X166

GERGONNE POINT OF EXCENTRAL TRIANGLE

X166 = X7-of-excentral triangle
< 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) { 165, 168 }: Central Line L(165, 168) { 167, 188 }: Central Line L(167, 188) { 1, 1488 }: Central Line L(1, 1488) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_266 = ISOGONAL CONJUGATE OF X(188) (0.6074, 0.2891) X_57 = ISOGONAL CONJUGATE OF X(9) (0.6138, 0.3110) X_165 = CENTROID OF THE EXCENTRAL TRIANGLE (0.4762, 0.0583) X_168 = MITTENPUNKT OF EXCENTRAL TRIANGLE (0.4565, 0.0098) X_167 = NAGEL POINT OF EXCENTRAL TRIANGLE (0.4425, -0.0480) X_188 = 2nd MID-ARC POINT OF ANTICOMPLEMENTARY TRIANGLE (0.5245, 0.1802) X_1 = INCENTER (0.5714, 0.2333) X_1488 = 2nd STEVANOVIC POINT (0.5582, 0.2148) X_166 = GERGONNE POINT OF EXCENTRAL TRIANGLE (0.5156, 0.1554) A B C X266 X57 X165 X168 X167 X188 X1 X1488 X166 (a, b, c) = (5, 6, 7) | AB = 1

X166 (GERGONNE POINT OF EXCENTRAL TRIANGLE) 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 \(f(A,B,C) = (\tan A/2)/(\cos B/2 + \cos C/2 - \cos A/2) - (\tan B/2)/(\cos C/2 + \cos A/2 - \cos B/2) - (\tan C/2)/(\cos A/2 + \cos B/2 - \cos C/2)\)
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): \(4.147361892877\)
▸ Incident Central Lines3 lines

{1, 1488}, {165, 168}, {167, 188}

▸ Functional Relations3 relations