A Theorem about Simultaneous Orthological and Homological Triangles

TL;DR

This paper proves a new geometric theorem establishing that if two isogonal points produce pedal triangles with certain homological properties, then the original triangle maintains homology with both pedal triangles.

Contribution

It introduces a novel theorem linking isogonal points, pedal triangles, and homological relationships within a triangle.

Findings

If two isogonal points have pedal triangles with a homological relation to the original triangle, then the second pedal triangle is also homological.

The theorem generalizes the understanding of homological and orthological triangle configurations.

Provides a new geometric condition connecting pedal triangles and homology in triangle geometry.

Abstract

In this paper we prove that if $P_1, P_2$ are isogonal points in the triangle $ABC$, and if $A_1B_1C_1$ and $A_2B_2C_2$ are their corresponding pedal triangles such that the triangles $ABC$ and $A_1B_1C_1$ are homological (the lines $AA_1, BB_1, CC_1$ are concurrent), then the triangles $ABC$ and $A_2B_2C_2$ are also homological.