X₅₇

ISOGONAL CONJUGATE OF X₉

X₅₇ is the isogonal conjugate of X₉.

5, 6, 7 3, 4, 5 Kepler Kimberling

X₅₇ (ISOGONAL CONJUGATE OF X₉) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {497, 3474}

• Related Centers: X₄₉₇ (CROSSPOINT OF GERGONNE POINT AND NAGEL POINT), X₃₄₇₄ (INTERSECTION OF LINES X)

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

Barycentrics: \(R_{a} - r : R_{b} - r : R_{c} - r\), where \(R_{a}\), \(R_{b}\), \(R_{c}\) are the exradii

Search Value (6, 9, 13): \(0.380455291518\)

▸ Alternate Coordinates14 forms

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

• Trilinears: \(1 + \cos B + \cos C - \cos A\)

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

• Trilinears: \(\cos^{2}(B/2) + \cos^{2}(C/2) - \cos^{2}(A/2) ::\)

• Trilinears: \(S_{A} - bc : S_{B} - ca : S_{C} - ab : :\)

• Trilinears: \(\csc A - \cot A : :\)

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

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

• Trilinears: \(\cot A' : :\), where J_aJ_bJ_c is the excentral triangle

• Trilinears: \(|AA'|/|AX(1)| : |BB'|/|BX(1)| : |CC'|/|CX(1)|\), where J_aJ_bJ_c is the excentral triangle

• Trilinears: \(R_{a} : R_{b} : R_{c}\), where \(R_{a}\), \(R_{b}\), \(R_{c}\) are the exradii

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

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

• Barycentrics: \(\operatorname{area}(A'BC) : :\), where J_aJ_bJ_c = excentral triangle

▸ Linear Combinations1 combination

• X₅₇ = \((R_{a} + R_{b} + R_{c})\) X₁ - \(r\) J_a - \(r\) J_b - \(r\) J_c, where J_a, J_b, J_c are excenters, and \(R_{a}\), \(R_{b}\), \(R_{c}\) are the exradii

X₅₇ lies on the Thomson cubic.

▸ Incident Central Lines47 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}

▸ Functional Relations61 relations

• X₅₇ = midpoint of X₄₉₇ and X₃₄₇₄

• X₅₇ = reflection of X_i in X_j for these (i,j): (1, 999), (200, 1376)

• X₅₇ = isogonal conjugate of X₉

• X₅₇ = isotomic conjugate of X₃₁₂

• X₅₇ = complement of X₃₂₉

• X₅₇ = anticomplement of X₃₄₅₂

• X₅₇ = circumcircle-inverse of X₂₀₇₈

• X₅₇ = incircle-inverse of X₃₆₆₀

• X₅₇ = Bevan-circle-inverse of X₁₁₅₅

• X₅₇ = trilinear product of PU(46)

• X₅₇ = antigonal conjugate of polar conjugate of X₃₇₇₆₉

• X₅₇ = trilinear pole of PU(96) (line X₅₁₃X₆₆₃, the polar of X₃₁₈ wrt polar circle, and the Monge line of the mixtilinear incircles)

• X₅₇ = barycentric product of PU(94)

• X₅₇ = pole wrt polar circle of trilinear polar of X₃₁₈

• X₅₇ = X₄₈-isoconjugate (polar conjugate) of X₃₁₈

• X₅₇ = X₆-isoconjugate of X₈

• X₅₇ = X₇₅-isoconjugate of X₄₁

• X₅₇ = X₉₂-isoconjugate of X₂₁₂

• X₅₇ = 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)

• X₅₇ = cevapoint of X_i and X_j for these (i,j): (6, 56), (19, 208)

• X₅₇ = X_i-cross conjugate of X_j for these (i,j): (6, 1), (19, 84), (56, 269), (65, 7)

• X₅₇ = 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)

• X₅₇ = 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)

• X₅₇ = crossdifference of every pair of points on line X₆₅₀X₆₆₃

• X₅₇ = X_i-Hirst inverse of X_j for these (i,j): (1, 241), (7, 1447), (56, 1429), (105, 1462), (910, 1419)

• X₅₇ = perspector of ABC and unary cofactor triangle of 6th mixtilinear triangle

• X₅₇ = perspector of ABC and antipedal triangle of X₄₀

• X₅₇ = homothetic center of: ABC; orthic triangle of intouch triangle; tangential triangle of excentral triangle

• X₅₇ = X₂₅-of-excentral-triangle

• X₅₇ = X₂₅-of-intouch-triangle

• X₅₇ = pole wrt Bevan circle of antiorthic axis

• X₅₇ = perspector of Bevan circle

• X₅₇ = perspector of circumconic centered at X₂₂₃

• X₅₇ = center of circumconic that is locus of trilinear poles of lines passing through X₂₂₃

• X₅₇ = perspector of pedal and antipedal (or anticevian) triangles of X₁

• X₅₇ = perspector of ABC and medial triangle of pedal triangle of X₈₄

• X₅₇ = inverse-in-circumconic-centered-at-X₉ of X₃₉₁₁

• X₅₇ = orthocorrespondent of X₁

• X₅₇ = Danneels point of X₇

• X₅₇ = vertex conjugate of X₅₅ and X₅₇

• X₅₇ = perspector of ABC and extraversion triangle of X₉

• X₅₇ = trilinear product of extraversions of X₉

• X₅₇ = SS(A→J_a) of X₆₃, where J_aJ_bJ_c is the excentral triangle

• X₅₇ = Cundy-Parry Phi transform of X₄₀

• X₅₇ = Cundy-Parry Psi transform of X₈₄

• X₅₇ = perspector of ABC and cross-triangle of Gemini triangles 9 and 10

• X₅₇ = perspector of ABC and cross-triangle of ABC and Gemini triangle 9

• X₅₇ = perspector of ABC and cross-triangle of ABC and Gemini triangle 10

• X₅₇ = barycentric product of vertices of Gemini triangle 9

• X₅₇ = barycentric product of vertices of Gemini triangle 10

• X₅₇ = perspector of ABC and tangential triangle, wrt Gemini triangle 2, of {ABC, Gemini 2}-circumconic

• X₅₇ = perspector of Gemini triangle 36 and cross-triangle of ABC and Gemini triangle 36

• X₅₇ = perspector of ABC and unary cofactor triangle of Gemini triangle 36

• X₅₇ = 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)

• X₅₇ = 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)

• X₅₇ = {X₁,X₃}-harmonic conjugate of X₃₆₀₁

• X₅₇ = {X₁,X₄₀}-harmonic conjugate of X₁₆₉₇

• X₅₇ = {X₂,X₆₃}-harmonic conjugate of X₉

• X₅₇ = {X₅₅,X₅₆}-harmonic conjugate of X₁₆₁₇

• X₅₇ = {X₅₆,X₆₅}-harmonic conjugate of X₁

• X₅₇ = {X₃₅₁₃,X₃₅₁₄}-harmonic conjugate of X₁

Notes

Let J_a, J_b, J_c be the excenters and I the incenter of ABC. Let K_a be the symmedian point of JbJcI, and define K_b, K_c cyclically. Then K_aK_bK_c is perspective to ABC at X₅₇. (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 X₅₇. (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 X₅₇. (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 X₅₇. (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 X₅₇. (Randy Hutson, September 14, 2016)

Let H_aH_bH_c be the orthic triangle. Let L_a be the reflection of line H_bH_c in the internal angle bisector of 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. Triangle A″B″C″ is homothetic to ABC, with center of homothety X₅₇. (Randy Hutson, September 14, 2016)

Let O_a be the circle passing through B and C, and tangent to the incircle. Define O_b and O_c cyclically. Let A' be the point of tangency of O_a and the incircle, and define B' and C' cyclically. Triangle A'B'C' is perspective to the intouch triangle at X₅₇. Also, X₅₇ is the radical center of circles O_a, O_b, O_c. (Randy Hutson, July 31 2018)

Let T_aT_bT_c be the intouch triangle. Let A″ be the trilinear product of the circumcircle intercepts of line T_bT_c, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₅₇. (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 X₅₇. (Randy Hutson, January 15, 2019)

Let V_a, V_b, V_c be the antipodes of V=X₄₀ in the circles (VBC), (VCA), (VAB), respectively. The lines AV_a, BV_b, CV_c concur in X₅₇. (Angel Montesdeoca, October 14, 2019)

Let DEF be the intouch triangle. Let I_a be the internal bisector of angle BAC, and let D' be the point, other than D, where the line through D parallel to I_a meets the incircle. Let A' be the point, other than A, where AD' meets the incircle. Let L_a be the radical axis of the circumcircles of triangles A'BF and A’CE, and define L_b and L_c cyclically. The lines L_a, L_b, L_c concur in X₅₇. See X₅₇ (X(57)c.png). (Angel Montesdeoca, December 21, 2019)

In the plane of a triangle ABC, let

• A'B'C' = excentral triangle;

• B_a = reflection of B in A', and define C_b and A_c cyclically;

• C_a = reflection in C in A', and define A_b and B_c cyclically;

• V_a = A_cB_c∩C_bA_b, and define V_b and V_c cyclically.

The triangle V_aV_bV_c is perspective to ABC, and the perspector is X₅₇. (Dasari Naga Vijay Krishna, June 23, 2021)

X₅₇ is the perspector of the intouch triangle and excentral triangle.

X₅₇ is the {X₂,X₇}-harmonic conjugate of X₂₂₆.