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X(3), X(54), X(7691)

infinite points of K005

vertices of the CircumNormal triangle N1N2N3

vertices of the CircumTangential triangle T1T2T3

Geometric properties :

See K735, a similar cubic where a generalization is given, and also Table 25.

K1268 meets the Napoleon cubic K005 at three points on the line at infinity and six other points on the rectangular hyperbola (H) passing through X(3), X(54) and X(2), X(2574), X(2575).

K1268 meets the perpendicular bisectors of ABC again at three pairs of points {A1,A2}, {B1,B2}, {C1,C2}, which lie on a same circle (C) with center O and radius R / √3.

The CircumTangential-isogonal transform of K1268 is a stelloid with radial center X(549), whose asymptotes are parallel to those of K003. It passes through O and X(5).

Note that the CircumNormal-isogonal transform of K1268 is the reflection of the previous stelloid about O. Hence, its radial center is X(8703) and it passes through O and X(550).