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X(4), X(20), X(279), X(280), X(9292)

vertices of cevian triangle of X(69)

projections of X(20) on the altitudes

infinite points of pK(X6, X3146)

points of pK(X6, X12279) on (O)

Any line through X(20) (de Longchamps point) meets the Darboux cubic at two isogonal conjugate points M, N and the Lucas cubic at two cyclocevian conjugate points P, Q. The barycentric product R of P and Q also lies on the same line and the locus of R is the de Longchamps cubic. (see Hyacinthos #6622-23)

This cubic is not a pivotal isocubic (although it intersects the sidelines of ABC at the vertices of a cevian triangle) nor a strophoid (although it has a node with two perpendicular tangents parallel to the asymptotes of the rectangular circum-hyperbola through X(6) = K).

K041 has three (not all real) asymptotes concurring at X(376).

An easy construction is the following :

the trilinear polar of any point S on the line GK envelopes the Kiepert hyperbola and meets the lines joining X(20) to the vertices of cevian triangle of X(69) at Qa, Qb, Qc forming a triangle in perspective with ABC. The perspector is a point on the de Longchamps cubic.

K041 is also :

• the Lemoine generalized cubic K(X20) and it is an example of such cubic with three (not always real) concurring asymptotes at X(376), the reflection of G in O.

• psK+(X20, X69, X4) in Pseudo-Pivotal Cubics and Poristic Triangles. See also CL067.