X1111

TRILINEAR SQUARE OF X514

Let A7B7C7 and A8B8C8 be the Gemini triangles 7 and 8. Let A' be the trilinear product A7*A8 and define B' and C' cyclically. The lines AA', BB', CC' concur in X1111. (Randy Hutson, November 30, 2018)
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 1, 85 }: Central Line L(1, 85) { 7, 80 }: Central Line L(7, 80) { 76, 334 }: Central Line L(76, 334) { 269, 273 }: Central Line L(269, 273) { 75, 537 }: Central Line L(75, 537) { 348, 499 }: Central Line L(348, 499) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1110 = TRILINEAR SQUARE OF X(101) (0.5057, 0.3942) X_1 = INCENTER (0.5714, 0.2333) X_85 = ISOTOMIC CONJUGATE OF X(9) (0.5610, 0.2421) X_7 = GERGONNE POINT (0.5903, 0.2769) X_80 = REFLECTION OF INCENTER IN FEUERBACH POINT (0.8980, -0.1000) X_76 = 3rd BROCARD POINT (0.4567, 0.1388) X_334 = ISOTOMIC CONJUGATE OF X(238) (1.0545, 0.0278) X_269 = ISOGONAL CONJUGATE OF X(200) (0.6369, 0.3847) X_273 = ISOGONAL CONJUGATE OF X(212) (0.6191, 0.4329) X_75 = ISOTOMIC CONJUGATE OF INCENTER (0.4988, 0.1682) X_537 = DIRECTION OF VECTOR AX + BX + CX, where X = X(38) (-0.0190, 0.3866) X_348 = ISOTOMIC CONJUGATE OF X(281) (0.4854, 0.1069) X_499 = {X(1),X(2)}-HARMONIC CONJUGATE OF X(498) (0.4458, 0.1102) X_1111 = TRILINEAR SQUARE OF X(514) (0.7470, 0.0849) A B C X1110 X1 X85 X7 X80 X76 X334 X269 X273 X75 X537 X348 X499 X1111 (a, b, c) = (5, 6, 7) | AB = 1

X1111 (TRILINEAR SQUARE OF X514) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = [(b - c)/a]^{2}\)
Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = af(a,b,c)\)
Search Value (6, 9, 13): \(2.389406980359\)
▸ Alternate Coordinates1 form

  • Trilinears: squared distance from A to line X1X6 : :

X1111 lies on the inellipse centered at X10.

▸ Incident Central Lines7 lines

{1, 85}, {7, 80}, {75, 537}, {76, 334}, {269, 273}, {348, 499}, {918, 1086}

▸ Functional Relations9 relations

Notes

Let A7B7C7 and A8B8C8 be Gemini triangles 7 and 8, resp. Let A' be the intersection of the tangent to the {ABC, Gemini 7}-circumconic at A7 and the tangent to the {ABC, Gemini 8}-circumconic at A8. Define B' and C' cyclically. The lines AA', BB', CC' concur in X1111. (Randy Hutson, January 15, 2019)