, {1020, 6354}, {1021, 1146}, {1023, 3943}, {1026, 3932}, {1043, 4397}, {1098, 7253}, {1100, 4988}, {1101, 110}, {1102, 4143}, {1106, 7250}, {1109, 23105}, {1110, 4557}, {1113, 2589}, {1114, 2588}, {1125, 30591}, {1155, 30574}, {1172, 3064}, {1197, 40627}, {1252, 1018}, {1253, 4524}, {1255, 31010}, {1262, 1020}, {1290, 5620}, {1303, 9251}, {1304, 36119}, {1319, 30572}, {1325, 21180}, {1326, 9508}, {1331, 72}, {1332, 306}, {1333, 649}, {1376, 21052}, {1407, 7216}, {1408, 43924}, {1412, 3669}, {1414, 7}, {1415, 1400}, {1429, 7212}, {1434, 24002}, {1437, 1459}, {1444, 4025}, {1449, 4841}, {1459, 18210}, {1461, 1427}, {1468, 8672}, {1474, 6591}, {1492, 40718}, {1501, 1924}, {1509, 7199}, {1575, 21053}, {1576, 31}, {1577, 338}, {1580, 804}, {1621, 4151}, {1624, 774}, {1625, 1953}, {1632, 17871}, {1633, 3914}, {1634, 38}, {1707, 3566}, {1734, 21045}, {1740, 23301}, {1743, 14321}, {1749, 45147}, {1755, 3569}, {1757, 18004}, {1758, 18006}, {1760, 33294}, {1769, 42759}, {1780, 15313}, {1783, 1826}, {1790, 905}, {1812, 6332}, {1813, 1214}, {1817, 14837}, {1821, 43665}, {1822, 2575}, {1823, 2574}, {1897, 41013}, {1910, 2395}, {1914, 21832}, {1917, 9426}, {1919, 3121}, {1923, 688}, {1924, 1084}, {1927, 881}, {1929, 18014}, {1930, 23285}, {1931, 2786}, {1933, 5027}, {1953, 12077}, {1955, 6130}, {1957, 16229}, {1958, 30476}, {1959, 2799}, {1962, 6367}, {1964, 3005}, {1966, 14295}, {1967, 882}, {1973, 2489}, {1975, 17893}, {1978, 27801}, {1983, 2245}, {1992, 14207}, {2148, 2623}, {2149, 4559}, {2150, 7252}, {2151, 6137}, {2152, 6138}, {2153, 20578}, {2154, 20579}, {2159, 2433}, {2166, 10412}, {2167, 15412}, {2169, 23286}, {2172, 2485}, {2173, 1637}, {2176, 21834}, {2177, 4770}, {2185, 4560}, {2193, 652}, {2194, 663}, {2206, 667}, {2210, 4455}, {2234, 9148}, {2244, 14420}, {2245, 2610}, {2247, 1640}, {2251, 14407}, {2284, 3930}, {2287, 3239}, {2290, 2081}, {2299, 18344}, {2303, 6590}, {2308, 4983}, {2315, 686}, {2326, 17926}, {2328, 3900}, {2329, 4140}, {2349, 2394}, {2360, 6129}, {2363, 4581}, {2396, 46238}, {2407, 14206}, {2420, 2173}, {2421, 1959}, {2427, 21801}, {2576, 8106}, {2577, 8105}, {2580, 2593}, {2581, 2592}, {2588, 39240}, {2589, 39241}, {2605, 2611}, {2608, 2614}, {2612, 2606}, {2613, 2607}, {2616, 8901}, {2617, 5}, {2624, 2088}, {2626, 18114}, {2632, 5489}, {2640, 10278}, {2642, 1648}, {2643, 8029}, {2644, 148}, {2648, 18013}, {2651, 2785}, {2701, 2652}, {2702, 9278}, {2715, 1910}, {2964, 1510}, {2966, 1821}, {3051, 2084}, {3052, 4729}, {3125, 21131}, {3158, 44729}, {3216, 31946}, {3218, 4707}, {3219, 7265}, {3222, 18832}, {3233, 1099}, {3248, 8034}, {3257, 4080}, {3265, 17879}, {3284, 2631}, {3285, 1635}, {3286, 2254}, {3309, 21945}, {3501, 21958}, {3565, 8769}, {3566, 17876}, {3570, 3948}, {3573, 740}, {3616, 4815}, {3658, 1737}, {3666, 21124}, {3670, 21121}, {3699, 3701}, {3724, 42666}, {3725, 42661}, {3733, 244}, {3736, 1491}, {3737, 11}, {3747, 4155}, {3786, 4522}, {3794, 3810}, {3799, 3773}, {3882, 1211}, {3888, 2887}, {3909, 3454}, {3915, 4139}, {3938, 4808}, {3939, 210}, {3952, 1089}, {4008, 30735}, {4017, 1365}, {4033, 28654}, {4040, 2486}, {4041, 4092}, {4069, 6057}, {4079, 21833}, {4100, 32320}, {4117, 23099}, {4131, 17216}, {4184, 1734}, {4225, 21189}, {4226, 1733}, {4228, 21185}, {4230, 240}, {4236, 1738}, {4238, 1861}, {4240, 1784}, {4242, 860}, {4243, 1736}, {4246, 1785}, {4267, 17420}, {4273, 4893}, {4282, 654}, {4394, 21950}, {4427, 4647}, {4436, 21020}, {4467, 17886}, {4482, 4377}, {4512, 4843}, {4551, 12}, {4552, 6358}, {4553, 15523}, {4554, 349}, {4556, 81}, {4557, 756}, {4558, 63}, {4559, 2171}, {4560, 4858}, {4561, 20336}, {4563, 304}, {4564, 4552}, {4565, 57}, {4567, 190}, {4570, 100}, {4571, 3710}, {4573, 85}, {4574, 3949}, {4575, 3}, {4576, 1930}, {4577, 3112}, {4578, 4082}, {4579, 1215}, {4584, 335}, {4585, 3936}, {4587, 3694}, {4589, 334}, {4590, 799}, {4591, 88}, {4592, 69}, {4593, 308}, {4594, 7018}, {4596, 1268}, {4599, 83}, {4600, 668}, {4601, 1978}, {4602, 1502}, {4603, 257}, {4604, 30588}, {4609, 1928}, {4610, 274}, {4611, 1760}, {4612, 333}, {4614, 5936}, {4615, 20568}, {4616, 1088}, {4620, 4554}, {4622, 903}, {4623, 310}, {4625, 6063}, {4627, 25430}, {4628, 18098}, {4629, 1255}, {4630, 46289}, {4631, 28660}, {4632, 32018}, {4633, 40023}, {4636, 21}, {4637, 279}, {4638, 30575}, {4639, 18895}, {4649, 4824}, {4653, 4777}, {4658, 4802}, {4705, 21043}, {4756, 4066}, {4767, 4125}, {4781, 4714}, {4877, 4820}, {5009, 659}, {5053, 21894}, {5118, 2234}, {5127, 8674}, {5208, 23877}, {5235, 4791}, {5333, 4823}, {5375, 21090}, {5379, 1897}, {5384, 4613}, {5440, 14429}, {5467, 896}, {5468, 14210}, {5546, 9}, {5994, 2154}, {5995, 2153}, {6003, 8286}, {6011, 41501}, {6064, 7257}, {6065, 4069}, {6149, 526}, {6163, 21093}, {6331, 1969}, {6516, 307}, {6529, 6520}, {6563, 17881}, {6577, 40504}, {6578, 40438}, {6727, 6728}, {6733, 6724}, {6742, 6757}, {6758, 42005}, {7035, 27808}, {7045, 4566}, {7054, 1021}, {7070, 14308}, {7113, 21828}, {7122, 7234}, {7136, 7137}, {7192, 1111}, {7199, 23989}, {7202, 21141}, {7203, 1358}, {7234, 21725}, {7239, 16886}, {7252, 2170}, {7253, 24026}, {7254, 3942}, {7256, 341}, {7257, 3596}, {7259, 346}, {7260, 44187}, {7289, 21107}, {7305, 7255}, {7340, 4625}, {7341, 7203}, {7450, 1735}, {7480, 36063}, {7760, 20953}, {7768, 18076}, {8025, 4978}, {8061, 39691}, {8115, 2583}, {8116, 2582}, {8632, 39786}, {8690, 34860}, {8750, 1824}, {8771, 14341}, {8822, 17896}, {9145, 36263}, {9217, 9396}, {9218, 2640}, {9247, 3049}, {9266, 21100}, {9273, 37140}, {9274, 36069}, {9306, 17478}, {9395, 9293}, {9406, 14398}, {9417, 2491}, {9426, 4117}, {10420, 36053}, {10425, 8773}, {10458, 784}, {10471, 35559}, {11110, 7650}, {13138, 39130}, {13397, 23604}, {13398, 921}, {13486, 79}, {13588, 17072}, {13614, 14302}, {14206, 41079}, {14208, 339}, {14210, 35522}, {14213, 18314}, {14570, 14213}, {14574, 560}, {14586, 2148}, {14587, 36134}, {14588, 20903}, {14589, 21013}, {14838, 8287}, {14868, 20315}, {14966, 1755}, {15329, 1725}, {15352, 6521}, {15507, 42767}, {15792, 8043}, {15958, 2169}, {16049, 21186}, {16468, 4806}, {16562, 45801}, {16563, 18310}, {16568, 9979}, {16679, 40471}, {16680, 23668}, {16695, 3123}, {16696, 16892}, {16702, 4750}, {16704, 3762}, {16705, 4509}, {16706, 27712}, {16726, 6545}, {16727, 23100}, {16751, 116}, {16777, 4838}, {16948, 3667}, {17103, 4374}, {17104, 2605}, {17126, 4761}, {17127, 4170}, {17136, 18698}, {17139, 36038}, {17185, 3910}, {17187, 2530}, {17194, 6362}, {17197, 40166}, {17206, 15413}, {17467, 11123}, {17469, 7927}, {17515, 44428}, {17763, 18003}, {17780, 3992}, {17799, 9479}, {17872, 12075}, {17930, 18032}, {17932, 336}, {17934, 20947}, {17938, 1967}, {17939, 17954}, {17940, 1929}, {17941, 1966}, {17942, 1758}, {17943, 1757}, {17944, 17763}, {17945, 17958}, {17954, 18015}, {17955, 18007}, {17959, 18012}, {17960, 18005}, {18020, 811}, {18028, 18008}, {18042, 31296}, {18047, 3963}, {18062, 39998}, {18155, 34387}, {18161, 21117}, {18163, 21120}, {18164, 21104}, {18165, 21118}, {18167, 21110}, {18176, 21114}, {18178, 21119}, {18180, 21102}, {18183, 21125}, {18184, 21133}, {18191, 21132}, {18197, 21138}, {18198, 21115}, {18199, 21139}, {18200, 7200}, {18201, 21145}, {18205, 20505}, {18206, 918}, {18210, 21134}, {18211, 23764}, {18266, 17990}, {18268, 3572}, {18315, 2167}, {18604, 4091}, {18726, 23749}, {18756, 9402}, {18792, 3837}, {18829, 1934}, {18831, 40440}, {19215, 17431}, {19216, 17432}, {19555, 14316}, {20769, 24459}, {20948, 23962}, {20981, 16592}, {21362, 4415}, {21789, 2310}, {21858, 21720}, {21877, 21056}, {22003, 42708}, {22382, 16573}, {23067, 201}, {23090, 34591}, {23181, 44706}, {23189, 7004}, {23226, 22094}, {23343, 3994}, {23344, 21805}, {23348, 17955}, {23357, 163}, {23582, 823}, {23695, 2798}, {23703, 40663}, {23832, 4695}, {23845, 4642}, {23861, 23928}, {23889, 524}, {23964, 24019}, {23995, 1576}, {23996, 41167}, {23997, 511}, {23999, 6528}, {24000, 107}, {24001, 46106}, {24006, 2970}, {24018, 15526}, {24019, 393}, {24037, 670}, {24039, 3266}, {24041, 99}, {24530, 27710}, {25577, 21951}, {26714, 2186}, {26856, 40213}, {26885, 21831}, {27644, 3835}, {27834, 4052}, {28162, 31503}, {28471, 35347}, {28606, 23879}, {30528, 36102}, {30728, 42712}, {31614, 24037}, {31623, 46110}, {31855, 21714}, {31998, 20939}, {32230, 36126}, {32320, 37754}, {32640, 2159}, {32641, 2250}, {32652, 2357}, {32656, 228}, {32660, 1409}, {32661, 48}, {32671, 34079}, {32674, 1880}, {32676, 25}, {32678, 1989}, {32680, 94}, {32692, 2168}, {32713, 1096}, {32717, 37132}, {32729, 923}, {32739, 213}, {32911, 4129}, {33295, 3766}, {33296, 20906}, {33514, 3113}, {33628, 4394}, {33760, 44445}, {33766, 31290}, {33946, 20234}, {33948, 42714}, {34016, 18160}, {34054, 18010}, {34055, 4580}, {34071, 16606}, {34072, 251}, {34073, 28658}, {34476, 4782}, {34537, 4602}, {34586, 42768}, {34594, 596}, {35049, 38340}, {35057, 6741}, {35192, 9404}, {35193, 35057}, {35200, 14380}, {35278, 4008}, {35280, 3755}, {35281, 3753}, {35324, 17438}, {35325, 17442}, {35326, 21808}, {35327, 1962}, {35338, 3925}, {35342, 1213}, {35623, 29017}, {36034, 74}, {36036, 290}, {36037, 38955}, {36046, 43717}, {36047, 43707}, {36049, 1903}, {36050, 15232}, {36051, 35364}, {36052, 3657}, {36053, 15328}, {36057, 10099}, {36059, 73}, {36060, 10097}, {36061, 265}, {36062, 14220}, {36066, 18827}, {36069, 759}, {36084, 98}, {36085, 671}, {36086, 13576}, {36104, 6531}, {36105, 35142}, {36114, 1300}, {36119, 18808}, {36126, 1093}, {36129, 6344}, {36131, 8749}, {36133, 3228}, {36134, 54}, {36142, 111}, {36145, 2165}, {36147, 14624}, {36148, 2963}, {36149, 34288}, {36263, 3906}, {36277, 1499}, {36289, 5996}, {36797, 318}, {36830, 16562}, {36841, 18750}, {36860, 6382}, {37128, 4444}, {37129, 35353}, {37134, 1916}, {37135, 11599}, {37140, 24624}, {37141, 8808}, {37204, 40016}, {37205, 40013}, {37212, 6539}, {37216, 5485}, {37582, 2457}, {37732, 8819}, {38340, 43682}, {38470, 34920}, {38814, 21196}, {38832, 4083}, {39026, 22321}, {39054, 21221}, {39295, 32680}, {39298, 2580}, {39299, 2581}, {39673, 6005}, {39949, 40086}, {40091, 4145}, {40214, 14838}, {40302, 40345}, {40438, 4608}, {40501, 21728}, {40521, 6535}, {40592, 21192}, {40731, 3805}, {40773, 824}, {41206, 1952}, {41337, 2227}, {41405, 21888}, {41610, 4468}, {41614, 14209}, {41629, 4462}, {41676, 20883}, {41679, 1748}, {42028, 4801}, {42081, 1649}, {43076, 13476}, {43187, 46273}, {43356, 39708}, {43359, 17038}, {43754, 293}, {44119, 8678}, {44179, 6563}, {44706, 6368}, {44769, 2349}, {44770, 8767}, {46134, 20571}, {46148, 3954}, {46177, 21804}, {46254, 6331}, {46289, 18105}
X(662) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 2640, 2643}, {2, 8287, 31278}, {2, 21221, 8287}, {2, 31297, 21221}, {2, 40214, 2185}, {6, 757, 33766}, {6, 21783, 21755}, {19, 44179, 18041}, {44, 16702, 1931}, {48, 75, 18042}, {48, 1958, 75}, {75, 20941, 20902}, {99, 645, 190}, {100, 110, 643}, {249, 14838, 39054}, {304, 2172, 34065}, {326, 610, 1760}, {409, 2646, 40430}, {560, 1740, 33760}, {651, 4565, 1414}, {661, 24041, 2644}, {922, 2234, 1580}, {1100, 1963, 40438}, {1582, 1964, 82}, {1959, 2173, 16568}, {1959, 16568, 18722}, {2617, 4575, 162}, {2640, 2643, 897}, {2643, 17467, 1}, {2643, 42081, 17467}, {3733, 4557, 1634}, {4436, 35327, 3573}, {4552, 4560, 14570}, {4557, 4579, 765}, {4565, 5546, 4558}, {4596, 4629, 37212}, {5949, 7332, 34989}, {8287, 21221, 31175}, {8287, 40539, 2}, {17197, 18645, 86}, {17882, 20902, 75}, {19298, 19299, 2247}, {19642, 25533, 11}, {21221, 40539, 31278}, {31175, 31278, 8287}, {31297, 40539, 31175}, {39339, 42081, 897}


X(663) = CROSSDIFFERENCE OF X(2) AND X(7)

Trilinears    a(b - c)(b + c - a) : :
X(663) = 2 X[1] + X[4724], X[1] + 2 X[4794], 3 X[551] - X[23789], X[649] - 4 X[1960], X[649] + 2 X[4775], 3 X[649] - 2 X[4834], X[649] - 3 X[8643], 2 X[649] - 5 X[8656], 4 X[650] - X[4814], 2 X[650] + X[4895], 4 X[650] + X[4959], 3 X[667] - X[4834], 2 X[667] - 3 X[8643], 4 X[667] - 5 X[8656], 3 X[1635] - X[4729], 3 X[1635] - 4 X[6050], 3 X[1946] - 2 X[6139], 2 X[1960] + X[4775], 6 X[1960] - X[4834], 4 X[1960] - 3 X[8643], 8 X[1960] - 5 X[8656], 4 X[2605] - X[43924], X[3241] + 2 X[45673], 3 X[3251] + X[4705], 2 X[3251] + X[4893], 5 X[3616] - 2 X[24720], 2 X[3669] - 3 X[14413], 4 X[3716] - X[4474], 2 X[3737] + X[42312], 2 X[4040] + X[4449], X[4041] + 2 X[4162], 2 X[4041] + X[4959], 4 X[4162] + X[4814], 4 X[4162] - X[4959], X[4367] - 3 X[25569], X[4449] + 4 X[4794], X[4498] + 2 X[4879], 2 X[4528] - 3 X[44729], 3 X[4543] - 4 X[4546], 2 X[4705] - 3 X[4893], X[4724] - 4 X[4794], X[4729] - 4 X[6050], 3 X[4775] + X[4834], 2 X[4775] + 3 X[8643], 4 X[4775] + 5 X[8656], X[4814] + 2 X[4895], 2 X[4834] - 9 X[8643], 4 X[4834] - 15 X[8656], 2 X[6332] - 3 X[14432], 6 X[8643] - 5 X[8656], 3 X[14430] - 4 X[20317], 4 X[14837] - 3 X[30574], X[17072] - 3 X[45316], 4 X[21260] - 5 X[30835], X[21302] - 6 X[45316], 7 X[31207] - 8 X[31288], 3 X[38314] - 2 X[45667]

X(663) lies on these lines: 1, 514}, {2, 17072}, {6, 9029}, {8, 4147}, {29, 46110}, {31, 2423}, {33, 23615}, {36, 39476}, {41, 884}, {42, 1643}, {56, 2424}, {78, 44448}, {86, 4406}, {101, 919}, {105, 12032}, {106, 840}, {109, 1262}, {110, 2701}, {187, 237}, {200, 4543}, {212, 27780}, {220, 38379}, {238, 37998}, {513, 855}, {521, 17420}, {522, 1944}, {523, 10149}, {551, 23789}, {644, 40499}, {650, 861}, {652, 1951}, {656, 15313}, {657, 853}, {659, 4083}, {660, 9266}, {661, 810}, {664, 14727}, {672, 9320}, {676, 7178}, {692, 32675}, {693, 29051}, {788, 4455}, {814, 4010}, {830, 14349}, {834, 4057}, {875, 23493}, {899, 10006}, {905, 2254}, {928, 4091}, {1015, 35505}, {1019, 6005}, {1193, 1491}, {1201, 2530}, {1400, 9245}, {1575, 40464}, {1577, 29066}, {1618, 39026}, {1633, 46153}, {1635, 4729}, {1734, 3887}, {1912, 23572}, {1919, 2484}, {2082, 6170}, {2170, 38365}, {2176, 21791}, {2320, 23838}, {2440, 7084}, {2499, 7180}, {2517, 8062}, {2520, 4813}, {2533, 4874}, {2785, 4142}, {2821, 22091}, {3022, 7117}, {3241, 45673}, {3248, 42084}, {3616, 24720}, {3667, 21173}, {3676, 8713}, {3700, 4990}, {3716, 3907}, {3720, 4379}, {3768, 9010}, {3801, 29082}, {3810, 3904}, {3835, 4107}, {3938, 6546}, {3939, 5548}, {3960, 4905}, {3961, 10196}, {4024, 21831}, {4063, 4401}, {4069, 30727}, {4122, 29074}, {4163, 4521}, {4170, 29013}, {4336, 42462}, {4369, 24666}, {4374, 17215}, {4378, 6372}, {4382, 29070}, {4491, 9002}, {4502, 20981}, {4528, 44729}, {4564, 9323}, {4666, 21183}, {4707, 20517}, {4800, 29236}, {4804, 23882}, {4810, 29238}, {4843, 4976}, {4844, 30116}, {4922, 29324}, {4978, 29186}, {4992, 24719}, {5272, 44432}, {6003, 21189}, {6006, 18450}, {6182, 21127}, {6366, 21120}, {6373, 23464}, {6586, 19624}, {6589, 17966}, {6614, 24016}, {7190, 30181}, {7191, 44435}, {7265, 29062}, {7649, 14775}, {7662, 17478}, {8750, 14776}, {9032, 23650}, {9313, 21003}, {9817, 14476}, {10015, 28473}, {10459, 29298}, {10570, 23104}, {11193, 14547}, {11707, 35051}, {11708, 35052}, {11712, 39046}, {11918, 35072}, {14077, 23057}, {14430, 20317}, {14837, 28292}, {15283, 21260}, {16466, 22154}, {18000, 21748}, {19372, 44928}, {20295, 31291}, {21146, 29246}, {21204, 29820}, {21261, 30764}, {21300, 25128}, {21343, 29226}, {22383, 23568}, {23056, 24012}, {23226, 34948}, {23345, 41436}, {24533, 28255}, {24675, 28758}, {24749, 25666}, {25259, 29037}, {25299, 30023}, {25901, 25924}, {27416, 27419}, {28373, 28398}, {28468, 44433}, {28521, 44429}, {30804, 43041}, {31207, 31288}, {38314, 45667}, {38357, 38983}, {38986, 40623}

X(663) = midpoint of X(i) and X(j) for these {i,j}: {1, 4040}, {650, 4162}, {659, 4879}, {667, 4775}, {2530, 6161}, {4041, 4895}, {4449, 4724}, {4814, 4959}, {17418, 42312}, {20295, 31291}
X(663) = reflection of X(i) in X(j) for these {i,j}: {2, 45316}, {8, 4147}, {649, 667}, {652, 21789}, {667, 1960}, {1459, 2605}, {1734, 14838}, {2254, 905}, {2517, 8062}, {2533, 4874}, {3700, 4990}, {4017, 6129}, {4040, 4794}, {4041, 650}, {4063, 4401}, {4163, 4521}, {4391, 3716}, {4449, 1}, {4474, 4391}, {4498, 659}, {4707, 20517}, {4724, 4040}, {4813, 4983}, {4814, 4041}, {4895, 4162}, {4905, 3960}, {4959, 4895}, {7178, 676}, {17418, 3737}, {18344, 17115}, {20979, 4455}, {21118, 21185}, {21300, 25128}, {21301, 3835}, {21302, 17072}, {23655, 810}, {24719, 4992}, {42649, 8648}, {42666, 42653}, {43924, 1459}
X(663) = isogonal conjugate of X(664)
X(663) = isotomic conjugate of X(4572)
X(663) = complement of X(21302)
X(663) = anticomplement of X(17072)
X(663) = polar conjugate of the isotomic conjugate of X(652)
X(663) = psi-transform of X(41157)
X(663) = X(3500)-anticomplementary conjugate of X(150)
X(663) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 38991}, {40505, 124}, {40523, 3452}
X(663) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 2170}, {2, 38991}, {6, 14936}, {29, 21044}, {33, 2310}, {41, 38365}, {55, 3271}, {56, 7117}, {99, 23640}, {100, 2347}, {101, 41}, {108, 1400}, {109, 6}, {110, 21748}, {162, 1195}, {190, 20665}, {282, 3119}, {513, 649}, {522, 652}, {643, 2269}, {644, 1334}, {650, 657}, {651, 1200}, {664, 2082}, {692, 42}, {927, 20459}, {932, 20460}, {934, 1475}, {1415, 40957}, {1897, 40968}, {2191, 244}, {2192, 3270}, {2222, 2183}, {2424, 665}, {2701, 1951}, {2720, 1404}, {3433, 20974}, {3445, 1015}, {3699, 9}, {3737, 650}, {3903, 23544}, {3939, 55}, {4069, 3683}, {4551, 2264}, {4557, 20967}, {4626, 57}, {4636, 284}, {7073, 4516}, {7252, 3063}, {7257, 3691}, {8059, 604}, {8750, 31}, {8761, 39687}, {10570, 1146}, {14733, 1055}, {21789, 1946}, {23351, 6139}, {23838, 654}, {26700, 2260}, {29055, 1193}, {32652, 40958}, {32653, 32}, {32674, 607}, {32676, 40976}, {34435, 20982}, {34921, 7113}, {35348, 22108}, {36082, 48}, {36086, 672}, {36127, 19}, {40519, 23638}, {40523, 16588}
X(663) = X(i)-cross conjugate of X(j) for these (i,j): {512, 18344}, {810, 1946}, {2488, 513}, {3022, 607}, {3063, 649}, {3271, 55}, {3709, 650}, {4516, 42}, {6139, 23351}, {8641, 657}, {14936, 6}
X(663) = cevapoint of X(i) and X(j) for these (i,j): {512, 810}, {522, 25128}, {3063, 8641}
X(663) = crosspoint of X(i) and X(j) for these (i,j): {1, 101}, {6, 109}, {9, 3699}, {19, 36127}, {21, 644}, {33, 8750}, {55, 3939}, {56, 32674}, {57, 4626}, {86, 4603}, {100, 23617}, {108, 1172}, {190, 7033}, {282, 8059}, {284, 4636}, {513, 650}, {514, 26721}, {522, 3064}, {664, 7131}, {692, 2194}, {934, 1170}, {1318, 5548}, {3737, 7252}, {10570, 32653}, {11075, 34921}
X(663) = crosssum of X(i) and X(j) for these (i,j): {1, 514}, {2, 522}, {6, 23865}, {7, 3676}, {8, 6332}, {9, 4105}, {10, 7265}, {57, 43924}, {65, 3669}, {77, 4025}, {100, 651}, {109, 1813}, {145, 30719}, {190, 43290}, {192, 24749}, {223, 8058}, {513, 3752}, {521, 1214}, {523, 17056}, {525, 18641}, {649, 7032}, {650, 14100}, {661, 23668}, {663, 2082}, {693, 1441}, {905, 1071}, {1212, 3900}, {1317, 30725}, {1459, 26934}, {1537, 10015}, {1577, 23555}, {2785, 39035}, {3008, 38371}, {3649, 7178}, {3738, 16586}, {3835, 23669}, {3870, 31605}, {3911, 39771}, {3960, 11570}, {4391, 23528}, {4449, 9312}, {4453, 41801}, {4551, 4552}, {5586, 30723}, {6366, 35110}, {6608, 42449}, {7180, 39780}, {10052, 21188}, {14208, 18697}, {15185, 43049}, {16006, 41800}, {16585, 35057}, {17496, 37558}, {21183, 36595}, {23615, 31648}, {30724, 34502}, {30726, 39777}, {35312, 35338}, {39775, 43041}, {39782, 43052}
X(663) = trilinear pole of line {3271, 6139}
X(663) = crossdifference of every pair of points on line {2, 7}
X(663) = bicentric difference of PU(i) for i in (18, 49, 93, 115)
X(663) = PU(18)-harmonic conjugate of X(1400)
X(663) = PU(49)-harmonic conjugate of X(672)
X(663) = PU(93)-harmonic conjugate of X(41)
X(663) = trilinear pole of PU(103) (line X(3271)X(6139))
X(663) = PU(115)-harmonic conjugate of X(2082)
X(663) = perspector of hyperbola {A,B,C,X(6),X(9)}
X(663) = intersection of trilinear polars of X(6) and X(9)
X(663) = pole wrt polar circle of line X(75)X(225)
X(663) = X(92)-isoconjugate of X(1813)
X(663) = Parry-circle-inverse of X(5075)
X(663) = center of circle {X(1),X(15),X(16)}} (or V(X(1)))
X(663) = barycentric product of Feuerbach hyperbola intercepts of antiorthic axis
X(663) = trilinear product of Feuerbach hyperbola intercepts of Lemoine axis
X(663) = intersection of perspectrices of [ABC and Gemini triangle 35] and [ABC and Gemini triangle 36]
X(663) = X(i)-isoconjugate of X(j) for these (i,j): {1, 664}, {2, 651}, {3, 18026}, {4, 6516}, {6, 4554}, {7, 100}, {8, 934}, {9, 658}, {10, 1414}, {21, 4566}, {31, 4572}, {34, 4561}, {36, 35174}, {37, 4573}, {42, 4625}, {55, 4569}, {56, 668}, {57, 190}, {59, 693}, {63, 653}, {65, 99}, {69, 108}, {73, 811}, {75, 109}, {76, 1415}, {77, 1897}, {78, 36118}, {81, 4552}, {85, 101}, {86, 4551}, {92, 1813}, {105, 883}, {110, 1441}, {112, 1231}, {158, 6517}, {162, 307}, {163, 349}, {181, 4623}, {200, 4626}, {210, 4616}, {219, 13149}, {220, 36838}, {222, 6335}, {223, 44327}, {225, 4592}, {226, 662}, {241, 666}, {256, 6649}, {264, 36059}, {269, 3699}, {273, 1331}, {274, 4559}, {278, 1332}, {279, 644}, {286, 23067}, {304, 32674}, {312, 1461}, {319, 26700}, {320, 2222}, {321, 4565}, {322, 8059}, {326, 36127}, {329, 37141}, {331, 906}, {333, 1020}, {341, 6614}, {345, 32714}, {346, 4617}, {347, 13138}, {348, 1783}, {354, 6606}, {377, 13395}, {388, 1310}, {479, 4578}, {497, 8269}, {513, 4998}, {514, 4564}, {518, 927}, {522, 7045}, {527, 37139}, {552, 40521}, {553, 37212}, {604, 1978}, {643, 3668}, {645, 1427}, {646, 1407}, {648, 1214}, {650, 1275}, {655, 3218}, {660, 1447}, {661, 4620}, {670, 1402}, {672, 34085}, {673, 1025}, {692, 6063}, {738, 6558}, {765, 3676}, {789, 1469}, {799, 1400}, {813, 10030}, {823, 40152}, {831, 7247}, {894, 37137}, {898, 43037}, {903, 23703}, {905, 46102}, {908, 37136}, {919, 40704}, {932, 3212}, {940, 32038}, {1014, 3952}, {1016, 3669}, {1018, 1434}, {1042, 7257}, {1086, 31615}, {1088, 3939}, {1119, 4571}, {1121, 23890}, {1122, 8706}, {1155, 35157}, {1252, 24002}, {1262, 4391}, {1284, 4589}, {1292, 6604}, {1293, 39126}, {1305, 3868}, {1317, 4618}, {1319, 4555}, {1334, 4635}, {1397, 6386}, {1403, 18830}, {1408, 27808}, {1409, 6331}, {1412, 4033}, {1423, 4598}, {1428, 4583}, {1429, 4562}, {1432, 18047}, {1439, 36797}, {1442, 6742}, {1445, 37206}, {1446, 5546}, {1449, 4624}, {1462, 42720}, {1463, 8709}, {1465, 13136}, {1476, 21272}, {1492, 7179}, {1509, 21859}, {1633, 8817}, {1758, 35154}, {1799, 46152}, {1847, 4587}, {1880, 4563}, {1909, 29055}, {1945, 15418}, {1969, 32660}, {1981, 40843}, {2003, 15455}, {2006, 4585}, {2078, 35171}, {2099, 4597}, {2149, 3261}, {2171, 4610}, {2185, 4605}, {2223, 46135}, {2254, 39293}, {2283, 2481}, {2284, 34018}, {2285, 37215}, {2321, 4637}, {2338, 24015}, {2346, 35312}, {2397, 34051}, {2398, 43736}, {2406, 36100}, {2550, 6183}, {2652, 17933}, {2720, 3262}, {2742, 38468}, {3057, 6613}, {3112, 46153}, {3219, 38340}, {3257, 3911}, {3263, 32735}, {3451, 21580}, {3573, 7233}, {3598, 37223}, {3649, 4596}, {3666, 6648}, {3671, 4614}, {3674, 36147}, {3732, 7131}, {3779, 34083}, {3903, 7176}, {3912, 36146}, {4017, 4600}, {4025, 7012}, {4032, 4603}, {4059, 8708}, {4076, 43932}, {4077, 4570}, {4105, 24011}, {4130, 23586}, {4146, 6733}, {4163, 24013}, {4357, 36098}, {4397, 7339}, {4556, 6358}, {4558, 40149}, {4567, 7178}, {4579, 7249}, {4584, 16609}, {4586, 7146}, {4595, 7153}, {4601, 7180}, {4604, 5219}, {4606, 21454}, {4612, 6354}, {4619, 4858}, {4621, 41777}, {4622, 40663}, {4654, 37211}, {4705, 7340}, {4968, 29279}, {5061, 35147}, {5172, 35156}, {5221, 32042}, {5228, 32041}, {5252, 13396}, {5376, 30725}, {5377, 43042}, {5378, 43041}, {5379, 17094}, {5380, 7181}, {5382, 30719}, {5383, 43051}, {5385, 43052}, {5389, 7214}, {5435, 27834}, {6012, 30617}, {6135, 13453}, {6136, 13436}, {6180, 30610}, {6332, 7128}, {6528, 22341}, {6540, 32636}, {6734, 36048}, {7035, 43924}, {7115, 15413}, {7132, 33946}, {7175, 27805}, {7182, 8750}, {7235, 36066}, {7265, 35049}, {7316, 42721}, {7672, 43349}, {8047, 40577}, {8680, 41206}, {8685, 33930}, {8687, 20911}, {8707, 24471}, {9357, 10001}, {9436, 36086}, {10509, 35341}, {13486, 40999}, {13577, 40576}, {14612, 39957}, {14733, 30806}, {14942, 41353}, {17080, 44765}, {17090, 29227}, {17097, 17136}, {17791, 34921}, {17924, 44717}, {18033, 34067}, {18743, 38828}, {18816, 23981}, {19604, 43290}, {20567, 32739}, {20924, 32675}, {20930, 36082}, {21362, 40420}, {21453, 35338}, {22464, 36037}, {24027, 35519}, {24029, 34234}, {30456, 44326}, {30545, 34071}, {31618, 35326}, {32018, 36075}, {32693, 34284}, {33864, 36094}, {34855, 36802}, {35516, 36040}, {36049, 40702}, {37138, 40719}, {37143, 37787}
X(663) = barycentric product X(i)*X(j) for these {i,j}: {1, 650}, {3, 3064}, {4, 652}, {6, 522}, {7, 657}, {8, 649}, {9, 513}, {10, 7252}, {11, 101}, {19, 521}, {21, 661}, {25, 6332}, {28, 8611}, {29, 647}, {31, 4391}, {32, 35519}, {33, 905}, {37, 3737}, {41, 693}, {42, 4560}, {44, 23838}, {48, 44426}, {55, 514}, {56, 3239}, {57, 3900}, {58, 3700}, {59, 42462}, {60, 4024}, {63, 18344}, {64, 14331}, {65, 1021}, {73, 17926}, {74, 14400}, {75, 3063}, {78, 6591}, {79, 9404}, {80, 654}, {81, 4041}, {84, 14298}, {85, 8641}, {86, 3709}, {88, 4895}, {89, 4814}, {92, 1946}, {100, 2170}, {106, 1639}, {108, 34591}, {109, 1146}, {110, 21044}, {111, 14432}, {115, 4636}, {124, 32653}, {158, 36054}, {184, 46110}, {188, 6729}, {190, 3271}, {200, 3669}, {210, 1019}, {212, 17924}, {213, 18155}, {219, 7649}, {220, 3676}, {225, 23090}, {226, 21789}, {244, 644}, {256, 3287}, {259, 6728}, {261, 4079}, {266, 6730}, {269, 4130}, {279, 4105}, {281, 1459}, {282, 6129}, {283, 2501}, {284, 523}, {291, 4435}, {292, 3716}, {294, 2254}, {312, 667}, {314, 798}, {318, 22383}, {332, 2489}, {333, 512}, {346, 43924}, {515, 2432}, {518, 1024}, {520, 8748}, {525, 2299}, {527, 23351}, {579, 23289}, {604, 4397}, {607, 4025}, {643, 3125}, {645, 3122}, {646, 3248}, {651, 2310}, {653, 3270}, {656, 1172}, {658, 3022}, {659, 4876}, {662, 4516}, {664, 14936}, {665, 14942}, {669, 28660}, {672, 885}, {673, 926}, {676, 2338}, {692, 4858}, {728, 43932}, {739, 14430}, {749, 4501}, {810, 31623}, {812, 7077}, {813, 4124}, {822, 1896}, {875, 3975}, {876, 3684}, {884, 3912}, {893, 3907}, {900, 2316}, {901, 4530}, {909, 2804}, {918, 2195}, {934, 3119}, {941, 17418}, {1002, 45755}, {1014, 4171}, {1015, 3699}, {1018, 18191}, {1022, 3689}, {1027, 3693}, {1036, 6590}, {1039, 2522}, {1043, 7180}, {1086, 3939}, {1110, 40166}, {1121, 6139}, {1126, 4976}, {1155, 23893}, {1170, 6608}, {1174, 6362}, {1178, 4140}, {1252, 21132}, {1253, 24002}, {1262, 23615}, {1292, 38375}, {1293, 4534}, {1318, 6544}, {1320, 1635}, {1331, 8735}, {1333, 4086}, {1334, 7192}, {1357, 6558}, {1395, 15416}, {1400, 7253}, {1407, 4163}, {1414, 36197}, {1415, 24026}, {1431, 4529}, {1432, 4477}, {1434, 4524}, {1436, 8058}, {1458, 28132}, {1461, 4081}, {1491, 2344}, {1577, 2194}, {1638, 4845}, {1647, 5548}, {1751, 8676}, {1783, 7004}, {1813, 42069}, {1826, 23189}, {1857, 4091}, {1897, 7117}, {1919, 3596}, {1924, 40072}, {1960, 4997}, {1973, 35518}, {1980, 28659}, {2053, 3835}, {2090, 45878}, {2125, 17427}, {2149, 42455}, {2150, 4036}, {2160, 35057}, {2161, 3738}, {2162, 4147}, {2163, 4944}, {2175, 3261}, {2183, 43728}, {2185, 4705}, {2189, 4064}, {2192, 14837}, {2193, 24006}, {2204, 14208}, {2212, 15413}, {2226, 4543}, {2258, 23880}, {2265, 46041}, {2269, 4581}, {2287, 4017}, {2291, 6366}, {2298, 17420}, {2311, 4010}, {2318, 17925}, {2319, 4083}, {2320, 4893}, {2321, 3733}, {2325, 23345}, {2328, 7178}, {2332, 17094}, {2334, 4765}, {2339, 8678}, {2342, 10015}, {2346, 21127}, {2348, 35355}, {2364, 4777}, {2423, 6735}, {2424, 40869}, {2425, 15633}, {2484, 30479}, {2488, 32008}, {2590, 3308}, {2591, 3307}, {2605, 7110}, {2643, 4612}, {2785, 17963}, {2968, 32674}, {2969, 4587}, {3049, 44130}, {3120, 5546}, {3121, 7257}, {3208, 43931}, {3224, 25128}, {3254, 22108}, {3445, 4521}, {3451, 42337}, {3572, 3685}, {3680, 4394}, {3688, 10566}, {3692, 43923}, {3710, 43925}, {3717, 43929}, {3732, 14935}, {3768, 36798}, {3904, 6187}, {4009, 23892}, {4069, 16726}, {4076, 21143}, {4092, 4556}, {4162, 8056}, {4451, 20981}, {4455, 36800}, {4474, 30650}, {4515, 7203}, {4517, 4817}, {4518, 8632}, {4522, 40746}, {4526, 37129}, {4542, 4638}, {4546, 40151}, {4557, 17197}, {4603, 40608}, {4617, 24010}, {4619, 5532}, {4626, 35508}, {4724, 40779}, {4750, 5547}, {4768, 9456}, {4775, 30608}, {4778, 34820}, {4790, 4866}, {4825, 30607}, {4834, 42030}, {4885, 9439}, {4913, 25426}, {4919, 6164}, {4939, 34080}, {4953, 38828}, {4959, 26745}, {4977, 33635}, {4979, 32635}, {4985, 28615}, {5060, 18013}, {5075, 17947}, {5089, 23696}, {5452, 26721}, {5514, 8059}, {5540, 42552}, {6059, 30805}, {6065, 6545}, {6066, 23100}, {6169, 42341}, {6182, 39273}, {6364, 13456}, {6365, 13427}, {6373, 36799}, {6506, 36082}, {6557, 8643}, {6589, 10570}, {6603, 35348}, {6607, 10509}, {6614, 23970}, {6615, 23617}, {6740, 21828}, {7072, 21188}, {7073, 14838}, {7091, 40137}, {7106, 8062}, {7108, 21761}, {7118, 17896}, {7131, 17115}, {7152, 14302}, {7155, 20979}, {7160, 14300}, {7162, 13401}, {7255, 20684}, {7707, 10495}, {8602, 30201}, {8638, 18031}, {8640, 27424}, {8648, 18359}, {8750, 26932}, {9447, 40495}, {10397, 40836}, {10482, 21104}, {10492, 16012}, {10579, 14282}, {10581, 21453}, {11125, 15627}, {11934, 44178}, {12032, 28143}, {12077, 35196}, {14304, 32677}, {14392, 34056}, {14395, 36119}, {14399, 44693}, {14413, 41798}, {14418, 36125}, {14571, 37628}, {14733, 33573}, {15313, 39943}, {15420, 40976}, {15997, 45877}, {17435, 36086}, {18000, 40882}, {18101, 46148}, {18108, 33299}, {21172, 30457}, {21185, 40141}, {21362, 40528}, {21390, 40505}, {21666, 32660}, {21739, 42657}, {21807, 39177}, {23104, 23979}, {23493, 27527}, {24012, 36838}, {24225, 40523}, {28291, 43960}, {29226, 36630}, {30731, 43922}, {31182, 33963}, {32641, 35015}, {32739, 34387}, {35072, 36127}, {36049, 38357}, {36050, 38345}, {39924, 45902}, {39956, 42312}, {42064, 43050}
X(663) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 4554}, {2, 4572}, {6, 664}, {8, 1978}, {9, 668}, {11, 3261}, {19, 18026}, {21, 799}, {25, 653}, {29, 6331}, {31, 651}, {32, 109}, {33, 6335}, {34, 13149}, {41, 100}, {42, 4552}, {48, 6516}, {55, 190}, {56, 658}, {57, 4569}, {58, 4573}, {60, 4610}, {81, 4625}, {101, 4998}, {105, 34085}, {109, 1275}, {110, 4620}, {172, 6649}, {181, 4605}, {184, 1813}, {200, 646}, {210, 4033}, {212, 1332}, {213, 4551}, {219, 4561}, {220, 3699}, {244, 24002}, {269, 36838}, {283, 4563}, {284, 99}, {312, 6386}, {314, 4602}, {333, 670}, {480, 6558}, {512, 226}, {513, 85}, {514, 6063}, {521, 304}, {522, 76}, {523, 349}, {560, 1415}, {577, 6517}, {604, 934}, {607, 1897}, {608, 36118}, {643, 4601}, {644, 7035}, {647, 307}, {649, 7}, {650, 75}, {652, 69}, {654, 320}, {656, 1231}, {657, 8}, {659, 10030}, {661, 1441}, {665, 9436}, {667, 57}, {669, 1400}, {672, 883}, {673, 46135}, {692, 4564}, {693, 20567}, {788, 7146}, {798, 65}, {810, 1214}, {812, 18033}, {834, 33949}, {872, 21859}, {884, 673}, {885, 18031}, {904, 37137}, {905, 7182}, {919, 39293}, {926, 3912}, {1014, 4635}, {1015, 3676}, {1021, 314}, {1024, 2481}, {1027, 34018}, {1036, 37215}, {1098, 4631}, {1106, 4617}, {1110, 31615}, {1146, 35519}, {1172, 811}, {1174, 6606}, {1253, 644}, {1333, 1414}, {1334, 3952}, {1364, 30805}, {1395, 32714}, {1397, 1461}, {1400, 4566}, {1402, 1020}, {1407, 4626}, {1408, 4637}, {1412, 4616}, {1415, 7045}, {1438, 927}, {1456, 24015}, {1459, 348}, {1475, 35312}, {1639, 3264}, {1802, 4571}, {1918, 4559}, {1919, 56}, {1924, 1402}, {1936, 15418}, {1946, 63}, {1960, 3911}, {1973, 108}, {1974, 32674}, {1977, 43924}, {1980, 604}, {2053, 4598}, {2161, 35174}, {2170, 693}, {2175, 101}, {2185, 4623}, {2192, 44327}, {2193, 4592}, {2194, 662}, {2195, 666}, {2200, 23067}, {2204, 162}, {2206, 4565}, {2207, 36127}, {2208, 37141}, {2212, 1783}, {2223, 1025}, {2251, 23703}, {2254, 40704}, {2258, 32038}, {2287, 7257}, {2291, 35157}, {2299, 648}, {2310, 4391}, {2311, 4589}, {2316, 4555}, {2319, 18830}, {2321, 27808}, {2328, 645}, {2330, 18047}, {2332, 36797}, {2334, 4624}, {2340, 42720}, {2342, 13136}, {2344, 789}, {2347, 21272}, {2361, 4585}, {2364, 4597}, {2432, 34393}, {2483, 7247}, {2484, 388}, {2488, 142}, {2489, 225}, {2605, 17095}, {3022, 3239}, {3049, 73}, {3051, 46153}, {3056, 33946}, {3057, 21580}, {3063, 1}, {3064, 264}, {3119, 4397}, {3121, 4017}, {3122, 7178}, {3125, 4077}, {3208, 36863}, {3239, 3596}, {3248, 3669}, {3249, 1357}, {3250, 7179}, {3261, 41283}, {3270, 6332}, {3271, 514}, {3287, 1909}, {3309, 21609}, {3310, 22464}, {3451, 6613}, {3572, 7233}, {3669, 1088}, {3684, 874}, {3685, 27853}, {3688, 4568}, {3689, 24004}, {3699, 31625}, {3700, 313}, {3709, 10}, {3716, 1921}, {3733, 1434}, {3737, 274}, {3738, 20924}, {3768, 43037}, {3900, 312}, {3904, 40075}, {3907, 1920}, {3939, 1016}, {4017, 1446}, {4024, 34388}, {4041, 321}, {4079, 12}, {4083, 30545}, {4086, 27801}, {4091, 7055}, {4105, 346}, {4130, 341}, {4140, 1237}, {4147, 6382}, {4148, 4087}, {4162, 18743}, {4171, 3701}, {4367, 7196}, {4369, 7205}, {4391, 561}, {4394, 39126}, {4397, 28659}, {4435, 350}, {4455, 16609}, {4477, 17787}, {4501, 3760}, {4507, 7201}, {4516, 1577}, {4517, 3807}, {4524, 2321}, {4526, 6381}, {4531, 7239}, {4543, 36791}, {4546, 44723}, {4556, 7340}, {4560, 310}, {4612, 24037}, {4617, 24011}, {4636, 4590}, {4705, 6358}, {4775, 5219}, {4814, 4671}, {4820, 30596}, {4827, 4673}, {4832, 3671}, {4834, 4654}, {4858, 40495}, {4876, 4583}, {4895, 4358}, {4976, 1269}, {5060, 17933}, {5075, 17950}, {5546, 4600}, {6065, 6632}, {6129, 40702}, {6139, 527}, {6169, 14727}, {6186, 38340}, {6187, 655}, {6332, 305}, {6362, 1233}, {6371, 3674}, {6373, 43040}, {6586, 33298}, {6591, 273}, {6602, 4578}, {6608, 1229}, {6614, 23586}, {6615, 26563}, {6729, 4146}, {7004, 15413}, {7063, 4079}, {7064, 4103}, {7073, 15455}, {7077, 4562}, {7083, 3732}, {7104, 29055}, {7117, 4025}, {7118, 13138}, {7180, 3668}, {7234, 4032}, {7252, 86}, {7253, 28660}, {7336, 23100}, {7649, 331}, {8611, 20336}, {8632, 1447}, {8638, 672}, {8640, 1423}, {8641, 9}, {8642, 1445}, {8643, 5435}, {8645, 37787}, {8646, 2285}, {8648, 3218}, {8653, 5257}, {8676, 18134}, {8735, 46107}, {8748, 6528}, {8750, 46102}, {9247, 36059}, {9404, 319}, {9439, 30610}, {9447, 692}, {9448, 32739}, {9449, 46177}, {9454, 2283}, {10581, 4847}, {11124, 20881}, {11193, 18151}, {11934, 20927}, {14298, 322}, {14331, 14615}, {14400, 3260}, {14407, 40663}, {14413, 37780}, {14427, 4723}, {14430, 35543}, {14432, 3266}, {14575, 32660}, {14776, 39294}, {14827, 3939}, {14936, 522}, {14942, 36803}, {17412, 10527}, {17418, 34284}, {17420, 20911}, {17424, 4862}, {17425, 5231}, {17926, 44130}, {17963, 35154}, {18000, 11608}, {18105, 18097}, {18155, 6385}, {18191, 7199}, {18265, 813}, {18344, 92}, {20229, 35338}, {20665, 3888}, {20967, 3882}, {20979, 3212}, {20980, 9312}, {20981, 7176}, {21044, 850}, {21123, 3665}, {21127, 20880}, {21132, 23989}, {21143, 1358}, {21748, 17136}, {21758, 1443}, {21761, 1943}, {21789, 333}, {21828, 41804}, {22383, 77}, {23090, 332}, {23189, 17206}, {23224, 7183}, {23289, 40011}, {23351, 1121}, {23572, 17082}, {23615, 23978}, {23838, 20568}, {24012, 4130}, {25128, 6374}, {28660, 4609}, {32656, 44717}, {32739, 59}, {33525, 6734}, {33635, 6540}, {34068, 37139}, {34591, 35518}, {34858, 37136}, {35057, 33939}, {35508, 4163}, {35518, 40364}, {35519, 1502}, {36054, 326}, {36197, 4086}, {38347, 20954}, {38365, 17494}, {38367, 8850}, {38986, 43051}, {38991, 21302}, {39201, 40152}, {40972, 4553}, {40982, 17906}, {41339, 42719}, {42069, 46110}, {42312, 18135}, {42462, 34387}, {42649, 17483}, {42657, 17484}, {43923, 1847}, {43924, 279}, {43931, 7209}, {43932, 23062}, {44426, 1969}, {45755, 4441}, {46110, 18022}
X(663) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 4794, 4724}, {2, 21302, 17072}, {649, 1960, 8656}, {649, 8643, 667}, {650, 4435, 45755}, {650, 4895, 4814}, {667, 1946, 8648}, {667, 1960, 8643}, {667, 8641, 1946}, {667, 8643, 8656}, {669, 2978, 649}, {1919, 4079, 2484}, {1960, 4775, 649}, {2488, 8641, 649}, {3063, 3709, 657}, {3250, 8632, 649}, {4041, 4162, 4959}, {4564, 36086, 9323}, {5638, 5639, 5075}


X(664) = TRILINEAR POLE OF LINE X(2)X(7)

Trilinears    1/[a(b - c)(b + c - a)] : :
X(664) = 3 X[2] - 4 X[17044], 6 X[2] - 5 X[31640], 9 X[2] - 8 X[40483], X[150] - 3 X[38941], 3 X[1121] - 4 X[1146], 3 X[1121] - 8 X[17044], 3 X[1121] - 5 X[31640], X[1121] - 4 X[35110], 3 X[1121] - 2 X[39351], X[1121] + 2 X[39357], 9 X[1121] - 16 X[40483], 4 X[1146] - 5 X[31640], X[1146] - 3 X[35110], 2 X[1146] + 3 X[39357], 3 X[1146] - 4 X[40483], 3 X[1275] - X[35157], 4 X[1323] - 3 X[17078], 2 X[1565] - 3 X[38941], 2 X[9436] - 3 X[17078], 3 X[10001] - 2 X[35157], 8 X[17044] - 5 X[31640], 2 X[17044] - 3 X[35110], 4 X[17044] - X[39351], 4 X[17044] + 3 X[39357], 3 X[17044] - 2 X[40483], 5 X[31640] - 12 X[35110], 5 X[31640] - 2 X[39351], 5 X[31640] + 6 X[39357], 15 X[31640] - 16 X[40483], 2 X[31852] - 3 X[38690], 6 X[35110] - X[39351], 2 X[35110] + X[39357], 9 X[35110] - 4 X[40483], X[39351] + 3 X[39357], 3 X[39351] - 8 X[40483], 9 X[39357] + 8 X[40483]

X(664) lies on the Steiner circumellipse and these lines: {1, 85}, {2, 1121}, {6, 30627}, {7, 528}, {8, 348}, {9, 31169}, {10, 17095}, {12, 17084}, {20, 7973}, {30, 5195}, {34, 18156}, {37, 35144}, {40, 7183}, {42, 7196}, {43, 25721}, {56, 3212}, {57, 3227}, {65, 1434}, {69, 347}, {73, 290}, {75, 77}, {76, 10571}, {78, 16284}, {86, 1411}, {92, 6505}, {99, 109}, {100, 658}, {101, 514}, {106, 21208}, {108, 1310}, {110, 1305}, {118, 18328}, {141, 17086}, {145, 279}, {150, 952}, {163, 4237}, {175, 490}, {176, 489}, {190, 644}, {192, 6180}, {200, 31627}, {220, 3177}, {221, 1975}, {222, 24282}, {223, 312}, {224, 1847}, {225, 8773}, {226, 671}, {239, 241}, {269, 1120}, {273, 20930}, {274, 37558}, {278, 18134}, {292, 39940}, {304, 7210}, {307, 319}, {316, 38945}, {320, 22464}, {322, 326}, {325, 35149}, {333, 1214}, {345, 18623}, {350, 1457}, {355, 17181}, {385, 17966}, {388, 20539}, {394, 6360}, {515, 4872}, {516, 10697}, {517, 5088}, {518, 14189}, {519, 1323}, {522, 1275}, {523, 35154}, {524, 17950}, {527, 34926}, {536, 6610}, {553, 41823}, {604, 18825}, {648, 653}, {655, 4604}, {663, 14727}, {668, 1026}, {670, 4572}, {673, 2170}, {693, 35174}, {728, 2124}, {738, 2136}, {740, 5018}, {765, 31605}, {789, 29055}, {811, 6528}, {835, 29279}, {874, 18830}, {889, 43924}, {892, 4620}, {914, 17923}, {944, 17170}, {948, 17316}, {1015, 24281}, {1016, 30719}, {1018, 1025}, {1020, 3882}, {1042, 35159}, {1043, 1231}, {1088, 3870}, {1100, 41246}, {1125, 25723}, {1212, 32008}, {1215, 40723}, {1220, 40845}, {1223, 42449}, {1284, 35166}, {1292, 6183}, {1319, 1447}, {1331, 36048}, {1362, 14839}, {1376, 31526}, {1388, 17090}, {1397, 18824}, {1400, 3228}, {1402, 18826}, {1403, 17082}, {1407, 3210}, {1415, 4586}, {1418, 4852}, {1419, 3729}, {1427, 1999}, {1429, 35172}, {1432, 14970}, {1435, 8897}, {1443, 17160}, {1445, 3759}, {1446, 34772}, {1456, 3685}, {1458, 10030}, {1461, 4605}, {1469, 43096}, {1482, 17753}, {1575, 43062}, {1633, 16680}, {1788, 17081}, {1818, 33677}, {1897, 13149}, {1944, 6510}, {1992, 12848}, {2006, 25529}, {2082, 6167}, {2114, 17755}, {2222, 13396}, {2340, 40864}, {2669, 7235}, {2893, 6356}, {2898, 25568}, {2966, 32660}, {3174, 23062}, {3188, 3868}, {3243, 42309}, {3244, 10481}, {3501, 34497}, {3509, 6647}, {3616, 31994}, {3623, 43983}, {3665, 10944}, {3668, 3879}, {3669, 4562}, {3671, 41813}, {3674, 7247}, {3676, 4555}, {3693, 40872}, {3752, 40420}, {3873, 23839}, {3880, 34855}, {3888, 3903}, {3911, 35168}, {3912, 35158}, {3935, 37780}, {3936, 37798}, {3939, 6606}, {3952, 4624}, {3957, 21453}, {3996, 7182}, {4017, 35147}, {4025, 7045}, {4059, 11011}, {4077, 35156}, {4089, 7972}, {4318, 40704}, {4334, 32921}, {4350, 17158}, {4352, 37614}, {4384, 31225}, {4393, 5228}, {4449, 46135}, {4482, 4568}, {4511, 30806}, {4513, 25242}, {4534, 26007}, {4569, 4626}, {4589, 18829}, {4597, 23703}, {4598, 37137}, {4614, 4616}, {4645, 35150}, {4664, 8545}, {4861, 20880}, {4904, 43057}, {4911, 45287}, {4971, 40892}, {5176, 33864}, {5219, 35170}, {5252, 7179}, {5263, 43099}, {5289, 30946}, {5543, 20057}, {5845, 20096}, {5930, 7270}, {6063, 43093}, {6332, 46102}, {6335, 24035}, {6354, 17778}, {6357, 35161}, {6376, 37694}, {6540, 33948}, {6542, 35163}, {6554, 26658}, {6558, 42720}, {6559, 9502}, {6571, 8706}, {6603, 10025}, {6613, 6614}, {6645, 40765}, {6739, 23674}, {7056, 17784}, {7080, 34060}, {7117, 24499}, {7146, 41245}, {7175, 35143}, {7178, 35148}, {7181, 35153}, {7185, 30617}, {7190, 17393}, {7268, 21454}, {7354, 33867}, {7763, 34030}, {9086, 14733}, {9259, 21138}, {9263, 32029}, {9296, 42336}, {9318, 17439}, {9358, 17069}, {9362, 43051}, {10405, 27541}, {10509, 15185}, {10725, 31851}, {11054, 37856}, {11109, 20926}, {11185, 34029}, {12513, 36638}, {12635, 36854}, {13395, 13397}, {14100, 23618}, {14513, 44435}, {14723, 44408}, {14829, 17080}, {15343, 21297}, {15903, 17719}, {16090, 44150}, {16585, 40435}, {16586, 34234}, {16609, 35173}, {16777, 26125}, {17056, 17947}, {17075, 17295}, {17093, 36845}, {17094, 35169}, {17233, 28739}, {17234, 37800}, {17285, 28780}, {17317, 21617}, {17352, 44355}, {17743, 41771}, {17752, 28391}, {17791, 46141}, {17864, 37157}, {18031, 39046}, {18138, 34045}, {18359, 40612}, {18990, 33865}, {20050, 32003}, {20220, 21588}, {20247, 38859}, {20248, 38869}, {20533, 39063}, {20565, 46138}, {21294, 36154}, {21296, 36640}, {21605, 34039}, {21609, 34036}, {23890, 35342}, {24002, 35171}, {24028, 35046}, {24291, 32942}, {24349, 41354}, {24618, 45749}, {25082, 28961}, {25083, 40863}, {25930, 30854}, {26006, 37774}, {26015, 37757}, {26526, 27006}, {26653, 26690}, {27191, 37771}, {28660, 40611}, {29052, 34083}, {30379, 37756}, {30545, 32926}, {30567, 36636}, {30573, 39293}, {30726, 34024}, {31343, 43290}, {31604, 34247}, {31631, 40149}, {31648, 36956}, {31721, 38314}, {31852, 38690}, {31997, 37523}, {32025, 40999}, {32040, 44553}, {32851, 34050}, {34085, 37138}, {34342, 41556}, {35961, 42079}, {35962, 41553}, {36098, 37215}, {36276, 42046}, {36595, 39704}, {36802, 41075}, {37211, 38340}, {37686, 43039}, {38459, 38460}, {39350, 44351}, {40859, 43059}, {45798, 46108}

X(664) = midpoint of X(2) and X(39357)
X(664) = reflection of X(i) in X(j) for these {i,j}: {2, 35110}, {85, 36905}, {150, 1565}, {190, 40865}, {333, 39035}, {1121, 2}, {1146, 17044}, {1944, 6510}, {1952, 1214}, {3509, 6647}, {3732, 101}, {8777, 6505}, {9436, 1323}, {10001, 1275}, {10025, 6603}, {10725, 31851}, {14942, 1}, {17947, 17056}, {18328, 118}, {31648, 36956}, {39351, 1146}, {40862, 6610}
X(664) = reflection of X(927) in the Soddy line
X(664) = isogonal conjugate of X(663)
X(664) = isotomic conjugate of X(522)
X(664) = complement of X(39351)
X(664) = anticomplement of X(1146)
X(664) = polar conjugate of X(3064)
X(664) = cevapoint of X(i) and X(j) for these (i,j): {1, 514}, {2, 522}, {6, 23865}, {7, 3676}, {8, 6332}, {9, 4105}, {10, 7265}, {57, 43924}, {65, 3669}, {77, 4025}, {100, 651}, {109, 1813}, {145, 30719}, {190, 43290}, {192, 24749}, {223, 8058}, {513, 3752}, {521, 1214}, {523, 17056}, {525, 18641}, {649, 7032}, {650, 14100}, {661, 23668}, {663, 2082}, {693, 1441}, {905, 1071}, {1212, 3900}, {1317, 30725}, {1459, 26934}, {1537, 10015}, {1577, 23555}, {2785, 39035}, {3008, 38371}, {3649, 7178}, {3738, 16586}, {3835, 23669}, {3870, 31605}, {3911, 39771}, {3960, 11570}, {4391, 23528}, {4449, 9312}, {4453, 41801}, {4551, 4552}, {5586, 30723}, {6366, 35110}, {6608, 42449}, {7180, 39780}, {10052, 21188}, {14208, 18697}, {15185, 43049}, {16006, 41800}, {16585, 35057}, {17496, 37558}, {21183, 36595}, {23615, 31648}, {30724, 34502}, {30726, 39777}, {35312, 35338}, {39775, 43041}, {39782, 43052}
X(664) = crosspoint of X(i) and X(j) for these (i,j): {99, 811}, {4554, 4569}, {4573, 4625}
X(664) = crosssum of X(i) and X(j) for these (i,j): {512, 810}, {522, 25128}, {3063, 8641}
X(664) = trilinear pole of line {2, 7}
X(664) = crossdifference of every pair of points on line {3271, 6139}
X(664) = Steiner-circumellipse-antipode of X(1121)
X(664) = Steiner-circumellipse-X(1)-antipode of X(2481)
X(664) = trilinear product of circumcircle intercepts of line X(7)X(8)
X(664) = pole wrt polar circle of trilinear polar of X(3064)
X(664) = crossdifference of PU(103)
X(664) = Jerabek image of X(7)
X(664) = X(i)-isoconjugate of X(j) for these (i,j): {1, 663}, {2, 3063}, {3, 18344}, {4, 1946}, {6, 650}, {7, 8641}, {8, 667}, {9, 649}, {11, 692}, {19, 652}, {21, 512}, {25, 521}, {29, 810}, {31, 522}, {32, 4391}, {33, 1459}, {37, 7252}, {41, 514}, {42, 3737}, {48, 3064}, {55, 513}, {56, 3900}, {57, 657}, {58, 4041}, {60, 4705}, {65, 21789}, {80, 8648}, {81, 3709}, {100, 3271}, {101, 2170}, {105, 926}, {106, 4895}, {108, 3270}, {109, 2310}, {110, 4516}, {163, 21044}, {184, 44426}, {200, 43924}, {210, 3733}, {212, 7649}, {213, 4560}, {219, 6591}, {220, 3669}, {244, 3939}, {259, 6729}, {269, 4105}, {281, 22383}, {284, 661}, {292, 4435}, {294, 665}, {312, 1919}, {314, 669}, {333, 798}, {393, 36054}, {480, 43932}, {518, 884}, {523, 2194}, {525, 2204}, {560, 35519}, {604, 3239}, {607, 905}, {643, 3122}, {644, 1015}, {645, 3121}, {646, 1977}, {647, 1172}, {651, 14936}, {654, 2161}, {656, 2299}, {659, 7077}, {672, 1024}, {693, 2175}, {739, 4526}, {764, 6065}, {822, 8748}, {875, 3685}, {885, 2223}, {890, 36798}, {893, 3287}, {902, 23838}, {904, 3907}, {906, 8735}, {919, 17435}, {923, 14432}, {934, 3022}, {1014, 4524}, {1019, 1334}, {1021, 1400}, {1027, 2340}, {1036, 8678}, {1037, 17115}, {1055, 23893}, {1084, 4631}, {1106, 4163}, {1110, 21132}, {1146, 1415}, {1155, 23351}, {1156, 6139}, {1170, 10581}, {1174, 21127}, {1253, 3676}, {1260, 43923}, {1261, 6363}, {1318, 3251}, {1320, 1960}, {1333, 3700}, {1357, 4578}, {1397, 4397}, {1402, 7253}, {1407, 4130}, {1409, 17926}, {1412, 4171}, {1417, 4528}, {1431, 4477}, {1436, 14298}, {1461, 3119}, {1474, 8611}, {1633, 14935}, {1635, 2316}, {1638, 18889}, {1639, 9456}, {1769, 2342}, {1783, 7117}, {1812, 2489}, {1824, 23189}, {1857, 23224}, {1880, 23090}, {1896, 39201}, {1911, 3716}, {1918, 18155}, {1924, 28660}, {1973, 6332}, {1974, 35518}, {1980, 3596}, {2053, 4083}, {2087, 5548}, {2149, 42462}, {2150, 4024}, {2155, 14331}, {2159, 14400}, {2160, 9404}, {2163, 4814}, {2182, 2432}, {2185, 4079}, {2192, 6129}, {2193, 2501}, {2195, 2254}, {2206, 4086}, {2208, 8058}, {2212, 4025}, {2218, 8676}, {2258, 17418}, {2279, 45755}, {2287, 7180}, {2311, 21832}, {2319, 20979}, {2320, 4775}, {2328, 4017}, {2339, 2484}, {2341, 21828}, {2344, 3250}, {2346, 2488}, {2352, 23289}, {2356, 23696}, {2364, 4893}, {2424, 41339}, {2481, 8638}, {2520, 37741}, {2590, 2591}, {2605, 7073}, {2638, 36127}, {2643, 4636}, {2648, 5075}, {2651, 18000}, {2804, 34858}, {2982, 33525}, {3049, 31623}, {3065, 42657}, {3124, 4612}, {3125, 5546}, {3248, 3699}, {3254, 8645}, {3261, 9447}, {3423, 6182}, {3433, 11934}, {3445, 4162}, {3446, 11193}, {3467, 42649}, {3572, 3684}, {3680, 8643}, {3688, 18108}, {3689, 23345}, {3693, 43929}, {3694, 43925}, {3738, 6187}, {3766, 18265}, {4009, 23349}, {4076, 8027}, {4124, 34067}, {4131, 6059}, {4147, 7121}, {4449, 9439}, {4501, 30651}, {4521, 38266}, {4530, 32665}, {4531, 7255}, {4534, 34080}, {4546, 16945}, {4557, 18191}, {4565, 36197}, {4571, 42067}, {4581, 20967}, {4617, 35508}, {4623, 7063}, {4626, 24012}, {4790, 34820}, {4820, 34819}, {4845, 14413}, {4858, 32739}, {4876, 8632}, {4900, 8656}, {4919, 9262}, {4944, 28607}, {4976, 28615}, {4979, 33635}, {5040, 11609}, {5547, 14419}, {6186, 35057}, {6366, 34068}, {6373, 8851}, {6601, 8642}, {6614, 24010}, {7004, 8750}, {7105, 21761}, {7118, 14837}, {7129, 10397}, {7155, 8640}, {8646, 30479}, {8647, 35355}, {8749, 14395}, {8752, 14418}, {9247, 46110}, {9426, 40072}, {9448, 40495}, {10099, 37908}, {10566, 40972}, {11124, 18771}, {11604, 42670}, {14399, 15627}, {14775, 23207}, {14776, 35014}, {14827, 24002}, {15997, 45878}, {16686, 42552}, {21645, 43743}, {21758, 36910}, {21759, 27527}, {23615, 24027}, {23845, 40528}, {23865, 40505}, {23990, 40166}, {25128, 34248}, {32652, 38357}, {32653, 38345}, {32674, 34591}, {33573, 36141}, {36059, 42069}
X(664) = barycentric product X(i)*X(j) for these {i,j}: {1, 4554}, {6, 4572}, {7, 190}, {8, 658}, {9, 4569}, {10, 4573}, {12, 4610}, {37, 4625}, {56, 1978}, {57, 668}, {59, 3261}, {63, 18026}, {65, 799}, {69, 653}, {73, 6331}, {75, 651}, {76, 109}, {77, 6335}, {78, 13149}, {85, 100}, {86, 4552}, {92, 6516}, {99, 226}, {101, 6063}, {108, 304}, {110, 349}, {142, 6606}, {162, 1231}, {200, 36838}, {210, 4635}, {225, 4563}, {257, 6649}, {261, 4605}, {264, 1813}, {269, 646}, {273, 1332}, {274, 4551}, {278, 4561}, {279, 3699}, {305, 32674}, {307, 648}, {308, 46153}, {310, 4559}, {312, 934}, {313, 4565}, {314, 1020}, {319, 38340}, {320, 655}, {321, 1414}, {322, 37141}, {331, 1331}, {333, 4566}, {341, 4617}, {345, 36118}, {346, 4626}, {347, 44327}, {348, 1897}, {388, 37215}, {479, 6558}, {514, 4998}, {518, 34085}, {522, 1275}, {523, 4620}, {527, 35157}, {552, 4103}, {553, 6540}, {561, 1415}, {604, 6386}, {643, 1446}, {644, 1088}, {645, 3668}, {660, 10030}, {662, 1441}, {666, 9436}, {670, 1400}, {672, 46135}, {673, 883}, {692, 20567}, {693, 4564}, {765, 24002}, {789, 7146}, {811, 1214}, {813, 18033}, {835, 33949}, {873, 21859}, {918, 39293}, {927, 3912}, {932, 30545}, {1014, 4033}, {1016, 3676}, {1025, 2481}, {1026, 34018}, {1111, 31615}, {1254, 4631}, {1262, 35519}, {1284, 4639}, {1292, 21609}, {1305, 18134}, {1358, 6632}, {1402, 4602}, {1412, 27808}, {1423, 18830}, {1427, 7257}, {1429, 4583}, {1434, 3952}, {1442, 15455}, {1443, 36804}, {1447, 4562}, {1458, 36803}, {1461, 3596}, {1469, 37133}, {1476, 21580}, {1783, 7182}, {1847, 4571}, {1909, 37137}, {1920, 29055}, {1937, 15418}, {1969, 36059}, {2052, 6517}, {2149, 40495}, {2171, 4623}, {2222, 20924}, {2283, 18031}, {2321, 4616}, {2406, 34393}, {3212, 4598}, {3218, 35174}, {3262, 37136}, {3263, 36146}, {3452, 6613}, {3570, 7233}, {3616, 4624}, {3649, 4632}, {3669, 7035}, {3671, 4633}, {3674, 8707}, {3701, 4637}, {3718, 32714}, {3732, 8817}, {3882, 31643}, {3903, 7196}, {3911, 4555}, {3926, 36127}, {4017, 4601}, {4024, 7340}, {4025, 46102}, {4032, 4594}, {4077, 4567}, {4130, 24011}, {4163, 23586}, {4357, 6648}, {4391, 7045}, {4556, 34388}, {4576, 18097}, {4578, 23062}, {4585, 18815}, {4586, 7179}, {4589, 16609}, {4592, 40149}, {4597, 5219}, {4600, 7178}, {4607, 43037}, {4615, 40663}, {4619, 34387}, {4621, 7185}, {4654, 32042}, {5930, 44326}, {6168, 14727}, {6528, 40152}, {6604, 37206}, {6742, 17095}, {7012, 15413}, {7128, 35518}, {7153, 36863}, {7176, 27805}, {7249, 18047}, {8709, 43040}, {9312, 30610}, {10436, 32038}, {11608, 17933}, {13136, 22464}, {13138, 40702}, {14612, 39712}, {17950, 35154}, {18022, 32660}, {18816, 24029}, {20568, 23703}, {20911, 36098}, {21105, 31619}, {21272, 40420}, {23067, 44129}, {25716, 42343}, {26700, 33939}, {27818, 43290}, {27834, 39126}, {30806, 37139}, {31618, 35338}, {31625, 43924}, {32008, 35312}, {32028, 43948}, {32041, 40719}, {32675, 40075}, {32739, 41283}, {33298, 43190}, {34019, 42301}, {35171, 37787}, {35341, 42311}, {36086, 40704}, {36796, 41353}, {41206, 44150}, {42719, 43736}, {44717, 46107}
X(664) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 650}, {2, 522}, {3, 652}, {4, 3064}, {6, 663}, {7, 514}, {8, 3239}, {9, 3900}, {10, 3700}, {11, 42462}, {12, 4024}, {19, 18344}, {20, 14331}, {21, 1021}, {29, 17926}, {30, 14400}, {31, 3063}, {34, 6591}, {35, 9404}, {36, 654}, {37, 4041}, {40, 14298}, {41, 8641}, {42, 3709}, {44, 4895}, {45, 4814}, {48, 1946}, {55, 657}, {56, 649}, {57, 513}, {58, 7252}, {59, 101}, {63, 521}, {65, 661}, {69, 6332}, {72, 8611}, {73, 647}, {75, 4391}, {76, 35519}, {77, 905}, {81, 3737}, {85, 693}, {86, 4560}, {88, 23838}, {92, 44426}, {99, 333}, {100, 9}, {101, 55}, {102, 2432}, {105, 1024}, {107, 8748}, {108, 19}, {109, 6}, {110, 284}, {112, 2299}, {142, 6362}, {145, 4521}, {162, 1172}, {163, 2194}, {169, 11934}, {171, 3287}, {174, 6728}, {181, 4079}, {188, 6730}, {190, 8}, {192, 4147}, {194, 25128}, {200, 4130}, {210, 4171}, {220, 4105}, {222, 1459}, {223, 6129}, {225, 2501}, {226, 523}, {238, 4435}, {239, 3716}, {241, 2254}, {249, 4636}, {255, 36054}, {264, 46110}, {266, 6729}, {269, 3669}, {273, 17924}, {274, 18155}, {278, 7649}, {279, 3676}, {283, 23090}, {284, 21789}, {304, 35518}, {307, 525}, {312, 4397}, {320, 3904}, {321, 4086}, {329, 8058}, {331, 46107}, {332, 15411}, {333, 7253}, {344, 44448}, {346, 4163}, {347, 14837}, {348, 4025}, {349, 850}, {354, 21127}, {388, 6590}, {390, 14330}, {513, 2170}, {514, 11}, {519, 1639}, {521, 34591}, {522, 1146}, {523, 21044}, {524, 14432}, {527, 6366}, {536, 14430}, {553, 4977}, {579, 8676}, {603, 22383}, {604, 667}, {643, 2287}, {644, 200}, {645, 1043}, {646, 341}, {648, 29}, {649, 3271}, {650, 2310}, {651, 1}, {652, 3270}, {653, 4}, {655, 80}, {657, 3022}, {658, 7}, {660, 4876}, {661, 4516}, {662, 21}, {663, 14936}, {666, 14942}, {668, 312}, {670, 28660}, {672, 926}, {673, 885}, {677, 2338}, {692, 41}, {693, 4858}, {738, 43932}, {748, 4501}, {765, 644}, {799, 314}, {811, 31623}, {812, 4124}, {813, 7077}, {823, 1896}, {874, 3975}, {883, 3912}, {894, 3907}, {899, 4526}, {900, 4530}, {901, 2316}, {905, 7004}, {906, 212}, {908, 2804}, {919, 2195}, {927, 673}, {932, 2319}, {934, 57}, {940, 17418}, {1001, 45755}, {1014, 1019}, {1016, 3699}, {1018, 210}, {1019, 18191}, {1020, 65}, {1023, 3689}, {1025, 518}, {1026, 3693}, {1038, 2522}, {1042, 7180}, {1055, 6139}, {1071, 40628}, {1086, 21132}, {1088, 24002}, {1111, 40166}, {1125, 4976}, {1146, 23615}, {1156, 23893}, {1212, 6608}, {1214, 656}, {1215, 4140}, {1231, 14208}, {1252, 3939}, {1262, 109}, {1284, 21832}, {1305, 1751}, {1310, 2339}, {1317, 6544}, {1319, 1635}, {1323, 1638}, {1331, 219}, {1332, 78}, {1334, 4524}, {1357, 21143}, {1358, 6545}, {1365, 21131}, {1381, 2591}, {1382, 2590}, {1397, 1919}, {1400, 512}, {1401, 21123}, {1402, 798}, {1403, 20979}, {1404, 1960}, {1405, 4775}, {1407, 43924}, {1409, 810}, {1412, 3733}, {1414, 81}, {1415, 31}, {1416, 43929}, {1420, 4394}, {1423, 4083}, {1427, 4017}, {1428, 8632}, {1429, 659}, {1434, 7192}, {1435, 43923}, {1438, 884}, {1441, 1577}, {1442, 14838}, {1443, 3960}, {1445, 3309}, {1446, 4077}, {1447, 812}, {1457, 3310}, {1458, 665}, {1459, 7117}, {1460, 2484}, {1461, 56}, {1462, 1027}, {1464, 21828}, {1465, 1769}, {1469, 3250}, {1475, 2488}, {1492, 2344}, {1633, 2082}, {1697, 40137}, {1698, 4820}, {1708, 15313}, {1734, 38358}, {1743, 4162}, {1751, 23289}, {1783, 33}, {1790, 23189}, {1804, 4091}, {1813, 3}, {1814, 23696}, {1897, 281}, {1943, 8062}, {1950, 21761}, {1978, 3596}, {1981, 243}, {1983, 2361}, {1996, 30181}, {2002, 8760}, {2003, 2605}, {2078, 22108}, {2082, 17115}, {2099, 4893}, {2124, 17427}, {2149, 692}, {2171, 4705}, {2222, 2161}, {2254, 17435}, {2283, 672}, {2284, 2340}, {2285, 8678}, {2291, 23351}, {2293, 10581}, {2325, 4528}, {2329, 4477}, {2397, 6735}, {2398, 40869}, {2399, 15633}, {2406, 515}, {2701, 17963}, {2720, 909}, {3064, 42069}, {3160, 7658}, {3161, 4546}, {3212, 3835}, {3218, 3738}, {3219, 35057}, {3239, 4081}, {3257, 1320}, {3261, 34387}, {3309, 38375}, {3333, 14300}, {3338, 13401}, {3361, 4790}, {3452, 42337}, {3485, 45745}, {3570, 3685}, {3573, 3684}, {3616, 4765}, {3649, 4988}, {3661, 4522}, {3662, 3810}, {3663, 21120}, {3665, 16892}, {3666, 17420}, {3667, 4534}, {3668, 7178}, {3669, 244}, {3671, 4841}, {3674, 3004}, {3676, 1086}, {3679, 4944}, {3685, 4148}, {3686, 4990}, {3689, 14427}, {3699, 346}, {3718, 15416}, {3732, 497}, {3752, 6615}, {3782, 21119}, {3807, 3790}, {3875, 20317}, {3882, 960}, {3888, 3061}, {3900, 3119}, {3911, 900}, {3939, 220}, {3950, 44729}, {3952, 2321}, {3982, 28175}, {4017, 3125}, {4024, 4092}, {4025, 26932}, {4031, 28209}, {4032, 2533}, {4033, 3701}, {4040, 38347}, {4041, 36197}, {4069, 4515}, {4076, 6558}, {4077, 16732}, {4091, 1364}, {4103, 6057}, {4105, 35508}, {4114, 28213}, {4115, 4046}, {4130, 24010}, {4163, 23970}, {4296, 16612}, {4306, 43060}, {4308, 43061}, {4350, 43049}, {4357, 3910}, {4358, 4768}, {4359, 4985}, {4363, 4474}, {4369, 4459}, {4370, 4543}, {4380, 4965}, {4383, 42312}, {4391, 24026}, {4427, 3686}, {4436, 3691}, {4462, 4939}, {4499, 4051}, {4512, 4827}, {4521, 4953}, {4551, 37}, {4552, 10}, {4553, 33299}, {4554, 75}, {4555, 4997}, {4556, 60}, {4557, 1334}, {4558, 283}, {4559, 42}, {4561, 345}, {4562, 4518}, {4563, 332}, {4564, 100}, {4565, 58}, {4566, 226}, {4567, 643}, {4568, 3703}, {4569, 85}, {4570, 5546}, {4571, 3692}, {4572, 76}, {4573, 86}, {4574, 2318}, {4575, 2193}, {4578, 728}, {4579, 2329}, {4585, 4511}, {4587, 1260}, {4588, 2364}, {4589, 36800}, {4592, 1812}, {4595, 27538}, {4597, 30608}, {4598, 7155}, {4600, 645}, {4601, 7257}, {4602, 40072}, {4604, 2320}, {4605, 12}, {4606, 4866}, {4607, 36798}, {4610, 261}, {4612, 1098}, {4616, 1434}, {4617, 269}, {4619, 59}, {4620, 99}, {4624, 5936}, {4625, 274}, {4626, 279}, {4636, 7054}, {4637, 1014}, {4638, 1318}, {4654, 4802}, {4752, 3711}, {4756, 4007}, {4767, 4873}, {4781, 3707}, {4848, 14321}, {4858, 42455}, {4998, 190}, {5219, 4777}, {5221, 4813}, {5226, 28161}, {5228, 4724}, {5257, 4843}, {5298, 4984}, {5382, 31343}, {5435, 3667}, {5440, 14418}, {5540, 11193}, {5546, 2328}, {5750, 29278}, {5930, 6587}, {6049, 31182}, {6063, 3261}, {6135, 13456}, {6136, 13427}, {6163, 4919}, {6168, 42341}, {6180, 4449}, {6183, 39273}, {6331, 44130}, {6332, 2968}, {6335, 318}, {6357, 11125}, {6358, 4036}, {6366, 33573}, {6386, 28659}, {6510, 14414}, {6516, 63}, {6517, 394}, {6540, 4102}, {6544, 4542}, {6545, 7336}, {6558, 5423}, {6603, 14392}, {6604, 4468}, {6606, 32008}, {6610, 14413}, {6613, 40420}, {6614, 1407}, {6632, 4076}, {6648, 1220}, {6649, 894}, {6733, 259}, {6742, 7110}, {7012, 1783}, {7035, 646}, {7045, 651}, {7055, 30805}, {7078, 10397}, {7081, 4529}, {7113, 8648}, {7115, 8750}, {7125, 23224}, {7128, 108}, {7146, 1491}, {7153, 43931}, {7175, 4367}, {7176, 4369}, {7178, 3120}, {7179, 824}, {7180, 3122}, {7181, 4750}, {7182, 15413}, {7183, 4131}, {7185, 3776}, {7192, 17197}, {7196, 4374}, {7201, 4490}, {7203, 16726}, {7210, 16757}, {7217, 21110}, {7223, 4379}, {7233, 4444}, {7243, 4408}, {7251, 21122}, {7265, 6741}, {7339, 1461}, {7340, 4610}, {7649, 8735}, {8012, 6607}, {8058, 5514}, {8059, 1436}, {8269, 7131}, {8270, 2509}, {8545, 14077}, {8694, 34820}, {8701, 33635}, {8709, 36799}, {8750, 607}, {8804, 14308}, {9268, 5548}, {9312, 4885}, {9316, 20980}, {9358, 9355}, {9436, 918}, {9454, 8638}, {10001, 39351}, {10015, 35015}, {10030, 3766}, {10436, 23880}, {10481, 21104}, {10521, 23729}, {10566, 18101}, {10571, 6589}, {10578, 14282}, {11608, 18013}, {12848, 28292}, {13138, 282}, {13149, 273}, {13397, 39943}, {13444, 18888}, {13588, 21388}, {13589, 1731}, {14027, 14442}, {14543, 950}, {14544, 40942}, {14547, 33525}, {14594, 2345}, {14612, 5263}, {14733, 2291}, {14837, 38357}, {14942, 28132}, {15386, 32653}, {15413, 17880}, {15419, 17219}, {15439, 2259}, {16603, 4122}, {16609, 4010}, {16826, 4913}, {16885, 4959}, {16888, 3801}, {17074, 21173}, {17075, 20517}, {17076, 21178}, {17077, 8714}, {17078, 4453}, {17079, 21183}, {17080, 21189}, {17081, 3798}, {17082, 21191}, {17083, 21194}, {17084, 21196}, {17085, 21200}, {17086, 4142}, {17088, 21205}, {17089, 21204}, {17091, 21206}, {17092, 4905}, {17093, 31605}, {17094, 4466}, {17095, 4467}, {17096, 17205}, {17114, 23751}, {17136, 5745}, {17218, 16759}, {17439, 11124}, {17496, 34589}, {17752, 30584}, {17761, 42454}, {17780, 2325}, {17923, 44428}, {17933, 40882}, {17942, 5060}, {17950, 2785}, {17966, 5075}, {18026, 92}, {18047, 7081}, {18134, 20294}, {18315, 35196}, {18623, 21172}, {18626, 21175}, {18627, 21176}, {18830, 27424}, {19297, 42657}, {19804, 4811}, {20567, 40495}, {20616, 21727}, {21007, 38365}, {21105, 46101}, {21147, 6588}, {21173, 11998}, {21189, 38345}, {21272, 3452}, {21362, 3057}, {21454, 4778}, {21580, 20895}, {21773, 42649}, {21859, 756}, {22003, 21677}, {22341, 822}, {22464, 10015}, {23067, 71}, {23113, 22072}, {23187, 38344}, {23346, 1055}, {23353, 2202}, {23363, 23640}, {23586, 4626}, {23703, 44}, {23706, 14571}, {23845, 2347}, {23865, 38991}, {23890, 1155}, {23891, 4009}, {23971, 6614}, {23973, 43035}, {23978, 23104}, {23981, 2183}, {23984, 36127}, {23987, 8755}, {24002, 1111}, {24004, 4723}, {24011, 36838}, {24013, 4617}, {24027, 1415}, {24029, 517}, {24037, 4631}, {24041, 4612}, {24215, 20508}, {25268, 6736}, {25716, 31287}, {25722, 40465}, {26700, 2160}, {26942, 4064}, {27339, 28623}, {27805, 4451}, {27808, 30713}, {27834, 3680}, {27853, 4087}, {28292, 43960}, {28387, 4139}, {28850, 28143}, {29055, 893}, {29227, 36630}, {30239, 8602}, {30379, 2826}, {30545, 20906}, {30719, 3756}, {30720, 6555}, {30725, 1647}, {30730, 4082}, {31231, 4926}, {31286, 24840}, {31526, 21195}, {31605, 4904}, {31615, 765}, {32038, 31359}, {32042, 42030}, {32577, 17424}, {32636, 4979}, {32641, 2342}, {32652, 7118}, {32660, 184}, {32669, 34858}, {32674, 25}, {32675, 6187}, {32676, 2204}, {32693, 2258}, {32714, 34}, {32735, 1438}, {32739, 2175}, {32939, 20293}, {33296, 27527}, {33298, 25259}, {33946, 3705}, {33949, 45746}, {33951, 4514}, {33952, 3966}, {34056, 35348}, {34071, 2053}, {34085, 2481}, {34234, 43728}, {34393, 2399}, {34921, 19302}, {35049, 13486}, {35154, 17947}, {35157, 1121}, {35174, 18359}, {35307, 21807}, {35310, 21039}, {35312, 142}, {35326, 2293}, {35338, 1212}, {35341, 3059}, {35342, 3683}, {35519, 23978}, {36040, 32677}, {36048, 2982}, {36049, 2192}, {36054, 2638}, {36059, 48}, {36075, 2308}, {36082, 2164}, {36086, 294}, {36098, 2298}, {36099, 1039}, {36118, 278}, {36127, 393}, {36141, 34068}, {36146, 105}, {36538, 29362}, {36589, 23884}, {36797, 2322}, {36802, 6559}, {36838, 1088}, {36863, 4110}, {37136, 104}, {37137, 256}, {37138, 40779}, {37139, 1156}, {37141, 84}, {37143, 3254}, {37206, 6601}, {37212, 32635}, {37215, 30479}, {37771, 21201}, {37787, 3887}, {37789, 2827}, {37798, 21180}, {37800, 21185}, {38340, 79}, {38459, 43050}, {38828, 3445}, {39126, 4462}, {39293, 666}, {39294, 1309}, {39771, 35092}, {39780, 40627}, {40117, 7008}, {40131, 6182}, {40149, 24006}, {40152, 520}, {40166, 1090}, {40573, 14775}, {40576, 169}, {40577, 5540}, {40615, 23760}, {40617, 23764}, {40622, 23775}, {40663, 4120}, {40702, 17896}, {40719, 4762}, {40999, 7265}, {41003, 21124}, {41206, 37142}, {41246, 29051}, {41264, 9313}, {41350, 45902}, {41353, 241}, {41526, 8640}, {41572, 28473}, {41777, 3777}, {41803, 21198}, {41804, 4707}, {41808, 21192}, {42462, 5532}, {42717, 44694}, {42720, 3717}, {43035, 676}, {43037, 4728}, {43038, 14475}, {43040, 3837}, {43041, 27918}, {43048, 24457}, {43051, 3123}, {43069, 43073}, {43192, 7707}, {43290, 3161}, {43760, 35355}, {43924, 1015}, {44326, 5931}, {44327, 280}, {44717, 1331}, {44724, 30720}, {44765, 10570}, {45204, 14284}, {45875, 15997}, {45876, 2090}, {46102, 1897}, {46110, 21666}, {46135, 18031}, {46148, 3688}, {46152, 17442}, {46153, 39}, {46177, 16588}
X(664) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1111, 24203}, {1, 9312, 85}, {2, 1146, 31640}, {2, 39351, 1146}, {7, 17089, 1358}, {8, 348, 33298}, {8, 3160, 348}, {65, 7176, 1434}, {78, 34059, 40702}, {99, 4573, 1414}, {100, 934, 6516}, {100, 35312, 658}, {145, 279, 6604}, {150, 38941, 1565}, {220, 3177, 32024}, {269, 3875, 39126}, {651, 4552, 190}, {668, 4561, 3699}, {934, 4566, 658}, {1121, 31640, 1146}, {1146, 17044, 2}, {1146, 35110, 17044}, {1146, 39351, 1121}, {1214, 1943, 333}, {1317, 43038, 7}, {1319, 43037, 1447}, {1323, 9436, 17078}, {1358, 43038, 17089}, {1441, 1442, 86}, {1897, 36118, 18026}, {2082, 6167, 7131}, {2099, 7223, 7}, {2170, 9317, 673}, {3160, 25718, 8}, {3674, 10106, 7247}, {4551, 14594, 3699}, {4566, 17136, 6516}, {9259, 21138, 26273}, {9312, 25716, 1}, {17044, 39351, 31640}, {17136, 21272, 100}, {17136, 35312, 934}, {17439, 21139, 9318}, {17880, 18689, 75}, {18047, 33946, 190}, {20057, 32086, 5543}, {21272, 35312, 4566}, {30725, 45273, 651}, {35110, 39357, 1121}, {41801, 41803, 903}


X(665) = CROSSDIFFERENCE OF X(2) AND X(11)

Trilinears    a[(a - b)2(a + b - c) - (c - a)2(c + a - b)]
Barycentrics    a^2*(b - c)*(a*b - b^2 + a*c - c^2) : :
X(665) = 3 X[1635] - X[21832], 3 X[1638] - X[20507], 3 X[1638] - 2 X[20520], 2 X[3766] - 3 X[45338], X[3768] - 3 X[14407], 5 X[4699] - 3 X[21433], 5 X[4699] - 9 X[27344], 7 X[4751] - 3 X[21606], 3 X[14407] + X[21143], X[21433] - 3 X[27344]

X(665) lies on these lines: {1, 4435}, {2, 3766}, {6, 22108}, {37, 900}, {39, 22092}, {101, 109}, {106, 2291}, {187, 237}, {241, 514}, {244, 866}, {292, 659}, {513, 3709}, {522, 21348}, {573, 2815}, {657, 20980}, {661, 2530}, {672, 14411}, {743, 761}, {784, 6590}, {798, 6371}, {802, 21191}, {834, 4832}, {911, 1438}, {918, 16728}, {926, 42079}, {1155, 9321}, {1213, 2511}, {1333, 42741}, {1362, 34905}, {1459, 3063}, {1639, 21894}, {1642, 2254}, {1734, 22229}, {1841, 39534}, {2092, 2642}, {2178, 39200}, {2195, 43929}, {2277, 28284}, {2345, 26078}, {2440, 32658}, {2483, 3733}, {2498, 3941}, {2516, 29226}, {2605, 21007}, {2609, 15586}, {3261, 17066}, {3287, 21173}, {3290, 26275}, {3512, 21391}, {3572, 6373}, {3700, 8714}, {3768, 6085}, {3798, 25098}, {3803, 4790}, {3835, 25084}, {3837, 40549}, {4017, 21127}, {4083, 4394}, {4139, 17458}, {4151, 4976}, {4374, 21225}, {4391, 22222}, {4449, 45755}, {4486, 24287}, {4524, 23655}, {4699, 21433}, {4751, 21606}, {4785, 45658}, {4985, 21960}, {6165, 21830}, {6363, 20979}, {9508, 30665}, {17263, 28779}, {17302, 27012}, {20678, 21003}, {21261, 25381}, {21742, 21758}, {21796, 24118}, {23744, 23798}, {24462, 41531}, {25594, 27293}, {26242, 44433}, {29350, 30234}, {30572, 42462}, {30764, 30836}, {31207, 31208}, {41163, 41190}, {41164, 41189}, {41165, 41186}, {41166, 41191}

X(665) = midpoint of X(i) and X(j) for these {i,j}: {649, 3250}, {659, 876}, {798, 21123}, {3768, 21143}, {4374, 21225}
X(665) = reflection of X(i) in X(j) for these {i,j}: {3261, 17066}, {3709, 6586}, {3837, 40549}, {4526, 37}, {20507, 20520}, {23744, 23798}, {45338, 2}
X(665) = isogonal conjugate of X(666)
X(665) = isotomic conjugate of X(36803)
X(665) = complement of X(3766)
X(665) = Parry-circle inverse of X(5098)
X(665) = tripolar centroid of X(1002)
X(665) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 38989}, {32, 35119}, {100, 20542}, {101, 20333}, {291, 21252}, {292, 116}, {660, 2887}, {692, 17793}, {813, 141}, {1252, 27854}, {1911, 11}, {1922, 1086}, {4557, 45162}, {4562, 626}, {4583, 21235}, {4584, 21240}, {5378, 21260}, {7077, 124}, {14598, 1015}, {17938, 6682}, {18265, 1146}, {18267, 39786}, {18268, 17761}, {18897, 6377}, {20964, 2679}, {23990, 27929}, {32739, 17755}, {34067, 10}, {36081, 21238}
X(665) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 38989}, {6, 35505}, {7, 15615}, {56, 38363}, {100, 20455}, {101, 20662}, {105, 3271}, {241, 3675}, {292, 1015}, {651, 1362}, {659, 6373}, {660, 40730}, {876, 512}, {884, 20980}, {911, 7117}, {919, 6}, {927, 21746}, {1026, 20683}, {2254, 926}, {2283, 2223}, {2284, 672}, {2424, 663}, {5089, 17435}, {27918, 6165}, {34018, 4014}, {34067, 39}, {35185, 2175}, {37137, 34253}, {40519, 20860}, {42720, 518}, {43736, 3022}, {43760, 1357}
X(665) = X(i)-cross conjugate of X(j) for these (i,j): {15615, 7}, {35505, 6}
X(665) = crosspoint of X(i) and X(j) for these (i,j): {2, 660}, {6, 919}, {100, 2991}, {105, 651}, {110, 37128}, {241, 2283}, {518, 42720}, {649, 3572}, {672, 2284}, {694, 37137}, {1026, 18206}
X(665) = crosssum of X(i) and X(j) for these (i,j): {2, 918}, {6, 659}, {100, 2284}, {105, 43929}, {190, 3570}, {294, 885}, {385, 3287}, {513, 3290}, {514, 3008}, {518, 650}, {522, 40869}, {523, 2238}, {812, 27942}, {926, 16588}, {1027, 18785}, {14942, 28132}
X(665) = trilinear pole of line {15615, 35505}
X(665) = crossdifference of every pair of points on line {2, 11}
X(665) = tripolar centroid of X(1002)
X(665) = polar conjugate of isogonal conjugate of X(23225)
X(665) = perspector of hyperbola {A,B,C,X(6),X(7)}}
X(665) = X(i)-cross conjugate of X(j) for these (i,j): {15615, 7}, {35505, 6}
X(i)-isoconjugate of X(j) for these (i,j): {1, 666}, {2, 36086}, {8, 36146}, {9, 927}, {31, 36803}, {41, 46135}, {55, 34085}, {57, 36802}, {75, 919}, {76, 32666}, {83, 35333}, {99, 18785}, {100, 673}, {101, 2481}, {105, 190}, {109, 36796}, {294, 664}, {312, 32735}, {344, 36041}, {514, 5377}, {646, 1416}, {650, 39293}, {651, 14942}, {658, 28071}, {660, 6654}, {662, 13576}, {668, 1438}, {692, 18031}, {885, 4564}, {898, 36816}, {934, 6559}, {1016, 1027}, {1024, 4998}, {1026, 6185}, {1292, 31638}, {1332, 36124}, {1462, 3699}, {1783, 31637}, {1814, 1897}, {2195, 4554}, {2398, 9503}, {3008, 39272}, {3112, 46163}, {3939, 34018}, {4441, 36138}, {4561, 8751}, {4614, 14625}, {6335, 36057}, {6632, 43921}, {7035, 43929}, {7045, 28132}, {9310, 14727}, {14947, 34906}, {20927, 35185}, {21615, 32724}, {23696, 46102}, {28420, 36111}
X(665) = barycentric product X(i)*X(j) for these {i,j}: {1, 2254}, {6, 918}, {7, 926}, {11, 2283}, {42, 23829}, {55, 43042}, {58, 4088}, {81, 24290}, {100, 3675}, {104, 42758}, {105, 3126}, {241, 650}, {244, 1026}, {264, 23225}, {512, 30941}, {513, 518}, {514, 672}, {521, 1876}, {522, 1458}, {523, 3286}, {647, 15149}, {649, 3912}, {651, 17435}, {652, 5236}, {659, 22116}, {660, 38989}, {661, 18206}, {663, 9436}, {666, 35505}, {667, 3263}, {693, 2223}, {798, 18157}, {812, 3252}, {876, 8299}, {883, 3271}, {885, 1362}, {900, 34230}, {905, 5089}, {919, 35094}, {1015, 42720}, {1019, 3930}, {1022, 14439}, {1025, 2170}, {1027, 4712}, {1086, 2284}, {1459, 1861}, {1769, 36819}, {1818, 7649}, {2310, 41353}, {2340, 3676}, {2356, 4025}, {2428, 4904}, {3063, 40704}, {3261, 9454}, {3445, 4925}, {3572, 17755}, {3669, 3693}, {3717, 43924}, {3733, 3932}, {3766, 40730}, {3900, 34855}, {4238, 18210}, {4437, 43929}, {4778, 14626}, {6063, 8638}, {6591, 25083}, {7192, 20683}, {7199, 39258}, {8632, 40217}, {9309, 42341}, {9318, 34905}, {9455, 40495}, {15615, 46135}, {17094, 37908}, {17924, 20752}, {22383, 46108}, {23770, 34159}, {23773, 34071}, {32641, 42770}, {35293, 35348}
X(665) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 36803}, {6, 666}, {7, 46135}, {31, 36086}, {32, 919}, {55, 36802}, {56, 927}, {57, 34085}, {109, 39293}, {241, 4554}, {512, 13576}, {513, 2481}, {514, 18031}, {518, 668}, {560, 32666}, {604, 36146}, {649, 673}, {650, 36796}, {657, 6559}, {663, 14942}, {667, 105}, {672, 190}, {692, 5377}, {798, 18785}, {918, 76}, {926, 8}, {1026, 7035}, {1357, 43930}, {1362, 883}, {1397, 32735}, {1458, 664}, {1459, 31637}, {1642, 42722}, {1818, 4561}, {1876, 18026}, {1919, 1438}, {1964, 35333}, {1977, 43929}, {2223, 100}, {2254, 75}, {2283, 4998}, {2284, 1016}, {2340, 3699}, {2356, 1897}, {3051, 46163}, {3063, 294}, {3126, 3263}, {3248, 1027}, {3252, 4562}, {3263, 6386}, {3271, 885}, {3286, 99}, {3669, 34018}, {3675, 693}, {3693, 646}, {3768, 36816}, {3912, 1978}, {3930, 4033}, {3932, 27808}, {4088, 313}, {4832, 14625}, {5089, 6335}, {6184, 42720}, {8027, 43921}, {8299, 874}, {8632, 6654}, {8638, 55}, {8641, 28071}, {9309, 14727}, {9436, 4572}, {9454, 101}, {9455, 692}, {9502, 42719}, {14439, 24004}, {14936, 28132}, {15149, 6331}, {15615, 926}, {17435, 4391}, {17755, 27853}, {18157, 4602}, {18206, 799}, {20683, 3952}, {20752, 1332}, {20958, 35313}, {22116, 4583}, {22383, 1814}, {23225, 3}, {23829, 310}, {24290, 321}, {30941, 670}, {34159, 35574}, {34230, 4555}, {34855, 4569}, {35505, 918}, {37908, 36797}, {38989, 3766}, {39258, 1018}, {39686, 2284}, {40730, 660}, {42079, 1026}, {42720, 31625}, {42758, 3262}, {42771, 2804}, {43042, 6063}, {43929, 6185}
X(665) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {649, 1960, 8658}, {649, 5029, 8632}, {649, 8632, 8659}, {650, 43060, 7180}, {657, 43924, 20980}, {1635, 14413, 1643}, {1635, 21828, 3310}, {1638, 20507, 20520}, {1960, 8659, 8632}, {5029, 8632, 1960}, {5638, 5639, 5098}, {8632, 8659, 8658}, {14407, 21143, 3768}, {21173, 21390, 3287}, {43047, 43050, 30725}


X(666) = TRILINEAR POLE OF LINE X(2)X(11)

Trilinears    bc/[(a - b)2(a + b - c) - (c - a)2(c + a - b)]
Barycentrics    (a - b)*(a - c)*(a^2 + b^2 - a*c - b*c)*(a^2 - a*b - b*c + c^2) : :
X(666) = 3 X[2] - 4 X[40540], 3 X[18821] - 4 X[35094], X[18821] - 4 X[35113], 3 X[18821] - 2 X[39353], X[18821] + 2 X[39363], 3 X[18821] - 8 X[40540], 5 X[29590] - X[40868], X[35094] - 3 X[35113], 2 X[35094] + 3 X[39363], 6 X[35113] - X[39353], 2 X[35113] + X[39363], 3 X[35113] - 2 X[40540], X[39353] + 3 X[39363], X[39353] - 4 X[40540], 3 X[39363] + 4 X[40540]

X(666) lies on the Steiner circumellipse and these lines: {1, 35167}, {2, 18821}, {6, 35119}, {8, 35158}, {10, 35163}, {37, 35152}, {99, 919}, {100, 31150}, {101, 514}, {105, 898}, {144, 30228}, {190, 522}, {220, 35120}, {238, 33674}, {239, 294}, {290, 15628}, {519, 1121}, {527, 673}, {644, 668}, {645, 670}, {648, 5379}, {650, 4998}, {651, 1275}, {659, 34067}, {671, 5080}, {693, 5375}, {742, 43099}, {883, 41075}, {885, 2398}, {889, 5381}, {894, 40754}, {993, 31169}, {1026, 1027}, {1252, 17494}, {1438, 3226}, {1462, 40400}, {1494, 15627}, {1757, 33676}, {1783, 17924}, {1814, 1944}, {2311, 3509}, {2338, 3912}, {2341, 14616}, {2397, 2402}, {2401, 32641}, {2414, 32644}, {3008, 9436}, {3241, 35168}, {3758, 35175}, {3872, 35957}, {4416, 35150}, {4453, 37143}, {4552, 6606}, {4555, 5376}, {4573, 35326}, {4581, 6648}, {4582, 6635}, {4585, 35171}, {4586, 5384}, {4597, 5385}, {4762, 40865}, {5081, 36124}, {5382, 6631}, {5383, 18830}, {5386, 43097}, {5387, 24004}, {5388, 46132}, {5389, 43098}, {5452, 6063}, {6542, 35141}, {6654, 16834}, {7760, 35172}, {8751, 46108}, {14827, 21218}, {15629, 34393}, {16588, 40419}, {17350, 34361}, {18031, 43093}, {26777, 43986}, {28132, 35157}, {29574, 35153}, {29590, 40868}, {31633, 40166}, {39272, 42720}, {40863, 46137}

X(666) = midpoint of X(i) and X(j) for these {i,j}: {2, 39363}, {239, 10025}
X(666) = reflection of X(i) in X(j) for these {i,j}: {2, 35113}, {927, 34906}, {3912, 40869}, {9436, 3008}, {18821, 2}, {35094, 40540}, {39353, 35094}
X(666) = isogonal conjugate of X(665)
X(666) = isotomic conjugate of X(918)
X(666) = complement of X(39353)
X(666) = anticomplement of X(35094)
X(666) = X(92)-isoconjugate of X(23225)
X(666) = Steiner-circumellipse-X(1)-antipode of X(35167)
X(666) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {5377, 20552}, {32666, 39353}
X(666) = X(i)-Ceva conjugate of X(j) for these (i,j): {34085, 927}, {39293, 14942}
X(666) = X(i)-cross conjugate of X(j) for these (i,j): {239, 1016}, {294, 5377}, {518, 4998}, {644, 39272}, {660, 8709}, {883, 30610}, {885, 2481}, {918, 2}, {926, 40419}, {2284, 100}, {2348, 4076}, {2398, 664}, {2402, 34018}, {3573, 99}, {3766, 83}, {4435, 1}, {10025, 1275}, {14942, 39293}, {28132, 14942}, {30807, 46102}, {32922, 7035}, {36086, 927}, {36236, 4555}, {43929, 105}, {46163, 919}
X(666) = cevapoint of X(i) and X(j) for these (i,j): {2, 918}, {6, 659}, {100, 2284}, {105, 43929}, {190, 3570}, {294, 885}, {385, 3287}, {513, 3290}, {514, 3008}, {518, 650}, {522, 40869}, {523, 2238}, {812, 27942}, {926, 16588}, {1027, 18785}, {14942, 28132}
X(666) = trilinear pole of line {2, 11}
X(666) = crossdifference of every pair of points on line {15615, 35505}
X(666) = X(i)-isoconjugate of X(j) for these (i,j): {1, 665}, {6, 2254}, {31, 918}, {41, 43042}, {57, 926}, {58, 24290}, {85, 8638}, {92, 23225}, {101, 3675}, {109, 17435}, {213, 23829}, {241, 663}, {244, 2284}, {512, 18206}, {513, 672}, {514, 2223}, {518, 649}, {650, 1458}, {652, 1876}, {657, 34855}, {659, 3252}, {661, 3286}, {667, 3912}, {669, 18157}, {693, 9454}, {798, 30941}, {810, 15149}, {812, 40730}, {813, 38989}, {875, 17755}, {905, 2356}, {909, 42758}, {1015, 1026}, {1019, 20683}, {1024, 1362}, {1025, 3271}, {1027, 6184}, {1333, 4088}, {1438, 3126}, {1459, 5089}, {1635, 34230}, {1818, 6591}, {1861, 22383}, {1919, 3263}, {1946, 5236}, {2170, 2283}, {2340, 3669}, {2424, 9502}, {3063, 9436}, {3248, 42720}, {3261, 9455}, {3310, 36819}, {3572, 8299}, {3693, 43924}, {3733, 3930}, {4712, 43929}, {4790, 14626}, {4925, 38266}, {5091, 34905}, {7192, 39258}, {7649, 20752}, {8632, 22116}, {9315, 42341}, {14439, 23345}, {14936, 41353}, {15615, 34085}, {20662, 35355}, {32666, 35094}, {35505, 36086}, {37136, 42771}
X(666) = barycentric product X(i)*X(j) for these {i,j}: {6, 36803}, {7, 36802}, {8, 927}, {9, 34085}, {55, 46135}, {75, 36086}, {76, 919}, {99, 13576}, {100, 2481}, {101, 18031}, {105, 668}, {190, 673}, {294, 4554}, {308, 46163}, {312, 36146}, {390, 41075}, {522, 39293}, {561, 32666}, {644, 34018}, {646, 1462}, {651, 36796}, {658, 6559}, {664, 14942}, {693, 5377}, {799, 18785}, {885, 4998}, {1027, 7035}, {1275, 28132}, {1376, 14727}, {1438, 1978}, {1814, 6335}, {1897, 31637}, {2195, 4572}, {3112, 35333}, {3596, 32735}, {4076, 43930}, {4561, 36124}, {4562, 6654}, {4569, 28071}, {4607, 36816}, {4633, 14625}, {5388, 29956}, {6185, 42720}, {9503, 42719}, {14267, 35574}, {21615, 36138}, {31625, 43929}, {31638, 37206}
X(666) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2254}, {2, 918}, {6, 665}, {7, 43042}, {10, 4088}, {37, 24290}, {55, 926}, {59, 2283}, {86, 23829}, {99, 30941}, {100, 518}, {101, 672}, {105, 513}, {108, 1876}, {109, 1458}, {110, 3286}, {145, 4925}, {184, 23225}, {190, 3912}, {294, 650}, {513, 3675}, {517, 42758}, {518, 3126}, {644, 3693}, {648, 15149}, {650, 17435}, {651, 241}, {653, 5236}, {659, 38989}, {660, 22116}, {662, 18206}, {664, 9436}, {665, 35505}, {668, 3263}, {673, 514}, {692, 2223}, {765, 1026}, {799, 18157}, {813, 3252}, {884, 3271}, {885, 11}, {901, 34230}, {906, 20752}, {918, 35094}, {919, 6}, {927, 7}, {934, 34855}, {1016, 42720}, {1018, 3930}, {1023, 14439}, {1024, 2170}, {1026, 4712}, {1027, 244}, {1252, 2284}, {1331, 1818}, {1332, 25083}, {1376, 42341}, {1416, 43924}, {1438, 649}, {1462, 3669}, {1738, 20504}, {1783, 5089}, {1814, 905}, {1897, 1861}, {2175, 8638}, {2195, 663}, {2283, 1362}, {2284, 6184}, {2402, 4904}, {2481, 693}, {3570, 17755}, {3573, 8299}, {3699, 3717}, {3835, 23773}, {3939, 2340}, {3952, 3932}, {3971, 21959}, {4427, 4966}, {4554, 40704}, {4557, 20683}, {4562, 40217}, {4564, 1025}, {4579, 4447}, {4998, 883}, {5377, 100}, {5379, 4238}, {6335, 46108}, {6559, 3239}, {6654, 812}, {7045, 41353}, {8638, 15615}, {8694, 14626}, {8750, 2356}, {8751, 6591}, {9319, 34905}, {10015, 42770}, {10099, 18210}, {13576, 523}, {14267, 23770}, {14625, 4841}, {14727, 32023}, {14942, 522}, {18031, 3261}, {18785, 661}, {21450, 14347}, {23696, 7004}, {28071, 3900}, {28132, 1146}, {31637, 4025}, {31638, 4468}, {32658, 22383}, {32666, 31}, {32735, 56}, {32739, 9454}, {34018, 24002}, {34067, 40730}, {34085, 85}, {34906, 9318}, {35185, 3433}, {35313, 3035}, {35333, 38}, {36037, 36819}, {36041, 2191}, {36057, 1459}, {36086, 1}, {36124, 7649}, {36138, 2279}, {36146, 57}, {36796, 4391}, {36802, 8}, {36803, 76}, {36816, 4728}, {39272, 1280}, {39293, 664}, {40724, 4458}, {41934, 43929}, {42720, 4437}, {43042, 3323}, {43290, 4899}, {43921, 764}, {43929, 1015}, {43930, 1358}, {46135, 6063}, {46149, 2530}, {46163, 39}
X(666) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 39353, 35094}, {885, 35313, 5377}, {35094, 35113, 40540}, {35094, 39353, 18821}, {35094, 40540, 2}, {35113, 39363, 18821}


X(667) = CROSSDIFFERENCE OF X(2) AND X(37)

Trilinears    a2(b - c) : :
Barycentrics    a3(b - c) : :
X(667) = X[1] + 2 X[4782], 6 X[2] - 5 X[31251], 3 X[2] - 4 X[31288], 3 X[2] + X[31291], 3 X[351] - 2 X[42653], 3 X[351] - X[42661], X[649] + 2 X[1960], 2 X[649] + X[4775], X[649] + 3 X[8643], X[649] + 5 X[8656], 2 X[659] + X[4378], 2 X[663] + X[4834], X[663] - 3 X[8643], X[663] - 5 X[8656], 2 X[905] - 3 X[14419], X[905] - 3 X[30234], 4 X[1125] - X[24719], 3 X[1635] - X[4041], 4 X[1635] - X[4825], 4 X[1960] - X[4775], 4 X[1960] + X[4834], 2 X[1960] - 3 X[8643], 2 X[1960] - 5 X[8656], X[2530] + 2 X[3803], X[2530] - 3 X[14419], X[2530] - 6 X[30234], 3 X[3251] - 2 X[4162], 3 X[3733] + X[4491], X[3801] - 3 X[4809], 2 X[3803] + 3 X[14419], X[3803] + 3 X[30234], 4 X[4041] - 3 X[4825], 3 X[4057] - X[4491], X[4367] + 2 X[4401], X[4378] + 4 X[4401], 4 X[4394] - X[4730], X[4705] - 4 X[6050], X[4775] - 6 X[8643], X[4775] - 10 X[8656], X[4784] + 2 X[4794], 3 X[4809] - 2 X[20517], X[4834] + 6 X[8643], X[4834] + 10 X[8656], X[4879] - 3 X[25569], 3 X[8643] - 5 X[8656], 4 X[21260] - 3 X[31149], 4 X[21260] - 5 X[31251], 2 X[21260] + X[31291], 2 X[21301] - 3 X[31149], 2 X[21301] - 5 X[31251], X[21301] - 4 X[31288], X[21302] - 5 X[27013], 3 X[31149] - 5 X[31251], 3 X[31149] - 8 X[31288], 3 X[31149] + 2 X[31291], 5 X[31251] - 8 X[31288], 5 X[31251] + 2 X[31291], 4 X[31288] + X[31291]

X(667) = radical center of the circumcircle, Brocard circle, and the circle with (diameter = segment X(1)X(3)) (Wilson Stothers, 3/31/2003)

X(667) lies on these lines: {1, 4063}, {2, 21260}, {3, 1083}, {6, 9010}, {10, 21724}, {21, 29150}, {23, 9980}, {25, 18344}, {28, 17924}, {31, 8027}, {36, 238}, {55, 3251}, {56, 764}, {75, 21440}, {100, 898}, {101, 813}, {104, 2726}, {105, 14665}, {106, 2382}, {110, 2703}, {111, 35107}, {171, 38238}, {187, 237}, {197, 11124}, {213, 875}, {274, 23807}, {514, 659}, {522, 8045}, {644, 8671}, {650, 4705}, {656, 832}, {668, 932}, {676, 29162}, {692, 1110}, {693, 18108}, {750, 14474}, {784, 4560}, {788, 798}, {814, 1577}, {830, 1491}, {834, 2605}, {838, 1980}, {876, 37609}, {885, 37254}, {891, 4449}, {900, 4990}, {956, 30583}, {958, 20317}, {993, 4448}, {999, 14421}, {1001, 6008}, {1022, 37587}, {1125, 24719}, {1262, 32735}, {1379, 11651}, {1380, 11652}, {1415, 32666}, {1459, 6371}, {1635, 4041}, {1734, 9508}, {1979, 9265}, {2163, 23345}, {2217, 42455}, {2254, 6004}, {2300, 18002}, {2483, 6586}, {2484, 3709}, {2533, 29066}, {2787, 4391}, {2826, 44811}, {2832, 23765}, {2975, 4462}, {3271, 22096}, {3398, 39519}, {3508, 21390}, {3667, 39210}, {3700, 29232}, {3716, 6002}, {3762, 29324}, {3766, 26277}, {3777, 3960}, {3835, 25537}, {3900, 4394}, {4010, 29013}, {4086, 6133}, {4122, 29062}, {4155, 4501}, {4160, 4490}, {4170, 29328}, {4223, 40551}, {4369, 29051}, {4474, 29268}, {4477, 43061}, {4557, 40522}, {4707, 29082}, {4724, 6372}, {4729, 4895}, {4761, 29366}, {4784, 4794}, {4785, 45316}, {4790, 16874}, {4791, 29344}, {4800, 29178}, {4810, 29270}, {4822, 4979}, {4823, 29033}, {4879, 25569}, {4978, 29362}, {5251, 45666}, {5253, 19947}, {6003, 39225}, {6085, 22379}, {6363, 7250}, {6373, 20459}, {7117, 23646}, {7178, 29240}, {7255, 14296}, {7265, 29078}, {7497, 39536}, {7662, 23882}, {8034, 40956}, {8652, 29341}, {9029, 22108}, {9048, 23087}, {9218, 17940}, {9269, 37602}, {10006, 33849}, {11171, 39502}, {11193, 20988}, {11247, 20831}, {14431, 28475}, {14434, 17125}, {16158, 29298}, {17072, 19522}, {17126, 43928}, {17166, 17494}, {17197, 24234}, {17989, 29058}, {18102, 18107}, {18173, 18197}, {20456, 20962}, {20982, 22373}, {21007, 21788}, {21146, 29186}, {21302, 27013}, {21348, 21389}, {21385, 29226}, {22224, 22381}, {22386, 39025}, {23394, 29198}, {23656, 23657}, {23770, 34958}, {24286, 37575}, {24698, 25356}, {25259, 29090}, {26148, 27345}, {26275, 29126}, {27673, 27677}, {28401, 30094}, {28468, 30580}, {28473, 44805}, {34467, 38389}, {39476, 42325}, {40519, 40521}

X(667) = midpoint of X(i) and X(j) for these {i,j}: {1, 4063}, {649, 663}, {659, 4367}, {669, 8639}, {905, 3803}, {1019, 4040}, {3733, 4057}, {4449, 4498}, {4729, 4895}, {4775, 4834}, {4822, 4979}, {17166, 17494}, {21301, 31291}
X(667) = reflection of X(i) in X(j) for these {i,j}: {3, 39227}, {650, 6050}, {659, 4401}, {663, 1960}, {764, 3669}, {1491, 14838}, {1577, 4874}, {1734, 9508}, {2530, 905}, {3230, 42655}, {3777, 3960}, {3801, 20517}, {4063, 4782}, {4086, 6133}, {4142, 13246}, {4378, 4367}, {4705, 650}, {4775, 663}, {4834, 649}, {8640, 8637}, {14419, 30234}, {17072, 31286}, {21260, 31288}, {21301, 21260}, {23224, 34948}, {23770, 34958}, {24698, 25356}, {31149, 2}, {42661, 42653}
X(667) = reflection of X(667) in the Lemoine axis
X(667) = isogonal conjugate of X(668)
X(667) = isotomic conjugate of X(6386)
X(667) = complement of X(21301)
X(667) = anticomplement of X(21260)
X(667) = circumcircle-inverse of X(1083)
X(667) = Parry-circle-inverse of X(5040)
X(667) = Parry-isodynamic-circle-inverse of X(5163)
X(667) = polar conjugate of the isotomic conjugate of X(22383)
X(667) = tangential-isogonal conjugate of X(23402)
X(667) = X(i)-Ceva conjugate of X(j) for these (i,j): {6, 1977}, {25, 3271}, {28, 3125}, {31, 3248}, {56, 1015}, {99, 20963}, {100, 6}, {101, 213}, {104, 2087}, {109, 20228}, {110, 2300}, {190, 21757}, {513, 22383}, {649, 3063}, {662, 1197}, {692, 31}, {789, 23660}, {813, 21760}, {898, 3230}, {932, 1}, {934, 16502}, {1333, 3121}, {1415, 32}, {1436, 14936}, {1576, 40956}, {1919, 8640}, {2162, 21762}, {2176, 23573}, {2217, 2170}, {2248, 21755}, {2353, 23646}, {3435, 7117}, {3437, 20982}, {3444, 3124}, {3733, 649}, {3903, 23525}, {4557, 2308}, {4598, 21759}, {4623, 81}, {6186, 3122}, {7087, 20974}, {7152, 39687}, {7169, 3270}, {7255, 514}, {8693, 40728}, {8709, 20669}, {16695, 23572}, {18108, 513}, {23345, 21758}, {23349, 890}, {23355, 8632}, {26703, 17435}, {29227, 2176}, {29351, 16971}, {30664, 16782}, {32665, 2251}, {34067, 2223}, {34080, 41}, {34183, 35505}, {34443, 8054}, {34444, 38346}, {34445, 23470}, {34594, 1100}, {36614, 23560}, {38470, 16470}, {40519, 42}, {40770, 9427}, {43350, 1449}
X(667) = X(i)-cross conjugate of X(j) for these (i,j): {6, 9299}, {669, 1919}, {798, 649}, {890, 23349}, {1977, 6}, {3121, 1333}, {3122, 1402}, {3248, 31}, {8027, 3248}, {20979, 8640}, {21762, 2162}, {21835, 893}, {22096, 1397}, {23560, 36614}, {38367, 8632}, {38986, 1}
X(667) = cevapoint of X(i) and X(j) for these (i,j): {6, 1979}, {649, 20979}, {669, 798}, {3248, 8027}
X(667) = crosspoint of X(i) and X(j) for these (i,j): {6, 100}, {31, 692}, {56, 1415}, {58, 101}, {81, 4623}, {86, 4598}, {99, 40408}, {109, 3451}, {110, 1169}, {513, 6591}, {649, 43924}, {919, 15382}, {932, 7121}, {16945, 34080}, {32735, 41934}
X(667) = crosssum of X(i) and X(j) for these (i,j): {1, 4063}, {2, 513}, {6, 21005}, {8, 4391}, {10, 514}, {42, 20979}, {69, 35518}, {75, 693}, {100, 1332}, {120, 918}, {190, 3699}, {192, 25142}, {210, 4140}, {312, 4397}, {313, 35519}, {321, 4036}, {341, 15416}, {350, 27855}, {512, 16589}, {522, 3452}, {523, 1211}, {525, 21530}, {536, 14434}, {650, 3056}, {661, 3728}, {812, 17793}, {850, 1234}, {891, 13466}, {900, 16594}, {1016, 11607}, {1233, 3261}, {1577, 4647}, {1654, 24381}, {1909, 2533}, {2397, 15632}, {3762, 4738}, {3789, 4762}, {3900, 6554}, {3952, 4033}, {3954, 4705}, {4024, 21728}, {4083, 6376}, {4462, 44720}, {4651, 20954}, {4791, 4793}, {4824, 27495}, {6084, 40609}, {7192, 16709}, {7199, 16748}, {8027, 31645}, {20983, 30473}, {24533, 28369}, {28195, 28651}, {29226, 40598}
X(667) = trilinear pole of line {890, 1977}
X(667) = crossdifference of every pair of points on line {2, 37}
X(667) = intersection of tangents to circumcircle at intersections with line X(1)X(6)
X(667) = bicentric difference of PU(i) for i in (9, 26)
X(667) = PU(9)-harmonic conjugate of X(213)
X(667) = trilinear product of PU(25)
X(667) = PU(26)-harmonic conjugate of X(3230)
X(667) = vertex conjugate of PU(26)
X(667) = trilinear pole of PU(42)
X(667) = pole, wrt circumcircle, of line X(1)X(6)
X(667) = inverse-in-Parry-circle of X(5040)
X(667) = homothetic center of 4th Euler triangle and extraversion triangle of X(8)
X(667) = inverse-in-Parry-isodynamic-circle of X(5163); see X(2)
X(667) = polar conjugate of isotomic conjugate of X(22383)
X(667) = X(i)-isoconjugate of X(j) for these (i,j): {1, 668}, {2, 190}, {4, 4561}, {6, 1978}, {7, 3699}, {8, 664}, {9, 4554}, {10, 99}, {31, 6386}, {37, 799}, {42, 670}, {43, 18830}, {55, 4572}, {57, 646}, {58, 27808}, {59, 35519}, {63, 6335}, {65, 7257}, {69, 1897}, {71, 6331}, {72, 811}, {75, 100}, {76, 101}, {78, 18026}, {81, 4033}, {83, 4568}, {85, 644}, {86, 3952}, {87, 36863}, {88, 24004}, {92, 1332}, {108, 3718}, {109, 3596}, {110, 313}, {112, 40071}, {162, 20336}, {163, 27801}, {192, 4598}, {200, 4569}, {210, 4625}, {213, 4602}, {226, 645}, {238, 4583}, {239, 4562}, {257, 18047}, {264, 1331}, {273, 4571}, {274, 1018}, {279, 6558}, {291, 874}, {292, 27853}, {304, 1783}, {305, 8750}, {306, 648}, {307, 36797}, {308, 46148}, {310, 4557}, {312, 651}, {314, 4551}, {318, 6516}, {319, 6742}, {321, 662}, {322, 13138}, {329, 44327}, {330, 4595}, {331, 4587}, {333, 4552}, {334, 3573}, {335, 3570}, {341, 934}, {344, 37206}, {345, 653}, {346, 658}, {349, 5546}, {350, 660}, {391, 4624}, {513, 7035}, {514, 1016}, {519, 4555}, {522, 4998}, {523, 4600}, {536, 4607}, {561, 692}, {594, 4610}, {643, 1441}, {649, 31625}, {655, 32851}, {661, 4601}, {666, 3912}, {672, 36803}, {673, 42720}, {677, 35517}, {689, 21035}, {693, 765}, {726, 8709}, {728, 36838}, {740, 4589}, {756, 4623}, {789, 984}, {813, 1921}, {823, 3998}, {831, 33941}, {835, 5224}, {850, 4570}, {869, 46132}, {870, 3799}, {873, 40521}, {883, 14942}, {889, 899}, {892, 4062}, {894, 27805}, {897, 42721}, {898, 6381}, {901, 3264}, {903, 17780}, {906, 1969}, {908, 13136}, {927, 3717}, {932, 6376}, {985, 4505}, {1023, 20568}, {1025, 36796}, {1026, 2481}, {1043, 4566}, {1086, 6632}, {1088, 4578}, {1110, 40495}, {1125, 6540}, {1213, 4632}, {1215, 4594}, {1222, 21272}, {1230, 4629}, {1252, 3261}, {1264, 36127}, {1265, 36118}, {1266, 6079}, {1268, 4427}, {1269, 8701}, {1275, 3239}, {1310, 4385}, {1414, 3701}, {1415, 28659}, {1427, 7258}, {1434, 30730}, {1446, 7259}, {1447, 36801}, {1492, 33931}, {1502, 32739}, {1509, 4103}, {1577, 4567}, {1647, 6635}, {1698, 32042}, {1738, 35574}, {1813, 7017}, {1821, 42717}, {1826, 4563}, {1909, 3903}, {1918, 4609}, {2171, 4631}, {2238, 4639}, {2276, 37133}, {2284, 18031}, {2295, 7260}, {2321, 4573}, {2340, 46135}, {2345, 37215}, {2349, 42716}, {2397, 34234}, {2398, 18025}, {2414, 31638}, {2415, 31227}, {3112, 4553}, {3218, 36804}, {3219, 15455}, {3222, 21080}, {3226, 23354}, {3227, 23891}, {3250, 5388}, {3257, 4358}, {3262, 36037}, {3263, 36086}, {3661, 4586}, {3662, 4621}, {3663, 8706}, {3668, 7256}, {3676, 4076}, {3679, 4597}, {3682, 6528}, {3687, 6648}, {3692, 13149}, {3693, 34085}, {3700, 4620}, {3729, 30610}, {3730, 31624}, {3732, 30701}, {3762, 5376}, {3766, 5378}, {3783, 41072}, {3797, 37207}, {3807, 14621}, {3835, 5383}, {3882, 30710}, {3888, 7033}, {3911, 4582}, {3935, 35171}, {3939, 6063}, {3943, 4615}, {3948, 4584}, {3954, 4593}, {3963, 4603}, {3992, 4622}, {3995, 37205}, {4024, 4590}, {4025, 15742}, {4028, 35136}, {4036, 24041}, {4039, 18829}, {4064, 18020}, {4079, 34537}, {4082, 4616}, {4115, 32014}, {4150, 44766}, {4357, 8707}, {4359, 37212}, {4360, 8050}, {4373, 43290}, {4384, 32041}, {4389, 9059}, {4391, 4564}, {4397, 7045}, {4417, 44765}, {4441, 37138}, {4451, 6649}, {4462, 5382}, {4511, 35174}, {4515, 4635}, {4556, 28654}, {4559, 28660}, {4565, 30713}, {4574, 44129}, {4576, 18082}, {4577, 15523}, {4579, 7018}, {4585, 18359}, {4592, 41013}, {4596, 4647}, {4604, 4671}, {4605, 7058}, {4606, 19804}, {4612, 6358}, {4613, 30966}, {4617, 30693}, {4618, 4738}, {4619, 23978}, {4626, 5423}, {4628, 8024}, {4633, 5257}, {4634, 21805}, {4636, 34388}, {4638, 36791}, {4664, 37209}, {4705, 24037}, {4728, 5381}, {4752, 20569}, {4756, 30598}, {4767, 39704}, {4791, 5385}, {4847, 6606}, {4858, 31615}, {5297, 35181}, {5379, 14208}, {5380, 14210}, {6012, 33937}, {6332, 46102}, {6382, 34071}, {6541, 17930}, {6542, 35148}, {6550, 42372}, {6613, 6736}, {6630, 6631}, {6633, 35168}, {6634, 42555}, {6745, 35157}, {7012, 35518}, {7128, 15416}, {7239, 38810}, {7289, 42384}, {8652, 30596}, {8693, 21615}, {8708, 20888}, {8804, 44326}, {9065, 14620}, {9295, 9362}, {9296, 9361}, {9436, 36802}, {10009, 43077}, {11599, 17934}, {11679, 32038}, {14594, 30479}, {16086, 35169}, {16606, 36860}, {17233, 43190}, {17264, 37143}, {17295, 43191}, {17708, 21094}, {17743, 33946}, {17763, 35147}, {17787, 37137}, {18022, 32656}, {18743, 27834}, {18891, 34067}, {19799, 36099}, {20881, 31628}, {20911, 36147}, {20943, 29227}, {20947, 37135}, {21043, 31614}, {21089, 37880}, {21362, 32017}, {21580, 23617}, {21814, 37204}, {23067, 44130}, {23343, 31002}, {24029, 36795}, {25268, 40420}, {25728, 42343}, {27818, 30720}, {28605, 37211}, {28606, 37218}, {29574, 35177}, {29615, 35180}, {29616, 32040}, {30625, 42303}, {31343, 39126}, {31618, 35341}, {32018, 35342}, {32028, 36954}, {32680, 42701}, {32931, 35008}, {33948, 43531}, {34075, 35543}, {34832, 35572}, {36084, 42703}, {36085, 42713}, {36100, 42718}, {36101, 42719}, {36570, 42380}, {37129, 41314}, {37130, 42723}, {37131, 42722}, {37216, 42724}, {37764, 46143}, {38340, 42033}, {38828, 44723}, {40087, 40519}, {41267, 42371}, {44717, 46110}
X(667) = barycentric product X(i)*X(j) for these {i,j}: {1, 649}, {3, 6591}, {4, 22383}, {6, 513}, {7, 3063}, {9, 43924}, {11, 1415}, {19, 1459}, {21, 7180}, {25, 905}, {27, 810}, {28, 647}, {31, 514}, {32, 693}, {34, 652}, {37, 3733}, {39, 18108}, {41, 3676}, {42, 1019}, {44, 23345}, {48, 7649}, {50, 43082}, {55, 3669}, {56, 650}, {57, 663}, {58, 661}, {65, 7252}, {72, 43925}, {74, 14399}, {75, 1919}, {76, 1980}, {80, 21758}, {81, 512}, {82, 21123}, {86, 798}, {87, 20979}, {88, 1960}, {89, 4775}, {99, 3121}, {100, 1015}, {101, 244}, {104, 3310}, {105, 665}, {106, 1635}, {108, 7117}, {109, 2170}, {110, 3125}, {111, 14419}, {112, 18210}, {163, 3120}, {184, 17924}, {190, 3248}, {213, 7192}, {219, 43923}, {220, 43932}, {222, 18344}, {228, 17925}, {238, 3572}, {239, 875}, {241, 884}, {251, 2530}, {256, 20981}, {266, 6729}, {269, 657}, {274, 669}, {277, 8642}, {278, 1946}, {279, 8641}, {284, 4017}, {286, 3049}, {291, 8632}, {292, 659}, {310, 1924}, {330, 8640}, {393, 23224}, {517, 2423}, {518, 43929}, {520, 5317}, {521, 608}, {522, 604}, {523, 1333}, {525, 2203}, {536, 23349}, {560, 3261}, {593, 4705}, {603, 3064}, {644, 1357}, {651, 3271}, {654, 1411}, {656, 1474}, {662, 3122}, {668, 1977}, {672, 1027}, {676, 911}, {692, 1086}, {694, 4164}, {738, 4105}, {739, 891}, {741, 21832}, {753, 14438}, {757, 4079}, {759, 21828}, {764, 1252}, {765, 21143}, {788, 14621}, {812, 1911}, {813, 27846}, {822, 8747}, {825, 4475}, {834, 2214}, {840, 1643}, {849, 4024}, {869, 4817}, {876, 1914}, {890, 3227}, {893, 4367}, {898, 1646}, {899, 23892}, {900, 9456}, {901, 2087}, {902, 1022}, {904, 4369}, {906, 2969}, {909, 1769}, {910, 2424}, {919, 3675}, {923, 4750}, {926, 1462}, {932, 6377}, {934, 14936}, {985, 3250}, {1014, 3709}, {1016, 8027}, {1021, 1042}, {1023, 43922}, {1024, 1458}, {1054, 9262}, {1055, 35348}, {1084, 4623}, {1096, 4091}, {1101, 21131}, {1106, 3239}, {1110, 6545}, {1111, 32739}, {1118, 36054}, {1126, 4979}, {1170, 2488}, {1171, 4983}, {1219, 8662}, {1280, 8659}, {1332, 42067}, {1334, 7203}, {1356, 4631}, {1395, 6332}, {1397, 4391}, {1400, 3737}, {1402, 4560}, {1404, 23838}, {1407, 3900}, {1408, 3700}, {1410, 17926}, {1412, 4041}, {1413, 14298}, {1417, 1639}, {1426, 23090}, {1427, 21789}, {1431, 3287}, {1436, 6129}, {1437, 2501}, {1438, 2254}, {1444, 2489}, {1455, 2432}, {1461, 2310}, {1472, 6590}, {1491, 40746}, {1501, 40495}, {1575, 23355}, {1576, 16732}, {1577, 2206}, {1638, 34068}, {1647, 32665}, {1783, 3937}, {1824, 7254}, {1880, 23189}, {1918, 7199}, {1922, 3766}, {1929, 5029}, {1931, 18001}, {1964, 10566}, {1967, 4107}, {1973, 4025}, {1974, 15413}, {1979, 9267}, {2006, 8648}, {2053, 43051}, {2148, 21102}, {2149, 21132}, {2155, 21172}, {2159, 11125}, {2160, 2605}, {2162, 4083}, {2163, 4893}, {2165, 34948}, {2175, 24002}, {2176, 43931}, {2194, 7178}, {2204, 17094}, {2207, 4131}, {2208, 14837}, {2210, 4444}, {2217, 6589}, {2218, 43060}, {2220, 40086}, {2221, 8678}, {2226, 3251}, {2251, 6548}, {2279, 4724}, {2284, 43921}, {2287, 7250}, {2291, 14413}, {2298, 6371}, {2300, 4581}, {2328, 7216}, {2334, 4790}, {2350, 4040}, {2353, 16757}, {2384, 14421}, {2427, 15635}, {2623, 18180}, {2643, 4556}, {2787, 17961}, {3022, 4617}, {3119, 6614}, {3123, 34071}, {3126, 41934}, {3223, 23572}, {3230, 43928}, {3249, 7035}, {3270, 32714}, {3420, 40134}, {3435, 6588}, {3444, 31947}, {3445, 4394}, {3451, 6615}, {3500, 23655}, {3551, 23472}, {3667, 38266}, {3756, 34080}, {3768, 37129}, {3798, 38252}, {3803, 39951}, {3835, 7121}, {3837, 34077}, {3942, 8750}, {3960, 6187}, {4010, 18268}, {4057, 39798}, {4063, 40148}, {4086, 16947}, {4128, 4603}, {4130, 7023}, {4162, 40151}, {4163, 7366}, {4374, 7104}, {4378, 30650}, {4449, 9315}, {4455, 37128}, {4466, 32676}, {4491, 39981}, {4516, 4565}, {4521, 16945}, {4557, 16726}, {4559, 18191}, {4567, 8034}, {4594, 21755}, {4598, 38986}, {4638, 42084}, {4777, 28607}, {4784, 25426}, {4785, 40735}, {4802, 34819}, {4833, 28658}, {4834, 25417}, {4840, 28625}, {4977, 28615}, {5006, 18015}, {5040, 17946}, {5376, 8661}, {5381, 33917}, {6085, 40400}, {6139, 34056}, {6164, 9259}, {6186, 14838}, {6335, 22096}, {6363, 23617}, {6373, 20332}, {6385, 9426}, {6610, 23351}, {7004, 32674}, {7050, 8712}, {7139, 21184}, {7234, 40432}, {7255, 16584}, {8056, 8643}, {8615, 16612}, {8638, 34018}, {8639, 37870}, {8645, 34578}, {8647, 37626}, {8650, 34892}, {8656, 39963}, {8660, 36805}, {8735, 36059}, {9002, 40401}, {9178, 16702}, {9247, 46107}, {9263, 9299}, {9265, 38238}, {9297, 18825}, {9309, 20980}, {9468, 14296}, {9506, 38348}, {9508, 17962}, {10015, 34858}, {13476, 21007}, {13486, 20982}, {14422, 28317}, {14578, 39534}, {14597, 14775}, {16082, 23220}, {16606, 16695}, {16696, 18105}, {16892, 46289}, {17212, 40729}, {17217, 21759}, {17435, 32735}, {17921, 22381}, {18002, 19623}, {18197, 23493}, {18757, 21196}, {18830, 21762}, {19945, 34075}, {20517, 40145}, {21003, 39979}, {21035, 39179}, {21191, 34248}, {21448, 30234}, {21839, 43926}, {21907, 42670}, {22086, 36125}, {23524, 25576}, {23979, 42455}, {24027, 42462}, {27918, 34067}, {29226, 36614}, {30691, 32726}, {32641, 42753}, {32669, 35015}, {32702, 35014}, {40143, 42653}, {40763, 45882}, {41799, 45878}, {41933, 42757}
X(667) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1978}, {2, 6386}, {6, 668}, {25, 6335}, {28, 6331}, {31, 190}, {32, 100}, {37, 27808}, {41, 3699}, {42, 4033}, {48, 4561}, {55, 646}, {56, 4554}, {57, 4572}, {58, 799}, {60, 4631}, {81, 670}, {86, 4602}, {100, 31625}, {101, 7035}, {105, 36803}, {110, 4601}, {163, 4600}, {184, 1332}, {187, 42721}, {213, 3952}, {237, 42717}, {238, 27853}, {244, 3261}, {274, 4609}, {284, 7257}, {292, 4583}, {351, 42713}, {512, 321}, {513, 76}, {514, 561}, {522, 28659}, {523, 27801}, {560, 101}, {593, 4623}, {604, 664}, {608, 18026}, {647, 20336}, {649, 75}, {650, 3596}, {652, 3718}, {656, 40071}, {657, 341}, {659, 1921}, {661, 313}, {663, 312}, {665, 3263}, {669, 37}, {688, 3954}, {692, 1016}, {693, 1502}, {739, 889}, {741, 4639}, {764, 23989}, {788, 3661}, {798, 10}, {810, 306}, {812, 18891}, {834, 33935}, {838, 32782}, {849, 4610}, {869, 3807}, {872, 4103}, {875, 335}, {876, 18895}, {884, 36796}, {890, 536}, {891, 35543}, {902, 24004}, {904, 27805}, {905, 305}, {985, 37133}, {1015, 693}, {1019, 310}, {1027, 18031}, {1084, 4705}, {1086, 40495}, {1106, 658}, {1110, 6632}, {1178, 7260}, {1201, 21580}, {1253, 6558}, {1333, 99}, {1357, 24002}, {1395, 653}, {1397, 651}, {1398, 13149}, {1402, 4552}, {1407, 4569}, {1408, 4573}, {1412, 4625}, {1415, 4998}, {1416, 34085}, {1437, 4563}, {1459, 304}, {1462, 46135}, {1472, 37215}, {1474, 811}, {1492, 5388}, {1495, 42716}, {1501, 692}, {1576, 4567}, {1635, 3264}, {1911, 4562}, {1914, 874}, {1917, 32739}, {1918, 1018}, {1919, 1}, {1922, 660}, {1923, 46148}, {1924, 42}, {1946, 345}, {1960, 4358}, {1964, 4568}, {1973, 1897}, {1974, 1783}, {1977, 513}, {1979, 9296}, {1980, 6}, {2084, 15523}, {2162, 18830}, {2170, 35519}, {2175, 644}, {2176, 36863}, {2194, 645}, {2203, 648}, {2204, 36797}, {2205, 4557}, {2206, 662}, {2208, 44327}, {2209, 4595}, {2210, 3570}, {2223, 42720}, {2251, 17780}, {2276, 4505}, {2328, 7258}, {2423, 18816}, {2483, 33941}, {2484, 4385}, {2488, 1229}, {2489, 41013}, {2530, 8024}, {2605, 33939}, {3049, 72}, {3051, 4553}, {3063, 8}, {3120, 20948}, {3121, 523}, {3122, 1577}, {3124, 4036}, {3125, 850}, {3221, 22028}, {3230, 41314}, {3248, 514}, {3249, 244}, {3250, 33931}, {3251, 36791}, {3261, 1928}, {3270, 15416}, {3271, 4391}, {3288, 42711}, {3310, 3262}, {3569, 42703}, {3572, 334}, {3669, 6063}, {3676, 20567}, {3709, 3701}, {3733, 274}, {3737, 28660}, {3766, 44169}, {3768, 6381}, {3803, 40022}, {3937, 15413}, {3960, 40075}, {4017, 349}, {4025, 40364}, {4040, 18152}, {4041, 30713}, {4057, 18140}, {4063, 40087}, {4079, 1089}, {4083, 6382}, {4105, 30693}, {4107, 1926}, {4117, 4079}, {4162, 44723}, {4164, 3978}, {4367, 1920}, {4391, 40363}, {4435, 4087}, {4444, 44172}, {4455, 3948}, {4491, 18145}, {4556, 24037}, {4560, 40072}, {4623, 44168}, {4705, 28654}, {4724, 21615}, {4775, 4671}, {4782, 10009}, {4809, 30874}, {4813, 30596}, {4817, 871}, {4826, 4066}, {4834, 28605}, {4979, 1269}, {4983, 1230}, {5006, 17935}, {5029, 20947}, {5040, 17790}, {5317, 6528}, {6186, 15455}, {6187, 36804}, {6363, 26563}, {6371, 20911}, {6377, 20906}, {6586, 33932}, {6591, 264}, {7023, 36838}, {7032, 33946}, {7104, 3903}, {7109, 40521}, {7117, 35518}, {7121, 4598}, {7122, 18047}, {7180, 1441}, {7192, 6385}, {7234, 3963}, {7250, 1446}, {7252, 314}, {7366, 4626}, {7649, 1969}, {8027, 1086}, {8034, 16732}, {8054, 20949}, {8578, 20940}, {8618, 42723}, {8630, 2276}, {8632, 350}, {8635, 17289}, {8636, 32777}, {8637, 28606}, {8638, 3693}, {8639, 31993}, {8640, 192}, {8641, 346}, {8642, 344}, {8643, 18743}, {8644, 42724}, {8645, 17264}, {8646, 2345}, {8648, 32851}, {8650, 37756}, {8653, 42712}, {8654, 17279}, {8655, 4687}, {8656, 30829}, {8660, 16610}, {8662, 3672}, {9002, 33934}, {9247, 1331}, {9297, 712}, {9299, 9295}, {9426, 213}, {9447, 3939}, {9454, 1026}, {9455, 2284}, {9456, 4555}, {9459, 1023}, {9491, 21877}, {9494, 21814}, {10566, 18833}, {11125, 46234}, {14270, 42701}, {14296, 14603}, {14399, 3260}, {14407, 3992}, {14419, 3266}, {14436, 4439}, {14438, 35548}, {14575, 906}, {14598, 813}, {14599, 3573}, {14621, 46132}, {14827, 4578}, {14936, 4397}, {15413, 40050}, {15451, 42698}, {16692, 32026}, {16695, 31008}, {16732, 44173}, {16757, 40073}, {16947, 1414}, {17494, 40088}, {17924, 18022}, {17961, 35147}, {18002, 11611}, {18108, 308}, {18210, 3267}, {18268, 4589}, {18344, 7017}, {18897, 34067}, {20228, 21272}, {20906, 40367}, {20979, 6376}, {20981, 1909}, {20983, 30473}, {21007, 17143}, {21122, 1760}, {21123, 1930}, {21131, 23994}, {21143, 1111}, {21191, 18837}, {21385, 40089}, {21755, 2533}, {21758, 320}, {21760, 23354}, {21762, 4083}, {21828, 35550}, {21832, 35544}, {21835, 21051}, {22096, 905}, {22383, 69}, {22386, 25098}, {23099, 21833}, {23224, 3926}, {23225, 25083}, {23227, 22095}, {23345, 20568}, {23349, 3227}, {23355, 32020}, {23470, 23794}, {23472, 24524}, {23503, 21080}, {23560, 29226}, {23572, 17149}, {23655, 17786}, {23892, 31002}, {24002, 41283}, {26884, 15418}, {28607, 4597}, {28615, 6540}, {30234, 11059}, {32718, 5381}, {32719, 5376}, {32739, 765}, {32740, 5380}, {34077, 8709}, {34819, 32042}, {34858, 13136}, {34948, 7763}, {34952, 42700}, {36054, 1264}, {38346, 20954}, {38348, 18035}, {38367, 17793}, {38832, 36860}, {38986, 3835}, {38996, 22322}, {39201, 3998}, {40495, 40362}, {40728, 3799}, {40746, 789}, {40935, 7239}, {41267, 35309}, {41280, 1415}, {41935, 4618}, {42067, 17924}, {42653, 42710}, {42658, 42699}, {42660, 42704}, {42662, 42709}, {42664, 42714}, {42670, 32849}, {43082, 20573}, {43923, 331}, {43924, 85}, {43925, 286}, {43929, 2481}, {43931, 6383}
X(667) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 21260, 31251}, {2, 21301, 21260}, {2, 26823, 27168}, {2, 31291, 21301}, {75, 21613, 21440}, {100, 9266, 1016}, {351, 42661, 42653}, {649, 669, 8640}, {649, 1960, 4775}, {649, 5029, 3250}, {649, 8635, 8646}, {649, 8642, 8641}, {649, 8643, 663}, {649, 8655, 669}, {649, 8656, 1960}, {663, 1946, 8641}, {663, 8643, 1960}, {663, 8648, 1946}, {663, 8656, 8643}, {669, 8646, 8641}, {669, 8654, 8642}, {798, 1919, 3063}, {898, 1016, 9266}, {1960, 8648, 42670}, {2530, 14419, 905}, {3733, 16695, 1019}, {3801, 4809, 20517}, {3803, 30234, 905}, {4775, 42670, 8641}, {5638, 5639, 5040}, {8630, 8632, 8640}, {8635, 8655, 8642}, {8636, 8637, 42670}, {8636, 8639, 8646}, {8637, 8639, 4834}, {8641, 8662, 8640}, {8645, 8650, 890}, {8648, 8656, 8642}, {8656, 8660, 8640}, {20979, 20981, 20980}, {20979, 23572, 21763}, {21260, 21301, 31149}, {21260, 31288, 2}, {21301, 31288, 31251}, {26984, 27046, 2}, {28255, 28373, 21260}, {29458, 29770, 24601}, {31149, 31251, 21260}, {31288, 31291, 31149}, {42667, 42668, 42670}


X(668) = TRILINEAR POLE OF LINE X(2)X(37)

Trilinears    1/[a2(b - c)] : :
Barycentrics    b*c*(a - b)*(a - c) : :
X(668) = 3 X[2] - 4 X[27076], 6 X[2] - 5 X[27195], 3 X[2] + X[31298], X[2] - 4 X[36524], 9 X[2] - 8 X[40479], 2 X[350] - 3 X[18145], 2 X[889] - 3 X[9296], X[889] - 3 X[31625], 4 X[1015] - 3 X[3227], X[1015] - 3 X[13466], 4 X[1015] - 5 X[27195], 2 X[1015] + X[31298], X[1015] - 6 X[36524], 2 X[1015] + 3 X[39360], 3 X[1015] - 4 X[40479], 5 X[1698] - 4 X[40533], 3 X[3227] - 2 X[9263], X[3227] - 4 X[13466], 3 X[3227] - 8 X[27076], 3 X[3227] - 5 X[27195], 3 X[3227] + 2 X[31298], X[3227] - 8 X[36524], X[3227] + 2 X[39360], 9 X[3227] - 16 X[40479], 3 X[3807] - 4 X[4103], 3 X[3807] - 2 X[4568], 3 X[3807] - X[33946], 4 X[4103] - X[33946], 4 X[4422] - 3 X[24508], 4 X[6381] - 3 X[18145], X[9263] - 6 X[13466], X[9263] - 4 X[27076], 2 X[9263] - 5 X[27195], X[9263] - 12 X[36524], X[9263] + 3 X[39360], 3 X[9263] - 8 X[40479], 2 X[9267] - 3 X[14441], 3 X[13466] - 2 X[27076], 12 X[13466] - 5 X[27195], 6 X[13466] + X[31298], 2 X[13466] + X[39360], 9 X[13466] - 4 X[40479], 8 X[27076] - 5 X[27195], 4 X[27076] + X[31298], X[27076] - 3 X[36524], 4 X[27076] + 3 X[39360], 3 X[27076] - 2 X[40479], 5 X[27195] + 2 X[31298], 5 X[27195] - 24 X[36524], 5 X[27195] + 6 X[39360], 15 X[27195] - 16 X[40479], X[31298] + 12 X[36524], X[31298] - 3 X[39360], 3 X[31298] + 8 X[40479], 4 X[36524] + X[39360], 9 X[36524] - 2 X[40479], 9 X[39360] + 8 X[40479]

As the trilinear product of Steiner circumellipse antipodes, X(668) lies on conic {A,B,C,X(668),X(789)}} with center X(6376) and perspector X(75). (Randy Hutson, July 11, 2019)

X(668) lies on the Steiner circumellipse and these lines: {1, 3226}, {2, 1015}, {6, 18825}, {8, 76}, {9, 17786}, {10, 274}, {30, 16085}, {31, 18824}, {37, 3228}, {39, 21226}, {42, 18826}, {43, 17149}, {65, 35159}, {69, 150}, {72, 290}, {75, 537}, {80, 313}, {81, 40603}, {85, 9578}, {86, 40039}, {99, 100}, {101, 789}, {110, 839}, {141, 40857}, {145, 18135}, {148, 6653}, {183, 956}, {190, 646}, {194, 20671}, {200, 25297}, {210, 1920}, {213, 3225}, {226, 35176}, {239, 25298}, {257, 3954}, {264, 46133}, {279, 6552}, {304, 341}, {305, 10327}, {306, 30973}, {310, 4651}, {312, 1121}, {315, 3436}, {316, 5080}, {320, 3264}, {321, 671}, {322, 3718}, {325, 17757}, {330, 40598}, {335, 3125}, {350, 519}, {385, 5291}, {495, 37664}, {513, 889}, {514, 3807}, {517, 4087}, {518, 1921}, {523, 35147}, {524, 17790}, {527, 40875}, {538, 17759}, {561, 3681}, {594, 3770}, {644, 666}, {645, 648}, {651, 6648}, {660, 37133}, {662, 37218}, {664, 1026}, {667, 932}, {670, 4553}, {673, 29479}, {689, 29071}, {692, 4577}, {693, 4555}, {718, 4093}, {730, 3783}, {740, 35166}, {756, 18059}, {758, 35544}, {850, 35156}, {870, 36480}, {883, 4566}, {891, 1978}, {892, 3908}, {894, 35143}, {898, 9067}, {906, 2966}, {934, 6613}, {984, 43096}, {1078, 2975}, {1089, 17762}, {1107, 25102}, {1125, 25303}, {1146, 4437}, {1211, 17946}, {1221, 3728}, {1234, 40422}, {1237, 3678}, {1264, 14615}, {1265, 46137}, {1268, 16709}, {1269, 5564}, {1292, 35574}, {1310, 8707}, {1330, 32031}, {1423, 45242}, {1494, 20336}, {1500, 1655}, {1577, 6631}, {1654, 3963}, {1698, 31997}, {1757, 1966}, {1930, 20955}, {1965, 3961}, {1975, 5687}, {1992, 41316}, {2170, 18061}, {2176, 29425}, {2238, 40859}, {2241, 16916}, {2242, 16997}, {2275, 27091}, {2276, 43095}, {2295, 17499}, {2321, 35144}, {2345, 34283}, {2533, 4589}, {2664, 19567}, {2864, 29337}, {2895, 28654}, {3216, 18148}, {3230, 10027}, {3240, 30964}, {3241, 18146}, {3260, 46141}, {3261, 35171}, {3262, 46136}, {3263, 4723}, {3293, 33296}, {3294, 29699}, {3403, 5223}, {3416, 43099}, {3434, 11185}, {3452, 32017}, {3617, 34284}, {3626, 20888}, {3632, 3760}, {3661, 3765}, {3662, 44353}, {3688, 9230}, {3701, 33939}, {3717, 33677}, {3729, 4110}, {3730, 29697}, {3759, 18044}, {3780, 17034}, {3792, 35539}, {3869, 28659}, {3879, 17195}, {3888, 6373}, {3900, 14727}, {3907, 40499}, {3912, 3975}, {3926, 7080}, {3932, 35152}, {3934, 26801}, {3948, 6542}, {3978, 20683}, {3992, 14210}, {4001, 19811}, {4086, 35154}, {4090, 41318}, {4154, 23648}, {4253, 29400}, {4358, 17310}, {4360, 18133}, {4361, 18144}, {4369, 9362}, {4377, 4690}, {4384, 20917}, {4385, 33935}, {4397, 35157}, {4404, 46143}, {4416, 17787}, {4422, 24508}, {4427, 29340}, {4462, 32028}, {4463, 46140}, {4479, 4677}, {4485, 5692}, {4506, 4715}, {4552, 32038}, {4576, 8050}, {4578, 6606}, {4592, 36050}, {4596, 4623}, {4597, 17136}, {4598, 32039}, {4607, 37209}, {4647, 18032}, {4664, 9331}, {4668, 32104}, {4671, 31172}, {4696, 20911}, {4705, 7260}, {4710, 33082}, {4756, 6540}, {4783, 24715}, {4791, 6633}, {4899, 10030}, {4998, 6516}, {5209, 17731}, {5224, 10468}, {5277, 6645}, {5283, 41838}, {5303, 43459}, {5383, 20980}, {5552, 7763}, {5641, 42703}, {6382, 24282}, {6384, 16569}, {6385, 22271}, {6554, 30701}, {6604, 42020}, {7095, 34016}, {7200, 7245}, {7270, 40071}, {7278, 41875}, {7321, 32097}, {7752, 11681}, {7757, 17756}, {7769, 27529}, {8024, 33091}, {8300, 24294}, {9055, 21138}, {9059, 13396}, {9267, 14441}, {9651, 33823}, {9708, 16992}, {10009, 24349}, {10527, 32832}, {11054, 37857}, {11998, 28798}, {14206, 35161}, {14208, 35169}, {14568, 17737}, {14759, 18743}, {14829, 24618}, {14951, 17760}, {16086, 44150}, {16100, 21530}, {16502, 26687}, {16549, 29691}, {16552, 29381}, {16589, 32009}, {16594, 36805}, {16604, 25107}, {16829, 21264}, {16885, 29542}, {16974, 30141}, {17033, 20457}, {17135, 18152}, {17158, 33780}, {17160, 39995}, {17165, 40087}, {17213, 21041}, {17277, 18040}, {17282, 44359}, {17287, 20891}, {17288, 20892}, {17295, 18137}, {17296, 20923}, {17298, 30090}, {17312, 29982}, {17316, 30830}, {17349, 32012}, {17377, 18147}, {17390, 25660}, {17448, 26959}, {17684, 31456}, {17751, 28660}, {17755, 21232}, {17761, 30997}, {17784, 32815}, {17789, 35163}, {17791, 35550}, {18035, 20716}, {18037, 20496}, {18057, 32926}, {18064, 32911}, {18067, 32920}, {18149, 24003}, {18150, 27191}, {18153, 36845}, {18206, 29511}, {18823, 42713}, {19565, 21830}, {19581, 41531}, {19870, 28653}, {19974, 20333}, {20055, 31060}, {20669, 39914}, {20691, 25264}, {20963, 41240}, {21220, 26072}, {21238, 24437}, {21384, 27424}, {21587, 35149}, {21814, 43094}, {21839, 35146}, {22275, 40072}, {23632, 27035}, {24191, 32030}, {24482, 25333}, {24487, 37129}, {24517, 41683}, {24656, 31996}, {24722, 25382}, {25277, 34086}, {25294, 34088}, {26563, 33940}, {26757, 26770}, {26759, 27040}, {26794, 26795}, {27096, 27109}, {27133, 27134}, {27525, 32831}, {27855, 30583}, {28809, 29616}, {29349, 36216}, {29383, 46196}, {29420, 29421}, {29433, 29447}, {29441, 29482}, {29445, 29471}, {29501, 29528}, {29526, 29543}, {29731, 29732}, {30092, 36854}, {30116, 37632}, {30225, 39351}, {30578, 36791}, {30631, 33064}, {30632, 33065}, {30695, 32034}, {30713, 33066}, {30728, 32040}, {30730, 32041}, {30807, 35158}, {30893, 44153}, {31645, 36957}, {32099, 34282}, {33090, 39998}, {33769, 40088}, {33930, 33937}, {33931, 33936}, {34018, 40609}, {34085, 37223}, {35142, 41013}, {35150, 46238}, {35164, 35517}, {35174, 35519}, {35958, 40844}, {36912, 40833}, {37676, 41232}, {37686, 45751}, {40301, 40339}, {40495, 46135}, {40883, 44664}, {42696, 44147}

X(668) = midpoint of X(i) and X(j) for these {i,j}: {2, 39360}, {8, 17794}, {9263, 31298}
X(668) = reflection of X(i) in X(j) for these {i,j}: {1, 17793}, {2, 13466}, {194, 20671}, {274, 39028}, {291, 10}, {350, 6381}, {1015, 27076}, {3227, 2}, {4568, 4103}, {9263, 1015}, {9296, 31625}, {13466, 36524}, {16100, 21530}, {17946, 1211}, {19565, 21830}, {24722, 25382}, {29811, 29547}, {31645, 36957}, {32020, 6376}, {32035, 76}, {33946, 4568}, {39925, 16589}
X(668) = isogonal conjugate of X(667)
X(668) = isotomic conjugate of X(513)
X(668) = complement of X(9263)
X(668) = anticomplement of X(1015)
X(668) = polar conjugate of X(6591)
X(668) = cyclocevian conjugate of X(8047)
X(668) = anticomplement of the isogonal conjugate of X(1016)
X(668) = complement of the isogonal conjugate of X(9265)
X(668) = anticomplement of the isotomic conjugate of X(31625)
X(668) = complement of the isotomic conjugate of X(9295)
X(668) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {9, 17036}, {59, 3210}, {100, 4440}, {101, 9263}, {190, 149}, {644, 39351}, {646, 33650}, {662, 17154}, {668, 150}, {692, 21224}, {765, 2}, {1016, 8}, {1018, 148}, {1023, 39349}, {1026, 39353}, {1110, 194}, {1252, 192}, {1262, 17480}, {1275, 36845}, {1978, 21293}, {3257, 20042}, {3573, 39362}, {3699, 37781}, {3799, 39345}, {3952, 21221}, {4033, 3448}, {4076, 329}, {4557, 21220}, {4564, 145}, {4567, 1}, {4570, 17147}, {4590, 17140}, {4600, 75}, {4601, 17135}, {4620, 3873}, {4752, 39364}, {4998, 7}, {5376, 519}, {5377, 239}, {5378, 6542}, {5379, 3187}, {5381, 29824}, {5382, 3621}, {5383, 10453}, {5384, 4393}, {5385, 3241}, {6064, 21273}, {6065, 3177}, {6551, 21222}, {6632, 513}, {6635, 21297}, {7012, 30699}, {7035, 69}, {7045, 4452}, {7128, 11851}, {9268, 17495}, {15742, 5905}, {23990, 17486}, {24037, 17143}, {24041, 4360}, {27808, 21294}, {31615, 522}, {31625, 6327}, {35309, 39346}, {35342, 39348}, {37212, 44006}, {44724, 8055}, {46102, 12649}
X(668) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 9296}, {9265, 10}, {9267, 116}, {9295, 2887}, {9299, 1086}, {9361, 141}
X(668) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 9296}, {190, 36863}, {670, 1978}, {799, 190}, {889, 41314}, {1016, 17143}, {1978, 646}, {4572, 4554}, {4598, 18830}, {4600, 314}, {4601, 321}, {7035, 75}, {7257, 4561}, {7260, 4568}, {24037, 33775}, {31625, 2}, {36803, 42720}, {44168, 32026}
X(668) = X(i)-cross conjugate of X(j) for these (i,j): {2, 31625}, {8, 1016}, {69, 4998}, {72, 4567}, {75, 7035}, {100, 6335}, {190, 4554}, {319, 4600}, {321, 4601}, {329, 1275}, {512, 32009}, {513, 2}, {514, 274}, {517, 5376}, {522, 32017}, {523, 30710}, {661, 1221}, {693, 75}, {812, 32020}, {834, 37870}, {850, 40422}, {891, 3227}, {900, 36805}, {1577, 32018}, {1655, 44168}, {1978, 18830}, {2517, 34258}, {2533, 10}, {2895, 4590}, {3309, 32019}, {3436, 46102}, {3681, 765}, {3699, 646}, {3762, 20568}, {3765, 5388}, {3766, 2481}, {3770, 34537}, {3799, 32041}, {3869, 4564}, {3888, 30610}, {3900, 30701}, {3909, 662}, {3952, 190}, {4010, 335}, {4033, 1978}, {4036, 321}, {4063, 18140}, {4083, 1}, {4106, 7}, {4132, 1255}, {4140, 1920}, {4145, 39698}, {4391, 76}, {4397, 312}, {4427, 15455}, {4462, 85}, {4463, 5379}, {4468, 31618}, {4499, 42343}, {4553, 100}, {4872, 39293}, {6084, 34018}, {7192, 1268}, {8027, 36957}, {9400, 30571}, {9443, 39959}, {10327, 15742}, {14434, 536}, {14923, 5382}, {15416, 304}, {15632, 2397}, {17217, 40418}, {17494, 308}, {17762, 24037}, {17780, 36804}, {18155, 1240}, {20293, 333}, {20294, 31623}, {20295, 86}, {20296, 348}, {20906, 7033}, {20929, 46254}, {20949, 3112}, {20952, 38810}, {20954, 310}, {20979, 31008}, {20983, 6}, {21146, 27483}, {21272, 664}, {21287, 18020}, {21297, 903}, {21301, 83}, {21302, 32008}, {21304, 40415}, {21613, 38847}, {22319, 18785}, {22322, 18098}, {23354, 4562}, {23794, 6384}, {23813, 4373}, {23819, 1088}, {24381, 3963}, {24719, 14621}, {25142, 192}, {25300, 33296}, {25312, 4595}, {27855, 350}, {29226, 330}, {30709, 671}, {32937, 5383}, {33066, 4620}, {33948, 32042}, {35518, 3596}, {35519, 314}, {37781, 31619}, {41314, 889}, {42720, 36803}, {43067, 5936}
X(668) = cevapoint of X(i) and X(j) for these (i,j): {1, 4063}, {2, 513}, {6, 21005}, {8, 4391}, {10, 514}, {42, 20979}, {69, 35518}, {75, 693}, {100, 1332}, {120, 918}, {190, 3699}, {192, 25142}, {210, 4140}, {312, 4397}, {313, 35519}, {321, 4036}, {341, 15416}, {350, 27855}, {512, 16589}, {522, 3452}, {523, 1211}, {525, 21530}, {536, 14434}, {650, 3056}, {661, 3728}, {812, 17793}, {850, 1234}, {891, 13466}, {900, 16594}, {1016, 11607}, {1233, 3261}, {1577, 4647}, {1654, 24381}, {1909, 2533}, {2397, 15632}, {3762, 4738}, {3789, 4762}, {3900, 6554}, {3952, 4033}, {3954, 4705}, {4024, 21728}, {4083, 6376}, {4462, 44720}, {4651, 20954}, {4791, 4793}, {4824, 27495}, {6084, 40609}, {7192, 16709}, {7199, 16748}, {8027, 31645}, {20983, 30473}, {24533, 28369}, {28195, 28651}, {29226, 40598}
X(668) = crosspoint of X(i) and X(j) for these (i,j): {2, 9295}, {190, 4598}, {670, 799}, {1978, 4572}
X(668) = crosssum of X(i) and X(j) for these (i,j): {6, 1979}, {649, 20979}, {669, 798}, {3248, 8027}
X(668) = trilinear pole of line {2, 37}
X(668) = crossdifference of every pair of points on line {890, 1977}
X(668) = Steiner-circumellipse-antipode of X(3227)
X(668) = trilinear product of PU(24)
X(668) = barycentric product of PU(41)
X(668) = crossdifference of PU(42)
X(668) = trilinear product of intercepts of Steiner circumellipse and Nagel line
X(668) = pole wrt polar circle of trilinear polar of X(6591) (line X(3125)X(3271))
X(668) = perspector of hyperbola {A,B,C,PU(41)}}
X(668) = trilinear product of vertices of Gemini triangle 5
X(668) = trilinear product of vertices of Gemini triangle 6
X(668) = trilinear product of vertices of Gemini triangle 29
X(668) = trilinear product of vertices of Gemini triangle 30
X(668) = areal center of cevian triangles of PU(27)
X(668) = areal center of cevian triangles of PU(41)
X(668) = Steiner-circumellipse-X(1)-antipode of X(3226)
X(668) = Steiner-circumellipse-X(6)-antipode of X(18825)
X(668) = X(i)-isoconjugate of X(j) for these (i,j): {1, 667}, {2, 1919}, {6, 649}, {19, 22383}, {25, 1459}, {27, 3049}, {28, 810}, {31, 513}, {32, 514}, {34, 1946}, {41, 3669}, {42, 3733}, {48, 6591}, {55, 43924}, {56, 663}, {57, 3063}, {58, 512}, {66, 21122}, {71, 43925}, {75, 1980}, {81, 798}, {86, 669}, {87, 8640}, {100, 3248}, {101, 1015}, {106, 1960}, {109, 3271}, {110, 3122}, {163, 3125}, {184, 7649}, {190, 1977}, {212, 43923}, {213, 1019}, {238, 875}, {244, 692}, {251, 21123}, {269, 8641}, {274, 1924}, {284, 7180}, {292, 8632}, {310, 9426}, {521, 1395}, {522, 1397}, {523, 2206}, {560, 693}, {593, 4079}, {603, 18344}, {604, 650}, {608, 652}, {647, 1474}, {656, 2203}, {657, 1407}, {659, 1911}, {661, 1333}, {662, 3121}, {665, 1438}, {672, 43929}, {727, 6373}, {739, 3768}, {741, 4455}, {764, 1110}, {765, 8027}, {788, 985}, {812, 1922}, {822, 5317}, {849, 4705}, {870, 8630}, {876, 2210}, {884, 1458}, {890, 37129}, {893, 20981}, {899, 23349}, {902, 23345}, {904, 4367}, {905, 1973}, {909, 3310}, {923, 14419}, {926, 1416}, {932, 38986}, {977, 8636}, {1016, 3249}, {1022, 2251}, {1027, 2223}, {1042, 21789}, {1084, 4610}, {1086, 32739}, {1096, 23224}, {1106, 3900}, {1120, 8660}, {1178, 7234}, {1252, 21143}, {1253, 43932}, {1261, 42336}, {1326, 18001}, {1331, 42067}, {1357, 3939}, {1400, 7252}, {1402, 3737}, {1408, 4041}, {1411, 8648}, {1412, 3709}, {1415, 2170}, {1417, 4895}, {1461, 14936}, {1472, 8678}, {1501, 3261}, {1576, 3120}, {1635, 9456}, {1646, 34075}, {1647, 32719}, {1769, 34858}, {1790, 2489}, {1897, 22096}, {1914, 3572}, {1917, 40495}, {1918, 7192}, {1927, 14296}, {1964, 18108}, {1967, 4164}, {1974, 4025}, {2087, 32665}, {2159, 14399}, {2161, 21758}, {2162, 20979}, {2163, 4775}, {2175, 3676}, {2183, 2423}, {2191, 8642}, {2194, 4017}, {2200, 17925}, {2205, 7199}, {2207, 4091}, {2208, 6129}, {2209, 43931}, {2221, 2484}, {2297, 8662}, {2328, 7250}, {2333, 7254}, {2350, 21007}, {2422, 17209}, {2530, 46289}, {2605, 6186}, {2969, 32656}, {3009, 23355}, {3022, 6614}, {3051, 10566}, {3124, 4556}, {3224, 23572}, {3230, 23892}, {3250, 40746}, {3445, 8643}, {3675, 32666}, {3700, 16947}, {3766, 14598}, {3937, 8750}, {4057, 40148}, {4083, 7121}, {4105, 7023}, {4107, 9468}, {4117, 4623}, {4130, 7366}, {4162, 16945}, {4369, 7104}, {4394, 38266}, {4444, 14599}, {4475, 34069}, {4570, 8034}, {4598, 21762}, {4603, 21755}, {4750, 32740}, {4782, 40735}, {4786, 39238}, {4813, 34819}, {4817, 40728}, {4893, 28607}, {4979, 28615}, {5029, 17962}, {5040, 17954}, {5331, 8639}, {6377, 34071}, {6544, 41935}, {6545, 23990}, {6548, 9459}, {7117, 32674}, {7255, 40935}, {8054, 40519}, {8578, 34179}, {8637, 43531}, {8650, 34893}, {8655, 10013}, {8656, 41436}, {8661, 9268}, {8735, 32660}, {8747, 39201}, {8752, 22086}, {9008, 14623}, {9247, 17924}, {9259, 9262}, {9299, 9359}, {9315, 20980}, {9447, 24002}, {10547, 21108}, {11125, 40352}, {14413, 34068}, {14574, 21207}, {14575, 46107}, {16695, 23493}, {16892, 46288}, {17187, 18105}, {18197, 21759}, {18200, 40729}, {18210, 32676}, {18263, 27929}, {18265, 43041}, {18267, 27855}, {18268, 21832}, {19945, 32718}, {21110, 44167}, {21131, 23357}, {21172, 33581}, {21178, 40146}, {21190, 22262}, {21814, 39179}, {21828, 34079}, {23220, 36123}, {23225, 36124}, {23344, 43922}, {23979, 42462}, {27846, 34067}, {30691, 34078}, {30805, 36417}, {35519, 41280}
X(668) = barycentric product X(i)*X(j) for these {i,j}: {1, 1978}, {6, 6386}, {7, 646}, {8, 4554}, {9, 4572}, {10, 799}, {12, 4631}, {37, 670}, {42, 4602}, {69, 6335}, {72, 6331}, {75, 190}, {76, 100}, {81, 27808}, {85, 3699}, {86, 4033}, {92, 4561}, {99, 321}, {101, 561}, {109, 28659}, {110, 27801}, {162, 40071}, {192, 18830}, {213, 4609}, {226, 7257}, {239, 4583}, {264, 1332}, {274, 3952}, {290, 42717}, {291, 27853}, {304, 1897}, {305, 1783}, {306, 811}, {308, 4553}, {310, 1018}, {312, 664}, {313, 662}, {314, 4552}, {319, 15455}, {320, 36804}, {322, 44327}, {330, 36863}, {331, 4571}, {334, 3570}, {335, 874}, {341, 658}, {345, 18026}, {346, 4569}, {349, 643}, {350, 4562}, {513, 31625}, {514, 7035}, {518, 36803}, {523, 4601}, {536, 889}, {594, 4623}, {644, 6063}, {645, 1441}, {648, 20336}, {651, 3596}, {653, 3718}, {660, 1921}, {666, 3263}, {671, 42721}, {689, 3954}, {692, 1502}, {693, 1016}, {740, 4639}, {765, 3261}, {789, 3661}, {813, 18891}, {835, 33935}, {839, 32782}, {850, 4567}, {870, 3807}, {873, 4103}, {883, 36796}, {892, 42713}, {898, 35543}, {903, 24004}, {906, 18022}, {932, 6382}, {984, 37133}, {1026, 18031}, {1088, 6558}, {1089, 4610}, {1111, 6632}, {1215, 7260}, {1222, 21580}, {1229, 6606}, {1230, 4596}, {1231, 36797}, {1237, 4603}, {1240, 3882}, {1252, 40495}, {1265, 13149}, {1269, 37212}, {1275, 4397}, {1331, 1969}, {1414, 30713}, {1415, 40363}, {1446, 7256}, {1491, 5388}, {1494, 42716}, {1577, 4600}, {1909, 27805}, {1920, 3903}, {1928, 32739}, {2276, 46132}, {2321, 4625}, {2397, 18816}, {2481, 42720}, {2966, 42703}, {3112, 4568}, {3222, 22028}, {3227, 41314}, {3257, 3264}, {3262, 13136}, {3266, 5380}, {3267, 5379}, {3573, 18895}, {3668, 7258}, {3681, 31624}, {3693, 46135}, {3701, 4573}, {3717, 34085}, {3797, 41072}, {3908, 40826}, {3939, 20567}, {3943, 4634}, {3948, 4589}, {3963, 4594}, {3992, 4615}, {3998, 6528}, {4024, 24037}, {4036, 4590}, {4064, 46254}, {4076, 24002}, {4082, 4635}, {4086, 4620}, {4358, 4555}, {4359, 6540}, {4385, 37215}, {4391, 4998}, {4427, 32018}, {4441, 32041}, {4505, 14621}, {4551, 28660}, {4557, 6385}, {4559, 40072}, {4563, 41013}, {4564, 35519}, {4570, 20948}, {4579, 44187}, {4584, 35544}, {4585, 20566}, {4586, 33931}, {4593, 15523}, {4595, 6384}, {4597, 4671}, {4598, 6376}, {4607, 6381}, {4612, 34388}, {4618, 36791}, {4621, 33930}, {4624, 4673}, {4626, 30693}, {4632, 4647}, {4705, 34537}, {4767, 20569}, {5224, 37218}, {5383, 20906}, {5386, 35548}, {5389, 35549}, {5423, 36838}, {6516, 7017}, {6742, 33939}, {7018, 18047}, {7033, 33946}, {7235, 36806}, {8026, 32039}, {8050, 18140}, {8706, 26563}, {8707, 20911}, {8750, 40364}, {9059, 33934}, {9295, 9296}, {10030, 36801}, {11611, 17935}, {14727, 40883}, {15413, 15742}, {17170, 42384}, {17264, 35171}, {17780, 20568}, {17790, 35147}, {18021, 21859}, {18025, 42719}, {18821, 42722}, {18831, 42698}, {18833, 46148}, {20899, 35572}, {20947, 35148}, {21035, 37204}, {21272, 32017}, {21615, 37138}, {21814, 42371}, {23354, 32020}, {23891, 31002}, {25278, 42343}, {28605, 32042}, {30596, 37211}, {31615, 34387}, {32849, 35156}, {32851, 35174}, {33932, 43190}, {34067, 44169}, {34393, 42718}, {35139, 42701}, {35169, 42709}, {35179, 42724}, {35518, 46102}, {36802, 40704}, {36860, 42027}, {40014, 43290}, {42700, 46134}, {42723, 43093}
X(668) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 649}, {2, 513}, {3, 22383}, {4, 6591}, {6, 667}, {7, 3669}, {8, 650}, {9, 663}, {10, 661}, {21, 7252}, {28, 43925}, {30, 14399}, {31, 1919}, {32, 1980}, {36, 21758}, {37, 512}, {38, 21123}, {42, 798}, {43, 20979}, {44, 1960}, {45, 4775}, {55, 3063}, {57, 43924}, {59, 1415}, {63, 1459}, {65, 7180}, {69, 905}, {71, 810}, {72, 647}, {75, 514}, {76, 693}, {78, 652}, {81, 3733}, {83, 18108}, {85, 3676}, {86, 1019}, {88, 23345}, {92, 7649}, {94, 43082}, {99, 81}, {100, 6}, {101, 31}, {104, 2423}, {105, 43929}, {107, 5317}, {108, 608}, {109, 604}, {110, 1333}, {112, 2203}, {141, 2530}, {145, 4394}, {162, 1474}, {163, 2206}, {171, 20981}, {188, 6729}, {190, 1}, {192, 4083}, {200, 657}, {210, 3709}, {213, 669}, {218, 8642}, {219, 1946}, {220, 8641}, {226, 4017}, {228, 3049}, {238, 8632}, {239, 659}, {244, 21143}, {264, 17924}, {274, 7192}, {278, 43923}, {279, 43932}, {281, 18344}, {286, 17925}, {291, 3572}, {292, 875}, {294, 884}, {304, 4025}, {305, 15413}, {306, 656}, {310, 7199}, {312, 522}, {313, 1577}, {314, 4560}, {315, 16757}, {318, 3064}, {319, 14838}, {320, 3960}, {321, 523}, {322, 14837}, {326, 4091}, {329, 6129}, {330, 43931}, {333, 3737}, {334, 4444}, {335, 876}, {341, 3239}, {344, 3309}, {345, 521}, {346, 3900}, {349, 4077}, {350, 812}, {385, 4164}, {394, 23224}, {512, 3121}, {513, 1015}, {514, 244}, {517, 3310}, {518, 665}, {519, 1635}, {521, 7117}, {522, 2170}, {523, 3125}, {524, 14419}, {525, 18210}, {527, 14413}, {528, 1643}, {536, 891}, {545, 14421}, {556, 6728}, {561, 3261}, {594, 4705}, {612, 2484}, {643, 284}, {644, 55}, {645, 21}, {646, 8}, {648, 28}, {649, 3248}, {650, 3271}, {651, 56}, {653, 34}, {655, 1411}, {658, 269}, {660, 292}, {661, 3122}, {662, 58}, {664, 57}, {666, 105}, {667, 1977}, {670, 274}, {673, 1027}, {677, 911}, {692, 32}, {693, 1086}, {728, 4105}, {739, 23349}, {740, 21832}, {752, 14438}, {756, 4079}, {758, 21828}, {765, 101}, {789, 14621}, {799, 86}, {811, 27}, {812, 27846}, {813, 1911}, {823, 8747}, {824, 4475}, {835, 2214}, {850, 16732}, {870, 4817}, {874, 239}, {883, 241}, {889, 3227}, {891, 1646}, {894, 4367}, {898, 739}, {899, 3768}, {900, 2087}, {901, 9456}, {903, 1022}, {905, 3937}, {906, 184}, {908, 1769}, {918, 3675}, {927, 1462}, {932, 2162}, {934, 1407}, {984, 3250}, {1015, 8027}, {1016, 100}, {1018, 42}, {1020, 1042}, {1022, 43922}, {1023, 902}, {1025, 1458}, {1026, 672}, {1043, 1021}, {1086, 764}, {1089, 4024}, {1109, 21131}, {1110, 32739}, {1111, 6545}, {1121, 35348}, {1125, 4979}, {1191, 8662}, {1212, 2488}, {1213, 4983}, {1229, 6362}, {1230, 30591}, {1231, 17094}, {1252, 692}, {1259, 36054}, {1267, 6364}, {1269, 4978}, {1275, 934}, {1278, 29226}, {1279, 8659}, {1293, 38266}, {1310, 2221}, {1331, 48}, {1332, 3}, {1376, 20980}, {1414, 1412}, {1415, 1397}, {1427, 7250}, {1434, 7203}, {1441, 7178}, {1444, 7254}, {1461, 1106}, {1492, 40746}, {1502, 40495}, {1575, 6373}, {1577, 3120}, {1621, 21007}, {1633, 16502}, {1646, 33917}, {1698, 4813}, {1734, 20974}, {1740, 23572}, {1743, 8643}, {1757, 5029}, {1783, 25}, {1792, 23090}, {1812, 23189}, {1813, 603}, {1824, 2489}, {1897, 19}, {1909, 4369}, {1918, 1924}, {1920, 4374}, {1921, 3766}, {1930, 16892}, {1966, 4107}, {1969, 46107}, {1978, 75}, {1981, 1430}, {1992, 30234}, {1993, 34948}, {1997, 30198}, {2087, 8661}, {2098, 17424}, {2172, 21122}, {2176, 8640}, {2205, 9426}, {2238, 4455}, {2273, 8636}, {2276, 788}, {2284, 2223}, {2287, 21789}, {2295, 7234}, {2321, 4041}, {2323, 8648}, {2325, 4895}, {2345, 8678}, {2397, 517}, {2398, 910}, {2401, 15635}, {2406, 1455}, {2533, 16592}, {2703, 17961}, {2895, 31947}, {3059, 10581}, {3112, 10566}, {3125, 8034}, {3126, 35505}, {3161, 4162}, {3175, 4139}, {3212, 43051}, {3219, 2605}, {3227, 43928}, {3230, 890}, {3234, 42077}, {3239, 2310}, {3246, 8658}, {3248, 3249}, {3257, 106}, {3261, 1111}, {3262, 10015}, {3263, 918}, {3264, 3762}, {3421, 40134}, {3436, 6588}, {3452, 6615}, {3501, 23655}, {3550, 23472}, {3570, 238}, {3573, 1914}, {3596, 4391}, {3616, 4790}, {3618, 3803}, {3621, 2516}, {3661, 1491}, {3662, 3777}, {3666, 6371}, {3668, 7216}, {3669, 1357}, {3672, 8712}, {3679, 4893}, {3681, 6586}, {3682, 822}, {3685, 4435}, {3687, 17420}, {3693, 926}, {3699, 9}, {3700, 4516}, {3701, 3700}, {3702, 4976}, {3710, 8611}, {3718, 6332}, {3728, 40627}, {3729, 4449}, {3732, 614}, {3739, 6372}, {3752, 6363}, {3759, 4401}, {3760, 4382}, {3761, 4379}, {3762, 1647}, {3765, 4874}, {3766, 27918}, {3797, 30665}, {3799, 2276}, {3807, 984}, {3835, 3123}, {3864, 30671}, {3868, 43060}, {3869, 6589}, {3875, 4498}, {3882, 1193}, {3886, 45755}, {3888, 2275}, {3900, 14936}, {3903, 893}, {3908, 574}, {3912, 2254}, {3920, 2483}, {3926, 4131}, {3932, 24290}, {3935, 22108}, {3939, 41}, {3943, 4730}, {3948, 4010}, {3950, 4729}, {3952, 37}, {3954, 3005}, {3963, 2533}, {3971, 21834}, {3975, 3716}, {3978, 14296}, {3990, 39201}, {3992, 4120}, {3995, 4132}, {3998, 520}, {4009, 4526}, {4010, 39786}, {4024, 2643}, {4025, 3942}, {4033, 10}, {4036, 115}, {4037, 4155}, {4040, 38346}, {4043, 4151}, {4044, 4804}, {4053, 42666}, {4062, 2642}, {4063, 8054}, {4064, 3708}, {4066, 4838}, {4069, 1334}, {4076, 644}, {4082, 4171}, {4083, 6377}, {4086, 21044}, {4103, 756}, {4110, 4147}, {4115, 1962}, {4125, 4931}, {4130, 3022}, {4140, 40608}, {4163, 3119}, {4169, 21805}, {4261, 838}, {4358, 900}, {4359, 4977}, {4360, 4063}, {4363, 4378}, {4370, 3251}, {4374, 7200}, {4384, 4724}, {4385, 6590}, {4387, 4501}, {4391, 11}, {4393, 4782}, {4397, 1146}, {4404, 21950}, {4406, 7208}, {4411, 4403}, {4417, 21189}, {4420, 9404}, {4422, 6161}, {4426, 8633}, {4427, 1100}, {4436, 20963}, {4437, 3126}, {4441, 4762}, {4462, 3756}, {4463, 2485}, {4467, 7202}, {4482, 750}, {4487, 14425}, {4494, 4474}, {4495, 4508}, {4498, 17477}, {4505, 3661}, {4511, 654}, {4515, 4524}, {4551, 1400}, {4552, 65}, {4553, 39}, {4554, 7}, {4555, 88}, {4556, 849}, {4557, 213}, {4558, 1437}, {4559, 1402}, {4560, 18191}, {4561, 63}, {4562, 291}, {4563, 1444}, {4564, 109}, {4565, 1408}, {4566, 1427}, {4567, 110}, {4568, 38}, {4569, 279}, {4570, 163}, {4571, 219}, {4572, 85}, {4573, 1014}, {4574, 228}, {4576, 16696}, {4578, 220}, {4579, 172}, {4582, 1320}, {4583, 335}, {4584, 741}, {4585, 36}, {4586, 985}, {4587, 212}, {4588, 28607}, {4589, 37128}, {4592, 1790}, {4594, 40432}, {4595, 43}, {4596, 1171}, {4597, 89}, {4598, 87}, {4600, 662}, {4601, 99}, {4602, 310}, {4603, 1178}, {4604, 2163}, {4605, 1254}, {4606, 2334}, {4607, 37129}, {4609, 6385}, {4610, 757}, {4612, 60}, {4613, 40747}, {4617, 7023}, {4618, 2226}, {4619, 24027}, {4620, 1414}, {4621, 983}, {4623, 1509}, {4625, 1434}, {4626, 738}, {4628, 46289}, {4631, 261}, {4632, 40438}, {4636, 2150}, {4639, 18827}, {4647, 4988}, {4664, 29350}, {4671, 4777}, {4673, 4765}, {4687, 6005}, {4699, 29198}, {4705, 3124}, {4715, 14422}, {4723, 1639}, {4728, 19945}, {4738, 6544}, {4742, 4773}, {4752, 2177}, {4756, 16777}, {4767, 45}, {4768, 4530}, {4781, 16666}, {4812, 29025}, {4847, 21127}, {4850, 9002}, {4858, 21132}, {4873, 4814}, {4885, 4014}, {4945, 23352}, {4975, 4984}, {4980, 28175}, {4986, 6546}, {4997, 23838}, {4998, 651}, {5224, 14349}, {5235, 4833}, {5257, 4822}, {5280, 8635}, {5283, 2978}, {5291, 5040}, {5308, 7659}, {5333, 4840}, {5360, 2491}, {5375, 16686}, {5376, 901}, {5377, 919}, {5378, 813}, {5379, 112}, {5380, 111}, {5381, 898}, {5382, 1293}, {5383, 932}, {5384, 825}, {5385, 4588}, {5386, 753}, {5387, 2748}, {5388, 789}, {5389, 755}, {5391, 6365}, {5423, 4130}, {5440, 22086}, {5468, 16702}, {5526, 8645}, {5546, 2194}, {5839, 6050}, {6063, 24002}, {6064, 4612}, {6079, 40400}, {6099, 32655}, {6163, 9259}, {6331, 286}, {6332, 7004}, {6335, 4}, {6374, 23807}, {6376, 3835}, {6381, 4728}, {6382, 20906}, {6386, 76}, {6516, 222}, {6517, 7125}, {6540, 1255}, {6542, 9508}, {6544, 42084}, {6554, 17115}, {6558, 200}, {6574, 7050}, {6591, 42067}, {6603, 6139}, {6604, 43049}, {6606, 1170}, {6614, 7366}, {6630, 6164}, {6631, 1054}, {6632, 765}, {6634, 6163}, {6635, 5376}, {6648, 961}, {6649, 7175}, {6651, 38348}, {6742, 2160}, {7012, 32674}, {7017, 44426}, {7027, 6730}, {7035, 190}, {7045, 1461}, {7080, 14298}, {7081, 3287}, {7192, 16726}, {7199, 17205}, {7234, 21755}, {7239, 3778}, {7256, 2287}, {7257, 333}, {7258, 1043}, {7259, 2328}, {7260, 32010}, {7265, 2611}, {7270, 16612}, {8026, 23886}, {8033, 17212}, {8050, 39798}, {8620, 9297}, {8640, 21762}, {8652, 34819}, {8701, 28615}, {8706, 23617}, {8707, 2298}, {8709, 20332}, {8750, 1973}, {9059, 40401}, {9260, 9283}, {9263, 38238}, {9265, 9299}, {9266, 1979}, {9268, 32665}, {9272, 41461}, {9278, 18001}, {9282, 9262}, {9295, 9267}, {9296, 9263}, {9362, 9359}, {10015, 42753}, {10030, 43041}, {10196, 45233}, {10327, 2509}, {10527, 13401}, {10580, 17410}, {11607, 5375}, {11611, 18015}, {11679, 17418}, {13136, 104}, {13138, 1436}, {13149, 1119}, {13466, 14434}, {14206, 11125}, {14208, 4466}, {14210, 4750}, {14213, 21102}, {14434, 39011}, {14543, 1104}, {14570, 18180}, {14589, 20958}, {14594, 2285}, {14829, 21173}, {14942, 1024}, {15413, 1565}, {15416, 2968}, {15418, 5088}, {15455, 79}, {15523, 8061}, {15631, 16725}, {15632, 23980}, {15742, 1783}, {16082, 43933}, {16284, 7658}, {16560, 8578}, {16610, 6085}, {16670, 8656}, {16777, 4834}, {16784, 8650}, {16826, 4784}, {17103, 18200}, {17143, 17494}, {17149, 21191}, {17160, 21385}, {17205, 8042}, {17217, 16742}, {17233, 1734}, {17234, 4905}, {17261, 4879}, {17263, 42325}, {17264, 3887}, {17277, 4040}, {17279, 6004}, {17289, 830}, {17335, 4794}, {17487, 9269}, {17740, 9001}, {17752, 24533}, {17756, 9010}, {17762, 21196}, {17776, 15313}, {17780, 44}, {17786, 17072}, {17787, 3907}, {17788, 4142}, {17789, 4458}, {17790, 2787}, {17792, 45902}, {17796, 42670}, {17863, 29162}, {17864, 21117}, {17886, 21141}, {17893, 21137}, {17906, 1828}, {17924, 2969}, {17934, 1931}, {17935, 19623}, {17944, 5006}, {18026, 278}, {18047, 171}, {18052, 23790}, {18062, 17200}, {18098, 18105}, {18134, 23800}, {18135, 4106}, {18137, 8714}, {18138, 23785}, {18140, 20295}, {18142, 23798}, {18143, 23789}, {18144, 23815}, {18145, 21297}, {18147, 29013}, {18148, 23803}, {18149, 21211}, {18150, 23814}, {18151, 21201}, {18152, 20954}, {18153, 23819}, {18154, 23823}, {18155, 17197}, {18156, 3798}, {18157, 23829}, {18159, 21204}, {18697, 21124}, {18698, 23755}, {18738, 23806}, {18740, 12047}, {18743, 3667}, {18747, 23782}, {18750, 21172}, {18816, 2401}, {18830, 330}, {19799, 23874}, {19804, 4778}, {20234, 3801}, {20235, 21107}, {20236, 21118}, {20237, 21119}, {20332, 23355}, {20336, 525}, {20431, 20504}, {20433, 20505}, {20434, 20506}, {20435, 20507}, {20436, 20508}, {20437, 20509}, {20438, 20510}, {20439, 20511}, {20440, 20512}, {20441, 20513}, {20442, 20514}, {20443, 20515}, {20444, 20517}, {20445, 20516}, {20446, 20518}, {20447, 20519}, {20448, 20520}, {20449, 20521}, {20450, 20522}, {20451, 20523}, {20452, 20524}, {20453, 20525}, {20454, 20526}, {20568, 6548}, {20627, 21110}, {20641, 21178}, {20663, 38367}, {20693, 17990}, {20752, 23225}, {20769, 22384}, {20879, 21103}, {20880, 21104}, {20881, 21105}, {20882, 21106}, {20883, 21108}, {20884, 21109}, {20886, 21111}, {20887, 21112}, {20889, 21113}, {20890, 21114}, {20893, 21115}, {20894, 21116}, {20895, 21120}, {20896, 21121}, {20898, 21126}, {20899, 21128}, {20900, 21129}, {20901, 21133}, {20902, 21134}, {20903, 21135}, {20904, 21136}, {20906, 21138}, {20907, 21139}, {20908, 21140}, {20909, 21142}, {20910, 21144}, {20911, 3004}, {20912, 21145}, {20913, 21146}, {20914, 21174}, {20915, 21175}, {20916, 21176}, {20917, 24720}, {20919, 21179}, {20920, 21180}, {20921, 7661}, {20922, 21182}, {20924, 4453}, {20925, 21183}, {20926, 21184}, {20927, 21185}, {20928, 21186}, {20929, 21187}, {20930, 21188}, {20931, 21190}, {20932, 21192}, {20933, 21193}, {20934, 21194}, {20935, 21195}, {20936, 21197}, {20937, 21198}, {20938, 21199}, {20939, 21200}, {20940, 21202}, {20941, 21203}, {20942, 4962}, {20944, 21205}, {20945, 21206}, {20947, 2786}, {20948, 21207}, {20949, 21208}, {20951, 21209}, {20952, 21210}, {20954, 17761}, {20955, 21212}, {20956, 21181}, {20979, 38986}, {21035, 2084}, {21272, 3752}, {21362, 1201}, {21414, 23735}, {21425, 21125}, {21436, 23748}, {21438, 23772}, {21580, 3663}, {21581, 23801}, {21596, 23799}, {21604, 4425}, {21609, 31605}, {21611, 24225}, {21802, 8664}, {21805, 14407}, {21810, 42661}, {21814, 688}, {21816, 8663}, {21833, 22260}, {21839, 351}, {21859, 181}, {21873, 42653}, {21874, 8651}, {21877, 3221}, {21883, 9402}, {22003, 2650}, {22011, 40471}, {22028, 23301}, {22128, 22379}, {22279, 14991}, {22370, 22090}, {22383, 22096}, {23067, 1409}, {23113, 22344}, {23343, 3230}, {23344, 2251}, {23354, 1575}, {23581, 23726}, {23703, 1404}, {23704, 8647}, {23845, 20228}, {23891, 899}, {23978, 42455}, {24002, 1358}, {24004, 519}, {24026, 42462}, {24029, 1457}, {24037, 4610}, {24039, 6629}, {24041, 4556}, {24199, 23738}, {24358, 24286}, {24443, 23751}, {24524, 31286}, {24589, 28209}, {24622, 23774}, {25142, 40610}, {25259, 17463}, {25268, 3057}, {25272, 21342}, {25278, 31287}, {25280, 25666}, {25292, 40464}, {25577, 23524}, {25660, 29150}, {26242, 9313}, {26611, 42757}, {27396, 8676}, {27644, 16695}, {27801, 850}, {27805, 256}, {27808, 321}, {27834, 3445}, {27853, 350}, {27855, 35119}, {28605, 4802}, {28606, 834}, {28653, 15309}, {28654, 4036}, {28659, 35519}, {28660, 18155}, {28742, 44319}, {28974, 28473}, {29227, 36614}, {29477, 23725}, {29611, 2526}, {30473, 21260}, {30566, 24457}, {30568, 42312}, {30596, 4823}, {30610, 9309}, {30693, 4163}, {30695, 17427}, {30710, 4581}, {30713, 4086}, {30720, 3158}, {30728, 4512}, {30729, 3683}, {30730, 210}, {30731, 3689}, {30758, 28846}, {30806, 1638}, {30807, 676}, {30829, 6006}, {30963, 4785}, {30966, 4481}, {31008, 17217}, {31130, 30520}, {31615, 59}, {31628, 18771}, {31633, 40577}, {31993, 8672}, {31997, 4932}, {32018, 4608}, {32028, 3315}, {32038, 959}, {32041, 1002}, {32042, 25417}, {32094, 3722}, {32641, 34858}, {32656, 9247}, {32674, 1395}, {32714, 1398}, {32739, 560}, {32777, 832}, {32779, 9013}, {32849, 8674}, {32851, 3738}, {32911, 4057}, {32925, 17458}, {32926, 21389}, {32937, 21348}, {33116, 6003}, {33296, 18197}, {33297, 16751}, {33775, 17161}, {33854, 21003}, {33891, 3808}, {33908, 14474}, {33930, 3776}, {33931, 824}, {33932, 25259}, {33934, 44435}, {33935, 45746}, {33939, 4467}, {33946, 982}, {33948, 28606}, {33951, 7191}, {33952, 17017}, {34018, 43930}, {34067, 1922}, {34071, 7121}, {34284, 43067}, {34387, 40166}, {34525, 46006}, {34537, 4623}, {35147, 17946}, {35148, 1929}, {35156, 21907}, {35157, 34056}, {35160, 37626}, {35171, 34578}, {35174, 2006}, {35181, 34914}, {35309, 21035}, {35312, 1418}, {35338, 1475}, {35341, 2293}, {35342, 2308}, {35518, 26932}, {35519, 4858}, {35538, 20908}, {35550, 4707}, {35574, 2991}, {36037, 909}, {36038, 42754}, {36049, 2208}, {36086, 1438}, {36101, 2424}, {36106, 913}, {36118, 1435}, {36146, 1416}, {36795, 43728}, {36796, 885}, {36797, 1172}, {36801, 4876}, {36802, 294}, {36803, 2481}, {36804, 80}, {36805, 23836}, {36807, 35355}, {36838, 479}, {36860, 33296}, {36863, 192}, {37129, 23892}, {37133, 870}, {37135, 17962}, {37137, 1431}, {37138, 2279}, {37141, 1413}, {37205, 39949}, {37206, 2191}, {37210, 34916}, {37212, 1126}, {37218, 43531}, {37593, 4832}, {37659, 44408}, {37680, 4491}, {37756, 2832}, {37758, 2827}, {37783, 42741}, {37788, 2826}, {37857, 2837}, {38810, 7255}, {38828, 16945}, {38853, 16692}, {39011, 14441}, {39028, 27854}, {39044, 4375}, {39126, 30719}, {39293, 36146}, {39294, 36110}, {40013, 40086}, {40071, 14208}, {40087, 20949}, {40089, 21606}, {40117, 7151}, {40166, 7336}, {40447, 14775}, {40495, 23989}, {40499, 20665}, {40521, 1500}, {40603, 31946}, {40704, 43042}, {40728, 8630}, {40790, 45882}, {40865, 5091}, {40872, 9511}, {40883, 42341}, {41013, 2501}, {41314, 536}, {41315, 28582}, {41316, 28475}, {41798, 23351}, {42005, 17422}, {42012, 17412}, {42014, 17425}, {42029, 28147}, {42033, 35057}, {42034, 28161}, {42066, 17411}, {42343, 41439}, {42698, 6368}, {42699, 8057}, {42700, 924}, {42701, 526}, {42702, 39469}, {42703, 2799}, {42704, 8675}, {42711, 23878}, {42712, 4843}, {42713, 690}, {42714, 23879}, {42716, 30}, {42717, 511}, {42718, 515}, {42719, 516}, {42720, 518}, {42721, 524}, {42722, 528}, {42723, 674}, {42724, 1499}, {43069, 43070}, {43077, 40735}, {43290, 1743}, {44140, 23882}, {44327, 84}, {44426, 8735}, {44448, 38375}, {44717, 36059}, {44720, 4521}, {44765, 2217}, {45791, 6607}, {45876, 41799}, {46102, 108}, {46135, 34018}, {46148, 1964}
X(668) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 6376, 18140}, {2, 1015, 27195}, {2, 9263, 1015}, {2, 31298, 9263}, {8, 76, 17143}, {10, 1909, 274}, {43, 17149, 34020}, {75, 18159, 1111}, {101, 4482, 18047}, {190, 3732, 33951}, {190, 4033, 646}, {190, 4595, 1018}, {304, 341, 33932}, {313, 319, 314}, {341, 16284, 304}, {350, 6381, 18145}, {664, 3699, 4561}, {799, 7257, 99}, {874, 4505, 646}, {1015, 9263, 3227}, {1015, 13466, 27076}, {1015, 27076, 2}, {1018, 23891, 4595}, {1089, 17762, 33775}, {1107, 25102, 27020}, {1111, 4738, 4986}, {1111, 4986, 75}, {1111, 18159, 20568}, {1500, 1655, 32026}, {1909, 25280, 10}, {1978, 3952, 36863}, {3216, 18148, 29454}, {3227, 27195, 1015}, {3263, 30806, 20924}, {3570, 18047, 101}, {3632, 3760, 17144}, {3679, 3761, 75}, {3761, 43270, 20568}, {3799, 27853, 36863}, {3807, 33946, 4568}, {3952, 23354, 3799}, {3992, 14210, 20947}, {4103, 4568, 3807}, {4696, 20911, 33941}, {4723, 30806, 3263}, {5080, 20553, 316}, {6376, 24524, 1}, {9263, 27076, 27195}, {9263, 29547, 29491}, {9263, 39360, 31298}, {13466, 31298, 27195}, {13466, 39360, 3227}, {17144, 20943, 3760}, {17149, 25287, 43}, {18061, 35957, 2170}, {20955, 33938, 1930}, {21226, 26752, 39}, {25472, 25685, 2}, {27076, 31298, 3227}, {27295, 30026, 30866}, {27808, 33948, 36863}


X(669) = CROSSDIFFERENCE OF X(2) AND X(39)

Trilinears    a3(b2 - c2) : :
Barycentrics    a4(b2 - c2) : :
X(669) = 6 X[2] - 5 X[31279], 3 X[2] + X[31299], 3 X[2] - 4 X[44451], 3 X[351] - 2 X[647], 3 X[351] - X[3005], 3 X[351] + 2 X[3804], 3 X[351] - 4 X[8651], 3 X[351] + X[8664], 6 X[351] - X[8665], X[647] - 3 X[8644], 2 X[647] + X[8664], 4 X[647] - X[8665], 4 X[647] - 3 X[17414], X[850] - 3 X[4108], 2 X[887] - 3 X[9491], X[887] - 3 X[9494], 3 X[1637] - 2 X[12075], 5 X[1656] - 4 X[39511], 4 X[2501] - 3 X[8029], X[3005] + 2 X[3804], X[3005] - 6 X[8644], X[3005] - 4 X[8651], 2 X[3005] - 3 X[17414], 4 X[3265] - 3 X[14424], X[3288] + 2 X[14318], X[3804] + 3 X[8644], X[3804] + 2 X[8651], 4 X[3804] + X[8665], 4 X[3804] + 3 X[17414], 4 X[5113] - 3 X[9210], X[5996] - 3 X[15724], 2 X[6563] - 3 X[11123], 3 X[8644] - 2 X[8651], 6 X[8644] + X[8664], 12 X[8644] - X[8665], 4 X[8644] - X[17414], 4 X[8651] + X[8664], 8 X[8651] - X[8665], 8 X[8651] - 3 X[17414], 4 X[8653] - X[8663], 2 X[8664] + X[8665], 2 X[8664] + 3 X[17414], X[8665] - 3 X[17414], 3 X[9131] - X[41298], 3 X[9147] - X[31296], 3 X[9148] - 4 X[30476], 4 X[9489] - 3 X[9491], 2 X[9489] - 3 X[9494], 2 X[11176] - 3 X[15724], 2 X[14270] - 3 X[34952], 4 X[14270] - 3 X[39201], 3 X[14420] - 2 X[33294], 4 X[23301] - 3 X[31176], 4 X[23301] - 5 X[31279], 2 X[23301] + X[31299], X[23301] - 3 X[45317], 3 X[31176] - 5 X[31279], 3 X[31176] + 2 X[31299], 3 X[31176] - 2 X[44445], 3 X[31176] - 8 X[44451], X[31176] - 4 X[45317], 5 X[31279] + 2 X[31299], 5 X[31279] - 2 X[44445], 5 X[31279] - 8 X[44451], 5 X[31279] - 12 X[45317], X[31299] + 4 X[44451], X[31299] + 6 X[45317], 3 X[34952] - X[42660], 3 X[39201] - 2 X[42660], X[44445] - 4 X[44451], X[44445] - 6 X[45317], 2 X[44451] - 3 X[45317]

X(669) lies on the Kiepert parabola and these lines: {1, 18197}, {2, 23301}, {3, 1499}, {6, 9009}, {10, 21726}, {22, 6563}, {23, 385}, {25, 878}, {31, 875}, {32, 23099}, {55, 25819}, {75, 21441}, {99, 886}, {100, 9362}, {110, 805}, {111, 5970}, {112, 9091}, {157, 40914}, {160, 7712}, {184, 30451}, {187, 237}, {353, 9212}, {511, 30219}, {513, 27469}, {514, 23786}, {525, 2528}, {661, 4455}, {684, 924}, {688, 864}, {690, 2525}, {804, 850}, {865, 1648}, {881, 3051}, {1252, 34067}, {1383, 9178}, {1402, 7180}, {1491, 27648}, {1510, 6132}, {1576, 23357}, {1598, 39533}, {1634, 14607}, {1637, 12075}, {1656, 39511}, {1977, 4117}, {1995, 8371}, {2106, 3733}, {2308, 23570}, {2353, 5489}, {2373, 2868}, {2422, 46319}, {2451, 3221}, {2485, 2514}, {2487, 20470}, {2489, 44099}, {2513, 3050}, {2930, 30220}, {2937, 8151}, {3124, 15630}, {3222, 4609}, {3233, 7468}, {3265, 3566}, {3398, 39518}, {3455, 14443}, {3700, 17989}, {3800, 20854}, {3806, 7927}, {4010, 27731}, {4083, 23506}, {4128, 22386}, {4132, 4782}, {4151, 4401}, {4367, 5937}, {4467, 23401}, {4724, 8672}, {4897, 16678}, {5020, 14341}, {5026, 38998}, {5466, 14002}, {5468, 41337}, {5940, 35296}, {5996, 11176}, {6041, 22260}, {6130, 30735}, {6636, 10190}, {6753, 17994}, {7203, 16878}, {7253, 23400}, {7492, 9168}, {8617, 44007}, {8704, 11616}, {8723, 35268}, {9125, 39477}, {9131, 41298}, {9137, 20403}, {9148, 30476}, {9213, 9999}, {9313, 43060}, {10278, 13595}, {10279, 13621}, {10280, 18369}, {11171, 39501}, {11182, 37338}, {11183, 14096}, {11328, 34290}, {11450, 26881}, {11479, 44931}, {11634, 34245}, {11641, 24974}, {12106, 16220}, {13564, 32204}, {14417, 32478}, {15080, 39495}, {20965, 38237}, {20976, 38241}, {20981, 23467}, {21051, 27045}, {21301, 27293}, {21448, 34204}, {21763, 23574}, {23964, 32696}, {25862, 45882}, {28255, 30023}, {28373, 28470}, {32320, 39469}, {32473, 45687}, {34347, 34396}, {36182, 40871}, {37777, 41357}, {41939, 45900}

X(669) = midpoint of X(i) and X(j) for these {i,j}: {647, 3804}, {3005, 8664}, {5027, 14318}, {18105, 21006}, {31299, 44445}
X(669) = reflection of X(i) in X(j) for these {i,j}: {2, 45317}, {3, 5926}, {351, 8644}, {647, 8651}, {887, 9489}, {2514, 2485}, {3005, 647}, {3049, 9426}, {3288, 5027}, {5996, 11176}, {8639, 667}, {8664, 3804}, {8665, 3005}, {9178, 46001}, {9491, 9494}, {17414, 351}, {17415, 2491}, {18117, 6140}, {23301, 44451}, {30735, 6130}, {31176, 2}, {39201, 34952}, {42660, 14270}, {44445, 23301}
X(669) = isogonal conjugate of X(670)
X(669) = isotomic conjugate of X(4609)
X(669) = complement of X(44445)
X(669) = anticomplement of X(23301)
X(669) = circumcircle-inverse of X(5108)
X(669) = Parry-circle-inverse of X(5027)
X(669) = Stammler-circle-inverse of X(18346)
X(669) = Parry-isodynamic-circle-inverse of X(5106)
X(669) = tangential-isogonal conjugate of X(7669)
X(669) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {163, 32747}, {662, 32548}, {2998, 21294}, {3222, 6327}, {3223, 3448}, {3224, 21221}, {4599, 19562}, {34248, 148}
X(669) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 38996}, {163, 6665}, {6573, 21238}, {6664, 21253}
X(669) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 21762}, {2, 38996}, {6, 9427}, {25, 3124}, {31, 1977}, {32, 1084}, {56, 21755}, {98, 2086}, {99, 6}, {100, 21753}, {107, 42295}, {110, 3051}, {512, 3049}, {667, 798}, {670, 9490}, {689, 20965}, {805, 32748}, {931, 1185}, {932, 23551}, {1402, 3121}, {1576, 32}, {1634, 5007}, {1919, 1924}, {2353, 20975}, {2373, 1648}, {2980, 115}, {3222, 2}, {3455, 21906}, {3903, 23546}, {4557, 213}, {4559, 21751}, {6573, 3589}, {7087, 20982}, {7954, 16285}, {8640, 23503}, {8709, 2238}, {9150, 3231}, {9426, 9491}, {10425, 3289}, {14574, 40981}, {16277, 39691}, {17938, 237}, {18105, 512}, {26714, 18899}, {32696, 2211}, {32729, 14567}, {32734, 217}, {34067, 41333}, {34207, 3269}, {34444, 1015}, {34445, 23554}, {39644, 44114}, {40519, 20970}
X(669) = X(i)-cross conjugate of X(j) for these (i,j): {688, 512}, {1084, 32}, {2491, 881}, {3221, 9491}, {4117, 7109}, {9427, 6}, {9494, 9426}, {23099, 1084}, {23216, 1501}
X(669) = cevapoint of X(i) and X(j) for these (i,j): {6, 9431}, {512, 3221}, {688, 9494}, {1084, 23099}
X(669) = crosspoint of X(i) and X(j) for these (i,j): {6, 99}, {32, 1576}, {110, 251}, {213, 4557}, {512, 2489}, {667, 1919}, {691, 15387}, {14573, 14574}, {32729, 41936}
X(669) = crosssum of X(i) and X(j) for these (i,j): {1, 18197}, {2, 512}, {6, 21006}, {69, 3267}, {75, 7199}, {76, 850}, {99, 4563}, {110, 4611}, {126, 690}, {141, 523}, {274, 7192}, {513, 3739}, {514, 3741}, {520, 6389}, {522, 21246}, {525, 1368}, {650, 42397}, {668, 1978}, {669, 9490}, {693, 20911}, {740, 27854}, {799, 7257}, {804, 39080}, {826, 21248}, {888, 35073}, {1269, 3261}, {2396, 15631}, {2786, 20339}, {3051, 9491}, {3221, 6374}, {3794, 4560}, {6333, 32458}, {8672, 10472}, {9402, 34021}, {23285, 42554}, {23610, 31646}, {35522, 36792}
X(669) = trilinear pole of line {887, 1084}
X(669) = crossdifference of every pair of points on line {2, 39}
X(669) = circumcircle-pole of line X(2)X(6)
X(669) = trilinear pole of line X(887)X(1084)
X(669) = complement of isotomic conjugate of isogonal conjugate of X(21006)
X(669) = perspector of hyperbola {A,B,C,X(6),X(32)}}, which is the locus of barycentric product of circumcircle-X(512)-antipodes
X(669) = perspector of the tangential triangle and the tangential triangle, wrt the tangential triangle, of the Stammler hyperbola
X(669) = X(650)-of-the-tangential-triangle
X(669) = radical center of {circumcircle, 1st Brocard circle, 1st anti-Brocard circle}
X(669) = radical center of {circumcircle, anti-Brocard circle, anti-McCay circumcircle}
X(669) = trilinear pole of PU(91)
X(669) = barycentric product of PU(105)
X(669) = bicentric difference of PU(148)
X(669) = PU(148)-harmonic conjugate of X(2)
X(669) = pole wrt circumcircle of line X(2)X(6)
X(669) = complement of polar conjugate of isogonal conjugate of X(22159)
X(669) = X(i)-isoconjugate of X(j) for these (i,j): {1, 670}, {2, 799}, {6, 4602}, {7, 7257}, {8, 4625}, {10, 4623}, {21, 4572}, {31, 4609}, {38, 689}, {39, 37204}, {58, 6386}, {63, 6331}, {69, 811}, {75, 99}, {76, 662}, {81, 1978}, {85, 645}, {86, 668}, {92, 4563}, {100, 310}, {101, 6385}, {109, 40072}, {110, 561}, {112, 40364}, {141, 4593}, {162, 305}, {163, 1502}, {190, 274}, {226, 4631}, {239, 4639}, {249, 20948}, {264, 4592}, {279, 7258}, {286, 4561}, {304, 648}, {312, 4573}, {314, 664}, {321, 4610}, {325, 36036}, {326, 6528}, {330, 36860}, {332, 18026}, {333, 4554}, {336, 877}, {341, 4616}, {346, 4635}, {349, 4612}, {350, 4589}, {514, 4601}, {519, 4634}, {523, 24037}, {525, 46254}, {643, 6063}, {646, 1434}, {651, 28660}, {661, 34537}, {666, 18157}, {671, 24039}, {693, 4600}, {757, 27808}, {789, 30966}, {798, 44168}, {805, 1926}, {823, 3926}, {850, 24041}, {873, 3952}, {874, 18827}, {880, 1581}, {886, 2234}, {892, 14210}, {894, 7260}, {1016, 7199}, {1019, 31625}, {1043, 4569}, {1088, 7256}, {1101, 44173}, {1102, 15352}, {1109, 31614}, {1269, 4596}, {1310, 44154}, {1332, 44129}, {1414, 3596}, {1509, 4033}, {1576, 1928}, {1577, 4590}, {1634, 18833}, {1821, 2396}, {1909, 4594}, {1920, 4603}, {1921, 4584}, {1930, 4577}, {1934, 17941}, {1959, 43187}, {1964, 42371}, {1966, 18829}, {1969, 4558}, {2407, 33805}, {2421, 46273}, {2617, 34384}, {2966, 46238}, {3112, 4576}, {3116, 9063}, {3222, 17149}, {3261, 4567}, {3264, 4622}, {3265, 23999}, {3266, 36085}, {3570, 40017}, {3736, 46132}, {3882, 40827}, {3978, 37134}, {4077, 6064}, {4086, 7340}, {4176, 36126}, {4358, 4615}, {4359, 4632}, {4391, 4620}, {4481, 5388}, {4551, 18021}, {4555, 30939}, {4556, 27801}, {4562, 30940}, {4565, 28659}, {4570, 40495}, {4575, 18022}, {4583, 33295}, {4598, 31008}, {4599, 8024}, {4611, 46244}, {4633, 19804}, {4998, 18155}, {5209, 35147}, {5468, 46277}, {5546, 20567}, {6335, 17206}, {6516, 44130}, {6540, 16709}, {6632, 16727}, {7035, 7192}, {7182, 36797}, {7239, 7307}, {7799, 32680}, {8033, 27805}, {8707, 16739}, {10159, 18062}, {11059, 37216}, {11794, 33764}, {14208, 18020}, {15418, 35145}, {15455, 34016}, {16609, 36806}, {17446, 35567}, {17708, 20944}, {17871, 42297}, {17930, 20947}, {17932, 40703}, {17934, 18032}, {18023, 23889}, {18024, 23997}, {18140, 37205}, {18156, 35136}, {18206, 36803}, {18695, 18831}, {18750, 44326}, {18830, 33296}, {20641, 44766}, {20939, 37880}, {27853, 37128}, {30736, 36133}, {32676, 40050}, {33809, 36953}, {33946, 38810}, {34594, 40087}, {35544, 36066}, {37133, 40773}, {40088, 43076}, {44179, 46134}, {44769, 46234}
X(669) = barycentric product X(i)*X(j) for these {i,j}: {1, 798}, {3, 2489}, {4, 3049}, {6, 512}, {10, 1919}, {19, 810}, {25, 647}, {31, 661}, {32, 523}, {37, 667}, {39, 18105}, {41, 4017}, {42, 649}, {50, 15475}, {51, 2623}, {55, 7180}, {56, 3709}, {58, 4079}, {65, 3063}, {74, 14398}, {75, 1924}, {76, 9426}, {82, 2084}, {83, 688}, {98, 2491}, {99, 1084}, {100, 3121}, {101, 3122}, {106, 14407}, {110, 3124}, {111, 351}, {112, 20975}, {115, 1576}, {163, 2643}, {181, 7252}, {184, 2501}, {187, 9178}, {213, 513}, {220, 7250}, {228, 6591}, {232, 878}, {237, 2395}, {248, 17994}, {249, 22260}, {251, 3005}, {263, 3288}, {292, 4455}, {308, 9494}, {321, 1980}, {338, 14574}, {385, 881}, {393, 39201}, {418, 15422}, {511, 2422}, {514, 1918}, {520, 2207}, {525, 1974}, {526, 11060}, {560, 1577}, {574, 46001}, {604, 4041}, {645, 1356}, {650, 1402}, {656, 1973}, {657, 1042}, {663, 1400}, {670, 9427}, {690, 32740}, {691, 21906}, {692, 3125}, {693, 2205}, {694, 5027}, {729, 888}, {739, 14404}, {755, 14428}, {788, 40747}, {799, 4117}, {804, 9468}, {805, 2086}, {822, 1096}, {826, 46288}, {842, 6041}, {843, 9171}, {850, 1501}, {865, 9091}, {872, 1019}, {875, 2238}, {876, 41333}, {879, 2211}, {882, 1691}, {887, 3228}, {893, 7234}, {922, 23894}, {923, 2642}, {932, 21835}, {941, 8639}, {1015, 4557}, {1018, 3248}, {1027, 39258}, {1106, 4171}, {1169, 42661}, {1171, 8663}, {1245, 2484}, {1252, 8034}, {1253, 7216}, {1333, 4705}, {1334, 43924}, {1383, 17414}, {1395, 8611}, {1397, 3700}, {1407, 4524}, {1409, 18344}, {1415, 4516}, {1427, 8641}, {1459, 2333}, {1495, 2433}, {1499, 39238}, {1500, 3733}, {1637, 40352}, {1645, 9150}, {1648, 32729}, {1649, 41936}, {1692, 35364}, {1824, 22383}, {1880, 1946}, {1911, 21832}, {1917, 20948}, {1922, 4010}, {1923, 18070}, {1976, 3569}, {1977, 3952}, {1989, 14270}, {2028, 41881}, {2029, 41880}, {2054, 5029}, {2088, 14560}, {2165, 34952}, {2175, 7178}, {2179, 2616}, {2200, 7649}, {2206, 4024}, {2281, 8678}, {2334, 4832}, {2351, 6753}, {2353, 2485}, {2394, 9407}, {2421, 15630}, {2451, 9292}, {2492, 3455}, {2519, 15369}, {2533, 7104}, {2715, 44114}, {2799, 14601}, {2971, 4558}, {2987, 42663}, {2998, 9491}, {3050, 27375}, {3108, 8664}, {3114, 9006}, {3120, 32739}, {3199, 23286}, {3221, 3224}, {3223, 23503}, {3225, 9429}, {3265, 36417}, {3267, 44162}, {3269, 32713}, {3271, 4559}, {3407, 17415}, {3444, 42653}, {3447, 8574}, {3457, 6137}, {3458, 6138}, {3572, 3747}, {3690, 43925}, {3708, 32676}, {3804, 39951}, {3903, 21755}, {4049, 9459}, {4077, 9447}, {4083, 21759}, {4155, 18268}, {4367, 40729}, {4563, 42068}, {4573, 7063}, {4574, 42067}, {4580, 27369}, {4590, 23099}, {4630, 39691}, {4729, 38266}, {4730, 9456}, {4770, 28607}, {4775, 28658}, {4834, 28625}, {4983, 28615}, {5106, 14606}, {5113, 46286}, {5191, 14998}, {5291, 18002}, {5466, 14567}, {5489, 41937}, {5638, 5639}, {6140, 14579}, {6187, 21828}, {6331, 23216}, {6378, 16695}, {6524, 32320}, {6529, 34980}, {6531, 39469}, {6562, 10318}, {6587, 33581}, {6784, 26714}, {7109, 7192}, {7121, 21834}, {7212, 18265}, {7255, 21815}, {8029, 23357}, {8061, 46289}, {8105, 42667}, {8106, 42668}, {8541, 30491}, {8640, 16606}, {8644, 21448}, {8651, 8770}, {8665, 39955}, {8749, 9409}, {8754, 32661}, {8789, 14295}, {8791, 42659}, {8882, 15451}, {8884, 42293}, {9033, 40354}, {9218, 19610}, {9233, 44173}, {9247, 24006}, {9297, 27810}, {9299, 21893}, {9402, 40770}, {9418, 43665}, {9462, 9489}, {10097, 44102}, {10412, 19627}, {10566, 41267}, {11166, 11186}, {14273, 14908}, {14380, 14581}, {14569, 46088}, {14573, 18314}, {14575, 14618}, {14582, 34397}, {14586, 41221}, {14593, 30451}, {14599, 35352}, {14600, 16230}, {14642, 44705}, {14910, 21731}, {15321, 37085}, {15387, 21905}, {15412, 40981}, {17735, 18001}, {17961, 17989}, {17962, 17990}, {17963, 17992}, {17965, 17995}, {17966, 18000}, {18108, 21814}, {19626, 35522}, {20578, 34394}, {20579, 34395}, {20683, 43929}, {20979, 23493}, {21758, 34857}, {21830, 23355}, {22105, 41272}, {22227, 29227}, {23105, 23963}, {23610, 34537}, {23878, 46319}, {27374, 39182}, {30442, 32319}, {30735, 40823}, {32641, 42752}, {32696, 41172}, {33294, 40146}, {34067, 39786}, {34079, 42666}, {34212, 42671}, {34288, 42660}, {41489, 42658}
X(669) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 4602}, {2, 4609}, {6, 670}, {25, 6331}, {31, 799}, {32, 99}, {37, 6386}, {41, 7257}, {42, 1978}, {82, 37204}, {83, 42371}, {99, 44168}, {110, 34537}, {115, 44173}, {163, 24037}, {184, 4563}, {213, 668}, {237, 2396}, {251, 689}, {351, 3266}, {512, 76}, {513, 6385}, {523, 1502}, {525, 40050}, {560, 662}, {604, 4625}, {647, 305}, {649, 310}, {650, 40072}, {656, 40364}, {661, 561}, {663, 28660}, {667, 274}, {688, 141}, {692, 4601}, {729, 886}, {798, 75}, {804, 14603}, {810, 304}, {850, 40362}, {872, 4033}, {875, 40017}, {881, 1916}, {882, 18896}, {887, 538}, {888, 30736}, {904, 7260}, {922, 24039}, {1084, 523}, {1106, 4635}, {1253, 7258}, {1333, 4623}, {1356, 7178}, {1397, 4573}, {1400, 4572}, {1402, 4554}, {1500, 27808}, {1501, 110}, {1576, 4590}, {1577, 1928}, {1645, 9148}, {1691, 880}, {1911, 4639}, {1917, 163}, {1918, 190}, {1919, 86}, {1922, 4589}, {1924, 1}, {1927, 37134}, {1973, 811}, {1974, 648}, {1976, 43187}, {1977, 7192}, {1980, 81}, {2084, 1930}, {2086, 14295}, {2175, 645}, {2194, 4631}, {2200, 4561}, {2205, 100}, {2206, 4610}, {2207, 6528}, {2209, 36860}, {2211, 877}, {2395, 18024}, {2422, 290}, {2484, 44154}, {2485, 40073}, {2489, 264}, {2491, 325}, {2492, 40074}, {2501, 18022}, {2531, 7794}, {2623, 34384}, {2643, 20948}, {2971, 14618}, {3005, 8024}, {3049, 69}, {3050, 33769}, {3051, 4576}, {3063, 314}, {3121, 693}, {3122, 3261}, {3124, 850}, {3125, 40495}, {3221, 6374}, {3248, 7199}, {3249, 17205}, {3267, 40360}, {3288, 20023}, {3407, 9063}, {3700, 40363}, {3709, 3596}, {3747, 27853}, {3774, 4505}, {3804, 40022}, {4010, 44169}, {4017, 20567}, {4041, 28659}, {4079, 313}, {4117, 661}, {4455, 1921}, {4557, 31625}, {4705, 27801}, {4826, 30596}, {5027, 3978}, {7063, 3700}, {7104, 4594}, {7109, 3952}, {7178, 41283}, {7180, 6063}, {7234, 1920}, {7252, 18021}, {8027, 16727}, {8029, 23962}, {8034, 23989}, {8630, 40773}, {8639, 34284}, {8640, 31008}, {8644, 11059}, {8660, 16711}, {8663, 1230}, {8664, 39998}, {8789, 805}, {9006, 3094}, {9171, 45809}, {9178, 18023}, {9233, 1576}, {9247, 4592}, {9407, 2407}, {9418, 2421}, {9419, 15631}, {9426, 6}, {9427, 512}, {9429, 698}, {9431, 9428}, {9447, 643}, {9448, 5546}, {9456, 4634}, {9468, 18829}, {9489, 7757}, {9491, 194}, {9494, 39}, {11060, 35139}, {14270, 7799}, {14295, 18901}, {14398, 3260}, {14404, 35543}, {14407, 3264}, {14428, 35549}, {14567, 5468}, {14573, 18315}, {14574, 249}, {14575, 4558}, {14598, 4584}, {14600, 17932}, {14601, 2966}, {14602, 17941}, {14604, 17938}, {14618, 44161}, {14827, 7256}, {15451, 28706}, {15475, 20573}, {15630, 43665}, {17414, 9464}, {17415, 3314}, {17938, 39292}, {17994, 44132}, {18105, 308}, {19626, 691}, {19627, 10411}, {20968, 4611}, {20975, 3267}, {21051, 40367}, {21751, 3888}, {21755, 4374}, {21759, 18830}, {21762, 17217}, {21828, 40075}, {21832, 18891}, {21835, 20906}, {21837, 33932}, {21906, 35522}, {22096, 15419}, {22260, 338}, {23099, 115}, {23216, 647}, {23357, 31614}, {23503, 17149}, {23610, 3124}, {25423, 10010}, {27369, 41676}, {32320, 4176}, {32676, 46254}, {32696, 41174}, {32739, 4600}, {32740, 892}, {33581, 44326}, {33875, 23342}, {34952, 7763}, {34980, 4143}, {35352, 44170}, {36417, 107}, {37085, 7768}, {38996, 44445}, {39201, 3926}, {39238, 35179}, {39469, 6393}, {40146, 44766}, {40351, 1304}, {40354, 16077}, {40373, 32661}, {40747, 46132}, {40823, 35575}, {40935, 33946}, {40981, 14570}, {41221, 15415}, {41267, 4568}, {41280, 4565}, {41331, 1634}, {41333, 874}, {41993, 20578}, {41994, 20579}, {42068, 2501}, {42659, 37804}, {42660, 32833}, {42661, 1228}, {44112, 15418}, {44162, 112}, {44173, 40359}, {46001, 40826}, {46104, 42395}, {46288, 4577}, {46289, 4593}
X(669) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 23301, 31279}, {2, 26148, 31003}, {2, 31299, 44445}, {2, 44445, 23301}, {75, 21614, 21441}, {351, 3005, 647}, {351, 3804, 8665}, {351, 8664, 3005}, {351, 42652, 3231}, {647, 3005, 17414}, {647, 8644, 8651}, {647, 8651, 351}, {647, 8664, 8665}, {649, 663, 2978}, {649, 8640, 890}, {649, 8655, 667}, {667, 8640, 649}, {667, 8641, 8646}, {667, 8642, 8654}, {887, 9489, 9491}, {887, 9494, 9489}, {1995, 44823, 8371}, {3804, 8644, 647}, {3804, 8651, 3005}, {4057, 23865, 21005}, {4455, 7234, 661}, {5638, 5639, 5027}, {5996, 15724, 11176}, {6137, 6138, 9210}, {7492, 9168, 44822}, {8644, 42659, 34952}, {8651, 8664, 17414}, {8665, 17414, 3005}, {14270, 42660, 39201}, {18773, 18774, 887}, {20979, 23655, 20983}, {23301, 44445, 31176}, {23301, 44451, 2}, {23301, 45317, 44451}, {24533, 25537, 2}, {24533, 28401, 31003}, {25473, 25686, 2}, {27016, 27077, 2}, {31176, 31279, 23301}, {31299, 44451, 31176}, {31299, 45317, 31279}, {34952, 42660, 14270}, {42667, 42668, 351}, {44445, 44451, 31279}


X(670) = TRILINEAR POLE OF LINE X(2)X(39)

Trilinears    1/[a3(b2 - c2)] : :
Barycentrics    (a^2 - b^2)*b^2*(a^2 - c^2)*c^2 : :
X(670) = 6 X[2] - 5 X[31639], 3 X[2] - 4 X[36950], 9 X[2] - 8 X[40478], X[193] - 3 X[25319], 2 X[886] - 3 X[9428], X[886] - 3 X[44168], 4 X[1084] - 3 X[3228], 4 X[1084] - 5 X[31639], X[1084] - 3 X[35073], 2 X[1084] + 3 X[39361], 3 X[1084] - 4 X[40478], 3 X[3228] - 2 X[25054], 3 X[3228] - 5 X[31639], X[3228] - 4 X[35073], 3 X[3228] - 8 X[36950], X[3228] + 2 X[39361], 9 X[3228] - 16 X[40478], 5 X[3618] - 3 X[25318], 2 X[25054] - 5 X[31639], X[25054] - 6 X[35073], X[25054] - 4 X[36950], X[25054] + 3 X[39361], 3 X[25054] - 8 X[40478], 5 X[31639] - 12 X[35073], 5 X[31639] - 8 X[36950], 5 X[31639] + 6 X[39361], 15 X[31639] - 16 X[40478], 3 X[35073] - 2 X[36950], 2 X[35073] + X[39361], 9 X[35073] - 4 X[40478], 4 X[36950] + 3 X[39361], 3 X[36950] - 2 X[40478], 9 X[39361] + 8 X[40478]

X(670) lies on the Steiner circumellipse and these lines: {1, 18826}, {2, 1084}, {4, 36892}, {6, 3225}, {7, 35159}, {30, 16084}, {37, 34021}, {39, 43094}, {58, 18824}, {67, 7768}, {69, 290}, {75, 18827}, {76, 338}, {81, 18825}, {85, 35176}, {86, 3226}, {99, 804}, {110, 689}, {112, 35567}, {126, 14948}, {141, 308}, {183, 43664}, {190, 799}, {192, 34022}, {193, 25319}, {264, 35142}, {274, 3227}, {300, 11118}, {301, 11117}, {304, 35145}, {305, 1494}, {310, 903}, {312, 35144}, {313, 35162}, {314, 2481}, {315, 2882}, {316, 34171}, {320, 18891}, {325, 46142}, {350, 35166}, {351, 23356}, {385, 35524}, {393, 6338}, {511, 14603}, {512, 886}, {523, 18829}, {524, 3978}, {536, 30938}, {561, 14616}, {645, 666}, {646, 32041}, {648, 2421}, {660, 37204}, {662, 4586}, {664, 4572}, {668, 4553}, {683, 41521}, {693, 35147}, {702, 40858}, {805, 9063}, {827, 6572}, {850, 892}, {873, 40087}, {877, 6528}, {888, 4576}, {889, 7192}, {925, 42297}, {931, 35565}, {1086, 40017}, {1121, 28660}, {1218, 10472}, {1241, 21248}, {1350, 8920}, {1368, 16098}, {1509, 40034}, {1576, 17941}, {1613, 19562}, {1921, 35173}, {1965, 24425}, {2086, 13518}, {2396, 14570}, {2407, 9211}, {2966, 3267}, {2979, 40362}, {2998, 32746}, {3229, 6379}, {3260, 5641}, {3261, 35148}, {3262, 35151}, {3263, 35152}, {3264, 35153}, {3266, 7840}, {3589, 25326}, {3596, 35141}, {3618, 25318}, {3888, 46132}, {4033, 4562}, {4108, 9066}, {4360, 34086}, {4361, 6383}, {4363, 6382}, {4495, 5209}, {4554, 32038}, {4555, 4634}, {4573, 6648}, {4589, 41072}, {4616, 6613}, {5152, 7669}, {5468, 35138}, {6389, 42407}, {6393, 44137}, {6540, 27808}, {6606, 7256}, {7253, 14727}, {7779, 35540}, {8024, 43098}, {8264, 32747}, {9766, 14772}, {10008, 44144}, {10010, 14937}, {11059, 11163}, {12220, 40360}, {14221, 35139}, {14574, 33515}, {14615, 35140}, {14728, 23105}, {15164, 46166}, {15165, 46167}, {15526, 32458}, {16741, 35155}, {17731, 35165}, {18022, 44134}, {18026, 46152}, {18157, 35167}, {18816, 40072}, {18822, 30939}, {19585, 41331}, {19623, 35531}, {19643, 20345}, {20911, 40827}, {21001, 40162}, {22456, 35575}, {23285, 31998}, {24039, 35180}, {24524, 27891}, {25534, 32020}, {26714, 35566}, {30737, 46145}, {30940, 35172}, {30966, 43096}, {33297, 43093}, {33875, 44371}, {34064, 34088}, {34203, 35179}, {34384, 46138}, {34393, 46359}, {35149, 35516}, {35150, 35517}, {35154, 35519}, {35156, 40495}, {39939, 39968}, {40773, 43095}

X(670) = midpoint of X(i) and X(j) for these {i,j}: {2, 39361}, {69, 25332}
X(670) = reflection of X(i) in X(j) for these {i,j}: {2, 35073}, {6, 39080}, {694, 141}, {1084, 36950}, {3228, 2}, {3978, 30736}, {9428, 44168}, {14948, 126}, {16098, 1368}, {25054, 1084}, {25326, 3589}, {40874, 30938}
X(670) = isogonal conjugate of X(669)
X(670) = isotomic conjugate of X(512)
X(670) = polar conjugate of X(2489)
X(670) = antigonal image of X(14948)
X(670) = cyclocevian conjugate of X(35511)
X(670) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {99, 21220}, {249, 17486}, {662, 25054}, {670, 21221}, {799, 148}, {1101, 8264}, {4590, 192}, {4593, 25047}, {4600, 1655}, {4601, 1654}, {4602, 3448}, {4609, 21294}, {4610, 9263}, {4623, 4440}, {4631, 39351}, {6064, 3177}, {7340, 3210}, {18020, 21216}, {23995, 40382}, {23999, 6392}, {24037, 2}, {24039, 39356}, {24041, 194}, {31614, 4560}, {34537, 8}, {37204, 25051}, {39292, 17493}, {44168, 6327}, {46254, 193}
X(670) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 9428}, {46274, 2887}
X(670) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 9428}, {689, 99}, {4590, 33769}, {24037, 18021}, {34537, 76}, {42371, 4609}, {43187, 880}, {44168, 2}
X(670) = X(i)-cross conjugate of X(j) for these (i,j): {2, 44168}, {69, 4590}, {75, 31625}, {76, 34537}, {99, 6331}, {314, 4601}, {315, 18020}, {511, 39292}, {512, 2}, {520, 42407}, {523, 308}, {668, 799}, {688, 39968}, {693, 40827}, {826, 1241}, {850, 76}, {874, 36803}, {888, 3228}, {1370, 23582}, {1510, 42332}, {1978, 4602}, {2979, 249}, {3221, 6}, {3265, 40830}, {3267, 1502}, {3268, 40832}, {3766, 40017}, {3888, 662}, {4374, 75}, {4427, 31624}, {4576, 99}, {6088, 18818}, {6373, 37128}, {6563, 276}, {7192, 274}, {7950, 1239}, {8672, 1218}, {8711, 3108}, {9402, 37}, {12220, 250}, {14221, 6035}, {14295, 290}, {15631, 2396}, {16175, 34539}, {17135, 1016}, {17137, 4998}, {17143, 7035}, {17159, 1268}, {17166, 30710}, {17217, 86}, {20245, 1275}, {20295, 32014}, {21295, 15455}, {21300, 333}, {21301, 14534}, {21305, 40415}, {30737, 41174}, {31296, 44165}, {33297, 4600}, {35522, 18023}, {35614, 4564}, {42331, 42333}, {44173, 34384}, {44445, 83}
X(670) = cevapoint of X(i) and X(j) for these (i,j): {1, 18197}, {2, 512}, {6, 21006}, {69, 3267}, {75, 7199}, {76, 850}, {99, 4563}, {110, 4611}, {126, 690}, {141, 523}, {274, 7192}, {513, 3739}, {514, 3741}, {520, 6389}, {522, 21246}, {525, 1368}, {650, 42397}, {668, 1978}, {669, 9490}, {693, 20911}, {740, 27854}, {799, 7257}, {804, 39080}, {826, 21248}, {888, 35073}, {1269, 3261}, {2396, 15631}, {2786, 20339}, {3051, 9491}, {3221, 6374}, {3794, 4560}, {6333, 32458}, {8672, 10472}, {9402, 34021}, {23285, 42554}, {23610, 31646}, {35522, 36792}
X(670) = crosspoint of X(i) and X(j) for these (i,j): {2, 46274}, {99, 3222}, {689, 42371}
X(670) = crosssum of X(i) and X(j) for these (i,j): {6, 9431}, {512, 3221}, {688, 9494}, {1084, 23099}
X(670) = trilinear pole of line {2, 39}
X(670) = crossdifference of every pair of points on line {887, 1084}
X(670) = Steiner-circumellipse-antipode of X(3228)
X(670) = Steiner-circumellipse-X(1)-antipode of X(18826)
X(670) = Steiner-circumellipse-X(6)-antipode of X(3225)
X(670) = X(i)-isoconjugate of X(j) for these (i,j): {1, 669}, {2, 1924}, {6, 798}, {10, 1980}, {19, 3049}, {25, 810}, {31, 512}, {32, 661}, {37, 1919}, {41, 7180}, {42, 667}, {48, 2489}, {75, 9426}, {82, 688}, {99, 4117}, {101, 3121}, {163, 3124}, {213, 649}, {251, 2084}, {351, 923}, {513, 1918}, {514, 2205}, {523, 560}, {604, 3709}, {643, 1356}, {647, 1973}, {656, 1974}, {662, 1084}, {663, 1402}, {692, 3122}, {799, 9427}, {804, 1927}, {811, 23216}, {822, 2207}, {850, 1917}, {872, 3733}, {875, 3747}, {881, 1580}, {882, 1933}, {887, 37132}, {904, 7234}, {922, 9178}, {1018, 1977}, {1019, 7109}, {1042, 8641}, {1096, 39201}, {1101, 22260}, {1106, 4524}, {1109, 14574}, {1110, 8034}, {1245, 8646}, {1253, 7250}, {1333, 4079}, {1397, 4041}, {1400, 3063}, {1414, 7063}, {1501, 1577}, {1576, 2643}, {1645, 36133}, {1755, 2422}, {1910, 2491}, {1911, 4455}, {1922, 21832}, {1964, 18105}, {1967, 5027}, {2159, 14398}, {2175, 4017}, {2179, 2623}, {2200, 6591}, {2206, 4705}, {2258, 8639}, {2281, 2484}, {2333, 22383}, {2395, 9417}, {2433, 9406}, {2501, 9247}, {2616, 40981}, {2618, 14573}, {2624, 11060}, {2631, 40354}, {2642, 32740}, {2971, 4575}, {3005, 46289}, {3112, 9494}, {3113, 9006}, {3125, 32739}, {3221, 34248}, {3223, 9491}, {3224, 23503}, {3248, 4557}, {3288, 3402}, {3572, 41333}, {4010, 14598}, {4077, 9448}, {4086, 41280}, {4592, 42068}, {4826, 34819}, {7178, 9447}, {7216, 14827}, {8029, 23995}, {8061, 46288}, {8630, 40718}, {8640, 23493}, {8651, 38252}, {9233, 20948}, {9429, 43761}, {9456, 14407}, {14208, 44162}, {14567, 23894}, {14575, 24006}, {15630, 23997}, {18001, 18266}, {18070, 41331}, {18108, 41267}, {18892, 35352}, {20975, 32676}, {20979, 21759}, {20981, 40729}, {21835, 34071}, {21906, 36142}, {23099, 24041}, {23610, 24037}, {24018, 36417}, {36051, 42663}, {39258, 43929}
X(670) = barycentric product X(i)*X(j) for these {i,j}: {1, 4602}, {6, 4609}, {38, 37204}, {39, 42371}, {69, 6331}, {75, 799}, {76, 99}, {81, 6386}, {85, 7257}, {86, 1978}, {100, 6385}, {110, 1502}, {112, 40050}, {141, 689}, {162, 40364}, {163, 1928}, {190, 310}, {249, 44173}, {264, 4563}, {274, 668}, {290, 2396}, {304, 811}, {305, 648}, {308, 4576}, {312, 4625}, {313, 4610}, {314, 4554}, {321, 4623}, {325, 43187}, {333, 4572}, {338, 31614}, {341, 4635}, {350, 4639}, {512, 44168}, {523, 34537}, {538, 886}, {561, 662}, {643, 20567}, {645, 6063}, {651, 40072}, {664, 28660}, {693, 4601}, {805, 14603}, {850, 4590}, {873, 4033}, {874, 40017}, {880, 1916}, {892, 3266}, {1088, 7258}, {1269, 4632}, {1414, 28659}, {1441, 4631}, {1509, 27808}, {1576, 40362}, {1577, 24037}, {1634, 40016}, {1909, 7260}, {1920, 4594}, {1921, 4589}, {1926, 37134}, {1930, 4593}, {1969, 4592}, {2421, 18024}, {3094, 9063}, {3096, 6572}, {3222, 6374}, {3261, 4600}, {3264, 4615}, {3267, 18020}, {3596, 4573}, {3926, 6528}, {3978, 18829}, {4176, 15352}, {4358, 4634}, {4552, 18021}, {4558, 18022}, {4561, 44129}, {4565, 40363}, {4567, 40495}, {4577, 8024}, {4583, 30940}, {4584, 18891}, {4611, 40421}, {4620, 35519}, {5468, 18023}, {5546, 41283}, {6333, 41174}, {6384, 36860}, {6393, 22456}, {6656, 35567}, {7035, 7199}, {7192, 31625}, {7763, 46134}, {7769, 46139}, {7788, 9211}, {7799, 35139}, {9146, 40826}, {9150, 30736}, {9170, 45809}, {9428, 46274}, {9464, 35138}, {9865, 41073}, {10010, 25424}, {10411, 20573}, {11059, 35179}, {11794, 33769}, {14208, 46254}, {14295, 39292}, {14570, 34384}, {14574, 40359}, {14615, 44326}, {17708, 40074}, {17932, 44132}, {17938, 18901}, {17941, 18896}, {18827, 27853}, {18830, 31008}, {18831, 28706}, {20775, 42395}, {20948, 24041}, {23342, 34087}, {24039, 46277}, {26235, 42367}, {30941, 36803}, {30966, 37133}, {31624, 33297}, {32661, 44161}, {35137, 39998}, {35540, 41209}, {35575, 40822}, {36036, 46238}, {36841, 41530}, {37205, 40087}, {37215, 44154}, {40073, 44766}, {40773, 46132}, {41760, 42297}
X(670) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 798}, {2, 512}, {3, 3049}, {4, 2489}, {6, 669}, {7, 7180}, {8, 3709}, {10, 4079}, {21, 3063}, {30, 14398}, {31, 1924}, {32, 9426}, {38, 2084}, {39, 688}, {58, 1919}, {63, 810}, {69, 647}, {75, 661}, {76, 523}, {81, 667}, {83, 18105}, {85, 4017}, {86, 649}, {94, 15475}, {95, 2623}, {98, 2422}, {99, 6}, {100, 213}, {101, 1918}, {107, 2207}, {110, 32}, {112, 1974}, {115, 22260}, {126, 21905}, {141, 3005}, {162, 1973}, {163, 560}, {183, 3288}, {190, 42}, {193, 8651}, {194, 3221}, {230, 42663}, {239, 4455}, {249, 1576}, {261, 7252}, {264, 2501}, {274, 513}, {279, 7250}, {286, 6591}, {287, 878}, {290, 2395}, {297, 17994}, {298, 6137}, {299, 6138}, {300, 20578}, {301, 20579}, {304, 656}, {305, 525}, {310, 514}, {311, 12077}, {312, 4041}, {313, 4024}, {314, 650}, {315, 2485}, {316, 2492}, {317, 6753}, {320, 21828}, {321, 4705}, {323, 14270}, {325, 3569}, {326, 822}, {328, 14582}, {332, 652}, {333, 663}, {338, 8029}, {341, 4171}, {343, 15451}, {346, 4524}, {350, 21832}, {385, 5027}, {391, 8653}, {394, 39201}, {476, 11060}, {511, 2491}, {512, 1084}, {513, 3121}, {514, 3122}, {519, 14407}, {523, 3124}, {524, 351}, {525, 20975}, {536, 14404}, {538, 888}, {542, 6041}, {543, 9171}, {561, 1577}, {598, 46001}, {599, 17414}, {643, 41}, {645, 55}, {646, 210}, {648, 25}, {651, 1402}, {658, 1042}, {662, 31}, {664, 1400}, {668, 37}, {669, 9427}, {671, 9178}, {689, 83}, {690, 21906}, {691, 32740}, {692, 2205}, {693, 3125}, {694, 881}, {754, 14428}, {789, 40747}, {798, 4117}, {799, 1}, {804, 2086}, {805, 9468}, {811, 19}, {823, 1096}, {827, 46288}, {850, 115}, {873, 1019}, {874, 2238}, {877, 232}, {880, 385}, {886, 3228}, {888, 1645}, {892, 111}, {894, 7234}, {932, 21759}, {940, 8639}, {1010, 2484}, {1016, 4557}, {1018, 872}, {1019, 3248}, {1026, 39258}, {1043, 657}, {1078, 3050}, {1084, 23610}, {1086, 8034}, {1088, 7216}, {1211, 42661}, {1213, 8663}, {1230, 6367}, {1269, 4988}, {1273, 2081}, {1296, 39238}, {1304, 40354}, {1306, 26454}, {1307, 26461}, {1310, 2281}, {1331, 2200}, {1332, 228}, {1333, 1980}, {1414, 604}, {1434, 43924}, {1444, 22383}, {1494, 2433}, {1502, 850}, {1509, 3733}, {1576, 1501}, {1577, 2643}, {1613, 9491}, {1625, 40981}, {1632, 42295}, {1633, 21750}, {1634, 3051}, {1645, 33918}, {1655, 9402}, {1698, 4826}, {1740, 23503}, {1812, 1946}, {1897, 2333}, {1916, 882}, {1920, 2533}, {1921, 4010}, {1928, 20948}, {1930, 8061}, {1969, 24006}, {1975, 2451}, {1978, 10}, {1992, 8644}, {1993, 34952}, {2287, 8641}, {2303, 8646}, {2395, 15630}, {2396, 511}, {2407, 1495}, {2420, 9407}, {2421, 237}, {2489, 42068}, {2501, 2971}, {2533, 21823}, {2617, 2179}, {2715, 14601}, {2799, 44114}, {2895, 42653}, {2966, 1976}, {3049, 23216}, {3051, 9494}, {3094, 17415}, {3117, 9006}, {3124, 23099}, {3222, 3224}, {3229, 9429}, {3231, 887}, {3233, 9408}, {3260, 1637}, {3261, 3120}, {3263, 24290}, {3264, 4120}, {3265, 3269}, {3266, 690}, {3267, 125}, {3268, 2088}, {3329, 14318}, {3448, 8574}, {3570, 3747}, {3573, 41333}, {3580, 21731}, {3589, 8664}, {3596, 3700}, {3616, 4832}, {3618, 3804}, {3681, 21837}, {3699, 1334}, {3709, 7063}, {3718, 8611}, {3732, 40934}, {3733, 1977}, {3741, 40627}, {3763, 8665}, {3766, 39786}, {3799, 3774}, {3882, 3725}, {3888, 16584}, {3903, 40729}, {3909, 40986}, {3926, 520}, {3936, 42666}, {3948, 4155}, {3952, 1500}, {3964, 32320}, {3978, 804}, {4033, 756}, {4036, 21833}, {4043, 21727}, {4083, 21835}, {4143, 2972}, {4226, 1692}, {4230, 2211}, {4235, 44102}, {4240, 14581}, {4358, 4730}, {4359, 4983}, {4367, 21755}, {4369, 4128}, {4374, 16592}, {4391, 4516}, {4397, 36197}, {4427, 20970}, {4436, 21753}, {4467, 20982}, {4552, 181}, {4553, 21814}, {4554, 65}, {4556, 2206}, {4557, 7109}, {4558, 184}, {4560, 3271}, {4561, 71}, {4563, 3}, {4565, 1397}, {4567, 692}, {4568, 21035}, {4569, 1427}, {4570, 32739}, {4572, 226}, {4573, 56}, {4575, 9247}, {4576, 39}, {4577, 251}, {4584, 1911}, {4585, 3724}, {4589, 292}, {4590, 110}, {4592, 48}, {4593, 82}, {4594, 893}, {4596, 28615}, {4597, 28658}, {4598, 23493}, {4599, 46289}, {4600, 101}, {4601, 100}, {4602, 75}, {4603, 904}, {4609, 76}, {4610, 58}, {4611, 206}, {4612, 2194}, {4615, 106}, {4616, 1407}, {4620, 109}, {4622, 9456}, {4623, 81}, {4625, 57}, {4631, 21}, {4632, 1126}, {4633, 2334}, {4634, 88}, {4635, 269}, {4637, 1106}, {4639, 291}, {4671, 4770}, {4998, 4559}, {5118, 33875}, {5224, 42664}, {5235, 4775}, {5333, 4834}, {5388, 4613}, {5467, 14567}, {5468, 187}, {5546, 2175}, {5562, 42293}, {5641, 14998}, {6035, 842}, {6063, 7178}, {6064, 5546}, {6189, 5638}, {6190, 5639}, {6331, 4}, {6333, 41172}, {6335, 1824}, {6374, 23301}, {6376, 21834}, {6382, 21051}, {6385, 693}, {6386, 321}, {6393, 684}, {6516, 1409}, {6528, 393}, {6542, 17990}, {6636, 37085}, {6650, 18001}, {6656, 2514}, {7035, 1018}, {7058, 21789}, {7180, 1356}, {7192, 1015}, {7199, 244}, {7253, 14936}, {7254, 22096}, {7256, 220}, {7257, 9}, {7258, 200}, {7259, 1253}, {7260, 256}, {7304, 16695}, {7307, 7255}, {7340, 4565}, {7757, 9009}, {7760, 21006}, {7763, 924}, {7769, 1510}, {7779, 5113}, {7788, 9210}, {7799, 526}, {7840, 9208}, {8024, 826}, {8033, 4367}, {8041, 2531}, {8115, 42667}, {8116, 42668}, {8781, 35364}, {8795, 15422}, {8842, 39680}, {9035, 865}, {9063, 3114}, {9146, 574}, {9150, 729}, {9170, 843}, {9182, 2502}, {9211, 14458}, {9293, 19610}, {9296, 21893}, {9428, 25054}, {9463, 9489}, {9464, 3906}, {9723, 30451}, {10009, 4806}, {10015, 42752}, {10330, 5007}, {10411, 50}, {10425, 32654}, {11054, 6088}, {11059, 1499}, {11064, 9409}, {11163, 11186}, {11794, 27375}, {13149, 1426}, {14208, 3708}, {14210, 2642}, {14221, 2493}, {14543, 40984}, {14568, 2872}, {14570, 51}, {14574, 9233}, {14586, 14573}, {14588, 20976}, {14590, 34397}, {14603, 14295}, {14607, 5106}, {14615, 6587}, {14618, 8754}, {14966, 9418}, {14999, 5191}, {15066, 42660}, {15164, 8106}, {15165, 8105}, {15352, 6524}, {15411, 3270}, {15413, 18210}, {15418, 851}, {15419, 3937}, {15423, 6754}, {15466, 44705}, {15631, 11672}, {16077, 8749}, {16084, 9035}, {16237, 44084}, {16695, 21762}, {16703, 2530}, {16704, 1960}, {16705, 6371}, {16709, 4979}, {16711, 6085}, {16712, 9002}, {16713, 2488}, {16726, 8027}, {16727, 764}, {16729, 3251}, {16741, 14419}, {16748, 6372}, {16887, 21123}, {17096, 1357}, {17103, 20981}, {17139, 3310}, {17140, 14991}, {17205, 21143}, {17206, 1459}, {17217, 6377}, {17402, 34394}, {17403, 34395}, {17708, 3455}, {17731, 5029}, {17790, 17989}, {17925, 42067}, {17929, 17961}, {17930, 17962}, {17931, 17963}, {17932, 248}, {17933, 17966}, {17934, 17735}, {17935, 5291}, {17936, 17967}, {17937, 17968}, {17938, 8789}, {17941, 1691}, {17946, 18002}, {17947, 18000}, {17948, 17993}, {17950, 17992}, {17952, 17999}, {17953, 17991}, {18020, 112}, {18021, 4560}, {18022, 14618}, {18023, 5466}, {18024, 43665}, {18026, 1880}, {18033, 7212}, {18047, 20964}, {18062, 17469}, {18135, 4139}, {18140, 4132}, {18143, 40471}, {18145, 4145}, {18152, 4151}, {18155, 2170}, {18157, 2254}, {18160, 2611}, {18197, 38986}, {18314, 41221}, {18600, 6363}, {18743, 4729}, {18827, 3572}, {18829, 694}, {18830, 16606}, {18831, 8882}, {18833, 18070}, {18837, 20910}, {18878, 14910}, {18895, 35352}, {18906, 45907}, {19583, 2519}, {19623, 5040}, {19804, 4822}, {20023, 23878}, {20477, 30442}, {20567, 4077}, {20573, 10412}, {20948, 1109}, {21006, 38996}, {21207, 21131}, {21272, 21796}, {21580, 4642}, {21604, 23928}, {21615, 4804}, {22151, 42659}, {22254, 20403}, {22329, 9135}, {22456, 6531}, {23092, 22386}, {23106, 33915}, {23181, 217}, {23285, 39691}, {23342, 3231}, {23354, 21830}, {23357, 14574}, {23582, 32713}, {23794, 22215}, {23878, 6784}, {23889, 922}, {23895, 3457}, {23896, 3458}, {23962, 23105}, {23997, 9417}, {23999, 24019}, {24004, 21805}, {24037, 662}, {24039, 896}, {24041, 163}, {25054, 38237}, {26235, 12073}, {26714, 46319}, {27164, 2978}, {27644, 8640}, {27801, 4036}, {27808, 594}, {27853, 740}, {27854, 38978}, {28659, 4086}, {28660, 522}, {28706, 6368}, {29226, 22227}, {30441, 32319}, {30450, 14593}, {30508, 2029}, {30509, 2028}, {30530, 37841}, {30596, 4838}, {30730, 7064}, {30736, 9148}, {30786, 10097}, {30939, 1635}, {30940, 659}, {30941, 665}, {30966, 3250}, {31008, 4083}, {31614, 249}, {31623, 18344}, {31624, 15320}, {31625, 3952}, {31632, 9218}, {31998, 20998}, {32036, 21461}, {32037, 21462}, {32042, 28625}, {32458, 41167}, {32661, 14575}, {32713, 36417}, {32715, 40351}, {32729, 19626}, {32833, 8675}, {32937, 22229}, {33295, 8632}, {33296, 20979}, {33297, 6586}, {33513, 33631}, {33514, 18898}, {33769, 31296}, {33799, 39024}, {33946, 3778}, {34016, 2605}, {34211, 42671}, {34245, 2030}, {34254, 8673}, {34284, 8672}, {34384, 15412}, {34386, 23286}, {34537, 99}, {34574, 41936}, {34760, 17964}, {35136, 8770}, {35137, 3108}, {35138, 1383}, {35139, 1989}, {35140, 34212}, {35146, 14606}, {35148, 2054}, {35179, 21448}, {35278, 40825}, {35319, 27374}, {35325, 27369}, {35356, 5008}, {35360, 3199}, {35519, 21044}, {35522, 1648}, {35538, 21053}, {35543, 14431}, {35549, 14420}, {35550, 2610}, {35575, 40799}, {36036, 1910}, {36066, 18268}, {36085, 923}, {36212, 39469}, {36792, 1649}, {36793, 5489}, {36797, 607}, {36803, 13576}, {36804, 34857}, {36827, 41272}, {36841, 154}, {36860, 43}, {36863, 20691}, {37128, 875}, {37133, 40718}, {37134, 1967}, {37204, 3112}, {37205, 40148}, {37215, 1245}, {37669, 42658}, {37779, 6140}, {37783, 42670}, {37804, 9517}, {37880, 9217}, {39290, 40355}, {39291, 34238}, {39292, 805}, {39295, 14560}, {39298, 44123}, {39299, 44124}, {39998, 7927}, {40017, 876}, {40022, 3800}, {40034, 31946}, {40050, 3267}, {40071, 4064}, {40072, 4391}, {40073, 33294}, {40074, 9979}, {40075, 4707}, {40087, 4129}, {40339, 40345}, {40362, 44173}, {40364, 14208}, {40495, 16732}, {40773, 788}, {40822, 30735}, {40826, 8599}, {40827, 4581}, {40832, 15328}, {40850, 17997}, {40866, 44127}, {40882, 5075}, {41174, 685}, {41209, 733}, {41259, 25423}, {41337, 32748}, {41488, 39182}, {41610, 8642}, {41629, 8643}, {41676, 1843}, {41678, 44079}, {41679, 44077}, {42308, 32695}, {42367, 39389}, {42370, 45773}, {42371, 308}, {42405, 8884}, {42717, 5360}, {42720, 20683}, {42721, 21839}, {43187, 98}, {43188, 9292}, {43754, 14600}, {44129, 7649}, {44130, 3064}, {44132, 16230}, {44133, 9209}, {44137, 6130}, {44146, 14273}, {44154, 6590}, {44173, 338}, {44326, 64}, {44327, 2357}, {44363, 42651}, {44372, 39527}, {44723, 44729}, {44766, 2353}, {44767, 39644}, {44769, 40352}, {45215, 40951}, {45792, 16186}, {45809, 8371}, {46134, 2165}, {46139, 2963}, {46148, 41267}, {46234, 36035}, {46254, 162}, {46277, 23894}
X(670) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 1084, 31639}, {2, 25054, 1084}, {69, 1502, 33769}, {76, 45809, 18023}, {141, 9230, 308}, {799, 36860, 190}, {1084, 25054, 3228}, {1084, 35073, 36950}, {1084, 36950, 2}, {3228, 31639, 1084}, {18896, 42844, 694}, {25054, 36950, 31639}, {35073, 39361, 3228}, {36792, 45809, 76}


X(671) = TRILINEAR POLE OF LINE X(2)X(523)

Trilinears    1/[a(2a2 - b2 - c2)] : :
Barycentrics    (a^2 + b^2 - 2*c^2)*(a^2 - 2*b^2 + c^2) : :
X(671) = 5 X[2] - 4 X[620], 3 X[2] - 4 X[5461], 7 X[2] - 8 X[6722], 3 X[2] + X[8596], 2 X[2] - 3 X[9166], 7 X[2] - 6 X[9167], 4 X[2] - 5 X[14061], 5 X[2] - 6 X[14971], 5 X[2] - 2 X[15300], 5 X[2] - X[20094], 9 X[2] - 8 X[22247], 11 X[2] - 10 X[31274], 13 X[2] - 8 X[35022], 7 X[2] + X[35369], 7 X[2] - 4 X[36521], X[2] - 4 X[36523], 4 X[2] - 3 X[41134], X[2] - 3 X[41135], X[2] + 8 X[41147], 23 X[2] - 32 X[41148], 7 X[2] - 16 X[41154], 2 X[4] + X[38664], X[4] - 4 X[38734], 4 X[5] - 3 X[23234], 4 X[5] - X[23235], 3 X[6] - 2 X[8787], 3 X[6] - X[10488], 4 X[6] - X[45018], X[20] - 4 X[11623], X[98] + 2 X[6321], 2 X[98] + X[10723], 3 X[98] - 2 X[14830], 5 X[98] - 2 X[38741], X[99] - 4 X[115], X[99] + 2 X[148], 5 X[99] - 8 X[620], 3 X[99] - 4 X[2482], 3 X[99] - 8 X[5461], 7 X[99] - 16 X[6722], 3 X[99] - 2 X[8591], 3 X[99] + 2 X[8596], X[99] - 3 X[9166], 7 X[99] - 12 X[9167], 2 X[99] - 5 X[14061], 5 X[99] - 12 X[14971], 5 X[99] - 4 X[15300], 5 X[99] - 2 X[20094], 9 X[99] - 16 X[22247], 11 X[99] - 20 X[31274], 13 X[99] - 16 X[35022], 7 X[99] + 2 X[35369], 7 X[99] - 8 X[36521], X[99] - 8 X[36523], 2 X[99] - 3 X[41134], X[99] - 6 X[41135], X[99] + 16 X[41147], 23 X[99] - 64 X[41148], 7 X[99] - 32 X[41154], 2 X[114] - 3 X[3545], 2 X[115] + X[148], 5 X[115] - 2 X[620], 3 X[115] - X[2482], 3 X[115] - 2 X[5461], 7 X[115] - 4 X[6722], 6 X[115] - X[8591], 6 X[115] + X[8596], 4 X[115] - 3 X[9166], 7 X[115] - 3 X[9167], 8 X[115] - 5 X[14061], 5 X[115] - 3 X[14971], 5 X[115] - X[15300], 10 X[115] - X[20094], 9 X[115] - 4 X[22247], 11 X[115] - 5 X[31274], 13 X[115] - 4 X[35022], 14 X[115] + X[35369], 7 X[115] - 2 X[36521], 8 X[115] - 3 X[41134], 2 X[115] - 3 X[41135], X[115] + 4 X[41147], 23 X[115] - 16 X[41148], 7 X[115] - 8 X[41154], X[147] - 3 X[3839], 5 X[148] + 4 X[620], 3 X[148] + 2 X[2482], 3 X[148] + 4 X[5461], 7 X[148] + 8 X[6722], 3 X[148] + X[8591], 3 X[148] - X[8596], 2 X[148] + 3 X[9166], 7 X[148] + 6 X[9167], 4 X[148] + 5 X[14061], 5 X[148] + 6 X[14971], 5 X[148] + 2 X[15300], 5 X[148] + X[20094], 9 X[148] + 8 X[22247], 11 X[148] + 10 X[31274], 13 X[148] + 8 X[35022], 7 X[148] - X[35369], 7 X[148] + 4 X[36521], X[148] + 4 X[36523], 4 X[148] + 3 X[41134], X[148] + 3 X[41135], X[148] - 8 X[41147], 23 X[148] + 32 X[41148], 7 X[148] + 16 X[41154], 2 X[187] - 3 X[8859], 4 X[230] - 3 X[26613], X[376] - 3 X[14651], 2 X[376] - 3 X[34473], 2 X[381] - 3 X[14639], 3 X[381] - 2 X[22566], X[381] - 3 X[38732], 5 X[381] - 3 X[38743], X[385] - 4 X[32457], 4 X[547] - 3 X[15561], 2 X[547] - 3 X[38229], 4 X[549] - 3 X[21166], 2 X[549] - 3 X[38224], 2 X[551] - 3 X[38220], 4 X[597] - 3 X[5182], 2 X[597] - 3 X[6034], 2 X[618] - 3 X[22490], 2 X[619] - 3 X[22489], 6 X[620] - 5 X[2482], 3 X[620] - 5 X[5461], 7 X[620] - 10 X[6722], 12 X[620] - 5 X[8591], 12 X[620] + 5 X[8596], 8 X[620] - 15 X[9166], 14 X[620] - 15 X[9167], 16 X[620] - 25 X[14061], 2 X[620] - 3 X[14971], 4 X[620] - X[20094], 9 X[620] - 10 X[22247], 22 X[620] - 25 X[31274], 13 X[620] - 10 X[35022], 28 X[620] + 5 X[35369], 7 X[620] - 5 X[36521], X[620] - 5 X[36523], 16 X[620] - 15 X[41134], 4 X[620] - 15 X[41135], X[620] + 10 X[41147], 23 X[620] - 40 X[41148], 7 X[620] - 20 X[41154], 5 X[631] - 2 X[10992], 5 X[631] - 8 X[20398], 2 X[892] - 3 X[39061], 7 X[2482] - 12 X[6722], 2 X[2482] + X[8596], 4 X[2482] - 9 X[9166], 7 X[2482] - 9 X[9167], 8 X[2482] - 15 X[14061], 5 X[2482] - 9 X[14971], 5 X[2482] - 3 X[15300], 10 X[2482] - 3 X[20094], 3 X[2482] - 4 X[22247], 11 X[2482] - 15 X[31274], 13 X[2482] - 12 X[35022], 14 X[2482] + 3 X[35369], 7 X[2482] - 6 X[36521], X[2482] - 6 X[36523], 8 X[2482] - 9 X[41134], 2 X[2482] - 9 X[41135], X[2482] + 12 X[41147], 23 X[2482] - 48 X[41148], 7 X[2482] - 24 X[41154], 5 X[3091] - 2 X[14981], X[3146] + 2 X[10991], 7 X[3523] - 10 X[38740], 3 X[3524] - 4 X[6036], 3 X[3524] - X[13172], 5 X[3618] - 2 X[14928], 7 X[3832] - 4 X[38745], X[4590] - 4 X[31644], 3 X[4590] - 4 X[44397], 3 X[5032] - 2 X[5477], 3 X[5054] - 2 X[33813], 3 X[5055] - X[13188], 11 X[5056] - 8 X[20399], 13 X[5067] - 10 X[38751], 5 X[5071] - 6 X[23514], 5 X[5076] + 4 X[38627], 13 X[5079] - 4 X[38628], 3 X[5215] - 2 X[32456], 2 X[5459] - 3 X[5470], 2 X[5459] + X[22578], 2 X[5460] - 3 X[5469], 2 X[5460] + X[22577], 7 X[5461] - 6 X[6722], 4 X[5461] - X[8591], 4 X[5461] + X[8596], 8 X[5461] - 9 X[9166], 14 X[5461] - 9 X[9167], 16 X[5461] - 15 X[14061], 10 X[5461] - 9 X[14971], 10 X[5461] - 3 X[15300], 20 X[5461] - 3 X[20094], 3 X[5461] - 2 X[22247], 22 X[5461] - 15 X[31274], 13 X[5461] - 6 X[35022], 28 X[5461] + 3 X[35369], 7 X[5461] - 3 X[36521], X[5461] - 3 X[36523], 16 X[5461] - 9 X[41134], 4 X[5461] - 9 X[41135], X[5461] + 6 X[41147], 23 X[5461] - 24 X[41148], 7 X[5461] - 12 X[41154], X[5463] - 3 X[5469], X[5464] - 3 X[5470], 3 X[5469] + X[22577], 3 X[5470] + X[22578], X[5984] + 2 X[39838], 4 X[6036] - X[13172], X[6054] - 3 X[14639], 3 X[6054] - 4 X[22566], X[6054] - 6 X[38732], 5 X[6054] - 6 X[38743], 2 X[6055] - 3 X[14651], 4 X[6055] - 3 X[34473], 2 X[6071] + X[31513], 4 X[6321] - X[10723], 3 X[6321] + X[14830], 5 X[6321] + X[38741], 2 X[6390] - 3 X[41133], 24 X[6722] - 7 X[8591], 24 X[6722] + 7 X[8596], 16 X[6722] - 21 X[9166], 4 X[6722] - 3 X[9167], 32 X[6722] - 35 X[14061], 20 X[6722] - 21 X[14971], 20 X[6722] - 7 X[15300], 40 X[6722] - 7 X[20094], 9 X[6722] - 7 X[22247], 44 X[6722] - 35 X[31274], 13 X[6722] - 7 X[35022], 8 X[6722] + X[35369], 2 X[6722] - 7 X[36523], 32 X[6722] - 21 X[41134], 8 X[6722] - 21 X[41135], X[6722] + 7 X[41147], 23 X[6722] - 28 X[41148], 3 X[7615] - X[9890], 3 X[7799] - 4 X[22110], 3 X[7809] - 2 X[7840], 3 X[7809] - 4 X[31173], X[7809] - 4 X[39563], 2 X[7813] - 3 X[41136], X[7840] - 3 X[14041], X[7840] - 6 X[39563], X[7983] - 4 X[11599], X[7983] + 2 X[13178], 2 X[8352] + X[11054], 4 X[8355] - 3 X[41133], 2 X[8591] - 9 X[9166], 7 X[8591] - 18 X[9167], 4 X[8591] - 15 X[14061], 5 X[8591] - 18 X[14971], 5 X[8591] - 6 X[15300], 5 X[8591] - 3 X[20094], 3 X[8591] - 8 X[22247], 11 X[8591] - 30 X[31274], 13 X[8591] - 24 X[35022], 7 X[8591] + 3 X[35369], 7 X[8591] - 12 X[36521], X[8591] - 12 X[36523], 4 X[8591] - 9 X[41134], X[8591] - 9 X[41135], X[8591] + 24 X[41147], 23 X[8591] - 96 X[41148], 7 X[8591] - 48 X[41154], 3 X[8593] - 4 X[8787], 3 X[8593] - 2 X[10488], 2 X[8596] + 9 X[9166], 7 X[8596] + 18 X[9167], 4 X[8596] + 15 X[14061], 5 X[8596] + 18 X[14971], 5 X[8596] + 6 X[15300], 5 X[8596] + 3 X[20094], 3 X[8596] + 8 X[22247], 11 X[8596] + 30 X[31274], 13 X[8596] + 24 X[35022], 7 X[8596] - 3 X[35369], 7 X[8596] + 12 X[36521], X[8596] + 12 X[36523], 4 X[8596] + 9 X[41134], X[8596] + 9 X[41135], X[8596] - 24 X[41147], 23 X[8596] + 96 X[41148], 7 X[8596] + 48 X[41154], X[8597] + 4 X[32457], 2 X[8598] - 3 X[26613], 2 X[8724] - 3 X[23234], 8 X[8787] - 3 X[45018], 3 X[8859] - X[9855], X[9114] - 3 X[22489], X[9116] - 3 X[22490], 4 X[9164] - 3 X[14588], 8 X[9164] - 15 X[40429], 7 X[9166] - 4 X[9167], 6 X[9166] - 5 X[14061], 5 X[9166] - 4 X[14971], 15 X[9166] - 4 X[15300], 15 X[9166] - 2 X[20094], 27 X[9166] - 16 X[22247], 33 X[9166] - 20 X[31274], 39 X[9166] - 16 X[35022], 21 X[9166] + 2 X[35369], 21 X[9166] - 8 X[36521], 3 X[9166] - 8 X[36523], 3 X[9166] + 16 X[41147], 69 X[9166] - 64 X[41148], 21 X[9166] - 32 X[41154], 24 X[9167] - 35 X[14061], 5 X[9167] - 7 X[14971], 15 X[9167] - 7 X[15300], 30 X[9167] - 7 X[20094], 27 X[9167] - 28 X[22247], 33 X[9167] - 35 X[31274], 39 X[9167] - 28 X[35022], 6 X[9167] + X[35369], 3 X[9167] - 2 X[36521], 3 X[9167] - 14 X[36523], 8 X[9167] - 7 X[41134], 2 X[9167] - 7 X[41135], 3 X[9167] + 28 X[41147], 69 X[9167] - 112 X[41148], 3 X[9167] - 8 X[41154], 3 X[9180] - 2 X[14443], 4 X[9293] + 5 X[42345], 2 X[9293] + X[42553], X[9862] + 2 X[39809], 2 X[9875] + X[9884], X[9875] + 2 X[12258], 3 X[9877] - 2 X[9890], 2 X[9880] + X[12243], 4 X[9880] + X[38664], X[9884] - 4 X[12258], 3 X[10304] - 2 X[38738], 4 X[10488] - 3 X[45018], X[10722] + 2 X[12188], X[10722] - 4 X[22515], X[10723] + 4 X[11632], 3 X[10723] + 4 X[14830], 5 X[10723] + 4 X[38741], X[10754] + 2 X[11646], X[10992] - 4 X[20398], 6 X[11539] - 5 X[38750], 2 X[11599] + X[13178], 3 X[11632] - X[14830], 5 X[11632] - X[38741], 2 X[11711] - 3 X[25055], 4 X[11725] - 3 X[38314], 2 X[12042] + X[38733], 4 X[12100] - 5 X[38739], X[12117] + 2 X[12355], X[12188] + 2 X[22515], X[12243] + 4 X[38734], 4 X[12833] - X[31878], X[13174] - 3 X[19875], 3 X[14041] - 2 X[31173], 25 X[14061] - 24 X[14971], 25 X[14061] - 8 X[15300], 25 X[14061] - 4 X[20094], 45 X[14061] - 32 X[22247], 11 X[14061] - 8 X[31274], 65 X[14061] - 32 X[35022], 35 X[14061] + 4 X[35369], 35 X[14061] - 16 X[36521], 5 X[14061] - 16 X[36523], 5 X[14061] - 3 X[41134], 5 X[14061] - 12 X[41135], 5 X[14061] + 32 X[41147], 115 X[14061] - 128 X[41148], 35 X[14061] - 64 X[41154], 3 X[14269] - 2 X[22505], 3 X[14568] - 2 X[22329], 2 X[14588] - 5 X[40429], 9 X[14639] - 4 X[22566], 5 X[14639] - 2 X[38743], X[14692] - 6 X[23046], 5 X[14830] - 3 X[38741], 3 X[14971] - X[15300], 6 X[14971] - X[20094], 27 X[14971] - 20 X[22247], 33 X[14971] - 25 X[31274], 39 X[14971] - 20 X[35022], 42 X[14971] + 5 X[35369], 21 X[14971] - 10 X[36521], 3 X[14971] - 10 X[36523], 8 X[14971] - 5 X[41134], 2 X[14971] - 5 X[41135], 3 X[14971] + 20 X[41147], 69 X[14971] - 80 X[41148], 21 X[14971] - 40 X[41154], 9 X[15300] - 20 X[22247], 11 X[15300] - 25 X[31274], 13 X[15300] - 20 X[35022], 14 X[15300] + 5 X[35369], 7 X[15300] - 10 X[36521], X[15300] - 10 X[36523], 8 X[15300] - 15 X[41134], 2 X[15300] - 15 X[41135], X[15300] + 20 X[41147], 23 X[15300] - 80 X[41148], 7 X[15300] - 40 X[41154], 2 X[15301] - 5 X[31275], X[15342] - 4 X[16278], 2 X[15686] - 3 X[38742], 5 X[15692] - 6 X[38737], 5 X[15693] - 6 X[26614], 5 X[15694] - 6 X[34127], 7 X[15702] - 12 X[38735], 7 X[15702] - 6 X[38748], 4 X[17948] - 3 X[39061], 3 X[19570] - X[44367], 4 X[19662] - 3 X[21356], 5 X[19708] - 4 X[38736], 9 X[20094] - 40 X[22247], 11 X[20094] - 50 X[31274], 13 X[20094] - 40 X[35022], 7 X[20094] + 5 X[35369], 7 X[20094] - 20 X[36521], X[20094] - 20 X[36523], 4 X[20094] - 15 X[41134], X[20094] - 15 X[41135], X[20094] + 40 X[41147], 23 X[20094] - 160 X[41148], 7 X[20094] - 80 X[41154], X[20774] - 4 X[39120], 2 X[22110] - 3 X[33228], 44 X[22247] - 45 X[31274], 13 X[22247] - 9 X[35022], 56 X[22247] + 9 X[35369], 14 X[22247] - 9 X[36521], 2 X[22247] - 9 X[36523], 32 X[22247] - 27 X[41134], 8 X[22247] - 27 X[41135], X[22247] + 9 X[41147], 23 X[22247] - 36 X[41148], 7 X[22247] - 18 X[41154], 3 X[22510] - 2 X[45879], 3 X[22511] - 2 X[45880], 2 X[22566] - 9 X[38732], 10 X[22566] - 9 X[38743], 3 X[22571] - X[22997], 3 X[22572] - X[22998], 3 X[23234] - X[23235], X[31173] - 3 X[39563], 65 X[31274] - 44 X[35022], 70 X[31274] + 11 X[35369], 35 X[31274] - 22 X[36521], 5 X[31274] - 22 X[36523], 40 X[31274] - 33 X[41134], 10 X[31274] - 33 X[41135], 5 X[31274] + 44 X[41147], 115 X[31274] - 176 X[41148], 35 X[31274] - 88 X[41154], 3 X[31644] - X[44397], 3 X[31998] - 4 X[35087], 3 X[31998] - 2 X[44372], 2 X[32459] - 3 X[41139], 2 X[33376] - 3 X[42062], 2 X[33377] - 3 X[42063], 4 X[34200] - 3 X[38731], 56 X[35022] + 13 X[35369], 14 X[35022] - 13 X[36521], 2 X[35022] - 13 X[36523], 32 X[35022] - 39 X[41134], 8 X[35022] - 39 X[41135], X[35022] + 13 X[41147], 23 X[35022] - 52 X[41148], 7 X[35022] - 26 X[41154], 3 X[35297] - 4 X[44401], X[35369] + 4 X[36521], X[35369] + 28 X[36523], 4 X[35369] + 21 X[41134], X[35369] + 21 X[41135], X[35369] - 56 X[41147], 23 X[35369] + 224 X[41148], X[35369] + 16 X[41154], X[36521] - 7 X[36523], 16 X[36521] - 21 X[41134], 4 X[36521] - 21 X[41135], X[36521] + 14 X[41147], 23 X[36521] - 56 X[41148], X[36521] - 4 X[41154], 16 X[36523] - 3 X[41134], 4 X[36523] - 3 X[41135], X[36523] + 2 X[41147], 23 X[36523] - 8 X[41148], 7 X[36523] - 4 X[41154], 2 X[37461] - 3 X[38227], 3 X[38335] - X[38744], X[38664] + 8 X[38734], 5 X[38732] - X[38743], 4 X[39806] - X[39837], X[41134] - 4 X[41135], 3 X[41134] + 32 X[41147], 69 X[41134] - 128 X[41148], 21 X[41134] - 64 X[41154], 3 X[41135] + 8 X[41147], 69 X[41135] - 32 X[41148], 21 X[41135] - 16 X[41154], 23 X[41147] + 4 X[41148], 7 X[41147] + 2 X[41154], 14 X[41148] - 23 X[41154], 5 X[42345] - 2 X[42553]

Let A'B'C' be the 4th Brocard triangle. Let A″ be the pole, wrt the polar circle, of line B'C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(671). (Randy Hutson, November 30, 2015)

Let A'B'C' be the X(2)-Fuhrmann triangle. Let A″ be the reflection of A in B'C', and define B″ and C″ cyclically. A″B″C″ is inversely similar to ABC, with similitude center X(671). (Randy Hutson, November 30, 2015)

Let Pa be the perspector of the A-Neuberg circle, and define Pb and Pc cyclically. The lines APa, BPb, CPc concur in X(671). (Randy Hutson, November 30, 2015)

Let A' be the reflection in BC of the A-vertex of the antipedal triangle of X(2), and define B' and C' cyclically. The circumcircles of A'BC, B'CA, and C'AB concur at X(671). (Randy Hutson, November 30, 2015)

Let A'B'C' be the 1st Brocard triangle. Let A″ be the reflection of A in line B'C', and define B″ and C″ cyclically. Let A''' be the reflection of A' in line BC, and define B''' and C''' cyclically. Let A* = B″B'''∩C″C''', and define B*and C* cyclically. The lines AA*, BB*, CC* concur in X(671), which is also X(2)-of-A*B*C*. (Randy Hutson, July 11, 2019)

If you have The Geometer's Sketchpad, you can view X(671).

X(671) lies on the Kiepert circumhyperbola, Steiner circumellipse, the cubics K008, K025, K063, K185, K209, K239, K240, K241, K248, K273, K298, K299, K300, K301, K302, K303a, K303b, K304, K305, K353, K370, K565, K595, K715, K751, K809, K854, K914, K1153, the curves Q001, Q044, Q094, Q096, the Darboux septic, and these lines: {1, 9875}, {2, 99}, {3, 7607}, {4, 542}, {5, 7608}, {6, 598}, {8, 35177}, {10, 190}, {11, 12348}, {12, 12349}, {13, 531}, {14, 530}, {17, 5459}, {18, 5460}, {20, 11623}, {23, 3455}, {25, 9876}, {30, 98}, {32, 9878}, {39, 8786}, {55, 12326}, {56, 18969}, {69, 5485}, {75, 35181}, {76, 338}, {83, 597}, {94, 35139}, {110, 5465}, {114, 3545}, {125, 10556}, {140, 10185}, {141, 10302}, {147, 3839}, {183, 5077}, {187, 8587}, {193, 23334}, {194, 7775}, {226, 664}, {230, 8598}, {262, 381}, {265, 34174}, {275, 18831}, {290, 10097}, {297, 16077}, {298, 42035}, {299, 42036}, {303, 36775}, {315, 2996}, {316, 524}, {317, 34163}, {321, 668}, {325, 5503}, {371, 35699}, {372, 35698}, {376, 6055}, {382, 6179}, {384, 7817}, {385, 3849}, {395, 8595}, {396, 8594}, {402, 12347}, {458, 43530}, {485, 489}, {486, 490}, {493, 12352}, {494, 12353}, {511, 19905}, {512, 35146}, {513, 35155}, {514, 35153}, {519, 7983}, {523, 9180}, {525, 5641}, {527, 11608}, {528, 10769}, {532, 11602}, {533, 11603}, {536, 11611}, {538, 1916}, {546, 7858}, {547, 15561}, {549, 21166}, {551, 38220}, {616, 43543}, {617, 43542}, {618, 9116}, {619, 9114}, {625, 39785}, {631, 10992}, {666, 5080}, {690, 5466}, {698, 10290}, {729, 2086}, {754, 11606}, {804, 3228}, {843, 16341}, {850, 10561}, {858, 15398}, {886, 3978}, {889, 17790}, {903, 2786}, {923, 4586}, {1003, 5152}, {1007, 9741}, {1078, 7748}, {1084, 14700}, {1121, 2785}, {1131, 13639}, {1132, 13759}, {1153, 8589}, {1300, 35191}, {1327, 45420}, {1328, 45421}, {1446, 4569}, {1494, 2394}, {1499, 9487}, {1509, 23903}, {1641, 22254}, {1649, 9183}, {1975, 7870}, {1989, 39295}, {1995, 2936}, {2023, 42849}, {2052, 6528}, {2373, 39413}, {2407, 45774}, {2408, 2793}, {2444, 43668}, {2479, 24008}, {2480, 24007}, {2592, 15165}, {2593, 15164}, {2783, 10711}, {2784, 10710}, {2787, 3227}, {2792, 10709}, {2794, 3424}, {2795, 6175}, {2797, 10714}, {2798, 10715}, {2854, 16175}, {2986, 18878}, {3023, 11238}, {3027, 11237}, {3058, 13183}, {3068, 13908}, {3069, 13968}, {3091, 14981}, {3096, 18840}, {3124, 9169}, {3146, 10991}, {3316, 45509}, {3317, 45508}, {3329, 8592}, {3363, 15048}, {3399, 6248}, {3406, 12203}, {3407, 5309}, {3413, 6189}, {3414, 6190}, {3448, 10555}, {3523, 38740}, {3524, 6036}, {3534, 12042}, {3582, 10089}, {3584, 10086}, {3618, 14928}, {3629, 33698}, {3656, 7970}, {3767, 33007}, {3815, 20112}, {3830, 10722}, {3832, 38745}, {3845, 6033}, {3972, 5939}, {4080, 4555}, {4444, 18827}, {4562, 43534}, {4563, 14916}, {4590, 31644}, {4597, 16826}, {4762, 35152}, {4785, 35165}, {5007, 14042}, {5025, 7801}, {5032, 5477}, {5054, 11668}, {5055, 11669}, {5056, 20399}, {5064, 5186}, {5067, 38751}, {5071, 10155}, {5076, 38627}, {5079, 38628}, {5134, 37854}, {5189, 15899}, {5215, 32456}, {5286, 5395}, {5319, 14068}, {5392, 46134}, {5434, 13182}, {5471, 41745}, {5472, 41746}, {5476, 11170}, {5478, 41043}, {5479, 41042}, {5480, 14485}, {5523, 37765}, {5569, 17004}, {5597, 12345}, {5598, 12346}, {5976, 11287}, {5978, 31693}, {5979, 31694}, {5980, 11295}, {5981, 11296}, {5984, 39838}, {5989, 7884}, {6071, 31513}, {6094, 37859}, {6103, 40890}, {6108, 35932}, {6109, 35931}, {6114, 9762}, {6115, 9760}, {6390, 8355}, {6392, 7877}, {6539, 6540}, {6625, 33770}, {6648, 7316}, {6655, 7810}, {6656, 10159}, {6658, 7755}, {6669, 43548}, {6670, 43549}, {6777, 12817}, {6778, 12816}, {6787, 34383}, {7375, 43565}, {7376, 43564}, {7388, 10194}, {7389, 10195}, {7610, 7771}, {7668, 36165}, {7669, 37915}, {7739, 33016}, {7745, 20583}, {7746, 33274}, {7747, 34604}, {7751, 9939}, {7752, 33006}, {7753, 32528}, {7756, 34506}, {7758, 32996}, {7759, 14062}, {7763, 32984}, {7764, 32993}, {7765, 16044}, {7768, 33229}, {7770, 43527}, {7772, 33018}, {7777, 8176}, {7780, 33256}, {7781, 32966}, {7782, 11149}, {7783, 39565}, {7796, 14063}, {7799, 8781}, {7802, 33192}, {7803, 18841}, {7804, 8289}, {7811, 33017}, {7813, 41136}, {7818, 43688}, {7821, 14045}, {7825, 7917}, {7828, 8369}, {7832, 8360}, {7839, 39590}, {7842, 17129}, {7843, 14044}, {7847, 8359}, {7850, 15533}, {7851, 33237}, {7856, 14035}, {7857, 32985}, {7861, 7944}, {7865, 8782}, {7872, 31276}, {7880, 14046}, {7898, 17131}, {7903, 20105}, {7924, 9466}, {7926, 22253}, {7933, 17130}, {7937, 21358}, {7942, 8366}, {8179, 11171}, {8182, 17008}, {8288, 40877}, {8353, 13468}, {8356, 11168}, {8430, 23878}, {8584, 45103}, {8667, 11057}, {8703, 38730}, {8785, 20977}, {8982, 12297}, {9112, 41621}, {9113, 41620}, {9139, 34150}, {9141, 17708}, {9148, 17993}, {9164, 14588}, {9213, 34312}, {9227, 14724}, {9293, 42345}, {9302, 19924}, {9479, 43098}, {9698, 33024}, {9766, 11055}, {9862, 15682}, {9879, 13207}, {9909, 39832}, {10153, 27088}, {10187, 11290}, {10188, 11289}, {10304, 38738}, {10562, 31296}, {10733, 34211}, {10989, 34320}, {11001, 38749}, {11078, 36316}, {11092, 36317}, {11114, 38557}, {11140, 46139}, {11165, 15814}, {11172, 14907}, {11184, 31859}, {11539, 38750}, {11656, 17702}, {11711, 25055}, {11725, 38314}, {12040, 37647}, {12100, 38739}, {12154, 40671}, {12155, 40672}, {12156, 14537}, {12296, 14244}, {12833, 31878}, {13083, 46054}, {13084, 46053}, {13087, 35830}, {13088, 35831}, {13174, 19875}, {13180, 34612}, {13181, 34606}, {13189, 45701}, {13190, 45700}, {13273, 14612}, {13640, 13676}, {13760, 13796}, {14023, 33279}, {14269, 14488}, {14692, 23046}, {14712, 33689}, {14904, 40707}, {14905, 40706}, {14957, 36827}, {14995, 23348}, {15301, 31275}, {15534, 17503}, {15535, 20126}, {15638, 17952}, {15686, 38742}, {15692, 38737}, {15693, 26614}, {15694, 34127}, {15702, 38735}, {16041, 32833}, {16042, 34013}, {16277, 34603}, {16508, 16509}, {16922, 31652}, {17005, 39601}, {17132, 34899}, {17556, 38499}, {17577, 45964}, {17950, 35157}, {18026, 40149}, {18316, 41626}, {18800, 18842}, {18822, 40459}, {19108, 32788}, {19109, 32787}, {19662, 21356}, {19696, 35007}, {19708, 38736}, {20975, 38526}, {21849, 39846}, {22037, 35162}, {22510, 45879}, {22511, 45880}, {22735, 41143}, {23870, 43092}, {23871, 43091}, {23905, 25536}, {23942, 34016}, {24624, 36085}, {25221, 25222}, {25328, 41498}, {31160, 41142}, {31450, 33009}, {31862, 46023}, {31863, 46024}, {31998, 35087}, {32458, 32836}, {32459, 41139}, {32552, 33607}, {32553, 33606}, {32808, 42023}, {32809, 42024}, {32832, 33215}, {32982, 43681}, {32986, 42850}, {33220, 44536}, {33264, 39652}, {33285, 46236}, {33513, 39284}, {33602, 36327}, {33603, 35749}, {33604, 36331}, {33605, 35750}, {34171, 34574}, {34200, 38731}, {34289, 40814}, {34581, 37860}, {34664, 40448}, {35140, 43673}, {35142, 44427}, {35297, 44401}, {37461, 38227}, {37856, 38945}, {37940, 39854}, {37948, 39831}, {38335, 38744}, {38888, 38946}, {39229, 46082}, {39230, 46083}, {39806, 39837}, {39828, 44837}, {39842, 40178}, {40016, 42371}, {41022, 41047}, {41023, 41046}, {42738, 44552}, {44146, 46105}

X(671) = midpoint of X(i) and X(j) for these {i,j}: {1, 9875}, {2, 148}, {3, 12355}, {4, 12243}, {316, 11054}, {385, 8597}, {895, 14833}, {3534, 38733}, {3543, 11177}, {3830, 12188}, {5463, 22577}, {5464, 22578}, {6321, 11632}, {6777, 35752}, {6778, 36330}, {8591, 8596}, {9862, 15682}, {9879, 13207}, {9882, 9883}, {10754, 11161}, {14712, 40246}, {31695, 31696}, {33432, 33433}
X(671) = reflection of X(i) in X(j) for these {i,j}: {1, 12258}, {2, 115}, {4, 9880}, {98, 11632}, {99, 2}, {110, 5465}, {115, 36523}, {316, 8352}, {325, 37350}, {376, 6055}, {691, 16092}, {843, 16341}, {892, 17948}, {1649, 9183}, {2482, 5461}, {3534, 12042}, {3830, 22515}, {5182, 6034}, {5463, 5460}, {5464, 5459}, {5466, 18007}, {5978, 31693}, {5979, 31694}, {6033, 3845}, {6054, 381}, {6390, 8355}, {6722, 41154}, {7799, 33228}, {7809, 14041}, {7840, 31173}, {7970, 3656}, {8591, 2482}, {8593, 6}, {8594, 396}, {8595, 395}, {8598, 230}, {8724, 5}, {9114, 619}, {9116, 618}, {9144, 16278}, {9166, 41135}, {9855, 187}, {9877, 7615}, {9880, 38734}, {9881, 10}, {9884, 1}, {9885, 33476}, {9886, 33477}, {10488, 8787}, {10722, 3830}, {10753, 20423}, {11001, 38749}, {11006, 125}, {11152, 39}, {11161, 11646}, {12117, 3}, {12177, 5476}, {12347, 402}, {12356, 10054}, {12357, 10070}, {13640, 13676}, {13760, 13796}, {14041, 39563}, {14639, 38732}, {15300, 620}, {15342, 9144}, {15561, 38229}, {15682, 39809}, {16508, 16509}, {19911, 7617}, {20094, 15300}, {20126, 15535}, {21166, 38224}, {22568, 33461}, {22570, 33460}, {23235, 8724}, {27088, 43291}, {32469, 32447}, {33342, 13968}, {33343, 13908}, {34473, 14651}, {35705, 3363}, {35751, 32553}, {35931, 6109}, {35932, 6108}, {36329, 32552}, {36521, 6722}, {37785, 22574}, {37786, 22573}, {38664, 12243}, {38730, 8703}, {38748, 38735}, {39785, 625}, {39846, 21849}, {41042, 5479}, {41043, 5478}, {41134, 9166}, {44372, 35087}, {45018, 8593}
X(671) = reflection of X(9140) in the Fermat line
X(671) = isogonal conjugate of X(187)
X(671) = isotomic conjugate of X(524)
X(671) = anticomplement of X(2482)
X(671) = circumcircle-inverse of X(11643
X(671) = 3rd-Lemoine-circle-(Ehrmann)-inverse of X(895)
X(671) = polar-circle-inverse of X(5095)
X(671) = orthoptic-circle-of-Steiner-inellipe-inverse of X(9172)
X(671) = circumcircle-of-anticomplementary-triangle-inverse of X(32255)
X(671) = polar conjugate of X(468)
X(671) = antigonal image of X(2)
X(671) = symgonal image of X(2)
X(671) = cyclocevian conjugate of X(13574)
X(671) = psi-transform of X(9169)
X(671) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {897, 14360}, {923, 8591}, {10630, 8}, {15398, 4329}, {34539, 21295}, {34574, 7192}, {36142, 44010}, {41936, 192}
X(671) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 39061}, {46275, 2887}
X(671) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 39061}, {892, 5466}, {18023, 30786}, {46111, 17983}
X(671) = X(i)-cross conjugate of X(j) for these (i,j): {6, 10415}, {67, 2373}, {69, 44182}, {111, 17983}, {316, 83}, {524, 2}, {690, 99}, {858, 264}, {895, 30786}, {1499, 39296}, {1648, 523}, {4442, 75}, {5466, 892}, {5468, 40429}, {8352, 598}, {8586, 7608}, {9134, 35136}, {9140, 1494}, {9979, 648}, {10097, 691}, {10561, 34574}, {10754, 8781}, {11054, 10302}, {11646, 98}, {14263, 10630}, {15638, 2408}, {17162, 1268}, {17491, 86}, {17497, 274}, {20977, 6}, {21298, 40415}, {23061, 40410}, {24724, 14621}, {25328, 18019}, {30709, 668}, {31125, 18023}, {33906, 37880}, {34290, 18829}, {39232, 110}, {41724, 95}, {42344, 42345}, {44915, 468}, {45291, 4590}, {46154, 111}
X(671) = cevapoint of X(i) and X(j) for these (i,j): {2, 524}, {4, 5523}, {6, 23}, {111, 895}, {115, 690}, {523, 1648}, {2408, 15638}, {5468, 14588}, {13492, 14262}, {31125, 46154}, {31644, 45291}, {39229, 39230}, {42344, 42553}
X(671) = crosspoint of X(i) and X(j) for these (i,j): {2, 46275}, {18023, 46111}
X(671) = crosssum of X(i) and X(j) for these (i,j): {6, 46276}, {351, 21906}, {14567, 23200}
X(671) = trilinear pole of line {2, 523}
X(671) = crossdifference of every pair of points on line {351, 39689}
X(671) = the point of intersection, other than A, B, and C, of the Steiner circumellipse and the Kiepert hyperbola
X(671) = Kiepert-hyperbola-antipode of X(2)
X(671) = Steiner- circumellipse- antipode of X(99)
X(671) = projection from Steiner inellipse to Steiner circumellipse of X(115)
X(671) = antigonal image of X(2)
X(671) = syngonal conjugate of X(2)
X(671) = pole wrt polar circle of trilinear polar of X(468)
X(671) = Kirikami concurrent circles image of X(13)
X(671) = Kirikami concurrent circles image of X(14)
X(671) = X(3) of anti-McCay triangle
X(671) = crossdifference of PU(107)
X(671) = McCay-to-Artzt similarity image of X(6)
X(671) = perspector of ABC and 1st Brocard triangle of anti-McCay triangle
X(671) = Cundy-Parry Phi transform of X(7607)
X(671) = Cundy-Parry Psi transform of X(576)
X(671) = Steiner-circumellipse-X(1)-antipode of X(35180)
X(671) = Steiner-circumellipse-X(3)-antipode of X(35178)
X(671) = Steiner-circumellipse-X(4)-antipode of X(648)
X(671) = Steiner-circumellipse-X(6)-antipode of X(35138)
X(671) = perspector of hyperbola {A,B,C,X(892),PU(180)}}
X(671) = X(i)-isoconjugate of X(j) for these (i,j): {1, 187}, {2, 922}, {6, 896}, {19, 3292}, {31, 524}, {32, 14210}, {41, 7181}, {42, 16702}, {48, 468}, {58, 21839}, {63, 44102}, {75, 14567}, {92, 23200}, {101, 14419}, {110, 2642}, {111, 42081}, {163, 690}, {213, 6629}, {351, 662}, {512, 23889}, {560, 3266}, {604, 3712}, {661, 5467}, {669, 24039}, {692, 4750}, {798, 5468}, {810, 4235}, {897, 39689}, {904, 7267}, {923, 2482}, {1101, 1648}, {1333, 4062}, {1415, 14432}, {1580, 18872}, {1649, 36142}, {1755, 5967}, {1910, 9155}, {1911, 4760}, {1918, 16741}, {1919, 42721}, {1967, 5026}, {1973, 6390}, {2148, 41586}, {2157, 6593}, {2159, 5642}, {2173, 9717}, {2206, 42713}, {2234, 41309}, {2624, 14559}, {4575, 14273}, {4933, 28607}, {4938, 34819}, {5095, 36060}, {5477, 36051}, {7813, 46289}, {9177, 36150}, {9247, 44146}, {9406, 36890}, {14417, 32676}, {14424, 34072}, {21906, 24041}, {24038, 32740}, {30605, 34073}, {32459, 38252}, {32678, 44814}
X(671) = barycentric product X(i)*X(j) for these {i,j}: {1, 46277}, {3, 46111}, {4, 30786}, {6, 18023}, {69, 17983}, {75, 897}, {76, 111}, {83, 31125}, {99, 5466}, {264, 895}, {290, 5968}, {298, 36307}, {299, 36310}, {304, 36128}, {305, 8753}, {308, 46154}, {316, 10415}, {325, 9154}, {523, 892}, {561, 923}, {598, 42008}, {599, 18818}, {648, 14977}, {670, 9178}, {691, 850}, {693, 5380}, {799, 23894}, {1236, 10422}, {1494, 9214}, {1502, 32740}, {1577, 36085}, {1969, 36060}, {2408, 35179}, {3260, 9139}, {3266, 10630}, {3596, 7316}, {5547, 6063}, {5641, 16092}, {6330, 36894}, {6331, 10097}, {8430, 43187}, {9170, 18007}, {9180, 34760}, {9213, 35139}, {10416, 14364}, {11123, 14728}, {14246, 18019}, {14609, 34087}, {14908, 18022}, {15398, 44146}, {17948, 18823}, {19626, 40362}, {20948, 36142}, {23105, 45773}, {23288, 35138}, {27808, 43926}, {32729, 44173}, {34574, 35522}, {39061, 46275}, {40016, 41272}, {40826, 42007}
X(671) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 896}, {2, 524}, {3, 3292}, {4, 468}, {5, 41586}, {6, 187}, {7, 7181}, {8, 3712}, {10, 4062}, {23, 6593}, {25, 44102}, {30, 5642}, {31, 922}, {32, 14567}, {37, 21839}, {67, 14357}, {69, 6390}, {74, 9717}, {75, 14210}, {76, 3266}, {81, 16702}, {86, 6629}, {94, 43084}, {98, 5967}, {99, 5468}, {110, 5467}, {111, 6}, {115, 1648}, {141, 7813}, {148, 11053}, {184, 23200}, {187, 39689}, {193, 32459}, {230, 5477}, {239, 4760}, {264, 44146}, {274, 16741}, {316, 7664}, {321, 42713}, {381, 32225}, {385, 5026}, {395, 9117}, {396, 9115}, {403, 12828}, {468, 5095}, {476, 14559}, {511, 9155}, {512, 351}, {513, 14419}, {514, 4750}, {522, 14432}, {523, 690}, {524, 2482}, {525, 14417}, {526, 44814}, {538, 45672}, {542, 45662}, {543, 1641}, {599, 39785}, {648, 4235}, {661, 2642}, {662, 23889}, {668, 42721}, {690, 1649}, {691, 110}, {694, 18872}, {729, 41309}, {799, 24039}, {804, 11183}, {826, 14424}, {850, 35522}, {858, 5181}, {892, 99}, {894, 7267}, {895, 3}, {896, 42081}, {897, 1}, {923, 31}, {1268, 31013}, {1296, 2434}, {1494, 36890}, {1499, 9125}, {1503, 35282}, {1648, 23992}, {1649, 33915}, {1698, 4938}, {1992, 27088}, {2052, 37778}, {2408, 1499}, {2444, 8644}, {2482, 8030}, {2501, 14273}, {2854, 9177}, {3124, 21906}, {3228, 14608}, {3266, 36792}, {3267, 45807}, {3268, 45808}, {3616, 4831}, {3618, 3793}, {3679, 4933}, {3712, 7067}, {4232, 15471}, {4442, 16597}, {4777, 30605}, {4802, 30595}, {5169, 8262}, {5380, 100}, {5466, 523}, {5523, 1560}, {5547, 55}, {5968, 511}, {5969, 45330}, {6091, 3167}, {7181, 1366}, {7316, 56}, {7426, 15303}, {7469, 41606}, {8029, 33919}, {8371, 33921}, {8430, 3569}, {8591, 38239}, {8599, 23287}, {8753, 25}, {8859, 8787}, {8877, 19596}, {9139, 74}, {9154, 98}, {9178, 512}, {9180, 34763}, {9206, 5995}, {9207, 5994}, {9213, 526}, {9214, 30}, {9487, 37860}, {9979, 18311}, {10015, 42760}, {10097, 647}, {10159, 31068}, {10415, 67}, {10416, 11061}, {10422, 1177}, {10557, 14995}, {10558, 39231}, {10561, 2492}, {10562, 39232}, {10630, 111}, {11123, 33906}, {11643, 9716}, {13492, 10354}, {13574, 41498}, {14061, 45291}, {14210, 24038}, {14246, 23}, {14263, 3291}, {14443, 46049}, {14609, 3231}, {14908, 184}, {14977, 525}, {15398, 895}, {15638, 35133}, {15899, 2930}, {16092, 542}, {16741, 16733}, {17948, 543}, {17964, 2502}, {17983, 4}, {17993, 9171}, {18007, 8371}, {18023, 76}, {18818, 598}, {19330, 15141}, {19626, 1501}, {20099, 38304}, {21460, 2080}, {22258, 10417}, {22329, 18800}, {23288, 3906}, {23348, 9181}, {23870, 9204}, {23871, 9205}, {23894, 661}, {23992, 14444}, {25322, 40517}, {25423, 45680}, {30786, 69}, {31125, 141}, {32583, 9145}, {32729, 1576}, {32740, 32}, {33919, 14443}, {34169, 10418}, {34208, 5203}, {34320, 5648}, {34574, 691}, {34760, 9182}, {35179, 2418}, {36060, 48}, {36085, 662}, {36128, 19}, {36142, 163}, {36307, 13}, {36310, 14}, {36792, 23106}, {36821, 3229}, {36827, 1634}, {36877, 16317}, {36894, 441}, {37760, 41595}, {37777, 41616}, {37962, 41618}, {37980, 41612}, {39061, 8591}, {39169, 41615}, {39296, 6082}, {41272, 3051}, {41404, 46276}, {41511, 18876}, {41909, 34161}, {41936, 32740}, {42007, 574}, {42008, 599}, {43448, 24855}, {43926, 3733}, {44146, 34336}, {44182, 41909}, {46111, 264}, {46154, 39}, {46277, 75}
X(671) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 99, 41134}, {2, 115, 9166}, {2, 7620, 11185}, {2, 8591, 2482}, {2, 8596, 8591}, {2, 9166, 14061}, {2, 31125, 42008}, {2, 32480, 574}, {2, 41135, 115}, {2, 42008, 30786}, {5, 8724, 23234}, {6, 10488, 8787}, {6, 11317, 598}, {23, 10415, 10416}, {76, 7841, 7883}, {76, 7911, 32027}, {98, 6321, 10723}, {99, 115, 14061}, {99, 9166, 2}, {111, 148, 16093}, {111, 31125, 30786}, {111, 42008, 2}, {115, 148, 99}, {115, 2482, 5461}, {115, 14931, 7919}, {115, 15300, 14971}, {115, 36523, 41135}, {148, 8591, 8596}, {148, 36523, 9166}, {148, 41135, 2}, {230, 8598, 26613}, {376, 6055, 34473}, {376, 14651, 6055}, {574, 7617, 2}, {620, 14971, 2}, {620, 20094, 99}, {892, 17948, 39061}, {1975, 11318, 7870}, {2482, 5461, 2}, {2482, 8591, 99}, {2549, 7615, 2}, {3734, 14931, 99}, {5077, 40727, 183}, {5254, 8370, 7827}, {5309, 11361, 12150}, {5461, 8596, 99}, {5463, 5469, 5460}, {5464, 5470, 5459}, {5469, 22577, 5463}, {5470, 22578, 5464}, {5523, 37855, 37765}, {6054, 14639, 381}, {6390, 8355, 41133}, {6722, 9167, 2}, {6722, 35369, 99}, {6722, 36521, 9167}, {7610, 35955, 7771}, {7610, 44526, 35955}, {7618, 43620, 2}, {7620, 43448, 2}, {7746, 34504, 33274}, {7825, 20081, 7917}, {7827, 8370, 83}, {7840, 14041, 31173}, {7840, 31173, 7809}, {7841, 7883, 7911}, {7841, 34505, 76}, {7861, 17128, 7944}, {7870, 11318, 7899}, {7924, 9466, 31168}, {8591, 10487, 19911}, {8596, 41135, 5461}, {8787, 10488, 8593}, {8859, 9855, 187}, {9114, 22489, 619}, {9116, 22490, 618}, {9140, 10557, 5466}, {9875, 12258, 9884}, {9882, 19057, 8593}, {9883, 19058, 8593}, {9892, 9894, 7618}, {10054, 10070, 1}, {10630, 11054, 39061}, {10992, 20398, 631}, {11185, 43448, 7790}, {11599, 13178, 7983}, {11648, 18546, 2}, {12188, 22515, 10722}, {12356, 12357, 9884}, {14061, 41134, 2}, {14971, 15300, 620}, {16093, 30786, 99}, {19057, 19058, 6}, {22568, 22570, 9888}, {23234, 23235, 8724}, {24617, 31057, 30856}, {33006, 34511, 7752}, {34505, 44518, 7841}, {35087, 44372, 31998}, {36521, 36523, 41154}, {39103, 39105, 115}


X(672) = CROSSDIFFERENCE OF X(1) AND X(514)

Trilinears    a[b2 + c2 - a(b + c)] : :
Trilinears    a(as - Sω) : :
Trilinears    (cot A + cot B + cot C) csc A - (csc A + csc B + csc C) cot A : :
Barycentrics    a^2*(a*b - b^2 + a*c - c^2) : :
X(672) = X(672) = 2 X[3693] - 3 X[14439], X[3930] - 3 X[14439]

X(672) lies on the cubics K135, K155, K225, K382, K385, K506, K981, K982, K1243 K1244, the curve Q090, and these lines: {1, 1002}, {2, 7}, {3, 41}, {6, 31}, {8, 3501}, {10, 3691}, {11, 17747}, {19, 1851}, {21, 41239}, {34, 41320}, {35, 4251}, {36, 101}, {37, 38}, {39, 213}, {40, 2082}, {43, 165}, {44, 513}, {45, 4860}, {46, 169}, {48, 2911}, {56, 220}, {58, 4476}, {59, 1252}, {65, 1212}, {72, 1009}, {75, 20632}, {100, 3684}, {103, 919}, {105, 238}, {106, 6017}, {109, 43079}, {141, 24690}, {145, 3208}, {163, 5127}, {171, 5276}, {172, 22065}, {187, 2251}, {190, 350}, {191, 29633}, {192, 17027}, {193, 22370}, {194, 17033}, {198, 1473}, {210, 44798}, {219, 604}, {237, 20777}, {239, 2111}, {241, 9502}, {244, 3290}, {284, 1174}, {292, 3009}, {294, 9441}, {325, 4766}, {344, 30962}, {346, 10453}, {377, 26036}, {484, 5011}, {497, 41325}, {516, 13576}, {517, 2170}, {518, 3693}, {519, 1018}, {570, 22058}, {572, 3449}, {594, 33162}, {595, 5299}, {602, 22131}, {607, 1593}, {609, 4257}, {610, 37262}, {612, 16517}, {614, 16970}, {660, 36906}, {663, 9320}, {665, 14411}, {726, 17031}, {728, 6762}, {750, 5275}, {758, 24036}, {799, 41535}, {800, 22071}, {857, 26012}, {901, 2291}, {942, 16601}, {956, 4390}, {960, 39244}, {966, 25631}, {970, 23637}, {971, 44424}, {982, 26242}, {992, 28242}, {993, 16788}, {1015, 1149}, {1023, 24496}, {1100, 3748}, {1107, 2295}, {1108, 17452}, {1125, 3294}, {1146, 40663}, {1190, 37499}, {1201, 2176}, {1208, 7124}, {1213, 32781}, {1269, 41681}, {1282, 1757}, {1319, 6603}, {1362, 1458}, {1376, 37658}, {1405, 15288}, {1434, 32008}, {1438, 8647}, {1449, 10389}, {1457, 4559}, {1468, 5021}, {1477, 6078}, {1479, 17732}, {1500, 20963}, {1509, 46194}, {1621, 16503}, {1642, 3675}, {1655, 41240}, {1713, 8804}, {1723, 1766}, {1724, 4456}, {1731, 16548}, {1732, 8557}, {1736, 11028}, {1737, 5179}, {1738, 20605}, {1739, 16611}, {1761, 29850}, {1765, 10445}, {1778, 4269}, {1783, 2202}, {1788, 6554}, {1802, 37579}, {1818, 3286}, {1876, 5089}, {1901, 3136}, {1903, 41509}, {1911, 38874}, {1931, 2311}, {1953, 21853}, {1967, 2107}, {2054, 9506}, {2092, 21753}, {2099, 34522}, {2106, 37128}, {2110, 2117}, {2112, 2115}, {2171, 5173}, {2174, 5124}, {2178, 38269}, {2195, 41934}, {2201, 35993}, {2212, 45786}, {2223, 2340}, {2262, 21866}, {2264, 7964}, {2271, 31448}, {2273, 5019}, {2278, 34879}, {2284, 34230}, {2287, 13588}, {2292, 40978}, {2297, 15479}, {2300, 7109}, {2301, 7688}, {2317, 4268}, {2318, 2352}, {2321, 17135}, {2325, 29824}, {2329, 2975}, {2333, 4185}, {2345, 31330}, {2356, 37908}, {2382, 6016}, {2384, 6014}, {2670, 21838}, {2702, 28471}, {2957, 5536}, {3002, 8776}, {3003, 22059}, {3006, 4071}, {3008, 18785}, {3010, 46177}, {3057, 21872}, {3061, 3869}, {3097, 16468}, {3124, 20461}, {3126, 38379}, {3160, 34497}, {3169, 20012}, {3177, 3212}, {3195, 45141}, {3207, 5204}, {3240, 4266}, {3247, 29814}, {3252, 23612}, {3263, 17755}, {3271, 20863}, {3419, 11355}, {3446, 7113}, {3474, 5819}, {3496, 33950}, {3508, 17794}, {3510, 9359}, {3555, 3991}, {3583, 5134}, {3588, 21877}, {3589, 25349}, {3621, 4050}, {3634, 46196}, {3660, 43947}, {3666, 21840}, {3670, 16600}, {3678, 17746}, {3682, 19762}, {3685, 24727}, {3686, 4651}, {3690, 40956}, {3709, 5098}, {3729, 4441}, {3731, 10980}, {3741, 17355}, {3747, 16782}, {3758, 37632}, {3760, 29455}, {3778, 24513}, {3780, 20691}, {3781, 37507}, {3789, 5220}, {3792, 24484}, {3812, 21921}, {3831, 27040}, {3868, 25082}, {3874, 3970}, {3912, 18206}, {3915, 14974}, {3924, 16968}, {3934, 4721}, {3938, 16973}, {3950, 42057}, {3973, 16569}, {3985, 4358}, {3986, 25501}, {4032, 25001}, {4051, 14923}, {4059, 6706}, {4095, 4696}, {4153, 30171}, {4192, 5755}, {4210, 7293}, {4258, 5217}, {4262, 5010}, {4271, 16669}, {4284, 16470}, {4300, 10822}, {4368, 20372}, {4370, 34590}, {4414, 36404}, {4426, 13733}, {4438, 30751}, {4447, 19593}, {4465, 20530}, {4473, 30967}, {4513, 12513}, {4517, 21010}, {4641, 22097}, {4642, 41015}, {4672, 40718}, {4700, 19998}, {4735, 4749}, {4759, 24491}, {4805, 7761}, {4848, 41006}, {4857, 41326}, {4859, 31200}, {4872, 24712}, {4875, 5836}, {4878, 15624}, {4919, 38460}, {5036, 22354}, {5044, 25068}, {5060, 5546}, {5080, 26074}, {5088, 9317}, {5096, 20780}, {5164, 20982}, {5222, 37555}, {5227, 33171}, {5230, 5286}, {5247, 23640}, {5248, 16783}, {5254, 21935}, {5283, 17750}, {5338, 44103}, {5537, 18771}, {5709, 36670}, {5782, 16405}, {5783, 11358}, {5838, 9778}, {6205, 21373}, {6244, 18766}, {6656, 24995}, {6734, 21029}, {6735, 21013}, {6763, 17744}, {6817, 15656}, {7075, 37683}, {7076, 37386}, {7131, 7183}, {7146, 24635}, {7175, 37659}, {7181, 17044}, {7193, 19554}, {7201, 24554}, {7741, 24045}, {7778, 30816}, {8609, 18839}, {8616, 16779}, {8731, 11018}, {9263, 10027}, {9334, 16676}, {9343, 16671}, {9367, 28271}, {9605, 16466}, {10030, 18031}, {10916, 21073}, {11227, 25075}, {11509, 32561}, {12782, 16476}, {12915, 21809}, {14096, 22449}, {14964, 23531}, {15254, 28600}, {15629, 36040}, {16567, 24892}, {16583, 24443}, {16604, 28352}, {16609, 30807}, {16667, 42042}, {16673, 30350}, {16784, 40091}, {16850, 31445}, {16885, 37673}, {16972, 17017}, {16980, 23630}, {17032, 17379}, {17034, 25264}, {17046, 33839}, {17053, 28360}, {17081, 26658}, {17090, 20089}, {17120, 40721}, {17122, 37675}, {17137, 27109}, {17152, 30038}, {17165, 21101}, {17264, 41851}, {17275, 33074}, {17278, 31199}, {17279, 24691}, {17280, 31027}, {17281, 31136}, {17289, 30966}, {17303, 25611}, {17336, 30963}, {17339, 31028}, {17349, 19591}, {17351, 21264}, {17369, 30970}, {17443, 21863}, {17444, 21864}, {17499, 27020}, {17603, 30944}, {17751, 26770}, {17752, 21226}, {19346, 37504}, {20228, 22199}, {20244, 20257}, {20247, 25237}, {20457, 20669}, {20593, 20718}, {20594, 20719}, {20666, 20668}, {20769, 21495}, {20778, 20860}, {20785, 46148}, {20888, 29433}, {21078, 25078}, {21139, 43037}, {21232, 30806}, {21281, 30036}, {21369, 24259}, {21477, 23151}, {21760, 23579}, {21956, 33136}, {22079, 42447}, {23569, 23656}, {23622, 23988}, {24170, 33792}, {24214, 26978}, {24266, 28916}, {24549, 33819}, {24631, 26234}, {25427, 29578}, {26006, 43054}, {26043, 27091}, {26107, 27158}, {26244, 32918}, {27622, 28246}, {27919, 42720}, {28244, 28269}, {28535, 28875}, {30965, 33157}, {30969, 33115}, {31036, 41233}, {31508, 42043}, {33105, 37661}, {33888, 33889}, {37195, 40957}, {37548, 39247}, {39798, 40147}

X(672) = midpoint of X(1018) and X(45751)
X(672) = reflection of X(i) in X(j) for these {i,j}: {2170, 43065}, {3930, 3693}, {20347, 20335}, {21139, 43037}, {30806, 21232}
X(672) = isogonal conjugate of X(673)
X(672) = isotomic conjugate of X(18031)
X(672) = complement of X(20347)
X(672) = anticomplement of X(20335)
X(672) = X(i)-Hirst inverse of X(j) for these (i,j): (6,55), (1362,1458)
X(672) = X(232)-of-excentral-triangle
X(672) = trilinear pole of line X(665)X(926)
X(672) = crossdifference of PU(i) for i in (47, 51)
X(672) = bicentric sum of PU(49)
X(672) = PU(49)-harmonic conjugate of X(663)
X(672) = perspector of conic {A,B,C,X(1),X(101),PU(93)}
X(672) = perspector of 4th mixtilinear triangle and unary cofactor triangle of 5th mixtilinear triangle
X(672) = X(31)-complementary conjugate of X(39046)
X(672) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 42079}, {2, 39046}, {9, 9502}, {19, 17464}, {103, 55}, {238, 3009}, {241, 1458}, {291, 42}, {292, 3252}, {518, 2340}, {673, 1}, {813, 649}, {1024, 17439}, {1025, 2254}, {1861, 2356}, {2195, 9310}, {2283, 926}, {2284, 665}, {3286, 2223}, {3912, 1818}, {7096, 20589}, {9442, 2293}, {18206, 518}, {34085, 4449}, {36039, 652}, {36086, 663}, {39293, 46177}, {39979, 1193}, {43672, 14547}
X(672) = X(i)-cross conjugate of X(j) for these (i,j): {665, 2284}, {926, 2283}, {2223, 1458}, {6184, 3252}, {9454, 2356}, {20662, 6}, {20683, 518}, {39258, 2223}, {42079, 1}
X(672) = cevapoint of X(i) and X(j) for these (i,j): {1, 39341}, {6, 20672}, {9, 24578}, {238, 9441}, {17435, 24290}, {20683, 39258}
X(672) = crosspoint of X(i) and X(j) for these (i,j): {1, 673}, {6, 292}, {9, 2338}, {57, 1477}, {190, 5378}, {241, 518}, {1252, 6078}, {1861, 3912}, {2195, 9439}, {3286, 18206}, {4564, 36086}, {4570, 4584}
X(672) = crosssum of X(i) and X(j) for these (i,j): {1, 672}, {2, 239}, {6, 20470}, {7, 14189}, {9, 5853}, {57, 43035}, {105, 294}, {649, 27846}, {1086, 6084}, {1438, 36057}, {2170, 2254}, {3120, 21832}, {9312, 9436}, {13576, 18785}
X(672) = trilinear pole of line {665, 926}
X(672) = crossdifference of every pair of points on line {1, 514}
X(672) = X(i)-isoconjugate of X(j) for these (i,j): {1, 673}, {2, 105}, {4, 1814}, {6, 2481}, {7, 294}, {8, 1462}, {19, 31637}, {31, 18031}, {55, 34018}, {56, 36796}, {57, 14942}, {63, 36124}, {69, 8751}, {75, 1438}, {81, 13576}, {83, 46149}, {85, 2195}, {86, 18785}, {92, 36057}, {190, 1027}, {264, 32658}, {269, 6559}, {279, 28071}, {291, 6654}, {312, 1416}, {513, 666}, {514, 36086}, {516, 9503}, {518, 6185}, {522, 36146}, {644, 43930}, {648, 10099}, {650, 927}, {651, 885}, {653, 23696}, {663, 34085}, {664, 1024}, {667, 36803}, {668, 43929}, {693, 919}, {789, 29956}, {884, 4554}, {934, 28132}, {1016, 43921}, {1086, 5377}, {1292, 2402}, {1429, 33676}, {2111, 33674}, {2170, 39293}, {2191, 31638}, {2721, 38895}, {2991, 14267}, {3063, 46135}, {3261, 32666}, {3263, 41934}, {3512, 40724}, {3669, 36802}, {4391, 32735}, {4468, 36041}, {6084, 39272}, {6169, 9312}, {7261, 40754}, {10566, 35333}, {14727, 20980}, {26546, 35185}, {36816, 37129}
X(672) = barycentric product X(i)*X(j) for these {i,j}: {1, 518}, {3, 1861}, {4, 1818}, {6, 3912}, {7, 2340}, {8, 1458}, {9, 241}, {10, 3286}, {19, 25083}, {31, 3263}, {37, 18206}, {41, 40704}, {42, 30941}, {48, 46108}, {55, 9436}, {56, 3717}, {57, 3693}, {58, 3932}, {63, 5089}, {69, 2356}, {71, 15149}, {75, 2223}, {76, 9454}, {78, 1876}, {81, 3930}, {86, 20683}, {88, 14439}, {92, 20752}, {100, 2254}, {101, 918}, {105, 4712}, {110, 4088}, {190, 665}, {200, 34855}, {213, 18157}, {219, 5236}, {238, 22116}, {239, 3252}, {256, 4447}, {274, 39258}, {291, 8299}, {292, 17755}, {307, 37908}, {350, 40730}, {513, 1026}, {514, 2284}, {517, 36819}, {519, 34230}, {522, 2283}, {561, 9455}, {649, 42720}, {650, 1025}, {656, 4238}, {662, 24290}, {663, 883}, {664, 926}, {673, 6184}, {765, 3675}, {1126, 4966}, {1156, 35293}, {1293, 4925}, {1362, 14942}, {1438, 4437}, {1477, 40609}, {1642, 37131}, {1738, 34159}, {1914, 40217}, {2334, 4684}, {2338, 39063}, {2428, 4468}, {2481, 42079}, {2991, 17464}, {3126, 36086}, {3445, 4899}, {72) = 3509, 40781}, {3616, 14626}, {3900, 41353}, {3939, 43042}, {4557, 23829}, {4564, 17435}, {4572, 8638}, {4876, 34253}, {5378, 38989}, {7077, 39775}, {9315, 40883}, {9502, 36101}, {16728, 18785}, {18031, 39686}, {20662, 36807}, {31637, 42071}, {34337, 36057}, {34905, 40865}, {36037, 42758}, {39252, 40788}
X(672) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2481}, {2, 18031}, {3, 31637}, {6, 673}, {9, 36796}, {25, 36124}, {31, 105}, {32, 1438}, {41, 294}, {42, 13576}, {48, 1814}, {55, 14942}, {57, 34018}, {59, 39293}, {101, 666}, {109, 927}, {184, 36057}, {190, 36803}, {213, 18785}, {218, 31638}, {220, 6559}, {241, 85}, {518, 75}, {604, 1462}, {651, 34085}, {657, 28132}, {663, 885}, {664, 46135}, {665, 514}, {667, 1027}, {692, 36086}, {810, 10099}, {883, 4572}, {911, 9503}, {918, 3261}, {926, 522}, {1025, 4554}, {1026, 668}, {1110, 5377}, {1253, 28071}, {1362, 9436}, {1397, 1416}, {1415, 36146}, {1438, 6185}, {1458, 7}, {1818, 69}, {1861, 264}, {1876, 273}, {1914, 6654}, {1919, 43929}, {1946, 23696}, {1964, 46149}, {1973, 8751}, {2110, 33674}, {2175, 2195}, {2223, 1}, {2254, 693}, {2283, 664}, {2284, 190}, {2340, 8}, {2356, 4}, {2428, 37206}, {3063, 1024}, {3230, 36816}, {3248, 43921}, {3252, 335}, {3263, 561}, {3286, 86}, {3675, 1111}, {3693, 312}, {3717, 3596}, {3912, 76}, {3930, 321}, {3932, 313}, {3939, 36802}, {4088, 850}, {4238, 811}, {4447, 1909}, {4712, 3263}, {4966, 1269}, {5089, 92}, {5236, 331}, {6184, 3912}, {7077, 33676}, {8299, 350}, {8638, 663}, {9247, 32658}, {9436, 6063}, {9454, 6}, {9455, 31}, {9502, 30807}, {14439, 4358}, {14626, 5936}, {15149, 44129}, {16728, 18157}, {17435, 4858}, {17755, 1921}, {17798, 40724}, {18157, 6385}, {18206, 274}, {19554, 40754}, {20455, 1738}, {20662, 3008}, {20683, 10}, {20752, 63}, {20776, 1818}, {22116, 334}, {23225, 1459}, {23612, 4712}, {24290, 1577}, {25083, 304}, {25302, 34086}, {30941, 310}, {32739, 919}, {34230, 903}, {34253, 10030}, {34855, 1088}, {35293, 30806}, {36819, 18816}, {37908, 29}, {39046, 20347}, {39258, 37}, {39341, 33675}, {39775, 18033}, {40217, 18895}, {40704, 20567}, {40730, 291}, {40781, 40845}, {41353, 4569}, {42071, 1861}, {42079, 518}, {42341, 20907}, {42720, 1978}, {42758, 36038}, {42771, 35015}, {43924, 43930}, {46108, 1969}
X(672) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1475, 17474}, {1, 3730, 1334}, {1, 4253, 1475}, {2, 7, 30949}, {2, 144, 30946}, {2, 329, 30961}, {2, 5905, 30985}, {2, 6646, 31004}, {2, 17350, 24514}, {2, 20347, 20335}, {3, 218, 41}, {6, 31, 21764}, {6, 55, 2280}, {6, 71, 2269}, {6, 2276, 42}, {6, 17735, 1914}, {6, 21793, 5332}, {6, 42316, 55}, {7, 27626, 28351}, {9, 57, 40131}, {9, 63, 5282}, {9, 579, 1400}, {9, 17754, 2}, {10, 16552, 3691}, {35, 17745, 4251}, {36, 101, 1055}, {36, 5526, 101}, {37, 583, 2260}, {37, 24512, 3720}, {39, 213, 1193}, {40, 16572, 2082}, {42, 35270, 55}, {43, 1743, 37657}, {44, 910, 2348}, {44, 1155, 2246}, {44, 1575, 2238}, {44, 2245, 2183}, {44, 20331, 899}, {55, 42316, 41423}, {56, 220, 9310}, {57, 1423, 3598}, {63, 36276, 9318}, {65, 1212, 17451}, {72, 25066, 33299}, {75, 20646, 20632}, {101, 5030, 36}, {144, 27624, 1423}, {190, 37686, 350}, {219, 5120, 604}, {220, 5022, 56}, {292, 16514, 3009}, {329, 27659, 28387}, {484, 5540, 5011}, {573, 1743, 2347}, {583, 24512, 2350}, {910, 2348, 2246}, {942, 16601, 21808}, {1015, 3230, 1149}, {1025, 9436, 6168}, {1107, 2295, 10459}, {1108, 21871, 17452}, {1155, 2348, 910}, {1201, 23649, 2275}, {1202, 2269, 2280}, {1319, 6603, 17439}, {1334, 1475, 1}, {1334, 4253, 17474}, {1400, 8012, 40131}, {1447, 10025, 9318}, {1575, 2238, 899}, {1723, 1766, 40968}, {1737, 5179, 21044}, {1757, 2108, 3783}, {1783, 37305, 2202}, {1914, 17735, 902}, {2066, 5414, 1253}, {2174, 5124, 22054}, {2176, 2275, 1201}, {2183, 2272, 910}, {2223, 20683, 2340}, {2238, 20331, 1575}, {2280, 41423, 55}, {2323, 5053, 1404}, {2350, 40586, 3720}, {2590, 2591, 2246}, {2911, 36743, 48}, {3056, 36635, 20978}, {3501, 21384, 8}, {3729, 17026, 4441}, {3730, 4253, 1}, {3779, 20992, 2293}, {3869, 26690, 3061}, {3930, 14439, 3693}, {4251, 24047, 35}, {4438, 30953, 30751}, {4559, 43039, 1457}, {5030, 5526, 1055}, {5905, 27661, 28388}, {6203, 6204, 1445}, {6646, 27678, 28402}, {7113, 17796, 22356}, {14974, 16502, 3915}, {16549, 16552, 10}, {17137, 27109, 29960}, {17279, 24691, 30945}, {17279, 30945, 30821}, {17351, 21264, 24330}, {17756, 37657, 43}, {19037, 19038, 38293}, {21477, 23151, 25940}, {21872, 40133, 3057}, {24578, 39252, 239}, {27626, 44421, 7}


X(673) = TRILINEAR POLE OF LINE X(1)X(514)

Trilinears     bc/[b2 + c2 - a(b + c)]
Barycentrics    (a^2 + b^2 - a*c - b*c)*(a^2 - a*b - b*c + c^2) : :
X(673) = 3 X[2] + X[41845], 4 X[142] - 5 X[27191], 4 X[4422] - 5 X[18230], 3 X[5817] - 2 X[24828], 2 X[16593] + X[41845], 3 X[21168] - X[24817], 4 X[25351] - 3 X[38052], 3 X[27484] - X[33888]

X(673) lies on the conic {A,B,C,X(2),X(7)}}, the cubics K251, K323, K385, K506, K623, K980, K983, the curves CC9 and Q012, and these lines: {1, 17682}, {2, 11}, {4, 41785}, {6, 7}, {8, 4437}, {9, 75}, {10, 17681}, {19, 273}, {27, 162}, {56, 4209}, {57, 658}, {58, 29775}, {63, 37206}, {65, 27000}, {81, 39734}, {85, 2082}, {86, 142}, {87, 27498}, {88, 1638}, {101, 17761}, {103, 43672}, {144, 4373}, {150, 4904}, {169, 3673}, {218, 17753}, {226, 1174}, {234, 43192}, {238, 516}, {239, 335}, {241, 14189}, {272, 24606}, {277, 17170}, {310, 333}, {329, 42361}, {480, 24820}, {514, 30188}, {527, 666}, {537, 5223}, {545, 6172}, {655, 2161}, {664, 2170}, {668, 29479}, {675, 919}, {759, 4237}, {812, 1024}, {823, 8748}, {871, 37133}, {885, 900}, {893, 3752}, {897, 21180}, {909, 30379}, {910, 1447}, {918, 2400}, {927, 2291}, {954, 37502}, {958, 17691}, {971, 24813}, {1010, 16818}, {1027, 37129}, {1043, 29960}, {1111, 3732}, {1121, 4530}, {1125, 42335}, {1150, 37209}, {1155, 14197}, {1220, 17686}, {1268, 6666}, {1334, 32008}, {1416, 12573}, {1429, 9454}, {1434, 1475}, {1436, 1440}, {1465, 1945}, {1479, 17671}, {1492, 5135}, {1659, 31566}, {1714, 29461}, {1724, 29743}, {1733, 24846}, {1751, 15467}, {1760, 39943}, {1837, 26531}, {1861, 26001}, {1911, 27846}, {1981, 36123}, {2140, 4251}, {2160, 38340}, {2164, 7318}, {2246, 9318}, {2258, 2999}, {2259, 21617}, {2264, 41246}, {2280, 14828}, {2319, 4598}, {2329, 20257}, {2339, 19804}, {2345, 4422}, {2348, 10025}, {2364, 4604}, {2402, 6084}, {2432, 36100}, {2635, 23694}, {2669, 24378}, {2898, 5435}, {3216, 29434}, {3218, 37143}, {3243, 16834}, {3254, 5377}, {3286, 14953}, {3303, 27253}, {3451, 7175}, {3512, 16609}, {3570, 30997}, {3616, 17683}, {3684, 20335}, {3739, 24358}, {3871, 28742}, {3880, 40872}, {3885, 28961}, {3912, 5853}, {4008, 15299}, {4089, 34578}, {4253, 14377}, {4254, 25521}, {4312, 16468}, {4343, 45223}, {4361, 9055}, {4383, 39741}, {4393, 42871}, {4432, 16832}, {4534, 39351}, {4561, 18061}, {4599, 18087}, {4607, 24602}, {4649, 5542}, {4657, 25357}, {4859, 16779}, {5030, 17729}, {5086, 26526}, {5088, 43065}, {5220, 16816}, {5248, 17687}, {5253, 26964}, {5292, 29566}, {5299, 24790}, {5308, 8236}, {5698, 14267}, {5728, 36019}, {5759, 29243}, {5762, 24833}, {5817, 24828}, {6008, 24601}, {6650, 17768}, {6651, 15254}, {7008, 37279}, {7176, 40133}, {7384, 42356}, {7397, 35514}, {7402, 38149}, {7676, 8053}, {7677, 11349}, {7678, 44412}, {7770, 27299}, {8049, 32911}, {8257, 33794}, {8616, 31200}, {8751, 36099}, {9311, 9312}, {9328, 9329}, {9503, 43035}, {9669, 17675}, {10030, 20459}, {10099, 37142}, {11038, 17014}, {11375, 27183}, {11495, 20992}, {11683, 24633}, {12701, 27129}, {13390, 31565}, {13740, 30107}, {14923, 26653}, {14936, 24499}, {15481, 31349}, {15485, 31183}, {15570, 29584}, {15668, 28626}, {16060, 17030}, {16477, 30424}, {16484, 29571}, {16786, 17067}, {16826, 42819}, {16831, 38316}, {17050, 41239}, {17181, 20269}, {17284, 32941}, {17300, 20180}, {17308, 38200}, {17316, 20162}, {17370, 20195}, {17371, 28650}, {17379, 30712}, {17380, 20159}, {17680, 26561}, {17877, 21602}, {17963, 35466}, {20028, 27644}, {20059, 36606}, {20131, 26626}, {20132, 25557}, {20137, 29586}, {20147, 26806}, {20148, 39745}, {20153, 29592}, {20158, 39720}, {20332, 43929}, {20356, 25800}, {20367, 20605}, {20880, 33950}, {21044, 31640}, {21151, 37474}, {21168, 24817}, {21241, 30837}, {21264, 26244}, {23692, 36084}, {23696, 23707}, {24512, 40761}, {24584, 39732}, {24589, 37210}, {24610, 39746}, {24617, 24625}, {24619, 24624}, {24631, 40038}, {24632, 29756}, {24827, 31671}, {24880, 29564}, {25351, 29598}, {25532, 35342}, {26273, 27918}, {26801, 33826}, {26959, 33828}, {28916, 41228}, {29433, 29467}, {29437, 29483}, {29456, 29459}, {29473, 29742}, {29541, 40018}, {30332, 42318}, {30821, 32943}, {30946, 37658}, {32735, 36094}, {33150, 39723}, {36630, 40027}, {36662, 38037}, {36698, 43161}, {36799, 36803}, {36942, 38055}, {37207, 37686}, {37272, 42884}, {37650, 41325}, {38048, 39716}, {38941, 43057}

X(673) = midpoint of X(i) and X(j) for these {i,j}: {144, 4440}, {20533, 41845}
X(673) = reflection of X(i) in X(j) for these {i,j}: {7, 1086}, {190, 9}, {20533, 16593}, {31671, 24827}
X(673) = isogonal conjugate of X(672)
X(673) = isotomic conjugate of X(3912)
X(673) = complement of X(20533)
X(673) = anticomplement of X(16593)
X(673) = BSS(a^2→a) of X(98)
X(673) = antipode of X(7) in hyperbola {A,B,C,X(2),X(7)}}
X(673) = polar conjugate of X(1861)
X(673) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1280, 20552}, {39272, 20295}
X(673) = X(i)-complementary conjugate of X(j) for these (i,j): {2115, 3452}, {9499, 141}, {9500, 10}
X(673) = X(i)-Ceva conjugate of X(j) for these (i,j): {927, 885}, {2481, 14942}, {6185, 6654}, {18031, 31637}, {36802, 2402}, {39293, 36086}
X(673) = X(i)-cross conjugate of X(j) for these (i,j): {57, 9503}, {238, 86}, {239, 6654}, {291, 3226}, {294, 14942}, {516, 7}, {672, 1}, {812, 190}, {885, 927}, {1024, 36086}, {1438, 36124}, {1738, 75}, {2254, 664}, {3008, 2}, {4444, 35148}, {9436, 9311}, {9441, 21453}, {13576, 2481}, {18785, 105}, {20367, 81}, {20459, 6}, {20520, 43190}, {20605, 82}, {24618, 34234}, {24712, 1121}, {24715, 903}, {33854, 1220}, {36057, 31637}
X(673) = cevapoint of X(i) and X(j) for these (i,j): {1, 672}, {2, 239}, {6, 20470}, {7, 14189}, {9, 5853}, {57, 43035}, {105, 294}, {649, 27846}, {1086, 6084}, {1438, 36057}, {2170, 2254}, {3120, 21832}, {9312, 9436}, {13576, 18785}
X(673) = crosspoint of X(i) and X(j) for these (i,j): {57, 2111}, {291, 9442}, {2481, 34018}, {34085, 39293}
X(673) = crosssum of X(i) and X(j) for these (i,j): {1, 39341}, {6, 20672}, {9, 24578}, {238, 9441}, {17435, 24290}, {20683, 39258}
X(673) = trilinear pole of line {1, 514}
X(673) = crossdifference of every pair of points on line {665, 926}
X(673) = X(i)-isoconjugate of X(j) for these (i,j): {1, 672}, {2, 2223}, {3, 5089}, {4, 20752}, {6, 518}, {9, 1458}, {19, 1818}, {25, 25083}, {31, 3912}, {32, 3263}, {37, 3286}, {41, 9436}, {42, 18206}, {44, 34230}, {48, 1861}, {55, 241}, {56, 3693}, {57, 2340}, {58, 3930}, {59, 17435}, {63, 2356}, {75, 9454}, {76, 9455}, {81, 20683}, {86, 39258}, {100, 665}, {101, 2254}, {103, 9502}, {105, 6184}, {106, 14439}, {110, 24290}, {163, 4088}, {184, 46108}, {212, 5236}, {213, 30941}, {219, 1876}, {220, 34855}, {228, 15149}, {238, 3252}, {239, 40730}, {292, 8299}, {294, 1362}, {513, 2284}, {604, 3717}, {647, 4238}, {649, 1026}, {650, 2283}, {651, 926}, {657, 41353}, {663, 1025}, {667, 42720}, {673, 42079}, {692, 918}, {840, 1642}, {883, 3063}, {893, 4447}, {919, 3126}, {1214, 37908}, {1252, 3675}, {1280, 20662}, {1333, 3932}, {1438, 4712}, {1449, 14626}, {1814, 42071}, {1911, 17755}, {1914, 22116}, {1918, 18157}, {2175, 40704}, {2183, 36819}, {2210, 40217}, {2291, 35293}, {2414, 8642}, {2428, 3309}, {2481, 39686}, {2991, 20455}, {3290, 34159}, {4554, 8638}, {4899, 38266}, {4925, 34080}, {4966, 28615}, {5377, 35505}, {6168, 9439}, {6185, 23612}, {6335, 23225}, {7077, 34253}, {15344, 20728}, {17798, 40781}, {23102, 41934}, {32641, 42758}, {32658, 34337}
X(673) = barycentric product X(i)*X(j) for these {i,j}: {1, 2481}, {4, 31637}, {6, 18031}, {7, 14942}, {9, 34018}, {11, 39293}, {57, 36796}, {69, 36124}, {75, 105}, {76, 1438}, {85, 294}, {86, 13576}, {92, 1814}, {264, 36057}, {274, 18785}, {277, 31638}, {279, 6559}, {304, 8751}, {312, 1462}, {335, 6654}, {514, 666}, {522, 927}, {649, 36803}, {650, 34085}, {658, 28132}, {663, 46135}, {664, 885}, {668, 1027}, {693, 36086}, {811, 10099}, {884, 4572}, {919, 3261}, {1024, 4554}, {1088, 28071}, {1111, 5377}, {1416, 3596}, {1447, 33676}, {1969, 32658}, {1978, 43929}, {2195, 6063}, {2402, 37206}, {3112, 46149}, {3227, 36816}, {3676, 36802}, {3699, 43930}, {3912, 6185}, {4391, 36146}, {4449, 14727}, {7035, 43921}, {7261, 40724}, {9503, 30807}, {18026, 23696}, {29956, 37133}, {32666, 40495}, {32735, 35519}, {40754, 40845}
X(673) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 518}, {2, 3912}, {3, 1818}, {4, 1861}, {6, 672}, {7, 9436}, {8, 3717}, {9, 3693}, {10, 3932}, {19, 5089}, {25, 2356}, {27, 15149}, {31, 2223}, {32, 9454}, {34, 1876}, {37, 3930}, {42, 20683}, {44, 14439}, {48, 20752}, {55, 2340}, {56, 1458}, {57, 241}, {58, 3286}, {63, 25083}, {75, 3263}, {81, 18206}, {85, 40704}, {86, 30941}, {92, 46108}, {100, 1026}, {101, 2284}, {104, 36819}, {105, 1}, {106, 34230}, {109, 2283}, {145, 4899}, {162, 4238}, {171, 4447}, {190, 42720}, {213, 39258}, {238, 8299}, {239, 17755}, {244, 3675}, {269, 34855}, {274, 18157}, {278, 5236}, {291, 22116}, {292, 3252}, {294, 9}, {335, 40217}, {513, 2254}, {514, 918}, {518, 4712}, {523, 4088}, {560, 9455}, {649, 665}, {651, 1025}, {661, 24290}, {663, 926}, {664, 883}, {666, 190}, {672, 6184}, {884, 663}, {885, 522}, {910, 9502}, {919, 101}, {927, 664}, {934, 41353}, {1024, 650}, {1027, 513}, {1125, 4966}, {1155, 35293}, {1416, 56}, {1429, 34253}, {1438, 6}, {1447, 39775}, {1458, 1362}, {1462, 57}, {1738, 120}, {1769, 42758}, {1814, 63}, {1861, 34337}, {1911, 40730}, {2170, 17435}, {2195, 55}, {2223, 42079}, {2246, 1642}, {2254, 3126}, {2299, 37908}, {2334, 14626}, {2356, 42071}, {2402, 4468}, {2481, 75}, {3008, 16593}, {3290, 17464}, {3512, 40781}, {3616, 4684}, {3667, 4925}, {3676, 43042}, {3729, 40883}, {3912, 4437}, {4366, 27919}, {4449, 42341}, {4712, 23102}, {5377, 765}, {5853, 40609}, {6180, 6168}, {6559, 346}, {6654, 239}, {7192, 23829}, {7193, 20778}, {8751, 19}, {9453, 9451}, {9454, 39686}, {9503, 36101}, {10099, 656}, {13576, 10}, {14189, 36905}, {14197, 28850}, {14267, 1738}, {14625, 5257}, {14942, 8}, {18031, 76}, {18206, 16728}, {18785, 37}, {20470, 39046}, {20780, 20749}, {21051, 21959}, {21138, 23773}, {21956, 20482}, {23694, 2808}, {23696, 521}, {23770, 20504}, {27818, 10029}, {27846, 38989}, {28071, 200}, {28132, 3239}, {29956, 3250}, {31637, 69}, {31638, 344}, {32658, 48}, {32666, 692}, {32735, 109}, {33674, 17794}, {33676, 4518}, {34018, 85}, {34063, 25302}, {34085, 4554}, {34906, 40865}, {35333, 4553}, {36041, 1292}, {36057, 3}, {36086, 100}, {36111, 26706}, {36124, 4}, {36138, 8693}, {36146, 651}, {36796, 312}, {36802, 3699}, {36803, 1978}, {36816, 536}, {37206, 2414}, {39293, 4998}, {40724, 4645}, {40754, 3509}, {40761, 39252}, {41934, 1438}, {42079, 23612}, {42754, 42770}, {43035, 39063}, {43921, 244}, {43929, 649}, {43930, 3676}, {46135, 4572}, {46149, 38}, {46163, 46148}
X(673) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 39341, 42079}, {2, 20172, 5263}, {2, 20533, 16593}, {2, 31058, 30857}, {2, 41845, 20533}, {11, 26007, 2}, {75, 20445, 20431}, {101, 17761, 24203}, {105, 13576, 14942}, {142, 16503, 86}, {239, 335, 32029}, {294, 1462, 1814}, {335, 32029, 24841}, {335, 32096, 239}, {379, 5222, 41245}, {1001, 2550, 5263}, {1111, 5540, 3732}, {2170, 9317, 664}, {2280, 30949, 14828}, {2481, 31638, 6559}, {4000, 5819, 7}, {4366, 26582, 5263}, {4384, 17738, 17755}, {10707, 30857, 31058}, {10707, 31226, 30857}, {14936, 43063, 24499}, {16706, 20179, 86}, {17023, 24588, 16054}, {17026, 24586, 14829}, {17259, 20181, 2345}, {17686, 26965, 1220}, {17691, 27304, 958}, {17738, 17755, 190}, {30857, 31226, 2}, {39341, 42079, 37138}, {40565, 40566, 14942}


X(674) = CROSSDIFFERENCE OF X(6) AND X(514)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[b3 + c3 - a(b2 + c2)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(674) lies on the line at infinity.

X(674) lies on these (parallel) lines: 6,31   30,511   51,210

X(674) = isogonal conjugate of X(675)
X(674) = crossdifference of every pair of points on line X(6)X(514)


X(675) = TRILINEAR POLE OF LINE X(6)X(514)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc/[b3 + c3 - a(b2 + c2)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(675) lies on the circumcircle.

X(675) lies on these lines: 2,101   7,109   27,112   75,100   86,110   99,310   108,273   335,813   673,919   789,871   901,903   934,1088

X(675) = isogonal conjugate of X(674)
X(675) = isotomic conjugate of X(3006)
X(675) = inverse-in-orthoptic-circle-of-Steiner-inellipse of X(116)


X(676) = CROSSDIFFERENCE OF X(3) AND X(101)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b - c)[b3 + c3 - 2a3 + (b + c)(a2 - bc)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(676) = radical center of the circumcircle, nine-point circle, and incircle (Wilson Stothers, 3/31/2003)

X(676) lies on these lines: 11,244   105,659   230,231   928,942

X(676) = isogonal conjugate of X(677)
X(676) = crosspoint of X(105) and X(108)
X(676) = crosssum of X(i) and X(j) for these (i,j): (6,926), (518,521)
X(676) = crossdifference of every pair of points on line X(3)X(101)
X(676) = circumcenter of X(11)X(105)X(108)
X(676) = intersection of tangents to circumcircle at X(105) and X(108), and to nine-point-circle at X(11)
X(676) = pole of line X(25)X(105) wrt circumcircle
X(676) = perspector of circumconic centered at X(1566)
X(676) = center of circumconic that is locus of trilinear poles of lines passing through X(1566)
X(676) = X(2)-Ceva conjugate of X(1566)


X(677) = TRILINEAR POLE OF LINE X(3)X(101)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/{(b - c)[b3 + c3 - 2a3 + (b + c)(a2 - bc)]}
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(677) lies on the MacBeath circumconic and on these lines: 59,1813   103,901   518,1814   521,651   765,1332   883,2398   1252,1331   1815,2340   2323,2338

X(677) = isogonal conjugate of X(676)
X(677) = isotomic conjugate of isogonal conjugate of X(32642)


X(678) = CROSSPOINT OF X(1) AND X(44)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - 2a)2
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(678) lies on the incentral inellipse (a.k.a. Hofstadter ellipse E(1/2)).

X(678) lies on these lines: 1,88   44,902   45,55

X(678) = isogonal conjugate of X(679)
X(678) = X(1)-Ceva conjugate of X(44)
X(678) = crosspoint of X(1) and X(44)
X(678) = crosssum of X(i) and X(j) for these (i,j): (1,88), (244,1022)
X(678) = crossdifference of every pair of points on line X(88)X(1022)
X(678) = trilinear square of X(44)


X(679) = ISOGONAL CONJUGATE OF X(678)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/(b + c - 2a)2
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(679) lies on these lines: 44,88   320,519

X(679) = isogonal conjugate of X(678)
X(679) = isotomic conjugate of X(4738)
X(679) = trilinear pole of line X(88)X(1022)
X(679) = cevapoint of X(1) and X(88)
X(679) = X(1)-cross conjugate of X(88)


X(680) = CROSSDIFFERENCE OF X(6) AND X(158)

Trilinears        f(A,B,C): f(B,C,A) : f(C,A,B), where f(A,B,C) = cos2A (sin B cos2B - sin C cos2C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sin A f(A,B,C)

As the isogonal conjugate of a point on the circumcircle, X(680) lies on the line at infinity.

X(680) lies on this line: 30,511  

X(680) = isogonal conjugate of X(681)
X(680) = complementary conjugate of X(35970)
X(680) = X(4)-Ceva conjugate of X(35970)
X(680) = crossdifference of every pair of points on line X(6)X(158)


X(681) = TRILINEAR POLE OF LINE X(6)X(158)

Trilinears        f(A,B,C): f(B,C,A) : f(C,A,B), where f(A,B,C) = (sec2A)/(sin B cos2B - sin C cos2C)
Barycentrics  (sin A)f(A,B,C): (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(681) lies on the circumcircle.

X(681) lies on these lines: 110,823   1612,1924

X(681) = isogonal conjugate of X(680)
X(681) = isotomic conjugate of X(35521)
X(681) = anticomplement of X(35970)
X(681) = polar-circle-inverse of X(38969)


X(682) = POINT ARNEB

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sec B csc3C + sec C csc3B
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sin A f(A,B,C)

X(682) lies on these lines: 3,69   154,237   248,695

X(682) = isogonal conjugate of X(683)
X(682) = crosspoint of X(3) and X(32)
X(682) = crosssum of X(4) and X(76)


X(683) = ISOGONAL CONJUGATE OF X(682)

Trilinears        f(A,B,C): f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/[sec B csc3C + sec C csc3B]
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sin A f(A,B,C)

X(683) lies on this line: 25,305

X(683) = isogonal conjugate of X(682)
X(683) = isotomic conjugate of X(6467)
X(683) = cevapoint of X(4) and X(76)


X(684) = CROSSDIFFERENCE OF X(4) AND X(32)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sec B sin3 C - sec C sin3 B
Trilinears    a(b^2 + c^2 - a^2)(b^2 - c^2)(a^2b^2 + a^2c^2 - b^4 - c^4) : :

X(684) lies on these lines: 110,351   114,132   122,125   147,804   325,523   520,647   669,924   2491,3569

X(684) = isogonal conjugate of X(685)
X(684) = isotomic conjugate of X(22456)
X(684) = X(2)-Ceva conjugate of X(39000)
X(684) = perspector of hyperbola {A,B,C,X(3),X(511)}}
X(684) = crosspoint of X(99) and X(287)
X(684) = crosssum of X(i) and X(j) for these (i,j): (98,879), (232,512)
X(684) = crossdifference of every pair of points on line X(4)X(32)
X(684) = X(92)-isoconjugate of X(2715)


X(685) = TRILINEAR POLE OF LINE X(4)X(98)

Trilinears        f(A,B,C): f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/(sec B sin3 C - sec C sin3 B)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sin A f(A,B,C)

X(685) lies on these lines: 98,468   110,850   250,523   287,297

X(685) = isogonal conjugate of X(684)
X(685) = isotomic conjugate of X(6333)
X(685) = polar conjugate of X(2799)
X(685) = trilinear pole of line X(4)X(32)
X(685) = trilinear product of circumcircle intercepts of line X(31)X(92)
X(685) = vertex conjugate of MacBeath circumconic intercepts of Euler line


X(686) = CROSSDIFFERENCE OF X(4) AND X(110)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sec B csc(A - B) + sec C csc(A - C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sin A f(A,B,C)
Barycentrics    a^2 (b^2 - c^2) (a^2 - b^2 - c^2) (a^4 (b^2 + c^2) - 2 a^2 (b^4 - b^2 c^2 + c^4) + (b^2 - c^2)^2 (b^2 + c^2)) : :

X(686) lies on these lines: 115,125   184,351   520,647

X(686) = isogonal conjugate of X(687)
X(686) = X(2)-Ceva conjugate of X(39005)
X(686) = perspector of hyperbola {A,B,C,X(3),X(523)}}
X(686) = crossdifference of every pair of points on line X(4)X(110)
X(686) = areal center of cevian triangles of X(74) and X(110)


X(687) = TRILINEAR POLE OF LINE X(4)X(110)

Trilinears        f(A,B,C): f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/[sec B csc(A - B) + sec C csc(A - C)]
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sin A f(A,B,C)

X(687) lies on these lines: 107,250   249,648

X(687) = isogonal conjugate of X(686)
X(687) = isotomic conjugate of X(6334)


X(688) = CROSSDIFFERENCE OF X(6) AND X(76)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3(b4 - c4)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(688) lies on the line at infinity.

X(688) lies on these (parallel) lines: 6,882   30,511   669,864   798,872

X(688) = isogonal conjugate of X(689)
X(688) = crosssum of X(99) and X(670)
X(688) = complementary conjugate of X(35971)
X(688) = crossdifference of every pair of points on line X(6)X(76)


X(689) = TRILINEAR POLE OF LINE X(6)X(76)

Trilinears    1/[a3(b4 - c4)] : :

X(689) is the center of the bianticevian conic of X(1) and X(75). This conic is a rectangular hyperbola passing through X(1), X(75), the excenters and their isotomic conjugates. (Randy Hutson, July 11, 2019)

X(689) lies on the circumcircle and these lines: 1,719   2,733   6,703   75,745   76,755   82,715   83,729   110,670   111,308   251,699   662,787   741,873   799,813

X(689) = isogonal conjugate of X(688)
X(689) = isotomic conjugate of X(3005)
X(689) = anticomplement of X(35971)
X(689) = Ψ(X(32), X(2))
X(689) = Ψ(X(39), X(2))
X(689) = perspector of ABC and the tangential triangle, wrt the tangential triangle, of the bianticevian conic of X(2) and X(6) (see X(4577))


X(690) = CROSSDIFFERENCE OF LINE X(6) AND X(110)

Trilinears    bc(b2 - c2)(2a2 - b2 - c2) : :
Barycentrics    (cot B - cot C) (2 cot A - cot B - cot C) : :

Let NaNbNc and Na'Nb'Nc' be the outer and inner Napoleon triangles, respectively. Let A' be the isogonal conjugate of Na wrt Na'Nb'Nc', and define B' and C' cyclically. Let A″ be the isogonal conjugate of Na' wrt NaNbNc, and define B″ and C″ cyclically. The lines A'A″, B'B″, C'C″ concur in X(690). (Randy Hutson, November 30, 2015)

As the isogonal conjugate of a point on the circumcircle, X(690) lies on the line at infinity.

X(690) lies on these (parallel) lines: {2, 5465}, {3, 6334}, {4, 11005}, {5, 39509}, {6, 36880}, {10, 18004}, {13, 16255}, {14, 16256}, {30, 511}, {39, 2491}, {67, 10097}, {69, 13232}, {74, 98}, {76, 14295}, {83, 18010}, {99, 110}, {111, 39446}, {113, 114}, {115, 125}, {146, 147}, {148, 3448}, {182, 39499}, {226, 18006}, {265, 6321}, {321, 18003}, {351, 1641}, {399, 13188}, {599, 34206}, {620, 5113}, {669, 2525}, {671, 5466}, {684, 14981}, {685, 16077}, {691, 20404}, {850, 8599}, {882, 1916}, {887, 14824}, {895, 2408}, {905, 42653}, {1112, 5186}, {1177, 39840}, {1281, 44433}, {1495, 32120}, {1511, 33813}, {1539, 22505}, {1569, 9419}, {1962, 14413}, {2254, 2292}, {2378, 11612}, {2379, 11613}, {2395, 34246}, {2422, 14901}, {2435, 11744}, {2485, 7631}, {2489, 7651}, {2492, 14279}, {2510, 8574}, {2528, 8664}, {2530, 42661}, {2533, 7265}, {2650, 4895}, {2682, 5099}, {2931, 39828}, {2935, 39841}, {2936, 34519}, {2948, 13174}, {3005, 17436}, {3023, 3024}, {3027, 3028}, {3029, 3031}, {3044, 3047}, {3095, 45911}, {3111, 9828}, {3265, 8651}, {3426, 14220}, {3455, 42659}, {3743, 3960}, {3762, 4647}, {3801, 4170}, {3806, 8665}, {4010, 4707}, {4027, 13193}, {4049, 11599}, {4080, 18011}, {4088, 4730}, {4122, 4761}, {4444, 18009}, {4729, 4808}, {4736, 13277}, {4773, 8659}, {4846, 15453}, {4983, 21124}, {5026, 6593}, {5092, 14271}, {5095, 5477}, {5108, 9129}, {5181, 21905}, {5461, 9183}, {5467, 14559}, {5476, 11622}, {5485, 18012}, {5489, 9409}, {5503, 43674}, {5655, 8724}, {5916, 5917}, {5976, 38650}, {5987, 14318}, {5989, 38661}, {6027, 44051}, {6033, 7728}, {6036, 6699}, {6054, 10706}, {6055, 11656}, {6130, 11623}, {6132, 6140}, {6226, 7726}, {6227, 7725}, {6233, 6236}, {6319, 7732}, {6320, 7733}, {6323, 6325}, {6545, 30592}, {6721, 12900}, {6722, 6723}, {6771, 16233}, {6774, 16234}, {7472, 9181}, {7709, 15920}, {7731, 39837}, {7779, 13186}, {7893, 13237}, {7970, 7978}, {7983, 7984}, {8030, 9135}, {8591, 9131}, {8593, 41720}, {8663, 16892}, {8782, 13210}, {8980, 8994}, {8997, 8998}, {9080, 9187}, {9115, 30455}, {9117, 30454}, {9161, 43654}, {9178, 34898}, {9209, 22264}, {9507, 11725}, {9759, 9877}, {9860, 9904}, {9861, 9919}, {9862, 9984}, {9864, 12368}, {9880, 44203}, {9966, 16510}, {9970, 12177}, {10053, 10065}, {10069, 10081}, {10086, 10088}, {10089, 10091}, {10099, 10693}, {10113, 22515}, {10117, 32122}, {10118, 39851}, {10119, 39850}, {10168, 11621}, {10190, 36521}, {10264, 15535}, {10278, 36523}, {10293, 14380}, {10620, 12188}, {10663, 39829}, {10664, 39830}, {10681, 39858}, {10682, 39859}, {10721, 10722}, {10723, 10733}, {10752, 10753}, {10767, 10768}, {10769, 10778}, {10992, 30714}, {11061, 25052}, {11161, 13169}, {11257, 38520}, {11472, 15928}, {11579, 21732}, {11608, 18013}, {11611, 18015}, {11616, 12584}, {11632, 14849}, {11709, 11710}, {11711, 11720}, {11723, 11724}, {12041, 12042}, {12075, 33294}, {12121, 38730}, {12131, 12133}, {12168, 39803}, {12176, 12192}, {12178, 12327}, {12179, 12365}, {12180, 12366}, {12181, 12369}, {12182, 12371}, {12183, 12372}, {12184, 12373}, {12185, 12374}, {12186, 12377}, {12187, 12378}, {12189, 12381}, {12190, 12382}, {12228, 39805}, {12236, 39806}, {12258, 21181}, {12273, 39807}, {12284, 39808}, {12295, 39809}, {12302, 39812}, {12310, 13175}, {12319, 39813}, {12375, 35878}, {12376, 35879}, {12383, 13172}, {12494, 34113}, {12596, 39819}, {12661, 39821}, {12888, 39822}, {12891, 39823}, {12892, 39824}, {12893, 39825}, {12901, 39831}, {12902, 38733}, {12903, 13182}, {12904, 13183}, {13171, 39832}, {13173, 13204}, {13176, 13208}, {13177, 13209}, {13178, 13211}, {13179, 13212}, {13180, 13213}, {13181, 13214}, {13184, 13215}, {13185, 13216}, {13187, 14316}, {13189, 13217}, {13190, 13218}, {13198, 39834}, {13201, 39836}, {13202, 39838}, {13203, 39842}, {13233, 32305}, {13234, 45163}, {13248, 39848}, {13287, 39852}, {13288, 39853}, {13289, 39854}, {13293, 39860}, {13417, 39846}, {13642, 13643}, {13653, 13654}, {13761, 13762}, {13773, 13774}, {13967, 13969}, {13989, 13990}, {14061, 15059}, {14094, 23235}, {14337, 14339}, {14338, 14340}, {14345, 15131}, {14430, 21020}, {14606, 34364}, {14639, 14644}, {14643, 14850}, {14653, 16508}, {14670, 23108}, {14683, 19598}, {15035, 21166}, {15054, 38664}, {15055, 34473}, {15061, 38224}, {15070, 15099}, {15088, 15092}, {15303, 18311}, {15328, 34802}, {16003, 16220}, {16093, 32244}, {16111, 18556}, {16163, 38738}, {17708, 32729}, {17838, 39820}, {17847, 39849}, {18008, 40016}, {18309, 25562}, {18310, 19662}, {18932, 39804}, {18947, 39833}, {19055, 19059}, {19056, 19060}, {19108, 19110}, {19109, 19111}, {19138, 39811}, {19193, 39814}, {19208, 39843}, {19456, 39810}, {19469, 39815}, {19479, 39818}, {19482, 39826}, {19483, 39827}, {19504, 39839}, {19505, 39844}, {19506, 39847}, {19507, 39855}, {19508, 39856}, {20127, 38741}, {20397, 20398}, {21649, 39817}, {22089, 34952}, {22159, 37085}, {22504, 22583}, {22514, 22586}, {23306, 39816}, {23315, 39845}, {23514, 23515}, {23992, 41177}, {24978, 36253}, {24981, 30220}, {25315, 25320}, {25322, 25328}, {25325, 25329}, {25334, 25335}, {25556, 32135}, {31290, 40502}, {32111, 32112}, {32271, 33752}, {34127, 34128}, {34453, 34454}, {34801, 40048}, {34964, 39512}, {35824, 35826}, {35825, 35827}, {36165, 45284}, {36518, 36519}, {37853, 38747}, {38481, 38482}, {38497, 38498}, {38499, 38508}, {38555, 38556}, {38557, 38566}, {38626, 38627}, {38628, 38632}, {38633, 38634}, {38635, 38638}, {38641, 38642}, {38653, 38654}, {38723, 38731}, {38724, 38732}, {38725, 38735}, {38726, 38736}, {38727, 38737}, {38728, 38739}, {38729, 38740}, {38742, 38788}, {38743, 38789}, {38744, 38790}, {38745, 38791}, {38746, 38792}, {38748, 38793}, {38750, 38794}, {38751, 38795}, {39482, 39491}, {39495, 39501}, {39513, 39518}, {40486, 40513}, {41189, 41193}, {41190, 41192}, {43532, 43665}

X(690) = isogonal conjugate of X(691)
X(690) = isotomic conjugate of X(892)
X(690) = X(67)-Ceva conjugate of X(125)
X(690) = crosssum of X(i) and X(j) for these (i,j): (6,351), (187,512), (523,858)
X(690) = crossdifference of every pair of points on line X(6)X(110)
X(690) = orthopoint of X(542)
X(690) = X(2)-Ceva conjugate of X(23992)
X(690) = polar conjugate of the isotomic conjugate of X(14417)
X(690) = X(523)-of-1st-Brocard-triangle
X(690) = X(512)-of-4th-Brocard-triangle
X(690) = X(512)-of-orthocentroidal-triangle
X(690) = X(512)-of-X(4)-Brocard-triangle
X(690) = X(1499)-of-McCay-triangle
X(690) = X(1499)-of-anti-McCay-triangle
X(690) = crosspoint of X(I) and X(J) for these (I,J): (99,671), (110,1177)
X(690) = intersection of tangents to Steiner inellipse at X(115) and X(2482)
X(690) = crosspoint wrt medial triangle of X(115) and X(2482)
X(690) = trilinear pole of line X(1648)X(1649)
X(690) = barycentric product X(523)*X(524)
X(690) = 1st-Brocard-isogonal conjugate of X(1316)
X(690) = 1st-Brocard-isotomic conjugate of X(6)
X(690) = isogonal conjugate of the anticomplement of X(5099)
X(690) = isotomic conjugate of the anticomplement of X(23992)
X(690) = isotomic conjugate of the complement of X(39356)
X(690) = isotomic conjugate of the isogonal conjugate of X(351)
X(690) = isogonal conjugate of the isotomic conjugate of X(35522)
X(690) = isotomic conjugate of the polar conjugate of X(14273)
X(690) = polar conjugate of the isotomic conjugate of X(14417)
X(690) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 9144, 5465}, {2, 9185, 9189}, {3, 14270, 39477}, {4, 18331, 11005}, {4, 44427, 16230}, {10, 22037, 18004}, {74, 22265, 98}, {99, 15342, 110}, {115, 125, 15359}, {115, 15357, 125}, {125, 1637, 42736}, {125, 16278, 115}, {351, 1649, 9125}, {351, 11183, 45680}, {351, 14424, 14417}, {671, 5466, 18007}, {1637, 9148, 45688}, {1640, 3569, 1637}, {2254, 2292, 42666}, {2489, 7652, 7651}, {2682, 35582, 14114}, {3268, 9123, 9168}, {3268, 9147, 45687}, {3268, 39904, 5652}, {3569, 9148, 11182}, {4120, 30574, 14431}, {4750, 14432, 14419}, {5099, 38395, 2682}, {5113, 45693, 11176}, {5652, 9147, 5027}, {5653, 9138, 9208}, {6132, 11615, 6140}, {6321, 15545, 265}, {6699, 33511, 6036}, {7472, 14999, 9181}, {8552, 11615, 6132}, {9123, 9168, 45687}, {9125, 14417, 1649}, {9138, 14698, 14697}, {9138, 14932, 9185}, {9144, 11006, 2}, {9147, 9168, 9123}, {9148, 14420, 1637}, {9148, 36255, 125}, {9180, 14932, 5465}, {9185, 9191, 2}, {9185, 14698, 14932}, {9185, 14932, 14697}, {9194, 9195, 9189}, {9200, 9201, 1637}, {9200, 14447, 9201}, {9201, 14446, 9200}, {9204, 9205, 14417}, {9485, 44010, 9131}, {11176, 24284, 45693}, {13169, 14833, 11161}, {13291, 36255, 1637}, {13636, 13722, 8371}, {14270, 44826, 3}, {14419, 30595, 4750}, {14419, 30605, 14432}, {14420, 42734, 9200}, {14420, 42735, 9201}, {14420, 42737, 13291}, {14446, 14447, 1637}, {14643, 14850, 15561}, {14698, 39905, 5653}, {15357, 16278, 15359}, {30508, 30509, 9168}, {30595, 30605, 14419}, {31953, 42738, 1637}, {32193, 45689, 44564}, {32312, 32313, 351}, {42733, 42738, 115}, {42734, 42735, 9148}, {45327, 45336, 44564}, {45336, 45689, 45692}


X(691) = TRILINEAR POLE OF LINE X(6)X(110)

Trilinears    a/[(b2 - c2)(2a2 - b2 - c2)] : :

Let LA be the line of reflection of the line X(6)X(13) in line BC, and define LB and LC cyclically. Let A' = LB∩LC, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(691). (Randy Hutson, 9/23/2011)

Let A', B', C' be the intersections of line X(2)X(6) and lines BC, CA, AB, resp. The circumcircles of AB'C', BC'A', CA'B' concur in X(691). (Randy Hutson, December 26, 2015)

Let A1B1C1, A2B2C2 and A3B3C3 be the 1st, 2nd and 3rd Parry triangles. Let A' be the trilinear product A1*A2*A3, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(691). (Randy Hutson, February 10, 2016)

Let A'B'C' be the vertex-triangle of the 1st and 4th Brocard triangles. The lines AA', BB', CC' concur in X(691). (Randy Hutson, October 8, 2019)

X(691) lies on the circumcircle, the circle {X(2),X(3),X(6),X(111)}, and these lines: 3,842   6,843   23,111   30,98   74,511   99,523  110,249   112,250   182,2698   316,858   376,477   741,923   759,897   805,882

X(691) = reflection of X(i) in X(j) for these (i,j): (23,187), (316,858), (842,3)
X(691) = isogonal conjugate of X(690)
X(691) = isotomic conjugate of X(35522)
X(691) = cevapoint of X(6) and X(351)
X(691) = X(i)-cross conjugate of X(j) for these (i,j): (23,250), (187,249), (351,6)
X(691) = circumcircle-antipode of X(842)
X(691) = anticomplement of X(5099)
X(691) = intersection, other than A, B, C, of circumcircle and hyperbola {A,B,C,PU(2)}}
X(691) = intersection, other than A, B, C, of circumcircle and hyperbola {A,B,C,PU(62)}}
X(691) = Λ(X(115), X(125))
X(691) = Ψ(X(2), X(99))
X(691) = Ψ(X(650), X(21))
X(691) = reflection of X(99) in the Euler line
X(691) = reflection of X(110) in the Brocard axis
X(691) = crossdifference of every pair of points on line X(1648)X(1649)
X(691) = inverse-in-circle-O(15,16) of X(111)
X(691) = {X(15),X(16)}-harmonic conjugate of X(111)
X(691) = X(1577)-isoconjugate of X(187)
X(691) = 1st-Parry-to-ABC similarity image of X(23)
X(691) = X(843)-of-circumsymmedial-triangle
X(691) = perspector of circumsymmedial triangle and cross-triangle of ABC and circumcevian triangle of X(187)
X(691) = perspector of circummedial triangle and cross-triangle of ABC and circumcevian triangle of X(23)
X(691) = barycentric product X(99)*X(111) (circumcircle-X(2) antipodes)


X(692) = X(110)-CEVA CONJUGATE OF X(101)

Trilinears    a2/(b - c) : b2/(c - a) : c2/(a - b)
Barycentrics    a3/(b - c) : b3/(c - a) : c3/(a - b)

Let A'B'C' be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung). A'B'C' is homothetic to the anti-Mandart-incircle triangle at X(692). (Randy Hutson, January 15, 2019)

X(692) lies on these lines: 25,913   48,911   55,184   59,513   99,785   100,110   101,926   154,197   163,906   182,1001   206,219   213,923   667,1110   813,825

X(692) = isogonal conjugate of X(693)
X(692) = isotomic conjugate of complement of polar conjugate of isogonal conjugate of X(23191)
X(692) = isotomic conjugate of anticomplement of polar conjugate of isogonal conjugate of X(23228)
X(692) = complement of X(21293)
X(692) = anticomplement of X(21252)
X(692) = X(i)-Ceva conjugate of X(j) for these (i,j): (59,6), (110,101)
X(692) = cevapoint of X(110) and X(163)
X(692) = crosssum of X(i) and X(j) for these (i,j): (2,149), (513,905), (514,522), (764,1086)
X(692) = crossdifference of every pair of points on line X(918)X(1086)
X(692) = barycentric product of PU(i) for i in (26, 49)
X(692) = barycentric product of circumcircle intercepts of line X(1)X(6)
X(692) = barycentric product X(6)*X(100)
X(692) = trilinear pole of line X(31)X(32)
X(692) = X(92)-isoconjugate of X(905)
X(692) = trilinear product of circumcircle intercepts of line X(6)X(31)
X(692) = trilinear product X(6)*X(101)


X(693) = ISOTOMIC CONJUGATE OF X(100)

Trilinears    (b - c)/a2 : (c - a)/b2 : (a - b)/c2
Barycentrics   (b - c)/a : (c - a)/b : (a - b)/c
Barycentrics   |AP(1)| - |AU(1)| : :
Barycentrics    directed distance from A to line X(1)X(6) : :

X(693): Let A29B29C29 and A30B30C30 be Gemini triangles 29 and 30, resp. Let E29 and E30 be the {ABC, Gemini 29}-circumconic and {ABC, Gemini 30}-circumconic, resp. Let A' be the intersection of the tangent to E29 at A29 and the tangent to E30 at A30. Define B' and C' cyclically. The lines AA', BB', CC″ concur in X(693). (Randy Hutson, January 15, 2019)

X(693) lies on these lines: 2,650   76,764   100,927   320,350   321,824   325,523   514,661   649,812

X(693) = isogonal conjugate of X(692)
X(693) = isotomic conjugate of X(100)
X(693) = complement of X(17494)
X(693) = anticomplement of X(650)
X(693) = cevapoint of X(2) and X(149)
X(693) = X(i)-cross conjugate of X(j) for these (i,j): (11,2), (523,514)
X(693) = crosssum of X(i) and X(j) for these (i,j): (31,667), (42,663), (649,1475)
X(693) = crossdifference of every pair of points on line X(31)X(32)
X(693) = trilinear pole of line X(918)X(1086)
X(693) = pole wrt polar circle of trilinear polar of X(1783) (line X(19)X(25))
X(693) = polar conjugate of X(1783)
X(693) = X(6)-isoconjugate of X(101)
X(693) = complement of polar conjugate of isogonal conjugate of X(22160)
X(693) = perspector of side- and vertex-triangles of Gemini triangles 29 and 30


X(694) = ISOGONAL CONJUGATE OF X(385)

Trilinears    a/(a4 - b2c2) : :

Let A'B'C' be the 1st Brocard triangle. Let La be the trilinear polar of A', and define Lb and Lc cyclically. Let A″ = Lb∩Lc, and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X(694). (Randy Hutson, December 26, 2015)

Let A'B'C' be the 1st Brocard triangle. Let La be the trilinear polar, wrt A'B'C', of A, and define Lb, Lc cyclically. Let A″ = Lb∩Lc, and define B″, C″ cyclically. A'A″, B'B″, C'C″ concur in X(694). (Randy Hutson, December 26, 2015)

Let A'B'C' be the 1st Brocard triangle. Let A″ be the trilinear pole of line B'C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(385). Let A* be the trilinear pole of line B″C″, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(694). (Randy Hutson, December 26, 2015)

Let DEF and D'E'F' be the 1st and 2nd Sharygin triangles. Let A' be the barycentric product D*D', and define B' and C' cyclically. The lines AA', BB', CC' concur in X(694). (Randy Hutson, December 26, 2015)

X(694) lies on these lines: 6,1084   37,256   42,893   110,251   111,805   141,308   172,904   257,335   351,881   384,695   882,888

X(694) = reflection of X(i) in X(j) for these (i,j): (6,1084), (670,141)
X(694) = isogonal conjugate of X(385)
X(694) = isotomic conjugate of X(3978)
X(694) = anticomplement of X(39080)
X(694) = crossdifference of PU(185)
X(694) = perspector of conic {A,B,C,X(805),X(18829),X(36897)}}
X(694) = X(63)-isoconjugate of X(419)
X(694) = cevapoint of X(384) and X(385)
X(694) = X(i)-cross conjugate of X(j) for these (i,j): (446,232), (511,6)
X(694) = trilinear pole of PU(1) (line X(39)X(512))
X(694) = barycentric product of circumcircle intercepts of line PU(11), alias X(141)X(523))
X(694) = antipode of X(6) in hyperbola {A,B,C,X(2),X(6)}}
X(694) = perspector of circumconic centered at X(1567)
X(694) = center of circumconic that is locus of trilinear poles of lines passing through X(1567)
X(694) = X(2)-Ceva conjugate of X(39092)
X(694) = cevapoint of X(6) and X(2076)
X(694) = polar conjugate of isotomic conjugate of X(36214)
X(694) = perspector of ABC and unary cofactor triangle of 1st Brocard triangle
X(694) = perspector of ABC and unary cofactor triangle of 3rd Brocard triangle
X(694) = perspector of ABC and unary cofactor triangle of 1st Neuberg triangle
X(694) = center of the perspeconic of these triangles: 1st and 3rd Brocard
X(694) = perspector of ABC and vertex-triangle of anticevian triangles of PU(1)
X(694) = center of bicevian conic of PU(6)
X(694) = barycentric product of Steiner circumellipse intercepts of line PU(1)


X(695) = ISOGONAL CONJUGATE OF X(384)

Trilinears    a/(a4 + b2c2) : :

Let A'B'C' be the cross-triangle of the 1st and 2nd Neuberg triangles. Let A″ be the crossdifference of B' and C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(695). Let La be the trilinear polar of A', and define Lb and Lc cyclically. Let A* = Lb∩Lc, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(695). Also, X(695) is the orthologic center of ABC to A'B'C'. The reciprocal orthologic center is X(4). (Randy Hutson, July 31 2018)

X(695) lies on these lines: 69,194   99,711   248,682   384,694

X(695) = isogonal conjugate of X(384)
X(695) = isotomic conjugate of X(9230)
X(695) = complement of X(37889)
X(695) = anticomplement of X(37890)
X(695) = antigonal conjugate of isogonal conjugate of X(37896)
X(695) = isotomic conjugate of crosspoint of PU(11)
X(695) = cevapoint of PU(1)
X(695) = trilinear pole of line X(647)X(3221)


X(696) = EVEN (- 4, - 3) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -5(b -3 + c -3) - a -4(b -4 + c -4)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(696) lies on the line at infinity. The first trilinear coordinate has the form

am-1(bn + cn) - an-1(bm + cm).

If m and n are distinct integers, this form fits the definition of even polynomial center as in Clark Kimberling, "Functional equations associated with triangle geometry," Aequationes Mathematicae 45 (1993) 127-152. This form, perhaps appearing initially here (July 7, 2001) defines a triangle center for arbitrary distinct real numbers m and n. Selected even infinity and circumcircle points begin at X(696); odd ones begin at X(768).

Certain points of this type occur prior to this section. They are as follows:

     X(538) = even (- 2, 0) infinity point
     X(536) = even (- 1, 0) infinity point
     X(519) = even (0, 1) infinity point
     X(106) = even (0, 1) circumcircle point
     X(524) = even (0, 2) infinity point
     X(111) = even (0, 2) circumcircle point
     X(518) = even (1, 2) infinity point
     X(105) = even (1, 2) circumcircle point
     X(674) = even (2, 3) infinity point
     X(675) = even (2, 3) circumcircle point
     X(511) = even (2, 4) infinity point
     X(98) = even (2, 4) circumcircle point

X(696) lies on these lines: 30,511   313,561

X(696) = isogonal conjugate of X(697)\


X(697) = EVEN (- 4, - 3) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -5(b -3 + c -3) - a -4(b -4 + c -4)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(697) lies on the circumcircle. This is one of several points of the form given by first trilinear

1/[am-1(bn + cn) - an-1(bm + cm)],

hence the name "(m, n)-circumcircle point".

X(697) lies on this line: 100,560

X(697) = isogonal conjugate of X(696)
X(697) = isotomic conjugate of X(35523)


X(698) = EVEN (- 4, - 2) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -5(b -2 + c -2) - a -3(b -4 + c -4)
Barycentrics    a^2 b^4 + a^2 c^4 - b^4 c^2 - b^2 c^4 : :

As the isogonal conjugate of a point on the circumcircle, X(698) lies on the line at infinity.

X(698) lies on these (parallel) lines: 6,194  30,511   75,257  76,141

X(698) = isogonal conjugate of X(699)
X(698) = isotomic conjugate of X(3225)


X(699) = EVEN (- 4, - 2) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -5(b -2 + c -2) - a -3(b -4 + c -4)]
Barycentrics    a^2/(a^2 b^4 + a^2 c^4 - b^4 c^2 - b^2 c^4) : :

X(699) lies on the circumcircle and these lines: 32,99   172,932   251,689

X(699) = isogonal conjugate of X(698)
X(699) = isotomic conjugate of X(35524)
X(699) = X(385)-cross conjugate of X(251)
X(699) = trilinear pole of line X(6)X(3221)
X(699) = Ψ(X(6), X(3221))
X(699) = cevapoint of X(32) and X(1691)


X(700) = EVEN (- 4, - 1) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -5(b -1 + c -1) - a -2(b -4 + c -4)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(700) lies on the line at infinity.

X(700) lies on this line: 30,511   75,871

X(700) = isogonal conjugate of X(701)


X(701) = EVEN (- 4, - 1) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -5(b -1 + c -1) - a -2(b -4 + c -4)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(701) lies on the circumcircle.

X(701) lies on this line: 31,789

X(701) = isogonal conjugate of X(700)
X(701) = isotomic conjugate of X(35525)


X(702) = EVEN (- 4, 0) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -5(b0 + c0) - a -1(b -4 + c -4)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(702) lies on the line at infinity.

X(702) lies on these lines: 2,308   30,511

X(702) = isogonal conjugate of X(703)


X(703) = EVEN (- 4, 0) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -5(b0 + c0) - a -1(b -4 + c -4)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(703) lies on the circumcircle.

X(703) lies on this line: 6,689

X(703) = isogonal conjugate of X(702)
X(703) = isotomic conjugate of X(35526)


X(704) = EVEN (- 4, 1) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -5(b1 + c1) - a0(b -4 + c -4)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(704) lies on the line at infinity.

X(704) lies on this line: 30,511

X(704) = isogonal conjugate of X(705)


X(705) = EVEN (- 4, 1) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -5(b1 + c1) - a0(b -4 + c -4)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(705) lies on the circumcircle.

X(705) = isogonal conjugate of X(704)
X(705) = isotomic conjugate of X(35527)


X(706) = EVEN (- 4, 2) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -5(b2 + c2) - a1(b -4 + c -4)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(706) lies on the line at infinity.

X(706) lies on this line: 30,511

X(706) = isogonal conjugate of X(707)


X(707) = EVEN (- 4, 2) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -5(b2 + c2) - a1(b -4 + c -4)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(707) lies on the circumcircle.

X(707) = isogonal conjugate of X(706)
X(707) = isotomic conjugate of X(35528)


X(708) = EVEN (- 4, 3) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -5(b3 + c3) - a2(b -4 + c -4)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(708) lies on the line at infinity.

X(708) lies on this line: 30,511

X(708) = isogonal conjugate of X(709)


X(709) = EVEN (- 4, 3) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -5(b3 + c3) - a2(b -4 + c -4)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(709) lies on the circumcircle.

X(709) = isogonal conjugate of X(708)
X(709) = isotomic conjugate of X(35529)


X(710) = EVEN (- 4, 4) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -5(b4 + c4) - a3(b -4 + c -4)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(710) lies on the line at infinity.

X(710) lies on this line: 30,511

X(710) = isogonal conjugate of X(711)


X(711) = EVEN (- 4, 4) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -5(b4 + c4) - a3(b -4 + c -4)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(711) lies on the circumcircle.

X(711) lies on this line: 99,695

X(711) = isogonal conjugate of X(710)
X(711) = isotomic conjugate of X(35530)


X(712) = EVEN (- 3, - 2) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -4(b -2 + c -2) - a -3(b -3 + c -3)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(712) lies on the line at infinity.

X(712) lies on these lines: 30,511   76,321

X(712) = isogonal conjugate of X(713)


X(713) = EVEN (- 3, - 2) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -4(b -2 + c -2) - a -3(b -3 + c -3)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(713) lies on the circumcircle.

X(713) lies on these lines: 32,100   101,560

X(713) = isogonal conjugate of X(712)
X(713) = isotomic conjugate of X(35531)


X(714) = EVEN (- 3, - 1) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -4(b -1 + c -1) - a -2(b -3 + c -3)
Barycentrics    (b + c) (a^2 (b^2 - b c + c^2) - b^2 c^2) : :

As the isogonal conjugate of a point on the circumcircle, X(714) lies on the line at infinity.

X(714) lies on these lines: 30,511   38,75

X(714) = isogonal conjugate of X(715)


X(715) = EVEN (- 3, - 1) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -4(b -1 + c -1) - a -2(b -3 + c -3)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(715) lies on the circumcircle.

X(715) lies on these lines: 31,99   81,932   82,689   110,560

X(715) = isogonal conjugate of X(714)
X(715) = isotomic conjugate of X(35532)


X(716) = EVEN (- 3, 0) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -4(b0 + c0) - a -1(b -3 + c -3)
Barycentrics    a^3 b^3 + a^3 c^3 - 2 b^3 c^3 : :

As the isogonal conjugate of a point on the circumcircle, X(716) lies on the line at infinity.

X(716) lies on these lines: 2,561   30,511

X(716) = isogonal conjugate of X(717)


X(717) = EVEN (- 3, 0) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -4(b0 + c0) - a -1(b -3 + c -3)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(717) lies on the circumcircle.

X(717) lies on these lines: 6,789   560,825

X(717) = isogonal conjugate of X(716)
X(717) = isotomic conjugate of X(35533)


X(718) = EVEN (- 3, 1) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -4(b1 + c1) - a0(b -3 + c -3)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(718) lies on the line at infinity.

X(718) lies on these lines: 1,561   30,511

X(718) = isogonal conjugate of X(719)


X(719) = EVEN (- 3, 1) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -4(b1 + c1) - a0(b -3 + c -3)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(719) lies on the circumcircle.

X(719) lies on these lines: 1,689   560,827

X(719) = isogonal conjugate of X(718)
X(719) = isotomic conjugate of X(35534)


X(720) = EVEN (- 3, 2) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -4(b2 + c2) - a1(b -3 + c -3)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(720) lies on the line at infinity.

X(720) lies on these lines: 6,561   30,511

X(720) = isogonal conjugate of X(721)


X(721) = EVEN (- 3, 2) CIRCUMCIRCLE POINT

Trilinears         f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -4(b2 + c2) - a1(b -3 + c -3)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(721) lies on the circumcircle.

X(721) = isogonal conjugate of X(720)
X(721) = isotomic conjugate of X(35535)


X(722) = EVEN (- 3, 3) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -4(b3 + c3) - a2(b -3 + c -3)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(722) lies on the line at infinity.

X(722) lies on this line: 30,511

X(722) = isogonal conjugate of X(723)


X(723) = EVEN (- 3, 3) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -4(b3 + c3) - a2(b -3 + c -3)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(723) lies on the circumcircle.

X(723) = isogonal conjugate of X(722)
X(723) = isotomic conjugate of X(35536)


X(724) = EVEN (- 3, 4) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -4(b4 + c4) - a3(b -3 + c -3)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(724) lies on the line at infinity.

X(724) lies on this line: 30,511

X(724) = isogonal conjugate of X(725)


X(725) = EVEN (- 3, 4) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -4(b4 + c4) - a3(b -3 + c -3)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(725) lies on the circumcircle.

X(725) = isogonal conjugate of X(724)
X(725) = isotomic conjugate of X(35537)


X(726) = EVEN (- 2, -1) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -3(b -1 + c -1) - a -2(b -2 + c -2)
Barycentrics    a (b^2 + c^2) - b c (b + c) : :

As the isogonal conjugate of a point on the circumcircle, X(726) lies on the line at infinity.

X(726) lies on these (parallel) lines: 1,87   10,75   30,511  37,39   38,321   190,238   291,350   312,982

X(726) = isogonal conjugate of X(727)
X(726) = isotomic conjugate of X(3226)
X(726) = X(291)-Ceva conjugate of X(10)
X(726) = crosspoint of X(75) and X(335)


X(727) = EVEN (- 2, -1) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -3(b -1 + c -1) - a -2(b -2 + c -2)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(727) lies on the circumcircle.

X(727) lies on these lines: 1,932   31,43   32,101   58,99   789,985   934,1106

X(727) = isogonal conjugate of X(726)
X(727) = isotomic conjugate of X(35538)
X(727) = X(238)-cross conjugate of X(58)
X(727) = Ψ(X(190), X(6))


X(728) = INTERSECTION OF LINES X(8)X(9) AND X(57)X(345)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)3
Trilinears        cot3(A/2) : cot3(B/2) : cot3(C/2)     (M. Iliev, 4/12/07)

Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(728) lies on these lines: 8,9   40,1018   57,345   78,644   200,220

X(728) = isogonal conjugate of X(738)
X(728) = isotomic conjugate of X(23062)
X(728) = X(346)-Ceva conjugate of X(200)
X(728) = X(480)-cross conjugate of X(200)


X(729) = EVEN (- 2, 0) CIRCUMCIRCLE POINT

Trilinears    1/[a-3(b0 + c0) - a -1(b-2 + c -2)]
Trilinears    a/(2b^2c^2 - a^2b^2 - a^2c^2) : :

X(729) lies on the circumcircle and these lines: 6,99   32,110   83,689    100,213   187,805

X(729) = isogonal conjugate of X(538)
X(729) = isotomic conjugate of X(30736)
X(729) = Λ(X(2), X(39))
X(729) = Ψ(X(6), X(669))
X(729) = trilinear pole of line X(6)X(669)
X(729) = trilinear pole, wrt circumsymmedial triangle, of line X(2)X(6)
X(729) = eigencenter of dual of orthic triangle
X(729) = X(11636)-of-circumsymmedial-triangle
X(729) = Thomson-isogonal conjugate of X(32472)
X(729) = Lucas-isogonal conjugate of X(32472)
X(729) = barycentric product of circumcircle intercepts of line X(2)X(512)


X(730) = EVEN (- 2, 1) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -3(b1 + c1) - a0(b -2 + c -2)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(730) lies on the line at infinity.

X(730) lies on these (parallel) lines: 1,76   8,194   10,39  30,511  

X(730) = isogonal conjugate of X(731)


X(731) = EVEN (- 2, 1) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -3(b1 + c1) - a0(b -2 + c -2)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(731) lies on the circumcircle.

X(731) lies on these lines: 1,789   32,825   100,869

X(731) = isogonal conjugate of X(730)
X(731) = isotomic conjugate of X(35539)


X(732) = EVEN (- 2, 2) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -3(b2 + c2) - a1(b -2 + c -2)
Barycentrics    (b^2 + c^2) (a^4 - b^2 c^2) : :

As the isogonal conjugate of a point on the circumcircle, X(732) lies on the line at infinity.

X(732) lies on these (parallel) lines: 6,76   30,511   39,141   69,194

X(732) = isogonal conjugate of X(733)
X(732) = isotomic conjugate of X(14970)
X(732) = crossdifference of every pair of points on line X(6)X(688)
X(732) = X(39)-Hirst inverse of X(141)
X(732) = barycentric product X(141)*X(385)


X(733) = EVEN (- 2, 2) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-3(b2 + c2) - a1(b-2 + c -2)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(733) lies on the circumcircle and these lines: 2,689   32,827   39,83   100,893   101,904   110,251   755,882

X(733) = isogonal conjugate of X(732)
X(733) = isotomic conjugate of X(35540)
X(733) = orthoptic-circle-of-Steiner-inellipse-inverse of X(35971)
X(733) = trilinear pole of line X(6)X(688)
X(733) = Ψ(X(6), X(688))


X(734) = EVEN (- 2, 3) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -3(b3 + c3) - a2(b -2 + c -2)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(734) lies on the line at infinity.

X(734) lies on these lines: 30,511   31,76

X(734) = isogonal conjugate of X(735)


X(735) = EVEN (- 2, 3) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -3(b3 + c3) - a2(b -2 + c -2)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(735) lies on the circumcircle.

X(735) = isogonal conjugate of X(734)
X(735) = isotomic conjugate of X(35541)


X(736) = EVEN (- 2, 4) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -3(b4 + c4) - a3(b -2 + c -2)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
Barycentrics    a^6 (b^2 + c^2) - b^2 c^2 (b^4 + c^4) : :

As the isogonal conjugate of a point on the circumcircle, X(736) lies on the line at infinity.

X(736) lies on these (parallel) lines: 30,511   32,76   39,325   194,315

X(736) = isogonal conjugate of X(737)


X(737) = EVEN (- 2, 4) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -3(b4 + c4) - a3(b -2 + c -2)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(737) lies on the circumcircle.

X(737) = isogonal conjugate of X(736)
X(737) = isotomic conjugate of X(35542)


X(738) = ISOGONAL CONJUGATE OF X(728)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a) -3
Trilinears        tan3(A/2) : tan3(B/2) : tan3(C/2)     (M. Iliev, 4/12/07)

Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(738) lies on these lines: 9,348   56,269   57,279   77,951

X(738) = isogonal conjugate of X(728)
X(738) = isotomic conjugate of X(30693)
X(738) = X(479)-Ceva conjugate of X(269)


X(739) = EVEN (- 1, 0) CIRCUMCIRCLE POINT

Trilinears    1/[a-2(b0 + c0) - a -1(b -1 + c -1)] : :
Trilinears    a/(2 b c - c a - a b) : :

X(739) lies on the circumcircle and these lines: 6,100   31,101   81,99   108,608   109,604   813,902

X(739) = isogonal conjugate of X(536)
X(739) = isotomic conjugate of X(35543)
X(739) = trilinear pole of line X(6)X(667)
X(739) = Ψ(X(6), X(667))
X(739) = Ψ(X(190), X(1))
X(739) = Ψ(X(8), X(650))
X(739) = barycentric product of circumcircle intercepts of line X(2)X(513)
X(739) = trilinear pole wrt circumsymmedial triangle of line X(1)X(6)


X(740) = EVEN (- 1, 1) INFINITY POINT

Trilinears    a -2(b1 + c1) - a0(b -1 + c -1) : :
Barycentrics    (a^2 - b c) (b + c) : :

As the isogonal conjugate of a point on the circumcircle, X(740) lies on the line at infinity.

Let A'B'C' be the outer Garcia triangle and A″B″C″ the inner Garcia triangle. X(740) is the radical center of the circumcircles of AA'A″, BB'B″, CC'C″. (Randy Hutson, January 29, 2018)

X(740) lies on these (parallel) lines: 1,75   8,192   10,37   30,511   42,321   43,312   238,239   872,1089

X(740) = isogonal conjugate of X(741)
X(740) = isotomic conjugate of X(18827)
X(740) = crosspoint of X(239) and X(350)
X(740) = crosssum of X(58) and X(1326)
X(740) = crossdifference of every pair of points on line X(6)X(798)
X(740) = X(2)-Ceva conjugate of X(35068)
X(740) = X(10)-Hirst inverse of X(37)
X(740) = perspector of circumconic centered at X(35068)
X(740) = homothetic center of n(Incentral)*n(Medial) and n(Medial)*n(Incentral) triangles


X(741) = EVEN (- 1, 1) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -2(b1 + c1) - a0(b -1 + c -1)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(741) lies on the circumcircle and these lines: 1,99   21,932   31,110   42,81   58,101   86,789   107,1096   334,839   335,835   689,873   691,923   759,876   827,849   934,1042

X(741) = isogonal conjugate of X(740)
X(741) = isotomic conjugate of X(35544)
X(741) = trilinear pole of line X(6)X(798)
X(741) = Ψ(X(6), X(798))
X(741) = Ψ(X(190), X(37))
X(741) = Ψ(X(1), X(512))
X(741) = Λ(X(1), X(75))
X(741) = cevapoint of X(58) and X(1326)
X(741) = Conway-circle-inverse of X(38481)
X(741) = trilinear product of circumcircle intercepts of line X(1)X(512)


X(742) = EVEN (- 1, 2) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -2(b2 + c2) - a1(b -1 + c -1)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(742) lies on the line at infinity.

X(742) lies on these (parallel) lines: 6,75   30,511   37,141   69,192   320,335

X(742) = isogonal conjugate of X(743)
X(742) = crossdifference of every pair of points on line X(6)X(788)


X(743) = EVEN (- 1, 2) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -2(b2 + c2) - a1(b -1 + c -1)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(743) lies on the circumcircle and these lines: 2,789   31,825   101,869   665,761

X(743) = isogonal conjugate of X(742)
X(743) = isotomic conjugate of X(35545)
X(743) = trilinear pole of line X(6)X(788)
X(743) = Ψ(X(6), X(788))


X(744) = EVEN (- 1, 3) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -2(b3 + c3) - a2(b -1 + c -1)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(744) lies on the line at infinity.

X(744) lies on these lines: 30,511   31,75

X(744) = isogonal conjugate of X(745)


X(745) = EVEN (- 1, 3) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -2(b3 + c3) - a2(b -1 + c -1)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(745) lies on the circumcircle.

X(745) lies on these lines: 31,827   38,99   75,689

X(745) = isogonal conjugate of X(744)
X(745) = isotomic conjugate of X(35546)


X(746) = EVEN (- 1, 4) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -2(b4 + c4) - a3(b -1 + c -1)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(746) lies on the line at infinity.

X(746) lies on these (parallel) lines: 30,511   32,75   37,626   192,315

X(746) = isogonal conjugate of X(747)


X(747) = EVEN (- 1, 4) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -2(b4 + c4) - a3(b -1 + c -1)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(747) lies on the circumcircle.

X(747) = isogonal conjugate of X(746)
X(747) = isotomic conjugate of X(35547)


X(748) = INTERSECTION OF LINES X(2)X(31) AND X(9)X(38)

Trilinears        a2 - 2bc : b2 - 2ca : c2 - 2ab
Trilinears        a2 + rs - bcS :: b2 + rs - caS : c2 + rs - abS
Barycentrics  a3 - 2abc : b3 - 2abc : c3 - 2abc

X(748) lies on these lines: 1,756   2,31   5,602   9,38   11,212   21,978   42,1001   44,354   55,899   63,244   140,601   181,373   255,499   590,605   606,615

X(748) = isogonal conjugate of X(749)


X(749) = ISOGONAL CONJUGATE OF X(748)

Trilinears        1/(a2 - 2bc) : 1/(b2 - 2ca) : 1/(c2 - 2ab)
Trilinears        a2 + rs + bcS :: b2 + rs -+ caS : c2 + rs + abS
Barycentrics  a/(a2 - 2bc) : b/(b2 - 2ca) : c/(c2 - 2ab)

X(749) = isogonal conjugate of X(748)
X(749) = isotomic conjugate of X(3760)


X(750) = INTERSECTION OF LINES X(1)X(88) AND X(2)X(31)

Trilinears        a2 + 2bc : b2 + 2ca : c2 + 2ab
Barycentrics  a3 + 2abc : b3 + 2abc : c3 + 2abc

X(750) lies on these lines: 1,88   2,31   5,601   6,899   9,896   12,603   38,57   42,940   43,81   46,975   63,756   140,602   165,968   255,498   388,1106   590,606   605,615   902,1001   942,976

X(750) = isogonal conjugate of X(751)


X(751) = ISOGONAL CONJUGATE OF X(750)

Trilinears        1/(a2 + 2bc) : 1/(b2 + 2ca) : 1/(c2 + 2ab)
Barycentrics  a/(a2 + 2bc) : b/(b2 + 2ca) : c/(c2 + 2ab)

X(751) lies on this line: 519,984

X(751) = isogonal conjugate of X(750)
X(751) = isotomic conjugate of X(3761)


X(752) = EVEN (0, 3) INFINITY POINT

Trilinears    a -1(b3 + c3) - a2(b0 + c0) : :
Barycentrics      2 a^3 - b^3 - c^3 : :

As the isogonal conjugate of a point on the circumcircle, X(752) lies on the line at infinity.

X(752) lies on these (parallel) lines: 1,320   2,31   10,44   30,511

X(752) = isogonal conjugate of X(753)
X(752) = crossdifference of every pair of points on line X(6)X(3250)
X(752) = barycentric cube root of X(8032)


X(753) = EVEN (0, 3) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -1(b3 + c3) - a2(b0 + c0)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(753) lies on the circumcircle.

X(753) lies on these lines: 6,825   75,789   100,984

X(753) = isogonal conjugate of X(752)
X(753) = isotomic conjugate of X(35548)
X(753) = circumcircle-antipode of X(28467)
X(753) = trilinear pole of line X(6)X(3250)
X(753) = Ψ(X(6), X(3250))


X(754) = EVEN (0, 4) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -1(b4 + c4) - a3(b0 + c0)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(754) lies on the line at infinity.

X(754) lies on these (parallel) lines: 2,32   30,511   115,316   187,325   230,625

X(754) = isogonal conjugate of X(755)


X(755) = EVEN (0, 4) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -1(b4 + c4) - a3(b0 + c0)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(755) lies on the circumcircle.

X(755) lies on these lines: 6,827   39,110   76,689   99,141   733,882

X(755) = isogonal conjugate of X(754)
X(755) = isotomic conjugate of X(35549)
X(755) = Thomson-isogonal conjugate of X(32473)
X(755) = Lucas-isogonal conjugate of X(32473)


X(756) = CROSSPOINT OF X(10) AND X(37)

Trilinears        (b + c)2 : (c + a)2 : (a + b)2
Barycentrics  a(b + c)2 : b(c + a)2 : c(a + b)2

X(756) lies on these lines: 1,748   2,38   9,31   10,321   12,201   37,42   45,55   63,750   100,846   171,896   200,968   405,976

X(756) = isogonal conjugate of X(757)
X(756) = isotomic conjugate of X(873)
X(756) = X(37)-Ceva conjugate of (1500)
X(756) = crosspoint of X(10) and X(37)
X(756) = crosssum of X(i) and X(j) for these (i,j): (58,81), (60,593), (244,1019)
X(756) = trilinear square of X(37)


X(757) = ISOGONAL CONJUGATE OF X(756)

Trilinears        (b + c) -2 : (c + a) -2 : (a + b) -2
Barycentrics  a(b + c) -2 : b(c + a) -2 : c(a + b) -2

X(757) lies on these lines: 6,662   58,86   60,1014   81,593   171,319   763,849

X(757) = isogonal conjugate of X(756)
X(757) = isotomic conjugate of X(1089)
X(757) = cevapoint of X(58) and X(81)
X(757) = X(81)-cross conjugate of X(1509)
X(757) = trilinear square of X(81)


X(758) = EVEN (1, 3) INFINITY POINT

Trilinears    a0(b3 + c3) - a2(b1 + c1)
Trilinears    (b + c) cot A - b cot B - c cot C : :
Trilinears    (b + c) (b^2 + c^2 - a^2 - bc) : :

As the isogonal conjugate of a point on the circumcircle, X(758) lies on the line at infinity.

X(758) lies on these (parallel) lines: 1,21   8,79   10,12   30,511   36,214   46,78   57,997   100,484   354,392   386,986   942,960   982,995

X(758) = isogonal conjugate of X(759)
X(758) = isotomic conjugate of X(14616)
X(758) = X(1)-Ceva conjugate of X(214)
X(758) = X(2)-Ceva conjugate of X(35069)
X(758) = crosssum of X(523) and X(867)
X(758) = crossdifference of every pair of points on line X(6)X(661)
X(758) = crosspoint of X(1) and X(2948) wrt both the excentral and tangential triangles
X(758) = trilinear square root of X(4736)
X(758) = excentral-to-ABC barycentric image of X(30)
X(758) = excentral-isogonal conjugate of X(34196)
X(758) = perspector of circumconic centered at X(35069)


X(759) = EVEN (1, 3) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a0(b3 + c3) - a2(b1 + c1)]
Trilinears    1/((b + c) cot A - b cot B - c cot C) : :
Trilinears    1/((b + c) (b^2 + c^2 - a^2 - b c)) : :

X(759) lies on the circucircle and these lines: 1,60   10,21   19,112   28,108   31,994   37,101   58,65   75,99   82,827   91,925   107,158   214,662   270,933   484,901   691,897   741,876   833,1010   840,1019   934,1014

X(759) = isogonal conjugate of X(758)
X(759) = isotomic conjugate of X(35550)
X(759) = reflection of X(6011) in X(3)
X(759) = anticomplement of X(31845)
X(759) = trilinear pole of line X(6)X(661)
X(759) = Ψ(X(6), X(661))
X(759) = Ψ(X(1), X(523))
X(759) = trilinear product of circumcircle intercepts of line X(1)X(523)
X(759) = trilinear pole, wrt 2nd circumperp triangle, of line X(7)X(21)
X(759) = circumcircle-antipode of X(6011)
X(759) = Conway-circle-inverse of X(38482)


X(760) = EVEN (1, 4) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a0(b4 + c4) - a3(b1 + c1)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(760) lies on the line at infinity.

X(760) lies on these (parallel) lines: 1,32   8,315   10,626   30,511

X(760) = isogonal conjugate of X(761)
X(760) = crossdifference of every pair of points on line X(6)X(1491)


X(761) = EVEN (1, 4) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a0(b4 + c4) - a3(b1 + c1)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(761) lies on the circumcircle and these lines: 1,825   76,789   101,984   665,743

X(761) = isogonal conjugate of X(760)
X(761) = isotomic conjugate of X(35551)
X(761) = trilinear pole of line X(6)X(1491)
X(761) = Ψ(X(6), X(1491))


X(762) = TRILINEAR CUBE OF X(37)

Trilinears        (b + c)3 : (c + a)3 : (a + b)3
Barycentrics  a(b + c)3 : b(c + a)3 : c(a + b)3

X(762) lies on these lines: 210,213   594,1089

X(762) = isogonal conjugate of X(763)
X(762) = crosssum of X(593) and X(757)


X(763) = ISOGONAL CONJUGATE OF X(762)

Trilinears        (b + c) -3 : (c + a) -3 : (a + b) -3
Barycentrics  a(b + c) -3 : b(c + a) -3 : c(a + b) -3

X(763) lies on line 757,849

X(763) = isogonal conjugate of X(762)


X(764) = TRILINEAR CUBE OF X(513)

Trilinears        (b - c)3 : (c - a)3 : (a - b)3
Barycentrics  a(b - c)3 : b(c - a)3 : c(a - b)3

X(764) lies on the cubic K244 and these lines: 1,513   10,514   56,667   76,693

X(764) = isotomic conjugate of isogonal conjugate of X(8027)
X(764) = crosspoint of X(244) and X(513)
X(764) = crosssum of X(i) and X(j) for these (i,j): (100,765), (692,1252)
X(764) = crossdifference of every pair of points on line X(44)X(765)


X(765) = CEVAPOINT OF X(1) AND X(100)

Trilinears        (b - c) -2 : (c - a) -2 : (a - b) -2
Barycentrics  a(b - c) -2 : b(c - a) -2 : c(a - b) -2

X(765) lies on these lines: 1,1052   59,518   100,513   101,898   109,522   238,519   660,662   798,813

X(765) = isogonal conjugate of X(244)
X(765) = isotomic conjugate of X(1111)
X(765) = cevapoint of X(i) and X(j) for these (i,j): (1,100), (31,101)
X(765) = X(i)-cross conjugate of X(j) for these (i,j): (1,100), (9,190), (31,101)
X(765) = crosssum of X(1) and X(1052)
X(765) = crossdifference of every pair of points on line X(764)X(2087)
X(765) = trilinear pole of line X(100)X(101) (the tangent at X(100) to the circumconic centered at X(9); also the locus of trilinear poles of tangents at P to hyperbola {A,B,C,X(1),P}}, as P moves on line X(1)X(6))


X(766) = EVEN (3, 4) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b4 + c4) - a3(b3 + c3)

As the isogonal conjugate of a point on the circumcircle, X(766) lies on the line at infinity.

X(766) lies on these lines: 30,511   31,32

X(766) = isogonal conjugate of X(767)
X(766) = crossdifference of every pair of points on line X(6)X(693)


X(767) = EVEN (3, 4) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a2(b4 + c4) - a3(b3 + c3)]

X(767) lies on the circumcircle and these lines: 75,101   76,100   85,109   108,331   110,274   112,286   334,813   825,870

X(767) = isogonal conjugate of X(766)
X(767) = isotomic conjugate of X(35552)
X(767) = trilinear pole of line X(6)X(693)
X(767) = Ψ(X(6), X(693))


X(768) = ODD (- 4, - 3) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -5(b -3 - c -3) + a -4(b -4 - c -4)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(768) lies on the line at infinity. The first trilinear coordinate has the form

am-1(bn - cn) + an-1(bm - cm),

corresponding to an odd polynomial center in case m and n are distinct integers. See the note accompanying X(696), where even (m,n) infinity points and even (m,n) circumcircle points are introduced. [For nonzero n, "odd (m,n) circumcircle point" would be a misnomer (as the point is an even polynomial center); consequently, the prefix o- is used to distinguish this point from "even (m,n) circumcircle point" defined at X(696).] Certain points of these classes occur prior to this section. They are as follows:

     X(523) = odd (- 4, - 2) infinity point
     X(688) = odd (- 4, 0) infinity point
     X(689) = o-(- 4, 0) circumcircle point
     X(514) = odd (- 2, - 1) infinity point
     X(101) = o-(- 2, - 1) circumcircle point
     X(512) = odd (- 2, 0) infinity point
     X(99) = o-(- 2, 0) circumcircle point
     X(513) = odd (- 1, 0) infinity point
     X(100) = o-(- 1, 0) circumcircle point
     X(514) = odd (0, 1) infinity point
     X(101) = o-(0, 1) circumcircle point
     X(523) = odd (0, 2) infinity point
     X(110) = o-(0, 2) circumcircle point
     X(513) = odd (1, 2) infinity point
     X(100) = o-(1, 2) circumcircle point
     X(512) = odd (2, 4) infinity point
     X(99) = o-(2, 4) circumcircle point

X(768) lies on this line: 30,511

X(768) = isogonal conjugate of X(769)


X(769) = o-(- 4, - 3) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -5(b -3 - c -3) + a -4(b -4 - c -4)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(769) lies on the circumcircle. This is one of several points of the form given by first trilinear

1/[am-1(bn - cn) + an-1(bm - cm)],

hence the name "(m, n)-circumcircle point".

X(769) = isogonal conjugate of X(768)
X(769) = isotomic conjugate of X(35553)


X(770) = POINT ACAMAR

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos3B - cos3C
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(770) lies on this line: 44,513

X(770) = isogonal conjugate of X(771)
X(770) = crosssum of X(1) and X(770)
X(770) = crossdifference of every pair of points on line X(1)X(1092)


X(771) = ISOGONAL CONJUGATE OF X(770)

Trilinears    1/(cos3B - cos3C) : :
Barycentrics    a*(a - b)*(a - c)*(a + b - c)*(a - b + c)*(a^6 - a^5*b - a^4*b^2 + 2*a^3*b^3 - a^2*b^4 - a*b^5 + b^6 - 2*a^4*c^2 + 4*a^2*b^2*c^2 - 2*b^4*c^2 + a^2*c^4 + a*b*c^4 + b^2*c^4)*(a^6 - 2*a^4*b^2 + a^2*b^4 - a^5*c + a*b^4*c - a^4*c^2 + 4*a^2*b^2*c^2 + b^4*c^2 + 2*a^3*c^3 - a^2*c^4 - 2*b^2*c^4 - a*c^5 + c^6) : :

X(771) lies on this line: {823, 4558}

X(771) = isogonal conjugate of X(770)
X(771) = trilinear pole of line X(1)X(1092)
X(771) = X(770)-cross conjugate of X(1)
X(771) = cevapoint of X(1) and X(770)
X(771) = barycentric quotient X(6)/X(770)


X(772) = ODD (- 4, - 1) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -5(b -1 - c -1) + a -2(b -4 - c -4)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(772) lies on the line at infinity.

X(772) lies on this line: 30,511

X(772) = isogonal conjugate of X(773)


X(773) = o-(- 4, - 1) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -5(b -1 - c -1) + a -2(b -4 - c -4)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(773) lies on the circumcircle.

X(773) = isogonal conjugate of X(772)
X(773) = isotomic conjugate of X(35554)


X(774) = CROSSPOINT OF X(1) AND X(158)

Trilinears    cos2B + cos2C : :
Trilinears    1 - cos A cos(B - C) : :
Trilinears    2 a^2 (b^2 - c^2)^2 - a^4 (b^2 + c^2) - (b^2 - c^2)^2 (b^2 + c^2) : :

X(774) lies on these lines: 1,21   55,201   601,1060   602,1062   821,823   912,1066   938,986

X(774) = isogonal conjugate of X(775)
X(774) = crosspoint of X(1) and X(158)
X(774) = crosssum of X(i) and X(j) for these (i,j): (2,255), (31,610)
X(774) = {X(1),X(63)}-harmonic conjugate of X(1496)


X(775) = ISOGONAL CONJUGATE OF X(774)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/[cos2B + cos2C]
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(775) lies on these lines: 10,801   31,1097   158,255   225,412   662,820

X(775) = isogonal conjugate of X(774)
X(775) = cevapoint of X(1) and X(255)


X(776) = ODD (- 4, 1) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -5(b1 - c1) + a0(b -4 - c -4)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(776) lies on the line at infinity.

X(776) lies on this line: 30,511

X(776) = isogonal conjugate of X(777)


X(777) = o-(- 4, 1) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -5(b1 - c1) + a0(b -4 - c -4)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(777) lies on the circumcircle.

X(777) = isogonal conjugate of X(776)
X(777) = isotomic conjugate of X(35555)


X(778) = ODD (- 4, 2) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -5(b2 - c2) + a1(b -4 - c -4)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(778) lies on the line at infinity.

X(778) lies on this line: 30,511

X(778) = isogonal conjugate of X(779)


X(779) = o-(- 4, 2) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -5(b2 - c2) + a1(b -4 - c -4)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(779) lies on the circumcircle.

X(779) = isogonal conjugate of X(778)
X(779) = isotomic conjugate of X(35556)


X(780) = ODD (- 4, 3) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -5(b3 - c3) + a2(b -4 - c -4)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(780) lies on the line at infinity.

X(780) lies on this line: 30,511

X(780) = isogonal conjugate of X(781)


X(781) = o-(- 4, 3) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -5(b3 - c3) + a2(b -4 - c -4)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(781) lies on the circumcircle.

X(781) = isogonal conjugate of X(780)
X(781) = isotomic conjugate of X(35557)


X(782) = ODD (- 4, 4) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -5(b4 - c4) + a3(b -4 - c -4)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(782) lies on the line at infinity.

X(782) lies on this line: 30,511

X(782) = isogonal conjugate of X(783)
X(782) = crosssum of X(733) and X(881)


X(783) = o-(- 4, 4) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -5(b4 - c4) + a3(b -4 - c -4)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(783) lies on the circumcircle.

X(783) = isogonal conjugate of X(782)
X(783) = isotomic conjugate of X(35558)
X(783) = cevapoint of X(733) and X(881)


X(784) = ODD (- 3, - 2) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -4(b -2 - c -2) + a -3(b -3 - c -3)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(784) lies on the line at infinity.

X(784) lies on this line: 30,511

X(784) = isogonal conjugate of X(785)


X(785) = o-(- 3, - 2) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -4(b -2 - c -2) + a -3(b -3 - c -3)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(785) lies on the circumcicle and this line: 99,692

X(785) = isogonal conjugate of X(784)
X(785) = isotomic conjugate of X(35559)
X(785) = Ψ(X(76), X(37))


X(786) = ODD (- 3, - 1) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -4(b -1 - c -1) + a -2(b -3 - c -3)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(786) lies on the line at infinity.

X(786) lies on this line: 30,511

X(786) = isogonal conjugate of X(787)


X(787) = o-(- 3, - 1) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -4(b -1 - c -1) + a -2(b -3 - c -3)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(787) lies on the circumcircle.

X(787) lies on this line: 662,689

X(787) = isogonal conjugate of X(786)
X(787) = isotomic conjugate of X(35560)


X(788) = ODD (- 3, 0) INFINITY POINT

Trilinears    a2(b3 - c3) : :

As the isogonal conjugate of a point on the circumcircle, X(788) lies on the line at infinity.

X(788) lies on these (parallel) lines: 30,511   42,649   667,798

X(788) = isogonal conjugate of X(789)
X(788) = crossdifference of every pair of points on line X(6)X(75)
X(788) = ideal point of PU(8)


X(789) = o-(- 3, 0) CIRCUMCIRCLE POINT

Trilinears    a - 2/(b3 - c3) : :

As the trilinear product of Steiner circumellipse antipodes, X(789) lies on conic {A,B,C,X(668),X(789)}} with center X(6376) and perspector X(75). (Randy Hutson, July 11, 2019)

X(789) lies on the circumcircle and these lines: 1,731   2,743   6,717   31,701   75,753   76,761   86,741   100,874   101,668   106,870   110,799   112,811   190,813   675,871   727,985

X(789) = isogonal conjugate of X(788)
X(789) = isotomic conjugate of X(1491)
X(789) = intersection, other than A, B, C, of circumcircle and hyperbola {A,B,C,PU(6)}}
X(789) = trilinear pole of line X(6)X(75)
X(789) = Ψ(X(i),X(j)) for these (i,j): (1,76), (6,75), (31,2), (32,1)
X(789) = trilinear product of intercepts of Steiner circumellipse and line X(2)X(31)


X(790) = ODD (- 3, 1) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -4(b1 - c1) + a0(b -3 - c -3)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(790) lies on the line at infinity.

X(790) lies on this line: 30,511

X(790) = isogonal conjugate of X(791)


X(791) = o-(- 3, 1) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -4(b1 - c1) + a0(b -3 - c -3)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(791) lies on the circumcircle.

X(791) = isogonal conjugate of X(790)
X(791) = isotomic conjugate of X(35561)


X(792) = ODD (- 3, 2) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -4(b2 - c2) + a1(b -3 - c -3)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(792) lies on the line at infinity.

X(792) lies on this line: 30,511

X(792) = isogonal conjugate of X(793)


X(793) = o-(- 3, 2) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -4(b2 - c2) + a1(b -3 - c -3)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(793) lies on the circumcircle.

X(793) = isogonal conjugate of X(792)
X(793) = isotomic conjugate of X(35562)


X(794) = ODD (- 3, 3) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -4(b3 - c3) + a2(b -3 - c -3)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(794) lies on the line at infinity.

X(794) lies on this line: 30,511

X(794) = isogonal conjugate of X(795)


X(795) = o-(- 3, 3) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -4(b3 - c3) + a2(b -3 - c -3)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(795) lies on the circumcircle.

X(795) = isogonal conjugate of X(794)
X(795) = isotomic conjugate of X(35563)


X(796) = ODD (- 3, 4) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -4(b4 - c4) + a3(b -3 - c -3)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(796) lies on the line at infinity.

X(796) lies on this line: 30,511

X(796) = isogonal conjugate of X(797)


X(797) = o-(- 3, 4) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -4(b4 - c4) + a3(b -3 - c -3)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(797) lies on the circumcircle.

X(797) = isogonal conjugate of X(796)
X(797) = isotomic conjugate of X(35564)


X(798) = CROSSDIFFERENCE OF X(1) AND X(75)

Trilinears    a2(b2 - c2) : :
Trilinears    SBSB - SCSC : :
Trilinears    Area(BCP(1)) - Area(BCU(1)) : :
Barycentrics    a3(b2 - c2) : :

X(798) lies on these lines: 44,513   163,1101   667,788   688,872   765,813

X(798) = isogonal conjugate of X(799)
X(798) = isotomic conjugate of X(4602)
X(798) = complement of X(17217)
X(798) = X(163)-Ceva conjugate of X(31)
X(798) = crosspoint of X(31) and X(163)
X(798) = crosssum of X(i) and X(j) for these (i,j): (1,798), (38,661), (86,1019), (99,645), (190,668), (513,1107)
X(798) = crossdifference of every pair of points on line X(1)X(75)
X(798) = bicentric difference of PU(i) for i in (36, 85)
X(798) = PU(36)-harmonic conjugate of X(1964)
X(798) = PU(85)-harmonic conjugate of X(2667)
X(798) = perspector wrt excentral triangle of the bianticevian conic of X(1) and X(2)
X(798) = X(6)-isoconjugate of X(670)
X(798) = trilinear product of PU(105)
X(798) = X(2)-Ceva conjugate of X(38986)
X(798) = perspector of hyperbola {A,B,C,X(1),X(31)}}
X(798) = trilinear pole of line X(3121)X(4117) (the tangent to the inellipse that is the trilinear square of the Lemoine axis, at the trilinear square of X(512))


X(799) = ISOGONAL CONJUGATE OF X(798)

Trilinears    csc2A/(cos2B - cos2C) : :
Barycentrics    (csc A)/(cos2B - cos2C) : :
Barycentrics    bc/(b^2 - c^2) : :

Let La be the A-extraversion of line X(1)X(75), and define Lb and Lc cyclically. Let A' = Lb ∩ Lc, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(799). (Randy Hutson, January 29, 2018)

As the trilinear product of Steiner circumellipse antipodes, X(799) lies on conic {A,B,C,X(668),X(789)}} with center X(6376) and perspector X(75). (Randy Hutson, July 11, 2019)

X(799) lies on these lines: 2,873   63,561   75,897   88,274   99,100   110,789   162,811   190,670   310,333   645,651   689,813

X(799) = isogonal conjugate of X(798)
X(799) = complement of X(21220)
X(799) = anticomplement of X(16592)
X(799) = X(190)-cross conjugate of X(99)
X(799) = trilinear pole of line X(1)X(75)
X(799) = isotomic conjugate of X(661)
X(799) = perspector of ABC and the tangential triangle, wrt the excentral triangle, of the bianticevian conic of X(1) and X(2) (see X(99))
X(799) = X(6)-isoconjugate of X(512)
X(799) = trilinear product of circumcircle intercepts of line X(2)X(39)
X(799) = trilinear product X(6189)*X(6190) (the intercepts of Steiner circumellipse and line X(2)X(6))


X(800) = CROSSPOINT OF X(2) AND X(64)

Trilinears    sin A (cos2B + cos2C) : :
Barycentrics    sin2A (cos2B + cos2C) : :
Barycentrics    a^2 (a^4 (b^2 + c^2) - 2 a^2 (b^2 - c^2)^2 + (b^2 - c^2)^2 (b^2 + c^2)) : :
Barycentrics    (sin^2 A) (-2 + sin^2 B + sin^2 C) : :

X(800) lies on these lines: {3,6}, {19,1945}, {41,2197}, {53,115}, {232,1196}, {393,1093}, {1015,3554}, {1033,2207}, {1194,5304}, {1195,1409}, {1500,3553}, {1713,2238}, {2257,2277}, {2331,5336}, {3289,3787}

X(800) = isogonal conjugate of X(801)
X(800) = complement of X(14615)
X(800) = crosspoint of X(i) and X(j) for these (i,j): (2,64), (6,393)
X(800) = crosssum of X(i) and X(j) for these (i,j): (2,394), (6,20)
X(800) = perspector of circumconic centered at X(2883)
X(800) = center of circumconic that is locus of trilinear poles of lines passing through X(2883)
X(800) = X(2)-Ceva conjugate of X(2883)
X(800) = {X(3),X(6)}-harmonic conjugate of X(5065)


X(801) = ISOGONAL CONJUGATE OF X(800)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (csc A)/(cos2B + cos2C)
Barycentrics    1/(a^4 (b^2 + c^2) - 2 a^2 (b^2 - c^2)^2 + (b^2 - c^2)^2 (b^2 + c^2)) : :

X(801) lies on these lines: 4,1092   10,775

X(801) = isogonal conjugate of X(800)
X(801) = isotomic conjugate of X(13567)
X(801) = polar conjugate of X(235)
X(801) = trilinear pole of line X(523)X(2071) (the polar of X(20) wrt circumcircle)
X(801) = cevapoint of X(i) and X(j) for these (i,j): (2,394), (6,20)
X(801) = X(520)-cross conjugate of X(99)


X(802) = ODD (- 2, 1) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -3(b1 - c1) + a0(b -2 - c -2)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(802) lies on the line at infinity.

X(802) lies on this line: 30,511

X(802) = isogonal conjugate of X(803)


X(803) = o-(- 2, 1) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -3(b1 - c1) + a0(b -2 - c -2)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(803) lies on the circumcircle.

X(803) = isogonal conjugate of X(802)


X(804) = ODD (- 2, 2) INFINITY POINT

Trilinears    a -3(b2 - c2) + a1(b -2 - c -2) : :

As the isogonal conjugate of a point on the circumcircle, X(804) lies on the line at infinity.

X(804) lies on these (parallel) lines: 2,351   30,511   98,878   99,670   115,1084   147,684   669,850

X(804) = isogonal conjugate of X(805)
X(804) = isotomic conjugate of X(18829)
X(804) = crosspoint of X(98) and X(99)
X(804) = crosssum of X(i) and X(j) for these (i,j): (511,512), (694,882), (741,875)
X(804) = crossdifference of every pair of points on line X(6)X(694)
X(804) = X(2)-Ceva conjugate of X(35078)
X(804) = 1st-Brocard-isogonal conjugate of X(6)
X(804) = 1st-Brocard-isotomic conjugate of X(30229)
X(804) = X(512)-Hirst inverse of X(523)
X(804) = X(512) of 1st Brocard triangle
X(804) = ideal point of PU(133)
X(804) = perspector of circumconic centered at X(30229)


X(805) = o-(- 2, 2) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -3(b2 - c2) + a1(b -2 - c -2)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

Let A'B'C' be the vertex-triangle of the 2nd and 3rd Brocard triangles. The lines AA', BB', CC' concur in X(805). (Randy Hutson, March 25, 2016)

Let La, Lb, Lc be the lines through A, B, C, resp. parallel to the Brocard axis. Let Ma, Mb, Mc be the reflections of BC, CA, AB in La, Lb, Lc, resp. Let A' = Mb∩Mc, and define B' and C' cyclically. Triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. The centroid of A'B'C' is X(805); see Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, March 25, 2016)

Let La, Lb, Lc be the lines through A, B, C, resp. parallel to the Lemoine axis. Let Ma, Mb, Mc be the reflections of BC, CA, AB in La, Lb, Lc, resp. Let A' = Mb∩Mc, and define B' and C' cyclically. Triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. The centroid of A'B'C' is X(805); see Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, March 25, 2016)

X(805) lies on the circumcircle and these lines: 98,385   99,512   110,669   111,694   187,729   249,827   574,843   691,882   888,892

X(805) = isogonal conjugate of X(804)
X(805) = isotomic conjugate of X(14295)
X(805) = trilinear pole of line X(6)X(694)
X(805) = Ψ(X(6), X(694))
X(805) = cevapoint of X(511) and X(512)
X(805) = reflection of X(99) in the Brocard axis
X(805) = Brocard-circle-inverse of X(32531)
X(805) = X(9831)-of-circumsymmedial-triangle
X(805) = intersection of antipedal lines of X(98) and X(99)
X(805) = perspector of ABC and the triangle formed by reflecting line X(5)X(39) in the sides of ABC


X(806) = ODD (- 2, 3) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -3(b3 - c3) + a2(b -2 - c -2)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(806) lies on the line at infinity.

X(806) lies on this line: 30,511

X(806) = isogonal conjugate of X(807)


X(807) = o-(- 2, 3) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -3(b3 - c3) + a2(b -2 - c -2)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(807) lies on the circumcircle.

X(807) = isogonal conjugate of X(806)


X(808) = ODD (- 2, 4) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -3(b4 - c4) + a3(b -2 - c -2)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(808) lies on the line at infinity.

X(808) lies on this line: 30,511

X(808) = isogonal conjugate of X(809)


X(809) = o-(- 2, 4) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -3(b4 - c4) + a3(b -2 - c -2)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(809) lies on the circumcircle.

X(809) = isogonal conjugate of X(808)


X(810) = CROSSPOINT OF X(1) AND X(163)

Trilinears    sin 2A (cos2B - cos2C) : :
Trilinears    sin 2A (sin^2 B - sin^2 C) : :
Trilinears    a^2(b^2 - c^2)(b^2 + c^2 - a^2) : :
Trilinears    tan B cot C - tan C cot B : :
Barycentrics    sin2A cos A (cos2B - cos2C) : :

Let V = X(521)X(656) = isotomic conjugate of polar conjugate of antiorthic axis and let W = X(514)X(661) = polar conjugate of isotomic conjugate of antiorthic axis; then X(810 = V∩W. (Randy Hutson, December 26, 2015)

X(810) lies on these lines: 521,656   661,663   667,788

X(810) = isogonal conjugate of X(811)
X(810) = complement of X(21300)
X(810) = anticomplement of X(21259)
X(810) = crosspoint of X(1) and X(163)
X(810) = crosssum of X(162) and X(662)
X(810) = crossdifference of every pair of points on line X(19)X(27)
X(810) = X(92)-isoconjugate of X(662)
X(810) = trilinear product of PU(109)
X(810) = trilinear product of Jerabek hyperbola intercepts of Lemoine axis
X(810) = barycentric product of Jerabek hyperbola intercepts of antiorthic axis


X(811) = ISOGONAL CONJUGATE OF X(810)

Trilinears    (csc 2A)/(cos2B - cos2C) : :
Trilinears    (csc 2A)/(sin^2 B - sin^2 C) : :
Barycentrics    (sec A)/(cos2B - cos2C) : :
Barycentrics    (sec A)/(sin^2 B - sin^2 C) : :
Barycentrics    csc 2A csc(B - C) : :
Barycentrics    bc/[(b^2 - c^2)(b^2 + c^2 - a^2)] : :

Let La be the A-extraversion of line X(19)X(27), and define Lb, Lc cyclically. Let A' = Lb ∩ Lc, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(811). (Randy Hutson, January 29, 2018)

As the trilinear product of Steiner circumellipse antipodes, X(811) lies on conic {A,B,C,X(668),X(789)}} with center X(6376) and perspector X(75). (Randy Hutson, July 11, 2019)

X(811) lies on these lines: 1,336   75,1099   99,108   112,789   162,799   350,447   645,648   662,823

X(811) = isogonal conjugate of X(810)
X(811) = isotomic conjugate of X(656)
X(811) = complement of anticomplementary conjugate of X(21300)
X(811) = anticomplement of X(16573)
X(811) = trilinear pole of line X(19)X(27)
X(811) = polar conjugate of X(661)
X(811) = perspector of ABC and tangential triangle, wrt anticevian triangle of X(19), of bianticevian conic of X(1) and X(4)
X(811) = trilinear product X(4)*X(99)
X(811) = trilinear product X(2479)*X(2480)


X(812) = ODD (- 1, 1) INFINITY POINT

Trilinears    a -2(b1 - c1) + a0(b -1 - c -1) : :
Barycentrics    (b - c) (a^2 - b c) : :

As the isogonal conjugate of a point on the circumcircle, X(812) lies on the line at infinity.

X(812) lies on these (parallel) lines: 30,511   190,646   649,693   673,1024   903,1022   1015,1086

X(812) = isogonal conjugate of X(813)
X(812) = isotomic conjugate of X(4562)
X(812) = crosssum of X(649) and X(672)
X(812) = crossdifference of every pair of points on line X(6)X(292)
X(812) = X(2)-Ceva conjugate of X(35119)
X(812) = X(513)-Hirst inverse of X(514)
X(812) = ideal point of PU(i) for these i: 120, 122, 134
X(812) = perspector of circumconic centered at X(35119)
X(812) = polar conjugate of isogonal conjugate of X(22384)
X(812) = barycentric product X(239)*X(514)


X(813) = o-(- 1, 1) CIRCUMCIRCLE POINT

Trilinears      1/[a-2(b1 - c1) + a0(b-1 - c -1)] : :

X(813) lies on the circumcircle and these lines: 99,1016   100,649   101,667   103,295   105,238   106,292   163,827   190,789   334,767   335,675   644,932   689,799   692,825   739,902   765,798   898,1023   927,1025

X(813) = isogonal conjugate of X(812)
X(813) = trilinear pole of line X(6)X(292)
X(813) = Ψ(X(1), X(39))
X(813) = Ψ(X(2), X(38))
X(813) = Ψ(X(6), X(292))
X(813) = X(92)-isoconjugate of X(22384)
X(813) = barycentric product of circumcircle intercepts of line X(2)X(38)


X(814) = ODD (- 1, 2) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -2(b2 - c2) + a1(b -1 - c -1)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(814) lies on the line at infinity.

X(814) lies on this line: 30,511

X(814) = isogonal conjugate of X(815)


X(815) = o-(- 1, 2) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -2(b2 - c2) + a1(b -1 - c -1)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(815) lies on the circumcircle.

X(815) = isogonal conjugate of X(814)


X(816) = ODD (- 1, 3) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -2(b3 - c3) + a2(b -1 - c -1)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(816) lies on the line at infinity.

X(816) lies on this line: 30,511

X(816) = isogonal conjugate of X(817)


X(817) = o-(- 1, 3) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -2(b3 - c3) + a2(b -1 - c -1)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(817) lies on the circumcircle.

X(817) = isogonal conjugate of X(816)


X(818) = ODD (- 1, 4) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -2(b4 - c4) + a3(b -1 - c -1)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(818) lies on the line at infinity.

X(818) lies on this line: 30,511

X(818) = isogonal conjugate of X(819)


X(819) = o-(- 1, 4) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a -2(b4 - c4) + a3(b -1 - c -1)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(819) lies on the circumcircle.

X(819) = isogonal conjugate of X(818)


X(820) = CROSSPOINT OF X(1) AND X(255)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos2A (cos2B + cos2C)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(820) lies on these lines: 1,29   3,296   662,775   836,1100

X(820) = isogonal conjugate of X(821)
X(820) = crosspoint of X(1) and X(255)
X(820) = crosssum of X(1) and X(158)


X(821) = ISOGONAL CONJUGATE OF X(820)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sec2A /(cos2B + cos2C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(821) lies on these lines: 158,255   243,411   774,823

X(821) = isogonal conjugate of X(820)
X(821) = cevapoint of X(1) and X(158)

X(821) = polar conjugate of X(6508)

X(822) = CROSSDIFFERENCE OF X(1) AND X(29)

Trilinears    sec2B - sec2C : :
Trilinears    csc B/2 - csc C/2 : :
Trilinears    csc 2B - csc 2C : :
Trilinears    sin 2A (tan B - tan C) : :
Trilinears    a^2(b^2 - c^2)(b^2 + c^2 - a^2)^2 : :
Barycentrics    sin A (sec2B - sec2C) : :

X(822) lies on this line: 44,513

X(822) = isogonal conjugate of X(823)
X(822) = X(163)-Ceva conjugate of X(48)
X(822) = crosspoint of X(48) and X(163)
X(822) = crosssum of X(i) and X(j) for these (i,j): (1,822), (29,1021), (661,774)
X(822) = crossdifference of every pair of points on line X(1)X(29)
X(822) = perspector, wrt excentral triangle, of bianticevian conic of X(1) and X(4)
X(822) = X(92)-isoconjugate of X(162)
X(822) = bicentric difference of PU(83)
X(822) = polar conjugate of isotomic conjugate of isogonal conjugate of X(36126)
X(822) = PU(83)-harmonic conjugate of X(2658)
X(822) = X(63)-isoconjugate of X(36126)
X(822) = X(2)-Ceva conjugate of X(38985)
X(822) = perspector of hyperbola {A,B,C,X(1),X(48)}}


X(823) = ISOGONAL CONJUGATE OF X(822)

Trilinears    1/(sec2B - sec2C) : :
Trilinears    1/(csc B/2 - csc C/2) : :
Trilinears    1/(csc 2B - csc 2C) : :
Trilinears    b^2c^2/[(b^2 - c^2)(b^2 + c^2 - a^2)^2] : :

Let La be the A-extraversion of line X(1)X(29), and define Lb, Lc cyclically. Let A' = Lb ∩ Lc, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(823). (Randy Hutson, January 29, 2018)

X(823) lies on these lines: 100,107   110,681   158,897   264,379   648,651   662,811   774,821

X(823) = isogonal conjugate of X(822)
X(823) = isotomic conjugate of X(24018)
X(823) = anticomplement of X(16595)
X(823) = perspector of ABC and tangential triangle, wrt excentral triangle, of bianticevian conic of X(1) and X(4)
X(823) = trilinear pole of line X(1)X(29)
X(823) = pole wrt polar circle of trilinear polar of X(656) (line X(2631)X(2632), or PU(75))
X(823) = X(48)-isoconjugate (polar conjugate) of X(656)
X(823) = X(6)-isoconjugate of X(520)
X(823) = trilinear product of Steiner circumellipse intercepts of van Aubel line


X(824) = ODD (0, 3) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a -1(b3 - c3) + a2(b0 - c0)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(824) lies on the line at infinity.

X(824) lies on these lines: 30,511   321,693

X(824) = isogonal conjugate of X(825)
X(824) = isotomic conjugate of X(4586)
X(824) = crossdifference of every pair of points on line X(6)X(560)


X(825) = o-(0, 3) CIRCUMCIRCLE POINT

Trilinears    1/[a-1(b3 - c3) + a2(b0 - c0)] : :

Let Q be a point on line X(2)X(31) other than X(2). Let A'B'C' be the cevian triangle of Q. Let A″ be the {B,C}-harmonic conjugate of A'; equivalently, A″ = BC∩B'C'/ Define B″ and C″ cyclically. The circumcircles of AB″C″, BC″A″, CA″B″ concur in X(825). (Randy Hutson, December 29, 2015)

X(825) lies on the circumcircle and these lines: 1,761   6,753   31,743   32,731   99,163   103,572   105,985   560,717   692,813   767,870

X(825) = isogonal conjugate of X(824)
X(825) = intersection, other than A, B, C, of circumcircle and hyperbola {A,B,C,PU(12)}}
X(825) = trilinear pole of line X(6)X(560)
X(825) = Ψ(X(1), X(32))
X(825) = Ψ(X(2), X(31))
X(825) = Ψ(X(6), X(560))
X(825) = Ψ(X(76), X(1))
X(825) = barycentric product of circumcircle intercepts of line X(2)X(31)


X(826) = ODD (0, 4) INFINITY POINT

Trilinears    a -1(b4 - c4) + a3(b0 - c0) : :

As the isogonal conjugate of a point on the circumcircle, X(826) lies on the line at infinity.

X(826) lies on these (parallel) lines: 30,511   54,879   76,882

X(826) = isogonal conjugate of X(827)
X(826) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,15449), (6,125), (76,115)
X(826) = crossdifference of every pair of points on line X(6)X(22)
X(826) = isotomic conjugate of X(4577)
X(826) = perspector wrt anticomplementary triangle of the bianticevian conic of X(2) and X(6); see X(4577)
X(826) = ideal point of PU(137)
X(826) = bicentric difference of PU(137)


X(827) = o-(0, 4) CIRCUMCIRCLE POINT

Trilinears    1/[a-1(b4 - c4) + a3(b0 - c0)] : :

Let A', B', C' be the intersections of line X(23)X(385) and lines BC, CA, AB, resp. The circumcircles of AB'C', BC'A', CA'B' concur in X(827). Note: the line X(23)X(385) is the polar of X(2) wrt the circumcircle. (Randy Hutson, December 10, 2016)

X(827) lies on the circumcircle and these lines: 5,83   6,755   31,745   32,733   82,759   111,251   163,813   249,805   250,935   560,719   662,831   741,849

X(827) = isogonal conjugate of X(826)
X(827) = X(i)-cross conjugate of X(j) for these (i,j): (2,250), (32,249)
X(827) = trilinear pole of line X(6)X(22)
X(827) = Ψ(X(1), X(82))
X(827) = Ψ(X(2), X(32))
X(827) = Ψ(X(6), X(22))
X(827) = Ψ(X(76), X(6))
X(827) = center of bianticevian conic of X(1) and X(31)
X(827) = X(1577)-isoconjugate of X(39)
X(827) = perspector of circumcevian triangle of X(23) and cross-triangle of ABC and circummedial triangle
X(827) = isotomic conjugate of X(23285)
X(827) = barycentric product X(83)*X(110)
X(827) = barycentric quotient X(83)/X(850)
X(827) = barycentric product of circumcircle intercepts of line X(2)X(32)


X(828) = CROSSPOINT OF X(2) AND X(255)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin C sec2B + sin B sec2C
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sin A f(A,B,C)

X(828) = isogonal conjugate of X(829)
X(828) = crosspoint of X(2) and X(255)
X(828) = crosssum of X(6) and X(158)


X(829) = ISOGONAL CONJUGATE OF X(828)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/(sin C sec2B + sin B sec2C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(829) = isogonal conjugate of X(828)
X(829) = cevapoint of X(6) and X(158)


X(830) = ODD (1, 3) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a0(b3 - c3) + a2(b1 - c1)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(830) lies on the line at infinity.

X(830) lies on these lines: 30,511   661,1580

X(830) = isogonal conjugate of X(831)
X(830) = crossdifference of every pair of points on line X(6)X(38)


X(831) = o-(1, 3) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a0(b3 - c3) + a2(b1 - c1)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(831) lies on the circumcircle.

X(831) lies on this line: 662,827

X(831) = isogonal conjugate of X(830)
X(831) = trilinear pole of line X(6)X(38)
X(831) = Ψ(X(6), X(38))


X(832) = ODD (1, 4) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a0(b4 - c4) + a3(b1 - c1)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(832) lies on the line at infinity.

X(832) lies on these lines: 30,511   656,667

X(832) = isogonal conjugate of X(833)
X(832) = crossdifference of every pair of points on line X(6)X(977)


X(833) = o-(1, 4) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a0(b4 - c4) + a3(b1 - c1)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(833) lies on the circumcircle.

X(833) lies on these lines: 106,977   759,1010

X(833) = isogonal conjugate of X(832)
X(833) = trilinear pole of line X(6)X(977)
X(833) = Ψ(X(6), X(977))


X(834) = ODD (2, 3) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a1(b3 - c3) + a2(b2 - c2)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(834) lies on the line at infinity.

X(834) lies on this line: 30,511

X(834) = isogonal conjugate of X(835)
X(834) = crosssum of X(522) and X(958)
X(834) = crossdifference of every pair of points on line X(6)X(10)
X(834) = X(2)-Ceva conjugate of X(39016)
X(834) = perspector of hyperbola {A,B,C,X(2),X(58)}}
X(834) = barycentric square root of X(39016)


X(835) = o-(2, 3) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a1(b3 - c3) + a2(b2 - c2)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(835) lies on the circumcircle and these lines: 110,190   335,741

X(835) = isogonal conjugate of X(834)
X(835) = trilinear pole of line X(6)X(10)
X(835) = Ψ(X(6), X(10))
X(835) = cevapoint of X(522) and X(958)


X(836) = CROSSPOINT OF X(1) AND X(394)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin B sec2B + sin C sec2C
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sin A f(A,B,C)

X(836) lies on these lines: 1,393   37,73   820,1100

X(836) = isogonal conjugate of X(837)
X(836) = crosspoint of X(1) and X(394)
X(836) = crosssum of X(1) and X(393)


X(837) = ISOGONAL CONJUGATE OF X(836)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/(sin B sec2B + sin C sec2C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(837) lies on this line: 393,394

X(837) = isogonal conjugate of X(836)
X(837) = cevapoint of X(1) and X(393)


X(838) = ODD (3, 4) INFINITY POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b4 - c4) + a3(b3 - c3)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(838) lies on the line at infinity.

X(838) lies on this line: 30,511

X(838) = isogonal conjugate of X(839)
X(838) = crossdifference of every pair of points on line X(6)X(321)


X(839) = o-(3, 4) CIRCUMCIRCLE POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a2(b4 - c4) + a3(b3 - c3)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(839) lies on the circumcircle.

X(839) lies on these lines: 110,668   334,741

X(839) = isogonal conjugate of X(838)
X(839) = trilinear pole of line X(6)X(321)
X(839) = Ψ(X(6), X(321))


X(840) = ISOGONAL CONJUGATE OF X(528)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/(2ax - by - cz), x = x(A,B,C) = 1 - cos(B - C)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(840) is the perspector of ABC and the (degenerate) cross-triangle of the circumcevian triangles of X(3513) and X(3514). (Randy Hutson, June 7, 2019)

X(840) lies on the circumcircle and these lines: 6,919   7,927   36,101   55,901   100,518   105,513   106,663   109,902   759,1019   898,1083

X(840) = reflection of X(2742) in X(3)
X(840) = isogonal conjugate of X(528)
X(840) = trilinear pole of line X(6)X(665)
X(840) = Ψ(X(6), X(665))
X(840) = reflection of X(105) in line X(1)X(3)
X(840) = inverse-in-circle-O(3513,3514) of X(108)


X(841) = ISOGONAL CONJUGATE OF X(541)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/(2ax - by - cz), x = x(A,B,C) = 1/(cos A - 2 cos B cos C)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(841) lies on the circumcircle.

X(841) lies on this line: 376,476

X(841) = isogonal conjugate of X(541)
X(841) = trilinear pole, wrt circumcevian triangle of X(30), of line X(6)X(30)


X(842) = ISOGONAL CONJUGATE OF X(542)

Trilinears    a/(2ax - by - cz) : :, where x = x(A,B,C) = sec(A + ω)
Barycentrics    a^2/[2a^6 - 2a^4(b^2 + c^2) + a^2(b^4 + c^4) - (b^2 - c^2)^2(b^2 + c^2)] : :

Let L = X(74)X(98) or any line parallel to X(74)X(98). Let La be the line of reflection of L in line BC, and define Lb and Lc cyclically. Let A' = Lb∩Lc, and define B' and C' cyclically. Then the lines AA', BB', CC' concur in X(842). (Randy Hutson, February 10, 2016)

Let A'B'C' be the Artzt triangle. Let A″ be the reflection of A' in BC, and define B″ and C″ cyclically. Let A″' be the reflection of A in B'C', and define B″' and C″' cyclically. Let A* = B″B″'∩C″C″', and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(842). (Randy Hutson, December 10, 2016)

X(842) is the antipode of X(691) on the circumcircle.

X(842) lies on these lines: 2,476   3,691   4,935   23,110   30,99   74,512   98,523   107,468   111,647   112,186   858,925

X(842) = reflection of X(691) in X(3)
X(842) = isogonal conjugate of X(542)
X(842) = trilinear pole of line X(6)X(526)
X(842) = Λ(X(6), X(13))
X(842) = Ψ(X(6), X(526))
X(842) = intersection, other than A, B, and C, of the circumcircle and hyperbola {A,B,C,X(4),X(23)}}
X(842) = reflection of X(98) in the Euler line
X(842) = reflection of X(74) in the Brocard axis
X(842) = inverse-in-circle-O(15,16) of X(112)
X(842) = inverse-in-{circumcircle, nine-point circle}-inverter of X(3258)
X(842) = inverse-in-Moses-radical-circle of X(111)
X(842) = SR(P,U), where P and U are the circumcircle intercepts of the Fermat axis
X(842) = 2nd-Parry-to-ABC similarity image of X(23)
X(842) = 3rd-Parry-to-circumsymmedial similarity image of X(110)
X(842) = X(10787)-of-McCay-triangle
X(842) = X(2709)-of-circumsymmedial-triangle
X(842) = Cundy-Parry Phi transform of X(14246)
X(842) = Cundy-Parry Psi transform of X(14357)
X(842) = trilinear pole, wrt Thomson triangle, of line X(110)X(5085)
X(842) = trilinear pole, wrt circumcevian triangle of X(511), of Brocard axis
X(842) = orthoptic-circle-of-Steiner-circumellipse-inverse of X(14731)


X(843) = ISOGONAL CONJUGATE OF X(543)

Trilinears    a/(2ax - by - cz) : : , where x = x(a,b,c) = bc/(b2 - c2)

X(843) lies on the circumcircle.

X(843) lies on these lines: 6,691   99,525   110,187   111,512   574,805

X(843) = reflection of X(352) in X(187)
X(843) = isogonal conjugate of X(543)
X(843) = trilinear pole of line X(6)X(351)
X(843) = Ψ(X(2), X(690))
X(843) = Ψ(X(6), X(351))
X(843) = circumcircle-antipode of X(2709)
X(843) = reflection of X(2709) in X(3)
X(843) = reflection of X(111) in the Brocard axis
X(843) = inverse-in-circle-O(15,16) of X(110)
X(843) = 2nd-Parry-to-ABC similarity image of X(352)
X(843) = 3rd-Parry-to-circumsymmedial similarity image of X(111)
X(843) = X(9144)-of-4th-anti-Brocard-triangle
X(843) = cevapoint of Schoute circle intercepts with Lemoine axis
X(843) = X(691)-of-circumsymmedial-triangle
X(843) = barycentric product of circumcircle intercepts of line X(2)X(690)


X(844) = INTERSECTION OF LINES X(166)X(167) AND X(173)X(503)

Trilinears    - x + y + z : : , where x = x(A,B,C) = sin(A/2) sec2(A/2)

This point solves a problem posed by Antreas Hatzipolakis in Hyacinthos #2768, May 3, 2001.

X(844) lies on these lines: 166,167   173,503

X(844) = X(75)-of-excentral-triangle
X(844) = excentral isotomic conjugate of X(164)


X(845)  =  X(165)X(166)∩X(173)X(503)

Trilinears     - x + y + z : : , where x = x(A,B,C) = sin2(A/2) sec(A/2)

This point solves a problem posed by Antreas Hatzipolakis in Hyacinthos #2768, May 3, 2001.

Let Ia, Ib, Ic be the excenters of a triangle ABC. Let A' be the Clawson point of IaBC, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(845). (Angel Montesdeoca, August 12, 2018)

X(845) lies on these lines: {1,7370}, {164,362}, {165,166}, {167,7991}

X(845) = X(63)-of-excentral-triangle
X(845) = excentral-isogonal conjugate of X(173)


X(846) = 4th SHARYGIN POINT

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - a2 + b2 + c2 + bc + ca + ab
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a f(a,b,c)

This point solves a problem posed by Antreas Hatzipolakis in Hyacinthos #3001, June 11, 2001. For the construction as a Sharygin point, see the description at X(1281).

X(846) lies on these lines: 1,21   2,1054   6,1051   9,43   35,228   37,171   55,984   100,756   333,740   405,986   982,1001

X(846) = X(i)-Ceva conjugate of X(j) for these (i,j): (37,1), (171,43)
X(846) = crossdifference of every pair of points on line X(661)X(4367)
X(846) = homothetic center of the excentral triangle and the antipedal triangle, wrt the incentral triangle, of X(1)
X(846) = X(275)-of-excentral-triangle
X(846) = excentral polar conjugate of X(3)
X(846) = perspector of 1st Sharygin triangle and unary cofactor triangle of 2nd Sharygin triangle


X(847) = X(5)-CROSS CONJUGATE OF X(4)

Trilinears        sec A sec 2A : sec B sec 2B : sec C sec 2C
Barycentrics  tan A sec 2A : tan B sec 2B : tan C sec 2C
Barycentrics    1/(a^2 (a^2 - b^2 - c^2) (a^4 + b^4 + c^4 - 2 a^2 b^2 - 2 a^2 c^2)) : :

This point solves a problem posed by Antreas Hatzipolakis in Hyacinthos #3130, June 25, 2001; see also Jean-Pierre Ehrmann, #3135, June 26, 2001. The problem and solution may be stated as follows. Let ABC be a triangle, LA, LB, LC the perpendicular bisectors of sides BC, CA, AB, and AA', BB', CC' the altitudes of ABC, respectively. Let AB be the point of intersection of AA' and LB, and let AC be the point of intersection of AA' and LC. Let A″ be the point of intersection of BAB and CAC. Define B″ and C″ cyclically. Then triangle A″B″C″ is perspective to triangle ABC, with perspector X(847).

X(847) lies on the McCay orthic cubic; see McCay orthic cubic.

Let A'B'C' be the X(3)-cevian triangle of the orthic triangle of triangle ABC. The lines AA', BB', CC' concur in X(847). (Randy Hutson, 9/23/2011)

X(847) lies on these lines: 2,254   3,925   4,52   24,96   91,225   378,1105   403,1093

X(847) = isogonal conjugate of X(1147)
X(847) = isotomic conjugate of X(9723)
X(847) = polar conjugate of X(1993)
X(847) = barycentric product X(91)*X(92)
X(847) = X(5)-cross conjugate of X(4)
X(847) = cevapoint of X(485) and X(486)


X(848) = YIU ANGLE POINT

Trilinears    (csc A)/(cot A - cot A') : : , where A' = 2πa/(a + b + c)
Barycentrics    g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = 1/(cot A - cot A'), A' = 2πa/(a + b + c)

X(848) point is introduced by Paul Yiu in Hyacinthos #2704, April 7, 2001 (see also #2708, April 10, 2001) as the solution X of the equation

angle BXC : angle CXA : angle AXB = a : b : c.


X(849) =  X(249)-CEVA CONJUGATE OF X(163)

Trilinears     [a/(b + c)]2 : [b/(c + a)]2 : [c/(a + b)]2
Barycentrics   a[a/(b + c)]2 : b[b/(c + a)]2 : c[c/(a + b)]2

X(849) lies on these lines: 32,163   36,58   110,595   249,1110   741,827   757,763

X(849) = isogonal conjugate of X(1089)
X(849) = X(249)-Ceva conjugate of X(163)
X(849) = crosspoint of X(58) and X(501)
X(849) = crosssum of X(10) and X(502)
X(849) = trilinear square of X(58)


X(850) = BARYCENTRIC MULTIPLIER FOR KIEPERT HYPERBOLA

Trilinears    (b2 - c2)/a3 : :
Barycentrics    (b2 - c2)/a2 : :
Barycentrics    csc A sin(B - C) : :
Barycentrics    |AP(1)|^2 - |AU(1)|^2 : :

The barycentric product of X(850) and the circumcircle is the Kiepert hyperbola.

Let P1 and P2 be the two points on the de Longchamps line whose trilinear polars are parallel to the de Longchamps line. P1 and P2 lie on the Kiepert hyperbola and circle {X(2), X(98), X(99)}}. X(850) is the barycentric product P1*P2. (Randy Hutson, June 7, 2019)

X(850) lies on these lines: 2,647   99,476   110,685   297,525   316,512   325,523   340,520   669,804   670,892

X(850) = isotomic conjugate of X(110)
X(850) = complement of X(31296)
X(850) = anticomplement of X(647)
X(850) = anticomplementary conjugate of X(39352)
X(850) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,36901), (76,338), (99,311), (264,339)
X(850) = X(i)-cross conjugate of X(j) for these (i,j): (115,1502), (125,2), (338,76), (339,264)
X(850) = crosspoint of X(95) and X(99)
X(850) = crosssum of X(i) and X(j) for these (i,j): (32,669), (39,647), (51,512)
X(850) = crossdifference of every pair of points on line X(32)X(184)
X(850) = isogonal conjugate of X(1576)
X(850) = trilinear pole of line X(115)X(127) (polar of X(112) wrt polar circle)
X(850) = pole wrt polar circle of trilinear polar of X(112) (line X(6)X(25))
X(850) = polar conjugate of X(112)
X(850) = intersection of trilinear polars of X(300) and X(301)
X(850) = barycentric product X(76)*X(523)


X(851) = X(65)-HIRST INVERSE OF X(73)

Trilinears    sin 2B sin(C - A) + sin 2C sin(B - A) : :
Trilinears    (cos A + cos B + cos C) csc A - (csc A + csc B + csc C) cos A : :
Trilinears    (sec A + sec B + sec C) csc A - (csc A + csc B + csc C) sec A : :

As a point on the Euler line, X(851) has Shinagawa coefficients ($bcSBSC$,-$bc$S2).

X(851) lies on these lines: 2,3   42,65   43,46   44,513   226,228

X(851) = reflection of X(855) in X(859)
X(851) = isogonal conjugate of X(37142)
X(851) = complement of X(14956)
X(851) = cevapoint of X(1758) and X(2655)
X(851) = inverse-in-orthocentroidal-circle of X(1985)
X(851) = crosspoint of X(i) and X(j) for these {i,j}: {1, 37142}, {65, 2652}, {73, 2660}, {243, 1936}
X(851) = crosssum of X(i) and X(j) for these {i,j}: {1, 851}, {21, 2651}, {29, 2659}, {296, 1937}, {1021, 1984}
X(851) = crossdifference of every pair of points on line X(1)X(647)
X(851) = X(65)-Hirst inverse of X(73)


X(852) = X(2)-LINE CONJUGATE OF X(4)

Trilinears    sec C sin 2B sin(C - A) + sec B sin 2C sin(B - A) : :
Barycentrics    a^2*(a^2 - b^2 - c^2)*(a^6*b^2 - 2*a^4*b^4 + a^2*b^6 + a^6*c^2 + 2*a^4*b^2*c^2 - a^2*b^4*c^2 - 2*b^6*c^2 - 2*a^4*c^4 - a^2*b^2*c^4 + 4*b^4*c^4 + a^2*c^6 - 2*b^2*c^6) : :

As a point on the Euler line, X(852) has Shinagawa coefficients ((2E - F)F - S2,(E + F)F + S2).

X(852) lies on these lines: {2, 3}, {51, 6509}, {216, 373}, {511, 2972}, {520, 647}, {577, 5651}, {895, 1942}, {1503, 1624}, {1576, 15139}, {3168, 46717}, {3580, 35442}, {4993, 43975}, {5437, 26900}, {5462, 42441}, {5640, 30258}, {6090, 15905}, {6389, 40981}, {6688, 32078}, {6760, 43574}, {7308, 26901}, {8798, 45187}, {9155, 14961}, {9306, 23606}, {9475, 14580}, {11064, 23181}, {11412, 14059}, {11793, 31388}, {14152, 43598}, {14915, 40948}, {14919, 14941}, {15526, 41586}, {17825, 26898}, {18437, 18911}, {20977, 41172}, {22264, 39201}, {26880, 43650}, {32205, 46025}, {32428, 46106}, {34828, 35222}, {37648, 42353}, {38999, 43952}, {42453, 43988}, {44715, 46788}

X(852) = reflection of X(2972) in X(44436)
X(852) = X(2)-line conjugate of X(4)
X(852) = circumcircle-inverse of X(37926)
X(852) = isotomic conjugate of the polar conjugate of X(3331)
X(852) = crossdifference of every pair of points on line {4, 647}
X(852) = barycentric product X(69)*X(3331)
X(852) = X(i)-isoconjugate of X(j) for these (i,j): {92, 26717}, {525, 36139}, {14208, 32725}
X(852) = barycentric quotient X(i)/X(j) for these {i,j}: {184, 26717}, {3331, 4}, {32676, 36139}, {33571, 2972}
X(852) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 6638, 418}, {25, 6617, 426}, {51, 6509, 13409}, {401, 450, 41202}, {441, 468, 44888}, {441, 44892, 44891}, {441, 44894, 237}, {468, 44886, 237}, {468, 44888, 44891}, {1113, 1114, 37926}, {3284, 34147, 3292}, {6638, 38283, 2}, {44886, 44889, 468}, {44888, 44892, 468}, {44889, 44894, 44892}


X(853) = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(55)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = sec2(C/2) sin 2B sin(C - A) + sec2(B/2) sin 2C sin(B - A)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

As a point on the Euler line, X(853) has Shinagawa coefficients ([(2E-F)F-S2]S2-$bcSA2$F +$abSC$[(E+F)F+S2]-$ab$(E+F)S2, [(E+F)F+S2]S2+$bcSA2$(E+F) -$abSC$[(E+F)2+S2]+$ab$(3E+F)S2).

X(853) lies on these lines: 2,3   657,663

X(853) = crossdifference of every pair of points on line X(7)X(647)


X(854) = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(56)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = csc2(C/2) sin 2B sin(C - A) + csc2(B/2) sin 2C sin(B - A)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

As a point on the Euler line, X(854) has Shinagawa coefficients ([(2E-F)F-S2]S2-$abSC3$ -$abSASB$(E-2F) +$abSC$[(E+F)2-2S2], [(E+F)F+S2]S2-$abSASB$(E+F) +$abSC$S2).

X(854) lies on this line: 2,3

X(854) = crossdifference of every pair of points on line X(8)X(647)


X(855) = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(57)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = cot(C/2) sin 2B sin(C - A) + cot(B/2) sin 2C sin(B - A)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

As a point on the Euler line, X(855) has Shinagawa coefficients (ES2+$abSC2$+$abSASB$ -$abSC$(E+F), -3ES2-3$abSASB$ +2$ab$S2).

X(855) lies on these lines: 2,3   513,663

X(855) = reflection of X(851) in X(859)
X(855) = crossdifference of every pair of points on line X(9)X(647)


X(856) = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(63)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = tan C sin 2B sin(C - A) + tan B sin 2C sin(B - A)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

As a point on the Euler line, X(856) has Shinagawa coefficients ($abSASB$+$abSC$F-$ab$(E+F)F, -$abSASB$).

X(856) lies on these lines: 2,3   521,656

X(856) = crossdifference of every pair of points on line X(19)X(647)


X(857) = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(75)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = sin2C sin 2B sin(C - A) + sin2B sin 2C sin(B - A)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

As a point on the Euler line, X(857) has Shinagawa coefficients ($aSBSC$, - $a$S2).

X(857) lies on these lines: 2,3   514,661

X(857) = isotomic conjugate of X(37202)
X(857) = complement of X(14953)
X(857) = anticomplement of X(1375)
X(857) = inverse-in-orthocentroidal-circle of X(379)
X(857) = crossdifference of every pair of points on line X(31)X(647)


X(858) = COMPLEMENT OF X(23)

Trilinears    sin3C sin 2B sin(C - A) + sin3B sin 2C sin(B - A) : :
Barycentrics    a^4*b^2 - b^6 + a^4*c^2 - 2*a^2*b^2*c^2 + b^4*c^2 + b^2*c^4 - c^6 : :
Barycentrics    c^2*(a^4 - a^2*b^2 + b^4 - c^4) + b^2*(a^4 - b^4 - a^2*c^2 + c^4) : :
Barycentrics    Sin[2*B] + Sin[2*C] - 3*Tan[w] : :
X(858) = 3 X[2] - 4 X[5159], 3 X[2] + X[5189], 9 X[2] - X[20063], 3 X[2] - 5 X[30745], 9 X[2] - 5 X[37760], 9 X[2] - 4 X[37897], 9 X[2] - 2 X[37899], 6 X[2] - X[37900], 5 X[2] - X[37901], 5 X[2] - 2 X[37904], 5 X[2] - 3 X[37907], 7 X[2] - 3 X[37909], 15 X[2] - 4 X[37910], 9 X[2] - 8 X[37911], 3 X[2] + 2 X[46517], 4 X[3] - 3 X[44280], 4 X[5] - 3 X[403], 2 X[5] - 3 X[2072], 5 X[5] - 3 X[11563], X[5] - 3 X[37938], 7 X[5] - 3 X[43893], 3 X[5] - 2 X[44961], 7 X[5] - 6 X[46031], X[20] - 3 X[2071], X[20] + 3 X[3153], 2 X[20] - 3 X[16386], X[20] - 9 X[44450], X[23] - 4 X[5159], 2 X[23] - 3 X[7426], X[23] + 3 X[10989], 3 X[23] - X[20063], X[23] - 5 X[30745], 3 X[23] - 5 X[37760], 3 X[23] - 4 X[37897], 3 X[23] - 2 X[37899], 5 X[23] - 3 X[37901], 5 X[23] - 6 X[37904], 5 X[23] - 9 X[37907], 7 X[23] - 9 X[37909], 5 X[23] - 4 X[37910], 3 X[23] - 8 X[37911], X[23] + 2 X[46517], 3 X[125] - X[41586], 9 X[140] - 5 X[22248], 3 X[140] - 2 X[22249], 4 X[140] - 3 X[44214], 3 X[186] - 5 X[631], 3 X[381] - X[18325], 3 X[381] + X[35001], X[382] - 3 X[18403], X[382] + 3 X[18859], 5 X[403] - 4 X[11563], 3 X[403] - 2 X[11799], X[403] - 4 X[37938], 7 X[403] - 4 X[43893], 9 X[403] - 8 X[44961], 7 X[403] - 8 X[46031], 2 X[468] + X[5189], 4 X[468] - 3 X[7426], 2 X[468] + 3 X[10989], 6 X[468] - X[20063], 2 X[468] - 5 X[30745], 6 X[468] - 5 X[37760], 3 X[468] - 2 X[37897], 3 X[468] - X[37899], 4 X[468] - X[37900], 10 X[468] - 3 X[37901], 5 X[468] - 3 X[37904], 10 X[468] - 9 X[37907], 14 X[468] - 9 X[37909], 5 X[468] - 2 X[37910], 3 X[468] - 4 X[37911], 2 X[548] - 3 X[34152], 4 X[548] - 3 X[44246], 3 X[549] - 2 X[18571], 5 X[631] - 6 X[10257], 5 X[631] + 3 X[46450], 5 X[632] - 2 X[12105]

As a point on the Euler line, X(858) has Shinagawa coefficients (E - 2F,-2E - 2F).

Let P and P' be circumcircle antipodes. Let Q and Q' be the anticomplements of P and P', resp. The rectangular hyperbola passing through P, P', Q, Q' has center X(858) for all P. (Randy Hutson, March 29, 2020)

X(858) lies on the cubics K008, K288, K474, K475, K479, K794, and these lines: {2, 3}, {6, 16306}, {11, 3100}, {12, 4296}, {39, 15820}, {50, 230}, {51, 12058}, {52, 26879}, {53, 15355}, {66, 20806}, {67, 524}, {69, 23300}, {70, 15316}, {74, 16167}, {76, 15667}, {97, 23295}, {98, 2986}, {99, 37804}, {110, 1503}, {111, 24855}, {113, 14915}, {114, 3258}, {115, 3291}, {120, 5520}, {122, 42426}, {125, 511}, {126, 625}, {132, 16177}, {141, 7703}, {147, 14611}, {155, 11457}, {159, 28408}, {182, 14389}, {187, 40544}, {193, 18919}, {195, 43588}, {250, 37801}, {262, 16311}, {265, 37477}, {305, 7796}, {316, 691}, {323, 3448}, {325, 523}, {343, 2979}, {373, 19130}, {385, 16315}, {394, 1853}, {476, 1297}, {496, 9538}, {541, 44791}, {542, 3292}, {566, 3815}, {590, 11417}, {612, 37719}, {614, 9643}, {615, 11418}, {626, 30749}, {671, 15398}, {842, 925}, {930, 23319}, {1078, 11056}, {1086, 24164}, {1092, 14516}, {1147, 34224}, {1154, 20379}, {1180, 9607}, {1194, 7765}, {1209, 5447}, {1290, 26703}, {1294, 9060}, {1302, 2693}, {1304, 34168}, {1331, 41327}, {1350, 20300}, {1351, 26869}, {1352, 15066}, {1353, 11004}, {1495, 5972}, {1531, 2777}, {1533, 36518}, {1560, 5523}, {1568, 6000}, {1614, 9820}, {1619, 41736}, {1648, 20977}, {1843, 26156}, {1899, 1993}, {1994, 11245}, {2393, 5181}, {2452, 7774}, {2453, 7778}, {2549, 9745}, {2690, 30788}, {2694, 9058}, {2752, 13397}, {2770, 3565}, {2883, 12279}, {2886, 20243}, {2892, 11061}, {2972, 38552}, {3060, 13567}, {3101, 3925}, {3163, 41358}, {3164, 7777}, {3218, 26933}, {3219, 21015}, {3231, 39691}, {3284, 6103}, {3313, 6697}, {3314, 16313}, {3566, 14698}, {3574, 9729}, {3581, 15061}, {3589, 19121}, {3613, 14761}, {3788, 30747}, {3794, 26542}, {3818, 5651}, {3819, 43130}, {3917, 11649}, {3920, 15888}, {3933, 9464}, {4316, 5370}, {4324, 7302}, {4325, 5322}, {4329, 23305}, {4330, 5310}, {4576, 6393}, {5012, 23292}, {5103, 13518}, {5211, 31126}, {5254, 9465}, {5305, 5354}, {5318, 37776}, {5319, 5359}, {5321, 37775}, {5422, 44480}, {5446, 43817}, {5448, 10575}, {5449, 10625}, {5480, 5640}, {5562, 20299}, {5596, 28708}, {5642, 11645}, {5650, 24206}, {5654, 11456}, {5888, 32218}, {5971, 13219}, {6036, 22463}, {6090, 18440}, {6101, 13561}, {6106, 6109}, {6107, 6108}, {6146, 34148}, {6241, 22660}, {6243, 11660}, {6247, 12111}, {6340, 13575}, {6390, 14360}, {6515, 23291}, {6530, 46106}, {6688, 44300}, {6696, 11440}, {6698, 8262}, {6699, 32110}, {6723, 29317}, {6776, 37645}, {6795, 9744}, {6800, 46264}, {7191, 9630}, {7668, 22329}, {7689, 43607}, {7691, 32351}, {7706, 37470}, {7736, 16303}, {7750, 26233}, {7752, 11059}, {7779, 31372}, {7830, 15822}, {7832, 30785}, {7840, 39352}, {7849, 21248}, {7868, 16321}, {7912, 30793}, {7925, 16316}, {8024, 45201}, {8280, 35812}, {8281, 35813}, {8288, 15993}, {8550, 11422}, {8585, 18424}, {8680, 25344}, {8759, 43735}, {8791, 10317}, {8901, 43768}, {9076, 11635}, {9128, 10160}, {9189, 39509}, {9225, 11646}, {9300, 41335}, {9306, 11550}, {9479, 19375}, {9545, 31804}, {9681, 18289}, {9710, 29667}, {9711, 29679}, {9822, 46026}, {10102, 20187}, {10192, 26881}, {10264, 12219}, {10418, 40350}, {10423, 39436}, {10539, 16659}, {10564, 17702}, {10574, 12233}, {10627, 34826}, {10717, 22110}, {11002, 21850}, {11163, 16333}, {11174, 16324}, {11206, 41602}, {11411, 23307}, {11412, 12359}, {11420, 23302}, {11421, 23303}, {11441, 14216}, {11443, 23326}, {11449, 34782}, {11454, 23328}, {11580, 43291}, {11592, 13565}, {11605, 18876}, {11680, 23304}, {11750, 12038}, {11809, 29639}, {12022, 13352}, {12079, 31127}, {12121, 32227}, {12160, 26944}, {12163, 20302}, {12221, 23309}, {12222, 23310}, {12223, 23311}, {12224, 23312}, {12226, 21230}, {12272, 15583}, {12278, 41362}, {12370, 37495}, {12824, 32271}, {13009, 23313}, {13010, 23314}, {13292, 15134}, {13366, 33749}, {13367, 44829}, {13372, 14769}, {13391, 20396}, {13394, 15080}, {13398, 39119}, {13445, 15138}, {13470, 43394}, {13568, 43601}, {13573, 15262}, {13574, 34164}, {13754, 16003}, {13869, 29832}, {14156, 44407}, {14262, 32133}, {14538, 40709}, {14539, 40710}, {14567, 41939}, {14644, 43576}, {14852, 37483}, {14919, 17986}, {14927, 35260}, {14965, 35325}, {14981, 36212}, {15018, 18583}, {15043, 16227}, {15059, 15107}, {15132, 22115}, {15133, 15136}, {15140, 41595}, {15360, 44569}, {15448, 35904}, {15523, 21680}, {15606, 41590}, {15644, 32767}, {15880, 31401}, {15958, 46064}, {16103, 39356}, {16111, 33547}, {16163, 32743}, {16221, 31842}, {16266, 25738}, {16275, 33651}, {16319, 43460}, {16320, 30717}, {16325, 16990}, {16658, 46261}, {17134, 44412}, {17171, 26167}, {17712, 44516}, {17747, 24055}, {17834, 20303}, {18235, 27703}, {18279, 23097}, {18392, 23324}, {18396, 37497}, {18637, 21234}, {18669, 21017}, {18912, 36747}, {18914, 43590}, {18952, 36749}, {19122, 34774}, {19151, 45835}, {19379, 25329}, {19406, 23298}, {19407, 23299}, {19924, 32225}, {20403, 45689}, {20477, 23333}, {20481, 43620}, {21639, 32285}, {22112, 38317}, {22528, 23308}, {23325, 37480}, {23327, 41614}, {24284, 32120}, {24322, 24687}, {25320, 41617}, {25365, 31019}, {26875, 26951}, {26917, 41587}, {26958, 33586}, {28419, 36851}, {29323, 32237}, {29664, 39751}, {30741, 38514}, {31125, 46783}, {31383, 35264}, {31815, 37490}, {31843, 46439}, {32064, 37669}, {32068, 34565}, {32251, 37784}, {32300, 44102}, {34169, 41936}, {34237, 44766}, {34380, 37779}, {34545, 45298}, {34845, 37688}, {34989, 44401}, {35002, 38953}, {35259, 36990}, {35360, 44704}, {36983, 45014}, {37496, 38724}, {37649, 43815}, {37689, 41394}, {40111, 46114}, {41022, 41888}, {41023, 41887}, {41673, 44668}, {42329, 43461}, {43608, 44158}, {43650, 44491}, {44953, 46437}

X(858) = midpoint of (I) and X(j) for these (i,j): {2, 10989}, {3, 7574}, {4, 7464}, {20, 10296}, {23, 5189}, {67, 10510}, {186, 46450}, {265, 37477}, {316, 691}, {323, 3448}, {468, 46517}, {1370, 37980}, {2071, 3153}, {5002, 5003}, {5999, 36173}, {7468, 14957}, {14790, 45171}, {15133, 15136}, {17511, 36188}, {18325, 35001}, {18403, 18859}, {18572, 37950}, {23061, 41724}, {24322, 24687}, {25739, 43574}, {31726, 35452}, {35002, 38953}, {36185, 36186}

X(858) = reflection of X(i) in X(j) for these {i,j}: {3, 15122}, {4, 10297}, {22, 16387}, {23, 468}, {110, 11064}, {186, 10257}, {187, 40544}, {297, 37987}, {385, 16315}, {403, 2072}, {468, 5159}, {1495, 5972}, {1513, 36170}, {2070, 44452}, {2072, 37938}, {3580, 125}, {5099, 625}, {5112, 11007}, {5181, 19510}, {5189, 46517}, {5523, 38971}, {5913, 31655}, {7426, 2}, {7473, 441}, {7575, 140}, {8262, 6698}, {9128, 10160}, {10295, 3}, {11799, 5}, {15107, 32269}, {15360, 44569}, {16320, 44377}, {16386, 2071}, {18323, 18572}, {20063, 37899}, {21284, 16977}, {25338, 3628}, {31726, 23323}, {32110, 6699}, {32111, 113}, {32113, 141}, {32120, 24284}, {32123, 23306}, {32125, 23315}, {32217, 3589}, {32218, 34573}, {32220, 6}, {32223, 6723}, {32224, 16324}, {32225, 45311}, {36189, 21531}, {36196, 37350}, {37897, 37911}, {37899, 37897}, {37900, 23}, {37901, 37904}, {37924, 16619}, {37925, 37971}, {37927, 36157}, {37931, 16976}, {37932, 44907}, {37936, 44234}, {37947, 10096}, {37967, 25338}, {37969, 6676}, {37971, 44911}, {39356, 16103}, {40111, 46114}, {40112, 13857}, {41603, 15126}, {43893, 46031}, {44246, 34152}, {44264, 16239}, {44265, 549}, {44266, 547}, {44267, 546}

X(858) = isogonal conjugate of X(1177)
X(858) = isotomic conjugate of X(2373)
X(858) = complement of X(23)
X(858) = anticomplement of X(468)

X(858) = circumcircle-inverse of X(22)
X(858) = nine-point-circle-inverse of X(2)
X(858) = orthocentroidal-circle-inverse of X(1995)
X(858) = polar-circle-inverse of X(25)
X(858) = orthoptic-circle-of-Steiner-inellipse-inverse of X(3)
X(858) = orthoptic-circle-of-Steiner-circumellipse-inverse of X(20)
X(858) = 1st-Droz-Farney-circle-inverse of X(6644)
X(858) = 2nd-Droz-Farney-circle-inverse of X(44440)
X(858) = circumcircle-of-anticomplementary-triangle-inverse of X(1370)
X(858) = Steiner-circle-inverse of X(20)
X(858) = de-Longchamps circle inverse of X(2)
X(858) = nine-point-circle-of-medial-triangle-inverse of X(7499)
X(858) = Stammler-circle-inverse of X(44457)
X(858) = Stammler-circles-radical-circle-inverse of X(44458)
X(858) = circumcircle-of-inner-Napoleon-triangle-inverse of X(44461)
X(858) = circumcircle-of-outer-Napoleon-triangle-inverse of X(44465)
X(858) = {circumcircle, nine-point circle}-inverter-inverse of X(3)

X(858) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {31, 7665}, {48, 8591}, {63, 14360}, {75, 34518}, {111, 5905}, {671, 21270}, {691, 7253}, {810, 39356}, {892, 21300}, {895, 8}, {897, 4}, {923, 193}, {4575, 44010}, {5380, 20293}, {5547, 5942}, {7316, 12649}, {10097, 21221}, {14908, 192}, {14977, 21294}, {15398, 17491}, {17983, 5906}, {30786, 6327}, {32729, 17498}, {32740, 21216}, {34055, 25052}, {36060, 2}, {36085, 850}, {36128, 6515}, {36142, 525}, {46277, 11442}

X(858) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 6593}, {67, 10}, {656, 38971}, {935, 8062}, {2157, 2}, {3455, 37}, {8791, 226}, {9076, 1215}, {10415, 4892}, {14357, 16597}, {17708, 4369}, {18019, 2887}, {34897, 18589}, {36820, 19563}, {37221, 3934}, {46105, 20305}

X(858) = X(i)-Ceva conjugate of X(j) for these (i,j): {316, 524}, {691, 523}, {11605, 46442}, {17172, 18669}, {30786, 2}, {39436, 22}
X(858) = X(i)-cross conjugate of X(j) for these (i,j): {1560, 2}, {2393, 5523}, {15116, 39269}, {21017, 20884}

X(858) = cevapoint of X(i) and X(j) for these (i,j): {3, 15141}, {2393, 14961}
X(858) = crosspoint of X(i) and X(j) for these (i,j): {2, 18019}, {264, 671}, {892, 23582}
X(858) = crosssum of X(i) and X(j) for these (i,j): {3, 41615}, {6, 18374}, {184, 187}, {351, 3269}, {9426, 21906}
X(858) = trilinear pole of line {5181, 21109}
X(858) = crossdifference of every pair of points on line {32, 647}
X(858) = reflection of X(23) in the orthic axis
X(858) = homothetic center of complement of tangential triangle and anticomplement of orthic triangle
X(858) = X(125)-of-1st-anti-Brocard triangle
X(858) = radical trace of circumcircle and de Longchamps circle
X(858) = one of two harmonic traces of the power circles (X(2) is the other)
X(858) = homothetic center of X(2)- and X(4)-Ehrmann triangles; see X(25)
X(858) = Euler line intercept, other than X(858), of circle {X(403),X(858),PU(4)}}
X(858) = perspector of X(2)-Ehrmann triangle and cross-triangle of ABC and X(2)-Ehrmann triangle
X(858) = intersection of orthic axes of 1st and 2nd Ehrmann circumscribing triangles
X(858) = intersection of orthic axes of anticevian triangles of PU(4)

X(858) = X(i)-isoconjugate of X(j) for these (i,j): {1, 1177}, {19, 18876}, {31, 2373}, {32, 37220}, {560, 46140}, {647, 36095}, {656, 10423}, {896, 10422}, {1910, 36823}, {46165, 46289}

X(858) = barycentric product X(i)*X(j) for these {i,j}: {1, 20884}, {6, 1236}, {10, 17172}, {69, 5523}, {75, 18669}, {76, 2393}, {86, 21017}, {190, 21109}, {264, 14961}, {305, 14580}, {598, 19510}, {671, 5181}, {801, 41603}, {1560, 30786}, {2986, 12827}, {3267, 46592}, {3933, 21459}, {6331, 42665}, {22151, 39269}

X(858) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 2373}, {3, 18876}, {6, 1177}, {75, 37220}, {76, 46140}, {111, 10422}, {112, 10423}, {141, 46165}, {162, 36095}, {511, 36823}, {895, 41511}, {1236, 76}, {1560, 468}, {2393, 6}, {5181, 524}, {5523, 4}, {12827, 3580}, {14580, 25}, {14961, 3}, {15126, 26958}, {17172, 86}, {18669, 1}, {19510, 599}, {20410, 8744}, {20884, 75}, {21017, 10}, {21109, 514}, {21459, 32085}, {34158, 14908}, {39269, 46105}, {41603, 13567}, {42665, 647}, {46592, 112}

X(858) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3, 7495}, {2, 4, 1995}, {2, 20, 7493}, {2, 23, 468}, {2, 427, 5133}, {2, 1370, 22}, {2, 2071, 16387}, {2, 2475, 4239}, {2, 3146, 4232}, {2, 3151, 26252}, {2, 3152, 26254}, {2, 3153, 37980}, {2, 3543, 26255}, {2, 5133, 37990}, {2, 5169, 5}, {2, 5189, 23}, {2, 6636, 6676}, {2, 6655, 26257}, {2, 6872, 26256}, {2, 7378, 6997}, {2, 7379, 7474}, {2, 7386, 7485}, {2, 7391, 25}, {2, 7394, 5020}, {2, 7396, 1370}, {2, 7409, 7398}, {2, 7496, 140}, {2, 7500, 6353}, {2, 7533, 16042}, {2, 7570, 3628}, {2, 8889, 31236}, {2, 13595, 6677}, {2, 14790, 26284}, {2, 14807, 1114}, {2, 14808, 1113}, {2, 15246, 7499}, {2, 16063, 3}, {2, 17511, 36166}, {2, 20063, 37760}, {2, 24585, 1375}, {2, 26097, 25495}, {2, 26761, 27029}, {2, 26809, 26968}, {2, 30745, 5159}, {2, 30889, 46531}, {2, 30952, 37370}, {2, 31022, 857}, {2, 31074, 427}, {2, 31099, 4}, {2, 31100, 442}, {2, 31101, 1368}, {2, 31102, 440}, {2, 31103, 21530}, {2, 31104, 18641}, {2, 31105, 381}, {2, 31106, 405}, {2, 31107, 6656}, {2, 31114, 13371}, {2, 31857, 5169}, {2, 34609, 34603}, {2, 36163, 9832}, {2, 36173, 7471}, {2, 36174, 36168}, {2, 37353, 37439}, {2, 37444, 26283}, {2, 37456, 4228}, {2, 37901, 37907}, {2, 46336, 40916}, {2, 46517, 37900}, {3, 4, 38323}, {3, 1594, 13160}, {3, 5094, 2}, {3, 10295, 44280}, {3, 13371, 1594}, {3, 18281, 37118}, {3, 18569, 6240}, {3, 21284, 37978}, {3, 26283, 22}, {3, 31152, 16063}, {3, 34725, 37196}, {3, 37444, 12225}, {3, 37972, 21284}, {4, 3546, 17928}, {4, 16051, 2}, {4, 31099, 31133}, {5, 427, 5169}, {5, 1368, 30739}, {5, 1907, 3832}, {5, 5169, 5133}, {5, 11799, 403}, {5, 30739, 2}, {20, 631, 38444}, {20, 3153, 10296}, {20, 7493, 22}, {20, 30769, 2}, {20, 35497, 548}, {21, 30770, 2}, {22, 30744, 2}, {23, 468, 7426}, {23, 2071, 37929}, {23, 10989, 5189}, {23, 20063, 37899}, {23, 30745, 2}, {23, 37760, 37897}, {23, 37901, 37910}, {23, 37929, 22}, {23, 37978, 21284}, {24, 30802, 2}, {25, 382, 7519}, {25, 7391, 34603}, {25, 30771, 2}, {25, 34609, 7391}, {25, 37928, 23}, {26, 6640, 10018}, {26, 30803, 2}, {27, 30772, 2}, {28, 30773, 2}, {29, 30774, 2}, {50, 44529, 230}, {66, 20806, 46442}, {115, 39602, 5913}, {140, 5576, 14788}, {140, 7575, 44214}, {140, 10300, 43957}, {140, 11819, 45735}, {140, 37454, 2}, {140, 43957, 7496}, {343, 23332, 23293}, {376, 7577, 15760}, {376, 30775, 2}, {377, 30776, 2}, {378, 31180, 18531}, {381, 21312, 44440}, {381, 32216, 2}, {381, 35001, 18325}, {382, 7519, 34603}, {384, 30777, 2}, {394, 1853, 11442}, {427, 468, 37981}, {427, 1312, 46698}, {427, 1313, 46699}, {427, 1368, 2}, {427, 5094, 1594}, {427, 15809, 7378}, {427, 23335, 31099}, {427, 30739, 5}, {427, 37454, 5576}, {428, 6677, 13595}, {468, 5159, 2}, {468, 5189, 37900}, {468, 37897, 37760}, {468, 37899, 37897}, {468, 37981, 403}, {549, 39504, 37347}, {549, 44239, 10298}, {550, 10224, 10024}, {550, 44210, 7492}, {851, 37165, 46488}, {851, 46484, 46487}, {851, 46485, 46484}, {851, 46486, 37165}, {1092, 18381, 14516}, {1113, 1114, 22}, {1312, 1313, 2}, {1346, 10720, 1312}, {1347, 10719, 1313}, {1351, 26869, 37644}, {1368, 11585, 16051}, {1368, 13371, 5094}, {1368, 31074, 5133}, {1370, 7493, 20}, {1370, 16387, 16386}, {1370, 26283, 12225}, {1375, 46553, 33325}, {1594, 7495, 37990}, {1594, 10295, 403}, {1657, 10255, 15761}, {1899, 1993, 45968}, {1995, 7485, 17928}, {1995, 11413, 22}, {1995, 31133, 4}, {2041, 2042, 24}, {2071, 5999, 46620}, {2071, 7464, 11413}, {2071, 7574, 12225}, {2071, 10296, 20}, {2071, 10989, 1370}, {2071, 37980, 22}, {2072, 7426, 37990}, {2072, 7575, 14788}, {2072, 10295, 13160}, {2072, 11799, 5}, {2072, 25338, 34939}, {2072, 44265, 37347}, {2450, 36189, 403}, {2479, 2480, 35923}, {2979, 23293, 343}, {3060, 26913, 13567}, {3153, 7396, 10989}, {3153, 37444, 7574}, {3153, 44450, 2071}, {3154, 36170, 2}, {3520, 12605, 34005}, {3541, 6643, 7503}, {3548, 14790, 24}, {3575, 16196, 22467}, {3627, 44212, 10301}, {3628, 25338, 44282}, {3917, 21243, 37636}, {4232, 30552, 22}, {5000, 5001, 403}, {5004, 5005, 186}, {5020, 5064, 7394}, {5020, 31255, 2}, {5064, 31255, 5020}, {5094, 7396, 12225}, {5094, 16063, 7495}, {5094, 31152, 3}, {5117, 37190, 41237}, {5133, 7426, 403}, {5133, 7495, 13160}, {5159, 5189, 7426}, {5159, 10989, 37900}, {5159, 37897, 37911}, {5159, 46517, 23}, {5169, 31074, 31857}, {5169, 31857, 427}, {5189, 10989, 46517}, {5189, 30745, 468}, {5189, 37760, 20063}, {5480, 37648, 5640}, {5576, 7575, 403}, {6143, 7512, 7542}, {6353, 44442, 7500}, {6644, 31181, 31723}, {6644, 31723, 7576}, {6676, 7667, 6636}, {7378, 37962, 10151}, {7386, 8889, 2}, {7386, 16051, 3546}, {7391, 7492, 18565}, {7391, 7519, 382}, {7396, 16051, 11413}, {7396, 30769, 20}, {7426, 16386, 22}, {7426, 37900, 23}, {7471, 46620, 22}, {7485, 31236, 2}, {7495, 12225, 22}, {7499, 10691, 15246}, {7539, 16419, 2}, {7542, 32144, 6143}, {7553, 16238, 3518}, {7570, 33332, 5133}, {7574, 15122, 10295}, {7574, 18403, 18569}, {7574, 37938, 1594}, {7667, 37969, 44246}, {8613, 10486, 7493}, {8889, 37119, 5094}, {9140, 23061, 41724}, {10255, 15761, 35487}, {10257, 37984, 6803}, {10297, 11585, 2072}, {10300, 37454, 7496}, {10301, 44212, 14002}, {10415, 15899, 16092}, {10415, 34320, 15899}, {10989, 30745, 23}, {10989, 37938, 5133}, {11250, 18563, 35491}, {11284, 35001, 23}, {11318, 11336, 2}, {11412, 23294, 12359}, {11585, 23335, 4}, {11585, 31099, 5133}, {12084, 18404, 18560}, {12085, 18859, 7464}, {12088, 14940, 13383}, {13371, 15122, 2072}, {13371, 16063, 5133}, {13394, 44882, 15080}, {14791, 18281, 3}, {14807, 14808, 1370}, {14957, 33314, 297}, {15122, 37444, 16386}, {15154, 15155, 44457}, {15329, 46594, 22}, {15899, 40343, 34320}, {15899, 42008, 10415}, {16049, 26253, 22}, {16051, 31099, 1995}, {16063, 37119, 7485}, {16063, 37444, 1370}, {16239, 44264, 44234}, {16387, 37980, 7426}, {16976, 37931, 37941}, {18531, 44441, 378}, {18586, 18587, 44275}, {19772, 19773, 401}, {20063, 30745, 37911}, {20063, 37760, 23}, {20063, 37899, 37900}, {20063, 37911, 7426}, {20405, 20406, 10691}, {20408, 20409, 20}, {21284, 37972, 23}, {21531, 21536, 2450}, {23047, 31829, 34007}, {23335, 37938, 10297}, {24887, 24940, 2}, {27099, 27151, 2}, {27387, 27511, 33305}, {28021, 28085, 33302}, {28049, 28127, 33306}, {28241, 28355, 859}, {28413, 28702, 441}, {29386, 29443, 46497}, {29499, 29702, 46575}, {29554, 29753, 46574}, {29971, 30041, 46514}, {30739, 31857, 5133}, {30745, 46517, 7426}, {30771, 34609, 25}, {31074, 31101, 2}, {31101, 31857, 30739}, {33305, 46552, 13589}, {34152, 37969, 6636}, {34320, 42008, 16092}, {34725, 37196, 3146}, {37050, 37330, 2}, {37165, 46484, 851}, {37165, 46485, 46487}, {37165, 46486, 46492}, {37353, 37901, 11563}, {37454, 43957, 140}, {37760, 37897, 7426}, {37897, 37899, 23}, {37897, 37911, 468}, {37899, 37911, 37760}, {37901, 37907, 37904}, {37904, 37907, 7426}, {37904, 37910, 23}, {37911, 46517, 20063}, {37925, 37943, 37971}, {37929, 37980, 23}, {37967, 44282, 25338}, {37971, 44911, 37943}, {39504, 44265, 403}, {40343, 42008, 15899}, {44309, 44310, 16386}, {46484, 46485, 46491}, {46484, 46486, 46488}, {46485, 46486, 851}, {46485, 46490, 46489}, {46486, 46489, 46490}, {46487, 46488, 851}, {46487, 46491, 46484}, {46487, 46492, 46488}, {46488, 46491, 46487}, {46488, 46492, 37165}, {46489, 46490, 851}, {46491, 46492, 851}, {46550, 46555, 46549}, {46551, 46554, 46548}, {46698, 46699, 5133}


X(859) = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(81)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = (sin A + sin B) sin 2B sin(C - A) + (sin A + sin C) sin 2C sin(B - A)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

As a point on the Euler line, X(859) has Shinagawa coefficients ($aSA$ + $a$F,- $aSA$ - $a$(E + F)).

X(859) lies on these lines: 2,3   36,238   56,58   81,957   198,284   283,945   333,956

X(859) = midpoint of X(851) and X(855)
X(859) = isogonal conjugate of X(38955)
X(859) = trilinear pole of line X(3310)X(8677)
X(859) = crosspoint of X(i) and X(j) for these {i,j}: {58, 759}, {250, 36069}
X(859) = isogonal conjugate of anticomplement of X(34586)
X(859) = crosssum of X(i) and X(j) for these {i,j}: {10, 758}, {125, 6370}, {210, 3943}, {526, 6741}, {3738, 34589}
X(859) = crossdifference of every pair of points on line X(37)X(647)
X(859) = inverse-in-circumcircle of X(3109)


X(860) = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(92)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = (tan A + tan B) sin 2B sin(C - A) + (tan A + tan C) sin 2C sin(B - A)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

As a point on the Euler line, X(860) has Shinagawa coefficients ($abSC$F-$ab$(E+F)F, -$abSASB$+$ab$S2).

X(860) lies on these lines: 2,3   8,1068   10,201   34,997   240,522

X(860) = crossdifference of every pair of points on line X(48)X(647)
X(860) = inverse-in-polar-circle of X(3109)


X(861) = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(9)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = tan(C/2) sin 2B sin(C - A) + tan(B/2) sin 2C sin(B - A)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

As a point on the Euler line, X(861) has Shinagawa coefficients (ES2+2$abSC$F-$ab$[2(E+F)F-S2], -3ES2+$abSASB$-2$ab$S2).

X(861) lies on these lines: 2,3   650,663

X(861) = crossdifference of every pair of points on line X(57)X(647)


X(862) = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(19)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = cot C sin 2B sin(C - A) + cot B sin 2C sin(B - A)
Trilinears        g(a,b,c) : g(b,c,a) : g(c,a,b),
                        where g(a,b,c) = (a2 - bc)(b + c)/(b2 + c2 - a2)     (M. Iliev, 5/13/07)

Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(862) has Shinagawa coefficients ($ab$F, -$abSC$).

X(862) lies on these lines: 2,3   661,663

X(862) = crossdifference of every pair of points on line X(63)X(647)


X(863) = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(31)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = csc2C sin 2B sin(C - A) + csc2B sin 2C sin(B - A)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

As a point on the Euler line, X(863) has Shinagawa coefficients ($abSASB$+$abSC$F, -$abSC2$ -$ab$S2).

X(863) lies on these lines: 2,3   667,788

X(863) = crossdifference of every pair of points on line X(75)X(647)


X(864) = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(32)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = csc3C sin 2B sin(C - A) + csc3B sin 2C sin(B - A)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

As a point on the Euler line, X(864) has Shinagawa coefficients ((E + F)3F - (E - 2F)FS2 + S4, -(E + F)4 + 2(E2 - F2)S2 - S4).

X(864) lies on these lines: 2,3   669,688

X(864) = crossdifference of every pair of points on line X(76)X(647)


X(865) = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(512)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = sin 2B csc2C sin(C - A) csc(A - B) - sin 2C csc2B sin(B - A) csc(A - C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

As a point on the Euler line, X(865) has Shinagawa coefficients ((E + F)3F - (7E - 2F)FS2 + S4, -(E + F)4 + 2(E + F)(2E - F)S2 - S4).

X(865) lies on this line: 2,3   351,888

X(865) = crossdifference of every pair of points on line X(99)X(647)


X(866) = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(513)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = sin 2B sin(A - C)/(sin A - sin B) - sin 2C sin(A - B)/(sin A - sin C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

As a point on the Euler line, X(866) has Shinagawa coefficients ([(E+F)F+S2]S2+$abSC3$ -$abSB2SC$+$abSASB$(E+F) -$abSC$[(E+F)2-2S2]-3$ab$FS2, -[(E+F)2+S2]S2-$abSC$S2 +$abSB$S2+$ab$(E+F)S2).

X(866) lies on these lines: 2,3   244,665

X(866) = crossdifference of every pair of points on line X(100)X(647)


X(867) = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(514)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = sin C sin 2B sin(A - C)/(sin A - sin B) - sin B sin 2C sin(A - B)/(sin A - sin C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

As a point on the Euler line, X(867) has Shinagawa coefficients ((E-2F)S2+$abSC2$+$abSASB$ -$abSC$(E+F), -2(E+F)S2+$ab$S2).

X(867) lies on these lines: 2,3   11,244

X(867) = crossdifference of every pair of points on line X(101)X(647)


X(868) = CROSSPOINT OF X(98) AND X(523)

Trilinears    sin 2B sin(A - C) csc(A - B) - sin 2C sin(A - B) csc(A - C) : :
Barycentrics    (b2 - c2)2(a2b2 + a2c2 - b4 - c4) : :
Barycentrics    (SB - SC)^2*(SA^2 - SB*SC) : :

As a point on the Euler line, X(868) has Shinagawa coefficients (3(E + F)F - S2,(E + F)2 - 3S2).

X(868) lies on these lines: 2,3   115,125   127,136

X(868) = crosspoint of X(98) and X(523)
X(868) = crosssum of X(110) and X(511)
X(868) = crossdifference of every pair of points on line X(110)X(647)
X(868) = X(115)-Hirst inverse of X(125)
X(868) = intersection of trilinear polar and polar wrt polar circle of X(648)
X(868) = inverse-in-Hutson-Parry-circle of X(125)
X(868) = {X(13636),X(13722)}-harmonic conjugate of X(125)
X(868) = complementary conjugate of complement of X(35364)


X(869) = INTERSECTION OF LINES X(1)X(2) AND X(31)X(32)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = a2(b2 + c2 + bc)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(869) lies on these lines: 1,2   6,292   31,32   38,980   55,893   100,731   101,743   192,1045   210,1107

X(869) = isogonal conjugate of X(870)
X(869) = isotomic conjugate of X(871)
X(869) = crossdifference of every pair of points on line X(649)X(693)


X(870) = ISOGONAL CONJUGATE OF X(869)

Trilinears    1/[a2(b2 + c2 + bc)] : :

Let A'B'C' be the trilinear obverse triangle of X(2). Let LA be the line through A' parallel to BC, and define LB and LC cyclically. Let A″ = LB∩LC, and define B″ and C″ cyclically. Triangle A″B″C″ is homothetic to ABC at X(870). (Randy Hutson, November 30, 2018)

X(870) lies on these lines: 1,76   2,292   6,75   34,331   56,85   58,274   86,871   106,789   767,825

X(870) = isogonal conjugate of X(869)
X(870) = isotomic conjugate of X(984)
X(870) = trilinear pole of line X(649)X(693)


X(871) = ISOTOMIC CONJUGATE OF X(869)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = 1/[a4(b2 + c2 + bc)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(871) lies on these lines: 2,561   75,700   76,335   86,870   310,982   675,789

X(871) = isotomic conjugate of X(869)


X(872) = INTERSECTION OF LINES X(37)X(42) AND X(43)X(75)

Trilinears    [a(b + c)]2 : :

X(872) lies on these lines: 6,292   37,42   41,560   43,75   190,1045   386,984   688,798   740,1089

X(872) = isogonal conjugate of X(873)
X(872) = X(42)-Ceva conjugate of (1500)
X(872) = crosspoint of X(42) and X(213)
X(872) = crosssum of X(86) and X(274)
X(872) = crossdifference of every pair of points on line X(812)X(1019)
X(872) = trilinear square of X(42)


X(873) = ISOGONAL CONJUGATE OF X(872)

Trilinears    [a(b + c)] -2 : :

X(873) lies on these lines: 2,799   81,239   86,310   261,552   689,741

X(873) = isogonal conjugate of X(872)
X(873) = isotomic conjugate of X(756)
X(873) = cevapoint of X(86) and X(274)
X(873) = X(86)-cross conjugate of X(1509)
X(873) = trilinear pole of line X(812)X(1019)
X(873) = trilinear square of X(86)


X(874) = INTERSECTION OF LINES X(1)X(75) AND X(99)X(670)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = (a2u2 - bcvw)/[a2u(bv - cw)], u : v : w = X(1)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(874) lies on these lines: 1,75   99,670   100,789   190,646

X(874) = isogonal conjugate of X(875)
X(874) = isotomic conjugate of X(876)
X(874) = crossdifference of every pair of points on line X(798)X(1084)
X(874) = trilinear pole of line X(239)X(350)


X(875) = ISOGONAL CONJUGATE OF X(874)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = a2u(bv - cw)/(a2u2 - bcvw), u : v : w = X(1)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(875) lies on these lines: 1,512   31,669   42,649   213,667   291,659   295,926

X(875) = isogonal conjugate of X(874)
X(875) = crosssum of X(i) and X(j) for these (i,j): (239,659), (740,812)
X(875) = crossdifference of every pair of points on line X(239)X(350)
X(875) = trilinear pole of line X(798)X(1084)


X(876) = ISOTOMIC CONJUGATE OF X(874)

Trilinears    u(bv - cw)/(a2u2 - bcvw) : : , where u : v : w = X(1)

X(876) lies on these lines: 1,512   10,514   37,513   75,523   291,891   292,659   295,928   335,900   741,759

X(876) = reflection of X(659) in X(665)
X(876) = isogonal conjugate of X(3573)
X(876) = isotomic conjugate of X(874)
X(876) = crosssum of X(238) and X(659)
X(876) = trilinear pole of line X(244)X(661)


X(877) = INTERSECTION OF LINES X(4)X(69) AND X(99)X(112)

Trilinears    (a2u2 - bcvw)/[a2u(bv - cw)], where u : v : w = X(4)

Let T1 be the tangential triangle of hyperbola {A,B,C,X(99),PU(37)}}. Let T2 be the tangential triangle of hyperbola {A,B,C,X(4),X(112),PU(39)}}. Let A'B'C' be the vertex triangle of T1 and T2. The lines AA', BB', CC' concur in X(877). (Randy Hutson, December 29, 2015)

X(877) lies on these lines: 4,69   99,112

X(877) = isogonal conjugate of X(878)
X(877) = isotomic conjugate of X(879)
X(877) = trilinear pole of line X(232)X(297)


X(878) = ISOGONAL CONJUGATE OF X(877)

Trilinears    a2u(bv - cw)/(a2u2 - bcvw) : : , where u : v : w = X(4)

X(878) lies on these lines: 3,525   25,669   32,512   98,804   184,647

X(878) = isogonal conjugate of X(877)
X(878) = crossdifference of every pair of points on line X(232)X(297)
X(878) = intersection of lines PU(37) and PU(39)
X(878) = X(92)-isoconjugate of X(2421)


X(879) = ISOTOMIC CONJUGATE OF X(877)

Trilinears    u(bv - cw)/(a2u2 - bcvw) : : , where u : v : w = X(4)
Barycentrics    (b^2 - c^2) (b^2 + c^2 - a^2)/(b^4 + c^4 - a^2 b^2 - a^2 c^2) : :

X(879) lies on the Jerabek hyperbola and these lines: 3,525   4,512   6,523   54,826   66,924   67,526   69,520   74,98   287,895   2966,4266

X(879) = isogonal conjugate of X(4230)
X(879) = isotomic conjugate of X(877)
X(879) = crosssum of X(511) and X(684)
X(879) = crossdifference of every pair of points on line X(232)X(511)
X(879) = trilinear pole of line X(125)X(647)
X(879) = antigonal conjugate of X(35909)
X(879) = orthocenter of X(3)X(4)X(6)
X(879) = orthocenter of X(3)X(67)X(74)
X(879) = X(3)X(4)X(6)-isogonal conjugate of X(33752)


X(880) = INTERSECTION OF LINES X(6)X(76) AND X(99)X(670)

Trilinears    (a2u2 - bcvw)/[a2u(bv - cw)]: :, where u : v : w = X(6)

X(880) lies on these lines: 6,76   99,670   886,892

X(880) = isogonal conjugate of X(881)
X(880) = isotomic conjugate of X(882)
X(880) = crossdifference of every pair of points on line X(688)X(1084)
X(880) = P(1)U(11)∩U(1)P(11)


X(881) = ISOGONAL CONJUGATE OF X(880)

Trilinears    a2u(bv - cw)/(a2u2 - bcvw), where u : v : w = X(6)

X(881) lies on these lines: 39,512   351,694

X(881) = isogonal conjugate of X(880)
X(881) = crosssum of X(732) and X(804)
X(881) = trilinear pole of line X(688)X(1084)


X(882) = ISOTOMIC CONJUGATE OF X(880)

Trilinears    u(bv - cw)/(a2u2 - bcvw) : :, where u : v : w = X(6)

X(882) lies on these lines: 6,688   39,512   76,826   141,523   691,805   694,888   733,755

X(882) = isogonal conjugate of X(17941)
X(882) = isotomic conjugate of X(880)
X(882) = crosssum of X(385) and X(804)
X(882) = crossdifference of every pair of points on line X(385)X(732)
X(882) = barycentric product of Kiepert hyperbola intercepts of line PU(1)
X(882) = PU(1)∩PU(11)


X(883) = INTERSECTION OF LINES X(7)X(8) AND X(190)X(644)

Trilinears    (a2u2 - bcvw)/[a2u(bv - cw)] : : , where u : v : w = X(7)

X(883) lies on these lines: 7,8   190,644

X(883) = isogonal conjugate of X(884)
X(883) = isotomic conjugate of X(885)


X(884) = ISOGONAL CONJUGATE OF X(883)

Trilinears    a2u(bv - cw)/(a2u2 - bcvw) : : , where u : v : w = X(7)

X(884) lies on these lines: 21,885   31,649   41,663   55,650   56,667   105,659

X(884) = isogonal conjugate of X(883)
X(884) = crosssum of X(i) and X(j) for these (i,j): (518,918), (1025,1026)


X(885) = ISOTOMIC CONJUGATE OF X(883)

Trilinears    u(bv - cw)/(a2u2 - bcvw) : : , where u : v : w = X(7)
Barycentrics    (b - c) (a - b - c)/(b^2 + c^2 - a b - a c) : :

X(885) lies on these lines: 1,514   7,513   9,522   21,884   104,105   673,900   919,929

X(885) = isogonal conjugate of X(2283)
X(885) = isotomic conjugate of X(883)
X(885) = crosssum of X(672) and X(926)
X(885) = crossdifference of every pair of points on line X(672)X(1362)
X(885) = trilinear pole of line X(11)X(650)
X(885) = orthocenter of X(i)X(j)X(k) for thse (i,j,k): (1,4,9), (4,7,8)


X(886) = INTERSECTION OF LINES X(99)X(669) AND X(512)X(670)

Trilinears    (a2u2 - bcvw)/[a2u(bv - cw)] : : , where u : v : w = X(512)
Barycentrics    b^2 c^2/((b^2 - c^2) (a^2 b^2 + a^2 c^2 - 2 b^2 c^2)) : :

X(886) lies on the Steiner circumellipse and these lines: 99,669   512,670   538,3228   880,892

X(886) = isogonal conjugate of X(887)
X(886) = isotomic conjugate of X(888)
X(886) = anticomplement of X(39010)
X(886) = trilinear pole of line X(2)X(670)


X(887) = ISOGONAL CONJUGATE OF X(886)

Trilinears    a2u(bv - cw)/(a2u2 - bcvw) : : , where u : v : w = X(512)

X(887) lies on these lines: 99,670   187,237

X(887) = isogonal conjugate of X(886)
X(887) = anticomplement of complementary conjugate of X(39010)
X(887) = crosssum of X(i) and X(j) for these (i,j): (2,888), (512,538)
X(887) = crossdifference of every pair of points on line X(2)X(670)
X(887) = center of V(X(99)) = {15,16,99,729}}; see the preamble to X(6137)
X(887) = bicentric sum of PU(91)
X(887) = PU(91)-harmonic conjugate of X(9427)


X(888) = ISOTOMIC CONJUGATE OF X(886)

Barycentrics    a^2(b^2 - c^2)(a^2b^2 + a^2c^2 - 2b^2c^2) : :
Trilinears    u(bv - cw)/(a2u2 - bcvw) : : , where u : v : w = X(512)

As the isogonal conjugate of a point on the circumcircle, X(888) lies on the line at infinity.

X(888) lies on these lines: 30,511    351,865    694,882

X(888) = isogonal conjugate of X(9150)
X(888) = isotomic conjugate of X(886)
X(888) = crosssum of X(6) and X(887)
X(888) = crossdifference of every pair of points on line X(6)X(99)
X(888) = ideal point of PU(105)
X(888) = X(2)-Ceva conjugate of X(39010)
X(888) = perspector of hyperbola {A,B,C,X(2),X(512)}}
X(888) = barycentric square root of X(39010)


X(889) = INTERSECTION OF LINES X(99)X(898) AND X(190)X(649)

Trilinears    (a2u2 - bcvw)/[a2u(bv - cw)] : : , where u : v : w = X(513)
Barycentrics    b c/((b - c) (a b + a c - 2 b c)) : :

X(889) lies on the Steiner circumellipse, the hyperbola {A,B,C,PU(41)}}j, and these lines: 99,898   190,649   350,903   513,668

X(889) = isogonal conjugate of X(890)
X(889) = isotomic conjugate of X(891)
X(889) = anticomplement of X(39011)
X(889) = trilinear pole of line X(2)X(668)


X(890) = ISOGONAL CONJUGATE OF X(889)

Trilinears    a2u(bv - cw)/(a2u2 - bcvw) : : , where u : v : w = X(513)
Trilinears    a^2 (b - c) (a b + a c - 2 b c) : :

X(890) lies on these lines: 100,190   187,237

X(890) = isogonal conjugate of X(889)
X(890) = anticomplement of complementary conjugate of X(39011)
X(890) = crosssum of X(i) and X(j) for these (i,j): (2,891), (513,536)
X(890) = crossdifference of every pair of points on line X(2)X(668)
X(890) = center of circle V(X(100)) = {15,16,100,739}}; see the preamble to X(6137)


X(891) = ISOTOMIC CONJUGATE OF X(889)

Trilinears    u(bv - cw)/(a2u2 - bcvw) : :, where u : v : w = X(513)
Trilinears    (b - c) (a b + a c - 2 b c) : :

As the isogonal conjugate of a point on the circumcircle, X(891) lies on the line at infinity.

X(891) lies on these (parallel) lines: 1,659   30,511   244,665   291,876

X(891) = isogonal conjugate of X(898)
X(891) = isotomic conjugate of X(889)
X(891) = crosssum of X(6) and X(890)
X(891) = crossdifference of every pair of points on line X(6)X(100)
X(891) = bicentric difference of PU(27)
X(891) = ideal point of PU(i) for i in (27, 34)
X(891) = X(2)-Ceva conjugate of X(39011)
X(891) = perspector of hyperbola {A,B,C,X(2),X(513)}}
X(891) = barycentric square root of X(39011)


X(892) = ISOGONAL CONJUGATE OF X(351)

Trilinears    (a2u2 - bcvw)/[a2u(bv - cw)] : : , where u : v : w = X(523)

The osculating circle of the Steiner circumellipse at X(99) intersects the Steiner circumellipse in two points: X(99) and X(892). See Osculating circle of Steiner circumellipse at X(99). (Randy Hutson, November 5, 2021)

X(892) lies on the Steiner circumellipse and these lines: 99,523   111,381   290,895   316,524   670,850   805,888   880,886

X(892) = isogonal conjugate of X(351)
X(892) = isotomic conjugate of X(690)
X(892) = complement of X(39356)
X(892) = anticomplement of X(23992)
X(892) = Steiner-circumellipse-X(6)-antipode of X(35146)
X(892) = trilinear pole of line X(2)X(99)
X(892) = cevapoint of X(2) and X(690)


X(893) = X(238)-CROSS CONJUGATE OF X(292)

Trilinears    a/(a2 + bc) : :

Let A34B34C34 be Gemini triangle 34. Let A' be the perspector of conic {A,B,C,B34,C34}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(893). (Randy Hutson, January 15, 2019)

X(893) lies on these lines: 9,43   19,232   42,694   55,869   100,733   171,292   239,257

X(893) = isogonal conjugate of X(894)
X(893) = isotomic conjugate of X(1920)
X(893) = X(238)-cross conjugate of X(292)
X(893) = crosssum of X(9) and X(1045)
X(893) = X(239)-Hirst inverse of X(257)
X(893) = trilinear pole of line X(663)X(788)
X(893) = cevapoint of PU(8)
X(893) = perspector of ABC and unary cofactor triangle of N-obverse triangle of X(1)
X(893) = perspector of ABC and unary cofactor triangle of trilinear N-obverse triangle of X(2)
X(893) = perspector of ABC and unary cofactor triangle of Gemini triangle 3
X(893) = complement of X(30660)
X(893) = perspector of Gemini triangle 33 and cross-triangle of Gemini triangles 33 and 34


X(894) = ISOGONAL CONJUGATE OF X(893)

Trilinears    (a2 + bc)/a : :

X(894) lies on these lines: 1,87   2,7   6,75   8,193   10,1046   37,86   42,1045   65,257   72,1010   81,314   92,608   141,320   213,274   256,291   273,458   287,651   312,940   319,524   536,1100

X(894) = reflection of X(319) in X(594)
X(894) = isogonal conjugate of X(893)
X(894) = isotomic conjugate of X(257)
X(894) = complement of X(6646)
X(894) = anticomplement of X(4357)
X(894) = X(291)-Ceva conjugate of X(239)
X(894) = crossdifference of every pair of points on line X(663)X(788)
X(894) = intersection of tangents at PU(6) to hyperbola {A,B,C,X(789),PU(6)}
X(894) = crosspoint of PU(6)
X(894) = crosssum of PU(8)
X(894) = X(171)-Hirst inverse of X(385)
X(894) = perspector of Gemini triangle 4 and cross-triangle of ABC and Gemini triangle 4
X(894) = trilinear pole of perspectrix of ABC and Gemini triangle 3


X(895) = ISOGONAL CONJUGATE OF X(468)

Trilinears    a(b2 + c2 - a2)/(b2 + c2 - 2a2) : :

Let A″B″C″ be the 2nd Ehrmann triangle. Let A* be the cevapoint of B″ and C″, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(895). (Randy Hutson, November 18, 2015)

X(895) lies on the MacBeath circumconic, the Darboux septic, and these lines: 4,542   6,110   54,575   65,651   66,193   67,524   69,125   74,511   287,879   290,892

X(895) = midpoint of X(193) and X(3448)
X(895) = reflection of X(i) in X(j) for these (i,j): (69,125), (110,6)
X(895) = isogonal conjugate of X(468)
X(895) = anticomplement of X(5181)
X(895) = circumcircle-inverse of X(6091)
X(895) = trilinear pole of line X(3)X(647)
X(895) = antigonal image of X(69)
X(895) = syngonal conjugate of X(6)
X(895) = polar conjugate of X(37778)
X(895) = pole wrt polar circle of trilinear polar of X(37778) (line X(690)X(12828))
X(895) = MacBeath-circumconic-antipode of X(110)
X(895) = X(92)-isoconjugate of X(187)
X(895) = perspector of ABC and unary cofactor triangle of 4th Brocard triangle
X(895) = eigencenter of 2nd Ehrmann triangle
X(895) = perspector of 2nd Ehrmann triangle and cross-triangle of ABC and 2nd Ehrmann triangle


X(896) = INTERSECTION OF LINES X(1)X(21) AND X(9)X(750)

Trilinears    2a2 - b2 - c2 : :
Barycentrics    a(2a2 - b2 - c2) : :

X(896) lies on these lines: 1,21   9,750   44,513   57,748   162,240   171,756   238,244   518,902

X(896) = isogonal conjugate of X(897)
X(896) = complement of X(17491)
X(896) = anticomplement of X(4892)
X(896) = isotomic conjugate of isogonal conjugate of X(922)
X(896) = isotomic conjugate of complement of anticomplementary conjugate of X(21298)
X(896) = isotomic conjugate of anticomplement of complementary conjugate of X(21256)
X(896) = X(63)-isoconjugate of X(36128)
X(896) = crosssum of X(1) and X(896)
X(896) = crossdifference of every pair of points on line X(1)X(661)
X(896) = X(6)-isoconjugate of X(671)
X(896) = bicentric sum of PU(78)
X(896) = polar conjugate of isotomic conjugate of isogonal conjugate of X(36128)
X(896) = PU(78)-harmonic conjugate of X(661)


X(897) = ISOGONAL CONJUGATE OF X(896)

Trilinears    1/(2a2 - b2 - c2) : :

Let A1B1C1 and A3B3C3 be the 1st and 3rd Parry triangles. Let A' be the trilinear product A1*A3, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(897). (Randy Hutson, February 10, 2016)

Let A'B'C' and A″B″C″ be the 4th Brocard and 4th anti-Brocard triangles, resp. Let A* be the trilinear product A'*A″, and define B*, C* cyclically. The lines AA*, BB*, CC* concur in X(897). (Randy Hutson, March 21, 2019)

X(897) lies on these lines: 1,662   10,190   19,162   37,100   65,651   75,799   158,823   225,653   691,759

X(897) = isogonal conjugate of X(896)
X(897) = isotomic conjugate of X(14210)
X(897) = anticomplement of X(16597)
X(897) = trilinear product of circumcircle intercepts of line X(2)X(523)
X(897) = trilinear pole of line X(1)X(661)
X(897) = polar conjugate of isogonal conjugate of X(36060)
X(897) = X(6)-isoconjugate of X(524)


X(898) = ISOGONAL CONJUGATE OF X(891)

Trilinears    1/[(b - c)(2bc - ab - ac)] : :      (M. Iliev, 5/13/07)

X(898) lies on these lines:
99,889   100,667   101,765   105,666   106,238   813,1023   840,1083

X(898) = isogonal conjugate of X(891)
X(898) = intersection, other than A, B, C, of circumcircle and hyperbola {A,B,C,PU(26),PU(33)}}
X(898) = trilinear pole of line X(6)X(100)
X(898) = Ψ(X(1), X(190))
X(898) = Ψ(X(6), X(100))
X(898) = Ψ(X(650), X(8))
X(898) = trilinear product of circumcircle intercepts of line X(1)X(190)


X(899) = INTERSECTION OF LINES X(1)X(2) AND X(6)X(750)

Trilinears    1/b + 1/c - 2/a : 1/c + 1/a - 2/b : 1/a + 1/b - 2/c (Joe Goggins, 2002)
Barycentrics  a(1/b + 1/c - 2/a) : b(1/c + 1/a - 2/b) : c(1/a + 1/b - 2/c)

X(899) lies on these lines: 1,2   6,750   38,210   44,513   55,748   88,291   100,238   244,518

X(899) = isogonal conjugate of X(37129)
X(899) = isotomic conjugate of X(31002)
X(899) = complement of X(29824)
X(899) = anticomplement of X(4871)
X(899) = cevapoint of X(i) and X(j) for these {i,j}: {891, 19945}, {1646, 14404}
X(899) = trilinear pole of line X(891)X(3768)
X(899) = crosspoint of X(i) and X(j) for these {i,j}: {1, 37129}, {4607, 7035}
X(899) = crosssum of X(i) and X(j) for these {i,j}: {1, 899}, {513, 16507}, {3248, 3768}
X(899) = crossdifference of every pair of points on line X(1)X(649)
X(899) = bicentric sum of PU(58)
X(899) = PU(58)-harmonic conjugate of X(649)


X(900) = CROSSDIFFERENCE OF X(6) AND X(101)

Trilinears    (b - c)(b + c - 2a)/a : :
Barycentrics  (b - c)(b + c - 2a) : (c - a)(c + a - 2b) : (a - b)(a + b - 2c)

As the isogonal conjugate of a point that lies on the circumcircle, X(900) lies on the line at infinity.

Let Ia be the reflection of X(1) in the perpendicular bisector of BC, and define Ib and Ic cyclically; then X(900) = X(513) of IaIbIc. (Randy Hutson, February 10, 2016)

Let Ua be the line through X(80) perpendicular to the line AX(80), and define Ub and Uc cyclically. Let Va be the reflection of BC in Ua, and define Vb and Vc cyclically. The lines Va, Vb, Vc are parallel, and they concur in X(900). (Angel Montesdeoca, June 30, 2017)

X(900) lies on these (parallel) lines: 11,244   30,511   37,665   100,190   335,876   673,885   1635,1644

X(900) = isogonal conjugate of X(901)
X(900) = isotomic conjugate of X(4555)
X(900) = complementary conjugate of X(3259)
X(900) = X(80)-Ceva conjugate of X(11)
X(900) = X(2)-Ceva conjugate of X(35092)
X(900) = crosspoint of X(100) and X(104)
X(900) = crosssum of X(i) and X(j) for these (i,j): (55,654), (513,517), (649,902)
X(900) = crossdifference of every pair of points on line X(6)X(101)
X(900) = excentral-isogonal conjugate of X(34464)
X(900) = X(523)-of-Fuhrmann-triangle
X(900) = polar conjugate of isogonal conjugate of X(22086)
X(900) = trilinear pole of line X(1647)X(2087)
X(900) = ideal point of PU(i) for these i: 121, 123
X(900) = bicentric difference of PU(i) for these i: 121, 123
X(900) = barycentric product X(514)*X(519)
X(900) = perspector of circumconic centered at X(35092)


X(901) = ISOGONAL CONJUGATE OF X(900)

Trilinears    a/[(b - c)(b + c - 2a)] : :

Let LA be the line of reflection of line X(1)X(5) in line BC, and define LB and LC cyclically. Let A' = LB∩LC, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(901). (Randy Hutson, 9/23/2011)

Let La, Lb, Lc be the lines through A, B, C, resp. parallel to the antiorthic axis. Let Ma, Mb, Mc be the reflections of BC, CA, AB in La, Lb, Lc, resp. Let A' = Mb∩Mc, and define B' and C' cyclically. Triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. The centroid of A'B'C' is X(901); see Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, March 25, 2016)

Let A', B', C' be the intersections of the Nagel line and lines BC, CA, AB, resp. The circumcircles of AB'C', BC'A', CA'B' concur in X(901). (Randy Hutson, March 25, 2016)

X(901) lies on the circumcircle and these lines: 3,953   6,2384   36,106   55,840   59,109   88,105   100,513   101,649   104,517   484,759   675,903

X(901) = reflection of X(953) in X(3)
X(901) = isogonal conjugate of X(900)
X(901) = anticomplement of X(3259)
X(901) = isotomic conjugate of isogonal conjugate of X(32719)
X(901) = isotomic conjugate of complement of polar conjugate of isogonal conjugate of X(23184)
X(901) = isotomic conjugate of anticomplement of X(3310)
X(901) = Ψ(X(58), X(110))
X(901) = X(92)-isoconjugate of X(22086)
X(901) = X(36)-cross conjugate of X(59)
X(901) = Ψ(X(1),X(88))
X(901) = Ψ(X(2),X(45))
X(901) = Ψ(X(6),X(101))
X(901) = Ψ(X(8), X(11))
X(901) = trilinear pole, wrt circumtangential triangle, of line X(3)X(8)
X(901) = reflection of X(100) in line X(1)X(3)
X(901) = trilinear pole of line X(6)x(101)
X(901) = crossdifference of every pair of points on line X(1647)X(2087)
X(901) = intersection of antipedal lines of X(100) and X(104)
X(901) = trilinear product X(100)*X(106) (circumcircle-X(1) antipodes)
X(901) = Ψ(X(650),X(9))
X(901) = barycentric product of circumcircle intercepts of line X(2)X(45)


X(902) = CROSSPOINT OF X(6) AND X(106)

Trilinears    a(b + c - 2a) : b(c + a - 2b) : c(a + b - 2c) : :
Trilinears    1 - cos 2A - (1 + cos A)(cos B + cos C) : :
Barycentrics    a2(b + c - 2a) : b2(c + a - 2b) : c2(a + b - 2c)

X(902) lies on these lines: 1,89   6,31   35,595   36,106   44,678   100,238   109,840   165,614   187,237   518,896   739,813   750,1001

X(902) = isogonal conjugate of X(903)
X(902) = complement of X(21282)
X(902) = anticomplement of X(21241)
X(902) = X(106)-Ceva conjugate of X(6)
X(902) = crosspoint of X(6) and X(106)
X(902) = crosssum of X(i) and X(j) for these (i,j): (2,519), (88,1320), (900,1086)
X(902) = crossdifference of every pair of points on line X(2)X(514)
X(902) = trilinear pole of PU(99) (line X(1017)X(1960))
X(902) = inverse-in-Parry-isodynamic-circle of X(5029); see X(2)
X(902) = X(63)-isoconjugate of X(6336)
X(902) = perspector of hyperbola {A,B,C,X(6),X(101)}}
X(902) = {X(31),X(42)}-harmonic conjugate of X(2308)
X(902) = polar conjugate of isotomic conjugate of X(22356)


X(903) = X(1)-BLAIKIE TRANSFORM OF X(2)

Trilinears    bc/(b + c - 2a) : ca/(c + a - 2b) : ab/(a + b - 2c) : :
Barycentrics    1/(b + c - 2a) : 1/(c + a - 2b) : 1/(a + b - 2c) (Darij Grinberg, 12/28/02)

James Blaikie (1847-1929) proposed the following problem. Let O be any point in the plane of triangle ABC, and let any straight line g through O meet BC in P, CA in Q, AB in R; then, if points P', Q', R' be taken on the line so that PO = OP', QO = OQ', RO = OR'. Prove that AP', BQ', CR' concur.

Darij Grinberg introduces the term Blaikie point of O and g for the point Z of concurrence. If

              O = x : y : z and g = [k : l : m] (barycentric coordinates),

then Z has first barycentric 1/[k(y-z) - (ly-mz)]. Given a point S = u : v : w, Grinberg then defines the S-Blaikie transform of O as the Blaikie point of O and OS. The first barycentric of Z can be written as

              1/[yw(y+x) + zv(z+x) - yz(2u+v+w)].

Seet Blaikie theorem in barycentrics. (Darij Grinberg, 12/28/02)

X(903) lies on the Steiner circumellipse and these lines: 2,45   7,528   27,648   75,537   86,99   310,670   320,519   335,536   350,889   527,666   675,901   812,1022

X(903) = reflection of X(i) in X(j) for these (i,j): (2,1086), (190,2)
X(903) = isogonal conjugate of X(902)
X(903) = isotomic conjugate of X(519)
X(903) = complement of X(17487)
X(903) = X(i)-cross conjugate of X(j) for these (i,j): (320,86), (519,2)
X(903) = Steiner-circumellipse-antipode of X(190)
X(903) = anticomplement of X(4370)
X(903) = projection from Steiner inellipse to Steiner circumellipse of X(1086)
X(903) = antipode of X(2) in hyperbola {A,B,C,X(2),X(7)}
X(903) = trilinear pole of line X(2)X(514)
X(903) = pole wrt polar circle of trilinear polar of X(8756) (line X(4120)X(4895))
X(903) = X(19)-isoconjugate of X(22356)
X(903) = X(48)-isoconjugate (polar conjugate) of X(8756)
X(903) = crossdifference of PU(99)
X(903) = crossdifference of every pair of points on line X(1017)X(1960)


X(904) = X(238)-HIRST INVERSE OF X(256)

Trilinears    a2/(a2 + bc) : :

X(904) lies on these lines: 1,257   21,238   31,237   55,869   101,733   172,694

X(904) = isogonal conjugate of X(1909)
X(904) = X(238)-Hirst inverse of X(256)
X(904) = cevapoint of PU(9)


X(905) = POINT ACHERNAR

Trilinears    (b - c)cot A: (c - a)cot B: (a - b)cot C
Barycentrics    a(b - c)(b2 + c2 - a2) : :

X(905) lies on these lines: 36,238   241,514   441,525   521,656   1053,1054

X(905) = isogonal conjugate of X(1783)
X(905) = isotomic conjugate of X(6335)
X(905) = complement of X(4391)
X(904) = complement of anticomplementary conjugate of X(21226)
X(905) = X(92)-isoconjugate of X(692)
X(905) = trilinear pole of line X(1364)X(3270)
X(905) = crosssum of X(i) and X(j) for these (i,j): (6,650), (42,657), (513,614)
X(905) = crossdifference of every pair of points on line X(19)X(25)
X(905) = polar conjugate of isogonal conjugate of X(23224)


X(906) = INTERSECTION OF LINES X(100)X(112) AND X(101)X(109)

Trilinears    (cos A)/[wb2 - vc2 + a(wb - vc)], u : v : w = X(3)
Trilinears    (sin 2A)/(b - c) : :

Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a3(b2 + c2 - a2)/(b - c)

X(906) lies on these lines: 32,218   41,601   72,248   100,112   101,109   163,692   219,577   1331,4574

X(906) = isogonal conjugate of X(17924)
X(906) = X(92)-isoconjugate of X(513)
X(906) = trilinear pole of line X(48)X(184)
X(906) = crossdifference of every pair of points on line X(11)X(2969)
X(906) = barycentric product X(3)*X(100)
X(906) = barycentric product of circumcircle intercepts of line X(3)X(63)


X(907) = POINT ACRUX

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = a/[(b2 - c2)(3a2 + b2 + c2)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2/[(b2 - c2)(3a2 + b2 + c2)]

X(907) lies on the circumcircle and these lines: 98,620   111,1180   112,1634

X(907) = isogonal conjugate of X(3800)
X(907) = Λ(X(419), X(2501)) (line is radical axis of circumcircle and orthosymmedial circle)


X(908) = POINT ACUBENS

Trilinears    [(wb + vc)/a - v - w]/a, where u : v : w = X(3)
Barycentrics    cos B + cos C - 1 : :
Barycentrics    a^2 (b + c) - 2 a b c - (b - c)^2 (b + c) : :

Let La, Lb, Lc be the lines through A, B, C, resp. parallel to the antiorthic axis. Let Ma, Mb, Mc be the reflections of BC, CA, AB in La, Lb, Lc, resp. Let A' = Mb∩Mc, and define B' and C' cyclically. Triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. Let A″B″C″ be the reflection of A'B'C' in the antiorthic axis. The triangle A″B″C″ is homothetic to ABC, with center of homothety X(908); see Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, March 25, 2016)

X(908) lies on these lines: 1,998   2,7   4,78   5,72   8,946   10,994   11,518   12,960   80,519   92,264   100,516   119,517   153,515   214,535   224,1079   377,936   392,495   514,661

X(908) = reflection of X(1512) in X(119)
X(908) = isogonal conjugate of X(909)
X(908) = isotomic conjugate of X(34234)
X(908) = anticomplement of X(3911)
X(908) = complement of X(3218)
X(908) = crosssum of X(41) and X(902)
X(908) = crossdifference of every pair of points on line X(31)X(663)
X(908) = inverse-in-circumconic-centered-at-X(9) of X(63)
X(908) = trilinear pole of line X(1145)X(1769)
X(908) = {X(2),X(7)}-harmonic conjugate of X(3306)
X(908) = polar conjugate of X(36123)
X(908) = pole wrt polar circle of trilinear polar of X(36123) (line X(19)X(649))
X(908) = homothetic center of the complement of the excentral triangle and the anticomplement of the intouch triangle


X(909) = POINT ADHAFERA

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = a/[(wb + vc)/a - v - w], u : v : w = X(3)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(909) lies on these lines: 9,48   19,604   55,184   163,284   333,662

X(909) = isogonal conjugate of X(908)
X(909) = trilinear pole of line X(31)X(663)
X(909) = crossdifference of every pair of points on line X(1145)X(1769)
X(909) = barycentric product X(1)*X(104)
X(909) = trilinear product X(6)*X(104)
X(909) = X(92)-isoconjugate of X(22350)


X(910) = CROSSPOINT OF X(57) AND X(105)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = a[wb2 + vc2 - a(wb + vc)], u : v : w = X(3)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(910) lies on these lines: 3,169   6,57   9,165   19,25   32,1104   40,220   41,65   44,513   46,218   48,354   101,517   103,971   105,919   118,516   227,607   241,294

X(910) = reflection of X(1530) in X(118)
X(910) = isogonal conjugate of X(36101)
X(910) = X(294)-Ceva conjugate of X(6)
X(910) = crosspoint of X(57) and X(105)
X(910) = crosssum of X(i) and X(j) for these (i,j): (1,910), (9,518)
X(910) = crossdifference of every pair of points on line X(1)X(905)
X(910) = X(57)-Hirst inverse of X(1419)
X(910) = X(230)-of-excentral-triangle


X(911) = POINT ADHARA

Trilinears    a/[wb2 + vc2 - a(wb + vc)] : : , where u : v : w = X(3)
Barycentrics    a^3 / (2 a^3 - a^2 (b + c) - (b - c)^2 (b + c)) : :

X(911) lies on these lines: 3,101   41,603   48,692   56,607   241,294

X(911) = isogonal conjugate of X(30807)
X(911) = barycentric product X(1)*X(103)
X(911) = trilinear product X(6)*X(103)
X(911) = isogonal conjugate of isotomic conjugate of X(36101)
X(911) = isogonal conjugate of polar conjugate of X(36122)
X(911) = polar conjugate of isotomic conjugate of X(36056)


X(912) = X(104)-CEVA CONJUGATE OF X(3)

Trilinears    (a^2 - b^2 - c^2) (a^3 (b + c) - a^2 (b^2 + c^2) - a (b - c)^2 (b + c) + (b^2 - c^2)^2) : :

As the isogonal conjugate of a point on the circumcircle, X(912) lies on the line at infinity.

X(912) lies on these (parallel) lines: 1,90   3,63   5,226   30,511   38,1064   65,68   222,1060   601,976   774,1066   960,993

X(912) = isogonal conjugate of X(915)
X(912) = isotomic conjugate of polar conjugate of X(8609)
X(912) = X(104)-Ceva conjugate of X(3)
X(912) = crossdifference of every pair of points on line X(6)X(3657)
X(912) = X(19)-isoconjugate of X(2990)
X(912) = X(92)-isoconjugate of X(32655)


X(913) = POINT AGENA

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = a/[(wb + vc)/a - v - w], u : v : w = X(4)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(913) lies on these lines: 19,101   25,692   27,662   571,608

X(913) = isogonal conjugate of X(914)


X(914) = ISOGONAL CONJUGATE OF X(913)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = [(wb + vc)/a - v - w]/a, u : v : w = X(4)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(914) lies on these lines: 8,224   63,69   514,661

X(914) = isogonal conjugate of X(913)
X(914) = isotomic conjugate of X(37203)


X(915) = ISOGONAL CONJUGATE OF X(912)

Trilinears    (tan A)/(sin B (-1 + cos A + cos B) + sin C (-1 + cos A + cos C)) : :
Barycentrics    a/((a^2 - b^2 - c^2) (a^3 (b + c) - a^2 (b^2 + c^2) - a (b - c)^2 (b + c) + (b^2 - c^2)^2)) : :

X(915) lies on the circumcircle and these lines: 19,101   21,925   24,108   28,110   34,46   99,286   242,929

X(915) = isogonal conjugate of X(912)
X(915) = X(517)-cross conjugate of X(4)
X(915) = antipode of X(4) in hyperbola {A,B,C,X(4),X(19)}}
X(915) = inverse-in-polar-circle of X(119)
X(915) = trilinear pole of line X(6)X(3657)
X(915) = Ψ(X(6), X(3657))
X(915) = polar conjugate of isogonal conjugate of X(32655)
X(915) = polar conjugate of isotomic conjugate of X(2990)
X(915) = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {A,B,C,X(4),X(19)}}


X(916) = X(103)-CEVA CONJUGATE OF X(3)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = wb2 + vc2 - a(wb + vc), u : v : w = X(4)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(916) lies on the line at infinity.

X(916) lies on these (parallel) lines: 3,48   30,511   72,185   1037,1069

X(916) = isogonal conjugate of X(917)
X(916) = isotomic conjugate of polar conjugate of X(8608)
X(916) = X(19)-isoconjugate of X(2989)
X(916) = X(103)-Ceva conjugate of X(3)


X(917) = X(516)-CROSS CONJUGATE OF X(4)

Trilinears    (sec A)/((a - b) sin 2B + (a - c) sin 2C) : :
Barycentrics    1/((a^2 - b^2 - c^2) (a^3 (b^2 + c^2) - a^2 (b^3 + c^3) - a (b^2 - c^2)^2 + b^5 - b^3 c^2 - b^2 c^3 + c^5)) : :

Let A'B'C' be the 1st circumperp triangle. Let A″ be the pole, wrt the polar circle, of line B'C', and define B″and C″ cyclically. The lines A'A″, B'B″, C'C″ concur in X(917). (Randy Hutson, July 11, 2019)

X(917) lies on the circumcircle and these lines: 4,101   27,110   92,100   109,278

X(917) = reflection of X(1305) in X(3)
X(917) = reflection of X(4) in X(5190)
X(917) = isogonal conjugate of X(916)
X(917) = X(516)-cross conjugate of X(4)
X(917) = polar conjugate of isotomic conjugate of X(2989)
X(917) = X(63)-isoconjugate of X(8608)
X(917) = antipode of X(4) in hyperbola {A,B,C,X(4),X(27)}
X(917) = polar-circle-inverse of X(118)
X(917) = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {A,B,C,X(4),X(27)}


X(918) = ISOTOMIC CONJUGATE OF X(666)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = (w/b - v/c)/a2, u : v : w = X(100)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (b - c)(b2 + c2 -ab - ac)

As the isogonal conjugate of a point of the circumcircle, X(918) lies on the line at infinity.

X(918) lies on these (parallel) lines: 30,511   63,654   190,644   1086,1111

X(918) = isogonal conjugate of X(919)
X(918) = isotomic conjugate of X(666)
X(918) = crosssum of X(6) and X(665)
X(918) = crossdifference of every pair of points on line X(6)X(692)
X(918) = X(2)-Ceva conjugate of X(35094)
X(918) = X(514)-Hirst inverse of X(522)
X(918) = perspector of circumconic centered at X(35094)


X(919) = ISOGONAL CONJUGATE OF X(918)

Trilinears   a2/(w/b - v/c) : : , where u : v : w = X(100)

Let Q be a point on line X(2)X(11) other than X(2). Let A'B'C' be the cevian triangle of Q. Let A″ be the {B,C}-harmonic conjugate of A' (or equivalently, A″ = BC∩B'C'), and define B″, C″ cyclically. The circumcircles of AB″C″, BC″A″, CA″B″ concur in X(919). (Randy Hutson, February 10, 2016)

Let A', B', C' be the intersections of line X(1)X(6) and lines BC, CA, AB, resp. The circumcircles of AB'C', BC'A', CA'B' concur in X(919). (Randy Hutson, February 10, 2016)

X(919) lies on the circumcircle and these lines: 6,840   99,666   100,650   101,663   103,672   104,294   105,910   106,1055   109,649   673,675   885,929

X(919) = isogonal conjugate of X(918)
X(919) = trilinear pole of line X(6)X(692)
X(919) = Ψ(X(i), X(j)) for these (i,j): (1,41), (2,11), (6,692), (11,650), (76,8)
X(919) = isogonal conjugate of isotomic conjugate of trilinear pole of line X(2)X(11)
X(919) = circumcircle intercept, other than A, B, C, of conic {A,B,C,PU(95)}}
X(919) = inverse-in-Stevanovic-circle of X(100)
X(919) = trilinear product X(101)*X(105) (circumcircle-X(1) antipodes)
X(919) = barycentric product X(100)*X(105) (circumcircle-X(2) antipodes)


X(920) = X(158)-CEVA CONJUGATE OF X(1)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = cos2B + cos2C - cos2A\
Trilinears    SA - 2 R^2 : :
Trilinears    a^6 - 3 a^4 (b^2 + c^2) + a^2 (3 b^4 - 2 b^2 c^2 + 3 c^4) - (b^2 - c^2)^2 (b^2 + c^2) : :

X(920) lies on these lines: 1,21   4,46   4,78   9,498   19,91   57,499   158,921   201,601   243,1075

X(920) = isogonal conjugate of X(921)
X(920) = X(158)-Ceva conjugate of X(1)


X(921) = ISOGONAL CONJUGATE OF X(920)

Trilinears    1/(cos2B + cos2C - cos2A) : :
Trilinears    1/(a^6 - 3 a^4 (b^2 + c^2) + a^2 (3 b^4 - 2 b^2 c^2 + 3 c^4) - (b^2 - c^2)^2 (b^2 + c^2)) : :

X(921) lies on these lines: 19,47   46,225   63,91   158,920

X(921) = isogonal conjugate of X(920)
X(921) = X(255)-cross conjugate of X(1)
X(921) = intersection of tangents at X(46) and X(90) to Orthocubic K006


X(922) = POINT ALBALI

Trilinears    a2(b2 + c2 - 2a2) (M. Iliev, 5/13/07)

X(922) lies on these lines: 31,48   667,788

X(922) = isogonal conjugate of isotomic conjugate of X(896)
X(922) = isogonal conjugate of complement of anticomplementary conjugate of X(21298)
X(922) = isogonal conjugate of anticomplement of complementary conjugate of X(21256)
X(922) = complement of X(21298)
X(922) = anticomplement of X(21256)
X(922) = trilinear product of PU(107)


X(923) = POINT ALBIREO

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2/(b2 + c2 - 2a2)     (M. Iliev, 5/13/07)

Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(923) lies on these lines: 1,662   31,163   42,101   213,692   691,741

X(923) = isogonal conjugate of X(14210)
X(923) = barycentric product X(1)*X(111)
X(923) = trilinear product of circumcircle intercepts of Schoute circle
X(923) = trilinear product of circumcircle intercepts of line X(6)X(512)
X(923) = trilinear product X(6)*X(111)
X(923) = polar conjugate of isotomic conjugate of X(36060)


X(924) = CROSSDIFFERENCE OF X(5) AND X(6)

Trilinears    b cos(A - B) - c cos(A - C) : :
Trilinears    a (b^2 - c^2) (a^4 + b^4 + c^4 - 2 a^2 b^2 - 2 a^2 c^2) : :
Barycentrics    sec 2B - sec 2C : :

As the isogonal conjugate of a point on the circumcircle, X(924) lies on the line at infinity.

X(924) lies on these (parallel) lines: 30,511   50,647   66,879   669,684

X(924) = isogonal conjugate of X(925)
X(924) = isotomic conjugate of isogonal conjugate of X(34952)
X(924) = complementary conjugate of X(136)
X(924) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,39013), (4,136), (70,125)
X(924) = crosspoint of X(i) and X(j) for these (i,j): (4,110), (99,275)
X(924) = crosssum of X(i) and X(j) for these (i,j): (3,523), (216,512)
X(924) = crossdifference of every pair of points on line X(5)X(6)
X(924) = perspector of hyperbola {A,B,C,X(2),X(24)}}
X(924) = intersection of trilinear polars of X(2) and X(24)
X(924) = barycentric square root of X(39013)


X(925) = ANTICOMPLEMENT OF X(136)

Trilinears    (sin A)/(sec 2B - sec 2C) : :
Trilinears    1/(b cos(A - B) - c cos(A - C)) : :
Barycentrics    1/((b^2 - c^2) (a^4 + b^4 + c^4 - 2 a^2 b^2 - 2 a^2 c^2)) : :

X(925) lies on the circumcircle and these lines: 2,136   3,847   4,131   20,68   21,915   22,98   91,759   94,96   648,933   842,858

X(925) = reflection of X(i) in X(j) for these (i,j): (4,131), (1300,3)
X(925) = isogonal conjugate of X(924)
X(925) = isotomic conjugate of X(6563)
X(925) = trilinear pole of line X(5)X(6)
X(925) = concurrence of reflections in sides of ABC of line X(4)X(52)
X(925) = Ψ(X(1), X(91))
X(925) = Ψ(X(3), X(68))
X(925) = Ψ(X(4), X(52))
X(925) = Ψ(X(6), X(5))
X(925) = Λ(trilinear polar of X(24))
X(925) = inverse-in-polar-circle of X(135)
X(925) = anticomplement of X(136)
X(925) = X(26)-cross conjugate of X(250)
X(925) = Gibert-circumtangential conjugate of X(34952)


X(926) = CROSSDIFFERENCE OF X(6) AND X(7)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                         where f(a,b,c) = bw - cv, u : v : w = X(7)
Trilinears    a (a - b - c) (b - c) (a b + a c - b^2 - c^2) : :

As the isogonal conjugate of a point on the circumcircle, X(926) lies on the line at infinity.

X(926) lies on these (parallel) lines: 30,511   55,654   101,692   295,875   657,663   2170,2310

X(926) = isogonal conjugate of X(927)
X(926) = isotomic conjugate of anticomplement of X(39014)
X(926) = X(2)-Ceva conjugate of X(39014)
X(926) = perspector of hyperbola {A,B,C,X(2),X(55)}}
X(926) = barycentric square root of X(39014)
X(926) = crosspoint of X(i) and X(j) for these (i,j): (100,294), (101,103)
X(926) = crosssum of X(i) and X(j) for these (i,j): (241,513), (514,516), (523,857), (673,885)
X(926) = crossdifference of every pair of points on line X(6)X(7)


X(927) = ISOGONAL CONJUGATE OF X(926)

Trilinears    1/(bw - cv) : : , where u : v : w = X(7)

Let La, Lb, Lc be the lines through A, B, C, resp. parallel to the Gergonne line. Let Ma, Mb, Mc be the reflections of BC, CA, AB in La, Lb, Lc, resp. Let A' = Mb∩Mc, and define B' and C' cyclically. Triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. The centroid of A'B'C' is X(927); see Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, March 25, 2016)

Let A', B', C' be the intersections of line X(7)X(8) and lines BC, CA, AB, resp. The circumcircles of AB'C', BC'A', CA'B' concur in X(927). (Randy Hutson, March 25, 2016)

X(927) lies on the circumcircle and these lines: 7,840   100,693   101,514   103,516   109,658   813,1025

X(927) = isogonal conjugate of X(926)
X(927) = trilinear pole of line X(6)X(7)
X(927) = Ψ(X(1), X(85))
X(927) = Ψ(X(6), X(7))
X(927) = Ψ(X(41), X(1))
X(927) = Ψ(X(650), X(2))
X(927) = circumcircle-intercept, other than A, B, C, of conic {A,B,C,PU(57)}
X(927) = circumcircle-intercept, other than A, B, C, of conic {A,B,C,PU(96)}}
X(927) = intersection of antipedal lines of X(101) and X(103)


X(928) = CROSSDIFFERENCE OF X(6) AND X(11)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = bw - cv, u : v : w = X(11)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(928) lies on the line at infinity.

X(928) lies on these (parallel) lines: 30,511   101,109   102,103   116,124   117,118   151,152   295,876

X(928) = isogonal conjugate of X(929)
X(928) = crosssum of X(523) and X(851)
X(928) = crossdifference of every pair of points on line X(6)X(11)
X(928) = X(2)-Ceva conjugate of X(39017)
X(928) = perspector of hyperbola {A,B,C,X(2),X(59)}}
X(928) = barycentric square root of X(39017)


X(929) = ISOGONAL CONJUGATE OF X(928)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = 1/(bw - cv), u : v : w = X(11)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(929) lies on the circumcircle.

X(929) lies on these lines: 101,522   102,516   103,515   109,514   242,915   885,919

X(929) = isogonal conjugate of X(928)
X(929) = trilinear pole of line X(6)X(11)
X(929) = Ψ(X(6), X(11))


X(930) = ANTICOMPLEMENT OF X(137)

Trilinears    1/(bw - cv) : : , where u : v : w = X(17)
Barycentrics    1/((b^2 - c^2) (a^4 + b^4 + c^4 - 2 a^2 b^2 - 2 a^2 c^2 - b^2 c^2)) : :

X(930) lies on the circumcircle and these lines: 2,137   3,252   4,128   74,550

X(930) = reflection of X(i) in X(j) for these (i,j): (4,128), (1141,3), (1263,140)
X(930) = isogonal conjugate of X(1510)
X(930) = isotomic conjugate of isogonal conjugate of X(32737)
X(930) = anticomplement of X(137)
X(930) = X(523)-cross conjugate of X(1487)
X(930) = trilinear pole of line X(6)X(17) (Napoleon axis)
X(930) = Ψ(X(6), X(17))
X(930) = Ψ(X(49), X(3))
X(930) = X(74)-of-Lucas-triangle (defined at X(95))
X(930) = X(74)-of-circumorthic-triangle


X(931) = POINT ALCHIBA

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = 1/(bw - cv), u : v : w = X(21)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(931) lies on the circumcircle and these lines: 100,645   101,643   108,648   109,662   111,941

X(931) = isogonal conjugate of X(8672)
X(931) = trilinear pole of line X(6)X(21)
X(931) = Ψ(X(6), X(21))


X(932) = X(190)-CROSS CONJUGATE OF X(100)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = 1/(bw - cv), u : v : w = X(43)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(932) lies on the circumcircle and these lines: 1,727   21,741   81,715   87,106   105,330   172,699   644,813   667,668

X(932) = isogonal conjugate of X(4083)
X(932) = isotomic conjugate of X(20906)
X(932) = X(190)-cross conjugate of X(100)
X(932) = trilinear pole of line X(6)X(43)
X(932) = Ψ(X(1), X(87))
X(932) = Ψ(X(6), X(43))


X(933) = X(4)-CROSS CONJUGATE OF X(250)

Trilinears    1/(bw - cv) : : , where u : v : w = X(54)
Trilinears    sec A csc(2B - 2C) : :
Trilinears    a/[(b^2 - c^2)(b^2 + c^2 - a^2)(b^4 + c^4 - a^2b^2 - a^2c^2 - 2b^2c^2)] : :

Let A' = BC∩X(186)X(523), and define B' and C' cyclically. The circumcircles of AB'C', BC'A', CA'B' concur in X(933). Note: Line X(186)X(523) is the polar of X(4) wrt the circumcircle. (Randy Hutson, February 10, 2016)

Let P be a point on the circumcircle, and let

(Oab) = circle through A tangent to line BC at B
(Oac) = circle through A tangent to line BC at C
Ba = point other than A where the line PB meets (Oab)
Ca = point other than A where the line PC meets (Oac)
Hab = orthocenter of APBa
Hac = orthocenter of APCa

Define Hbc and Hca cyclically, and define Hba and Hcb cyclically. The points Hab, Hac, Hbc, Hba, Hca, Hcb lies on a hyperbola with center = midpoint(P and H), where H = X(4), the orthocenter. This hyperbola is rectangular if and only if P = X(933). This note is related to a problem posed in 2017 Olympiad of Rusanovsky Lyceum in Kyiv, Ukraine , IX-X Team p9. For details, see El centro X(933) y una hipérbola rectangular. (Angel Montesdeoca, July 3, 2021)

X(933) lies on the circumcircle and these lines: 4,137   54,74   98,275   250,476   270,759   648,925

X(933) = X(4)-cross conjugate of X(250)
X(933) = isogonal conjugate of X(6368)
X(933) = Λ(X(684), X(2525)); line X(684)X(2525) is the isotomic conjugate, wrt the MacBeath triangle, of the MacBeath inconic)
X(933) = Λ(X(2081), X(2600)) (line X(2081)X(2600) is the trilinear polar of X(5))
X(933) = trilinear pole of line X(6)X(24)
X(933) = concurrence of reflections of line X(4)X(54) in sides of ABC
X(933) = Ψ(XI),X(j)) for these (i,j): (3,54), (4,54), (5,2), (6,24), (69,54)
X(933) = inverse-in-polar-circle of X(137)
X(933) = X(1577)-isoconjugate of X(216)
X(933) = barycentric product X(110)*X(275)
X(933) = barycentric quotient X(275)/X(850)
X(933) = perspector of circumcevian triangle of X(186) and cross-triangle of ABC and circumorthic triangle
X(933) = the point of intersection, other than A, B, and C, of the circumcircle and conic {A,B,C,PU(61)}}
X(933) = center of the bianticevian conic of X(1) and X(47); i.e. the rectangular hyperbola {X(1), X(47), X(48), and vertices of their anticevian triangles}


X(934) = X(513)-CROSS CONJUGATE OF X(57)

Trilinears    1/(bw - cv) : : , where u : v : w = X(57)
Trilinears    1/[(b - c)(b + c - a)^2] : :

Let P be a point on line X(4)X(7) other than X(4). Let A' be the reflection of P in BC, and define B' and C' cyclically. The circumcircles of AA'B', BC'A', and CA'B' concur at X(934). (Randy Hutson, July 20, 2016)

X(934) lies on the circumcircle and these lines: 1,103   3,972   7,104   56,105   77,102   100,658   101,651   106,269   644,1025   675,1088   727,1106   741,1042   759,1014

X(934) = reflection of X(972) in X(3)
X(934) = isogonal conjugate of X(3900)
X(934) = isotomic conjugate of X(4397)
X(934) = X(513)-cross conjugate of X(57)

X(934) = perspector of ABC and triangle formed by line X(3)X(9) reflected in sides of ABC
X(934) = concurrence of reflections in sides of ABC of line X(4)X(7)
X(934) = Λ(X(650), X(663)) (Λ(trilinear polar of X(9))
X(934) = Λ(X(3064), X(3700)) (Λ(trilinear polar of X(281))
X(934) = trilinear pole of line X(6)X(57)
X(934) = polar-circle-inverse of X(38966)
X(934) = X(2)-Ceva conjugate of X(23971)
X(934) = Ψ(X(1), X(7))
X(934) = Ψ(X(3), X(77))
X(934) = Ψ(X(4), X(7))
X(934) = Ψ(X(6), X(57))
X(934) = Ψ(X(9), X(57))
X(934) = trilinear pole, wrt circumtangential triangle, of line X(3)X(9)
X(934) = trilinear product of intercepts of circumcircle and Soddy line
X(934) = eigencenter of 4th mixtilinear triangle
X(934) = intersection of antipedal lines of circumcircle intercepts of line X(3)X(9)

X(935) = POINT ALCOR

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = 1/(bw - cv), u : v : w = X(67)
Barycentrics    1/((b^2 - c^2) (b^2 + c^2 - a^2) (b^4 + c^4 - a^4 - b^2 c^2)) : :

X(935) lies on the circumcircle and these lines: 4,842   67,74   98,186   99,3267   110,525   111,468   112,523   250,827   378,477

X(935) = anticomplement of X(38971)
X(935) = X(63)-isoconjugate of X(2492)
X(935) = trilinear pole of line X(6)X(67) (the radical axis of the orthocentroidal circle and Dao-Moses-Telv circle)
X(935) = concurrence of reflections in sides of ABC of line X(4)X(67)
X(935) = polar conjugate of X(9979)
X(935) = Ψ(X(3), X(67))
X(935) = Ψ(X(4), X(67))
X(935) = Ψ(X(6), X(67))
X(935) = Ψ(X(23), X(2))
X(935) = Λ(trilinear polar of X(23))
X(935) = reflection of X(112) in the Euler line
X(935) = inverse-in-polar-circle of X(5099)


X(936) = INTERSECTION OF LINES X(1)X(2) AND X(3)X(9)

Trilinears  a3 - a2(b + c) - a(b - c)2 + (b + c)3 : :

X(936) lies on these lines: 1,2   3,9   40,960   56,210   57,72   63,404   165,411   223,1038   226,443   269,307   377,908   581,966   984,988

X(936) = isogonal conjugate of X(937)
X(936) = complement of X(938)
X(936) = exsimilicenter of hexyl and Spieker circles; the insimilicenter is X(9623)
X(936) = homothetic center of medial triangle and tangential triangle of the hexyl triangle


X(937) = ISOGONAL CONJUGATE OF X(936)

Trilinears    1/[a3 - a2(b + c) - a(b - c)2 + (b + c)3]

X(937) lies on these lines: 1,329   6,40   31,1103   34,196   56,223

X(937) = isogonal conjugate of X(936)


X(938) = ANTICOMPLEMENT OF X(936)

Trilinears         f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = bc[a4 - 2a3(b + c) - 4a2bc + (b - c)(b2 - c2)(2a - b - c)]
Trilinears    cos A' : :, where A' is the angle formed by the tangents from the A-excenter to the incircle

X(938) lies on these lines: 1,2   4,7   20,57   29,81   40,390   56,411   63,452   65,497   354,388   355,1056   517,1058   774,986   944,999

X(938) = isogonal conjugate of X(939)
X(938) = anticomplement of X(936)
X(938) = {X(2),X(145)}-harmonic conjugate of X(78)


X(939) = ISOGONAL CONJUGATE OF X(938)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = a/[a4 - 2a3(b + c) - 4a2bc + (b - c)(b2 - c2)(2a - b - c)]
Trilinears    sec A' : :, where A' is the angle formed by the tangents from the A-excenter to the incircle

X(939) lies on these lines: 3,269   34,55   56,212

X(939) = isogonal conjugate of X(938)


X(940) = POINT ALDEBARAN

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2 + a(b + c) + 2bc
Trilinears        as + bc : bs + ca : cs + ab
Barycentrics    a^2 + 4 R r : :

Let A' be the center of the conic through the contact points of the incircle and the A- excircle with the sidelines of ABC. Define B' and C' cyclically. The triangle A'B'C' is perspective to the 4th extouch triangle at X(940). See also X(6), X(25), X(218), X(222), X(1743). (Randy Hutson, July 23, 2015)

X(940) lies on these lines: 1,3   2,6   31,1001   37,63   42,750   58,405   72,975   222,226   312,894   386,474   387,443   518,612

X(940) = isogonal conjugate of X(941)
X(940) = isotomic conjugate of X(34258)
X(940) = complement of X(5739)
X(940) = anticomplement of X(5743)
X(940) = crosssum of X(11) and X(784)
X(940) = crossdifference of every pair of points on line X(512)X(650)
X(940) = {X(1),X(40)}-harmonic conjugate of X(37548)
X(940) = {X(2),X(6)}-harmonic conjugate of X(4383)


X(941) = ISOGONAL CONJUGATE OF X(940)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                         where f(a,b,c) = 1/[a2 + a(b + c) + 2bc]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(941) lies on these lines: 1,573   2,314   6,21   8,37   9,42   81,967   84,581   111,931

X(941) = isogonal conjugate of X(940)
X(941) = isotomic conjugate of X(34284)
X(941) = anticomplement of X(10472)
X(941) = polar conjugate of isotomic conjugate of X(34259)
X(941) = trilinear pole of line X(512)X(650)
X(941) = X(63)-isoconjugate of X(4185)


X(942) =  INCIRCLE-INVERSE OF X(36)

Trilinears    2abc + (b + c)(a - b + c)(a + b -c) : :
Trilinears    1 + cos B + cos C : :
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

Let A'B'C' be the incentral triangle of triangle ABC. Let LA be the line of reflection of line BC in line B'C', and define LB and LC cyclically. Let A'' = LB∩LC, and define B'' and C'' cyclically. The lines AA'', BB'', CC'' concur in X(942). (Randy Hutson, 9/23/2011)

Let I be the incenter of ABC. Let NA be the nine-point center of the triangle IBC, and define NB and NC cyclically. Let AA be the reflection of NA in the line AI, Let AB be the reflection of NA in the line BI, let Let AC be the reflection of NA in the line CI, and define BA, BB, and BC cyclically, and define CA, CB, and CC cyclically. Let OA be the nine-point center of the triangle AAABAC, let OB be the nine-point center of the triangle BABBBC, and let OC be the nine-point center of the triangle CACBCC. The circles OA, OB, OC concur in X(942). (Antreas Hatzipolakis and César Lozada, January 28, 2015, Hyacinthos 23074)

X(942) is the center of the conic which is the locus of poles, wrt the incircle, of tangents to the circumcircle. This conic has foci at X(1) and X(65). (Randy Hutson, July 20, 2016)

Let A'B'C' be the intouch triangle. Let A″ be the orthocenter of AB'C', and define B″ and C″ cyclically. The triangle A″B″C″ is homothetic to A'B'C', and the center of homothety is X(942). (Randy Hutson, July 20, 2016)

Let A'B'C' be the orthic triangle. Let A″ be the incenter of AB'C', and define B″ and C″ cyclically. The triangle A″B″C″ is homothetic to the intouch triangle, and the center of homothety is X(942). (Randy Hutson, July 20, 2016)

Let A' be the homothetic center of the orthic triangles of the intouch and A-extouch triangles, and define B' and C' cyclically. The triangle A'B'C' is perspective to the half-altitude triangle at X(942). In fact, A'B'C' is the cevian triangle of X(942), wrt the half-altitude triangle. (Randy Hutson, July 20, 2016)

A construction of X(942) is given at 24080. (Antreas Hatzipolakis, August 29, 2016)

X(942) lies on these lines: 1,3   2,72   4,7   5,226   6,169   8,443   10,141   11,113   28,60   30,553   34,222   37,579   42,1066   58,1104   63,405   78,474   212,582   238,1046   277,1002   279,955   284,501   355,388   496,946   750,976   758,960   962,1058   1042,1064

X(942) = midpoint of X(1) and X(65)
X(942) = reflection of X(960) in X(1125)
X(942) = isogonal conjugate of X(943)
X(942) = complement of X(72)
X(942) = anticomplement of X(5044)
X(942) = complementary conjugate of X(21530)
X(942) = incircle-inverse of X(36)
X(942) = Conway-circle-inverse of X(38483)
X(942) = X(1)-Ceva conjugate of X(500)
X(942) = crosspoint of X(i) and X(j) for these (i,j): (1,79), (2,286), (7,81)
X(942) = crosssum of X(i) and X(j) for these (i,j): (1,35), (6,228), (37,55)
X(942) = X(5)-of-the-intouch-triangle
X(942) = X(1112) of Fuhrmann triangle
X(942) = {X(1),X(40)}-harmonic conjugate of X(3295)
X(942) = X(3)-of-inverse-in-incircle-triangle
X(942) = X(4)-of-incircle-circles-triangle
X(942) = X(6756)-of-excentral-triangle
X(942) = excentral-to-intouch similarity image of X(3)
X(942) = Cundy-Parry Phi transform of X(55)
X(942) = Cundy-Parry Psi transform of X(7)
X(942) = QA-P23 (Inscribed Square Axes Crosspoint) of quadrangle ABCX(1) (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/51-qa-p23.html)


X(943) = ISOGONAL CONJUGATE OF X(942)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = 1/[2abc + (b + c)(a - b + c)(a + b - c)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(943) lies on these lines: 1,201   3,7   4,12   8,405   21,72   28,228   35,79   80,950   100,442   500,651   968,1039   1001,1058

X(943) = isogonal conjugate of X(942)
X(943) = cevapoint of X(i) and X(j) for these (i,j): (1,35), (6,228), (37,55)
X(943) = X(523)-cross conjugate of X(100)
X(943) = Cundy-Parry Phi transform of X(7)
X(943) = Cundy-Parry Psi transform of X(55)


X(944) = HOFSTADTER TRAPEZOID POINT

Trilinears    bc[3a4 - 2a3(b + c) + (b - c)2(2ab + 2ac - 2bc - b2 - c2 - 2a2)]
Trilinears    1 - cos A - cos B - cos C + cos B cos C
Trilinears    r - R cos B cos C : :
X(944) = 2 X(1) - X(4)

X(944) is the point in which the extended legs X(4)X(1) and X(3)X(8) of the trapezoid X(4)X(1)X(3)X(8) meet. The point is introduced in

Hofstadter, Douglas. R., "Discovery and dissection of a geometric gem," in Geometry Turned On! editors J. R. King and D. Schattschneider, Mathematical Association of America, Washington, D. C., 1997, 3-14.

The centroid of ABC is also the centroid of triangle X(4)X(8)X(944). (Darij Grinberg, August 22, 2002)

Let Ha be the hyperbola passing through A, and with foci at B and C. This is called the A-Soddy hyperbola in (Paul Yiu, Introduction to the Geometry of the Triangle, 2002; 12.4 The Soddy hyperbolas, p. 143). Let La be the polar of X(3) with respect to Ha. Define Lb and Lc cyclically. Let A' = Lb ∩ Lc, B' = Lc ∩ La, C' = La ∩ Lb. Triangle A'B'C' is homothetic to ABC, and its orthocenter is X(944). (Randy Hutson, January 29, 2018)

Let Ma be the polar of X(4) wrt the circle centered at A and passing through X(1), and define Mb, Mc cyclically. (Note: X(4) is the perspector of any circle centered at a vertex of ABC.) Let A″ = Mb ∩ Mc, and define B″ and C″ cyclically. Triangle A″B″C″ is homothetic to ABC, and its orthocenter is X(944). (Randy Hutson, January 29, 2018)

X(944) lies on these lines: 1,4   2,355   3,8   5,3616   10,631   20,145   30,962   40,376   48,281   80,499   84,1000   150,348   390,971   392,452   938,999   958,1006

X(944) = midpoint of X(20) and X(145)
X(944) = reflection of X(i) in X(j) for these (i,j): (4,1), (8,3), (355,1385), (962,1482)
X(944) = isogonal conjugate of X(945)
X(944) = anticomplement of X(355)
X(944) = crosspoint, wrt hexyl triangle, of X(1) and X(40)
X(944) = X(185)-of-hexyl-triangle
X(944) = outer-Garcia-to-inner-Garcia similarity image of X(4)
X(944) = {X(1),X(4)}-harmonic conjugate of X(5603)
X(944) = Ehrmann-mid-to-Johnson similarity image of X(1)
X(944) = X(4)-of-5th-mixtilinear-triangle


X(945) = ISOGONAL CONJUGATE OF X(944)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = a/[3a4 - 2a3(b + c) + (b - c)2(2ab + 2ac - 2bc - b2 - c2 - 2a2)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(945) lies on these lines: 78,517   283,859

X(945) = isogonal conjugate of X(944)
X(945) = X(3)-vertex conjugate of X(56)


X(946) = MIDPOINT OF X(1) AND X(4)

Trilinears    bc[a3(b + c) + (b - c)2(a2 - ab - ac - b2 - c2 - 2bc)] : :
Trilinears    r + 2 R cos B cos C : :
Trilinears    2 tan A + tan B + tan C + (sec A)(sin B + sin C) : :
X(946) = X(1) + X(4)

Let A' be the midpoint of BC, and define B' and C' cyclically. A″ be the midpoint of AX(1), and define B″ and C″ cyclically. Let A″'B″'C″' be the incentral triangle. Let A* be the orthocenter of A'A″A″', and define B* and C* cyclically. The lines A″A*, B″B*, C″C* concur in X(946). (Randy Hutson, November 18, 2015)

Let A' be the intersection of these three lines: the perpendicular from midpoint of segment CA to line BX(1), the perpendicular from midpoint of segment AB to line CX(1), and the perpendicular from midpoint of segment AX(1) to line BC. Define B' and C' cyclically. The circumcenter of A'B'C' is X(946). Note that A'B'C' is the complement of the excentral triangle; also A'B'C' is the extraversion triangle of X(10). (Randy Hutson, November 18, 2015)

Let A' be the midpoint of A and X(1), and define B' and C' cyclically. The orthocenter of A'B'C' is X(946). (Randy Hutson, November 18, 2015)

X(946) lies on these lines: 1,4   2,40   3,142   5,10   7,84   8,908   11,65   29,102   30,551   46,499   56,1012   79,104   165,631   238,580   355,381   392,442   496,942   546,952   951,1067

X(946) = midpoint of X(i) and X(j) for these (i,j): (1,4), (40,962)
X(946) = reflection of X(i) in X(j) for these (i,j): (3,1125), (10,5)
X(946) = isogonal conjugate of X(947)
X(946) = complement of X(40)
X(946) = incircle-inverse of X(1785)
X(946) = crosspoint of X(i) and X(j) for these (i,j): (2,309), (7,92)
X(946) = crosssum of X(48) and X(55)
X(946) = incenter-of-Euler-triangle
X(946) = X(4)-of-3rd-Euler-triangle
X(946) = X(20)-of-4th-Euler-triangle
X(946) = X(113)-of-Fuhrmann-triangle
X(946) = Johnson-isogonal conjugate of X(37823)
X(946) = center of conic that is the locus of orthopoles of lines passing through X(1)
X(946) = radical center of circles centered at excenters and internally tangent to nine-point circle
X(946) = harmonic center of 1st & 2nd Johnson-Yff circles
X(946) = X(5446)-of-excentral-triangle
X(946) = Johnson-to-Ehrmann-mid similarity image of X(1)


X(947) = ISOGONAL CONJUGATE OF X(946)

Barycentrics    a^2/(a^2*(b-c)^2+a^3*(b+c)-a*(b-c)^2*(b+c)-(b^2-c^2)^2) : :

Let PaPbPc be the cevian triangle of X(8). Let Pa', Pb' and Pc' be the reflections of Pa, Pb and Pc in the midsegments of ABC. ABC and Pa'Pb'Pc' are orthologic with center X(947). The reverse orthology center is X(40). (Ivan Pavlov, July 23, 2023)

X(947) lies on these lines: {1, 1753}, {3, 5399}, {6, 1622}, {29, 515}, {35, 1795}, {40, 77}, {48, 282}, {55, 1433}, {73, 102}, {78, 2975}, {84, 2187}, {101, 14872}, {219, 572}, {283, 4184}, {332, 33297}, {388, 54972}, {517, 7100}, {581, 1036}, {602, 5053}, {944, 10570}, {945, 10571}, {950, 1067}, {951, 1066}, {1065, 10106}, {1069, 10267}, {1385, 1807}, {1412, 3072}, {1437, 15626}, {1617, 11425}, {1790, 35995}, {1794, 15931}, {2817, 3466}, {5882, 51565}, {5907, 36942}, {11012, 40442}, {13346, 23853}, {13367, 20999}, {13598, 51621}, {16202, 38248}, {16980, 20838}, {23696, 48897}, {34043, 38674}, {35057, 37628}

X(947) = isogonal conjugate of X(946)
X(947) = cevapoint of X(48) and X(55)
X(947) = trilinear pole of line {652, 6586}
X(947) = X(i)-isoconjugate-of-X(j) for these {i, j}: {1, 946}, {2, 2262}, {4, 17102}, {56, 23528}, {57, 20262}, {77, 1856}, {85, 40957}, {92, 22063}, {189, 40943}, {273, 40945}, {40836, 52097}
X(947) = X(i)-vertex conjugate of X(j) for these {i, j}: {4, 58}, {54, 10308}, {84, 947}
X(947) = X(i)-Dao conjugate of X(j) for these {i, j}: {1, 23528}, {3, 946}, {5452, 20262}, {22391, 22063}, {32664, 2262}, {36033, 17102}
X(947) = X(i)-cross conjugate of X(j) for these {i, j}: {1208, 84}, {3900, 101}, {23224, 109}
X(947) = intersection, other than A, B, C, of circumconics {{A, B, C, X(1), X(3)}}, {{A, B, C, X(4), X(103)}}, {{A, B, C, X(6), X(84)}}, {{A, B, C, X(8), X(37741)}}, {{A, B, C, X(9), X(1753)}}, {{A, B, C, X(10), X(2708)}}, {{A, B, C, X(19), X(34447)}}, {{A, B, C, X(21), X(59)}}, {{A, B, C, X(28), X(6986)}}, {{A, B, C, X(34), X(3433)}}, {{A, B, C, X(35), X(517)}}, {{A, B, C, X(36), X(1385)}}, {{A, B, C, X(40), X(55)}}, {{A, B, C, X(46), X(10267)}}, {{A, B, C, X(48), X(2360)}}, {{A, B, C, X(54), X(58)}}, {{A, B, C, X(56), X(909)}}, {{A, B, C, X(60), X(1476)}}, {{A, B, C, X(63), X(40407)}}, {{A, B, C, X(64), X(2259)}}, {{A, B, C, X(65), X(10902)}}, {{A, B, C, X(71), X(39130)}}, {{A, B, C, X(73), X(515)}}, {{A, B, C, X(74), X(1389)}}, {{A, B, C, X(80), X(5399)}}, {{A, B, C, X(81), X(29053)}}, {{A, B, C, X(90), X(2316)}}, {{A, B, C, X(92), X(1796)}}, {{A, B, C, X(105), X(5481)}}, {{A, B, C, X(106), X(3417)}}, {{A, B, C, X(165), X(3295)}}, {{A, B, C, X(186), X(37158)}}, {{A, B, C, X(268), X(7078)}}, {{A, B, C, X(280), X(15629)}}, {{A, B, C, X(388), X(581)}}, {{A, B, C, X(484), X(37621)}}, {{A, B, C, X(501), X(39633)}}, {{A, B, C, X(511), X(29037)}}, {{A, B, C, X(727), X(3406)}}, {{A, B, C, X(937), X(3423)}}, {{A, B, C, X(942), X(15931)}}, {{A, B, C, X(943), X(1295)}}, {{A, B, C, X(944), X(10571)}}, {{A, B, C, X(950), X(1066)}}, {{A, B, C, X(952), X(2773)}}, {{A, B, C, X(953), X(37518)}}, {{A, B, C, X(999), X(7987)}}, {{A, B, C, X(1064), X(10106)}}, {{A, B, C, X(1155), X(34486)}}, {{A, B, C, X(1168), X(43078)}}, {{A, B, C, X(1173), X(10308)}}, {{A, B, C, X(1222), X(4570)}}, {{A, B, C, X(1297), X(1390)}}, {{A, B, C, X(1319), X(37561)}}, {{A, B, C, X(1326), X(2698)}}, {{A, B, C, X(1391), X(2745)}}, {{A, B, C, X(1411), X(34441)}}, {{A, B, C, X(1437), X(24027)}}, {{A, B, C, X(1457), X(5882)}}, {{A, B, C, X(1482), X(5010)}}, {{A, B, C, X(1617), X(8726)}}, {{A, B, C, X(1697), X(10310)}}, {{A, B, C, X(1790), X(2167)}}, {{A, B, C, X(1803), X(7128)}}, {{A, B, C, X(2065), X(12031)}}, {{A, B, C, X(2067), X(32556)}}, {{A, B, C, X(2077), X(3057)}}, {{A, B, C, X(2256), X(37504)}}, {{A, B, C, X(2291), X(41487)}}, {{A, B, C, X(2299), X(34429)}}, {{A, B, C, X(2328), X(3562)}}, {{A, B, C, X(2335), X(3346)}}, {{A, B, C, X(2364), X(7091)}}, {{A, B, C, X(2594), X(10693)}}, {{A, B, C, X(2646), X(11012)}}, {{A, B, C, X(2718), X(15381)}}, {{A, B, C, X(2723), X(40448)}}, {{A, B, C, X(2725), X(15380)}}, {{A, B, C, X(2733), X(15386)}}, {{A, B, C, X(2807), X(2811)}}, {{A, B, C, X(3062), X(3527)}}, {{A, B, C, X(3296), X(13404)}}, {{A, B, C, X(3303), X(35242)}}, {{A, B, C, X(3333), X(8273)}}, {{A, B, C, X(3359), X(11508)}}, {{A, B, C, X(3425), X(28476)}}, {{A, B, C, X(3427), X(51223)}}, {{A, B, C, X(3428), X(3601)}}, {{A, B, C, X(3445), X(44759)}}, {{A, B, C, X(3446), X(47645)}}, {{A, B, C, X(3532), X(28171)}}, {{A, B, C, X(3579), X(3746)}}, {{A, B, C, X(3612), X(11249)}}, {{A, B, C, X(3900), X(39558)}}, {{A, B, C, X(5045), X(35202)}}, {{A, B, C, X(5119), X(11248)}}, {{A, B, C, X(5217), X(7982)}}, {{A, B, C, X(5255), X(37619)}}, {{A, B, C, X(5285), X(37528)}}, {{A, B, C, X(5563), X(13624)}}, {{A, B, C, X(5697), X(26285)}}, {{A, B, C, X(5711), X(10434)}}, {{A, B, C, X(5903), X(32613)}}, {{A, B, C, X(6244), X(53053)}}, {{A, B, C, X(6502), X(32555)}}, {{A, B, C, X(6767), X(16192)}}, {{A, B, C, X(7161), X(29374)}}, {{A, B, C, X(7162), X(33635)}}, {{A, B, C, X(7280), X(10246)}}, {{A, B, C, X(7412), X(35995)}}, {{A, B, C, X(7688), X(37080)}}, {{A, B, C, X(7742), X(18443)}}, {{A, B, C, X(8071), X(37611)}}, {{A, B, C, X(9309), X(10305)}}, {{A, B, C, X(10269), X(37618)}}, {{A, B, C, X(10419), X(12030)}}, {{A, B, C, X(11009), X(33862)}}, {{A, B, C, X(11010), X(11849)}}, {{A, B, C, X(11270), X(14497)}}, {{A, B, C, X(13452), X(28157)}}, {{A, B, C, X(13472), X(28227)}}, {{A, B, C, X(13603), X(28201)}}, {{A, B, C, X(14483), X(28197)}}, {{A, B, C, X(14496), X(28177)}}, {{A, B, C, X(14547), X(21620)}}, {{A, B, C, X(14795), X(35004)}}, {{A, B, C, X(14798), X(34339)}}, {{A, B, C, X(15227), X(17501)}}, {{A, B, C, X(15446), X(36052)}}, {{A, B, C, X(15792), X(37508)}}, {{A, B, C, X(16615), X(16835)}}, {{A, B, C, X(18446), X(21147)}}, {{A, B, C, X(21842), X(32612)}}, {{A, B, C, X(22334), X(54446)}}, {{A, B, C, X(22765), X(37616)}}, {{A, B, C, X(22770), X(30282)}}, {{A, B, C, X(26286), X(37525)}}, {{A, B, C, X(28211), X(34567)}}, {{A, B, C, X(28291), X(34071)}}, {{A, B, C, X(29009), X(51449)}}, {{A, B, C, X(29056), X(51476)}}, {{A, B, C, X(32760), X(37562)}}, {{A, B, C, X(34250), X(43070)}}, {{A, B, C, X(35000), X(37563)}}, {{A, B, C, X(37531), X(40292)}}, {{A, B, C, X(37569), X(37601)}}, {{A, B, C, X(39578), X(50317)}}
X(947) = barycentric product X(i)*X(j) for these (i, j): {40396, 63}, {40417, 6}
X(947) = barycentric quotient X(i)/X(j) for these (i, j): {6, 946}, {9, 23528}, {31, 2262}, {48, 17102}, {55, 20262}, {184, 22063}, {607, 1856}, {2175, 40957}, {2187, 40943}, {40396, 92}, {40417, 76}, {52425, 40945}
X(947) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 34046, 44075}


X(948) = INTERSECTION OF LINES X(1)X(4) AND X(6)X(7)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = bcvw[a3 - a2(b + c) + a(b + c)2 - (b - c)(b2 - c2)],
                        where u : v : w = X(9)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(948) lies on these lines: 1,4   2,85   6,7   37,347   57,169   142,269   220,329   307,966   342,393

X(948) = isogonal conjugate of X(949)
X(948) = crossdifference of every pair of points on line X(652)X(926)


X(949) = ISOGONAL CONJUGATE OF X(948)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = au/[a3 - a2(b + c) + a(b + c)2 - (b - c)(b2 - c2)],
                        where u : v : w = X(9)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(949) lies on these lines: 1,607   2,294   3,41   6,77   48,1037   78,220

X(949) = isogonal conjugate of X(948)
X(949) = trilinear pole of line X(652)X(926)


X(950) = CROSSPOINT OF X(8) AND X(29)

Trilinears bc(b + c - a)[2a3 + (b + c)(a2 + (b - c)2)] : :

Let A'B'C' be the Gergonne line extraversion triangle, as defined at X(10180). Let La be the reflection of line BC in line B'C', and define Lb and Lc cyclically. Let A″ = Lb ∩ Lc, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(950). (Randy Hutson, January 29, 2018)

X(950) lies on these lines: 1,4   8,9   10,55   11,214   20,57   29,284   30,553   35,1006   65,516   72,519   80,943   142,377   145,329   281,380   389,517   440,1104   947,1067

X(950) = isogonal conjugate of X(951)
X(950) = crosspoint of X(i) and X(j) for these (i,j): (7,333), (8,29)
X(950) = crosssum of X(i) and X(j) for these (i,j): (55,1400), (56,73)
X(950) = X(185)-of-2nd-extouch-triangle
X(950) = X(10)-of-Mandart-incircle-triangle
X(950) = homothetic center of intangents triangle and reflection of extangents triangle in X(10)


X(951) = ISOGONAL CONJUGATE OF X(950)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = a/[(b + c - a)[2a3 + (b + c)(a2 + (b - c)2)]]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(951) lies on these lines: 29,226   56,219   57,78   73,284   77,738   946,1067   947,1066

X(951) = isogonal conjugate of X(950)
X(951) = cevapoint of X(56) and X(73)


X(952) = INTERSECTION OF X(1)X(5) AND X(3)X(8)

Trilinears    bc[2a4 - 2a3(b + c) - a2(b2 - 4bc + c2) + (2a - b - c)(b - c)(b2 - c2)] : :

As the isogonal conjugate of a point on the circumcircle, X(952) lies on the line at infinity.

Let Ia be the reflection of X(1) in the perpendicular bisector of BC, and define Ib and Ic cyclically. X(952) = X(517) of IaIbIc, which is the inner Garcia triangle.. (Randy Hutson, September 14, 2016)

Let A'B'C' be the outer Garcia triangle and A″B″C″ the inner Garcia triangle. Let A* be the isogonal conjugate, wrt A″B″C″, of A', and define B* and C* cyclically. The lines A″A*, B″B*, C″C* concur in X(952); see also X(355). (Randy Hutson, December 2, 2017)

X(952) lies on these (parallel) lines: 1,5   3,8   4,145   10,140   30,511   40,550   150,664   182,996   390,1000   546,946   547,551   572,594

X(952) = isogonal conjugate of X(953)
X(952) = crossdifference of every pair of points on line X(6)X(654)
X(952) = X(30)-of-Fuhrmann-triangle
X(952) = inner-Garcia-isogonal conjugate of X(40)
X(952) = X(5663)-of-excentral-triangle
X(952) = (inverse-in-incircle)-isogonal conjugate of X(33645)
X(952) = Cundy-Parry Psi transform of X(14260)


X(953) = ISOGONAL CONJUGATE OF X(952)

Trilinears    a/[2a4 - 2a3(b + c) - a2(b2 - 4bc + c2) + (2a - b - c)(b - c)(b2 - c2)] : :
Let L be the line tangent to the incircle and the nine-point circle; i.e., the line X(11)X(244), called the Feuerbach tangent line. Let LA be the reflection of L in BC, and define LB and LC cyclically. Let A'' = LB∩LC, and define B'' and C'' cyclically. The lines AA'', BB'', CC'' concur in X(953). (Randy Hutson, 9/23/2011)

X(953) = Miquel point of the sidelines of ABC and the Sherman line. (Angel Montesdeoca, July 24, 2019)

For a construction of X(953) involving the Euler line and X(110), see Antreas Hatzipolakis and César Lozada, Hyacinthos 26050.

X(953) lies on the circumcircle and these lines: 3,901   36,109   100,517   104,513   110,859

X(953) = reflection of X(901) in X(3)
X(953) = isogonal conjugate of X(952)
X(953) = anticomplement of X(31841)
X(953) = trilinear pole of line X(6)X(654)
X(953) = Ψ(X(6), X(654))
X(953) = trilinear pole wrt 2nd circumperp triangle of line X(1001)X(2801)
X(953) = X(476)-of-2nd-circumperp-triangle
X(953) = reflection of X(104) in line X(1)X(3)
X(953) = reflection of X(4) in the Sherman line
X(953) = Cundy-Parry Phi transform of X(14260)
X(953) = Cundy-Parry Psi transform of X(36944)


X(954) = INTERSECTION OF LINES X(1)X(6) AND X(3)X(7)

Trilinears     a5 + (b + c)[2a2(b2 + c2 - a2 + bc) - (b - c)2(2bc + ab + ac)] : :

X(954) lies on these lines: 1,6   3,7   4,390   10,480   21,144   55,226   142,474   971,1012   999,1006

X(954) = isogonal conjugate of X(955)


X(955) = ISOGONAL CONJUGATE OF X(954)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = 1/[a5 + (b + c)[2a2(b2 + c2 - a2 + bc) - (b - c)2(2bc + ab + ac)]]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(955) lies on these lines: 57,991   278,354   279,942

X(955) = isogonal conjugate of X(954)


X(956) = INTERSECTION OF LINES X(1)X(6) AND X(3)X(8)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = a3 - a(b - c)2 - 2bc(b + c)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(956) = 2X(1) - 3X(2) - (r/R)X(3)

X(956) lies on these lines: 1,6   2,495   3,8   10,56   21,145   55,519   63,517   183,668   210,997   333,859   388,442   452,1058

X(956) = reflection of X(55) in X(993)
X(956) = isogonal conjugate of X(957)
X(956) = {X(1),X(9)}-harmonic conjugate of X(392)
X(956) = X(55)-of-inner-Garcia-triangle


X(957) = ISOGONAL CONJUGATE OF X(956)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = 1/[a3 - a(b - c)2 - 2bc(b + c)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(957) lies on these lines: 2,392   57,995   81,859

X(957) = isogonal conjugate of X(956)


X(958) = INTERSECTION OF LINES X(1)X(6) AND X(2)X(12)

Trilinears    (b + c - a)(a2 + ab + ac + 2bc) : :
Barycentrics    b + c + a cos A : :
X(958) = X(1) - 3X(2) - (r/R)X(3)
X(958) = r*X(3) + 2R*X(10)
X(958) = X(1) + X(8) + (r/R)*X(3)

X(958) lies on these lines: 1,6   2,12   3,10   8,21   28,281   36,474   40,1012   48,965   63,65   78,210   104,631   198,966   243,318   452,497   944,1006

X(958) = isogonal conjugate of X(959)
X(958) = complement of X(388)
X(958) = crosssum of X(6) and X(1460)
X(958) = insimilicenter of circumcircle and Spieker circle
X(958) = {X(1),X(9)}-harmonic conjugate of X(960)
X(958) = homothetic center of ABC and cross-triangle of ABC and outer Johnson triangle


X(959) = ISOGONAL CONJUGATE OF X(958)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = 1/[(b + c - a)(a2 + ab + ac + 2bc)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(959) lies on these lines: 1,573   2,65   6,961   7,274   8,181   28,608   56,81   57,1042   193,330

X(959) = isogonal conjugate of X(958)


X(960) = INTERSECTION OF LINES X(1)X(6) AND X(5)X(10)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = (b + c - a)(b2 + c2 + ab + ac)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(960) lies on these lines: 1,6   2,65   3,997   5,10   8,210   12,908   19,965   21,60   36,191   40,936   46,474   55,78   56,63   113,123   221,1038   241,1042   329,388   758,942   912,993   978,986

X(960) = midpoint of X(1) and X(72)
X(960) = reflection of X(942) in X(1125)
X(960) = isogonal conjugate of X(961)
X(960) = isotomic conjugate of X(31643)
X(960) = complement of X(65)
X(960) = anticomplementary conjugate of X(442)
X(960) = crosspoint of X(i) and X(j) for these (i,j): (2,314), (8,21)
X(960) = crosssum of X(i) and X(j) for these (i,j): (6,1402), (56,65)
X(960) = anticomplement of X(3812)
X(960) = {X(1),X(9)}-harmonic conjugate of X(958)
X(960) = perspector of circumconic centered at X(2092)
X(960) = center of circumconic that is locus of trilinear poles of lines passing through X(2092)
X(960) = X(2)-Ceva conjugate of X(2092)
X(960) = X(11)-of-X(1)-Brocard-triangle
X(960) = X(3035)-of-inner-Garcia-triangle
X(960) = X(12241)-of-excentral-triangle
X(960) = polar conjugate of isogonal conjugate of X(22074)
X(960) = excentral-to-ABC barycentric image of X(10)


X(961) = ISOGONAL CONJUGATE OF X(960)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = 1/[(b + c - a)(b2 + c2 + ab + ac)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(961) lies on these lines: 1,572   2,12   6,959   57,1106   65,81   105,1104   108,429   274,1014

X(961) = isogonal conjugate of X(960)
X(961) = cevapoint of X(56) and X(65)
X(961) = X(523)-cross conjugate of X(108)
X(961) = X(92)-isoconjugate of X(22074)


X(962) = LONGUET-HIGGINS POINT

Trilinears    bc[a4 + 2a3(b + c) - 4a2bc - (b + c)(b - c)2(2a + b + c)] : :
Trilinears    sin B sin C + cos B + cos C - cos A - 1 : :
Trilinears    cos B + cos C + cos B cos C - 1 : :
Trilinears    1 - 2 cos2(B/2) cos2(C/2) : :
Trilinears    cos A' : :, where A' is the angle formed by the tangents from X(1) to the A-excircle

X(962) is shown in Michael S. Longuet-Higgins, "On the principal centers of a triangle," Elemente der Mathematik 56 (2001) 122-129, to complete a simple pattern of collinearities.

Let I = X(1) and H = X(4) in a triangle ABC. Let Lb be the perpendicular to B at I, and let Ic be the perpendicular to CI at I. Let Ab = Lb∩AH and Ac = Lc∩AH. Let U be the circle {I, Ab, Ac}}. Let Mb be the point, other than I, where BI meets U, and let Mc be the point, other than I, where CI meets U. Let ℓa be the line MbMc, and define ℓb and ℓc cyclically. The lines ℓa, ℓb, ℓc concur in X(962). (Angel Montesdeoca, March 12, 2020)

X(962) lies on these lines: 1,7   2,40   4,8   30,944   55,411   65,497   145,515   149,151   165,1125   278,412   382,952   392,443   484,499   942,1058

X(962) is the radical center of the circles centered at A, B, C, with respective
radii |CA| + |AB|, |AB| + |BC|, |BC| + |CA|. See

Floor van Lamoen, Problem 10734, American Mathematical Monthly 107 (2000) 658-659.

X(962) = reflection of X(i) in X(j) for these (i,j): (8,4), (20,1), (40,946), (944,1482)
X(962) = isogonal conjugate of X(963)
X(962) = anticomplement of X(40)
X(962) = isotomic conjugate of isogonal conjugate of X(20991)
X(962) = polar conjugate of isogonal conjugate of X(22124)
X(962) = X(309)-Ceva conjugate of X(2)
X(962) = {X(175),X(176)}-harmonic conjugate of X(77)
X(962) = X(3)-of-2nd-Conway-triangle
X(962) = perspector of 2nd Conway triangle and Gemini triangle 29
X(962) = mixtilinear-incentral-to-mixtilinear-excentral similarity image of X(7)


X(963) = ISOGONAL CONJUGATE OF X(962)

Trilinears    a/[a4 + 2a3(b + c) - 4a2bc - (b + c)(b - c)2(2a + b + c)] : :
Trilinears    sec A' : :, where A' is the angle formed by the tangents from X(1) to the A-excircle

X(963) lies on these lines: 3,200   33,56   48,220   55,603

X(963) = isogonal conjugate of X(962)
X(963) = X(92)-isoconjugate of X(22124)


X(964) = INTERCEPT OF EULER LINE AND LINE X(6)X(8)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = bc[a4 + (b + c)(a3 + ab2 + ac2 + abc + (b + c)(a2 + bc))]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As a point on the Euler line, X(964) has Shinagawa coefficients (2(E + F)2 + 2(E + F)*$bc$ + abc*$a$, 2S2).

X(964) lies on these lines: 1,321   2,3   6,8   10,31


X(965) = INTERSECTION OF LINES X(2)X(6) AND X(3)X(9)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = a4 - a3(b + c) - a2(b2 + c2) + a(b + c)3 + 2bc(b + c)2
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(965) lies on these lines: 2,6   3,9   10,219   19,960   37,78   48,958   284,405   474,579


X(966) = INTERSECTION OF LINES X(2)X(6) AND X(4)X(9)

Trilinears    bc[a2 - 2a(b + c) - (b + c)2] : :
Barycentrics    a^2 - 2s^2 : :\

X(966) lies on these lines: 2,6   4,9   8,37   45,346   198,958   307,948   374,3740   443,579   572,631   581,936

X(966) = isogonal conjugate of X(967)
X(966) = crossdifference of every pair of points on line X(512)X(1459)
X(966) = tripolar centroid of X(623)


X(967) = ISOGONAL CONJUGATE OF X(966)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = a/[a2 - 2a(b + c) - (b + c)2]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(967) lies on these lines: 3,42   25,58   27,393   37,63   81,941

X(967) = isogonal conjugate of X(966)
X(967) = trilinear pole of line X(512)X(1459)


X(968) = POINT ALDERAMIN

Trilinears    a2 - 2a(b + c) - (b + c)2 : :
Trilinears    ra2 - sS : rb2 - sS : rc2 - sS

X(968) lies on these lines: 1,21   9,42   19,25   35,975   45,210   165,750   200,756   614,1001   943,1039

X(968) = isogonal conjugate of X(969)
X(968) = crossdifference of every pair of points on line X(661)X(905)
X(968) = {X(1),X(63)}-harmonic conjugate of X(3158)


X(969) = INTERSECTION OF LINES X(10)X(69) AND X(19)X(81)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = 1/[a2 - 2a(b + c) - (b + c)2]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(969) lies on these lines: 7,225   10,69   19,81   37,63   65,77   158,286

X(969) = isogonal conjugate of X(968)
X(969) = trilinear pole of line X(661)X(905)


X(970) = CENTER OF THE APOLLONIUS CIRCLE

Trilinears    a[a3(b + c)2 + a(ab + ac - 2bc)(b2 + c2) - bc(b3 + c3) - a(b4 + c4) - (b5 + c5)] : :
Trilinears    (r2 - s2) cos A + 2rs sin A

The Apollonius circle is described at X(181) as the circle tangent to the three excircles and encompassing them. That X(970) is its center was noted on New Year's Day, 2002, by Paul Yiu. (Hyacinthos #4619-4623)

The Apollonius circle is the inverse-in-excircles-radical-circle of the nine-point circle. (Randy Hutson, December 10, 2016)

X(970) lies on these lines: 1,181   3,6   5,10   21,51   40,43   185,411

X(970) = complement of X(10441)
X(970) = {X(181),X(1682)}-harmonic conjugate of X(1)
X(970) = {X(371),X(372)}-harmonic conjugate of X(5019)
X(970) = inverse-in-excircles-radical-circle of X(3814)
X(970) = perspector of Apollonius triangle and cross-triangle of ABC and Apollonius triangle


X(971) = INTERSECTION OF LINES X(3)X(9) AND X(4)X(7)

Trilinears    a4(b + c) - 2a3(b2 + c2 - bc) + 2a(b - c)2(b2 + c2 + bc) - (b - c)2(b + c)3 : :

X(971) is the perspector of triangle ABC and the tangential triangle of the conic that passes through the points A, B, C, X(2), and X(9). (Randy Hutson, 9/23/2011)

As the isogonal conjugate of a point on the circumcircle, X(971) lies on the line at infinity.

X(971) lies on these (parallel) lines: 1,1419   3,9   4,7   5,142   6,990   20,72   30,511   33,222   37,991   103,910   165,210   390,944   954,1012

X(971) = isogonal conjugate of X(972)
X(971) = crosspoint of X(39144) and X(39145)
X(971) = crosssum of X(55) and X(910)
X(971) = crosssum of X(32622) and X(32623)
X(971) = intersection of trilinear polars of X(2) and X(9)
X(971) = X(3564)-of-excentral-triangle
X(971) = Cundy-Parry Phi transform of X(7367)
X(971) = Cundy-Parry Psi transform of X(14256)


X(972) = ISOGONAL CONJUGATE OF X(971)

Trilinears    1/[a4(b + c) - 2a3(b2 + c2 - bc) + 2a(b - c)2(b2 + c2 + bc) - (b - c)2(b + c)3] : :

X(972) lies on the circumcircle and these lines: 3,934   40,101   55,108   100,329   109,165

X(972) = reflection of X(934) in X(3)
X(972) = isogonal conjugate of X(971)
X(972) = cevapoint of X(32622) and X(32623)
X(972) = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {A,B,C,X(4),X(40),X(57)}}
X(972) = Cundy-Parry Phi transform of X(14256)
X(972) = Cundy-Parry Psi transform of X(7367)


X(973) = 1st EHRMANN POINT

Trilinears    u[a10 - 3a8s + a6(2b4 + 3b2c2 + 2c4) + a4s(2b4 - b2c2 + 2c4) - a2(b2 - c2)2(3b4 + 5b2c2 + s(b2 - c2)2(b4 - b2c2 + c4)] : : ,
                        where u : v : w = X(51) and s = b2 + c2

For constructions of X(973) and X(974), see Hyacinthos message 3695, Sept. 1, 2001, and related messages.

Let A'B'C' be the orthic triangle. X(973) is the radical center of the nine-point circles of AB'C', BC'A', CA'B'. (Randy Hutson, January 29, 2018)

X(973) lies on these lines: 5,51   6,24   68,568

X(973) = midpoint of X(52) and X(1209)
X(973) = crosssum of X(3) and X(1209)
X(973) = X(442)-of-orthic-triangle if ABC is acute


X(974) = 2nd EHRMANN POINT

Trilinears    [sa10 - a8(3b4 - 2b2c2 + 3c4) + a6s(2b4 - 3b2c2 + 2c4) + a4(b2 - c2)2(2b4 - 7b2c2 + 2c4) - 3a2s(b2 - c2)2(b4 - 3b2c2 + c4) + (b2 - c2)4(b4 + b2c2 + c4)] cos A, where s = b2 + c2

For constructions of X(973) and X(974), see Hyacinthos message 3695, Sept. 1, 2001, and related messages.

Let A'B'C' be the orthic triangle. Let La be the orthic axis of AB'C', and define Lb and Lc cyclically. Let A″ = Lb∩Lc. B″ = Lc∩La, C″ = La∩Lb. Triangle A″B″C″ is inversely similar to ABC, with similitude center X(6). The orthocenter of triangle A″B″C″ is X(974). (Randy Hutson, January 29, 2018)

The pedal triangles of circumcenter and orthocenter with respect to orthic and medial triangles, respectively, are perspective, with perspector X(974). (Angel Montesdeoca, August 29, 2019)

X(974) lies on these lines: 5,113   6,74

X(974) = midpoint of X(125) and X(185)
X(974) = reflection of X(1112) in X(389)
X(974) = crosssum of X(3) and X(113)
X(974) = X(1145)-of-orthic-triangle if ABC is acute


X(975) = INTERSECTION OF LINES X(1)X(2) AND X(3)X(37)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = a3 + a2(b + c) + a(b2 + c2 + 4bc) + (b + c)3
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(975) lies on these lines: 1,2   3,37   9,58   28,33   35,968   46,750   57,201   72,940   226,1038   312,1010


X(976) = INTERSECTION OF LINES X(1)X(2) AND X(3)X(38)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = a3 + (b + c)(b2 + c2)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(976) lies on these lines: 1,2   3,38   21,983   31,72   37,41   66,73   100,986   210,1104   244,474   404,982   405,756   601,912   750,942   1060,1066

X(976) = isogonal conjugate of X(977)


X(977) = ISOGONAL CONJUGATE OF X(976)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = 1/[a3 + (b + c)(b2 + c2)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(977) lies on these lines: 22,56   58,982   106,833

X(977) = isogonal conjugate of X(976)


X(978) = X(56)-CEVA CONJUGATE OF X(1)

Trilinears    a2(b + c) + a(b2 - bc + c2) - bc(b + c) : :
Trilinears    csc^2(B/2) + csc^2(C/2) - csc^2(A/2) : :

X(978) lies on these lines: 1,2   3,238   9,39   21,748   31,404   40,1050   46,1054   56,979   57,1046   58,87   72,982   171,474   266,361   631,1064   651,1106   960,986

X(978) = isogonal conjugate of X(979)
X(978) = X(56)-Ceva conjugate of X(1)
X(978) = X(1093)-of-excentral-triangle


X(979) = X(8)-CROSS CONJUGATE OF X(1)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = 1/[a2(b + c) + a(b2 - bc + c2) - bc(b + c)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(979) lies on these lines: 10,87   43,58   56,978

X(979) = isogonal conjugate of X(978)
X(979) = X(8)-cross conjugate of X(1)


X(980) = INTERSECTION OF LINES X(1)X(3) AND X(2)X(39)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = a2(b2 + bc + c2) + (b2 + c2)(bc + ca + ab)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(980) lies on these lines: 1,3   2,39   32,81   38,869   63,213

X(980) = isogonal conjugate of X(981)
X(980) = crossdifference of every pair of points on line X(650)X(669)


X(981) = ISOGONAL CONJUGATE OF X(980)

Trilinears    1/[a2(b2 + bc + c2) + (b2 + c2)(bc + ca + ab)] : :

X(981) lies on these lines: 6,314   8,213   21,32   256,573

X(981) = isogonal conjugate of X(980)
X(981) = trilinear pole of line X(650)X(669)


X(982) = INTERSECTION OF LINES X(1)X(3) AND X(2)X(38)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = b2 - bc + c2
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(982) lies on these lines: 1,3   2,38   7,256   43,518   58,977   63,238   72,978   81,985   222,613   226,262   240,278   257,330   310,871   312,726   404,976   758,995   846,1001

X(982) = isogonal conjugate of X(983)
X(982) = isotomic conjugate of X(7033)
X(982) = complement of X(32937)
X(982) = {X(1),X(40)}-harmonic conjugate of X(37588)
X(982) = crosspoint of X(i) and X(j) for these (i,j): (7,330), (81,310)


X(983) = ISOGONAL CONJUGATE OF X(982)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = 1/[b2 - bc + c2]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(983) lies on these lines: 1,182   7,171   8,238   21,976   55,256

X(983) = isogonal conjugate of X(982)
X(983) = isotomic conjugate of X(33930)


X(984) = MIDPOINT OF X(8) AND X(192)

Trilinears    b2 + bc + c2 : :

Let A15B15C15 and A16B16C16 be Gemini triangles 15 and 16, resp. Let LA be the tangent at A to conic {A,B15,C15,B16,C16}}, and define LB, LC cyclically. Let A' = LB∩LC, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(984). (Randy Hutson, January 15, 2019)

X(984) lies on these lines: 1,6   2,38   8,192   10,75   21,976   43,210   55,846   63,171   100,753   101,761   201,388   240,281   386,872   519,751   936,988

X(984) = midpoint of X(8) and X(192)
X(984) = reflection of X(i) in X(j) for these (i,j): (1,37), (75,10)
X(984) = isogonal conjugate of X(985)
X(984) = isotomic conjugate of X(870)
X(984) = complement of X(24349)
X(984) = anticomplement of X(24325)
X(984) = complement of polar conjugate of isogonal conjugate of X(22163)
X(984) = {X(1),X(9)}-harmonic conjugate of X(238)


X(985) = ISOGONAL CONJUGATE OF X(984)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = 1/[b2 + bc + c2]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(985) lies on these lines: 1,32   2,31   6,291   58,274   81,982   105,825   279,1106   727,789

X(985) = isogonal conjugate of X(984)
X(985) = isotomic conjugate of X(33931)


X(986) = INTERSECTION OF LINES X(1)X(3) AND X(8)X(38)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = a(b2 + bc + c2) + b3 + c3
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(986) lies on these lines: 1,3   4,240   6,1046   8,38   10,75   43,72   100,976   194,257   291,337   386,758   405,846   474,1054   774,938   960,978

X(986) = isogonal conjugate of X(987)


X(987) = ISOGONAL CONJUGATE OF X(986)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = 1/[a(b2 + bc + c2) + b3 + c3]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(987) lies on these lines: 3,256   4,171   7,1106   8,31   9,32   58,314

X(987) = isogonal conjugate of X(986)


X(988) = INTERSECTION OF LINES X(1)X(3) AND X(9)X(39)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = a3 - a2(b + c) - (3a + b + c)(b2 + c2)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(988) lies on these lines: 1,3   9,39   21,614   38,78   77,1106   84,256   404,612   936,984

X(988) = isogonal conjugate of X(989)
X(988) = {X(1),X(3)}-harmonic conjugate of X(37552)


X(989) = ISOGONAL CONJUGATE OF X(988)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = 1/[a3 - a2(b + c) - (3a + b + c)(b2 + c2)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(989) lies on these lines: 21,612   40,256   84,171

X(989) = isogonal conjugate of X(988)


X(990) = INTERSECTION OF LINES X(1)X(7) AND X(3)X(37)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = a5 - a4(b + c) - 2a3bc - a(b - c)2(b2 + c2) + (b - c)2(b + c)3
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(990) lies on these lines: 1,7   3,37   6,971   33,57   58,84   165,612   226,1040

X(990) = complement, wrt hexyl triangle, of X(12717)


X(991) = INTERSECTION OF LINES X(1)X(7) AND X(3)X(6)

Trilinears    a[a3(b + c) - a2(b2 - bc + c2) - a(b + c)(b2 + c2) + (b - c)(b3 - c3)] : :
Trilinears    sin(A + U) : : , where tan U = tan(A/2) tan(B/2) tan(C/2)     (Joe Goggins,11/26/08)
Barycentrics    (sin A)[cos(A - arccot(p))] : : , where p = (r + 4R)/s
X(991) = X(3) + ((cot ω)/p)X(6) = 3 X[3576] - X[6210]

The function p = (r + 4R)/s is the Tucker parameter for X(991); see the preamble to X(13323.)

X(991) lies on the cubic K382 and these lines: {1,7}, {2,5400}, {3,6}, {4,4648}, {9,1818}, {22,1790}, {35,255}, {36,1471}, {37,971}, {42,165}, {43,10164}, {45,5779}, {48,3220}, {51,4191}, {55,103}, {57,955}, {63,3190}, {73,1394}, {78,4416}, {81,1754}, {84,2335}, {212,2003}, {223,10383}, {241,5728}, {394,2328}, {550,5453}, {601,10902}, {741,6011}, {940,7580}, {942,1418}, {954,6180}, {968,1709}, {975,1490}, {984,2801}, {993,6518}, {995,1064}, {999,8147}, {1011,3917}, {1012,4653}, {1038,10393}, {1062,8555}, {1193,7987}, {1214,10391}, {1279,1385}, {1283,5197}, {1308,12032}, {1396,4219}, {1427,11018}, {1456,2646}, {1469,2223}, {1699,3720}, {1724,6986}, {1736,10394}, {1768,4414}, {1779,2979}, {2177,5537}, {2195,7295}, {2318,3929}, {2340,5223}, {2594,5217}, {2635,5219}, {2700,2701}, {2999,10857}, {3060,4210}, {3214,9588}, {3216,3523}, {3666,10167}, {3730,3781}, {3752,11227}, {3912,12618}, {3931,9943}, {4551,5218}, {4644,5759}, {4649,9441}, {4658,5706}, {4675,5805}, {5292,6908}, {5713,6851}, {6051,12688}

X(991) = reflection of X(573) in X(3)
X(991) = isogonal conjugate of polar conjugate of X(37448)
X(991) = crossdifference of every pair of points on line X(523)X(657)
X(991) = crosssum of X(i) and X(j) for these {i,j}: {1,4312}, {11,4724}
X(991) = inverse-in-Schoute-circle of X(4262)
X(991) = X(264)-of-2nd-circumperp-triangle
X(991) = perspector of excentral-hexyl ellipse wrt hexyl triangle
X(991) = intersection of Brocard axes of ABC and hexyl triangle
X(991) = {X(371),X(372)}-harmonic conjugate of X(4251)


X(992) = INTERSECTION OF LINES X(2)X(6) AND X(9)X(39)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = a3(b + c) + a2(b2 + c2) - abc(b + c) - bc(b + c)2
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(992) lies on these lines: 2,6   9,39   44,583   238,1009


X(993) = MIDPOINT OF X(1) AND X(63)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = a3 - a(b2 + c2) - bc(b + c)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(993) = X(1) - 3X(2) - 2(r/R)X(3)

X(993) lies on these lines: 1,21   2,36   3,10   8,35   9,48   32,1107   55,519   56,226   75,99   87,106   238,995   495,529   516,1012   527,551   912,960

X(993) = midpoint of X(i) and X(j) for these (i,j): (1,63), (55,956), (1012,3428)
X(993) = reflection of X(226) in X(1125)
X(993) = isogonal conjugate of X(994)
X(993) = complement of X(1478)


X(994) = ISOGONAL CONJUGATE OF X(993)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = 1/[a3 - a(b2 + c2) - bc(b + c)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(994) lies on these lines: 10,908   31,759   37,517   65,386   75,758

X(994) = isogonal conjugate of X(993)


X(995) = MIDPOINT OF X(1) AND X(43)

Trilinears    a(ab + ac - bc + b2 + c2) : :

See Dasari Naga Vijay Krishna, On a Conic Through Twelve Notable Points, Int. J. Adv. Math. and Mech. 7(2) (2019) 1-15.

X(995) lies on these lines: 1,2   3,595   6,101   31,36   56,58   57,957   238,993   581,1104   609,1055   758,982   991,1064

X(995) = midpoint of X(1) and X(43)
X(995) = isogonal conjugate of X(996)
X(995) = crossdifference of every pair of points on line X(649)X(900)
X(995) = {X(1),X(2)}-harmonic conjugate of X(30116)


X(996) = INTERSECTION OF LINES X(8)X(58) AND X(10)X(56)

Trilinears   1/[a(ab + ac - bc + b2 + c2)] : :

X(996) lies on these lines: 2,106   6,519   8,58   10,56   182,952

X(996) = isogonal conjugate of X(995)
X(996) = trilinear pole of line X(649)X(900)
X(996) = isotomic conjugate of X(4389)


X(997) = MIDPOINT OF X(1) AND X(200)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = a3 - a2(b + c) - a(b - c)2 + (b + c)(b2 + c2)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(997) lies on these lines: 1,2   3,960   9,48   21,90   34,860   36,63   46,404   55,392   56,72   57,758   65,474   141,1060   210,956   518,999

X(997) = midpoint of X(i) and X(j) for these (i,j): (1,200), (3421,3476)
X(997) = isogonal conjugate of X(998)


X(998) = ISOGONAL CONJUGATE OF X(997)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a3 - a2(b + c) - a(b - c)2 + (b + c)(b2 + c2)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(998) lies on these lines: 1,908   6,517   46,58   106,614

X(998) = isogonal conjugate of X(997)


X(999) = MIDPOINT OF X(1) AND X(57)

Trilinears    2 - cos A : 2 - cos B : 2 - cos C (Joe Goggins, 2002)
Trilinears    a(a^2 + 4bc - b^2 - c^2) : :

X(999) is the radical center of the mixtilinear incircles.

X(999) lies on these lines: 1,3   2,495   4,496   5,388   6,101   7,104   8,474   11,381   12,499   20,1058   30,497   63,392   77,1057   78,1059   81,859   145,404   329,405   376,390   518,997   527,551   601,1106   938,944   954,1006

X(999) is the {X(1),X(56)}-harmonic conjugate of X(3). For a list of other harmonic conjugates of X(999), click Tables at the top of this page.

X(999) = midpoint of X(1) and X(57)
X(999) = isogonal conjugate of X(1000)
X(999) = complement of X(3421)
X(999) = X(89)-Ceva conjugate of X(6)
X(999) = crossdifference of every pair of points on line X(650)X(900)
X(999) = {X(1),X(3)}-harmonic conjugate of X(3295)
X(999) = {X(55),X(56)}-harmonic conjugate of X(36)
X(999) = X(25)-of-incircle-circles-triangle
X(999) = X(1596)-of-excentral-triangle
X(999) = X(6644)-of-intouch-triangle
X(999) = homothetic center of outer Yff triangle and cross-triangle of ABC and 1st Johnson-Yff triangle
X(999) = homothetic center of anti-incircle-circles triangle and anti-tangential midarc triangle
X(999) = polar conjugate of isotomic conjugate of X(22129)


X(1000) = ISOGONAL CONJUGATE OF X(999)

Trilinears    1/[a(a2 + 4bc - b2 - c2)] : :
Trilinears    1/(2 - cos A) : :

Let A* be the parabola with focus A and directrix BC, and let A** be the polar of X(1) with respect to A*. Define B** and C** cyclically, and let A' = B**∩C**, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(1000); the centroid of A'B'C' is X(5657). (Randy Hutson, July 7, 2014)

X(1000) lies on these lines: 1,631   7,517   8,392   9,519   21,145   55,104   65,3296   79,388   80,497   84,944   390,952

X(1000) = isogonal conjugate of X(999)
X(1000) = X(45)-cross conjugate of X(2)
X(1000) = perspector of ABC and mid-triangle of excentral and extouch triangles
X(1000) = trilinear pole of line X(650)X(900)


End of PART 1

Introduction and Centers X(1) - X(1000) Centers X(1001) - X(3000) Centers X(3001) - X(5000)
Centers X(5001) - X(7000) Centers X(7001) - X(10000) Centers X(10001) - X(12000)
Centers X(12001) - X(14000) Centers X(14001) - X(16000) Centers X(16001) - X(18000)
Centers X(18001) - X(20000) Centers X(20001) - X(22000) Centers X(22001) - X(24000)
Centers X(24001) - X(26000) Centers X(26001) - X(28000) Centers X(28001) - X(30000)
Centers X(30001) - X(32000) Centers X(32001) - X(34000) Centers X(34001) - X(36000)
Centers X(36001) - X(38000) Centers X(38001) - X(40000) Centers X(40001) - X(42000)
Centers X(42001) - X(44000) Centers X(44001) - X(46000) Centers X(46001) - X(48000)
Centers X(48001) - X(50000) Centers X(50001) - X(52000) Centers X(52001) - X(54000)
Centers X(54001) - X(56000) Centers X(56001) - X(58000) Centers X(58001) - X(60000)
Centers X(60001) - X(62000) Centers X(62001) - X(64000) Centers X(64001) - X(66000)
Centers X(66001) - X(68000) Centers X(68001) - X(70000) Centers X(70001) - X(72000)