X1117

POINT ALRESCHA

X1117 on the Lester circle, the cubic K060, and these lines: {5,3470}, {30,5671}, {265,13582}, {11071,11581}
< 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) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1117 = POINT ALRESCHA (0.9978, 0.0022) A B C X1117 (a, b, c) = (5, 6, 7) | AB = 1

X1117 (POINT ALRESCHA) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = n(a,b,c)/d(a,b,c)\), n(a,b,c) = 5(\(S_{A})\)2[(\(S_{B})\)2 + (\(S_{C})\)2] - \(3a^{2}(S_{A})\)3 - 4(\(S_{B})\)2(\(S_{C})\)2 - \(S_{A}\)\(S_{B}\)\(S_{C}(2S_{A} - S_{B} - S_{C})\), \(d(a,b,c) = 2a[4(S_{A})^{2} - b^{2}c^{2}][b^{2}c^{2}(3a^{2} - 8S_{A}) + 8(S_{A})^{3}]\), \(S_{A} = (b^{2} + c^{2} - a^{2})/2\); \(S_{B} = (c^{2} + a^{2} - b^{2})/2\); \(S_{C} = (a^{2} + b^{2} - c^{2})/2\); (coordinates by Edward Brisse, Peter J. C. Moses).
Barycentrics: \((a^{2}-a b+b^{2}-c^{2}) (a^{2}+a b+b^{2}-c^{2}) (a^{2}-b^{2}-a c+c^{2}) (a^{2}-b^{2}+a c+c^{2}) (a^{6}-a^{4} b^{2}-a^{2} b^{4}+b^{6}-3 a^{4} c^{2}+a^{2} b^{2} c^{2}-3 b^{4} c^{2}+3 a^{2} c^{4}+3 b^{2} c^{4}-c^{6}) (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}-a^{4} c^{2}+a^{2} b^{2} c^{2}+3 b^{4} c^{2}-a^{2} c^{4}-3 b^{2} c^{4}+c^{6}) (a^{8}-4 a^{6} b^{2}+6 a^{4} b^{4}-4 a^{2} b^{6}+b^{8}-4 a^{6} c^{2}+a^{4} b^{2} c^{2}+a^{2} b^{4} c^{2}+2 b^{6} c^{2}+6 a^{4} c^{4}+a^{2} b^{2} c^{4}-6 b^{4} c^{4}-4 a^{2} c^{6}+2 b^{2} c^{6}+c^{8}) : :\)
Search Value (6, 9, 13): \(-0.202008677882\)

X1117 lies on the Lester circle. See Bernard Gibert’s message, Hyacinthos #5613, 5/31/02.

▸ Functional Relations8 relations