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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
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). It is actually psK(X20, X69, X4). See Pseudo-Pivotal Cubics and Poristic Triangles.
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.