  Given fixed distinct points $A, B, C, D$, we examine properties of the locus
of points $X$ for which $(XA, XC)$, $(XB, XD)$ are isogonal. This locus is a
cubic curve circumscribing $ABCD$. We characterize all possible such cubics
$\mathcal C \in \mathbb R^2$. These properties allow us to present
constructions involving these cubics, such as intersections and tangent lines,
using straightedge and compass.
