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Ga, Gb, Gc :vertices of the antimedial 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.

K211 is a nK with pole H and root X(264), the isotomic conjugate of O.

The tangents at A, B, C are the sidelines of the tangential triangle. They meet the cubic again on the trilinear polar of X(250), the isogonal conjugate of the Jerabek center X(125).

The complement of K211 is a cubic psK(X647, X2, ?).

Other locus properties :

1. Let P be a point and PaPbPc its cevian triangle, HaHbHc the orthic triangle (cevian triangle of H). The parallel through the point Pb to the sideline HaHb and the parallel through the point Pc to the sideline HaHc intersect at A'. The points B' and C' defined likewise. The triangles PaPbPc and A'B'C' are perspective if and only if P lies on on K211, together with the three circm-rectangular hyperbolas passing through Ga, Gb, Gc (Angel Montesdeoca, Hyacinthos #21806).