Constructions in the Locus of Isogonal Conjugates in a Quadrilateral

Abstract

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.

19 pages, 10 figures