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.

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 .

Cundy, H. M. and Parry, C. F. "Some Cubic Curves Associated with a Triangle." J. Geom.53, 41-66, 1995.Kimberling,
C. "Triangle Centers and Central Triangles." Congr. Numer.129,
1-295, 1998.Yff, P. "Two Families of Cubics Associated with a Triangle."
In MAA Notes, No. 34. Washington, DC: Math. Assoc. Amer., 1994.