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X(6), X(7), X(9), X(173), X(268), X(281), X(3161)

A', B', C' : midpoints of ABC

The locus of perspectors P of circum-conics having an axis passing through a given point Q is in general a cubic. This cubic is a circum-cubic if and only if Q is an in/excenter of ABC.

When Q = I (incenter), the cubic is K220. This is a nodal cubic with node X(9), the mittenpunkt. The nodal tangents are parallel to the asymptotes of the Feuerbach hyperbola. The tangents at A, B, C concur at X(55) not on the curve.

When P = X(9), the circum-conic is the one with center I.

If we replace the circum-conics by in-conics, we obtain a similar cubic which is K257.

K220 is a member of the class CL033 (Deléham cubics).

It is also psK(X55, X2, X6) in Pseudo-Pivotal Cubics and Poristic Triangles.

The isogonal transform of K220 is K577 and its isotomic transform is K1069.

Construction :

For any point Q on the line X(1)X(2), the trilinear polar of Q meets the lines X(9)A', X(9)B', X(9)C' at Qa, Qb, Qc. ABC and QaQbQc are perspective and the perspector is a point on K220. This gives a simple way for finding a lot of reasonably simple points on the curve.