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X(2), X(11058)

A', B', C' reflections of A, B, C in G

infinite points of K092

The locus of point P such that the sum of line angles (BC,AP)+(CA,BP)+(AB,CP) = t (mod. pi) is a K60+ cubic with three asymptotes concurring at G. When t = 0, the cubic is Kjp = K024 and with t = pi/2, it is the McCay cubic K003. Any other cubic belongs to the pencil generated by these two cubics and meets the circumcircle at A, B, C and three other points which are the vertices of an equilateral triangle.

In particular, the pencil contains one and only one K60++ which is K213. It is a central equilateral cubic with inflexional tangent at G passing through the Schoute center X(187).

See the page K092 for a description of X(11057).

See another characterization of K213 in table 22.


K213 also belongs to the pencil of equilateral cubics generated by K092 and K104.

All these cubics have the same points at infinity and the same tangent at G (when it is defined) which is the line through X(187). The last remaining base point is X(11058).

This pencil also contains :

• the cubic decomposed into the line at infinity and the circum-conic through X(2), X(67), X(599), X(11058) with perspector X(3906).

• a nodal cubic with node G.


The hessian of K213 is a central focal cubic with focus and center G.

This contains the isodynamic points X(15), X(16).