Constructions in the Locus of Isogonal Conjugates in a Quadrilateral

TL;DR

This paper studies the locus of points in a quadrilateral where pairs of lines are isogonal, revealing that this locus is a specific cubic curve and providing geometric constructions using classical tools.

Contribution

It characterizes all cubic curves associated with isogonal conjugates in a quadrilateral and offers geometric constructions involving these cubics.

Findings

The locus of points with isogonal pairs forms a cubic curve.

All such cubic curves can be characterized explicitly.

Constructive methods for intersections and tangents to these cubics are provided.

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.