X138

X111-OF-ORTHIC-TRIANGLE

X138 = X111-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_138 ]: Line [ H_a X_138 ] [ H_b X_138 ]: Line [ H_b X_138 ] [ H_c X_138 ]: Line [ H_c X_138 ] { 51, 130 }: Central Line L(51, 130) { 53, 137 }: Central Line L(53, 137) { 122, 233 }: Central Line L(122, 233) 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_51 = CENTROID OF ORTHIC TRIANGLE (0.6093, 0.3186) X_130 = -OF-ORTHIC-TRIANGLE (0.6536, 0.0131) X_53 = SYMMEDIAN POINT OF ORTHIC TRIANGLE (0.6164, 0.4390) X_137 = -OF-ORTHIC-TRIANGLE (0.7813, 0.1289) X_122 = -OF-MEDIAL-TRIANGLE (0.5195, -0.0036) X_233 = -CEVA CONJUGATE OF X(140) (0.5475, 0.2218) X_138 = -OF-ORTHIC-TRIANGLE (0.5825, 0.5027) A B C Ha Hb Hc X51 X130 X53 X137 X122 X233 X138 (a, b, c) = (5, 6, 7) | AB = 1

X138 (X111-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–X138, Hb–X138, Hc–X138 concurring at X138

Trilinears: \((v + w) \sec A : (w + u) \sec B : (u + v) \sec C\), where \(u = u(A,B,C) = (\sin 2A)/(2 \sin^{2} 2A - \sin^{2} 2B - \sin^{2} 2C)\), \(v = u(B,C,A)\), \(w = u(C,A,B)\)
Barycentrics: \((v + w) \tan A : (w + u) \tan B : (u + v) \tan C\)
Search Value (6, 9, 13): \(3.323272182949\)

X138 lies on the nine-point circle

▸ Incident Central Lines3 lines

{51, 130}, {53, 137}, {122, 233}

▸ Functional Relations2 relations