X1070

POINT ALKES

< 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, 4 }: Incenter–Orthocenter Line { 56, 1074 }: Central Line L(56, 1074) { 55, 1076 }: Central Line L(55, 1076) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1 = INCENTER (0.5714, 0.2333) X_4 = ORTHOCENTER (0.6122, 0.3958) X_56 = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.6316, 0.3438) X_1074 = POINT ALNITAK (0.5447, 0.2250) X_55 = INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.5490, 0.1921) X_1076 = POINT ALPHARD (0.3421, -0.3326) X_1070 = POINT ALKES (0.5823, 0.2764) A B C X1 X4 X56 X1074 X55 X1076 X1070 (a, b, c) = (5, 6, 7) | AB = 1

X1070 (POINT ALKES) 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 \(f(A,B,C) = 1 + \cos B \cos C (\cos B + \cos C)\)
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): \(0.998412379826\)
▸ Incident Central Lines3 lines

{1, 4}, {55, 1076}, {56, 1074}