Extreme Points on Circumconics Induced by Isogonal Conjugates in a Triangle

TL;DR

This paper explores properties of circumconics in triangles related to isogonal conjugates, focusing on tangent conics and the positions of extreme points on axes, leading to a general geometric result.

Contribution

It introduces new configurations involving isogonal conjugates and tangent conics, culminating in a general theorem about extreme points on circumconics.

Findings

Characterization of tangent conics at two points

Properties of isogonal conjugates in these configurations

A general theorem on extreme points of circumconics

Abstract

We first introduce a configuration of arbitrary isogonal conjugates related to a known property concerning the spiral center of two pairs of isogonal conjugates. We then consider a special case where two conics are tangent at exactly two points. Finally, we apply the discoveries made in both configurations to state a general result concerning the extreme points (those lying on either the major or minor axis) of certain circumconics of a triangle.