X2967

1st MACBEATH POINT

X2967 = reflection of X339 in X5
< 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) { 339, 5 }: Central Line L(339, 5) { 3, 112 }: Central Line L(3, 112) { 25, 110 }: Central Line L(25, 110) { 4, 147 }: Central Line L(4, 147) { 114, 132 }: Central Line L(114, 132) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_339 = ISOTOMIC CONJUGATE OF X(250) (0.6002, 0.0170) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_112 = Ψ(ORTHOCENTER, SYMMEDIAN POINT) (0.5162, 0.6121) X_25 = HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES (0.6329, 0.4497) X_110 = FOCUS OF KIEPERT PARABOLA (0.2516, 0.5478) X_4 = ORTHOCENTER (0.6122, 0.3958) X_147 = TARRY POINT OF ANTICOMPLEMENTARY TRIANGLE (-0.1660, 1.0561) X_114 = KIEPERT ANTIPODE (0.3616, 0.4140) X_132 = ∩X(4)X(32) (0.5642, 0.5039) X_2967 = 1st MACBEATH POINT (0.5121, 0.4808) A B C X339 X5 X3 X112 X25 X110 X4 X147 X114 X132 X2967 (a, b, c) = (5, 6, 7) | AB = 1

X2967 (1st MACBEATH POINT) 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 \(f(a,b,c) = (a^{2}\cos B \cos C - bc \cos^{2} A)^{2}(\sec A)\)
Barycentrics: \(\tan A \cos^{2}(A + \omega ) : :\)
Search Value (6, 9, 13): \(-3.903479118327\)
▸ Alternate Coordinates1 form

  • Barycentrics: \(a^{2} (b^{4} + c^{4} - a^{2} b^{2} - a^{2} c^{2})^{2} / (a^{2} - b^{2} - c^{2}) : :\)

▸ Incident Central Lines14 lines

{2, 1972}, {3, 112}, {4, 147}, {5, 339}, {25, 110}, {98, 648}, {114, 132}, {186, 2080}, {232, 511}, {250, 842}, {262, 264}, {324, 427}, {1312, 2592}, {1313, 2593}

▸ Functional Relations8 relations