X564

INTERSECTION OF LINES X1X_1048 AND X(47,91)

X564 = {X91,X92}-harmonic conjugate of X47
< 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) { 91, 92 }: Central Line L(91, 92) { 47, 91 }: Central Line L(47, 91) { 1, 1048 }: Central Line L(1, 1048) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_47 = -BETH CONJUGATE OF X(34) (0.7279, 0.4389) X_91 = ISOGONAL CONJUGATE OF X(47) (-0.0865, -0.0205) X_92 = CEVAPOINT OF INCENTER AND CLAWSON POINT (0.5949, 0.3639) X_1 = INCENTER (0.5714, 0.2333) X_1048 = POINT ALGIEBA (0.5862, 0.2176) X_564 = INTERSECTION OF LINES X(1)X(1048) AND X(47,91) (0.4993, 0.3100) A B C X47 X91 X92 X1 X1048 X564 (a, b, c) = (5, 6, 7) | AB = 1

X564 (INTERSECTION OF LINES X1X_1048 AND X(47,91)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(\cos(2B - 2C) : \cos(2C - 2A) : \cos(2A - 2B)\)
Barycentrics: \(\sin A \cos(2B - 2C) : \sin B \cos(2C - 2A) : \sin C \cos(2A - 2B)\)
Search Value (6, 9, 13): \(14.990638600206\)
▸ Alternate Coordinates1 form

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

▸ Incident Central Lines2 lines

{1, 1048}, {47, 91}

▸ Functional Relations1 relation

Notes

Clark Kimberling and Cyril Parry, “Products, Square Roots, and Layers in Triangle Geometry,” Mathematics and Informatics Quarterly 10 (2000) 9-22; on page 21, section 8.