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X(1), excenters

furthermore, each of the three cubics contains :

• one vertex X of the circumnormal triangle which is the intersection of the cubic with its real asymptote.
• its isogonal conjugate P which is the corresponding point at infinity of the McCay cubic and the pivot of the cubic.

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 C-vertices 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.