Information from Kimberling's Encyclopedia of Triangle Centers |
Trilinears (reasonable trilinears are sought)
Barycentrics (reasonable barycentrics are sought)
The center X for which the triangle XBC, XCA, XAB have equal Brocard angles. Peter Yff proved that X(368) 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).
Cyril Parry proved that X(368) 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.