All these cubics contains A, B, C, the circular points at infinity, G, X67, X524 (at infinity) and are tangent at G to the line G, X187 with the exception of K394.
Their real asymptotes are parallel to the line GK.
Their singular foci lie on a same line through G and the midpoint of O, X111.
This pencil contains :
– a cubic decomposed into the line at infinity and a circum conic passing through the points G, X67 and X599.
– a third pK which is K103 = pK(X67, X67).
– a central focal cubic with center G which is K065, the orthopivotal cubic O(X524).
– a nodal cubic with node G which is K394.
– two other focal cubics which are real when the triangle ABC is acutangle.
– two nKs with complicated equations.
See a generalization in CL051.