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X(2), X(2394) up to X(2419)

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 :

http://forumgeom.fau.edu/FG2001volume1/FG200115index.html

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 :

  1. 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.
  2. 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.
  3. (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).