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vertices of the orthic triangle

Let P be a point and PaPbPc its cevian triangle. The antiparallels through Pa, Pb, Pc to the sidelines of ABC concur if and only if P lies on K211. The locus of the point of concurrence is K212.

The tangents at A, B, C are parallel and perpendicular to the Brocard axis. The three real asymptotes are the parallels to the sidelines of the tangential triangle through the relative midpoints of ABC. They meet the curve again on L, the trilinear polar of the point [(b^2 - c^2) / (b^2 + c^2) : : ], passing through X(115), X(804), etc.

 

K212 is a psK cubic as in Pseudo-Pivotal Cubics and Poristic Triangles.