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X(99), X(110), X(648), X(2407)

Let P be a variable point. The perpendiculars dropped from X(110) onto the lines AP, BP, CP meet BC, CA, AB at Pa, Pb, Pc. These three points are collinear if and only if P lies on the rectangular circum-hyperbola (H) passing through X(110). They form a triangle perspective to ABC if and only if P lies on the cubic K255. The perspector lies on K256.

K256 is a pivotal cubic with rather complicated pole and pivot S on the Steiner ellipse.

This generalizes for any point Q instead of X(110) on the circumcircle and we always obtain a pivotal cubic with pivot on the Steiner ellipse.