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 
.
