X1064

INTERSECTION OF LINES X1X_4 AND X3X_31

X1064 = isogonal conjugate of X1065
< 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 { 3, 31 }: Central Line L(3, 31) { 102, 112 }: Central Line L(102, 112) { 104, 256 }: Central Line L(104, 256) { 40, 386 }: Central Line L(40, 386) { 42, 517 }: Central Line L(42, 517) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1065 = ISOGONAL CONJUGATE OF X(1064) (0.5777, 0.2607) X_1 = INCENTER (0.5714, 0.2333) X_4 = ORTHOCENTER (0.6122, 0.3958) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_31 = 2nd POWER POINT (0.6228, 0.3008) X_102 = Λ(INCENTER, ORTHOCENTER) (0.6364, -0.3897) X_112 = Ψ(ORTHOCENTER, SYMMEDIAN POINT) (0.5162, 0.6121) X_104 = ANTIPODE OF X(100) (0.8571, -0.2624) X_256 = 1st SHARYGIN POINT (0.5601, 0.2146) X_40 = BEVAN POINT (0.4286, -0.0292) X_386 = INVERSE-IN-BROCARD-CIRCLE OF X(58) (0.5912, 0.2526) X_42 = CROSSPOINT OF INCENTER AND SYMMEDIAN POINT (0.5880, 0.2495) X_517 = ISOGONAL CONJUGATE OF X(104) (0.8180, 0.7154) X_1064 = INTERSECTION OF LINES X(1)X(4) AND X(3)X(31) (0.5648, 0.2070) A B C X1065 X1 X4 X3 X31 X102 X112 X104 X256 X40 X386 X42 X517 X1064 (a, b, c) = (5, 6, 7) | AB = 1

X1064 (INTERSECTION OF LINES X1X_4 AND X3X_31) 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 \(f(A,B,C) = 1 + \cos A (\cos B + \cos C)\)
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): \(2.753108221769\)
▸ Incident Central Lines12 lines

{1, 4}, {3, 31}, {38, 912}, {40, 386}, {42, 517}, {102, 112}, {104, 256}, {238, 1006}, {631, 978}, {942, 1042}, {991, 995}, {1037, 1057}

▸ Functional Relations2 relations