X(6), X(7), X(69), X(264), X(9723), X(15394)
vertices of the cevian triangle of X(76).
The locus of the perspector P of inscribed conics such that one axis passes through a fixed point Q is in general a sextic which is the isogonal transform of a cubic. This cubic is a circum-cubic if and only if Q is the circumcenter O of ABC. In this case, the sextic decomposes into the line at infinity, the Steiner circum-ellipse and K257. The locus of the centers of the corresponding inscribed conics is K258.
K257 is a nodal K+ with node X(69), the isotomic conjugate of H. The nodal tangents are parallel to the asymptotes of the Jerabek hyperbola. The asymptotes concur at X(599), the reflection of K in G.
K257 meets the sidelines of ABC at the vertices of the cevian triangle of X(76). The tangents at A, B, C passes through O.
When P = X(69), the inscribed conic (C) is centered at O and its axes are parallel to the asymptotes of the Jerabek hyperbola. (C) contains X(125), the center of the Jerabek hyperbola and X(1565). The foci of (C) are obviously on the McCay cubic.
When P = X(6), X(7), X(264) the inscribed conics are the Brocard ellipse, the incircle, the MacBeath conic respectively.