X134

X107-OF-ORTHIC-TRIANGLE

X134 = X107-of-orthic triangle
< A B C >: Reference Triangle △ABC < H_a H_b H_c >: Orthic Triangle △H_a H_b H_c [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) [ H_a X_134 ]: Line [ H_a X_134 ] [ H_b X_134 ]: Line [ H_b X_134 ] [ H_c X_134 ]: Line [ H_c X_134 ] { 52, 128 }: Central Line L(52, 128) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) H_a = Altitude Foot H_a on BC (0.7053, 0.4559) H_b = Altitude Foot H_b on CA (0.5102, 0.4999) H_c = Altitude Foot H_c on AB (0.6122, 0.0000) X_52 = ORTHOCENTER OF ORTHIC TRIANGLE (0.7155, 0.4580) X_128 = -OF-ORTHIC-TRIANGLE (0.3310, 0.3690) X_134 = -OF-ORTHIC-TRIANGLE (0.7056, 0.4557) A B C Ha Hb Hc X52 X128 X134 (a, b, c) = (5, 6, 7) | AB = 1

X134 (X107-OF-ORTHIC-TRIANGLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

  • Derived Triangle: Orthic Triangle △HaHbHc — vertices Ha, Hb, Hc (Pedal triangle of Orthocenter X4 (altitude feet))

  • Concurrence Lines: Ha–X134, Hb–X134, Hc–X134 concurring at X134

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = (\sec A) u(A,B,C) [v(B,C,A) - v(C,B,A)]\), \(u(A,B,C) = (\sin 2A)[\sin^{2}(2B) - \sin^{2}(2C)][\sin^{2}(2B) + \sin^{2}(2C) - \sin^{2}(2A)]^{2}\), \(v(B,C,A) = \sin 2C [\sin^{2}(2A) - \sin^{2}(2B)][\sin^{2}(2A) + \sin^{2}(2B) - \sin^{2}(2C)]^{2}\)
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): \(-2.794052902662\)

X134 lies on the nine-point circle

▸ Incident Central Lines1 line

{52, 128}

▸ Functional Relations6 relations