X1092

TRILINEAR CUBE OF X3

X1092 = X92-isoconjugate of X393
< 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) { 1204, 3 }: Central Line L(1204, 3) { 3, 49 }: Central Line L(3, 49) { 54, 69 }: Central Line L(54, 69) { 20, 110 }: Central Line L(20, 110) { 68, 125 }: Central Line L(68, 125) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1204 = 3rd SARAGOSSA POINT OF X(64) (0.6497, 0.1923) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_49 = CENTER OF SINE-TRIPLE-ANGLE CIRCLE (0.6275, 0.1789) X_54 = KOSNITA POINT (0.5880, 0.2176) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.4122, 0.0654) X_20 = DE LONGCHAMPS POINT (0.3878, -0.1916) X_110 = FOCUS OF KIEPERT PARABOLA (0.2516, 0.5478) X_68 = PRASOLOV POINT (-0.0535, -0.0055) X_125 = CENTER OF JERABEK HYPERBOLA (0.6803, 0.0260) X_1092 = TRILINEAR CUBE OF X(3) (0.3503, 0.0118) A B C X1204 X3 X49 X54 X69 X20 X110 X68 X125 X1092 (a, b, c) = (5, 6, 7) | AB = 1

X1092 (TRILINEAR CUBE OF X3) 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 \(f(A,B,C) = \cos^{3} A\)
Barycentrics: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A,B,C) = (\sin A) f(A,B,C)\)
Search Value (6, 9, 13): \(4.727067351951\)
▸ Incident Central Lines11 lines

{2, 578}, {3, 49}, {4, 801}, {20, 110}, {24, 511}, {54, 69}, {68, 125}, {140, 343}, {156, 550}, {450, 1093}, {648, 1075}

▸ Functional Relations3 relations

Notes

Let A'B'C' be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung). A'B'C' is homothetic to the circumcevian triangle of X3 at X1092. (Randy Hutson, June 7, 2019)