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X(2), X(4), X(145), X(263), X(957), X(1992)

vertices of the pedal triangle of G

points at infinity of the Thomson cubic

common points of the circumcircle and pK(X6, X5640) where X5640 = X2-X51/\X6-X110

projections of G on the altitudes

Let M be the Miquel point of the quadrilateral formed by the sidelines of ABC and the trilinear polar L of P. Let M' be the second intersection of the line KM with the circumcircle. The Simson line of M' is perpendicular to L if and only if P lies on K015 (after Philippe Deléham). These lines are parallel if and only if P lies on K295.

K295 is the Lemoine generalized cubic K(X2) hence it is a nodal cubic with node G.

The nodal tangents are parallel to the asymptotes of the Kiepert hyperbola.

It has three real asymptotes parallel to those of the Thomson cubic.

It meets the Steiner ellipse again at three points and the tangents at these points are concurrent.

The isogonal transform of K295 is K297.

Locus properties :

  1. K295 = TC(X2) is a member of the class CL040 of Thomson centroidal cubics. It is the locus of point M such that the trilinear polar of M is perpendicular to the line GM.
  2. K295 is a tripolar centroidal cubic of the class CL045. It is the locus of M whose tripolar centroid lies on the orthic axis.
  3. K295 is the locus of the orthopivots of the orthopivotal cubics that meet the sidelines of ABC again at three points which are the vertices of a pedal triangle. This pedal triangle is that of a point on the Kiepert hyperbola. See K059 and K452 for example.
  4. see Q012, property 3 for another generalization.