The Lucas cubic is a pivotal isotomic cubic having pivot point at Kimberling center , the isogonal conjugate of the orthocenter, i.e., the locus of points such that the Cevian triangle of is the pedal triangle of some point .
The equation in trilinear coordinates is
Not only is the Lucas cubic invariant under isotomic conjugate, but also under cyclocevian conjugation.
When runs through the Lucas cubic, runs through the Darboux cubic.
The Lucas cubic passes through Kimberling centers for (triangle centroid ), 4 (orthocenter ), 7 (Gergonne point ), 8 (Nagel point ), 20 (de Longchamps point ), 69 (symmedian point of the anticomplementary triangle), 189, 253, 329, 1032, and 1034.