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The Droussent cubic K008 and the Droussent medial cubic K043 are two circular pivotal cubics which generate a pencil of cubics we call the Droussent pencil.

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.