X592

VAN LAMOEN CIRCUMCENTERS POINT

Let P be the point where the line through X6 parallel to line CA meets line BC, and let Q be the point where the line through X6 parallel to line AB meets line BC. Let X = X182, and let A' be the circumcenter of the triangle PQX. Define B' and C' cyclically. The lines AA', BB', CC' concur in X592. (Floor van Lamoen, 1/4/01)
< 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_592 = VAN LAMOEN CIRCUMCENTERS POINT (0.5761, 0.2269) A B C X592 (a, b, c) = (5, 6, 7) | AB = 1

X592 (VAN LAMOEN CIRCUMCENTERS POINT) 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 \(f(A,B,C) = 1/[\cos A + 2 \cos(B - \omega ) \cos(C - \omega )]\)
Barycentrics: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A,B,C) = (\sin A)f(A,B,C)\)
Search Value (6, 9, 13): \(2.043062756727\)