X129

X98-OF-ORTHIC-TRIANGLE

X129 = X98-of-orthic 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) { 130, 5 }: Central Line L(130, 5) { 51, 137 }: Central Line L(51, 137) { 52, 139 }: Central Line L(52, 139) { 115, 389 }: Central Line L(115, 389) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_130 = -OF-ORTHIC-TRIANGLE (0.6536, 0.0131) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_51 = CENTROID OF ORTHIC TRIANGLE (0.6093, 0.3186) X_137 = -OF-ORTHIC-TRIANGLE (0.7813, 0.1289) X_52 = ORTHOCENTER OF ORTHIC TRIANGLE (0.7155, 0.4580) X_139 = -OF-ORTHIC-TRIANGLE (0.7000, 0.4596) X_115 = CENTER OF KIEPERT HYPERBOLA (0.7507, 0.0838) X_389 = CENTER OF THE TAYLOR CIRCLE (0.6078, 0.2800) X_129 = -OF-ORTHIC-TRIANGLE (0.4587, 0.4847) A B C X130 X5 X51 X137 X52 X139 X115 X389 X129 (a, b, c) = (5, 6, 7) | AB = 1

X129 (X98-OF-ORTHIC-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 \(f(A,B,C) = (\sec A)(\sin 2A)(\sin 2B + \sin 2C) s(A,B,C) t(A,B,C)\), \(s(A,B,C) = \sin^{4}(2B) + \sin^{4}(2C) - \sin^{2}(2A) \sin^{2}(2B) - \sin^{2}(2A) \sin^{2}(2C)\), \(t(A,B,C) = \sin^{4}(2A) + \sin^{2}(2A) u(A,B,C) + v(A,B,C)\), \(u(A,B,C) = \sin 2B \sin 2C - \sin^{2}(2B) - \sin^{2}(2C)\), \(v(A,B,C) = (\sin 2B \sin 2C)(\sin 2B - \sin 2C)^{2}\)
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): \(3.956071987550\)

X129 lies on the nine-point circle

▸ Incident Central Lines4 lines

{5, 130}, {51, 137}, {52, 139}, {115, 389}

▸ Functional Relations7 relations