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see the general equation of Kc'(P) in CL018.

X(2), X(4), X(5), X(51), X(262)

infinite points of the McCay cubic K003

points of pK(X6, X51) on the circumcircle

Ga, Gb, Gc : vertices of the pedal triangle of G

other points below

K412 is the only equilateral cubic of the class CL018. It is the locus of the points of concurrence of the asymptotes of all pK+ with pivot the orthocenter H. The locus of the poles of these pK+ is K208 = Kw(X4).

In other words, for each point W on K208, the cubic pK(W, H) is a pK+ with asymptotes concurring at X on K412.

K412 has three real asymptotes parallel to those of the McCay cubic and concurring at the homothetic E = X(14845) of X(51) under h(X5, 1/3).

K412 is spK(X3, X5892) as in CL055.


K412 meets the nine point circle at six points :

• A1, A2, A3 which are the images of the vertices of the circum-normal triangle N1N2N3 under h(H,1/2). These points lie on the parallels at X(5) to the asymptotes of the McCay cubic. They are the contacts of the Steiner deltoid H3 with the nine point circle

• A', B', C' which are the intersections – apart X(125) – with the rectangular hyperbola (H) passing through X(2), X(4), X(6), X(51), X(125), X(1209), X(1640), X(1853), X(2574), X(2575).

The orthocenter of A'B'C' is X(51), the centroid of the orthic triangle.

K412 is actually the McCay cubic for this triangle A'B'C' hence it contains the in/excenters of A'B'C'.


Since K003 and K412 have already seven known common points, they must meet again at two points which lie on the line X(3)X(51) and on the rectangular circum-hyperbola which is its isogonal transform.