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

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

HOW TO USE ETC

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. 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.

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

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

NOTATION AND COORDINATES

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

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

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

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

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

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

Introduced on May 2, 2019: Writer

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

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

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

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

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

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

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

E = (S_B + S_C)(S_C + S_A)(S_A + S_B)/S², so that E = (abc/S)² = 4R²
F = S_AS_BS_C/S², so that F = (a² + b² + c²)/2 - 4R² = S_ω - 4R²

Examples:
X(2) has Shinagawa coefficients (1, 0); i.e., X(2) = 1*S² + 0*S_BS_C
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, $aS_BS_C$ abbreviates aS_BS_C + bS_CS_A + cS_AS_B.

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

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 giving by |GX| = 2v|GO|/(3u + v), where |GO| = (E - 8F)^1/2/6. Then the equation |GX| = 2v|GO|/(3u + v) can be used to obtain these combos:

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

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

Introduced on November 1, 2011: Combos

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

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

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

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

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

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

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

S = 2*area(ABC)
S_A = (b² + c² - a²)/2 = bc cos A
S_B = (c² + a² - b²)/2 = ca cos B
S_C = (a² + b² - c²)/2 = ab cos C
S_ω = S cot ω
s = (a + b + c)/2
s_a = (b + c - a)/2
s_b = (c + a - b)/2
s_c = (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 to X(3663) and X(3739).

X(1) = INCENTER

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 r_a, r_b, r_c, then 1/r = 1/r_a +1/r_b + 1/r_c. Moreover,

area(ABC) = sqrt(r*r_a*r_b*r_c) and r_a + r_b + r_c = 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 O_A be the circle tangent to side BC at its midpoint and to the circumcircle on the side of BC opposite A. Define O_B and O_C cyclically. Let L_A be the external tangent to circles O_B and O_C that is nearest to O_A. Define L_B and L_C cyclically. Let A' = L_B ∩L_C, 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'A", BB'B", CC'C". (Randy Hutson, December 10, 2016)

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   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   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)
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) = 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(2) = CENTROID

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

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) = 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) = 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) = 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)

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)

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) = barycentric product of PU(i) for these i 3, 35
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(3) = CIRCUMCENTER

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 AA", 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)

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

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) = inverse-in-nine-point-circle of X(2072)
X(3) = inverse-in-orthocentroidal-circle of X(5)
X(3) = inverse-in-1st-Lemoine-circle of X(2456)
X(3) = inverse-in-2nd-Lemoine-circle of X(1570)
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) = 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) = {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), 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X(4) = ORTHOCENTER

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)

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)

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

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) = 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(5) = NINE-POINT CENTER

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'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'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)

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 AA", 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 AA", 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 AA", 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)

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) = 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) = QA-P32 center (Centroid of the Circumcenter Quadrangle) of quadrangle ABCX(4) (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/60-qa-p32.html)

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

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.

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'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 L_A be the line through A parallel to B'C', and define L_B and L_C cyclically. Let A" = L_B∩L_C, and define B", C" cyclically. The lines A'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 AA", 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) lies on 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)

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)

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) = crosssum of X(5408) and X(5409)
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) = QA-P23 (Inscribed Square Axes Crosspoint) of quadrangle ABCX(2); see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/51-qa-p23.html

X(7) = GERGONNE POINT

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).

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) = endo-homothetic center of Ehrmann vertex-triangle and Ehrmann mid-triangle; the homothetic center is X(3818)

X(8) = NAGEL POINT

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 L_A be the line through A₂₈ parallel to BC, and define L_B and L_C cyclically. Let A'₂₈ = L_B∩L_C, and define B'₂₈, C'₂₈ cyclically. Triangle A'₂₈B'₂₈C'₂₈ is homothetic to ABC at X(8). (Randy Hutson, November 30, 2018)

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

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) = 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)X9k) 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) = isoconjugate of X(i) and X(j) for these {i,j}: {1,56}, {2,604}, {3,34}, {4,603}, {6,57}, {7,31}, {8,1106}, {9,1407}, {10,1408}, {12,849}, {19,222}, {21,1042}, {25,77}, {27,1409}, {28,73}, {29,1410}, {32,85}, {33,7053}, {36,1411}, {37,1412}, {40,1413}, {41,279}, {42,1014}, {48,278}, {54,1393}, {55,269}, {58,65}, {59,244}, {60,1254}, {63,608}, {64,1394}, {69,1395}, {71,1396}, {75,1397}, {78,1398}, {79,1399}, {81,1400}, {82,1401}, {84,221}, {86,1402}, {87,1403}, {88,1404}, {89,1405}, {90,1406}, {101,3669}, {102,1455}, {103,1456}, {104,1457}, {105,1458}, {106,1319}, {108,1459}, {109,513}, {110,4017}, {154,8809}, {158,7335}, {163,7178}, {171,1431}, {172,1432}, {181,757}, {184,273}, {189,2199}, {198,1422}, {200,7023}, {208,1433}, {212,1119}, {213,1434}, {219,1435}, {220,738}, {223,1436}, {225,1437}, {226,1333}, {241,1438}, {255,1118}, {259,7370}, {267,8614}, {270,1425}, {281,7099}, {282,6611}, {283,1426}, {284,1427}, {291,1428}, {292,1429}, {296,1430}, {307,2203}, {326,7337}, {331,9247}, {346,7366}, {347,2208}, {348,1973}, {388,1472}, {393,7125}, {479,1253}, {512,1414}, {514,1415}, {518,1416}, {519,1417}, {552,872}, {560,6063}, {593,2171}, {607,7177}, {614,1037}, {643,7250}, {649,651}, {650,1461}, {657,4617}, {658,3063}, {661,4565}, {662,7180}, {663,934}, {664,667}, {669,4625}, {672,1462}, {692,3676}, {727,1463}, {741,1284}, {756,7341}, {759,1464}, {765,1357}, {798,4573}, {893,7175}, {896,7316}, {904,7176}, {909,1465}, {923,7181}, {937,1466}, {939,1467}, {951,1104}, {959,1468}, {961,1193}, {983,7248}, {985,1469}, {998,1470}, {1002,1471}, {1015,4564}, {1019,4559}, {1020,7252}, {1027,2283}, {1035,3345}, {1036,4320}, {1041,1473}, {1073,3213}, {1086,2149}, {1088,2175}, {1089,7342}, {1096,1804}, {1098,7143}, {1101,1365}, {1110,1358}, {1149,8686}, {1170,1475}, {1174,1418}, {1191,7091}, {1201,1476}, {1214,1474}, {1245,5323}, {1262,2170}, {1279,1477}, {1420,3445}, {1421,3446}, {1423,2162}, {1424,3224}, {1439,2299}, {1440,2187}, {1441,2206}, {1442,6186}, {1443,6187}, {1447,1911}, {1453,2213}, {1576,4077}, {1616,2137}, {1617,2191}, {1631,7213}, {1769,2720}, {1790,1880}, {1795,1875}, {1813,6591}, {1919,4554}, {1922,10030}, {1974,7182}, {1980,4572}, {2003,2160}, {2006,7113}, {2099,2163}, {2114,9500}, {2150,6354}, {2159,6357}, {2176,7153}, {2194,3668}, {2207,7183}, {2210,7233}, {2212,7056}, {2218,4306}, {2221,2285}, {2260,2982}, {2263,3423}, {2275,7132}, {2279,5228}, {2291,6610}, {2306,2307}, {2310,7339}, {2324,6612}, {2334,3361}, {2353,7210}, {2362,6502}, {3121,4620}, {3212,7121}, {3248,4998}, {3271,7045}, {3451,3752}, {3709,4637}, {3733,4551}, {3777,8685}, {3900,6614}, {3911,9456}, {3937,7012}, {3942,7115}, {4252,5665}, {4296,8615}, {4557,7203}, {4626,8641}, {4822,5545}, {5018,8852}, {5546,7216}, {6129,8059}, {6180,9315}, {7011,7129}, {7013,7151}, {7051,7052}, {7054,7147}, {7084,7195}, {7104,7196}, {7117,7128}, {7122,7249}, {9309,9316}, {9363,9435}, {9364,9432}

X(9) = MITTENPUNKT

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, 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.

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 Reznick, July 1, 2019)

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(13), 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, called the Gergonne line and denoted by L(55), 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)

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}, {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}, {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}, {1020, 6356}, {1021, 3700}, {1026, 9319}, {1030, 2932}, {1040, 5452}, {1046, 10381}, {1050, 2275}, {1054, 3038}, {1055, 18450}, {1071, 8726}, {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}, {1385, 20818}, {1389, 1953}, {1392, 4861}, {1404, 30318}, {1405, 2171}, {1418, 7271}, {1424, 25918}, {1435, 17917}, {1441, 25001}, {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}, {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}, {1836, 3925}, {1837, 6598}, {1848, 21062}, {1863, 28120}, {1868, 4185}, {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}, {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}, {2272, 5658}, {2293, 2340}, {2295, 4274}, {2305, 5277}, {2308, 5311}, {2312, 4220}, {2317, 22356}, {2318, 2335}, {2319, 7081}, {2320, 2364}, {2326, 11107}, {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}, {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}, {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}, {3486, 6737}, {3503, 25994}, {3525, 26877}, {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}, {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}, {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}, {3984, 16865}, {3989, 17017}, {3997, 30116}, {3998, 16368}, {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}, {4180, 4182}, {4189, 4855}, {4191, 22060}, {4255, 8951}, {4260, 16850}, {4268, 7113}, {4269, 25516}, {4272, 5312}, {4295, 19855}, {4297, 10864}, {4301, 6766}, {4304, 11111}, {4310, 16020}, {4313, 11106}, {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}, {4798, 6707}, {4869, 21296}, {4872, 24694}, {4929, 9053}, {4967, 26998}, {4974, 32921}, {4981, 24552}, {4997, 30608}, {5046, 21029}, {5057, 30311}, {5082, 10624}, {5087, 5536}, {5088, 27472}, {5110, 5529}, {5122, 16417}, {5124, 7280}, {5128, 5177}, {5153, 5313}, {5217, 7285}, {5232, 7291}, {5256, 16579}, {5266, 30435}, {5267, 22054}, {5274, 24386}, {5286, 13161}, {5290, 25466}, {5297, 9330}, {5307, 22001}, {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}, {6245, 6865}, {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}, {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}, {7283, 9534}, {7293, 7485}, {7377, 24702}, {7595, 8231}, {7670, 11691}, {7673, 14923}, {7678, 11680}, {7679, 11681}, {7736, 24239}, {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}, {8941, 31459}, {9028, 26130}, {9342, 9352}, {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}, {9614, 24390}, {9616, 30354}, {9619, 31456}, {9624, 31458}, {9785, 21627}, {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}, {10638, 19551}, {10645, 11791}, {10646, 11790}, {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}, {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, 28616}, {14621, 31323}, {14829, 18743}, {14963, 22073}, {14996, 17021}, {14997, 17012}, {15066, 22128}, {15487, 26034}, {15496, 22080}, {15507, 31394}, {15669, 24315}, {15934, 16857}, {16058, 20760}, {16286, 22458}, {16367, 20769}, {16374, 23206}, {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}, {16887, 30110}, {17002, 26247}, {17046, 17671}, {17050, 17753}, {17063, 18193}, {17107, 24797}, {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}, {17444, 21398}, {17499, 26110}, {17542, 24473}, {17550, 24995}, {17605, 31245}, {17614, 20991}, {17625, 25893}, {17687, 25500}, {17691, 25242}, {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}, {18758, 23863}, {19261, 23169}, {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}, {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}, {21026, 31134}, {21066, 26793}, {21090, 24298}, {21281, 30030}, {21286, 26581}, {21367, 32782}, {21368, 24611}, {21405, 27390}, {21759, 23566}, {21827, 23543}, {21832, 24121}, {21956, 32865}, {22003, 24435}, {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}, {24398, 27918}, {24512, 26102}, {24542, 33122}, {24547, 24612}, {24586, 30758}, {24603, 28827}, {24633, 24993}, {24690, 30822}, {24935, 25669}, {25010, 30312}, {25057, 31171}, {25343, 30779}, {25457, 25682}, {25570, 25571}, {25660, 29456}, {25690, 25693}, {25842, 25856}, {25850, 25858}, {25958, 29873}, {25960, 33119}, {25961, 33067}, {26006, 26668}, {26035, 31339}, {26068, 27020}, {26107, 26959}, {26724, 33146}, {26730, 26731}, {26756, 27073}, {26919, 26940}, {26933, 30739}, {27036, 27102}, {27044, 27136}, {27108, 27514}, {27507, 28789}, {28604, 29576}, {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}, {30416, 30429}, {30417, 30430}, {30676, 30701}, {30695, 31994}, {31316, 31343}, {31408, 31533}, {31547, 31565}, {31548, 31566}, {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}

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}, {40, 11372}, {63, 8545}, {72, 5728}, {100, 1156}, {190, 673}, {329, 12848}, {346, 5838}, {1001, 5220}, {2262, 21871}, {2294, 3958}, {2550, 5698}, {2951, 3062}, {3059, 14100}, {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}, {31547, 31565}, {31548, 31566}

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}, {15185, 5572}, {15298, 15296}, {15299, 15297}, {16593, 4422}, {17151, 4361}, {17314, 3950}, {17668, 15587}, {18443, 6883}, {20195, 18230}, {31657, 140}, {31671, 18482}

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) = 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}, {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}, {798, 17058}, {893, 17062}, {902, 1145}, {904, 24239}, {923, 17070}, {983, 17792}, {1106, 5573}, {1110, 3035}, {1174, 6706}, {1252, 21232}, {1253, 9}, {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}, {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}, {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}, {23990, 16578}, {32652, 8058}, {32666, 676}, {32739, 905}, {33299, 21248}

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

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

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

X(10) = SPIEKER CENTER

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 L_A be the line through A₂₀ parallel to BC, and define L_B and L_C cyclically. Let A'₂₀ = L_B∩L_C, 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)

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) = 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(11) = FEUERBACH POINT

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 L_A be the line through A parallel to X(1)X(3), and define L_B and L_C cyclically. Let M_A be the reflection of BC in L_A, and define M_B and M_C cyclically. Let A' = M_B∩M_C, 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)

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

X(11) is the {X(1),X(5)}-harmonic conjugate of X(12) and also the {X(5),X(12)}-harmonic conjugate of X(3614) . For a list of other harmonic conjugates of X(11), click Tables at the top of this page.

X(11) lies on the incentral circle, Mandart circle, cevian circle of every point on the Feuerbach hyperbola, and these lines:
1,5   2,55   3,499   4,56   7,658   8,1320   9,3254   10,121   13,202   14,203   28,1852   30,36   33,427   34,235   35,140   57,1360   65,117   68,1069   110,215   113,942   115,1015   118,226   124,1364   182,1848   133,1838   153,388   212,748   214,442   244,867   278,1857   325,350   381,999   403,1870   429,1104   485,1124   486,1335   498,1656   515,1319   516,1155   517,1737   518,908   523,1090   613,1352   650,1566   774,1393   944,1388   962,1788   971,1538   1012,1470   1040,1368   1111,1358   1146,1639   1193,1834   1427,1856   1428,1503   1455,1877   1500,1506   1697,1698

X(11) = midpoint of X(i) and X(j) for these (i,j): (1,80), (4,104), (5,1484), (9,3254), (100,149)
X(11) = reflection of X(i) in X(j) for these (i,j): (1,1387), (119,5), (214,1125), (1145,10), (1317,1), (1537,946)
X(11) = isogonal conjugate of X(59)
X(11) = isotomic conjugate of X(4998)
X(11) = inverse-in-Fuhrmann-circle of X(1837)
X(11) = complement of X(100)
X(11) = anticomplement of X(3035)
X(11) = complementary conjugate of X(513)
X(11) = circumcircle-inverse of X(14667)
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) = 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) = inverse-in-polar-circle of X(108)
X(11) = inverse-in-{circumcircle, nine-point circle}-inverter of X(105)
X(11) = inverse-in-Fuhrmann-circle of X(1837)
X(11) = inverse-in-excircles-radical-circle of X(3030)
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) = X(6)-isoconjugate of X(4564)
X(11) = orthic-isogonal conjugate of X(513)
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(12) = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11)

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 these lines:
1,5   2,56   3,498   4,55   7,1268   10,65   17,203   18,202   30,35   33,235   34,427   36,140   37,225   38,1393   40,1836   42,1834   54,215   57,1224   63,1454   71,1901   79,484   85,120   108,451   115,1500   116,1362   117,1364   121,1357   123,1359   124,1361   125,1425   141,1469   171,1399   172,230   201,756   208,1360   221,1853   228,407   281,1118   313,349   354,1210   377,1259   381,1479   431,1824   443,1466   474,1470   485,1124   486,1124   499,999   603,750   611,1352   908,960   942,1209   946,1532   968,1904   1015,1506   1038,1368   1091,1109   1125,1319   1213,1400   1452,1892   1594,1870   1697,1699   1861,1887   1877,1883

X(12) is the {X(1),X(5)}-harmonic conjugate of X(11). For a list of other harmonic conjugates of X(12), click Tables at the top of this page.

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

X(12) = isogonal conjugate of X(60)
X(12) = isotomic conjugate of X(261)
X(12) = complement of X(2975)
X(12) = circumcircle-inverse of X(32626)
X(12) = X(10)-Ceva conjugate of X(201)
X(12) = crosssum of X(58) and X(1437)
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(13) = 1st ISOGONIC CENTER (FERMAT POINT, TORRICELLI POINT)

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) + 3^1/2)|*((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 L_BC = L∩BC, and define L_CA and L_AB cyclically. Let L'_BC = L'∩BC, and define L'_CA and L'_AB cyclically. The lines L_BCL'_CA, L_CAL'_AB, L_ABL'_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)

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(14) = 2nd ISOGONIC CENTER

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) = 3^1/2(r² + 2rR + s²)*X(1) - 6r(3^1/2R + 2s)*X(2) + 2r(3^1/2r - 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) - 3^1/2)|((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.

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(15) = 1st ISODYNAMIC POINT

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) + 3^1/2)|*((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.
If you have GeoGebra, you can view 1st isodynamic point.

X(15) lies on the Parry circle, Neuberg cubic, and these lines:
1,1251   2,14   3,6   4,17   13,30   18,140   35,1250   36,202   55,203   298,533   303,316   395,549   397,550   532,616   628,636   1337,2981

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

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) = Brocard-circle-inverse of X(16)
X(15) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,202), (13,62), (74,16)
X(15) = crosspoint of X(i) and X(j) for these (i,j): (13,18), (298,470)
X(15) = crosssum of X(i) and X(j) for these (i,j): (15,62), (532,619)
X(15) = crossdifference of every pair of points on line X(395)X(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(16) = 2nd ISODYNAMIC POINT

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) - 3^1/2)|((E - 8F)S²)^1/2. (Kiminari Shinagawa, February 20, 2018)

X(16) lies on the Parry circle, Neuberg cubic, and these lines:
1,1250   2,13   3,6   4,18   14,30   17,140   36,203   55,202   299,532   302,316   358,1135   396,549   398,550   533,617   627,635   1338,3458

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

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) = Brocard-circle-inverse of X(15)
X(16) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,203), (14,61), (74,15)
X(16) = crosspoint of X(i) and X(j) for these (i,j): (14,17), (299,471)
X(16) = crosssum of X(i) and X(j) for these (i,j): (16,61), (533,618)
X(16) = crossdifference of every pair of points on line X(396)X(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(17) = 1st NAPOLEON POINT

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

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

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

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

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

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

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

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

X(18) = 2nd NAPOLEON POINT

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(19) = CLAWSON POINT

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.

Further information is available from
Paul Yiu's Website.

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 these lines:
1,28   2,534   3,1871   4,9   6,34   8,1891   25,33   27,63   31,204   41,1825   44,1828   45,1900   46,579   47,921   53,1846   56,207   57,196   64,1903   81,969   91,920   101,913   102,282   112,759   158,1712   162,897   163,563   208,225   219,517   220,1902   226,1763   232,444   273,653   294,1041   318,1840   379,1441   407,1865   429,1213   560,1910   604,909   672,1851   960,965   1158,1715   1212,1593   1405,1866   1449,1870   1581,1740   1598,1872   1633,1721   1707,1719   1708,1713   1743,1783   1836,1901   1837,1852

X(19) is the {X(607),X(608)}-harmonic conjugate of X(6). For a list of other harmonic conjugates of X(19), click Tables at the top of this page.

X(19) = isogonal conjugate of X(63)
X(19) = isotomic conjugate of X(304)
X(19) = circumcircle-inverse of X(32756)
X(19) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,204), (4,33), (27,4), (28,25), (57,208), (92,1), (196,207), (278,34)
X(19) = X(i)-cross conjugate of X(j) for these (i,j): (25,34), (31,1)
X(19) = crosspoint of X(i) and X(j) for these (i,j): (4,278), (27,28), (57,84), (92,158)
X(19) = crosssum of X(i) and X(j) for these (i,j): (1,610), (3,219), (9,40), (48,255), (71,72)
X(19) = crossdifference of every pair of points on line X(521)X(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

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) = DE LONGCHAMPS POINT

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.

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) = 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) = QA-P5 (Isotomic Center) of the incenter-excenters quadrangle (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/26-qa-p5.html)
X(20) = barycentric product X(i)*X(j) for these {i,j}: {63,1895}, {69,1249}, {75,610}, {76,154}, {85,7070}, {86,8804}, {99,6587}, {204,304}, {274,3198}, {305,3172}, {312,1394}, {333,5930}, {648,8057}, {801,2883}, {1032,6616}, {1097,2184}, {3213,3718}, {3344,6527}, {3926,6525}, {7156,7182}, {10152,11064}
X(20) = barycentric quotient X(i)/X(j) for these (i,j): (1,2184), (2,253), (3,1073), (4,459), (6,64), (31,2155), (57,8809), (112,1301), (154,6), (204,19), (216,8798), (333,5931), (393,6526), (610,1), (1249,4), (1394,57), (1498,3343), (1562,125), (1895,92), (2285,10375), (3079,1249), (3172,25), (3198,37), (3213,34), (3284,11589), (3344,3346), (5930,226), (6525,393), (6587,523), (7070,9), (7156,33), (8057,525), (8804,10)
X(20) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,7,11036), (1,1044,1042), (1,1742,4300), (1,1770,4295), (1,3100,9538), (1,4292,7), (1,4293,3600), (1,4294,390), (1,4297,5731), (1,4298,11037), (1,4299,4293), (1,4301,5734), (1,4302,4294), (1,4304,4313), (1,4311,4308), (1,4312,3671), (1,4316,4299), (1,4324,4302), (1,4325,4317), (1,4330,4309), (1,4333,1770), (1,4340,3945), (1,4355,5542), (1,5732,10884), (1,9589,4301), (1,10624,9785), (2,3,3523), (2,4,3091), (2,5,7486), (2,22,10565), (2,23,4232), (2,376,10304), (2,377,4208), (2,452,5129), (2,1370,7396), (2,2475,5177), (2,3091,5056), (2,3146,4), (2,3522,3), (2,3523,10303), (2,3543,3839), (2,3832,5), (2,4190,6904), (2,5046,6919), (2,5059,3146), (2,5068,3090), (2,6837,6884), (2,6838,6960), (2,6839,6993), (2,6847,6888), (2,6848,6979), (2,6872,452), (2,6890,6972), (2,6987,6992), (2,6995,7398), (2,7391,7378), (2,7408,6997), (2,7409,5133), (2,7500,6995), (2,11106,405), (3,4,2), (3,5,631), (3,22,7488), (3,26,186), (3,140,3524), (3,376,3522), (3,381,140), (3,382,5), (3,405,6986), (3,546,3525), (3,548,3528), (3,549,10299), (3,550,376), (3,1012,21), (3,1532,6921), (3,1593,7503), (3,1597,7395), (3,1656,549), (3,1885,6816), (3,2937,1658), (3,3079,2060), (3,3091,10303), (3,3146,3091), (3,3149,404), (3,3522,10304), (3,3526,3530), (3,3529,3146), (3,3534,550), (3,3543,5056), (3,3560,1006), (3,3575,6815), (3,3627,3090), (3,3830,1656), (3,3843,3526), (3,3845,3533), (3,3851,5054), (3,3853,5067), (3,5059,3543), (3,5073,381), (3,5076,3628), (3,6638,417), (3,6756,6803), (3,6823,7494), (3,6827,6926), (3,6831,6910), (3,6836,6890), (3,6840,6972), (3,6842,6954), (3,6850,6908), (3,6851,6847), (3,6868,6987), (3,6872,6992), (3,6882,6961), (3,6895,6888), (3,6905,4188), (3,6906,4189), (3,6907,6988), (3,6911,6940), (3,6914,6875), (3,6917,6889), (3,6923,6825), (3,6925,6838), (3,6928,6891), (3,6929,6967), (3,6934,4190), (3,6938,6872), (3,6985,6905), (3,7387,24), (3,7395,7485), (3,7488,10298), (3,7491,6827), (3,7517,6644), (3,7553,7401), (3,7580,411), (3,8727,6857), (3,9122,1817), (3,9818,7509), (3,9909,3515), (3,10323,6636), (3,10431,6837), (3,11001,5059), (3,11413,2071), (3,11414,22), (3,11479,7484), (3,11676,3552), (4,5,3832), (4,21,6837), (4,24,3089), (4,140,5068), (4,186,3542), (4,376,3), (4,377,6839), (4,378,3088), (4,404,6953), (4,411,6838), (4,443,6835), (4,550,3522), (4,631,5), (4,1006,6846), (4,1595,7409), (4,1656,3854), (4,1657,5059), (4,3079,6616), (4,3088,7378), (4,3090,381), (4,3091,3839), (4,3146,3543), (4,3147,403), (4,3520,3541), (4,3522,3523), (4,3523,5056), (4,3524,3090), (4,3525,3545), (4,3528,631), (4,3533,3851), (4,3537,6803), (4,3538,6804), (4,3542,6623), (4,3545,546), (4,3651,6908), (4,3855,3843), (4,4188,6979), (4,4189,6888), (4,5067,3855), (4,5084,6957), (4,6353,235), (4,6616,1559), (4,6756,7408), (4,6803,6997), (4,6815,7544), (4,6824,6870), (4,6825,6871), (4,6826,6894), (4,6827,5046), (4,6833,6844), (4,6836,6840), (4,6850,2475), (4,6851,6895), (4,6852,6866), (4,6853,6867), (4,6854,6849), (4,6856,7548), (4,6857,6828), (4,6865,2478), (4,6868,6872), (4,6875,6824), (4,6876,6825), (4,6878,6990), (4,6880,6941), (4,6889,6843), (4,6891,5187), (4,6897,6826), (4,6899,6827), (4,6902,6929), (4,6903,6928), (4,6905,6848), (4,6906,6847), (4,6908,5177), (4,6909,6890), (4,6916,377), (4,6926,6919), (4,6927,1532), (4,6935,6831), (4,6940,6964), (4,6942,6834), (4,6947,6893), (4,6948,4190), (4,6949,6968), (4,6950,6833), (4,6951,6917), (4,6954,5141), (4,6961,5154), (4,6977,6830), (4,6986,6886), (4,6987,452), (4,6988,2476), (4,7386,6816), (4,7401,7394), (4,7412,4194), (4,7487,6995), (4,7512,3547), (4,7714,5198), (4,8889,7507), (4,10299,1656), (4,10304,10303), (4,10323,7400), (4,10996,6815), (4,11001,3529), (4,11111,6912), (5,140,5070), (5,382,4), (5,548,3), (5,550,548), (5,631,2), (5,3526,5067), (5,3530,3526), (5,3627,3861), (5,3832,3091), (5,3843,3855), (5,3845,3859), (5,3853,3843), (5,3859,3851), (5,3861,381), (5,5070,3090), (5,7486,5056), (5,9715,7493), (7,3188,279), (7,4313,1), (8,9778,40), (8,10430,9799), (8,10538,280), (11,5204,7288), (12,5217,5218), (21,377,2), (21,6839,6884), (21,7411,3), (22,858,7493), (22,1370,2), (22,2071,10298), (22,11413,3), (24,3089,4232), (24,6643,2), (24,7387,23), (25,1885,4), (25,7386,2), (25,7667,7386), (27,464,2), (32,2549,5286), (32,5286,5304), (32,7756,2549), (32,7765,5319), (35,1478,3085), (35,3085,5281), (35,10483,1478), (36,1479,3086), (36,3086,5265), (40,84,63), (40,3101,9537), (40,5881,11362), (55,7354,388), (56,6284,497), (57,950,938), (65,5918,9943), (68,11457,3448), (69,6527,253), (76,5188,6194), (98,5171,7793), (99,315,3926), (99,7802,315), (100,3436,7080), (140,381,3090), (140,546,10109), (140,3090,2), (140,3627,381), (140,3861,5), (140,5073,4), (140,8703,3), (145,9965,3868), (154,5895,2883), (165,5691,10), (175,176,347), (187,7748,3767), (226,3601,5703), (235,3515,6353), (315,8721,147), (316,7782,7763), (355,3579,5657), (355,5657,3617), (371,1587,7585), (371,6560,1587), (372,1588,7586), (372,6561,1588), (376,631,3528), (376,1370,2071), (376,1657,3146), (376,3146,3523), (376,3524,8703), (376,3528,548), (376,3529,4), (376,5059,3091), (376,6240,7400), (376,6851,4189), (376,6869,4190), (376,6916,7411), (376,6938,6987), (376,11541,140), (377,1012,6837), (377,6837,6993), (377,10431,4), (378,6240,4), (378,10323,3), (381,3090,5068), (381,3524,2), (381,3627,4), (381,5068,3091), (381,5070,5), (381,5073,3627), (381,8703,3524), (381,10109,3545), (382,548,631), (382,550,3528), (382,631,3832), (382,3526,3843), (382,3528,2), (382,3530,3855), (382,3843,3853), (382,5070,3861), (384,7791,2), (384,7833,7791), (390,3600,1), (394,1498,11441), (404,2478,2), (404,11114,2478), (405,443,2), (405,6835,6886), (405,11111,11106), (405,11112,443), (411,6836,2), (411,6840,6960), (411,6909,3), (411,6943,6962), (427,7494,2), (428,7484,7392), (428,11403,4), (440,7490,2), (442,6857,2), (442,8727,6828), (443,6912,6886), (443,11111,405), (452,6904,2), (474,5084,2), (474,11113,5084), (485,6200,9540), (485,9540,8972), (487,638,1271), (488,637,1270), (489,490,69), (498,3585,10590), (499,3583,10591), (546,549,1656), (546,1656,3545), (546,3545,3854), (546,3830,4), (546,10299,2), (547,3858,5072), (548,3528,3522), (548,3853,3530), (549,1656,3525), (549,3545,2), (549,3830,3545), (550,1657,4), (550,3146,10304), (550,3529,2), (550,3627,8703), (550,5059,3523), (550,6240,6636), (550,11001,3146), (574,7747,2548), (578,10984,5012), (631,3528,3), (631,3832,7486), (631,3855,5067), (631,5067,3526), (632,3850,5055), (858,7493,2), (858,10296,3153), (944,2096,1071), (946,3576,3616), (958,11495,5584), (962,5731,1), (962,5734,4301), (962,10884,11036), (991,3332,3945), (1006,6826,2), (1011,6817,2), (1012,6916,2), (1042,3000,1044), (1092,6759,110), (1113,1114,2071), (1131,8972,485), (1147,1614,9544), (1151,3070,3068), (1152,3071,3069), (1155,1837,1788), (1352,3098,10519), (1352,10519,3620), (1368,6353,2), (1368,9909,6353), (1370,7493,858), (1385,5603,3622), (1478,3085,5261), (1479,3086,5274), (1482,7967,3623), (1490,6282,78), (1532,6922,4193), (1583,6805,2), (1584,6806,2), (1585,1589,2), (1586,1590,2), (1587,9541,371), (1593,3575,4), (1593,7503,7527), (1593,10996,2), (1595,7399,5133), (1597,3537,2), (1597,6756,4), (1598,3538,2), (1610,1633,3556), (1656,3525,2), (1656,3830,546), (1656,10109,3090), (1657,3534,3), (1657,8703,11541), (1658,2937,7556), (1699,7987,1125), (1764,10454,10449), (1764,10463,10461), (1836,2646,3485), (1853,8567,6696), (1975,7750,69), (2041,2042,4), (2043,2044,376), (2045,2046,3533), (2060,3146,1559), (2071,7488,3), (2071,10296,858), (2077,6256,5552), (2475,4189,2), (2475,6847,3091), (2475,6895,4), (2475,6906,6888), (2476,6910,2), (2478,3149,6953), (2479,2480,441), (2549,5319,7765), (3053,5254,7735), (3060,10574,389), (3088,6636,3523), (3088,7400,2), (3090,3524,140), (3090,3529,11541), (3090,11541,3627), (3091,3523,2), (3091,3543,4), (3091,4208,6993), (3091,6992,5129), (3091,7486,5), (3091,10304,3523), (3098,9873,2896), (3100,4296,1), (3146,3522,2), (3146,3523,3839), (3146,3528,7486), (3146,3854,3830), (3146,7411,4208), (3146,10304,5056), (3146,11413,7396), (3147,11585,2), (3149,6865,2), (3151,7560,2), (3152,7538,2), (3153,10298,2), (3474,3486,65), (3520,7512,3), (3522,3529,3543), (3522,3543,10303), (3522,3854,10299), (3522,5059,4), (3523,3543,3091), (3523,10304,3), (3524,3529,5073), (3524,3627,5068), (3524,11541,4), (3525,3545,1656), (3525,3830,3854), (3525,10299,549), (3526,3530,631), (3526,3843,5), (3526,3853,3855), (3526,5067,2), (3528,3529,382), (3528,3832,3523), (3528,3855,3530), (3529,3534,3522), (3529,11001,1657), (3530,3843,5067), (3530,3853,5), (3530,3855,2), (3533,3628,2), (3533,5071,3628), (3534,11001,2), (3541,3547,2), (3542,3546,2), (3543,7396,3153), (3543,10304,2), (3545,3854,3091), (3545,10299,3525), (3548,7505,2), (3552,6655,2), (3560,6826,6846), (3560,6897,2), (3567,5446,11002), (3575,7503,7544), (3583,7280,499), (3585,5010,498), (3601,9579,226), (3616,9812,946), (3627,8703,140), (3627,11541,3146), (3628,3845,3851), (3628,3851,5071), (3628,3859,5), (3628,5054,3533), (3651,6851,2), (3651,6906,3), (3734,7830,7800), (3839,5056,3091), (3839,10303,5056), (3843,3853,4), (3843,3855,3832), (3845,5054,5071), (3845,5076,4), (3851,5054,3628), (3851,5076,3845), (3853,5067,3832), (3855,5067,5), (3868,11220,1071), (3911,9581,5704), (3917,5907,11444), (3917,11381,5907), (3972,7847,7803), (4188,5046,2), (4188,6926,3523), (4189,6895,6847), (4189,6908,3523), (4190,6868,6992), (4190,6872,2), (4190,6987,3523), (4191,6818,2), (4193,6921,2), (4195,4201,2), (4197,10883,5), (4208,6837,5056), (4292,4297,10884), (4292,4304,1), (4292,4313,11036), (4293,4294,1), (4293,4302,390), (4294,4299,3600), (4295,4305,1), (4298,4314,1), (4299,4302,1), (4299,4309,4317), (4299,4317,4325), (4299,4324,4294), (4301,9589,962), (4302,4309,4330), (4302,4316,4293), (4302,4317,4309), (4304,5732,5731), (4308,9785,1), (4309,4317,1), (4309,4330,4294), (4311,10624,1), (4316,4324,1), (4316,4330,4325), (4317,4325,4293), (4319,4320,1), (4324,4325,4330), (4325,4330,1), (4345,6049,1), (4348,7221,1), (4351,4354,1), (4652,6734,5744), (5004,5005,25), (5013,7745,7736), (5046,6848,3091), (5046,6905,6979), (5054,5071,2), (5054,5076,3851), (5056,10303,2), (5059,5068,5073), (5068,11541,3543), (5073,8703,3090), (5077,7866,8357), (5085,5480,3618), (5128,5727,4848), (5175,5744,6734), (5177,6888,5056), (5189,7492,2), (5218,5229,12), (5225,7288,11), (5261,5281,3085), (5265,5274,3086), (5318,11480,11488), (5319,7765,5286), (5321,11481,11489), (5432,10895,10588), (5433,10896,10589), (5446,9730,3567), (5447,5891,7999), (5550,9779,8227), (5550,10248,9779), (5584,6253,2550), (5587,6684,9780), (5806,11227,5439), (5878,6759,5656), (5881,11362,8), (5882,7982,3241), (6143,6639,2), (6143,7552,6639), (6225,11206,1498), (6459,6460,6), (6560,9541,7585), (6636,7391,2), (6643,7387,3089), (6644,7517,3518), (6676,8889,2), (6756,7395,6997), (6781,7756,32), (6803,7395,2), (6815,7503,2), (6824,6889,2), (6824,6917,6843), (6825,6833,2), (6827,6848,6919), (6827,6905,2), (6827,6970,6963), (6827,6985,6848), (6830,6842,5141), (6830,6954,2), (6831,6907,2476), (6831,6988,2), (6832,6989,2), (6833,6923,6871), (6834,6891,2), (6834,6928,5187), (6835,6986,2), (6836,6925,4), (6836,6962,6943), (6836,7580,6838), (6837,6839,3091), (6838,6840,3091), (6838,6890,2), (6842,6977,2), (6843,6870,3091), (6844,6871,3091), (6845,6937,5), (6846,6894,3091), (6847,6850,5177), (6847,6908,2), (6848,6926,2), (6850,6851,4), (6850,6892,6937), (6850,6906,2), (6853,6862,2), (6854,6883,2), (6856,7483,2), (6863,6952,2), (6868,6869,4), (6868,6885,6936), (6868,6934,2), (6868,6948,3), (6869,6938,3146), (6869,6948,6934), (6872,6904,5129), (6875,6917,2), (6875,6951,6889), (6876,6950,3), (6878,6881,2), (6880,6882,2), (6882,6941,5154), (6884,6993,5056), (6885,6930,5), (6885,6936,2), (6892,6937,2), (6893,6911,6964), (6893,6940,2), (6895,6908,3091), (6899,6905,6926), (6899,6985,2), (6902,6924,2), (6903,6942,6891), (6904,6992,10303), (6905,6963,6970), (6905,7491,5046), (6906,6937,6892), (6907,6935,2), (6909,6925,2), (6909,6932,6966), (6911,6947,2), (6912,6986,405), (6914,6917,6824), (6914,6951,2), (6916,10431,6839), (6919,6979,5056), (6922,6927,2), (6923,6950,2), (6924,6929,6944), (6925,6966,6932), (6928,6942,2), (6930,6948,6955), (6930,6955,2), (6932,6943,5), (6932,6966,2), (6934,6936,6885), (6934,6938,4), (6934,6987,6904), (6935,6988,6910), (6936,6955,631), (6938,6948,2), (6938,6955,6930), (6941,6961,2), (6943,6962,2), (6944,6967,2), (6949,6958,2), (6960,6972,2), (6962,6966,631), (6963,6970,2), (6985,7491,4), (6997,7485,2), (7381,11340,2), (7383,7404,2), (7388,11291,2), (7389,11292,2), (7390,7406,3091), (7391,7400,3091), (7392,7484,2), (7395,7408,3091), (7396,7488,10303), (7396,10565,2), (7399,7409,3091), (7401,7509,2), (7411,10431,2), (7464,7512,3520), (7470,11676,3), (7484,11403,11479), (7502,11250,3), (7509,7553,7394), (7509,7576,7401), (7511,7549,7557), (7538,7560,7520), (7540,7550,7533), (7553,9818,4), (7555,7574,7552), (7576,9818,7394), (7689,11750,11457), (7714,10691,2), (7761,7816,7795), (7783,7823,7774), (8226,8728,6991), (8227,10165,5550), (8357,8369,7866), (8550,11477,1992), (8703,11541,5068), (9778,10430,63), (9825,11479,7392), (10267,10532,10587), (10269,10531,10586), (10304,10565,10298), (10310,11500,100), (11015,11220,944), (11111,11112,2), (11291,11292,7819), (11293,11294,2), (11413,11414,7488)

X(21) = SCHIFFLER POINT

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

Let A'B'C' be the incentral triangle of ABC, and let L_A be the reflection of line B'C' in line BC; define L_B and L_C cyclically. The triangle formed by the lines L_A, L_B, L_C 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.

The name of this point honors Kurt Schiffler.

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}, {36, 79}, {37, 172}, {40, 3577}, {42, 4281}, {44, 4273}, {45, 3285}, {51, 970}, {57, 4652}, {60, 960}, {65, 4640}, {71, 4269}, {72, 943}, {73, 651}, {75, 272}, {77, 1394}, {84, 285}, {85, 3188}, {90, 224}, {99, 105}, {101, 3294}, {104, 110}, {107, 1295}, {141, 4265}, {144, 954}, {145, 956}, {149, 2894}, {162, 3194}, {187, 5277}, {198, 5296}, {200, 4866}, {210, 4420}, {214, 501}, {219, 2335}, {238, 256}, {243, 1896}, {261, 314}, {268, 280}, {270, 1172}, {286, 1441}, {294, 1212}, {307, 2062}, {323, 5453}, {329, 5703}, {332, 1036}, {385, 1655}, {386, 1724}, {390, 6601}, {391, 4254}, {476, 2687}, {484, 3754}, {517, 1389}, {518, 2346}, {519, 3746}, {535, 5270}, {551, 5557}, {572, 1765}, {593, 6051}, {612, 989}, {614, 988}, {643, 1320}, {644, 1334}, {662, 1156}, {691, 2752}, {741, 932}, {748, 978}, {756, 5293}, {884, 885}, {902, 5255}, {915, 925}, {936, 3305}, {938, 5744}, {940, 4252}, {942, 3218}, {950, 5745}, {961, 1402}, {962, 3428}, {976, 983}, {986, 3924}, {987, 2206}, {992, 5110}, {999, 3296}, {1030, 1213}, {1038, 1041}, {1039, 1040}, {1060, 1063}, {1061, 1062}, {1064, 3073}, {1083, 3110}, {1104, 3666}, {1107, 1914}, {1155, 3812}, {1214, 1396}, {1251, 5240}, {1254, 1758}, {1261, 4723}, {1304, 2694}, {1319, 1408}, {1329, 5432}, {1330, 3936}, {1376, 5217}, {1392, 2098}, {1412, 1420}, {1453, 5256}, {1466, 5435}, {1470, 5555}, {1500, 5291}, {1610, 2217}, {1617, 3600}, {1682, 3271}, {1697, 3680}, {1698, 5010}, {1761, 2294}, {1936, 2654}, {1946, 4391}, {2077, 6684}, {2096, 5553}, {2276, 4426}, {2310, 2648}, {2320, 5289}, {2341, 5549}, {2344, 3061}, {2551, 5218}, {2782, 5985}, {2886, 6284}, {3006, 5015}, {3052, 5710}, {3053, 5275}, {3058, 3813}, {3060, 5752}, {3062, 5732}, {3085, 3436}, {3207, 5781}, {3208, 4390}, {3216, 4256}, {3220, 4357}, {3241, 3303}, {3244, 5288}, {3256, 4848}, {3304, 5558}, {3315, 3953}, {3333, 4666}, {3336, 5883}, {3337, 4973}, {3419, 5791}, {3427, 5731}, {3434, 4294}, {3487, 5905}, {3496, 5060}, {3555, 3957}, {3579, 3753}, {3585, 3822}, {3586, 5705}, {3589, 5096}, {3614, 6668}, {3617, 5687}, {3624, 5561}, {3681, 3811}, {3684, 3691}, {3689, 4662}, {3737, 6615}, {3757, 4968}, {3816, 5433}, {3833, 5131}, {3841, 4324}, {3886, 4483}, {3895, 4853}, {3920, 5266}, {3929, 3951}, {4084, 5425}, {4101, 4416}, {4255, 4383}, {4314, 4847}, {4423, 5204}, {4516, 4612}, {4520, 6603}, {4567, 5377}, {4646, 4689}, {4668, 4803}, {4867, 5424}, {5044, 5440}, {5294, 5314}, {5554, 5657}, {5686, 6600}

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

X(22) = EXETER POINT

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)

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

X(22) lies on these lines:
2,3   6,251   32,1194   35,612   36,614   51,182   56,977   69,159   76,1799   98,925   99,305   100,197   110,154   155,1614   157,183   160,325   161,343   184,511   187,1196   232,577   264,1629   347,1617   675,1305   991,1790   1184,1627   1294,1302   1486,1621   1602,1626

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) = inverse-in-circumcircle of X(858)
X(22) = anticomplement of X(427)
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) = complement of X(7391)
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) = exsimilicenter of circumcircle and tangential circle
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(23) = FAR-OUT POINT

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, Octobe3r 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.

X(23) lies on the Parry circle, anti-Brocard circle, anti-McCay circumcircle, and these lines:
2,3   6,353   51,575   52,1614   94,98   105,1290   110,323   111,187   143,1199   159,193   184,576   232,250   251,1194   324,1629   385,523   477,1302   895,1177   1196,1627   1297,1804

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

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   32,232   33,35   34,36   49,568   51,578   52,1147   56,1870   64,74   96,847   98,1289   107,1093   108,915   110,155   154,1181   182,1843   183,1235   184,389   185,1495   242,1602   254,393   264,1078   511,1092   573,1474   602,1395   944,1610   1063,1775   1112,1511   1192,1511   1324,1603   1385,1829

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) = 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) = inverse-in-circumcircle of X(403)
X(24) = inverse-in-orthocentroidal circle of X(1594)
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) = insimilicenter of circumcircle and tangential circle
X(24) = inverse-in-polar-circle of X(2072)
X(24) = homothetic center of tangential and circumorthic triangles
X(24) = homothetic center of orthic and Kosnita triangles
X(24) = X(i)-isoconjugate of X(j) for these (i,j): (75,2351), (91,3)

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

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)

X(25) lies on these lines:
1,1036   2,3   6,51   19,33   31,608   32,1184   34,56   35,1900   36,1878   40,1902   41,42   52,155   53,157   57,1473   58,967   64,1192   65,1452   76,1241   92,242   98,107   100,1862   105,108   110,1112   111,112   114,135   125,1853   132,136   143,156   182,3066  183,264   185,1498   221,1425   225,1842   226,1892   262,275   273,1447   286,1218   317,325   339,1289   343,1352   371,493   372,494   389,1181   393,1033   394,511   669,878   692,913   694,1613   842,1304   847,1179   941,1172   958,1891   999,1870   1001,1848   1073,1297   1096,1402   1235,1239   1300,1302   1324,1785   1376,1861   1470,1877   1503,1619   1604,1863   1631,1826   1726,1736   1730,1754

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) = inverse-in-circumcircle of X(468)
X(25) = inverse-in-orthocentroidal-circle 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) = 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) = 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) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,3,7484), (2,4,427), (2,5,7539), (3,4,1593), (3,5,7395), (4,5,7507), (4,24,3)

X(26) = CIRCUMCENTER OF THE TANGENTIAL TRIANGLE

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.

X(26) lies on these lines: 2,3   6,143   52,184   68,161   98,1286   154,155   206,511   1605,1607   1606,1608

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(27) = CEVAPOINT OF ORTHOCENTER AND CLAWSON CENTER

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).

X(27) lies on these lines:
2,3   6,1246   7,81   19,63   57,273   58,270   84,1896   86,1474   103,107   110,917   226,284   239,1829   243,1859   295,335   306,1043   393,967   579,1751   648,903   662,913   1014,1440   1088,1434   1268,1796   1719,1733   1730,1746   1770,1780

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(75)-isoconjugate of X(2200)
X(27) = crosspoint of X(4) and X(19) wrt excentral triangle
X(27) = trilinear product X(2)*X(28)

X(28) = CEVAPOINT OF X(19) AND X(25)

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) = 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) = inverse-in-circumcircle of X(2074)
X(28) = X(i)-Ceva conjugate of X(j) for these (i,j): (270,58), (286,81)
X(28) = cevapoint of X(i) and X(j) for these (i,j): (19,25), (34,56)
X(28) = X(i)-cross conjugate of X(j) for these (i,j): (19,27), (58,58)
X(28) = crossdifference of every pair of points on line X(647)X(656)
X(28) = X(4)-Hirst inverse of X(422)
X(28) = X(i)-beth conjugate of X(j) for these (i,j): (29,29), (107,28), (162,28), (270,28)

X(29) = CEVAPOINT OF INCENTER AND ORTHOCENTER

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

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

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

X(30) = EULER INFINITY POINT

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(31) = 2nd POWER POINT

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

X(32) = 3rd POWER POINT

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) = inverse-in-circumcircle of X(1691)
X(32) = inverse-in-Brocard-circle of X(39)
X(32) = inverse-in-1st-Lemoine-circle of X(1692)
X(32) = complement of X(315)
X(32) = anticomplement of X(626)
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) = barycentric square of X(6)
X(32) = barycentric product of (nonreal) circumcircle intercepts of the Lemoine axis

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

Barycentrics  sin A + tan A : sin B + tan B : sin C + tan C
                        = h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = tan A cos²(A/2)

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

Let L_A be the reflection of line BC in the internal angle bisector of angle A, and define L_B and L_C cyclically. Let DEF be the triangle formed by L_A, L_B, L_C. 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) = trilinear pole of line X(657)X(4041) (the polar of X(85) wrt polar circle)
X(33) = pole wrt polar circle of trilinear polar of X(85) (line X(522)X(693))
X(33) = polar conjugate of X(85)
X(33) = perspector of ABC and extraversion triangle of X(34)
X(33) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,19), (29,281), (318,9)
X(33) = X(i)-cross conjugate of X(j) for these (i,j): (41,9), (42,55)
X(33) = crosspoint of X(i) and X(j) for these (i,j): (1,282), (4,281)
X(33) = crosssum of X(i) and X(j) for these (i,j): (1,223), (3,222), (57,1394), (73,1214)
X(33) = crossdifference of every pair of points on line X(652)X(905)
X(33) = X(33)-beth conjugate of X(25)
X(33) = homothetic center of anti-excenters-incenter reflections triangle and anti-tangential midarc triangle

X(34) = X(4)-BETH CONJUGATE OF X(4)

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

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

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)

X(35) lies on these lines:
1,3   4,498   8,993   9,90   10,21   11,140   12,30   22,612   24,33   31,386   34,378   37,267   42,58   43,1011   47,212   71,284   72,191   73,74   79,226   172,187   228,846   255,991   376,388   404,1125   411,516   474,1001   495,550   496,549   497,499   500,1154   595,902   950,1006   968,975   1124,1152

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

X(35) is the {X(1),X(3)}-harmonic conjugate of X(36). For a list of other harmonic conjugates of X(35), click Tables at the top of this page.

X(35) = isogonal conjugate of X(79)
X(35) = isotomic conjugate of X(20565)
X(35) = homothetic center of Trinh triangle and anti-tangential midarc triangle
X(35) = inverse-in-circumcircle of X(484)
X(35) = X(500)-cross conjugate of X(1)
X(35) = crosssum of X(481) and X(482)
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) = {X(55),X(56)}-harmonic conjugate of X(3295)
X(35) = crossdifference of every pair of points on line X(650)X(4802)
X(35) = homothetic center of intangents and Kosnita triangles
X(35) = perspector of ABC and extraversion triangle of X(36)
X(35) = Hofstadter 3/2 point
X(35) = homothetic center of 2nd isogonal triangle of X(1) and cevian triangle of X(3); see X(36)
X(35) = insimilicenter of circumcircle and circumcircle of reflection triangle of X(1); exsimilicenter is X(36)
X(35) = Cundy-Parry Phi transform of X(5902)

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

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) = polar conjugate of isotomic conjugate of X(22128)
X(36) = complement of X(5080)
X(36) = anticomplement of X(3814)
X(36) = inverse-in-circumcircle of X(1)
X(36) = inverse-in-incircle of X(942)
X(36) = inverse-in-Bevan-circle of X(46)
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) = 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(37) = CROSSPOINT OF INCENTER AND CENTROID

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(r_a^2 + s^2), where r_a 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) = 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(38) = CROSSPOINT OF X(1) AND X(75)

Barycentrics  a(b² + c²) : b(c² + a²) : c(a² + b²)
                        = sin(A + ω) : sin(B + ω) : sin(C + ω)
X(38) = 3r² + 8rR - s²)*X(1) - 6rR*X(2) - 4r²*X(3)   (Peter Moses, April 2, 2013)

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

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

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

X(39) = BROCARD MIDPOINT

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)

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

X(40) = BEVAN POINT

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)

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

X(41) = X(6)-CEVA CONJUGATE OF X(31)

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) = 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) = perspector of unary cofactor triangles of Gemini triangles 1 and 13

X(42)  CROSSPOINT OF INCENTER AND SYMMEDIAN POINT

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

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

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

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

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

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

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

X(43) = X(6)-CEVA CONJUGATE OF X(1)

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

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

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

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

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

X(44) = X(6)-LINE CONJUGATE OF X(1)

X(44) lies on these lines: 1,6   2,89   10,752   31,210   51,209   65,374   88,679   181,375   190,239   193,344   214,1017   241,651   292,660   354,748   513,649   527,1086   583,992   678,902

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

X(44) = midpoint of X(i) and X(j) for these (i,j): (190,239), (3218,3257)
X(44) = reflection of X(1279) in X(238)
X(44) = isogonal conjugate of X(88)
X(44) = isotomic conjugate of X(20568)
X(44) = complement of X(320)
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) = X(6)-line conjugate of X(1)
X(44) = X(88)-cross conjugate of X(44)
X(44) = X(i)-beth conjugate of X(j) for these (i,j): (9,44), (644,44), (645,239), (44,44)
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) = crossdifference of PU(55)
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) = {X(6),X(9)}-harmonic conjugate of X(37)
X(44) = inverse-in-circumconic-centered-at-X(9) of X(1)
X(44) = trilinear pole of line line X(678)X(1635)

X(45) = X(9)-BETH CONJUGATE OF X(1)

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

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

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

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

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

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

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

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

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

X(48) = CROSSPOINT OF X(1) AND X(63)

X(48) = (r² + 4rR² + 4R² + s²)*X(1) - 6R(2R + r)*X(2) - 2(r² + 2rR - s²)*X(3)    (Peter Moses, April 2, 2013)

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

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

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

X(48) = isogonal conjugate of X(92)
X(48) = isotomic conjugate of X(1969)
X(48) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,31), (3,212), (63,255), (92,47), (284, 6)
X(48) = X(228)-cross conjugate of X(3)
X(48) = complement of X(21270)
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(49) = CENTER OF SINE-TRIPLE-ANGLE CIRCLE

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 P_A = reflection of P in line AQ, let Q_A = reflection of Q in line AP, and let M_A = midpoint of segment P_AQ_A. Define M_B and M_C cyclically. César Lozada found that if Q = isogonal conjugate of P, then the locus of P for which M_AM_BM_C 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 M_AM_BM_C 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) = X(4)-isoconjugate of X(2962)
X(49) = X(92)-isoconjugate of X(2963)
X(49) = eigencenter of cevian triangle of X(94)
X(49) = eigencenter of anticevian triangle of X(50)
X(49) = X(94)-Ceva conjugate of X(50)

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

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

X(51) = CENTROID OF ORTHIC TRIANGLE

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) = 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(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) = 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(52) = ORTHOCENTER OF ORTHIC TRIANGLE

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)

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) = 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(53) = SYMMEDIAN POINT OF ORTHIC TRIANGLE

Let A'B'C' be the Euler triangle. Let L_A be the trilinear polar of A', and define L_B and L_C cyclically. Let A" = L_B∩L_C, 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) = 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(54) = KOSNITA POINT

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 O_A the circumcenter of triangle BOC. Define O_B and O_C cyclically. Then the lines AO_A, BO_B, CO_C 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)

Let N_AN_BN_C be the reflection triangle of X(5). Let O_A be the circumcenter of AN_BN_C, and define O_B and O_C cyclically. Triangle O_AO_BO_C 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)

X(54) lies on the Napoleon cubic and these lines:
1,3460   2,68   3,97   4,184   5,49   6,24   12,215   36,73   39,248   51,288   64,378   69,95   71,572   72,1006   74,185   112,217   140,252   156,381   186,389   276,290   575,895   826,879   3336,3468

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

X(54) = midpoint of X(3) and X(195)
X(54) = reflection of X(195) in X(1493)
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) = X(i)-Ceva conjugate of X(j) for these (i,j): (5,2120), (95,97), (288,6)
X(54) = cevapoint of X(6) and X(184)
X(54) = X(i)-cross conjugate of X(j) for these (i,j): (3,96), (6,275), (186,74), (389,4), (523,110)
X(54) = crosspoint of X(i) and X(j) for these {i,j}: {4,3459}, {95,275}
X(54) = crosssum of X(i) and X(j) for these (i,j): (3,195), (51,216), (627,628)
X(54) = X(24)-of-intouch-triangle
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(55) = INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE)

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

X(56) = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE)

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)

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

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

X(57)  ISOGONAL CONJUGATE OF X(9)

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)

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) = inverse-in-circumcircle of X(2078)
X(57) = inverse-in-Bevan-circle of X(1155)
X(57) = complement of X(329)
X(57) = anticomplement of X(3452)
X(57) = trilinear product of PU(46)
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): (2,189), (7,279), (27,81), (85,273)
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)
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) = {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(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-incircle of X(3660)
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) = crosssum of X(2066) and X(5414)
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(58)  ISOGONAL CONJUGATE OF X(10)

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)

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) = inverse-in-circumcircle of X(1326)
X(58) = inverse-in-Brocard-circle of X(386)
X(58) = complement of X(1330)
X(58) = anticomplement of X(3454)
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) = X(i)-beth conjugate of X(j) for these (i,j): (21,21), (60,58), (110,58), (162,58), (643,58), (1098,283)
X(58) = barycentric product of PU(31)
X(58) = trilinear pole of line X(649)X(834)
X(58) = {X(1),X(31)}-harmonic conjugate of X(595)
X(58) = {X(21),X(283)}-harmonic conjugate of X(2328)
X(58) = X(42)-isoconjugate of X(75)
X(58) = X(71)-isoconjugate of X(92)
X(58) = X(101)-isoconjugate of X(1577)
X(58) = homothetic center of 2nd circumperp triangle and 'Hatzipolakis-Brocard triangle' (A'B'C' as defined at X(5429))
X(58) = trilinear product of vertices of 2nd circumperp triangle
X(58) = perspector of 2nd circumperp triangle and unary cofactor triangle of 1st circumperp triangle
X(58) = perspector of ABC and cross-triangle of ABC and 2nd circumperp triangle
X(58) = Cundy-Parry Phi transform of X(573)
X(58) = Cundy-Parry Psi transform of X(13478)
X(58) = perspector of ABC and unary cofactor triangle of Gemini triangle 11
X(58) = {X(1),X(21)}-harmonic conjugate of X(4653)

X(59)  ISOGONAL CONJUGATE OF X(11)

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) = 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) = 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(60) = ISOGONAL CONJUGATE OF X(12)

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) = 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(61) = ISOGONAL CONJUGATE OF X(17)

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(633) in X(635)
X(61) = isogonal conjugate of X(17)
X(61) = inverse-in-Brocard-circle of X(62)
X(61) = complement of X(633)
X(61) = anticomplement of X(635)
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) = 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) = reflection of X(62) in X(5007)
X(61) = Kosnita(X(15),X(3)) point
X(61) = Kosnita(X(15),X(15)) point

X(62) = ISOGONAL CONJUGATE OF X(18)

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(634) in X(636)
X(62) = isogonal conjugate of X(18)
X(62) = inverse-in-Brocard-circle of X(61)
X(62) = complement of X(634)
X(62) = anticomplement of X(636)
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) = 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) = reflection of X(61) in X(5007)
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(63) = ISOGONAL CONJUGATE OF X(19)

Let Oa be the A-extraversion of the Conway circle (the circle centered at the A-excenter and passing through A, with radius sqrt(r_a^2 + s^2), where r_a is the A-exradius). Let Pa be the perspector of Oa, and La the polar of Pa wrt Oa. Define Lb and Lc cyclically. Let A' = Lb∩Lc, B' = Lc∩La, C' = La∩Lb. Triangle A'B'C' is perspective to the excentral triangle at X(63). (Randy Hutson, February 10, 2016)

Let A'B'C' be the 2nd Brocard triangle. Let A" be the trilinear product B'*C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(63). (Randy Hutson, February 10, 2016)

Let A'B'C' be the hexyl triangle. Let A" be the trilinear pole of line B'C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(63). (Randy Hutson, February 10, 2016)

Let A'B'C' be the side-triangle of ABC and hexyl triangle. Let A" be the {B,C}-harmonic conjugate of A', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(63). (Randy Hutson, February 10, 2016)

Let A'B'C' be the excentral triangle. Let A" be the isotomic conjugate, wrt triangle A'BC, of X(1). Define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(63). (Randy Hutson, July 31 2018)

X(63) lies on these lines:
1,21   2,7   3,72   6,2221   8,20   10,46   19,27   33,1013   36,997   37,940   48,326   55,518   56,960   65,958   69,71   77,219   91,921   100,103   162,204   169,379   171,612   190,312   194,239   201,603   210,1004   212,1040   213,980   220,241   223,651   238,614   240,1096   244,748   304,1102   318,412   354,1001   392,999   404,936   405,942   452,938   484,535   517,956   544,1018   561,799   654,918   750,756

X(63) is the {X(9),X(57)}-harmonic conjugate of X(2). For a list of other harmonic conjugates of X(63), click Tables at the top of this page.

X(63) = reflection of X(i) in X(j) for these (i,j): (1,993), (1478,10)
X(63) = isogonal conjugate of X(19)
X(63) = isotomic conjugate of X(92)
X(63) = complement of X(5905)
X(63) = anticomplement of X(226)
X(63) = anticomplementary conjugate of X(2893)
X(63) = X(i)-Ceva conjugate of X(j) for these (i,j): (7,224), (69,78), (75,1), (304,326), (333,2), (348,77)
X(63) = cevapoint of X(i) and X(j) for these (i,j): (3,219), (9,40), (48,255), (71,72)
X(63) = X(i)-cross conjugate of X(j) for these (i,j): (3,77), (9,271), (48,1), (71,3), (72,69), (219,78), (255,326)
X(63) = crosspoint of X(i) and X(j) for these (i,j): (69,348), (75,304)
X(63) = crosssum of X(25) and X(607)
X(63) = crossdifference of every pair of points on line X(661)X(663)
X(63) = trilinear product X(2)*X(3)
X(63) = trilinear product of PU(22)
X(63) = bicentric sum of PU(i) for these i: 128, 129
X(63) = PU(128)-harmonic conjugate of X(661)
X(63) = midpoint of PU(129)
X(63) = {X(1),X(1707)}-harmonic conjugate of X(31)
X(63) = {X(2),X(9)}-harmonic conjugate of X(3305)
X(63) = {X(2),X(57)}-harmonic conjugate of X(3306)
X(63) = {X(92),X(1748)}-harmonic conjugate of X(19)
X(63) = trilinear pole of line X(521)X(656)
X(63) = pole wrt polar circle of trilinear polar of X(158)
X(63) = X(48)-isoconjugate (polar conjugate) of X(158)
X(63) = X(i)-isoconjugate of X(j) for these {i,j}: {4,6}, {31,92}, {75,1973}
X(63) = excentral isogonal conjugate of X(1742)
X(63) = homothetic center of excentral triangle and anticomplement of the intouch triangle
X(63) = X(161)-of-intouch-triangle
X(63) = X(184)-of-excentral-triangle
X(63) = inverse-in-circumconic-centered-at-X(9) of X(908)
X(63) = trilinear square of X(5374)
X(63) = perspector of excentral triangle and Gemini triangle 2
X(63) = homothetic center of excentral triangle and Gemini triangle 30
X(63) = perspector of ABC and cross-triangle of Gemini triangles 35 and 36
X(63) = perspector of ABC and cross-triangle of ABC and Gemini triangle 35
X(63) = perspector of ABC and cross-triangle of ABC and Gemini triangle 36
X(63) = barycentric product of vertices of Gemini triangle 35
X(63) = barycentric product of vertices of Gemini triangle 36
X(63) = X(i)-aleph conjugate of X(j) for these (i,j):
(2,1), (75,63), (92,920), (99,662), (174,978), (190,100), (333,411), (366,43), (514,1052), (556,40), (648,162), (664,651), (668,190), (670,799), (671,897), (903,88)
X(63) = X(i)-beth conjugate of X(j) for these (i,j):
(63,222), (190,63), (333,57), (345,345), (643,63), (645,312), (662,223)
X(63) = perspector of ABC and extraversion triangle of X(63), which is also the anticevian triangle of X(63)

X(64) = ISOGONAL CONJUGATE OF X(20)

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) = X(25)-cross conjugate of X(6)
X(64) = X(1)-beth conjugate of X(207)

X(65) = ORTHOCENTER OF THE INTOUCH TRIANGLE

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)

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) = inverse-in-incircle of X(1319)
X(65) = complement of X(3869)
X(65) = anticomplement of X(960)
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 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) = inverse-in-circumcircle of X(5172)
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(66) = ISOGONAL CONJUGATE OF X(22)

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) = cyclocevian conjugate of X(2998)
X(66) = anticomplement of X(206)
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(67) = ISOGONAL CONJUGATE OF X(23)

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) = inverse-in-circumcircle 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) = orthocenter of X(3)X(74)X(879)
X(67) = perspector of ABC and X(2)-Ehrmann triangle; see X(25)

X(68)  PRASOLOV POINT

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 given his 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) = 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(69) = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE

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) 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   225,2888   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.

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) = cyclocevian conjugate of X(253)
X(69) = complement of X(193)
X(69) = anticomplement of X(6)
X(69) = anticomplementary conjugate of X(2)
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): (3,3053), (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) = 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(70) = ISOGONAL CONJUGATE OF X(26)

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) = X(571)-crossconjugate of X(2)
X(70) = X(i)-isoconjugate of X(j) for these {i,j}: {{1,26}, {63,8746}
X(70) = reflection of the isogonal conjugate of X(2072) in X(125)
X(70) = X(125)-cevapoint of X(924)
X(70) = X(161)-crosssum of X(8553)
X(70) = barycentric product X(525) X(1288)

X(71) = ISOGONAL CONJUGATE OF X(27)

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) = 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(72) = ISOGONAL CONJUGATE OF X(28)

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 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), (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(73)  CROSSPOINT OF INCENTER AND CIRCUMCENTER

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) = 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(92)-isoconjugate of X(284)
X(73) = {X(1),X(1745)}-harmonic conjugate of X(4)

X(74)  ISOGONAL CONJUGATE OF EULER INFINITY POINT

X(74) = (r² + 2rR + s²)*X(1) - R(6r + 9R)*X(2) + (r² + 12rR + 18R² - 3s²)*X(3)    (Peter Moses, April 2, 2013)

As the isogonal conjugate of the point in which the Euler line meets the line at infinity, X(74) lies on the circumcircle.

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 ix X(74). Let A' be the point of intersection of the orthic axis and line BC, and define B' and C' cyclically. Let O_A be the circumcenter of AB'C', and define Let O_B and O_C cyclically; then the lines AO_A, BO_B, CO_C 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)

X(74) lies on the circumcircle, Jerabek hyperbola, Neuberg cubic, Darboux septic curve, and these lines:
1,3464   2,113   3,110   4,107   6,112   20,68   24,64   30,265   35,73   54,185   65,108   67,935   69,99   71,101   72,100   98,690   187,248   477,523   484,3465   511,691   512,842   550,930   1157,3484

X(74) = midpoint of X(20) and X(3448)
X(74) = reflection of X(i) in X(j) for these (i,j): (4,125), (110,3), (146,113), (399,1511)
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) = 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) = trilinear pole of line X(6)X(647)
X(74) = Ψ(X(6),X(647))
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) = inverse-in-polar-circle of X(133)
X(74) = trilinear pole wrt circumorthic triangle of van Aubel line
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) = 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(75)  ISOTOMIC CONJUGATE OF INCENTER

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 these lines:
1,86   2,37   6,239   7,8   9,190   10,76   19,27   21,272   31,82   32,746   38,310   42,1218   43,872   47,2216   48,336   72,1246   77,664   81,2214   87,3226   99,261   100,675   101,767   141,334   142,2321   144,391   149,2805   150,2893   158,240   183,1376   194,1107   219,1944   222,1943   225,264   234,556   244,1978   255,2190   257,698   269,1222   279,1219   280,309   298,1081   299,554   325,2886   491,1659   522,3261   523,876   537,668   538,1573   560,1580   689,745   700,971   728,1223   753,789   757,1468   758,994   775,1496   799,897   811,1099   901,2863   927,2751   934,2370940,1999   958,1975   982,1920   1088,3668   1089,1268   1150,3218&bsp;  1237,1240   1332,2989   1370,3434   1444,2217   1581,1934   1812,2219   1897,2000   1928,2085   1953,1959   2167,2168   2894,2897

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

X(75) = reflection of X(i) in X(j) for these (i,j): (192,37), (335,1086), (984,10)
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) = X(i)-Ceva conjugate of X(j) for these (i,j): (76,312), (274,2), (310,76), (314,69)
X(75) = cevapoint of X(i) and X(j) for these (i,j): (1,63), (2,8), (7,347), (10,321), (244,514)
X(75) = X(i)-cross conjugate of X(j) for these (i,j): (1,92), (2,85), (7,309), (8,312), (10,2), (38,1), (63,304), (244,514), (307,69), (321,76), (347,322), (522,190)
X(75) = crosspoint of X(i) and X(j) for these (i,j): (2,330), (274,310)
X(75) = crossdifference of every pair of points on line X(667)X(788)
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; https://bernard-gibert.pagesperso-orange.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(76) = 3rd BROCARD POINT

Let A' be the perspector of the A-McCay circle, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(76). (Randy Hutson, April 9, 2016)

X(76) is the vertex conjugate of the foci of the inellipse that is the barycentric square of the de Longchamps line. The center of the inellipse is X(626) and its Brianchon point (perspector) is X(1502). (Randy Hutson, October 15, 2018)

Let A'B'C' be the obverse triangle of X(1). Let A"B"C" be the N-obverse triangle of X(1). Let A* be the barycentric product A'*A", and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(76). (Randy Hutson, October 15, 2018)

X(76) lies on these lines:
1,350   2,39   3,98   4,69   5,262   6,83   7,1240   8,668   10,75   13,299   14,298   17,303   18,302   20,3424   22,1799   25,1241   31,734   32,384   37,1218   85,226   95,96   100,767   107,2366   110,2367   115,626   141,698   148,2896   182,3406   187,3552   192,1221   251,1239   257,1926   275,276   297,343   321,561   330,1015   331,1231   333,1751   334,1089   335,871   338,599  485,491   486,492   524,598   620,1569   689,755   691,2868   693,764   761,789   799,1150   826,882   940,1509   1003,3053   1007,3090   1131,1271   1132,1270   1229,1446   1423,3403   1501,3115   1670,1677   1671,1676   1698,3097   2001,2909   2319,3500   2394,3267   3224,3225   3492,3506   3496,3512   3497,3509

X(76) is the {X(2),X(194)}-harmonic conjugate of X(39). For a list of other harmonic conjugates of X(76), click Tables at the top of this page.

X(76) = reflection of X(194) in X(39)
X(76) = isogonal conjugate of X(32)
X(76) = isotomic conjugate of X(6)
X(76) = complement of X(194)
X(76) = anticomplement of X(39)
X(76) = anticomplementary conjugate of X(2896)
X(76) = X(i)-Ceva conjugate of X(j) for these (i,j): (308,2), (310,75)
X(76) = cevapoint of X(i) and X(j) for these (i,j): (2,69), (6,22), (75,312), (311,343), (313,321), (339,525)
X(76) = X(i)-cross conjugate of X(j) for these (i,j): (2,264), (69,305), (141,2), (321,75), (343,69), (525,99)
X(76) = crosssum of X(669) and X(1084)
X(76) = crossdifference of every pair of points on line X(669)X(688)
X(76) = X(i)-beth conjugate of X(j) for these (i,j): (76,85), (799,348)
X(76) = pole wrt polar circle of trilinear polar of X(25) (line X(512)X(1692))
X(76) = X(48)-isoconjugate (polar conjugate) of X(25)
X(76) = X(6)-isoconjugate of X(31)
X(76) = trilinear product of PU(i) for these i: 10, 86
X(76) = barycentric product of PU(11)
X(76) = antigonal image of X(1916)
X(76) = cevapoint of polar conjugates of PU(4)
X(76) = trilinear product of vertices of 1st Brocard triangle
X(76) = trilinear product of vertices of 1st anti-Brocard triangle
X(76) = X(2)-Ceva conjugate of X(6374)
X(76) = X(384)-of-5th-Brocard-triangle
X(76) = X(6)-of-6th-Brocard-triangle
X(76) = perspector of ABC and 1st Brocard triangle
X(76) = trilinear pole of de Longchamps line
X(76) = bicentric sum of PU(159)
X(76) = PU(159)-harmonic conjugate of X(9494)
X(76) = perspector of conic {{A,B,C,X(670),X(689),X(1978)}} (isotomic conjugate of Lemoine axis.)
X(76) = X(1916) of 1st Brocard triangle
X(76) = crosspoint of X(6) and X(22) wrt both the anticomplementary and tangential triangles
X(76) = inverse-in-circumcircle of X(5152)
X(76) = inverse-in-2nd-Brocard circle of X(99)
X(76) = X(3094)-of-1st anti-Brocard-triangle
X(76) = trilinear product of vertices of mid-triangle of 1st Brocard and 1st anti-Brocard triangles
X(76) = perspector of ABC and cross-triangle of ABC and 3rd Brocard triangle
X(76) = trilinear product of vertices of the three anti-altimedial triangles
X(76) = Cundy-Parry Phi transform of X(98)
X(76) = Cundy-Parry Psi transform of X(511)
X(76) = barycentric product X(99)*X(850)
X(76) = intersection, other than X(4), of P(1)- and U(1)-Fuhrmann circles (aka -Hagge circles)
X(76) = {X(7737),X(14023)}-harmonic conjugate of X(20065)
X(76) = intersection of lines PU(1) of 1st and 2nd Ehrmann circumscribing triangles
X(76) = trilinear cube of X(2)
X(76) = barycentric square of X(75)
X(76) = trilinear product of vertices of Gemini triangle 19

X(77) = ISOGONAL CONJUGATE OF X(33)

X(77) lies on these lines:
1,7   2,189   6,241   9,651   29,34   40,947   55,1037   56,1036   57,81   63,219   65,969   69,73   75,664   102,934   283,603   309,318   738,951   988,1106   999,1057

X(77) = isogonal conjugate of X(33)
X(77) = isotomic conjugate of X(318)
X(77) = X(i)-Ceva conjugate of X(j) for these (i,j): (85,57), (86,7), (348,63)
X(77) = cevapoint of X(i) and X(j) for these (i,j): (1,223), (3,222)
X(77) = X(i)-cross conjugate of X(j) for these (i,j): (3,63), (73,222)
X(77) = trilinear pole of line X(652)X(905)
X(77) = {X(175),X(176)}-harmonic conjugate of X(962)
X(77) = X(92)-isoconjugate of X(41)
X(77) = perspector of ABC and extraversion triangle of X(78)
X(77) = X(i)-beth conjugate of X(j) for these (i,j): (21,990), (69,69), (86,269), (99,75), (332,326), (336,77), (662,77), (664,77), (811,77)

X(78) = ISOGONAL CONJUGATE OF X(34)

If you have The Geometer's Sketchpad, you can view X(78).

X(78) lies on these lines:
1,2   3,63   4,908   9,21   20,329   29,33   37,965   38,988   40,100   46,758   55,960   56,480   57,404   69,73   101,205   207,653   210,958   212,283   220,949   226,377   271,394   273,322   280,282   345,1040   392,1057   474,942   517,945   644,728   999,1059

X(78) = isogonal conjugate of X(34)
X(78) = isotomic conjugate of X(273)
X(78) = X(i)-Ceva conjugate of X(j) for these (i,j): (69,63), (312,9), (332,345)
X(78) = X(i)-cross conjugate of X(j) for these (i,j): (3,271), (72,8), (212,9), (219,63)
X(78) = crosspoint of X(69) and X(345)
X(78) = crosssum of X(i) and X(j) for these (i,j): (25,608), (56,1406), (604,1395), (1042,1426)
X(78) = X(i)-beth conjugate of X(j) for these (i,j): (78,3), (643,40), (1043,1)
X(78) = trilinear pole of line X(521)X(652)
X(78) = {X(1),X(8)}-harmonic conjugate of X(3872)
X(78) = {X(2),X(145)}-harmonic conjugate of X(938)
X(78) = X(92)-isoconjugate of X(604)
X(78) = homothetic center of anticomplementary triangle and tangential triangle of the hexyl triangle
X(78) = perspector of ABC and extraversion triangle of X(77)

X(79) = ISOGONAL CONJUGATE OF X(35)

X(79) = (2r + 3R)*X(1) + 6r*X(2) - 6r*X(3)    (Peter Moses, April 2, 2013)

Let A' be the reflection of X(1) in sideline BC, and define B' and C' cyclically. Then the lines AA', BB', CC' concur in X(79). (Eric Danneels, Hyacinthos 7892, 9/13/03)

A'B'C' is also the reflection triangle of X(1). The lines AA', BB', CC' concur in X(79). (Randy Hutson, July 20, 2016)

Let P and Q be the intersections of line BC and circle {X(1),2r}. Let X = X(1). Let A' be the circumcenter of triangle PQX, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(79). (Compare to X(592), where the circle is the 1st Lemoine circle) (Randy Hutson, July 20, 2016)

Let A₂₅B₂₅C₂₅ be Gemini triangle 25. Let A' be the perspector of conic {{A,B,C,B₂₅,C₂₅}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(79). (Randy Hutson, January 15, 2019)

X(79) lies on these lines:
1,30   2,3647   8,758   9,46   12,484   21,36   33,1063   34,1061   35,226   57,90   65,80   104,946   314,320   388,1000

X(79) = reflection of X(191) in X(442)
X(79) = isogonal conjugate of X(35)
X(79) = isotomic conjugate of X(319)
X(79) = cevapoint of X(481) and X(482)
X(79) = crosssum of X(55) and X(1030)
X(79) = anticomplement of X(3647)
X(79) = X(2914) of Fuhrmann triangle
X(79) = antigonal image of X(3065)
X(79) = trilinear pole of line X(650)X(4802)
X(79) = perspector of ABC and extraversion triangle of X(80)
X(79) = Hofstadter -1/2 point
X(79) = trilinear pole of line X(650)X(4802)
X(79) = trilinear product of vertices of reflection triangle of X(1)
X(79) = X(6152)-of-excentral-triangle

X(80)  REFLECTION OF INCENTER IN FEUERBACH POINT

Let A' be the reflection in BC of the A-vertex of the excentral triangle, and define B' and C' cyclically. The circumcircles of A'BC, B'CA, and C'AB concur in X(80). Also, the lines AA', BB', CC' concur in X(80). (Randy Hutson, December 10, 2016)

Let A'B'C' be the Fuhrmann triangle. Let La be the line through A parallel to B'C', and define Lb and Lc cyclically. Let A" = Lb∩Lc, B" = Lc∩La, C" = La∩Lb. The lines A'A", B'B", C'C" concur in X(80). (Randy Hutson, December 10, 2016)

Let A'B'C' be the orthic triangle. Let La be the antiorthic axis of AB'C', and define Lb and Lc cyclically. Let A" = Lb∩Lc. B" = Lc∩La, C" = La∩Lb. The triangle A"B"C" is inversely similar to ABC, with similitude center X(9). The incenter of triangle A"B"C" is X(80). Also, the lines AA", BB", CC" concur in X(80).(Randy Hutson, December 10, 2016)

Let A'B'C' be the excentral triangle. Let A" be the isogonal conjugate, wrt A'BC, of A. Define B", C" cyclically. (A" is also the reflection of A' in BC, and cyclically for B" and C"). The lines AA", BB", CC" concur in X(80). (Randy Hutson, January 29, 2018)

Let A'B'C' be the excentral triangle. Let Oa be the A'-Johnson circle of triangle A'BC, and define Ob and Oc cyclically. X(80) is the radical center of Oa, Ob, Oc. (Randy Hutson, June 27, 2018)

X(80) lies on these lines:
1,5   2,214   7,150   8,149   9,528   10,21   30,484   33,1061   34,1063   36,104   40,90   46,84   65,79   313,314   497,1000   499,944   516,655   519,908   943,950

X(80) = midpoint of X(8) and X(149)
X(80) = reflection of X(i) in X(j) for these (i,j): (1,11), (100,10), (1317,1387)
X(80) = isogonal conjugate of X(36)
X(80) = isotomic conjugate of X(320)
X(80) = inverse-in-incircle of X(1387)
X(80) = inverse-in-Fuhrmann-circle of X(1)
X(80) = anticomplement of X(214)
X(80) = cevapoint of X(10) and X(519)
X(80) = X(i)-cross conjugate of X(j) for these (i,j): (44,2), (517,1)
X(80) = X(8)-beth conjugate of X(100)
X(80) = antigonal image of X(1)
X(80) = syngonal conjugate of X(10)
X(80) = X(186)-of-Fuhrmann triangle
X(80) = orthology center of ABC and Fuhrmann triangle
X(80) = reflection of any vertex of ABC in the corresponding side of the Fuhrmann triangle
X(80) = perspector of ABC and reflection of Fuhrmann triangle in X(11)
X(80) = trilinear pole of line X(37)X(650)
X(80) = inverse-in-circumconic-centered-at-X(1)-of-X(1807)
X(80) = perspector of ABC and extraversion triangle of X(79)
X(80) = X(1986)-of-excentral triangle
X(80) = perspector of ABC and mid-triangle of 1st and 2nd extouch triangles
X(80) = inner-Garcia-to-outer-Garcia similarity image of X(1)
X(80) = X(100)-of-outer-Garcia-triangle

X(81)  CEVAPOINT OF INCENTER AND SYMMEDIAN POINT

X(81) = (r² + 2rR + s²)*X(1) - 3rR*X(2) - 2r²*X(3)    (Peter Moses, April 2, 2013)

Let A'B'C' be the cevian triangle of X(1). Let A" be the symmedian point of triangle AB'C', and define B" and C" cyclically. Then the lines AA", BB", CC" concur in X(81). (Eric Danneels, Hyacinthos 7892, 9/13/03)

Let A'B'C' be the incentral triangle. Let L_A be the reflection of B'C' in the internal angle bisector of vertex angle A, and define L_B and L_C cyclically. Let A'' = L_B∩L_C, B'' = L_C∩L_A, C'' = L_A∩L_B. The lines AA'', BB'', CC'' concur in X(81). (Randy Hutson, 9/23/2011)

Let H* be the Stammler hyperbola. Let A'B'C' be the tangential triangle and A"B"C" be the excentral triangle. Let A* be the intersection of the tangents to H* at A' and A", and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(81). (Randy Hutson, February 10, 2016)

Let A'B'C' be the 2nd circumperp triangle. Let A" be the trilinear pole of line B'C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(81). (Randy Hutson, February 10, 2016)

Let A'B'C' be the anticomplement of the Feuerbach triangle. Let A" be BB'∩CC', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(81). (Randy Hutson, February 10, 2016)

Let Ba, Ca be the intersections of lines CA, AB, resp., and the antiparallel to BC through X(1). Define Cb, Ab, Ac, Bc cyclically. Triangles ABaCa, AbBCb, AcBcC are similar to each other and inversely similar to ABC. Let Sa be the similitude center of triangles AbBCb and AcBcC. Define Sb and Sc cyclically. The lines ASa, BSb, CSc concur in X(81). (Randy Hutson, February 10, 2016)

Let A₁₀B₁₀C₁₀ be Gemini triangle 10. Let A' be the perspector of conic {{A,B,C,B₁₀,C₁₀}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(81). (Randy Hutson, January 15, 2019)

Let A₁₁B₁₁C₁₁ be Gemini triangle 11. Let A' be the perspector of conic {{A,B,C,B₁₁,C₁₁}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(81). (Randy Hutson, January 15, 2019)

If you have The Geometer's Sketchpad, you can view X(81).

X(81) lies on these lines:
1,21   2,6   7,27   8,1010   19,969   28,60   29,189   32,980   42,100   43,750   55,1002   56,959   57,77   65,961   88,662   99,739   105,110   145,1043   226,651   239,274   314,321   377,387   386,404   411,581   593,757   715,932   859,957   941,967   982,985   1019,1022   1051,1054   1098,1104

X(81) = isogonal conjugate of X(37)
X(81) = isotomic conjugate of X(321)
X(81) = anticomplement of X(1211)
X(81) = X(i)-Ceva conjugate of X(j) for these (i,j): (7,229), (86,21), (286,28)
X(81) = cevapoint of X(i) and X(j) for these (i,j): (1,6), (57,222), (58,284)
X(81) = X(i)-cross conjugate of X(j) for these (i,j): (1,86), (3,272), (6,58), (57,27), (284,21)
X(81) = crosspoint of X(274) and X(286)
X(81) = crosssum of X(i) and X(j) for these (i,j): (1,846), (6,1030), (42,1334), (213,228)
X(81) = crossdifference of every pair of points on line X(512)X(661)
X(81) = X(i)-beth conjugate of X(j) for these (i,j): (333,333), (643,81), (645,81), (648,81), (662,81), (931,81)
X(81) = trilinear product of PU(31)
X(81) = intersection of tangents at X(1) and X(6) to the Stammler hyperbola
X(81) = crosspoint of X(1) and X(6) wrt both the excentral and tangential triangles
X(81) = trilinear pole of line X(36)X(238) (the polar of X(1) wrt the circumcircle)
X(81) = {X(1),X(31)}-harmonic conjugate of X(1621)
X(81) = X(6)-isoconjugate of X(10)
X(81) = X(92)-isoconjugate of X(228)
X(81) = perspector of ABC and cross-triangle of Gemini triangles 1 and 2
X(81) = barycentric product of vertices of Gemini triangle 1
X(81) = barycentric product of vertices of Gemini triangle 2
X(81) = barycentric product of vertices of Gemini triangle 3
X(81) = barycentric product of vertices of Gemini triangle 4
X(81) = perspector of Gemini triangles 2 and 7
X(81) = perspector of ABC and cross-triangle of ABC and Gemini triangle 1
X(81) = perspector of ABC and cross-triangle of ABC and Gemini triangle 2
X(81) = perspector of Gemini triangle 24 and cross-triangle of ABC and Gemini triangle 24
X(81) = perspector of Gemini triangle 28 and cross-triangle of ABC and Gemini triangle 28

X(82) = ISOGONAL CONJUGATE OF X(38)

Barycentrics  a/(b² + c²) : b/(c² + a²) : c/(a² + b²)

Let A'B'C' be the circummedial triangle. Let A" be the trilinear product B'*C', and define B", C" cyclically. A", B", C" are collinear on line X(798)X(812) (the trilinear polar of X(3112)). The lines AA", BB", CC" concur in X(82). (Randy Hutson, October 15, 2018)

X(82) lies on these lines: 1,560   10,83   31,75   37,251   58,596   689,715   759,827

X(82) = isogonal conjugate of X(38)
X(82) = isotomic conjugate of X(1930)
X(82) = cevapoint of X(1) and X(31)
X(82) = trilinear pole of line X(661)X(830)
X(82) = crossdifference of every pair of points on line X(2084)X(2530)
X(82) = perspector of ABC and extraversion triangle of X(82) (which is also the anticevian triangle of X(82))
X(82) = crosspoint of X(1) and X(31) wrt the excentral triangle

X(83)  CEVAPOINT OF CENTROID AND SYMMEDIAN POINT

Barycentrics  1/(b² + c²) : 1/(c² + a²) : 1/(a² + b²)

Let K denote the symmedian point, X(6). Let A'B'C' be the cevian triangle of K. Let K_A be K of the triangle AB'C'; let K_B be K of A'BC' and let K_C be K of A'B'C. The lines AK_A, BK_B, CK_C concur in X(83). (Randy Hutson, 9/23/2011)

Let A'B'C' be the 1st Brocard triangle. Let A" be the reflection of A' in BC, and define B" and C" cyclically. AA", BB", CC" concur in X(83). (Randy Hutson, December 26, 2015)

Let Ba, Ca be the intersections of lines CA, AB, resp., and the antiparallel to BC through X(2). Define Cb, Ab, Ac, Bc cyclically. Triangles ABaCa, AbBCb, AcBcC are similar to each other and inversely similar to ABC. Let Sa be the similitude center of triangles AbBCb and AcBcC. Define Sb and Sc cyclically. The lines ASa, BSb, CSc concur in X(83). (Randy Hutson, December 26, 2015)

Let (Oa) be the circle whose diameter is the orthogonal projections of PU(1) on line BC. Define (Ob) and (Oc) cyclically. X(83) is the radical center of circles (Oa), (Ob), (Oc). (Randy Hutson, December 26, 2015)

Let A'B'C' be the circummedial triangle. Let A" be the trilinear pole of line B'C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(83). (Randy Hutson, December 26, 2015)

X(83) lies on these lines:
2,32   3,262   4,182   5,98   6,76   10,82   17,624   18,623   39,99   213,239   217,287   275,297   597,671   689,729

X(83) = isogonal conjugate of X(39)
X(83) = isotomic conjugate of X(141)
X(83) = complement of X(2896)
X(83) = cevapoint of X(2) and X(6)
X(83) = X(i)-cross conjugate of X(j) for these (i,j): (2,308), (6,251), (512,99)
X(83) = trilinear pole of line X(23)X(385) (line is the polar of X(2) wrt the circumcircle, and also the anticomplement of the de Longchamps line, and also the polar of X(5) wrt {circumcircle, nine-point circle}-inverter)
X(83) = crossdifference of every pair of points on line X(688)X(3005)
X(83) = pole wrt polar circle of trilinear polar of X(427)
X(83) = X(48)-isoconjugate (polar conjugate) of X(427)
X(83) = perspector of ABC and medial triangle of 1st Brocard triangle
X(83) = crosspoint of X(2) and X(6) wrt both the anticomplementary and tangential triangles
X(83) = trilinear product of vertices of circummedial triangle
X(83) = midpoint of PU(137)
X(83) = bicentric sum of PU(i) for these i: 137, 141
X(83) = homothetic center of 5th anti-Brocard triangle and medial triangle
X(83) = X(8290)-of-1st-Brocard-triangle
X(83) = perspector of ABC and 1st Brocard triangle of medial triangle
X(83) = perspector of ABC and 1st Brocard triangle of 5th anti-Brocard triangle
X(83) = homothetic center of ABC and cross-triangle of ABC and 5th anti-Brocard triangle
X(83) = Cundy-Parry Phi transform of X(262)
X(83) = Cundy-Parry Psi transform of X(182)

X(84) = ISOGONAL CONJUGATE OF X(40)

Let A',B',C' be the excenters. The perpendiculars from B' to AB and from C' to AC meet in a point A". Points B" and C" are determined cyclically. The hexyl triangle, A"B"C", is perspective to ABC, and X(84) is the perspector.

Let A'B'C' be the extouch triangle. Let A" be the orthocenter of AB'C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(84). (Randy Hutson, September 14, 2016)

Let A1B1C1 be the 1st Conway triangle. Let A' be the crosspoint of B1 and C1, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(84). (Randy Hutson, December 2, 2017)

X(84) lies on the Darboux cubic, the circumellipse with center X(9), and these lines: 1,221   3,9   4,57   7,946   8,20   21,285   33,603   36,90   46,80   58,990   64,3353   171,989   256,988   294,580   309,314   581,941   944,1000   2130,3345   3346,3472   3347,3355

X(84) = reflection of X(i) in X(j) for these (i,j): (40,1158), (1490,3)
X(84) = isogonal conjugate of X(40)
X(84) = isotomic conjugate of X(322)
X(84) = X(i)-Ceva conjugate of X(j) for these (i,j): (189,282), (280,1)
X(84) = X(i)-cross conjugate of X(j) for these (i,j): (19,57), (56,1)
X(84) = X(280)-aleph conjugate of X(84)
X(84) = X(i)-beth conjugate of X(j) for these (i,j): (271,3), (280,280), (285,84)
X(84) = X(68)-of-the-hexyl-triangle. X(84) = trilinear pole of line X(650)X(1459)
X(84) = perspector of ABC and the reflection in X(9) of the antipedal triangle of X(9)
X(84) = Danneels point of X(110)
X(84) = trilinear product of vertices of hexyl triangle (i.e., the extraversions of X(40))
X(84) = hexyl-isotomic conjugate of X(12717)
X(84) = perspector of ABC and cross-triangle of extouch and Hutson-extouch triangles
X(84) = Cundy-Parry Phi transform of X(9)
X(84) = Cundy-Parry Psi transform of X(57)

X(85)  ISOTOMIC CONJUGATE OF X(9)

Let A₃₈B₃₈C₃₈ be Gemini triangle 38. Let A' be the perspector of conic {{A,B,C,B₃₈,C₃₈}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(85). (Randy Hutson, January 15, 2019)

X(85) lies on these lines:
1,664   2,241   7,8   12,120   29,34   56,870   57,274   76,226   92,331   109,767   150,355   264,309

X(85) = isogonal conjugate of X(41)
X(85) = isotomic conjugate of X(9)
X(85) = complement of X(3177)
X(85) = anticomplement of X(1212)
X(85) = X(274)-Ceva conjugate of X(348)
X(85) = cevapoint of X(i) and X(j) for these (i,j): (1,169), (2,7), (57,77), (92,342)
X(85) = X(i)-cross conjugate of X(j) for these (i,j): (2,75), (57,273), (92,309), (142,2), (226,7)
X(85) = X(i)-beth conjugate of X(j) for these (i,j): (76,76), (85,279), (99,1), (274,85), (668,85), (789,85), (799,85), (811,85)
X(85) = trilinear pole of line X(522)X(693) (the isotomic conjugate of the circumconic centered at X(1), conic {A,B,C,X(100),X(664),X(1120),X(1320)}; also the polar of X(33) wrt polar circle)
X(85) = pole wrt polar circle of trilinear polar of X(33) (line X(657)X(4041))
X(85) = polar conjugate of X(33)
X(85) = {X(7),X(8)}-harmonic conjugate of X(6604)
X(85) = trilinear square of X(508)
X(85) = trilinear product of vertices of Gemini triangle 9
X(85) = trilinear product of vertices of Gemini triangle 10

X(86)   CEVAPOINT OF INCENTER AND CENTROID

X(86) = 2(r² + 2rR + s²)*X(1) + 3(r² + s²)*X(2) - 4r²*X(3)    (Peter Moses, April 2, 2013)

Let A'B'C' be the anticomplement of the Feuerbach triangle. Let La be the tangent to the circumcircle at A', and define Lb and Lc cyclically. Let A" be the point where La is tangent to the Steiner circumellipse, and define B" and C" cyclically. Let A* = BB"∩CC", and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(86). (Randy Hutson, December 10, 2016)

Let A1B1C1 be the 1st Conway triangle. Let A' be the trilinear pole of line B1C1, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(86). (Randy Hutson, December 10, 2016)

Let A5B5C5 be the 5th Conway triangle. Let A' be the trilinear pole of line B5C5, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(86). (Randy Hutson, December 10, 2016)

Let A₃B₃C₃ and A₄B₄C₄ be Gemini triangles 3 and 4, resp. Let L_A be the tangent at A to conic {{A,B₃,C₃,B₄,C₄}}, and define L_B, L_C cyclically. Let A' = L_B∩L_C, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(86). (Randy Hutson, January 15, 2019)

Let A₂₁B₂₁C₂₁ be Gemini triangle 21. Let A' be the perspector of conic {{A,B,C,B₂₁,C₂₁}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(86). (Randy Hutson, January 15, 2019)

X(86) lies on these lines:
1,75   2,6   7,21   10,319   29,34   37,190   58,238   60,272   99,106   110,675   142,284   239,1100   269,1088   283,307   310,350   741,789   870,871

X(86) = isogonal conjugate of X(42)
X(86) = isotomic conjugate of X(10)
X(86) = complement of X(1654)
X(86) = anticomplement of X(1213)
X(86) = X(274)-Ceva conjugate of X(333)
X(86) = cevapoint of X(i) and X(j) for these (i,j): (1,2), (7,77), (21,81)
X(86) = crosssum of X(1) and X(1045)
X(86) = crossdifference of every pair of points on line X(512)X(798)
X(86) = X(i)-cross conjugate of X(j) for these (i,j): (1,81), (2,274), (7,286), (21,333), (58,27), (513,190)
X(86) = X(i)-beth conjugate of X(j) for these (i,j): (86,1014), (99,86), (261,86), (314,314), (645,86), (811,86)
X(86) = X(2)-Ceva conjugate of X(6626)
X(86) = intersection of tangents at X(1) and X(2) to the bianticevian conic of X(1) and X(2); see X(99)
X(86) = crosspoint of X(1) and X(2) wrt both the excentral and anticomplementary triangles
X(86) = trilinear pole of line X(239)X(514) (Lemoine axis of excentral triangle)
X(86) = pole wrt polar circle of trilinear polar of X(1826)
X(86) = X(48)-isoconjugate (polar conjugate)-of-X(1826)
X(86) = perspector of Gemini triangle 1 and cross-triangle of ABC and Gemini triangle 1
X(86) = perspector of ABC and cross-triangle of ABC and Gemini triangle 23
X(86) = perspector of ABC and cross-triangle of ABC and Gemini triangle 24
X(86) = perspector of ABC and cross-triangle of Gemini triangles 23 and 24
X(86) = perspector of ABC and Gemini triangle 25
X(86) = barycentric product of vertices of Gemini triangle 23
X(86) = barycentric product of vertices of Gemini triangle 24
X(86) = barycentric product of vertices of Gemini triangle 25

X(87)   X(2)-CROSS CONJUGATE OF X(1)

X(87) lies on these lines: 1,192   6,43   9,292   10,979   34,242   56,238   58,978   106,932

X(87) = isogonal conjugate of X(43)
X(87) = isotomic conjugate of X(6376)
X(87) = perspector of ABC and extraversion triangle of X(43)
X(87) = trilinear product of extraversions of X(43)
X(87) = cevapoint of X(2) and X(330)
X(87) = X(2)-cross conjugate of X(1)
X(87) = X(932)-beth conjugate of X(87)

X(88) = ISOGONAL CONJUGATE OF X(44)

Let A₉B₉C₉ be Gemini triangle 9. Let A' be the perspector of conic {{A,B,C,B₉,C₉}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(88). (Randy Hutson, January 15, 2019)

X(88) lies on these lines: 1,100   2,45   6,89   28,162   44,679   57,651   81,662   105,901   274,799   278,653   279,658   291,660

X(88) = isogonal conjugate of X(44)
X(88) = isotomic conjugate of X(4358)
X(88) = complement of X(30578)
X(88) = cevapoint of X(i) and X(j) for these (i,j): (1,44), (6,36)
X(88) = X(i)-cross conjugate of X(j) for these (i,j): (44,1), (517,7)
X(88) = X(i)-aleph conjugate of X(j) for these (i,j): (88,1), (679,88), (903,63), (1022,1052)
X(88) = X(333)-beth conjugate of X(190)

X(89) = ISOGONAL CONJUGATE OF X(45)

Let A₉B₉C₉ be the Gemini triangle 9. Let L_A be the line through A₉ parallel to BC, and define L_B and L_C cyclically. Let A'₉ = L_B∩L_C, and define B'₉, C'₉ cyclically. Triangle A'₉B'₉C'₉ is homothetic to ABC at X(89). (Randy Hutson, November 30, 2018)

X(89) lies on these lines: 1,902   2,44   6,88   649,1022

X(89) = isogonal conjugate of X(45)
X(89) = isotomic conjugate of X(4671)

X(90) = X(3)-CROSS CONJUGATE OF X(1)

X(90) = (r + R)²*X(1) - 6rR*X(2) - 2r(r - R)*X(3)    (Peter Moses, April 2, 2013)

X(90) lies on these lines: 1,155   4,46   9,35   21,224   33,47   36,84   40,80   57,79

X(90) = isogonal conjugate of X(46)
X(90) = isotomic conjugate of X(20930)
X(90) = X(3)-cross conjugate of X(1)
X(90) = perspector of ABC and extraversion triangle of X(46)
X(90) = trilinear product of the extraversions of X(46), which is also the cross-triangle of the orthic and excentral triangles
X(90) = trilinear product of PU(125)
X(90) = Cundy-Parry Phi transform of X(90)
X(90) = Cundy-Parry Psi transform of X(46)

X(91) = ISOGONAL CONJUGATE OF X(47)

X(91) lies on these lines: 19,920   31,1087   37,498   47,92   63,921   65,68   225,847   255,1109   759,925

X(91) = isogonal conjugate of X(47)
X(91) = X(48)-cross conjugate of X(92)
X(91) = trilinear product of X(485) and X(486)
X(91) = polar conjugate of X(1748)
X(91) = X(92)-isoconjugate of X(563)
X(91) = trilinear pole of line X(661)X(2618)
X(91) = perspector of ABC and extraversion triangle of X(91) (which is also the anticevian triangle of X(91))
X(91) = trilinear product of vertices of outer and inner Vecten triangles

X(92)  CEVAPOINT OF INCENTER AND CLAWSON POINT

Let L_A be the line through X(4) parallel to the internal bisector of angle A, and let
A' = BC∩L_A. Define B' and C' cyclically.

Alexei Myakishev, "The M-Configuration of a Triangle," Forum Geometricorum 3 (2003) 135-144,

proves that the lines AA', BB', CC' concur in X(92). He notes that another construction follows from Proposition 2 of the article: let A₁ be the midpoint of the arc BC of the circumcircle that passes through A, and let A₂ be the point, other than A, in which the A-altitude meets the circumcircle. Let A" = A₁A₂∩BC. Define B" and C" cyclically. Then the lines AA", BB", CC" concur in X(92).

Suppose that T = A'B'C' is a central triangle. Let A'' be the pole with respect to the polar circle of the line B'C', and define B'' and C'' cyclically. The appearance of T in the following list means that the lines AA'', BB'', CC'' concur in X(92): Feurerbach, incentral, excentral, extangents, Apollonius, mixtilinear excentral. (Randy Hutson, December 26, 2015)

X(92) lies on these lines:
1,29   2,273   4,8   7,189   10,1838   19,27   25,242   28,2975   31,162   33,1897   34,1220   38,240   40,412   47,91   48,2167   53,4415   55,243   56,1940   57,653   81,2995   85,331   100,917   108,1311   171,1430   226,342   239,607   255,1087   257,297   264,306   304,561   345,3262   388,1118   394,1943   406,1068   427,2969   429,3948   429,3948   459,1446   497,1857   518,1859   608,894   651,2988   823,2349   938,3176   942,1148   960,1882   984,1860   994,1845   1146,1952   1172,2997   1211,1865   1309,2717   1435,3306   1585,1659   1621,4183   1707,1733   1726,1746   1731,1751   1785,4656   1842,1891   1844,3874   1870,5136   1947,2994   1954,1955   1956,2632   1973,3112   2064,3596      2331,5256   2399,4391   3064,4468   4198,4968

X(92) = isogonal conjugate of X(48)
X(92) = isotomic conjugate of X(63)
X(92) = anticomplement of X(1214)
X(92) = anticomplementary conjugate of X(2897)
X(92) = X(i)-Ceva conjugate of X(j) for these (i,j): (85, 342), (264,318), (286,4), (331,273)
X(92) = cevapoint of X(i) and X(j) for these (i,j): (1,19), (4,281), (47,48), (196,278)
X(92) = X(i)-cross conjugate of X(j) for these (i,j): (1,75), (4,273), (19,158), (48,91), (226,2), (281,318)
X(92) = crosspoint of X(i) and X(j) for these (i,j): (85,309), (264,331)
X(92) = crossdifference of every pair of points on line X(810)X(822)
X(92) = X(275)-aleph conjugate of X(47)
X(92) = X(i)-beth conjugate of X(j) for these (i,j): (92,278), (312,329), (648,57)
X(92) = {X(19),X(63)}-harmonic conjugate of X(1748)
X(92) = barycentric product of PU(20)
X(92) = trilinear product of PU(i) for these i: 21, 45
X(92) = bicentric sum of PU(130)
X(92) = midpoint of PU(130)
X(92) = trilinear product X(2)*X(4)
X(92) = trilinear pole of line X(240)X(522) (polar of X(1) wrt polar circle)
X(92) = pole of antiorthic axis wrt polar circle
X(92) = X(6)-isoconjugate of X(3)
X(92) = X(48)-isoconjugate (polar conjugate) of X(1)
X(92) = X(91)-isoconjugate of X(563)
X(92) = inverse-in-Fuhrmann-circle of X(5174)
X(92) = perspector of ABC and extraversion triangle of X(92) (which is also the anticevian triangle of X(92))
X(92) = crosspoint of X(1) and X(19) wrt excentral triangle
X(92) = crosspoint of X(47) and X(48) wrt excentral triangle
X(92) = perspector of ABC and cross-triangle of Gemini triangles 37 and 38
X(92) = perspector of ABC and cross-triangle of ABC and Gemini triangle 37
X(92) = perspector of ABC and cross-triangle of ABC and Gemini triangle 38
X(92) = barycentric product of vertices of Gemini triangle 37
X(92) = barycentric product of vertices of Gemini triangle 38

X(93) = ISOGONAL CONJUGATE OF X(49)

Let O_AO_BO_C be the Kosnita triangle. Let A' be the pole wrt polar circle of line O_BO_C, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(93). (Randy Hutson, June 7, 2019)

X(93) lies on these lines: 4,562   49,94   186,252

X(93) = isogonal conjugate of X(49)
X(93) = X(50)-cross conjugate of X(94)
X(93) = polar conjugate of X(1994)
X(93) = X(2964)-isoconjugate of X(3)

X(94) = ISOGONAL CONJUGATE OF X(50)

Let A1B1C1 and A2B2C2 be the 1st and 2nd Ehrmann circumscribing triangles. Let A' be the trilinear pole of line A1A2, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(94). Let A1'B1'C1' and A2'B2'C2' be the 1st and 2nd Ehrmann inscribed triangles. Then X(94) is the radical center of nine-point circles of AA1'A2', BB1'B2', CC1'C2'. (Randy Hutson, June 27, 2018)

X(94) lies on the Kiepert hyperbola and these lines: 2,300   4,143   23,98   49,93   96,925   275,324

X(94) = isogonal conjugate of X(50)
X(94) = isotomic conjugate of X(323)
X(94) = cevapoint of X(49) and X(50)
X(94) = X(i)-cross conjugate of X(j) for these (i,j): (30,264), (50,93), (265,328)
X(94) = X(300)-Hirst inverse of X(301)
X(94) = trilinear pole of PU(5) (line X(5)X(523))
X(94) = pole wrt polar circle of trilinear polar of X(186)
X(94) = X(48)-isoconjugate (polar conjugate) of X(186)
X(94) = barycentric product X(476)*X(850)
X(94) = trilinear pole of PU(173)
X(94) = trilinear product X(74)*X(107) (circumcircle-X(4) antipodes)

X(95)  CEVAPOINT OF CENTROID AND CIRCUMCENTER

Let A'B'C' be the symmedial triangle. Let La be the reflection of line B'C' in line BC, and define Lb and Lc cyclically. Let A" = Lb∩Lc, and define B" and C" cyclically. The lines AA", BB", CC" concur in X(95). (Randy Hutson, August 19, 2015)

Let A' be the intersection, other than A, of the circumcircle and the branch of the Lucas cubic that contains A, and define B' and C' cyclically. The triangle A'B'C' is here introduced as the Lucas triangle (not to be confused with the Lucas central triangle). The vertices A', B', C' lie on the rectangular hyperbola {{X(2),X(20),X(54),X(69),X(110),X(2574),X(2575),X(2979)}}. (See https://bernard-gibert.pagesperso-orange.fr/Exemples/k007.html.) Also, X(95) is the trilinear product of the vertices of the Lucas triangle. (Randy Hutson, August 19, 2015)

X(95) lies on these lines:
2,97   3,264   54,69   76,96   99,311   140,340   141,287   160,327   183,305   216,648   307,320

X(95) = isogonal conjugate of X(51)
X(95) = isotomic conjugate of X(5)
X(95) = anticomplement of X(233)
X(95) = X(276)-Ceva conjugate of X(275)
X(95) = cevapoint of X(i) and X(j) for these (i,j): (2,3), (6,160), (54,97)
X(95) = X(i)-cross conjugate of X(j) for these (i,j): (2,276), (3,97), (54,275), (140,2), (340,1494)
X(95) = intersection of tangents at X(2) and X(3) to bianticevian conic of X(2) and X(3)
X(95) = crosspoint of X(2) and X(3) wrt both the anticomplementary triangle and anticevian triangle of X(3)
X(95) = trilinear pole of line X(323)X(401) (polar of X(53) wrt polar circle, and polar of X(69) wrt de Longchamps circle)
X(95) = pole wrt polar circle of trilinear polar of X(53)
X(95) = X(48)-isoconjugate (polar conjugate) of X(53)
X(95) = X(92)-isoconjugate of X(217)

X(96) = ISOGONAL CONJUGATE OF X(52)

Let A'B'C' be the reflection triangle. Let B_A and C_A be the orthogonal projections of B' and C' on line BC, resp. Let (O_A) be the circle with segment B_AC_A as diameter. Define (O_B) and (O_C) cyclically. X(96) is the radical center of circles (O_A), (O_B), (O_C). (Randy Hutson, June 7, 2019)

X(96) lies on these lines: 2,54   4,231   24,847   76,95   94,925

X(96) = isogonal conjugate of X(52)
X(96) = cevapoint of X(3) and X(68)
X(96) = X(3)-cross conjugate of X(54)
X(96) = polar conjugate of X(467)
X(96) = Cundy-Parry Phi transform of X(5392)
X(96) = Cundy-Parry Psi transform of X(571)

X(97) = ISOGONAL CONJUGATE OF X(53)

X(97) lies on these lines: 2,95   3,54   110,418   216,288   276,401

X(97) = isogonal conjugate of X(53)
X(97) = isotomic conjugate of X(324)
X(97) = X(95)-Ceva conjugate of X(54)
X(97) = X(3)-cross conjugate of X(95)
X(97) = cevapoint of X(3) and X(577)
X(97) = X(51)-isoconjugate of X(92)
X(97) = Cundy-Parry Phi transform of X(7592)

Centers 74, 98 - 112,
and others lie on the circumcircle. Mappings Λ and Ψ derived from such a point P for application to points X, are defined here:

Λ(P,X) = isogonal conjugate of the point where line PX meets the line at infinity.

Let Y = Λ(P,X), let Q = isogonal conjugate of P, and let Y and Z be the points where line YQ meets the circumcircle;
then Ψ(P,X) = Z.

X(98) = TARRY POINT

Let A' be the apex of the isosceles triangle BA'C constructed inward on BC such that ∠A'BC = ∠A'CB = ω. Define B' nad C' cyclically. Let Ha be the orthocenter of BA'C, and define Hb and Hc cyclically. The lines AHa, BHb, CHc concur in X(98). (Randy Hutson, July 20, 2016)

Let A' be the apex of the isosceles triangle BA'C constructed outward on BC such that ∠A'BC = ∠A'CB = ω/2. Define B' and C' cyclically. Let Oa be the circumcenter of BA'C, and define Ob and Oc cyclically. The lines AOa, BOb, COc concur in X(98). (Randy Hutson, July 20, 2016)

If you have The Geometer's Sketchpad, you can view X(98).

J. W. Clawson, "Points on the circumcircle," American Mathematical Monthly 32 (1925) 169-174.

Let A'B'C' be the 1st Brocard triangle. Let A" be the reflection of A in B'C', and define B" and C" cyclically. X(98) is the radical center of the circumcircles of AA'A", BB'B", CC'C". (Randy Hutson, November 18, 2015)

Let A'B'C' be the orthic triangle. Let La be the Lemoine axis of triangle AB'C', and define Lb and Lc cyclically. Let A" = Lb∩Lc, and define B" and C" cyclically. Triangle A"B"C" is inversely similar to ABC, with similicenter X(2). The lines AA", BB", CC" concur in X(98), which is X(4)-of-A"B"C". (Randy Hutson, July 31 2018)

Let P be a point on the Steiner circumellipse. Let A' be the orthocenter of BCP, and define B' and C' cyclically. Let Q be the centroid of A'B'C'. The locus of Q as P varies is an ellipse similar and orthogonal to the Steiner circumellipse, and also centered at X(2). When P = X(671), Q = X(98). See also X(6054) and X(24808). (Randy Hutson, October 15, 2018)

X(98) lies on these lines:
2,110   3,76   4,32   5,83   6,262   10,101   13,1080   14,383   20,148   22,925   23,94   25,107   30,671   100,228   109,171   111,1637   186,935   275,427   376,543   381,598   385,511   468,685   523,842   620,631   804,878

X(98) is the {X(2),X(147)}-harmonic conjugate of X(114). For a list of harmonic conjugates, click Tables at the top of this page.

X(98) = midpoint between X(20) and X(148)
X(98) = reflection of X(i) in X(j) for these (i,j): (4,115), (99,3), (147,114), (1513,230)
X(98) = isogonal conjugate of X(511)
X(98) = isotomic conjugate of X(325)
X(98) = complement of X(147)
X(98) = anticomplement of X(114)
X(98) = X(290)-Ceva conjugate of X(287)
X(98) = cevapoint of X(i) and X(j) for these (i,j): (2,385), (6,237)
X(98) = X(i)-cross conjugate of X(j) for these (i,j): (230,2), (237,6), (248,287), (446,511)
X(98) = crosssum of X(385) and X(401)
X(98) = X(2)-Hirst inverse of X(287)
X(98) = perspector of ABC and triangle formed by Lemoine axis (or PU(1) or PU(2)) reflected in sides of ABC
X(98) = Λ(X(4), X(69)) (the line that is the isotomic conjugate of the Jerabek hyperbola)
X(98) = trilinear pole of line X(6)X(523) (polar of X(297) wrt polar circle, and the radical axis of circles with segments X(13)X(16) and X(14)X(15) as diameters)
X(98) = pole wrt polar circle of trilinear polar of X(297) (line X(114)X(132))
X(98) = pole wrt {circumcircle, nine-point circle}-inverter of line X(115)X(125)
X(98) = X(48)-isoconjugate (polar conjugate) of X(297)
X(98) = X(6)-isoconjugate of X(1959)
X(98) = inverse-in-polar-circle of X(132)
X(98) = inverse-in-{circumcircle, nine-point circle}-inverter of X(125)
X(98) = inverse-in-circle-{{X(1687),X(1688),PU(1),PU(2)}} of X(2715)
X(98) = Ψ(X(6), X(523))
X(98) = Ψ(X(190), X(71))
X(98) = Kiepert-hyperbola antipode of X(4)
X(98) = reflection of X(842) in the Euler line
X(98) = reflection of X(2698) in the Brocard axis
X(98) = reflection of X(2699) in line X(1)X(3)
X(98) = X(129)-of-excentral-triangle
X(98) = X(130)-of-hexyl-triangle
X(98) = X(3)-of-1st-anti-Brocard-triangle
X(98) = perspector of ABC and 1st Neuberg triangle
X(98) = trilinear product of vertices of 1st Neuberg triangle
X(98) = orthocenter of X(13)X(14)X(2394)
X(98) = 2nd-Parry-to-ABC similarity image of X(2)
X(98) = trilinear product of PU(88)
X(98) = X(2456) of 6th Brocard triangle
X(98) = midpoint of PU(135)
X(98) = bicentric sum of PU(135)
X(98) = perspector of ABC and circumsymmedial triangle of Artzt triangle
X(98) = McCay-to-Artzt similarity image of X(381)
X(98) = circumcircle-antipode of X(99)
X(98) = the point of intersection, other than A, B, and C, of the circumcircle and Kiepert hyperbola
X(98) = Ψ(X(101), X(100)
X(98) = Λ(X(3), X(6))
X(98) = homothetic center of 5th anti-Brocard triangle and Euler triangle
X(98) = Thomson-isogonal conjugate of X(512)
X(98) = Lucas-isogonal conjugate of X(512)
X(98) = X(1380)-of-circummedial-triangle
X(98) = X(6233)-of-circumsymmedial-triangle
X(98) = Cundy-Parry Phi transform of X(76)
X(98) = Cundy-Parry Psi transform of X(32)
X(98) = perspector of ABC and vertex-triangle of reflection triangles of PU(1)
X(98) = X(2)-of-2nd-anti-Parry-triangle
X(98) = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {{A,B,C,X(3),X(25),X(32)}} (isogonal conjugate of line X(4)X(69))

X(99) = STEINER POINT

Let L_A be the reflection of the line X(3)X(6) in line BC, and define L_B and L_C cyclically. Let A' = L_B∩L_C, B' = L_C∩L_A, C' = L_A∩L_B. The lines AA', BB', CC' concur in X(99). (Randy Hutson, 9/23/2011)

X(99) is the only point on the circumcircle whose isotomic conjugate lies on the line at infinity. (Randy Hutson, 9/23/2011)

X(99) is the center of the bianticevian conic of X(1) and X(2), which is the rectangular hyperbola H that passes through these points: X(1), X(2), X(20), X(63), X(147), X(194), X(487), X(488), X(616), X(617), X(627), X(628), X(1764), X(2896), the excenters, the vertices of the anticomplementary triangle, and the extraversions of X(63). Also, H is the anticomplementary conjugate of line X(4)X(69), the anticomplementary isotomic conjugate of line X(2)X(6), the excentral isogonal conjugate of line X(40)X(511), and the excentral isotomic conjugate of line X(1045)X(2951); also, H is tangent to line X(1)X(75) at X(1), to line X(2)X(6) at X(2), and meets the line at infinity (and the Kiepert hyperbola, other than at X(2)) at X(3413) and X(3414). (Randy Hutson, December 26, 2015)

Let A'B'C' be the anticomplement of the Feuerbach triangle. Let A"B"C" be the tangential triangle of A'B'C'. Let A* be the cevapoint of B" and C", and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(99). (Randy Hutson, February 10, 2016)

Let La, Lb, Lc be the lines through A, B, C, resp. parallel to the Brocard axis. Let Ma, Mb, Mc be the reflections of BC, CA, AB in La, Lb, Lc, resp. Let A' = Mb∩Mc, and define B' and C' cyclically. Triangle A'B'C' is inversely similar to ABC with similarity ratio 3. Let A"B"C" be the reflection of A'B'C' in the Brocard axis. The triangle A"B"C" is homothetic to ABC, with center of homothety X(115) and centroid X(99). See Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, February 10, 2016)

Let A', B' and C' be the intersections of the de Longchamps line and lines BC, CA, AB, resp. The circumcircles of AB'C', BC'A', CA'B' concur in X(99). (Randy Hutson, February 10, 2016)

Let A', B', C' be the intersections of the Brocard axis and lines BC, CA, AB, resp. Let Oa, Ob, Oc be the circumcenters of AB'C', BC'A', CA'B', resp. The lines AOa, BOb, COc concur in X(99). (Randy Hutson, February 10, 2016)

Let A'B'C' be the 1st Brocard triangle. Let La be the line through A parallel to B'C', and define Lb and Lc cyclically. La, Lb, Lc concur in X(99). (Randy Hutson, February 10, 2016)

Let A' be the trilinear pole of the perpendicular bisector of BC, and define B' and C' cyclically. A'B'C' is also the anticomplement of the anticomplement of the midheight triangle. X(99) is the trilinear product A'*B'*C'. (Randy Hutson, January 29, 2018)

Let A'B'C' be the 1st Brocard triangle. Let A" be the reflection of A in line B'C', and define B", C" cyclically. Let A"' be the reflection of A' in line BC, and define B"', C"' cyclically. Let A* = B"B"'∩\C"C"', and define B*, C* cyclically. Triangle A*B*C* is homothetic to A'B'C' at X(99). (Randy Hutson, June 27, 2018)

See a construction: Ercole Suppa, Hyacinthos 29064.

If you have The Geometer's Sketchpad, you can view the following dynamic sketches:
X(99) and Steiner Circum-ellipse (showing X(99) and an area-ratio property)

For more about the Steiner circumellipse, visit MathWorld.

X(99) lies on these lines:
1,741   2,111   3,76   4,114   6,729   13,303   14,302   20,147   21,105   22,305   30,316   31,715   32,194   36,350   38,745   39,83   58,727   69,74   75,261   81,739   86,106   95,311   100,668   101,190   102,332   103,1043   104,314   108,811   109,643   110,690   112,648   141,755   163,825   187,385   249,525   264,378   286,915   298,531   299,530   310,675   476,850   512,805   523,691   524,843   666,919   669,886   670,804   692,785   695,711   813,1016   889,898   935,3267

X(99) is the {X(39),X(384)}-harmonic conjugate of X(83). For a list of other harmonic conjugates of X(99), click Tables at the top of this page.

X(99) = midpoint of X(i) and X(j) for these (i,j): (20,147), (616,617)
X(99) = reflection of X(i) in X(j) for these (i,j): (4,114), (13,619), (14,618), (98,3), (115,620), (148,115), (316,325), (385,187), (671,2)
X(99) = isogonal conjugate of X(512)
X(99) = isotomic conjugate of X(523)
X(99) = complement of X(148)
X(99) = anticomplement of X(115)
X(99) = cevapoint of X(i) and X(j) for these (i,j): (2,523), (3,525), (39,512), (100,190)
X(99) = X(1019)-cross conjugate of X(1509)
X(99) = crossdifference of every pair of points on line X(351)X(865)
X(99) = X(i)-cross conjugate of X(j) for these (i,j): (3,249), (22,250), (512,83), (523,2), (525,76)
X(99) = X(21)-beth conjugate of X(741)
X(99) = X(6)-of-1st-anti-Brocard-triangle
X(99) = X(381)-of-anti-McCay-triangle
X(99) = circumcircle-antipode of X(98)
X(99) = point of intersection, other than A, B, and C, of the circumcircle and Steiner ellipse
X(99) = Ψ(X(i), X(j) for these (i,j): (1,75), (2,39), (3,69), (4,69), (37,2), (51,5), (351,690)
X(99) = point of intersection, other than A, B, C, of the circumcircle and hyperbola {{A,B,C,PU(1)}}
X(99) = point of intersection, other than A, B, C, of the circumcircle and hyperbola {{A,B,C,PU(37)}}
X(99) = trilinear product of PU(90)
X(99) = similitude center of (equilateral) antipedal triangles of X(13) and X(14)
X(99) = Steiner-circumellipse-antipode of X(671)
X(99) = projection from Steiner inellipse to Steiner circumellipse of X(2482)
X(99) = trilinear pole of line X(2)X(6)
X(99) = pole wrt polar circle of trilinear polar of X(2501) (line X(115)X(2971))
X(99) = X(48)-isoconjugate (polar conjugate) of X(2501)
X(99) = X(6)-isoconjugate of X(661)
X(99) = X(1577)-isoconjugate of X(32)
X(99) = concurrence of reflections in sides of ABC of line X(4)X(69)
X(99) = Λ(X(1), X(512))
X(99) = isotomic conjugate wrt 1st Brocard triangle of X(76)
X(99) = perspector of ABC and the tangential triangle, wrt the anticomplementary triangle, of the bianticevian conic of X(1) and X(2)
X(99) = perspector of ABC and the tangential triangle, wrt the tangential triangle, of the Stammler hyperbola
X(99) = reflection of X(691) in the Euler line
X(99) = reflection of X(805) in the Brocard axis
X(99) = reflection of X(2703) in line X(1)X(3)
X(99) = reflection of X(316) in the de Longchamps line
X(99) = X(130)-of-excentral-triangle
X(99) = X(129)-of-hexyl-triangle
X(99) = inverse-in-polar-circle of X(5139)
X(99) = inverse-in-{circumcircle, nine-point circle}-inverter of X(126)
X(99) = inverse-in-2nd-Brocard-circle of X(76)
X(99) = trilinear product of vertices of circumcircle antipode of circumorthic triangle
X(99) = 1st-Parry-to-ABC similarity image of X(2)
X(99) = crossdifference of PU(105)
X(99) = X(1691) of 6th Brocard triangle
X(99) = eigencenter of circummedial triangle
X(99) = eigencenter of circumsymmedial triangle
X(99) = perspector of ABC and cross-triangle of circumcevian triangles of PU(1)
X(99) = X(98)-of-anti-Artzt-triangle
X(99) = X(2)-of-1st-anti-Parry-triangle
X(99) = Thomson-isogonal conjugate of X(511)
X(99) = Lucas-isogonal conjugate of X(511)
X(99) = X(1379)-of-circummedial-triangle
X(99) = X(6323)-of-circumsymmedial-triangle
X(99) = Kiepert image of X(2)
X(99) = Cundy-Parry Phi transform of X(14265)
X(99) = intersection of antipedal lines of X(1379) and X(1380)
X(99) = homothetic center of anticomplementary triangle and mid-triangle of antipedal triangles of X(13) and X(14)
X(99) = barycentric square root of X(4590)
X(99) = barycentric product of circumcircle intercepts of line X(2)X(39)
X(99) = perspector of ABC and vertex triangle of 1st and 2nd isodynamic-Dao triangles

X(100)  ANTICOMPLEMENT OF FEUERBACH POINT

X(100) = 2R*X(1) - 3R*X(2) + 2r*X(3)    (Peter Moses, April 2, 2013)

Let L_A be the reflection of the line X(1)X(3) in line BC, and define L_B and L_C cyclically. Let A' = L_B∩L_C, B' = L_C∩L_A, C' = L_A∩L_B. The lines AA', BB', CC' concur in X(100). (Randy Hutson, 9/23/2011)

Let Ia be the reflection of X(1) in the perpendicular bisector of BC, and define Ib and Ic cyclically; then X(100) = X(36)-of-IaIbIc. Also, let P be a point on line X(4)X(8) other than X(4). Let A' be the reflection of P in BC, and define B' and C' cyclically. The circumcircles of AB'C', BC'A', and CA'B' concur at X(100). (Randy Hutson, 9/5/2015)

Let IaIbIc be the excentral triangle. The Euler lines of triangles BCIa, CAIb, ABIc concur in X(100). (Randy Hutson, June 27, 2018)

X(100) lies on these lines:
1,88   2,11   3,8   4,119   6,739   7,1004   9,1005   10,21   20,153   22,197   25,1862   231,43   32,713   36,519   37,111   40,78   42,81   46,224   56,145   59,521   63,103   72,74   75,675   76,767   92,917   98,228   99,668   101,644   107,823   108,653   109,651   110,643   112,162   144,480   190,659   198,346   213,729   238,899   281,1013   329,972   442,943   484,758   513,765   516,908   517,953   518,840   522,655   560,697   594,1030   645,931   649,660   650,919   658,664   667,898   693,927   731,869   733,893   753,984   756,846   789,874   976,986   2859,3267

X(100) is the {X(10),X(35)}-harmonic conjugate of X(21). For a list of other harmonic conjugates of X(100), click Tables at the top of this page.

X(100) = midpoint of X(20) and X(153)
X(100) = reflection of X(i) in X(j) for these (i,j): (1,214), (4,119), (8,1145), (80,10), (104,3), (145,1317), (149,11), (962,1537), (1156,9), (1320,1), (1484,140)
X(100) = isogonal conjugate of X(513)
X(100) = isotomic conjugate of X(693)
X(100) = complement of X(149)
X(100) = anticomplement of X(11)
X(100) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,5375), (99,190)
X(100) = cevapoint of X(i) and X(j) for these (i,j): (1,513), (3,521), (10,522), (142,514), (442,523)
X(100) = X(i)-cross conjugate of X(j) for these (i,j): (3,59), (513,1), (521,8), (522,21)
X(100) = crosssum of X(i) and X(j) for these (i,j): (1,1054), (244,764), (512,661), (649,663)
X(100) = crossdifference of every pair of points on line X(244)X(665)
X(100) = circumcircle-antipode of X(104)
X(100) = Ψ(X(i),X(j)) for these (i,j): (1,2), 2,37), (3,63), (4,8), (6,1), (48,3), (68,72)
X(100) = X(1)-line conjugate of X(244)
X(100) = X(113)-of-the-hexyl-triangle.
X(100) = concurrence of reflections in sides of ABC of line X(4)X(8)
X(100) = perspector of Hutson-Moses hyperbola
X(100) = trilinear pole of line X(1)X(6) (and PU(28)) (van Aubel line of excentral triangle)
X(100) = trilinear product of PU(33)
X(100) = trilinear product of intercepts of circumcircle and Nagel line
X(100) = the point of intersection, other than A, B, and C, of the circumcircle and the circumellipse centered at X(1) (viz., {{A,B,C,X(100),X(664),X(1120),X(1320)}})
X(100) = the point of intersection, other than A, B, and C, of the circumcircle and the circumellipse centered at X(9) (viz., {{A,B,C,X(100),X(658),X(662),X(799),X(1821),X(2580),X(2581),PU(34)}})
X(100) = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {{A,B,C,PU(8)}}
X(100) = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {{A,B,C,PU(32)}}
X(100) = Collings transform of X(1)
X(100) = Collings transform of X(9)
X(100) = center of hyperbola passing through X(1), X(9), and the excenters
X(100) = X(125)-of-excentral-triangle
X(100) = trilinear pole wrt 1st circumperp triangle of line X(3)X(142)
X(100) = X(110)-of=1st-circumperp-triangle
X(100) = reflection of X(1290) in the Euler line
X(100) = reflection of X(2703) in the Brocard axis
X(100) = reflection of X(901) in line X(1)X(3)
X(100) = cevapoint of X(59) and inverse-in-circumcircle-of-X(59)
X(100) = X(i)-isoconjugate of X(j) for these (i,j): (6,514), (63,6591), (1333,1577)
X(100) = inverse-in-{circumcircle, nine-point circle}-inverter of X(120)
X(100) = exsimilicenter of circumcircle and AC-incircle
X(100) = X(i)-aleph conjugate of X(j) for these (i,j) (1,1052), (100,1), (190,63), (643,411), (666,673), (765,100), (1016,190)
X(100) = X(i)-beth conjugate of X(j) for these (i,j): (8,80), (21,106), (100,109), (333,673), (643,100), (765,100)
X(100) = the point of intersection, other than A, B, and C, of the circumcircle and ellipse {{A,B,C,PU(75)}}
X(100) = crossdifference of PU(27)
X(100) = Thomson-isogonal conjugate of X(517)
X(100) = Lucas-isogonal conjugate of X(517)
X(100) = homothetic center of 2nd Schiffler triangle and excenters-midpoints triangle
X(100) = Feuerbach image of X(2)
X(100) = Cundy-Parry Phi transform of X(14266)
X(100) = perspector of anti-Mandart-incircle and anticomplementary triangles
X(100) = intersection of antipedal lines of X(1381) and X(1382)
X(100) = eigencenter of Gemini triangle 2
X(100) = barycentric product of vertices of Gemini triangle 5
X(100) = barycentric product of vertices of Gemini triangle 6
X(100) = perspector of ABC and side-triangle of Gemini triangles 29 and 30
X(100) = homothetic center of 2nd Schiffler triangle and excenters-midpoints triangle
X(100) = barycentric product of vertices of Gemini triangle 29
X(100) = barycentric product of vertices of Gemini triangle 30
X(100) = intersection, other than A, B, C, of {ABC, Gemini 29}-circumconic and {ABC, Gemini 30}-circumconic
X(100) = barycentric product of circumcircle intercepts of line X(2)X(37)

X(101) = Ψ(INCENTER, SYMMEDIAN POINT)

Let L_A be the reflection of the line X(1)X(7) in line BC, and define L_B and L_C cyclically. Let A' = L_B∩L_C, B' = L_C∩L_A, C' = L_A∩L_B. The lines AA', BB', CC' concur in X(101). (Randy Hutson, 9/23/2011)

Let I_aI_bI_c be the excentral triangle. The Brocard axes of BCI_a, CAI_b, ABI_c concur in X(101). (Randy Hutson, February 10, 2016)

Let P be a point on line X(4)X(9) other than X(4). Let A' be the reflection of P in BC, and define B' and C' cyclically. The circumcircles of AA'B', BC'A', CA'B' concur in X(101). (Randy Hutson, February 10, 2016)

Let Q be a point on the Nagel line other than X(2). Let A'B'C' be the cevian triangle of Q. Let A" be the {B,C}-harmonic conjugate of A' (or equivalently, A" = BC∩B'C'), and define B" and C" cyclically. The circumcircles of AB"C", BC"A", CA"B" concur in X(101). (Randy Hutson, February 10, 2016)

Let A', B', C' be the intersections of the antiorthic axis and lines BC, CA, AB, resp. The circumcircles of AB'C', BC'A', CA'B' concur in X(101). (Randy Hutson, February 10, 2016)

X(101) is the perspector of the anticevian triangle of X(109) and the unary cofactor triangle of the intangents triangle. (Randy Hutson, January 29, 2018)

X(101) lies on these lines:
1,41   2,116   3,103   4,118   6,106   9,48   10,98   19,913   20,152   31,609   32,595   36,672   37,284   40,972   42,111   56,218   58,172   59,657   71,74   75,767   78,205   99,190   100,644   102,198   109,654   110,163   514,664   517,910   522,929   560,713   643,931   649,901   651,934   663,919   667,813   668,789   692,926   733,904   743,869   761,984   765,898

X(101) = midpoint of X(20) and X(152)
X(101) = reflection of X(i) in X(j) for these (i,j): (4,118), (103,3), (150,116)
X(101) = isogonal conjugate of X(514)
X(101) = isotomic conjugate of X(3261)
X(101) = complement of X(150)
X(101) = anticomplement of X(116)
X(101) = X(59)-Ceva conjugate of X(55)
X(101) = cevapoint of X(354) and X(513)
X(101) = X(i)-cross conjugate of X(j) for these (i,j): (55,59), (199,250)
X(101) = crosssum of X(i) and X(j) for these (i,j): (513,650), (523,661), (649,1459)
X(101) = crossdifference of every pair of points on line X(11)X(244)
X(101) = X(i)-aleph conjugate of X(j) for these (i,j): (100,165), (509,1052), (662,572), (664,169)
X(101) = X(i)-beth conjugate of X(j) for these (i,j): (21,105), (644,644)
X(101) = circumcircle-antipode of X(103)
X(101) = Ψ(X(i),X(j)) for these (i,j): (1,6), (2,1), (3,48), (4,9), (6,31), (7,2), (63,3), (69,63), (76,10)
X(101) = X(114)-of-the-hexyl-triangle
X(101) = trilinear product of PU(i) for these i: 26, 49
X(101) = barycentric product of PU(33)
X(101) = the point of intersection, other than A, B, C, of the circumcircle and hyperbola {{A,B,C,PU(9)}}
X(101) = the point of intersection, other than A, B, C, of conic {{A,B,C,X(1),PU(93)}}
X(101) = trilinear pole of line X(6)X(31) (the isogonal conjugate of the isotomic conjugate of the Nagel line)
X(101) = trilinear pole wrt 1st circumperp triangle of line X(9)X(165)
X(101) = X(99)-of -1st-circumperp-triangle
X(101) = crossdifference of PU(i) for these i: 121, 123
X(101) = concurrence of reflections of line X(4)X(9) in sides of ABC
X(101) = isogonal conjugate of isotomic conjugate of trilinear pole of Nagel line
X(101) = center of Kiepert hyperbola of excentral triangle (i.e. X(115) of excentral triangle)
X(101) = reflection of X(2690) in the Euler line
X(101) = reflection of X(2702) in the Brocard axis
X(101) = reflection of X(1308) in line X(1)X(3)
X(101) = reflection of X(5011) in antiorthic axis
X(101) = inverse-in-polar-circle of X(5190)
X(101) = inverse-in-{circumcircle, nine-point circle}-inverter of X(5513)
X(101) = X(6)-isoconjugate of X(693)
X(101) = X(92)-isoconjugate of X(1459)
X(101) = X(1577)-isoconjugate of X(58)
X(101) = eigencenter of 2nd circumperp triangle
X(101) = perspector of 3rd mixtilinear triangle and unary cofactor triangle of 5th mixtilinear triangle
X(101) = trilinear product of vertices of 1st circumperp triangle
X(101) = Thomson-isogonal conjugate of X(516)
X(101) = Lucas-isogonal conjugate of X(516)
X(101) = intersection of Lemoine axes of 1st & 2nd Montesdeoca bisector triangles
X(101) = focus of Yff parabola
X(101) = polar conjugate of isogonal conjugate of X(32656)
X(101) = trilinear product of circumcircle intercepts of line X(1)X(6)
X(101) = barycentric product of circumcircle intercepts of the Nagel line

X(102) = Λ(INCENTER, ORTHOCENTER)

X(102) lies on these lines:
1,108   2,117   3,109   4,124   19,282   29,107   40,78   73,947   77,934   99,332   101,198   103,928   110,283   112,284   226,1065   516,929

X(102) = midpoint of X(20) and X(153)
X(102) = reflection of X(i) in X(j) for these (i,j): (4,124), (109,3), (151,117)
X(102) = isogonal conjugate of X(515)
X(102) = complement of X(151)
X(102) = anticomplement of X(117)
X(102) = X(21)-beth conjugate of X(108)
X(102) = circumcircle-antipode of X(109)
X(102) = Λ(X(1), X(4))
X(102) = trilinear pole of line X(6)X(652)
X(102) = Ψ(X(6), X(652))
X(102) = trilinear pole wrt 2nd circumperp triangle of line X(971)X(1001)
X(102) = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {{A,B,C,X(4),X(58)}}
X(102) = reflection of X(2695) in the Euler line
X(102) = reflection of X(2708) in the Brocard axis
X(102) = reflection of X(2716) in line X(1)X(3)
X(102) = X(131)-of-excentral-triangle
X(102) = X(136)-of-hexyl-triangle
X(102) = X(925)-of-2nd-circumperp-triangle
X(102) = Thomson-isogonal conjugate of X(522)
X(102) = Lucas-isogonal conjugate of X(522)
X(102) = Cundy-Parry Phi transform of X(10571)
X(102) = Cundy-Parry Psi transform of X(10570)
X(102) = Ψ(X(19), X(650))

X(103) = ANTIPODE OF X(101)

Let A'B'C' be the excentral triangle. The Lemoine axes of triangles A'BC, B'CA, C'AB bound a triangle perspective to ABC at X(103). (Randy Hutson, June 27, 2018)

X(103) lies on these lines:
1,934   2,118   3,101   4,116   20,150   27,107   33,57   55,109   58,112   63,100   99,1043   102,928   295,813   376,544   515,929   516,927   572,825   672,919   910,971

X(103) = midpoint of X(20) and X(150)
X(103) = reflection of X(i) in X(j) for these (i,j): (4,116), (101,3), (152,118)
X(103) = isogonal conjugate of X(516)
X(103) = complement of X(152)
X(103) = anticomplement of X(118)
X(103) = X(21)-beth conjugate of X(934)

X(104) = ANTIPODE OF X(100)

Let L_A be the reflection of the line X(1)X(513) in line BC, and define L_B and L_C cyclically. Let A' = L_B∩L_C, B' = L_C∩L_A, C' = L_A∩L_B. The lines AA', BB', CC' concur in X(104). (Randy Hutson, 9/23/2011)

Let A', B', C' be the intersections of the antiorthic axis and lines BC, CA, AB, resp. Let Oa, Ob, Oc be the circumcenters of AB'C', BC'A', CA'B', resp. The lines AOa, BOb, COc conucr in X(104). (Randy Hutson, December 2, 2017)

Let A'B'C' be the excentral triangle (i.e. the antiorthic triangle). Let A"B"C" be the triangle bounded by the orthic axes of A'BC, B'CA, C'AB. Then A"B"C" is perspective to ABC at X(104); c.f. X(8068). (Randy Hutson, December 2, 2017)

Let A'B'C' be the excentral triangle. The de Longchamps lines of triangles A'BC, B'CA, C'AB bound a triangle perspective to ABC at X(104). (Randy Hutson, June 27, 2018)

Let A'B'C' be the excentral triangle. The Hatzipolakis axes of triangles A'BC, B'CA, C'AB bound a triangle perspective to ABC at X(104). (Randy Hutson, June 27, 2018)

X(104) lies on these lines:
1,109   2,119   3,8   4,11   7,934   9,48   20,149   21,110   28,107   36,80   55,1000   79,946   99,314   105,885   112,1108   256,1064   294,919   355,404   376,528   513,953   517,901   631,958

X(104) = midpoint of X(20) and X(149)
X(104) = reflection of X(i) in X(j) for these (i,j): (4,11), (100,3), (153,119), (1537,1387)
X(104) = isogonal conjugate of X(517)
X(104) = isotomic conjugate of X(3262)
X(104) = complement of X(153)
X(104) = anticomplement of X(119)
X(104) = cevapoint of X(i) and X(j) for these (i,j): (1,36), (44,55)
X(104) = X(21)-beth conjugate of X(109)
X(104) = circumcircle-antipode of X(100)
X(104) = point of intersection, other than A, B, and C, of the circumcircle and Feuerbach hyperbola
X(104) = Λ(X(1), X(3))
X(104) = Ψ(X(101), X(9))
X(104) = X(125)-of-the-hexyl-triangle
X(104) = X(113)-of-excentral-triangle
X(104) = X(110)-of-2nd-circumperp-triangle
X(104) = trilinear pole of line X(6)X(650)
X(104) = Ψ(X(6), X(650))
X(104) = Ψ(X(190), X(63))
X(104) = Feuerbach hyperbola antipode of X(4)
X(104) = trilinear pole wrt 2nd circumperp triangle of line X(3)X(142)
X(104) = reflection of X(2687) in the Euler line
X(104) = reflection of X(2699) in the Brocard axis
X(104) = reflection of X(953) in line X(1)X(3)
X(104) = crossdifference of every pair of points on line X(1769)X(3310)
X(104) = Thomson-isogonal conjugate of X(513)
X(104) = Lucas-isogonal conjugate of X(513)
X(104) = Cundy-Parry Phi transform of X(8)
X(104) = Cundy-Parry Psi transform of X(56)

X(105) = Λ(INCENTER, SYMMEDIAN POINT)

X(105) is the perspector of ABC and the (degenerate) side-triangle of the circumcevian triangles of X(3513) and X(3514). (Randy Hutson, June 7, 2019)

X(105) lies on these lines:
1,41   2,11   3,277   6,1002   21,99   25,108   28,112   31,57   56,279   81,110   88,901   104,885   106,1022   165,1054   238,291   330,932   513,840   644,1083   659,884   666,898   825,985   910,919   961,1104

X(105) = reflection of X(i) in X(j) for these (i,j): (644,1083), (1292,3)
X(105) = isogonal conjugate of X(518)
X(105) = anticomplement of X(120)
X(105) = cevapoint of X(1) and X(238)
X(105) = X(1)-Hirst inverse of X(294)
X(105) = X(i)-beth conjugate of X(j) for these (i,j): (21,101), (927,105)
X(105) = Λ(X(1), X(6))
X(105) = isotomic conjugate of X(3263)
X(105) = crossdifference of every pair of points on line X(665)X(1642)
X(105) = Ψ(X(i), X(j)) for these (i,j): (2,650), (6,513), (101,1), (190,9), (650,11)
X(105) = reflection of X(2752) in the Euler line
X(105) = reflection of X(2711) in the Brocard axis
X(105) = reflection of X(840) in line X(1)X(3)
X(105) = X(132)-of-excentral-triangle
X(105) = X(127)-of-hexyl-triangle
X(105) = X(6)-isoconjugate of X(3912)
X(105) = inverse-in-{circumcircle, nine-point circle}-inverter of X(11)
X(105) = trilinear pole of PU(i) for these i: 46, 54
X(105) = trilinear product of PU(96)
X(105) = bicentric sum of PU(142)
X(105) = Thomson-isogonal conjugate of X(3309)
X(105) = Lucas-isogonal conjugate of X(3309)
X(105) = Λ(X(1), X(6))
X(105) = Ψ(X(101), X(1))
X(105) = polar conjugate of isogonal conjugate of X(32658)

X(106) = Λ(INCENTER, CENTROID)

X(106) lies on these lines:
1,88   2,121   3,1293   6,101   34,108   36,901   56,109   58,110   86,99   87,932   105,1022   238,898   269,934   292,813   614,998   663,840   789,870   833,977   919,1055

X(106) = reflection of X(1293) in X(3)
X(106) = isogonal conjugate of X(519)
X(106) = isotomic conjugate of X(3264)
X(106) = anticomplement of X(121)
X(106) = X(36)-cross conjugate of X(58)
X(106) = X(i)-beth conjugate of X(j) for these (i,j): (21,100), (901,106)
X(106) = Λ(X(1), X(2))
X(106) = X(122)-of-hexyl-triangle
X(106) = trilinear pole of line X(6)X(649)
X(106) = Ψ(X(i), X(j)) for these (i,j): (6,649), (9,650), (101,6), (190,2)
X(106) = trilinear pole wrt 2nd circumperp triangle of line X(1)X(6)
X(106) = X(107) of 2nd circumperp triangle
X(106) = trilinear pole wrt circumsymmedial triangle of line X(6)X(31)
X(106) = reflection of X(2758) in the Euler line
X(106) = reflection of X(2712) in the Brocard axis
X(106) = reflection of X(2718) in line X(1)X(3)
X(106) = X(6)-isoconjugate of X(4358)
X(106) = X(133)-of-excentral triangle
X(106) = barycentric product of PU(50)
X(106) = trilinear product of PU(98)
X(106) = eigencenter of 1st circumperp triangle
X(106) = Thomson-isogonal conjugate of X(3667)
X(106) = Lucas-isogonal conjugate of X(3667)
X(106) = polar conjugate of isogonal conjugate of X(32659)

X(107) = Ψ(SYMMEDIAN POINT, ORTHOCENTER)

X(107) = center of the bianticevian conic of X(1) and X(4), the rectangular hyperbola passing through X(1), X(4), X(19), and the vertices of their anticevian triangles. This hyperbola is the excentral isogonal conjugate of line X(40)X(2939), the anticomplementary conjugate of line X(20)X(1330), and the anticomplementary isotomic conjugate of line X(1654)X(3164). (Randy Hutson, April 9, 2016)

X(107) lies on these lines:
2,122   3,1294   4,74   19,2249   20,3184   21,1295   23,2697   24,1093   25,98   27,103   28,104   29,102   51,275   100,823   109,162   110,648   111,393   158,759   186,477   250,687   450,511   468,842   741,1096 20,3184   21,1295   23,2697  

X(107) = reflection of X(i) in X(j) for these (i,j): (4,133), (1294,3)
X(107) = isogonal conjugate of X(520)
X(107) = isotomic conjugate of X(3265)
X(107) = anticomplement of X(122)
X(107) = cevapoint of X(4) and X(523)
X(107) = cevapoint of X(24007) and X(24008) (the Kiepert hyperbola intercepts of the orthic axis)
X(107) = X(i)-cross conjugate of X(j) for these (i,j): (24,250), (108,162), (523,4)
X(107) = trilinear pole of line X(4)X(6)
X(107) = Ψ(X(i),X(j)) for these (i,j): (1,29), (3,2), (6,4), (4,51), (54,4), (64,4), (65,4), (67,4), (69,4)
X(107) = intersection of reflections in sides of ABC of line X(4)X(51)
X(107) = reflection of X(1304) in the Euler line
X(107) = reflection of X(2713) in the Brocard axis
X(107) = reflection of X(2719) in line X(1)X(3)
X(107) = inverse-in-polar-circle of X(125)
X(107) = inverse-in-{circumcircle, nine-point circle}-inverter of X(132)
X(107) = pole wrt polar circle of trilinear polar of X(525) (line X(122)X(125))
X(107) = X(48)-isoconjugate (polar conjugate) of X(525)
X(107) = X(1577)-isoconjugate of X(577)
X(107) = crossdifference of every pair of points on line X(1636)X(2972)
X(107) = X(134)-of-excentral-triangle
X(107) = circumcircle intercept, other than A, B, C, of conic {{A,B,C,PU(157)}}
X(107) = Thomson-isogonal conjugate of X(6000)
X(107) = Lucas-isogonal conjugate of X(6000)
X(107) = barycentric product of Steiner circumellipse intercepts of van Aubel line

X(108) = Ψ(CIRCUMCENTER, INCENTER)

X(108) lies on these lines:
1,102   2,123   3,1295   4,11   7,1013   12,451   24,915   25,105   28,225   33,57   34,106   40,207   55,196   65,74   99,811   100,653   109,1020   110,162   204,223   273,675   318,404   331,767   388,406   429,961   608,739   648,931

X(108) = reflection of X(1295) in X(3)
X(108) = isogonal conjugate of X(521)
X(108) = anticomplement of X(123)
X(108) = X(162)-Ceva conjugate of X(109)
X(108) = cevapoint of X(i) and X(j) for these (i,j): (56,513), (429,523)
X(108) = X(513)-cross conjugate of X(4)
X(108) = crosspoint of X(107) and X(162)
X(108) = crosssum of X(520) and X(656)
X(108) = X(i)-beth conjugate of X(j) for these (i,j): (21,102), (162,108)
X(108) = point of intersection, other than A, B, and C, of the circumcircle and hyperbola {{A,B,C,PU(18)}}
X(108) = trilinear pole of line X(6)X(19) (the polar of X(4391) wrt polar circle)
X(108) = pole wrt polar circle of trilinear polar of X(4391) (line X(11)X(123))
X(108) = X(48)-isoconjugate (polar conjugate) of X(4391)
X(108) = X(1577)-isoconjugate of X(2193)
X(108) = concurrence of the reflections of line X(4)X(65) in the sidelines of ABC
X(108) = Ψ(X(i),Xj)) for these (i,j): (1,4), (3,1), (4,65), (6,19), (7,4), (8,4), (9,4), (29,1), (69,7), (80,4)
X(108) = reflection of X(2766) in the Euler line
X(108) = reflection of X(2714) in the Brocard axis
X(108) = reflection of X(2720) in line X(1)X(3)
X(108) = inverse-in-polar-circle of X(11)
X(108) = X(135)-of-excentral-triangle
X(108) = barycentric product of PU(76)
X(108) = trilinear product of PU(100)
X(108) = Thomson-isogonal conjugate of X(6001)
X(108) = Lucas-isogonal conjugate of X(6001)
X(108) = trilinear product of circumcircle intercepts of line X(1)X(4)

X(109) = Ψ(INCENTER, CIRCUMCENTER)

If the line X(1)X(4) is reflected in every side of triangle ABC, then the reflections concur in X(109). (Randy Hutson, 9/23/2011)

Let P be a point on line X(1)X(4) other than X(4). Let A' be the reflection of P in BC, and define B' and C' cyclically. The circumcircles of CA'B', BC'A', and CA'B' concur in X(109). (Randy Hutson, December 26, 2011)

Let Q be a point on line X(2)X(7) other than X(2). Let A'B'C' be the cevian triangle of Q. Let A" be the {B,C}-harmonic conjugate of A' (or equivalently, A" = BC∩B'C'), and define B" and C" cyclically. The circumcircles of AB"C", BC"A", CA"B" concur in X(109). (Randy Hutson, December 26, 2011)

Let A', B', C' be the Fuhrmann triangle. The circumcircles of AB'C', BC'A', CA'B' concur in X(109). (Randy Hutson, December 26, 2011)

Let A', B', C' be the intersections of the Gergonne line and lines BC, CA, AB, respectively. The circumcircles of AB'C', BC'A', CA'B' concur in X(109). (Randy Hutson, December 26, 2011)

Let P and Q be the points where the line tangent to the incircle at X(11) intersects the circumcircle. Let L(P) be the line through P, other than PQ, that is tangent to the incircle; let L(Q) be the line through Q, other than PQ, that is tangent to the incircle. Then X(109) = L(P)∩L(Q). (Piotr Ambroszczyk, December 21, 2016)

X(109) lies on these lines:
1,104   2,124   3,102   4,117   7,675   20,151   31,57   34,46   35,73   36,953   40,255   55,103   56,106   58,65   59,901   85,767   98,171   99,643   100,651   101,654   107,162   108,1020   112,163   165,212   191,201   278,917   284,296   478,573   579,608   604,739   649,919   658,927   662,931   840,902

X(109) = midpoint of X(20) and X(151)
X(109) = reflection of X(i) in X(j) for these (i,j): (4,117), (102,3)
X(109) = isogonal conjugate of X(522)
X(109) = anticomplement of X(124)
X(109) = X(i)-Ceva conjugate of X(j) for these (i,j): (59,56), (162,108)
X(109) = cevapoint of X(65) and X(513)
X(109) = X(i)-cross conjugate of X(j) for these (i,j): (56,59), (513,58)
X(109) = crosspoint of X(110) and X(162)
X(109) = crosssum of X(i) and X(j) for these (i,j): (523,656), (652,663)
X(109) = crossdifference of every pair of points on line X(11)X(1146)
X(109) = X(i)-aleph conjugate of X(j) for these (i,j): (100,1079), (162,580), (651,223)
X(109) = X(i)-beth conjugate of X(j) for these (i,j): (21,104), (59,109), (100,100), (110,109), (765,109), (901,109)
X(109) = circumcircle-antipode of X(102)
X(109) = trilinear product X(1381)*X(1382)
X(109) = isogonal conjugate of isotomic conjugate of trilinear pole of line X(2)X(7)
X(109) = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {A,B,C,PU(19)}
X(109) = trilinear pole of line X(6)X(41)
X(109) = trilinear pole wrt 1st circumperp triangle of line X(971)X(1158)
X(109) = X(925)-of-1st-circumperp-triangle
X(109) = X(136)-of-excentral-triangle
X(109) = X(131)-of-hexyl-triangle
X(109) = Ψ(X(i),X(j)) for these (i,j): (1,3), (2,7), (3,73), (4,1), (6,41), (7,20), (21,2), (69,73), (77,3)
X(109) = reflection of X(2689) in the Euler line
X(109) = reflection of X(2701) in the Brocard axis
X(109) = reflection of X(2222) in line X(1)X(3)
X(109) = X(6)-isoconjugate of X(4391)
X(109) = X(92)-isoconjugate of X(652)
X(109) = X(1577)-isoconjugate of X(284)
X(109) = barycentric product of PU(57)
X(109) = trilinear product of PU(102)
X(109) = Thomson isogonal conjugate of X(515)
X(109) = Lucas isogonal conjugate of X(515)
X(109) = isotomic conjugate of polar conjugate of X(32674)
X(109) = polar conjugate of isogonal conjugate of X(32660)
X(109) = intersection of antipedal lines of circumcircle intercepts of line X(3)X(10)
X(109) = barycentric product of circumcircle intercepts of line X(2)X(7)

X(110) = FOCUS OF KIEPERT PARABOLA

X(110) is the center of the Stammler hyperbola, SH, which is the rectangular hyperbola that passes through X(1), X(3), X(6), X(155), X(159), X(195), X(399), X(1498), X(2916), X(2917), X(2918), X(2929), X(2930), X(2931), X(2935), X(2948), X(3511), the excenters, and the vertices of the tangential triangle. SH is the bianticevian conic of X(1) and X(6) and the antipedal-anticevian conic of X(3). SH is the tangential isogonal conjugate of the Euler line, the tangential isotomic conjugate of the van Aubel line, the excentral isogonal conjugate of line X(30)X(40), and the excentral isotomic conjugate of line X(191)X(2938). SH is tangent to the Euler line at X(3) and meets the line at infinity (and the Jerabek hyperbola, other than at X(3) and X(6)) at X(2574) and X(2575). SH is the locus of a point P for which the P-Brocard triangle is perspective to ABC. (Randy Hutson, 9/23/2011, 1/29/2015)

J. W. Clawson, "Points on the circumcircle," American Mathematical Monthly 32 (1925) 169-174.

Roland H. Eddy and R. Fritsch, "The conics of Ludwig Kiepert: a comprehensive lesson in the geometry of the triangle," Mathematics Magazine 67 (1994) 188-205.

Benedetto Scimemi, "Paper-folding and Euler's Theorem Revisited," Forum Geometricorum.

Scimemi proves that if the Euler line is reflected in every side of triangle ABC, then the three reflections concur in X(110).

Seven constructions from Randy Hutson, January 29, 2015:

(1) Let P be a point, other than X(4), on Euler line. Let A' be the reflection of P in BC, and define B' and C' cyclically. The circumcircles of AA'B', BC'A', and CA'B' concur in X(110).

(2) Let Q be a point on line X(2)X(6) other than X(2). Let A'B'C' be the cevian triangle of Q. Let A" be the {B,C}-harmonic conjugate of A' (or equivalently, A" = BC∩B'C'), and define B", C" cyclically. The circumcircles of AB"C", BC"A", CA"B" concur in X(110).

(3) Let La, Lb, Lc be the lines through A, B, C, resp. parallel to the Euler line. Let Ma, Mb, Mc be the reflections of BC, CA, AB in La, Lb, Lc, resp. Let A' = Mb∩Mc, and define B', C' cyclically. Triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. Let A"B"C" be the reflection of A'B'C' in the Euler line. The triangle A"B"C" is homothetic to ABC, with center of homothety X(125) and centroid X(110). (See Hyacinthos #16741/16782, Sep 2008.)

(4) Let A', B', C' be the intersections of the Lemoine axis and lines BC, CA, AB, resp. The circumcircles of AB'C', BC'A', CA'B' concur in X(110).

(5) Let A', B', C' be the intersections of line X(36)X(238) and lines BC, CA, AB, resp. The circumcircles of AB'C', BC'A', CA'B' concur in X(110).

(6) Let A', B', C' be the intersections of the Euler line and lines BC, CA, AB, resp. Let Oa, Ob, Oc be the circumcenters of AB'C', BC'A', CA'B', resp. The lines AOa, BOb, COc concur in X(110).

(7) Let Na be the reflection of X(5) in the perpendicular bisector of BC, and define Nb, Nc cyclically. X(110) = X(2070) of NaNbNc.

Let A2B2C2 and A3B3C3 be the 2nd and 3rd Parry triangles. Let A' be the barycentric product A2*A3, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(110). (Randy Hutson, February 10, 2016)

Let A'B'C' be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung). A'B'C' is homothetic to the anticomplementary triangle at X(110). The nine-point circle of A'B'C' touches the circumcircle of ABC at X(110). Also, X(110) = X(125)-of-A'B'C'. (Randy Hutson, Novermber 2, 2016)

X(110): Let A' be the trilinear pole of the perpendicular bisector of BC, and define B', C' cyclically. A'B'C' is homothetic to the circumanticevian triangle of X(4) at X(110). X(110) is also the barycentric product A'*B'*C'. (Randy Hutson, January 29, 2018)

X(110) lies on the circumcircle, the Parry circle, and these lines:
1,60   2,98   3,74   4,113   5,49   6,111   11,215   20,146   21,104   22,154   23,323   24,155   27,917   28,915   30,477   31,593   32,729   39,755   58,106   65,229   67,141   69,206   81,105   86,675   97,418   99,690   100,643   101,163   102,283   107,648   108,162   143,195   187,352   190,835   249,512   250,520   251,694   274,767   324,436   351,526   353,574   373,575   376,541   476,523   525,935   560,715   595,849   668,839   669,805   670,689   681,823   685,850   789,799   859,953   2868,3266

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

X(110) = midpoint of X(i) and X(j) for these (i,j): (3,399), (20,146), (23,323), (1495,3292)
X(110) = reflection of X(i) in X(j) for these (i,j): (3,1511), (4,113), (23,1495), (67,141), (74,3), (265,5), (382,1539), (895,6), (1177,206)
X(110) = circumcircle-antipode of X(74)
X(110) = isogonal conjugate of X(523)
X(110) = isotomic conjugate of X(850)
X(110) = isogonal conjugate of the isotomic conjugate of X(99)
X(110) = inverse of X(2) in the Brocard circle
X(110) = complement of X(3448)
X(110) = anticomplement of X(125)
X(110) = X(i)-Ceva conjugate of X(j) for these (i,j): (249,6), (250,3)
X(110) = cevapoint of X(i) and X(j) for these (i,j): (3,520), (5,523), (6,512), (141,525) X(110) = crosssum of X(i) and X(j) for these (i,j): (2,148), (512,647), (520,647)
X(110) = crossdifference of every pair of points on line X(115)X(125)
X(110) = X(i)-Hirst inverse of X(j) for these (i,j): (1,245), (2,125), (3,246), (4,247)
X(110) = X(i)-beth conjugate of X(j) for these (i,j): (21,759), (643,643)
X(110) = X(23)-of-1st-Brocard triangle
X(110) = X(111)-of-circumsymmedial-triangle
X(110) = X(323)-of-orthocentroidal-triangle
X(110) = X(137)-of-excentral-triangle
X(110) = X(128)-of-hexyl-triangle
X(110) = trilinear pole of the Brocard axis
X(110) = trilinear pole of PU(29) (see ETC->Tables->Bicentric Pairs)
X(110) = perspector of ABC and vertex-triangle of anticevian triangles of X(3) and X(6)
X(110) = Johnson-circumconic antipode of X(265)
X(110) = MacBeath-circumconic antipode of X(895)
X(110) = perspector of conic {A,B,C,PU(2)}
X(110) = intersection of trilinear polars of P(2) and U(2)
X(110) = intersection of tangents to Steiner circumellipse at X(99) and X(648)
X(110) = crosspoint of X(99) and X(648)
X(110) = reflection of X(476) in the Euler line
X(110) = reflection of X(691) in the Brocard axis
X(110) = reflection of X(23) in the Lemoine axis
X(110) = reflection of X(1290) in line X(1)X(3)
X(110) = reflection of X(111) in line X(3)X(351)
X(110) = inverse-in-polar-circle of X(136)
X(110) = inverse-in-{circumcircle, nine-point circle}-inverter of X(114)
X(110) = inverse-in-Moses-radical-circle of X(2715)
X(110) = inverse-in-O(15,16) of X(843), where O(15,16) is the circle having segment X(15)X(16) as diameter
X(110) = X(i)-isoconjugate of X(j) for these (i,j): (6,1577), (92,647), (1577,6)
X(110) = perspector of circumorthic triangle and Johnson triangle (the reflection triangles of X(4) and X(3), resp.)
X(110) = trilinear product of vertices of circumtangential triangle
X(110) = {X(3),X(156)}-harmonic conjugate of X(1614)
X(110) = orthocentroidal-to-ABC similarity image of X(2)
X(110) = 4th-Brocard-to-circumsymmedial similarity image of X(2)
X(110) = isogonal conjugate of isotomic conjugate of trilinear pole of line X(2)X(6)
X(110) = the point of intersection, other than A, B, C, of the circumcircle and Johnson circumconic
X(110) = the point of intersection, other than A, B, C, of the circumcircle and MacBeath circumconic
X(110) = the point of intersection, other than A, B, C, of the circumcircle and circumconic {{A,B,C,PU(5)}}
X(110) = Collings transform of X(5)
X(110) = Collings transform of X(6)
X(110) = intersection of tangents at X(61) and X(62) to the Napoleon-Feuerbach cubic K005
X(110) = SR(PU(4))
X(110) = insimilicenter of nine-point circle and sine-triple-angle circle
X(110) = insimilicenter of circumcircle and nine-point circle of tangential triangle; the exsimilicenter is X(1614)
X(110) = X(7972)-of-Trinh-triangle
X(110) = Ψ(X(i), X(j)) for these (i,j): (1,21), (2,6), (3,49), (4,2), (5,51), (6,3), (19,1), (53,5), (54,3), (64,3), (66,3), (67,3), (68,3), (69,3), (73,3), (74,3), (75,1), (76,2), (115,125), (190,99)
X(110) = X(110)-of-1st-Parry-triangle
X(110) = X(74)-of-2nd-Parry-triangle
X(110) = center of similitude of ABC and 1st Parry triangle
X(110) = inverse-in-Parry-isodynamic-circle of X(111); see X(2)
X(110) = barycentric product of PU(i) for these i: 78, 145
X(110) = perspector of unary cofactor triangles of outer and inner Napoleon triangles
X(110) = X(6792)-of-4th-anti-Brocard-triangle
X(110) = X(111)-of-anti-Artzt-triangle
X(110) = perspector of circumcevian triangle of X(36) and cross-triangle of ABC and 2nd circumperp triangle
X(110) = perspector of circumcevian triangle of X(187) and cross-triangle of ABC and circumsymmedial triangle
X(110) = Kiepert image of X(3)
X(110) = Jerabek image of X(2)
X(110) = Cundy-Parry Phi transform of X(14264)
X(110) = endo-homothetic center of X(2)-altimedial and X(2)-anti-altimedial triangles
X(110) = endo-homothetic center of X(20)-altimedial and X(3)-anti-altimedial triangles
X(110) = Thomson isogonal conjugate of X(30)
X(110) = Lucas isogonal conjugate of X(30)
X(110) = center of the perspeconic of these triangles: inner and outer Napoleon
X(110) = intersection of antipedal lines of X(1113) and X(1114)
X(110) = X(104)-of-circumorthic-triangle if ABC is acute
X(110) = perspector, wrt circumorthic triangle, of polar circle
X(110) = trilinear product of circumcircle intercepts of line X(1)X(21)
X(110) = barycentric product of circumcircle intercepts of line X(2)X(6)
X(110) = barycentric product X(6)*X(99)
X(110) = barycentric product of Steiner circumellipse intercepts of Brocard axis
X(110) = Feuerbach point of the tangential triangle if ABC is acute; otherwise, a vertex of the Feuerbach triangle of the tangential triangle

X(111) = PARRY POINT

Let L be a line tangent to the Brocard circle. Let P be the trilinear pole of L, and let P' be the isogonal conjugate of P. As L varies, P' traces a parabola with focus at X(111). The parabola meets the line at infinity at X(524). Also, X(111) is the QA-P4 center (Isogonal Center) of quadrangle X(13)X(14)X(15)X(16) (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/25-qa-p4.html) Also, let A' = BC∩X(115)X(125), and define B' and C' cyclically. The circumcircles of AB'C', BC'A', CA'B' concur in X(111). (Randy Hutson, October 13, 2015)

Let A"B"C" be the 2nd Ehrmann triangle. Let Pa be the pole of line B"C" wrt the A-Ehrmann circle, and define Pb and Pc cyclically. Let Pa' be the pole of line BC wrt the A-Ehrmann circle, and define Pb' and Pc' cyclically. The lines APaPa', BPbPb', CPcPc' concur in X(111). Also, let A* be the trilinear pole of line B"C", and define B*, C* cyclically. The lines AA*, BB*, CC* concur in X(111). (Randy Hutson, November 18, 2015)

Let A1B1C1 and A3B3C3 be the 1st and 3rd Parry triangles. Let A' be the barycentric product A1*A3, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(111). (Randy Hutson, February 10, 2016)

Let A'B'C' and A"B"C" be the 4th Brocard and 4th anti-Brocard triangles, resp. Let A* be the barycentric product A'*A'', and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(111). (Randy Hutson, December 2, 2017)

X(111) lies on the Parry circle and these lines:
2,99   3,1296   6,110   23,187   25,112   37,100   42,101   98,1637   107,393   182,353   230,476   251,827   308,689   352,511   385,892   468,935   512,843   647,842   694,805   931,941

X(111) = reflection of X(1296) in X(3)
X(111) = isogonal conjugate of X(524)
X(111) = isotomic conjugate of X(3266)
X(111) = inverse-in-Brocard-circle of X(353)
X(111) = anticomplement of X(126)
X(111) = cevapoint of X(6) and X(187)
X(111) = X(i)-cross conjugate of X(j) for these (i,j): (23,251), (187,6), (351,110)
X(111) = crossdifference of every pair of points on line X(351)X(690)

X(112) = Ψ(ORTHOCENTER, SYMMEDIAN POINT)

If the line X(4)X(6) is reflected in every side of triangle ABC, then the reflections concur in X(112). (Randy Hutson, 9/23/2011)

Let P be a point on the van Aubel line other than X(4). Let A' be the reflection of P in BC, and define B' and C' cyclically. The circumcircles of AA'B', BC'A', and CA'B' concur at X(112). (Randy Hutson, December 26, 2015)

Let Q be a point on the Euler line other than X(2). Let A'B'C' be the cevian triangle of Q. Let A" be the {B,C}-harmonic conjugate of A' (or equivalently, A" = BC∩B'C'), and define B" and C" cyclically. The circumcircles of AB"C", BC"A", CA"B" concur in X(112). (Randy Hutson, December 26, 2015)

Let A', B', C' be the intersections of the orthic axis and lines BC, CA, AB, respectively. The circumcircles of AB'C', BC'A', CA'B' concur in X(112). (Randy Hutson, December 26, 2015)

Let A' be the reflection of X(6) in BC, and define B' and C' cyclically. The circumcircles of AB'C', BC'A', CA'B' concur in X(112). (Randy Hutson, December 26, 2015)

Let A'B'C' be the circummedial triangle. Let A" be the pole, wrt the polar circle, of line B'C', and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(112). (Randy Hutson, December 26, 2015)

Let L₁ be the line that is the barycentric product of the Euler line and P(4). Let L₂ be the line that is the barycentric product of the Euler line and U(4). Then X(112) = L₁nL₂. (Randy Hutson, July 11, 2019)

X(112) lies on these lines:
2,127   3,1297   4,32   6,74   19,759   25,111   27,675   28,105   33,609   50,477   54,217   58,103   99,648   100,162   102,284   104,1108   109,163   186,187   230,403   250,691   251,427   286,767   376,577   393,571   523,935   647,1304   789,811

X(112) = reflection of X(i) in X(j) for these (i,j): (4,132), (1297,3)
X(112) = isogonal conjugate of X(525)
X(112) = isotomic conjugate of X(3267)
X(112) = anticomplement of X(127)
X(112) = X(i)-Ceva conjugate of X(j) for these (i,j): (249,24), (250,25)
X(112) = cevapoint of X(i) and X(j) for these (i,j): (32,512), (427,523)
X(112) = X(i)-cross conjugate of X(j) for these (i,j): (25,250), (512,4), (523,251)
X(112) = crossdifference of every pair of points on line X(122)X(125)
X(112) = barycentric product of X(1113) and X(1114)
X(112) = isogonal conjugate of isotomic conjugate of trilinear pole of Euler line
X(112) = point of intersection, other than A, B, C, of circumcircle and hyperbola {{A,B,C,X(4),PU(39)}}
X(112) = trilinear pole of line X(6)X(25)
X(112) = X(647)-cross conjugate of X(6)
X(112) = pole wrt polar circle of trilinear polar of X(850) (line X(115)X(127))
X(112) = X(48)-isoconjugate (polar conjugate) of X(850)
X(112) = X(92)-isoconjugate of X(520)
X(112) = X(1577)-isoconjugate of X(3)
X(112) = trilinear pole wrt circumsymmedial triangle of line X(6)X(647)
X(112) = reflection of X(935) in the Euler line
X(112) = reflection of X(2715) in the Brocard axis
X(112) = reflection of X(2722) in line X(1)X(3)
X(112) = inverse-in-polar-circle of X(115)
X(112) = inverse-in-{circumcircle, nine-point circle}-inverter of X(1560)
X(112) = inverse-in-Moses-radical-circle of X(1304)
X(112) = inverse-in-[circle with diameter X(15)X(16) and center X(187)] of X(842)
X(112) = inverse-in-circle-{X(1687),X(1688),PU(1),PU(2)} of X(2698)
X(112) = X(139)-of-excentral-triangle
X(112) = barycentric product of PU(74)
X(112) = trilinear product of PU(108)
X(112) = eigencenter of circumnormal triangle
X(112) = Thomson-isogonal conjugate of X(1503)
X(112) = Lucas-isogonal conjugate of X(1503)
X(112) = perspector of circumcevian triangle of X(468) and cross-triangle of ABC and circumcevian triangle of X(25)
X(112) = trilinear product of circumcircle intercepts of line X(1)X(19)
X(112) = barycentric product X(3)*X(107)
X(112) = barycentric product X(4)*X(110)
X(112) = X(6)-of-1st-anti-orthosymmedial-triangle
X(112) = cevapoint of Jerabek hyperbola intercepts of Lemoine axis
X(112) = Ψ(X(i),X(j)) for these (i,j): (1,19), (2,3), (3,6), (4,6), (5,53), (6,25), (69,2), (76,4), (125,115)

Centers 113-139
lie on the nine-point circle.

Suppose that X is a point on the nine-point circle, and let X' be the reflection of X in the orthocenter, H. Then X is the anticenter of the cyclic quadrilateral ABCX'. Let H_A be the orthocenter of triangle BCX, Let H_B be the orthocenter of CAX, and let H_C be the orthocenter of triangle ABX. Then the quadrilateral HH_AH_BH_C is homothetic to and congruent to the cyclic quadrilateral ABCX', and X is the center of homothety. (Randy Hutson, 9/23/2011)

X(113) = JERABEK ANTIPODE

Let A'B'C' be the orthic triangle. Let L be the line through A' parallel to the Euler line, and define M and N cyclically. Let L' be the reflection of L in sideline BC, and define M' and N' cyclically. The lines L', M', N' concur in X(113). (Randy Hutson, August 26, 2014)

Let A'B'C' be the orthic triangle. Let M_A be the reflection of the orthic axis in line B'C', and define Let M_B and Let M_C cyclically. Let A'' = Let M_B∩M_C, and define B'' and C'' cyclically. The lines A'A'', B'B'' C'C'' concur in X(113). (Randy Hutson, August 26, 2014)

Let A'B'C' be the orthic triangle. Let N_A be the orthic axis of AB'C', and define N_B and N_C cyclically. Let A'' = N_B∩N_C, B'' = N_C∩A_C, C'' = N_A∩B_C. Then triangle A''B''C'' is inversely similar to ABC, with similitude center X(6), and the lines A'A'', B'B'', C'C'' concur in X(113). Also, X(113) = X(3)-of-A''B''C''. (Randy Hutson, August 26, 2014)

X(113) lies on the bicevian conic of X(2) and X(110) and on these lines:
2,74   3,122   4,110   5,125   6,13   11,942   52,135   114,690   123,960   127,141   137,546

X(113) = midpoint of X(i) and X(j) for these (i,j): (4,110), (74,146), (265,399), (1553,3258)
X(113) = reflection of X(i) in X(j) for these (i,j): (52,1112), (125,5)
X(113) = complementary conjugate of X(30)
X(113) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,30), (2,3003)
X(113) = crosspoint of X(4) and X(403)
X(113) = crossdifference of every pair of points on line X(526)X(686)
X(113) = nine-point-circle-antipode of X(125)
X(113) = X(74)-of-medial-triangle
X(113) = X(104)-of-orthic-triangle if ABC is acute
X(113) = X(186)-of-X(4)-Brocard-triangle
X(113) = center of rectangular circumhyperbola that passes through X(110)
X(113) = center of rectangular hyperbola {{X(3),X(4),X(110),X(155),X(1351),X(1352),X(2574,X(2575)}}
X(113) = perpsector of circumconic centered at X(3003)
X(113) = inverse-in-polar-circle-of X(1300)
X(113) = inverse-in-{circumcircle, nine-point circle}-inverter of X(1302)
X(113) = anticenter of cyclic quadrilateral ABCX(110)
X(113) = Λ(X(2),X(3))-with-respect-to-orthic-triangle
X(113) = barycentric product X(30)*X(3580)
X(113) = complement of X(74)
X(113) = reflection of X(3) in X(5972)
X(113) = antipode of X(3) in the bicevian conic of X(2) and X(110)
X(113) = antipode of X(52) in the Hatzipolakis-Lozada hyperbola
X(113) = orthopole of line X(3)X(523)
X(113) = perspector of Ehrmann mid-triangle and orthic triangle

X(114) = KIEPERT ANTIPODE

X(114) is the QA-P30 center (Reflection of QA-P2 in QA-P11) of quadrangle ABCX(2) (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/58-qa-p30.html)

Let A'B'C' be the orthic triangle. Let La be the Lemoine axis of triangle AB'C', and define Lb and Lc cyclically. Let A" = Lb∩Lc, and define B" and C" cyclically. Triangle A"B"C" is inversely similar to ABC, with similicenter X(2). The lines A'A", B'B", C'C" concur in X(114), which is X(3)-of-A"B"C".

X(114) lies on the bicevian conic of X(2) and X(99), and on these lines:
2,98   3,127   4,99   5,39   25,135   52,211   113,690   132,684   136,427   325,511   381,543

X(114) = isogonal conjugate of X(2065)
X(114) = midpoint of X(i) and X(j) for these (i,j): (4,99), (98,147)
X(114) = reflection of X(i) in X(j) for these (i,j): (3,620), (115,5)
X(114) = complementary conjugate of X(511)
X(114) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,230), (4,511)
X(114) = crosspoint of X(2) and X(325)
X(114) = orthojoin of X(230)
X(114) = nine-point-circle-antipode of X(115)
X(114) = X(98)-of-medial triangle
X(114) = X(103)-of-orthic triangle if ABC is acute
X(114) = perspector of circumconic centered at X(230)
X(114) = center of circumconic that is locus of trilinear poles of lines passing through X(230)
X(114) = center of rectangular hyperbola {{X(4),X(76),X(99),X(376),X(487),X(488)}}
X(114) = X(1513)-of-1st-Brocard-triangle
X(114) = inverse-in-polar-circle of X(3563)
X(114) = inverse-in-{circumcircle, nine-point circle}-inverter of X(110)
X(114) = center of inverse-in-{circumcircle, nine-point circle}-inverter of Brocard circle
X(114) = X(5)=of=1st=antiBrocard=triangle
X(114) = anticenter of cyclic quadrilateral ABCX(99)
X(114) = Λ(X(3), X(6)), wrt orthic triangle
X(114) = centroid of mid-triangle of X(15)- and X(16)-Fuhrmann triangles

X(115) = CENTER OF KIEPERT HYPERBOLA

The circumcircle of the incentral triangle intersects the nine-point circle at 2 points, X(11) and X(115), and X(115) lies on the incentral circle and the cevian circle of every point on the Kiepert hyperbola. Let A'B'C' be the orthic triangle. The Brocard axes of AB'C', BC'A', CA'B' concur in X(115). Let P be a point on the Brocard circle, and let L be the line tangent to the Brocard circle at P. Let P' be the trilinear pole of L, and let P" be the isotomic conjugate of P'. As P varies, P" traces an ellipse with center at X(115). (Randy Hutson, July 23, 2015)

Let La, Lb, Lc be the lines through A, B, C, resp. parallel to the Brocard axis. Let Ma, Mb, Mc be the reflections of BC, CA, AB in La, Lb, Lc, resp. Let A' = Mb∩Mc, and define B', C' cyclically. Triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. Let A"B"C" be the reflection of A'B'C' in the Brocard axis. The triangle A"B"C" is homothetic to ABC, with center of homothety X(115); see Hyacinthos #16741/16782, Sep 2008.

X(115) is the QA-P2 center (Euler-Poncelet Point) of quadrangle ABCX(2) (see http://www.chrisvantienhoven.nl/quadrangle-objects/10-mathematics/quadrangle-objects/12-qa-p2.html)

Let F be the Feuerbach point, X(11), and FaFbFc be the Feuerbach triangle (the extraversion triangle of X(11)). Let A' be the barycentric product F*Fa, and define B', C' cyclically. The lines AA', BB', CC' concur in X(115). (Randy Hutson, January 29, 2018)

If you have The Geometer's Sketchpad, you can view Kiepert Hyperbola, showing X(115).

Roland H. Eddy and R. Fritsch, "The conics of Ludwig Kiepert: a comprehensive lesson in the geometry of the triangle," Mathematics Magazine 67 (1994) 188-205.

X(115) lies on the nine-point circle, the Steiner inellipse, the bicevian conic of X(2) and X(98), and on these lines:
2,99   4,32   5,39   6,13   11,1015   30,187   50,231   53,133   76,626   116,1086   120,442   125,245   127,338   128,233   129,389   131,216   232,403   316,385   325,538   395,530   396,531   427,1560   593,1029   804,1084

X(115) = midpoint of X(i) and X(j) for these (i,j): (4,98), (13,14), (99,148), (316,385), (2009,2010)
X(115) = midpoint of PU(40)
X(115) = reflection of X(i) in X(j) for these (i,j): (99,620), (114,5), (187,230), (325,625)
X(115) = isogonal conjugate of X(249)
X(115) = isotomic conjugate of X(4590)
X(115) = complement of X(99)
X(115) = anticomplement of X(620)
X(115) = inverse-in-orthocentroidal-circle of X(6)
X(115) = complementary conjugate of X(512)
X(115) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,523), (4,512), (338,125)
X(115) = crosspoint of X(i) and X(j) for these (i,j): (2,523), (68,525), (3413,3414)
X(115) = crosssum of X(i) and X(j) for these (i,j): (6,110), (24,112), (163,849), (1379,1380)
X(115) = crossdifference of every pair of points on line X(110)X(351)
X(115) = X(2)-Hirst inverse of X(148)
X(115) = X(99)-of-medial triangle
X(115) = X(101)-of-orthic triangle if ABC is acute
X(115) = X(325)-of-1st-Brocard-triangle
X(115) = X(187)-of-4th-Brocard-triangle
X(115) = X(187)-of-orthocentroidal-triangle
X(115) = X(141)-of-1st-antiBrocard-triangle
X(115) = barycentric product X(11)*X(12)
X(115) = {X(5),X(39)}-harmonic conjugate of X(1506)
X(115) = projection from Steiner circumellipse to Steiner inellipse of X(671)
X(115) = center of similitude of incentral and Feuerbach triangles
X(115) = center of circumconic that is locus of trilinear poles of lines parallel to the orthic axis (i.e. lines that pass through X(523))
X(115) = perspector of circumconic centered at X(523) (parabola {{A,B,C,X(476),X(523),X(685)})
X(115) = trilinear pole wrt medial triangle of line X(2)X(6)
X(115) = inverse-in-circumcircle of X(2079)
X(115) = inverse-in-polar-circle of X(112)
X(115) = inverse-in-{circumcircle, nine-point circle}-inverter of X(111)
X(115) = inverse-in-Moses-radical-circle of X(3258)
X(115) = inverse-in-Steiner-circumellipse of X(148)
X(115) = inverse-in-excircles-radical-circle of X(5213)
X(115) = {X(99),X(671)}-harmonic conjugate of X(148)
X(115) = {X(6108),X(6109)}-harmonic conjugate of X(6055)
X(115) = inverse-in-circle-{{X(2),X(13),X(14),X(111),X(476)}} of X(1648)
X(115) = orthopole of Brocard axis
X(115) = orthic isogonal conjugate of X(512)
X(115) = incentral isogonal conjugate of X(512)
X(115) = perspector of orthic triangle and tangential triangle of hyperbola {{A,B,C,X(2),X(6)}}
X(115) = similitude center of (equilateral) pedal triangles of X(15) and X(16)
X(115) = exsimilicenter of Moses circle and the nine-point circle
X(115) = anticenter of cyclic quadrilateral ABCX(98)
X(115) = Λ(X(187), X(237))-wrt-orthic-triangle
X(115) = X(1101)-isoconjugate of X(2)
X(115) = harmonic center of nine-point circle and Gallatly circle
X(115) = perspector of medial triangle and Schroeter triangle
X(115) = trilinear pole of line X(1648)X(8029)
X(115) = barycentric square of X(523)
X(115) = inverse-in-Hutson-Parry-circle of X(1648)
X(115) = {X(13636),X(13722)}-harmonic conjugate of X(1648)
X(115) = homothetic center of medial triangle and mid-triangle of antipedal triangles of X(13) and X(14)
X(115) = homothetic center of Ehrmann vertex-triangle and mid-triangle of 1st and 2nd Kenmotu diagonals triangles
X(115) = centroid of reflection triangle of X(187)
X(115) = centroid of mid-triangle of 1st and 2nd isodynamic-Dao triangles
X(115) = centroid of mid-triangle of 3rd and 4th isodynamic-Dao triangles
X(115) = point of concurrence of cevian circles of the excenters
X(115) = inverse-in-circle-{{X(3102),X(3103),PU(1)}} of X(32448)

X(116) = MIDPOINT OF X(4) AND X(103)

Let A'B'C' be the orthic triangle. Let La be the Soddy line of triangle AB'C', and define Lb and Lc cyclically. Let A" = Lb∩Lc, and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(116), which is X(3)-of-A"B"C". (Randy Hutson, July 31 2018)

X(116) lies on the nine-point circle and these lines: 2,101   4,103   5,118   10,120   115,1086   119,142   121,141   124,928

X(116) = midpoint of X(i) and X(j) for these (i,j): (4,103), (101,150)
X(116) = reflection of X(118) in X(5)
X(116) = isotomic conjugate of isogonal conjugate of X(20974)
X(116) = complementary conjugate of X(514)
X(116) = X(4)-Ceva conjugate of X(514)
X(116) = complement of X(101)
X(116) = inverse-in-excircles-radical-circle of X(3034)
X(116) = inverse-in-orthoptic-circle-of-Steiner-inellipse of X(675)
X(116) = anticenter of cyclic quadrilateral ABCX(103)
X(116) = Λ(Gergonne line), wrt orthic triangle
X(116) = X(2)-Ceva conjugate of X(6586)
X(116) = X(101)-of-medial triangle.

X(117) = MIDPOINT OF X(4) AND X(109)

Barycentrics  h(a,b,c) : h(b,c,a) : h(c,a,b) where h(a,b,c) = af(a,b,c)

X(117) lies on the nine-point circle
X(117) = X(102)-of-medial triangle.

X(117) lies on these lines: 2,102   4,109   5,124   10,123   11,65   118,928   136,407

X(117) = midpoint of X(i) and X(j) for these (i,j): (4,109), (102,151)
X(117) = reflection of X(124) in X(5)
X(117) = complement of X(102)
X(117) = complementary conjugate of X(515)
X(117) = X(4)-Ceva conjugate of X(515)
X(117) = inverse-in-polar-circle of X(32706)
X(117) = anticenter of cyclic quadrilateral ABCX(109)
X(117) = Λ(X(1), X(4)), wrt orthic triangle

X(118) = MIDPOINT OF X(4) AND X(101)

Let A'B'C' be the orthic triangle. Let La be the Gergonne line of triangle AB'C', and define Lb and Lc cyclically. Let A" = Lb∩Lc, and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(118), which is X(3)-of-A"B"C". (Randy Hutson, July 31 2018)

X(118) lies on the nine-point circle and these lines: 2,103   4,101   5,116   11,226   117,928   122,440   136,430   381,544   516,910

X(118) = midpoint of X(i) and X(j) for these (i,j): (4,101), (103,152)
X(118) = reflection of X(116) in X(5)
X(118) = complementary conjugate of X(516)
X(118) = X(4)-Ceva conjugate of X(516)
X(118) = complement of X(103)
X(118) = inverse-in-polar-circle of X(917)
X(118) = anticenter of cyclic quadrilateral ABCX(101)
X(118) = X(103)-of-medial triangle.
X(118) = Λ(X(1), X(7)), wrt orthic triangle
X(118) = Λ(X(4), X(9)), wrt orthic triangle

X(119) = FEUERBACH ANTIPODE

Let Na = X(5)-BCX(1), Nb = X(5)-of-CAX(1), Nc = X(5)-of-ABX(1). Then X(119) = X(2071)-of-NaNbNc. (Randy Hutson, January 29, 2018)

Let A'B'C' be the orthic triangle. Let La be the antiorthic axis of AB'C', and define Lb and Lc cyclically. Let A" = Lb∩Lc, B" = Lc∩La, C" = La∩Lb. Then triangle A"B"C" is inversely similar to ABC, with similicenter X(9). The lines A'A", B'B", C'C" concur in X(119). Also, X(119) = X(3) of A"B"C". (Randy Hutson, January 29, 2018)

X(119) lies on these lines:
1,5   2,104   3,123   4,100   10,124   116,142   125,442   135,431   136,429   214,515   381,528   517,908

X(119) = midpoint of X(i) and X(j) for these (i,j): (4,100), (104,153)
X(119) = reflection of X(i) in X(j) for these (i,j): (11,5), (3,3035)
X(119) = complement of X(104)
X(119) = complementary conjugate of X(517)
X(119) = X(4)-Ceva conjugate of X(517)
X(119) = nine-point-circle-antipode of X(11)
X(119) = X(104)-of-medial triangle
X(119) = X(2072)-of-Fuhrmann-triangle
X(119) = inverse-in-polar-circle of X(915)
X(119) = anticenter of cyclic quadrilateral ABCX(100)
X(119) = Λ(X(1), X(3)), wrt orthic triangle
X(119) = Λ(X(4), X(8)), wrt orthic triangle
X(119) = center of rectangular circumhyperbola passing through isogonal and isotomic conjugates of X(3657)

X(120) = X(105)-OF-MEDIAL-TRIANGLE

X(120) lies on the nine-point circle and these lines: 2,11   10,116   12,85   115,442

X(120) = complementary conjugate of X(518)
X(120) = X(4)-Ceva conjugate of X(518)

X(121) = X(106)-OF-MEDIAL-TRIANGLE

Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b) where g(a,b,c) = (b + c - 2a)[b³ + c³ + a(b² + c²) - 2bc(b + c)]

X(121) lies on the nine-point circle
X(121) = X(106)-of-medial triangle.

X(121) lies on these lines: 2,106   10,11   116,141

X(121) = complementary conjugate of X(519)
X(121) = X(4)-Ceva conjugate of X(519)
X(121) = complement of X(106)
X(121) = inverse-in-excircles-radical-circle of X(3032)
X(121) = Λ(X(1), X(2)), wrt orthic triangle

X(122) = X(107)-OF-MEDIAL-TRIANGLE

Barycentrics    (b^2 - c^2)^2 (a^2 - b^2 - c^2)^2 (3 a^4 - 2 a^2 (b^2 + c^2) - (b^2 - c^2)^2) : :

X(122) lies on the nine-point circle, the cevian circle of X(20), and these lines: 2,107   3,113   5,133   118,440   125,684   138,233

X(122) = reflection of X(133) in X(5)
X(122) = isogonal conjugate of X(15384)
X(122) = complementary conjugate of X(520)
X(122) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,520), (253,525)
X(122) = crosssum of X(i) and X(j) for these (i,j): (64,1301), (112,154)
X(122) = crosspoint of X(253) and X(525)
X(122) = crossdifference of every pair of points on line X(112)X(1301)
X(122) = X(107)-of-medial triangle
X(122) = center of the rectangular hyperbola that passes through A, B, C, and X(20)
X(122) = X(1293)-of-orthic-triangle if ABC is acute
X(122) = complement of X(107)
X(122) = perspector of orthic triangle and tangential triangle of hyperbola {{A,B,C,X(2),X(3)}}
X(122) = inverse-in-polar-circle of X(1301)
X(122) = inverse-in-orthoptic-circle-of-Steiner-inellipse of X(1297)
X(122) = inverse-in-Moses-radical-circle of X(33504)
X(122) = X(2)-Ceva conjugate of X(6587)
X(122) = barycentric product X(20)*X(15526) = X(20)*X(525)^2

X(123) =  X(108)-OF-MEDIAL-TRIANGLE

X(123) lies on the nine-point circle and hese lines: 2,108   3,119   10,117   113,960

X(123) = complement of X(108)
X(123) = complementary conjugate of X(521)
X(123) = X(2)-Ceva conjugate of X(6588)
X(123) = X(4)-Ceva conjugate of X(521)
X(123) = X(108)-of-medial triangle

X(124) = X(109)-OF-MEDIAL-TRIANGLE

Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where
                        g(a,b,c) = (b + c - a)(b - c)² [(b + c)(b² + c² - a² - bc) + abc]

X(124) lies on the nine-point circle
X(124) = X(109)-of-medial triangle
X(124) = center of the rectangular hyperbola that passes through A, B, C, and X(58)

X(124) lies on these lines: 2,109   4,102   5,117   10,119   116,928

X(124) = midpoint of X(4) and X(102)
X(124) = reflection of X(117) in X(5)
X(124) = complementary conjugate of X(522)
X(124) = X(4)-Ceva conjugate of X(522)
X(124) = complement of X(109)
X(124) = crosssum of circumcircle intercepts of line X(3)X(10)
X(124) = orthopole of line X(3)X(10)
X(124) = anticenter of cyclic quadrilateral ABCX(102)
X(124) = inverse-in-orthoptic-circle-of-Steiner-inellipse of X(1311)
X(124) = X(2)-Ceva conjugate of X(6589)
X(124) = center of rectangular circumhyperbola that is isogonal conjugate of line X(3)X(10)

X(125) = CENTER OF JERABEK HYPERBOLA

Roland H. Eddy and R. Fritsch, "The conics of Ludwig Kiepert: a comprehensive lesson in the geometry of the triangle," Mathematics Magazine 67 (1994) 188-205.

X(125) is the pole of the Fermat axis with respect to the Dao-Moses-Telv circle. (Randy Hutson, December 14, 2014)

Let La, Lb, Lc be the lines through A, B, C, resp. parallel to the Euler line. Let Ma, Mb, Mc be the reflections of BC, CA, AB in La, Lb, Lc, resp. Let A' = Mb∩Mc, and define B', C' cyclically. Triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. Let A"B"C" be the reflection of A'B'C' in the Euler line. The triangle A"B"C" is homothetic to ABC, with center of homothety X(125); see Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, March 25, 2016)

Let A'B'C' be the orthic triangle. The Euler lines of AB'C', BC'A', CA'B' concur in X(125). (Randy Hutson, March 25, 2016)

X(125) lies on these curves: nine-point circle, orthic inconic, symmedial circle, Johnson circumconic of the medial triangle, cevian circle of every point on the Jerabek hyperbola, and bicevian conic of X(2) and X(72). X(125) also lies on these lines:
2,98   3,131   4,74   5,113   6,67   51,132   68,1092   69,895   115,245   119,442   122,684   126,141   128,140   136,338   381,541   511,858

X(125) = midpoint of X(i) and X(j) for these (i,j): (3,265), (4,74), (6,67), (110,3448)
X(125) = reflection of X(i) in X(j) for these (i,j): (113,5), (185,974), (1495,468), (1511,140), (1539,546)
X(125) = isogonal conjugate of X(250)
X(125) = isotomic conjugate of X(18020)
X(125) = inverse-in-Brocard-circle of X(184)
X(125) = complement of X(110)
X(125) = complementary conjugate of X(523)
X(125) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,523), (66,512), (68,520), (69,525), (338,115)
X(125) = crosspoint of X(i) and X(j) for these (i,j): (4,523), (69,525), (338,339)
X(125) = crosssum of X(i) and X(j) for these (i,j): (3,110), (25,112), (162,270), (1113,1114)
X(125) = crossdifference of every pair of points on the line X(110)X(112)
X(125) = X(115)-Hirst inverse of X(868)
X(125) = X(2)-line conjugate of X(110)
X(125) = orthopole of the Euler line
X(125) = perspector of orthic triangle and Schroeter triangle
X(125) = X(110)-of-medial triangle
X(125) = X(100)-of-orthic triangle, if ABC is acute
X(125) = X(858)-of-1st-Brocard triangle
X(125) = anticenter of cyclic quadrilateral ABCX(74)
X(125) = Λ(X(230), X(231)), wrt orthic triangle
X(125) = anticomplement of X(5972)
X(125) = pole of Fermat axis wrt Dao-Moses-Telv circle
X(125) = orthic-isogonal conjugate of X(523)
X(125) = perspector of circumconic centered at X(647)
X(125) = center of circumconic that is locus of trilinear poles of lines passing through X(647)
X(125) = X(2)-Ceva conjugate of X(647)
X(125) = trilinear pole wrt orthic triangle of van Aubel line
X(125) = inverse-in-polar-circle of X(107)
X(125) = inverse-in-{circumcircle, nine-point circle}-inverter of X(98)
X(125) = inverse-in-orthosymmedial-circle of X(51)
X(125) = centroid of (degenerate) pedal triangle of X(74)
X(125) = X(i)-isoconjugate of X(j) for these {i,j}: {4,1101}, {92,249}
X(125) = inverse-in-Hutson-Parry-circle of X(868)
X(125) = {X(13636),X(13722)}-harmonic conjugate of X(868)

X(126) = X(111)-OF-MEDIAL-TRIANGLE

Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b) where g(a,b,c) = (2a² - b² - c²)[b⁴ + c⁴ + a²(b² + c²) - 4b²c²]

X(126) lies on the nine-point circle
X(126) = X(111)-of-medial triangle.

X(126) lies on these lines: 2,99   125,141   625,858

X(126) = complement of X(111)
X(126) = complementary conjugate of X(524)
X(126) = X(4)-Ceva conjugate of X(524)
X(126) = perspector of circumconic centered at X(3291)
X(126) = center of circumconic that is locus of trilinear poles of lines passing through X(3291)
X(126) = X(2)-Ceva conjugate of X(3291)
X(126) = one of two intersections (X(3258) is the other) of the nine-point circle of ABC and the Parry circle of the X(2)-Brocard triangle
X(126) = inverse-in-polar-circle of X(2374)
X(126) = inverse-in-{circumcircle, nine-point circle}-inverter of X(99)
X(126) = Λ(X(2), X(6)), wrt orthic triangle

X(127) = X(112)-OF-MEDIAL-TRIANGLE

Let A'B'C' be the orthic triangle. Let La be the van Aubel line of triangle AB'C', and define Lb and Lc cyclically. Let A" = Lb∩Lc, and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(127), which is X(3)-of-A"B"C". (Randy Hutson, July 31 2018)

X(127) lies on the nine-point circle, the cevian circle of X(22), and on these lines: 2,112   3,114   5,132   113,141   115,338   133,381   125,140

X(127) = reflection of X(132) in X(5)
X(127) = complement of X(112)
X(127) = anticomplementary conjugate of X(525)
X(127) = X(4)-Ceva conjugate of X(525)
X(127) = X(1292)-of-orthic-triangle if ABC is acute
X(127) = perspector of circumconic centered at X(2485)
X(127) = center of circumconic that is locus of trilinear poles of lines passing through X(2485)
X(127) = X(2)-Ceva conjugate of X(2485)
X(127) = inverse-in-polar-circle of X(1289)
X(127) = inverse-in-orthoptic-circle-of-Steiner-inellipse of X(2373)
X(127) = X(112)-of-medial triangle
X(127) = center of the rectangular hyperbola that passes through A, B, C, and X(22)

X(128) = X(74)-OF-ORTHIC-TRIANGLE

Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B) where g(A,B,C) = (sin A) f(A,B,C)

X(128) lies on the nine-point circle
X(128) = X(74)-of-orthic triangle.

X(128) lies on these lines: 5,137   52,134   53,139   115,233   125,140

X(128) = reflection of X(137) in X(5)
X(128) = isogonal conjugate of X(15401)
X(128) = complement of X(1141)
X(128) = perspector of circumconic centered at X(231)
X(128) = center of circumconic that is locus of trilinear poles of lines passing through X(231)
X(128) = inverse-in-polar-circle of X(2383)
X(128) = X(2)-Ceva conjugate of X(231)
X(128) = orthojoin of X(231)

X(129) = X(98)-OF-ORTHIC-TRIANGLE

Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B) where g(A,B,C) = (sin A) f(A,B,C)

X(129) lies on the nine-point circle
X(129) = X(98)-of-orthic triangle.

X(129) lies on these lines: 5,130   51,137   52,139   115,389

X(129) = reflection of X(130) in X(5)
X(129) = complement of X(1298)
X(129) = complementary conjugate of X(32428)
X(129) = Λ(X(5), X(53)), wrt orthic triangle

X(130) = X(99)-OF-ORTHIC-TRIANGLE

Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(130 lies on the nine-point circle
X(130) = X(99)-of-orthic triangle
X(130) = center of the rectangular hyperbola that passes through A, B, C, and X(51)

X(130) lies on these lines: 5,129   51,138

X(130) = reflection of X(129) in X(5)
X(130) = complement of X(1303)
X(130) = trilinear pole wrt orthic triangle of line X(51)X(53)

X(131) = INTERSECTION OF LINES X(3)X(125) AND X(4)X(135)

Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(131) lies on the nine-point circle
X(131) = X(102)-of-orthic triangle if ABC is acute.

X(131) lies on these lines: 3,125   4,135   5,136   115,216

X(131) = reflection of X(136) in X(5)
X(131) = complement of X(1300)
X(131) = inverse-in-polar-circle of X(1299)
X(131) = Λ(X(4), X(52)), wrt orthic triangle

X(132) = X(2)X(107)∩X(4)X(32)

Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(132) lies on the nine-point circle and these lines: 2,107   4,32   5,127   25,136   51,125   114,684   137,428   147,648

X(132) = midpoint of X(4) and X(112)
X(132) = reflection of X(127) in X(5)
X(132) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,232), (4,1503)
X(132) = X(4)-line conjugate of X(248)
X(132) = crossdifference of every pair of points on line X(248)X(684)
X(132) = X(105)-of-orthic triangle if ABC is acute
X(132) = complement of X(1297)
X(132) = perspector of circumconic centered at X(232)
X(132) = center of rectangular hyperbola {{A,B,C,X(4),X(112),PU(39)}} that is locus of trilinear poles of lines passing through X(232)
X(132) = center of rectangular hyperbola {X(4),X(112),X(371),X(372),X(378),X(1064)}
X(132) = inverse-in-polar-circle of X(98)
X(132) = inverse-in-{circumcircle, nine-point circle}-inverter of X(107)
X(132) = anticenter of cyclic quadrilateral ABCX(112)
X(132) = Λ(X(4), X(6)), wrt orthic triangle
X(132) = orthopole of PU(37)
X(132) = isogonal conjugate of X(15407)

X(133) = INTERSECTION OF LINES X(4)X(74) AND X(5)X(122)

X(133) lies on the nine-point circle
X(133) = X(106)-of-orthic triangle is ABC is acute.

X(133) lies on these lines: 2,1294   4,74   5,122   53,115   127,381   136,235

X(133) = midpoint of X(4) and X(107)
X(133) = reflection of X(122) in X(5)
X(133) = isogonal conjugate of X(15404)
X(133) = complement of X(1294)
X(133) = perspector of circumconic centered at X(1990)
X(133) = center of circumconic that is locus of trilinear poles of lines passing through X(1990)
X(133) = X(2)-Ceva conjugate of X(1990)
X(133) = trilinear pole wrt Euler triangle of van Aubel line
X(133) = inverse-in-polar-circle of X(74)
X(133) = anticenter of cyclic quadrilateral ABCX(107)
X(133) = Λ(X(4), X(51)), wrt orthic triangle

X(134) = X(107)-OF-ORTHIC-TRIANGLE

Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(134) lies on the nine-point circle
X(134) = X(107)-of-orthic triangle
X(134) = center of the rectangular hyperbola that passes through A, B, C, and X(52)

X(134) lies on this line: 52,128

X(134) = trilinear pole wrt orthic triangle of line X(52)X(53)

X(135) = INTERSECTION OF LINE X(4)X(131) AND X(25)X(114)

Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where
           &nbnbsp;            g(A,B,C) = (tan A)[(sin 2B)/(sec 2C - sec 2A) + (sin 2C)/(sec 2A - sec 2B)]

X(135) lies on the nine-point circle
X(135) = X(108)-of-orthic-triangle if ABC is acute
X(135) = center of the rectangular hyperbola that passes through A, B, C, and X(24)

X(135) lies on these lines: 4,131   25,114   52,113   119,431

X(135) = inverse-in-polar-circle of X(925)

X(136) = INTERSECTION OF LINE X(4)X(110) AND X(25)X(132)

Barycentrics  af(A,B,C) : bf(B,C,A) : cf(C,A,B)

X(136) lies on the nine-point circle
X(136) =X(109)-of-orthic triangle if ABC is acute
X(136) = center of the rectangular hyperbola that passes through A, B, C, and X(93)

X(136) lies on these lines:
2,925   4,110   5,131   25,132   68,254   114,427   117,407   118,430   119,429   125,338   127,868   133,235

X(136) = reflection of X(131) in X(5)
X(136) = complement of X(925)
X(136) = complementary conjugate of X(924)
X(136) = X(254)-Ceva conjugate of X(523)
X(136) = perspector of circumconic centered at X(2501)
X(136) = center of circumconic that is locus of trilinear poles of lines passing through X(2501) (hyperbola {{A,B,C,X(4),X(93)}})
X(136) = X(2)-Ceva conjugate of X(2501)
X(136) = inverse-in-polar-circle of X(110)
X(136) = inverse-in-orthoptic-circle-of-Steiner-inellipse of X(3563)

X(137) = X(110)-OF-ORTHIC-TRIANGLE

Barycentrics  af(A,B,C) : bf(B,C,A) : cf(C,A,B)

X(137) lies on the nine-point circle
X(137) = X(110)-of-orthic triangle
X(137) = center of the rectangular hyperbola that passes through A, B, C, X(5), and X(53)

X(137) lies on the cevian circle of X(5) and these lines: 5,128   51,129   53,138   113,546   132,428

X(137) = reflection of X(128) in X(5)
X(137) = complement of X(930)
X(137) = X(4)-Ceva conjugate of X(1510)
X(137) = crosssum of X(252) and X(930)
X(137) = trilinear pole wrt orthic triangle of line X(5)X(53)
X(137) = inverse-in-polar-circle of X(933)

X(138) = X(111)-OF-ORTHIC-TRIANGLE

Barycentrics  (v + w) tan A : (w + u) tan B : (u + v) tan C

X(138) lies on the nine-point circle
X(138) = X(111)-of-orthic triangle

X(138) lies on these lines: 51,130   53,137   122,233

X(138) = Λ(X(51), X(53)), wrt orthic triangle

X(139) = X(112)-OF-ORTHIC-TRIANGLE

X(139) lies on the nine-point circle and these lines: 52,129   53,128

X(139) = inverse-in-polar-circle of X(32692)
X(139) = X(112)-of-orthic-triangle

Centers 140- 170
113- 127, 140- 143: centers of the medial triangle
128- 139: centers of the orthic triangle
144- 153: centers of the anticomplementary triangle
154- 157, 159- 163: centers of the tangential triangle
164- 170: centers of the excentral triangle

X(140) = MIDPOINT OF X(3) AND X(5)