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

Trilinears

\(1/(b + c - a) : 1/(c + a - b) : 1/(a + b - c)\)

\(tan(A/2) : tan(B/2) : tan(C/2)\)

\(1 + cos B + cos C - cos A\)

\(1 + sin(A/2)csc(B/2)csc(C/2) : :\)

\(cos2(B/2) + cos2(C/2) - cos2(A/2) ::\)

\(SA - bc : SB - ca : SC - ab : :\)

\(csc A - cot A : :\)

\((1 - cos A) csc A : :\)

\(b(cot B/2) + c(cot C/2) - a(cot A/2) : :\)

\(cot A' : :, where A'B'C' is the excentral triangle\)

\(|AA'|/|A:ref:`X(1) <X(1)>`| : |BB'|/|B:ref:`X(1) <X(1)>`| : |CC'|/|C:ref:`X(1) <X(1)>`|, where A'B'C' is the excentral triangle\)

\(Ra : Rb : Rc, where Ra, Rb, Rc are the exradii\)

Barycentrics

\(a/(b + c - a) : b/(c + a - b) : c/(a + b - c)\)

\(1 - cos A : 1 - cos B : 1 - cos C\)

\(area(A'BC) : : , where A'B'C' = excentral triangle\)

Notes

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

X(57) = midpoint of X(497) and X(3474)

X(57) = reflection of X(i) in X(j) for these (i,j): (1,999), (200,1376)

X(57) = isogonal conjugate of X(9)

X(57) = isotomic conjugate of X(312)

X(57) = complement of X(329)

X(57) = anticomplement of X(3452)

X(57) = circumcircle-inverse of X(2078)

X(57) = incircle-inverse of X(3660)

X(57) = Bevan-circle-inverse of X(1155)

X(57) = trilinear product of PU(46)

X(57) = antigonal conjugate of polar conjugate of X(37769)

X(57) = trilinear pole of PU(96) (line X(513)X(663), the polar of X(318) wrt polar circle, and the Monge line of the mixtilinear incircles)

X(57) = barycentric product of PU(94)

X(57) = pole wrt polar circle of trilinear polar of X(318)

X(57) = X(48)-isoconjugate (polar conjugate) of X(318)

X(57) = X(6)-isoconjugate of X(8)

X(57) = X(75)-isoconjugate of X(41)

X(57) = X(92)-isoconjugate of X(212)

X(57) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,223), (7,1), (27,278), (81,222), (85,77), (273,34), (279,269)

X(57) = cevapoint of X(i) and X(j) for these (i,j): (6,56), (19,208)

X(57) = X(i)-cross conjugate of X(j) for these (i,j): (6,1), (19,84), (56,269), (65,7)

X(57) = crosspoint of X(i) and X(j) for these (i,j): (1,3062), (2,189), (7,279), (27,81), (85,273), (1014,1434), (1659,13390)

X(57) = crosssum of X(i) and X(j) for these (i,j): (1,165), (6,198), (31,205), (37,71), (41,212), (55,220), (210,1334), (2066,5414)

X(57) = crossdifference of every pair of points on line X(650)X(663)

X(57) = X(i)-Hirst inverse of X(j) for these (i,j): (1,241), (7,1447), (56,1429), (105,1462), (910,1419)

X(57) = perspector of ABC and unary cofactor triangle of 6th mixtilinear triangle

X(57) = perspector of ABC and antipedal triangle of X(40)

X(57) = homothetic center of: ABC; orthic triangle of intouch triangle; tangential triangle of excentral triangle

X(57) = X(25)-of-excentral-triangle

X(57) = X(25)-of-intouch-triangle

X(57) = pole wrt Bevan circle of antiorthic axis

X(57) = perspector of Bevan circle

X(57) = perspector of circumconic centered at X(223)

X(57) = center of circumconic that is locus of trilinear poles of lines passing through X(223)

X(57) = perspector of pedal and antipedal (or anticevian) triangles of X(1)

X(57) = perspector of ABC and medial triangle of pedal triangle of X(84)

X(57) = inverse-in-circumconic-centered-at-X(9) of X(3911)

X(57) = orthocorrespondent of X(1)

X(57) = Danneels point of X(7)

X(57) = vertex conjugate of X(55) and X(57)

X(57) = perspector of ABC and extraversion triangle of X(9)

X(57) = trilinear product of extraversions of X(9)

X(57) = SS(A&srarr;A’) of X(63), where A’B’C’ is the excentral triangle

X(57) = Cundy-Parry Phi transform of X(40)

X(57) = Cundy-Parry Psi transform of X(84)

X(57) = perspector of ABC and cross-triangle of Gemini triangles 9 and 10

X(57) = perspector of ABC and cross-triangle of ABC and Gemini triangle 9

X(57) = perspector of ABC and cross-triangle of ABC and Gemini triangle 10

X(57) = barycentric product of vertices of Gemini triangle 9

X(57) = barycentric product of vertices of Gemini triangle 10

X(57) = perspector of ABC and tangential triangle, wrt Gemini triangle 2, of {ABC, Gemini 2}-circumconic

X(57) = perspector of Gemini triangle 36 and cross-triangle of ABC and Gemini triangle 36

X(57) = perspector of ABC and unary cofactor triangle of Gemini triangle 36

X(57) = X(i)-aleph conjugate of X(j) for these (i,j): (2,40), (7,57), (57,978), (174,1), (366,165), (507,503), (508,9), (509,43)

X(57) = X(i)-beth conjugate of X(j) for these (i,j): (2,2), (81,57), (88,57), (100,57), (110,31), (162,57), (190,57), (333,63), (648,92), (651,57), (653,57), (655,57), (658,57), (660,57), (662,57), (673,57), (771,57), (799,57), (823,57), (897, 57)

X(57) = {X(1),:ref:X(3) <X(3)>}-harmonic conjugate of X(3601)

X(57) = {X(1),:ref:X(40) <X(40)>}-harmonic conjugate of X(1697)

X(57) = {X(2),:ref:X(63) <X(63)>}-harmonic conjugate of X(9)

X(57) = {X(55),:ref:X(56) <X(56)>}-harmonic conjugate of X(1617)

X(57) = {X(56),:ref:X(65) <X(65)>}-harmonic conjugate of X(1)

X(57) = {X(3513),:ref:X(3514) <X(3514)>}-harmonic conjugate of X(1)