A pivotal isogonal cubic is a self-isogonal cubic that possesses a pivot point, i.e., in which points lying on the conic and their isogonal conjugates are collinear with a fixed point known as the pivot of the cubic.
Pivotal isogonal cubics pass through the vertices of the Cevian triangle of the pivot point.
Let the trilinear coordinates of be , then has trilinear coordinates , or equivalently . If the trilinear coordinates of are , then collinearity requires
so the self-isogonal cubic with pivot point has a trilinear equation of the form
The only self-isogonal triangle center is the incenter , which a self-isogonal cubic therefore must pass through. Self-isogonal cubics also pass through the excenters , , and .
The following table summarizes some named pivotal isogonal cubics together with their pivot points and parameters .
Kimberling (1998, p. 240) gives lists of triangle centers passing through the pivotal isogonal cubics generated by the following pivot points: , , , , , , , , , , , , , , , , , , , , and .