X₂₂₆

X₇-CEVA CONJUGATE OF X₆₅

X₂₂₆ = complement of X₆₃

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

X₂₂₆ (X₇-CEVA CONJUGATE OF X₆₅) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {993, 1125}

• Related Centers: X₉₉₃ (MIDPOINT OF X₁ AND X₆₃), X₁₁₂₅ (COMPLEMENT OF X₁₀)

Trilinears: \((\csc A)(\cos B + \cos C) : (\csc B)(\cos C + \cos A) : (\csc C)(\cos A + \cos B)\)

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

Search Value (6, 9, 13): \(1.006602901997\)

▸ Alternate Coordinates4 forms

• Trilinears: \(bc(b + c)/(b + c - a) : ca(c + a)/(c + a - b) : ab(a + b)/(a + b - c)\)

• Trilinears: \(\cos(angle BIC) : \cos(angle CIA) : \cos(angle AIB)\)

• Barycentrics: \(\operatorname{area}(A'BC) : :\), where A'B'C' is the 2nd extouch triangle

• Barycentrics: \(r_{A} + r : r_{B} + r : r_{C} + r\), where \(r_{A}\), \(r_{B}\), \(r_{C}\) are the exradii

▸ Incident Central Lines51 lines

{1, 4}, {2, 7}, {5, 912}, {10, 12}, {11, 118}, {13, 1082}, {14, 554}, {27, 284}, {29, 951}, {35, 79}, {36, 1006}, {37, 440}, {41, 379}, {46, 498}, {55, 516}, {56, 405}, {76, 85}, {78, 377}, {81, 651}, {83, 1429}, {86, 1412}, {92, 342}, {98, 109}, {102, 1065}, {175, 1131}, {176, 1132}, {196, 281}, {208, 406}, {222, 478}, {228, 851}, {262, 982}, {273, 469}, {306, 321}, {429, 1426}, {443, 936}, {452, 1420}, {474, 1466}, {481, 485}, {482, 486}, {495, 517}, {535, 551}, {664, 671}, {673, 1174}, {748, 1471}, {857, 1446}, {975, 1038}, {990, 1040}, {1029, 1442}, {1260, 1376}, {1284, 1402}, {1401, 1463}

▸ Functional Relations30 relations

• X₂₂₆ = reflection of X₉₉₃ in X₁₁₂₅

• X₂₂₆ = isogonal conjugate of X₂₈₄

• X₂₂₆ = isotomic conjugate of X₃₃₃

• X₂₂₆ = complement of X₆₃

• X₂₂₆ = X_i-Ceva conjugate of X_j for these (i,j): (7, 65), (349, 307)

• X₂₂₆ = cevapoint of X₃₇ and X₆₅

• X₂₂₆ = X_i-cross conjugate of X_j for these (i,j): (37, 10), (73, 307)

• X₂₂₆ = crosspoint of X₂ and X₉₂

• X₂₂₆ = crosssum of X_i and X_j for these (i,j): (6, 48), (41, 55)

• X₂₂₆ = crossdifference of every pair of points on line X₆₅₂X₆₆₃

• X₂₂₆ = X₆₃-of-medial-triangle

• X₂₂₆ = bicentric sum of PU(20)

• X₂₂₆ = midpoint of PU(20)

• X₂₂₆ = radical center of cevian circles of the extraversions of X₈

• X₂₂₆ = trilinear pole of line X₅₂₃X₆₅₆ (the polar of X₂₉ wrt polar circle)

• X₂₂₆ = pole wrt polar circle of trilinear polar of X₂₉ (the line X₂₄₃X₅₂₂)

• X₂₂₆ = X₄₈-isoconjugate (polar conjugate) of X₂₉

• X₂₂₆ = X₆-isoconjugate of X₂₁

• X₂₂₆ = X₁₈₄-of-2nd-extouch-triangle

• X₂₂₆ = {X₂,X₅₇}-harmonic conjugate of X₃₉₁₁

• X₂₂₆ = {X₉,X₅₇}-harmonic conjugate of X₁₇₀₈

• X₂₂₆ = homothetic center of intouch triangle and the complement of excentral triangle)

• X₂₂₆ = homothetic center of 3rd Euler tringle and inverse-in-incircle triangle

• X₂₂₆ = perspector of intouch triangle and Gergonne line extraversion triangle

• X₂₂₆ = perspector of 2nd extouch triangle and Gergonne line extraversion triangle

• X₂₂₆ = X_i-beth conjugate of X_j for these (i,j): (2, 226), (21, 1064), (100, 42), (190, 226), (312, 306), (321, 321), (335, 226), (835, 226)

• X₂₂₆ = barycentric product X₁₀₉*X₈₅₀

• X₂₂₆ = perspector of Gemini triangle 9 and cross-triangle of ABC and Gemini triangle 9

• X₂₂₆ = trilinear pole of perspectrix of ABC and Gemini triangle 10

• X₂₂₆ = perspector of Gemini triangle 40 and cross-triangle of ABC and Gemini triangle 40

Notes

X₂₂₆ is the homothetic center of the intouch triangle and the triangle formed by the lines of the external pairs of extouch points of the excircles. (Randy Hutson, 9/23/2011)

Let A' be the radical center of the incircle and the B- and C-excircles; define B' and C' cyclically. A'B'C' is also the complement of the excentral triangle, and the triangle formed by the radical axes of the incircle and each excircle. Then X₂₂₆ is the homothetic center of A'B'C' and the intouch triangle. (Randy Hutson, December 26, 2015)

Let A' be the homothetic center of the orthic triangles of the intouch and A-extouch triangles, and define B' and C' cyclically. Let A_aB_aC_a be the orthic triangle of the A-extouch triangle, and define A_bB_bC_b, and A_cB_cC_c cyclically. Let A″ be the centroid of A_aA_bA_c, and define B″ and C″ cyclically. Then A'B'C' and A″B″C″ are homothetic to each other and to the medial triangle and the orthic triangle of the intouch triangle at X₂₂₆. (Randy Hutson, December 26, 2015)

Let (A') be the pedal circle of the A-vertex of the hexyl triangle, and define (B') and (C') cyclically. Then X₂₂₆ is the radical center of circles (A'), (B'), (C'). (Randy Hutson, December 26, 2015)

Let I_aI_bI_c be the reflection triangle of X₁. Let A' be the trilinear pole of line I_bI_c, and define B', C' cyclically. The lines AA', BB', CC' concur in X₂₂₆. (Randy Hutson, July 20, 2016)

Let A₁₃B₁₃C₁₃ be Gemini triangle 13. Let A' be the perspector of conic {A, B, C,B₁₃,C₁₃}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₂₂₆. (Randy Hutson, January 15, 2019)

Let A₁₄B₁₄C₁₄ be Gemini triangle 14. Let A' be the center of conic {A, B, C,B₁₄,C₁₄}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₂₂₆. (Randy Hutson, January 15, 2019)