PART 1: | Introduction and Centers X(1) - X(1000) | PART 2: | Centers X(1001) - X(3000) | PART 3: | Centers X(3001) - X(5000) |
PART 4: | Centers X(5001) - X(7000) | PART 5: | Centers X(7001) - X(10000) | PART 6: | Centers X(10001) - X(12000) |
PART 7: | Centers X(12001) - X(14000) | PART 8: | Centers X(14001) - X(16000) | PART 9: | Centers X(16001) - X(18000) |
PART 10: | Centers X(18001) - X(20000) | PART 11: | Centers X(20001) - X(22000) | PART 12: | Centers X(22001) - X(24000) |
PART 13: | Centers X(24001) - X(26000) | PART 14: | Centers X(26001) - X(28000) | PART 15: | Centers X(28001) - X(30000) |
PART 16: | Centers X(30001) - X(32000) | PART 17: | Centers X(32001) - X(34000) | PART 18: | Centers X(34001) - X(36000) |
PART 19: | Centers X(36001) - X(38000) | PART 20: | Centers X(38001) - X(40000) | PART 21: | Centers X(40001) - X(42000) |
PART 22: | Centers X(42001) - X(44000) | PART 23: | Centers X(44001) - X(46000) | PART 24: | Centers X(46001) - X(48000) |
PART 25: | Centers X(48001) - X(50000) | PART 26: | Centers X(50001) - X(52000) | PART 27: | Centers X(52001) - X(54000) |
PART 28: | Centers X(54001) - X(56000) | PART 29: | Centers X(56001) - X(58000) | PART 30: | Centers X(58001) - X(60000) |
PART 31: | Centers X(60001) - X(62000) | PART 32: | Centers X(62001) - X(64000) | PART 33: | Centers X(64001) - X(66000) |
PART 34: | Centers X(66001) - X(68000) | PART 35: | Centers X(68001) - X(70000) | PART 36: | Centers X(70001) - X(72000) |
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).
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).
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.
Information for prospective authors in Geometry can be found in Information and Instructions.
[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.
Barycentric coordinates or Barycentrics, by Paris Pamfilos.
in International Journal of Computer Discovered Mathematics (IJCDM)
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=3001A New Electric Field Interpretation of Barycentric and Trilinear Coordinates, by Suren
Alphabetical Index of Terms in ETC, by César Lozada
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.
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.
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.
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) 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)
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
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, letLet 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))
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), (33,1753,1872), (35,36,1), (35,56,3295), (35,5010,5217), (35,5204,999), (35,5563,3746), (35,5584,6244), (36,55,999), (36,165,3428), (36,2078,5126), (36,3746,5563), (36,5010,55), (36,5217,3295), (39,187,32), (39,574,5013), (39,5008,5041), (39,5013,5024), (39,5023,1384), (39,5206,3053), (40,57,5709), (40,165,3579), (40,1385,1482), (40,3576,1), (41,672,218), (46,3612,1), (48,71,219), (50,566,6), (52,389,568), (52,569,6), (55,56,1), (55,165,6244), (55,3303,3746), (55,3304,3303), (55,5204,56), (55,5217,35), (55,5584,40), (56,1466,57), (56,3303,3304), (56,3304,5563), (56,5204,36), (56,5217,55), (56,5584,3428), (57,942,5708), (57,1420,1467), (57,3601,1), (58,386,6), (58,580,5398), (58,4256,386), (58,4257,4252), (58,4276,4267), (58,4278,3286), (61,62,6), (61,5238,15), (61,5351,16), (61,5864,1351), (62,5237,16), (62,5352,15), (62,5865,1351), (63,72,3927), (63,78,72), (63,3984,3951), (63,4652,3916), (63,4855,78), (63,5440,3940), (64,154,1498), (65,1155,46), (65,2646,1), (69,3926,3933), (69,6337,3926), (71,1818,3781), (72,78,3940), (72,3916,63), (72,5440,78), (73,255,3157), (73,603,222), (74,1511,399), (74,1614,6241), (76,99,1975), (76,1078,183), (78,1259,1260), (78,3916,3927), (78,3951,3984), (78,4652,63), (78,4855,5440), (84,936,5777), (84,5044,5779), (84,5438,5720), (99,1078,76), (99,5152,5989), (100,2975,8), (100,5303,2975), (101,3730,220), (104,5657,956), (110,1614,156), (140,376,382), (140,381,5070), (140,382,5055), (140,546,3628), (140,549,631), (140,550,4), (140,631,5054), (140,632,3525), (140,1368,3548), (140,1657,3851), (140,1658,6644), (140,3146,5079), (140,3522,1657), (140,3528,3534), (140,3529,5072), (140,3530,549), (140,3534,3843), (140,3627,3090), (140,3628,632), (140,3845,5067), (140,3853,547), (140,5428,1006), (140,6636,2937), (143,5946,3567), (155,1147,3167), (157,160,159), (165,5010,2077), (165,6282,3587), (171,5329,1460), (182,576,575), (182,578,569), (182,1160,6418), (182,1161,6417), (182,1350,1351), (182,5092,5085), (182,5171,32), (183,1975,76), (184,185,1181), (184,394,3167), (184,1092,1147), (184,1147,49), (184,1204,185), (184,3917,394), (184,5562,155), (185,1092,155), (185,3917,5562), (186,376,22), (186,378,25), (186,550,2937), (186,1593,3517), (186,3516,1598), (186,3520,4), (186,3651,2915), (187,574,6), (187,2021,1691), (187,5162,2076), (187,5188,5171), (187,5206,5023), (191,6326,5694), (198,1436,610), (199,1011,25), (199,3145,2915), (212,603,255), (212,4303,3157), (216,577,6), (216,3284,5158), (220,3207,101), (230,5254,3767), (232,1968,2207), (235,468,3542), (235,1885,4), (237,3148,25), (243,1940,158), (255,4303,222), (283,1790,1437), (284,579,6), (371,372,6), (371,1151,6221), (371,1152,3312), (371,1350,1161), (371,2459,6423), (371,3103,6422), (371,3311,6199), (371,3312,6417), (371,3594,6427), (371,6200,1151), (371,6395,6500), (371,6396,372), (371,6398,6418), (371,6409,6449), (371,6410,6398), (371,6411,6455), (371,6412,6450), (371,6419,3592), (371,6420,6419), (371,6425,6447), (371,6426,6428), (371,6449,6407), (371,6450,6395), (371,6452,6408), (371,6453,6425), (371,6454,6420), (371,6455,6445), (371,6481,6432), (371,6484,6429), (371,6486,6480), (371,6497,6446), (372,1151,3311), (372,1152,6398), (372,1350,1160), (372,2460,6424), (372,3102,6421), (372,3311,6418), (372,3312,6395), (372,3592,6428), (372,6199,6501), (372,6200,371), (372,6221,6417), (372,6396,1152), (372,6409,6221), (372,6410,6450), (372,6411,6449), (372,6412,6456), (372,6419,6420), (372,6420,3594), (372,6425,6427), (372,6426,6448), (372,6449,6199), (372,6450,6408), (372,6451,6407), (372,6453,6419), (372,6454,6426), (372,6456,6446), (372,6480,6431), (372,6485,6430), (372,6487,6481), (372,6496,6445), (376,549,381), (376,631,4), (376,1006,1012), (376,3090,3529), (376,3522,548), (376,3523,5), (376,3524,2), (376,3525,3146), (376,3526,5073), (376,3528,3522), (376,3530,1656), (376,5054,3830), (376,5067,5059), (378,2070,3830), (378,2937,5073), (378,3515,1598), (378,3520,3516), (378,6644,381), (381,382,4), (381,1656,5), (381,1657,382), (381,2070,25), (381,3526,1656), (381,5054,2), (381,5072,3091), (381,5079,5072), (382,631,5070), (382,1656,381), (382,3526,5), (382,3534,1657), (382,5054,1656), (382,5076,3627), (382,5079,546), (384,3552,1003), (384,5999,4), (386,573,970), (386,581,5396), (386,991,581), (386,4256,4255), (386,4257,58), (386,5752,5754), (388,3085,495), (388,5218,3085), (389,578,6), (394,1181,155), (394,3796,184), (394,5406,5408), (394,5407,5409), (404,1006,140), (404,4189,405), (404,6636,2915), (405,474,2), (405,1012,3560), (405,2915,25), (405,3149,5), (408,4189,6638), (411,1006,5), (411,3523,474), (411,4189,1012), (417,1593,6617), (418,6641,25), (426,3148,441), (426,6641,2), (427,3575,4), (428,1907,4), (454,3548,6617), (465,466,2), (468,1885,235), (474,1012,5), (474,3560,1656), (485,5418,590), (485,6560,3070), (486,5420,615), (486,6561,3071), (487,488,69), (489,492,637), (490,491,638), (497,3086,496), (498,1478,12), (498,4299,1478), (499,1479,11), (499,4302,1479), (500,582,6), (500,5396,581), (546,549,3525), (546,550,3529), (546,632,3090), (546,3090,5072), (546,3091,381), (546,3146,5076), (546,3525,1656), (546,3529,382), (546,3627,4), (546,3628,5), (546,5079,3851), (547,3543,381), (547,3845,3545), (547,3850,5), (547,3853,3850), (547,5067,1656), (548,549,4), (548,550,376), (548,631,382), (548,632,3529), (548,3523,381), (548,3524,1656), (548,3530,5), (548,5054,5073), (549,550,5), (549,1657,5070), (549,3522,382), (549,3528,1657), (549,3530,3523), (549,3534,5055), (549,3627,632), (549,3853,3533), (549,6636,2070), (550,631,381), (550,632,3627), (550,1656,5073), (550,1658,22), (550,3523,1656), (550,3524,3526), (550,3525,5076), (550,3526,3830), (550,3530,2), (550,3628,3146), (550,3850,5059), (550,5054,3843), (550,5498,3153), (551,5493,4301), (567,568,6), (567,3581,568), (568,6243,52), (569,578,567), (570,571,6), (572,573,6), (572,3430,581), (573,579,5755), (573,581,5752), (574,5171,3095), (574,5206,32), (574,5210,1384), (575,576,6), (577,578,2055), (577,5158,3284), (579,991,5751), 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(6494,6495,6), (6494,6499,6435), (6495,6498,6436), (6496,6497,6), (6496,6522,6519), (6497,6519,6522), (6498,6499,6), (6500,6501,6), (6519,6522,6), (6566,6567,1570), (6639,6640,2)
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
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
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) 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
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)
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)
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) 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 , 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, 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85}, {32555, 32556, 3}, {32784, 33159, 1698}, {32864, 32915, 17156}, {32917, 32931, 29828}, {33076, 33165, 3679}
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) 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.
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}
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, 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{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}}
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)
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)
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}
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}
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).
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)
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) 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}
Centers 20-30,
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) = 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)
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)
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)
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)
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)
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)
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)
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)
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) 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)
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)
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) = (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).
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)
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) = (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).
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) 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).
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}
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, 975}, {971, 15837}, {976, 4414}, {978, 8616}, {983, 15315}, {984, 3220}, {995, 3915}, {997, 4855}, {1005, 6745}, {1009, 29633}, {1010, 19842}, {1012, 5691}, {1018, 2329}, {1035, 34033}, {1036, 34446}, {1056, 3528}, {1058, 3524}, {1064, 22072}, {1066, 22053}, {1071, 1768}, {1089, 7081}, {1094, 3165}, {1095, 3166}, {1100, 5124}, {1110, 14887}, {1124, 1152}, {1145, 32157}, {1147, 6238}, {1151, 1335}, {1158, 10093}, {1191, 21000}, {1210, 4314}, {1212, 5540}, {1215, 24850}, {1220, 4234}, {1255, 6186}, {1297, 13311}, {1329, 11113}, {1386, 5096}, {1398, 11410}, {1425, 21663}, {1428, 5092}, {1442, 7279}, {1444, 3879}, {1447, 7264}, {1464, 18360}, {1468, 2177}, {1469, 3098}, {1475, 5030}, {1476, 13606}, {1486, 31318}, {1490, 1709}, {1498, 10060}, {1511, 3024}, {1587, 13905}, {1588, 13963}, {1593, 11398}, {1599, 3084}, {1600, 3083}, {1609, 3553}, {1626, 23206}, {1656, 9668}, {1657, 9654}, {1658, 8144}, {1699, 3149}, {1707, 7085}, {1708, 10399}, {1714, 13726}, {1718, 2955}, {1728, 10382}, {1734, 1946}, {1738, 7523}, {1742, 1745}, {1743, 4254}, {1749, 17637}, {1761, 22021}, {1776, 26878}, {1782, 18673}, {1784, 1940}, {1785, 7412}, {1788, 3488}, {1807, 10260}, {1816, 17188}, {1819, 2328}, {1824, 20832}, {1836, 11374}, {1837, 18395}, {1838, 4219}, {1858, 31837}, {1870, 3520}, {1891, 4227}, {1918, 3736}, {1935, 4551}, {1950, 2197}, {1975, 3761}, {2067, 6200}, {2071, 4296}, {2080, 10799}, {2136, 8668}, {2161, 26744}, {2174, 17454}, {2175, 3781}, {2178, 3247}, {2192, 17821}, {2209, 5145}, {2218, 16290}, {2222, 2687}, {2242, 5206}, {2252, 2302}, {2267, 4266}, {2270, 11434}, {2278, 2323}, {2280, 4253}, {2289, 15830}, {2291, 20219}, {2292, 15618}, {2293, 13329}, {2300, 5110}, {2309, 4279}, {2320, 21398}, {2324, 15817}, {2346, 5542}, {2361, 2964}, {2475, 3822}, {2477, 22115}, {2478, 26364}, {2548, 31497}, {2550, 6857}, {2551, 11111}, {2653, 20666}, {2664, 16372}, {2782, 15452}, {2800, 21740}, {2802, 4861}, {2886, 7483}, {2887, 25645}, {2901, 17763}, {2933, 3185}, {2937, 9628}, {2948, 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, 4210}, {3723, 21773}, {3724, 3743}, {3751, 12329}, {3753, 5426}, {3754, 27086}, {3767, 9598}, {3779, 5138}, {3792, 19624}, {3797, 30167}, {3814, 5046}, {3816, 13747}, {3817, 6915}, {3828, 16858}, {3864, 18265}, {3868, 4880}, {3869, 4867}, {3870, 4652}, {3877, 30144}, {3878, 4511}, {3881, 3957}, {3885, 22837}, {3887, 8648}, {3897, 14923}, {3898, 4881}, {3899, 5730}, {3911, 5442}, {3912, 21511}, {3920, 5322}, {3925, 6675}, {3944, 19548}, {3947, 33557}, {3961, 16064}, {3987, 34868}, {3991, 5525}, {4018, 16126}, {4044, 26243}, {4056, 7179}, {4067, 11684}, {4084, 34195}, {4191, 26102}, {4193, 31263}, {4203, 6685}, {4216, 10448}, {4218, 24443}, {4220, 30362}, {4225, 4653}, {4260, 19133}, {4261, 5301}, {4292, 7411}, {4295, 5703}, {4297, 6909}, {4300, 22350}, {4306, 9316}, {4313, 18391}, {4319, 17928}, {4323, 34632}, {4326, 15299}, {4333, 9579}, {4347, 17080}, {4366, 7824}, {4384, 16367}, {4386, 5283}, {4396, 7780}, {4413, 11108}, {4423, 16408}, {4428, 16371}, {4429, 19846}, {4432, 25079}, {4471, 7301}, {4533, 15481}, {4557, 24436}, {4570, 9273}, {4645, 25650}, {4647, 32932}, {4674, 8683}, {4714, 16824}, {4760, 16720}, {4781, 17164}, {4816, 8168}, {4848, 21161}, {4868, 17016}, {4870, 28198}, {4920, 33870}, {5009, 21035}, {5011, 17451}, {5013, 16502}, {5015, 30171}, {5023, 9331}, {5054, 11238}, {5070, 9671}, {5080, 15680}, {5082, 30478}, {5120, 16667}, {5160, 7575}, {5171, 10802}, {5174, 17515}, {5195, 17084}, {5219, 6985}, {5223, 6600}, {5252, 18481}, {5263, 19270}, {5265, 15692}, {5272, 7484}, {5274, 10303}, {5276, 25092}, {5284, 17531}, {5287, 11340}, {5291, 20691}, {5298, 12100}, {5300, 24587}, {5332, 7772}, {5370, 7492}, {5393, 16441}, {5405, 16440}, {5424, 17097}, {5427, 15174}, {5428, 10543}, {5434, 8703}, {5438, 31435}, {5473, 10062}, {5474, 10061}, {5475, 31501}, {5506, 5528}, {5531, 14872}, {5533, 6713}, {5541, 10914}, {5550, 17572}, {5552, 6872}, {5588, 11498}, {5589, 11497}, {5603, 6942}, {5690, 7508}, {5722, 24914}, {5727, 22760}, {5732, 10042}, {5775, 12536}, {5836, 19525}, {5840, 6842}, {5842, 6831}, {5881, 22758}, {5886, 6924}, {5887, 6326}, {5889, 9637}, {5951, 26700}, {5974, 11990}, {6000, 26888}, {6001, 33597}, {6019, 14650}, {6042, 35205}, {6146, 26956}, {6147, 11246}, {6221, 18996}, {6253, 8727}, {6256, 6938}, {6283, 12974}, {6285, 6759}, {6292, 8299}, {6396, 6502}, {6398, 18995}, {6405, 12975}, {6423, 31459}, {6533, 16823}, {6560, 31472}, {6642, 9817}, {6645, 13586}, {6656, 26629}, {6668, 17530}, {6693, 29631}, {6771, 13076}, {6774, 13075}, {6825, 10320}, {6834, 26333}, {6841, 18406}, {6853, 13199}, {6863, 10525}, {6880, 10531}, {6882, 8070}, {6883, 9581}, {6894, 12558}, {6907, 10523}, {6908, 10321}, {6911, 8227}, {6912, 19925}, {6913, 7989}, {6916, 10629}, {6918, 7988}, {6920, 10175}, {6921, 10200}, {6923, 10953}, {6934, 26332}, {6940, 10090}, {6977, 12116}, {7051, 10645}, {7080, 17576}, {7098, 15556}, {7160, 7284}, {7176, 7278}, {7191, 15246}, {7235, 24640}, {7270, 17512}, {7292, 7496}, {7320, 15180}, {7330, 17857}, {7352, 7689}, {7353, 7692}, {7356, 7691}, {7362, 7690}, {7416, 15622}, {7428, 23361}, {7489, 9956}, {7504, 20104}, {7506, 9673}, {7512, 22347}, {7580, 9612}, {7583, 13901}, {7584, 13958}, {7587, 30411}, {7588, 30423}, {7589, 30408}, {7675, 18412}, {7677, 30331}, {7713, 11383}, {7737, 9596}, {7745, 31460}, {7746, 9664}, {7747, 31476}, {7807, 26590}, {7992, 12330}, {8075, 30370}, {8076, 30420}, {8077, 11013}, {8167, 16862}, {8188, 11503}, {8189, 11504}, {8356, 26561}, {8572, 16486}, {8580, 13615}, {8583, 30294}, {8614, 23071}, {8669, 11688}, {8685, 29096}, {8725, 12944}, {8731, 33138}, {8981, 19030}, {9342, 17536}, {9538, 10298}, {9540, 13904}, {9574, 16780}, {9578, 11501}, {9599, 31401}, {9638, 11461}, {9645, 14070}, {9656, 17800}, {9657, 15696}, {9658, 12083}, {9665, 31455}, {9666, 32046}, {9708, 17571}, {9709, 16418}, {9712, 15076}, {9751, 10080}, {9780, 16865}, {9818, 19372}, {9821, 12837}, {9860, 12178}, {9875, 12326}, {9896, 12328}, {9897, 12331}, {9898, 12333}, {9899, 12335}, {9900, 12336}, {9901, 12337}, {9902, 12338}, {9903, 12339}, {9904, 12327}, {9905, 12341}, {9906, 12343}, {9907, 12344}, {9931, 9932}, {9938, 19471}, {9943, 17613}, {10037, 11414}, {10040, 11824}, {10041, 11825}, {10054, 12117}, {10055, 12118}, {10057, 12119}, {10059, 12120}, {10063, 11257}, {10064, 12122}, {10067, 12123}, {10068, 12124}, {10069, 34473}, {10077, 21157}, {10078, 21156}, {10079, 22712}, {10081, 15055}, {10089, 21166}, {10091, 15035}, {10106, 21578}, {10118, 13289}, {10122, 12432}, {10149, 15646}, {10179, 35271}, {10197, 17579}, {10282, 10535}, {10479, 32916}, {10527, 20075}, {10605, 19349}, {10610, 13079}, {10646, 19373}, {10647, 30433}, {10648, 30434}, {10651, 30300}, {10652, 30301}, {10707, 34649}, {10789, 11490}, {10806, 15868}, {10860, 30290}, {10877, 26316}, {10927, 26341}, {10928, 26348}, {10944, 34773}, {10947, 26492}, {10954, 11827}, {10974, 22080}, {11063, 21864}, {11101, 21674}, {11102, 33116}, {11110, 16828}, {11112, 25466}, {11114, 11681}, {11115, 26115}, {11171, 12836}, {11189, 11202}, {11194, 19704}, {11204, 32065}, {11219, 12750}, {11231, 17606}, {11239, 20076}, {11250, 32047}, {11263, 20292}, {11319, 26030}, {11329, 16831}, {11343, 17284}, {11349, 29571}, {11350, 17022}, {11375, 12699}, {11392, 18533}, {11430, 19365}, {11438, 19366}, {11446, 11449}, {11454, 19367}, {11468, 19368}, {11522, 22753}, {11545, 12104}, {11570, 18444}, {11571, 12515}, {11587, 35201}, {11715, 18861}, {11753, 35207}, {11762, 35208}, {11771, 35209}, {11780, 35210}, {11828, 11951}, {11829, 11952}, {11848, 11852}, {11909, 26451}, {12005, 26877}, {12054, 12835}, {12121, 12903}, {12185, 15561}, {12334, 12407}, {12340, 12408}, {12342, 12409}, {12350, 14830}, {12359, 12428}, {12373, 20127}, {12374, 14643}, {12556, 13128}, {12584, 32286}, {12608, 12775}, {12609, 16155}, {12619, 12743}, {12653, 22560}, {12680, 34862}, {12888, 12893}, {12901, 19469}, {12904, 15061}, {12910, 12972}, {12911, 12973}, {12940, 20427}, {12984, 19473}, {12985, 19474}, {13043, 13049}, {13044, 13050}, {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, 26040}, {16850, 19856}, {16857, 19876}, {16859, 19877}, {16915, 27255}, {16916, 27091}, {16917, 31996}, {16930, 27048}, {16931, 27026}, {16992, 32092}, {17023, 21495}, {17030, 17684}, {17104, 35193}, {17126, 19767}, {17316, 21508}, {17532, 34626}, {17534, 31253}, {17535, 19878}, {17558, 19855}, {17567, 26105}, {17588, 19874}, {17605, 22793}, {17638, 22935}, {17647, 24987}, {17660, 26201}, {17674, 24542}, {17692, 26752}, {17696, 27299}, {17717, 19543}, {17719, 24851}, {17729, 17758}, {17734, 21935}, {17760, 33952}, {17784, 19843}, {18180, 22300}, {18357, 31649}, {18491, 18492}, {18515, 18526}, {18518, 18761}, {18589, 24780}, {18915, 18931}, {18922, 18925}, {18954, 35243}, {18957, 35248}, {18958, 35241}, {18959, 35246}, {18960, 35247}, {18961, 35249}, {18962, 35250}, {18965, 35255}, {18966, 35256}, {18999, 19003}, {19000, 19004}, {19175, 19192}, {19182, 19185}, {19237, 29610}, {19314, 26241}, {19354, 19357}, {19370, 19454}, {19371, 19455}, {19434, 19440}, {19435, 19441}, {19472, 22978}, {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)
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)
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) 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) 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)
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)
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
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
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) 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) 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)
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) = (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)
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)
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)
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) = (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)
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)
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)
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) = 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) 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
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) 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) 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)
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) 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) 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)
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)
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)
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)
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)
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)
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)
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) 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) 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)
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) 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)
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}
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, 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