X368

EQUI-BROCARD CENTER

The center X for which the triangle XBC, XCA, XAB have equal Brocard angles. Peter Yff proved that X368 lies on the self-isogonal conjugate cubic with trilinear equation f(a,b,c)u + f(b,c,a)v + f(c,a,b)w = 0, where \(f(a,b,c) = bc(b^{2} - c^{2})\) and, for variable \(x : y : z\), the cubics u, v, w are given by u(x,y,z) = \(x(y^{2}\) + \(z^{2})\), v = u(y,z,x), w = u(z,x,y).
< 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_368 = EQUI-BROCARD CENTER (0.5443, 0.1976) A B C X368 (a, b, c) = (5, 6, 7) | AB = 1

X368 (EQUI-BROCARD CENTER) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Trilinears: (reasonable trilinears are sought)
Barycentrics: (reasonable barycentrics are sought)
Search Value (6, 9, 13): \(2.431598760008\)

Notes

Cyril Parry proved that X368 lies on the anticomplement of the Kiepert hyperbola, this anticomplement being given by the trilinear equation \(a^{2}(b^{2} - c^{2})\)\(x^{2}\) + \(b^{2}(c^{2} - a^{2})\)\(y^{2}\) + \(c^{2}(a^{2} - b^{2})\)\(z^{2}\) = 0.