X(2), X(3), X(4), X(13), X(14), X(67), X(1113), X(1114)
midpoints of ABC
Ga, Gb, Gc vertices of the antimedial triangle
A', B', C' projections of O on the sidelines of the antimedial triangle
points at infinity of the Thomson cubic
foci of the inconic with perspector X(76), center X(141)
Q050 is a circular quintic with singular focus G. It has three real asymptotes parallel to those of the Thomson cubic.
A, B, C are double points on the curve.
Q050 is the locus of point M such that G, M and the isogonal conjugate of M with respect to the circumcevian triangle of M (or the inverse in (O) of the isogonal conjugate of M) are collinear.
The isogonal transform of Q050 is Q136.
Let P be a point. The locus of point M such that P, M and the isogonal conjugate of M with respect to the circumcevian triangle of M are collinear is in general a circular circum-quintic Q(P) passing through H, P, the vertices of the cevian triangle of P, the infinite points of pK(X6, P), the foci of the inconic with perspector tgP and center cgP, the intersections of the circumcircle and the line OP when P ≠ O (when P = O, Q(P) splits into the circumcircle and the McCay cubic). A, B, C are three nodes on Q(P). For example, Q050 = Q(X2) and Q094 = Q(X6).
Special cases :
A pencil of related quintics