The Simson cubic is the locus of tripoles of the Simson lines of triangle ABC hence it is the dual of the Steiner deltoid H3. A study can be found at :
The Simson cubic is a special case of isotomic conico-pivotal isocubic. It is cK(#X2, X69). The line PQ (see below) envelopes the ellipse centered at K which is inscribed in the antimedial triangle. The contact conic is the circum-conic centered at X(216). The three real inflexion points lie on the trilinear polar of X(95). See Special isocubics, §8.
The isogonal transform of the Simson cubic is K162 = cK(#X6, X3) and its H-isoconjugate is K406.
The homothetic of K010 under h(G,1/4) is related to the class CL001 of isogonal central nK cubics.
Locus properties :
- Locus of point P such that the trilinear polars of P and its isotomic conjugate Q are perpendicular. The intersection of these two lines lies on the nine-point circle. Thus, K010 is a member of the class CL008 of cubics.
- Let M be the Miquel point of the quadrilateral formed by the sidelines of ABC and the trilinear polar L of P. The Simson line of M is parallel to L if and only if P lies on K010.
- (equivalently) The Newton line of the quadrilateral formed by the sidelines of ABC and the trilinear polar L of P is perpendicular to L if and only if P lies on K010 (Philippe Deléham).