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).
- The trilinear polar of a point P meets the sidelines of ABC at A', B', C'. Denote by Oa, Ob, Oc the centers of the circles AB'C', BC'A', CA'B' respectively. The triangles ABC and OaObOc are perspective and directly similar. They are homothetic if and only if P lies on K010 (Angel Montesdeoca, 2019-10-12).