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X(1), excenters furthermore, each of the three cubics contains :
more points and details below 

There are three isogonal circular pKs whose singular focus F lies on the real asymptote. These are the three McCay circular cubics K227A, K227B, K227C. See a general discussion on circular isogonal pKs in Special Isocubics §4.1.1 (downloads page). Any K227 must have its pivot P at infinity on the McCay cubic, and therefore its real asymptote is parallel at O to one of the asymptotes of the McCay cubic. It meets this asymptote at X, isogonal conjugate of P, which is a vertex of the circumnormal triangle and a point on the circumcircle and on the McCay cubic. The singular focus F is the antipode of X on the circumcircle. F is a vertex of the circumtangential triangle and lies on the circumcircle and on the Kjp cubic. The tangents at the four in/excenters I, Ia, Ib, Ic are obviously parallel to the asymptote and the parallels at A, B, C to this same asymptote meet the sidelines of ABC at three points A', B', C' on the curve. K227 is invariant under four inversions with pole one of the in/excenters, swapping one vertex of ABC and the corresponding in/excenter (for example, the inversion with pole I which swaps A and Ia, B and Ib, C and Ic). 

Peter Moses observes that each cubic K227 contains one vertex of each of the following triangles : • the three Morley triangles M1, M2, M3, • their isogonal conjugates or adjuncts A1, A2, A3, • the (light blue) circumnormal triangle. Those represented on the figure above correspond to the Cvertices on the cubic K227C. Recall that the singular focus F is a vertex of the circumtangential triangle. Note that the polar conic of F must split into the line at infinity and another line since F lies on the real asymptote. 
