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X(20)

Any inscribed conic with perspector P on the Steiner ellipse is a parabola touching the sidelines of ABC at the vertices of the cevian triangle T of P. When P traverses the Steiner ellipse, the locus of the circumcenter of T is the cubic K462. See also K461.

K462 is an acnodal cubic with singularity the de Longchamps point X(20).

It has three real asymptotes forming a triangle homothetic to the pedal triangle of the Lemoine point K. The center of homothety is X on the lines X(4)X(1384), X(5)X(3053), X(6)X(550), etc. The ratio of homothety is cot^2w where w is the Brocard angle.