The Lucas cubic is a pivotal isotomic cubic having pivot point at Kimberling center X_69, the isogonal conjugate of the orthocenter, i.e., the locus of points P such that the Cevian triangle of P is the pedal triangle of some point Q. The equation in trilinear coordinates is α cos A(b^2 β^2 - c^2 γ^2) + β cos B(c^2 γ^2 - a^2 α^2) + γ cos C(a^2 α^2 - b^2 β^2) = 0. Not only is the Lucas cubic invariant under isotomic conjugate, but also under cyclocevian conjugation. When P runs through the Lucas cubic, Q runs through the Darboux cubic.

Darboux cubic | pivotal isogonal cubic | triangle cubic

Édouard Lucas