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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 = X2X51/\X6X110 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 DelĂ©ham). 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 :

