X3564

1st MOSES INFINITY POINT

X3564 = crossdifference of every pair of points on line X6X924
< 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) { 5, 6 }: Nine-Point–Symmedian Line { 3, 69 }: Central Line L(3, 69) { 26, 159 }: Central Line L(26, 159) { 4, 193 }: Central Line L(4, 193) { 140, 141 }: Central Line L(140, 141) { 114, 230 }: Central Line L(114, 230) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_2 = CENTROID (0.5374, 0.2000) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.4122, 0.0654) X_26 = CIRCUMCENTER OF THE TANGENTIAL TRIANGLE (0.1382, -0.8447) X_159 = -OF-TANGENTIAL-TRIANGLE (0.4211, -0.7266) X_4 = ORTHOCENTER (0.6122, 0.3958) X_193 = -CEVA CONJUGATE OF X(2) (0.7878, 0.4690) X_140 = MIDPOINT OF X(3) AND X(5) (0.5281, 0.1755) X_141 = COMPLEMENT OF SYMMEDIAN POINT (0.5061, 0.1663) X_114 = KIEPERT ANTIPODE (0.3616, 0.4140) X_230 = -CEVA CONJUGATE OF X(114) (0.9746, 0.6698) X_3564 = 1st MOSES INFINITY POINT (1.0790, 0.4260) A B C X2 X5 X6 X3 X69 X26 X159 X4 X193 X140 X141 X114 X230 X3564 (a, b, c) = (5, 6, 7) | AB = 1

X3564 (1st MOSES INFINITY POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \((\cos A)(b^{2} \cos^{2} C + c^{2} \cos^{2} B - bc \cos A) : :\)
▸ Alternate Coordinates1 form

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

▸ Incident Central Lines38 lines

{2, 3167}, {3, 69}, {4, 193}, {5, 6}, {26, 159}, {30, 511}, {52, 1843}, {98, 325}, {110, 468}, {114, 230}, {115, 1570}, {125, 3292}, {140, 141}, {147, 385}, {156, 206}, {184, 343}, {265, 895}, {287, 441}, {323, 858}, {381, 1992}, {383, 3181}, {394, 1368}, {427, 1993}, {428, 3060}, {429, 3193}, {495, 611}, {496, 613}, {546, 576}, {547, 597}, {548, 3098}, {549, 599}, {550, 1350}, {1080, 3180}, {1483, 3242}, {1625, 2211}, {1994, 3410}, {2023, 2025}, {2930, 2931}

▸ Functional Relations6 relations

Notes

X3564 and X3566 are on the line at infinity. They may be regarded as directions in the plane of ABC, and as such are perpendicular. Continuing from X3563, the infinite point of the line O'O″ is X3564. (Randy Hutson, June 19, 2015)