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X(4), X(5), X(382)

reflections A', B', C' of A, B, C in the sidelines of ABC

Ix-anticevian points : see table 23

This cubic belongs to the pencil of cubics containing the Neuberg cubic, the Soddy cubic and the union of the three altitudes and also to the pencil of cubics containing the Napoleon cubic and the Kn cubic. See "Two Remarkable Pencils..." in the Downloads page.

It follows from the definitions found in this paper that this cubic is the locus of point P such that :

• the triangle formed by the nine-point centers of PBC, PCA, PAB is perspective with ABC.
• the triangle formed by the reflections of the circumcenters in the orthocenters of PBC, PCA, PAB is perspective with ABC. The perspector is also a point on the curve.

See also Hyacinthos #27709.