Triangles with Vertices Equidistant to a Pedal Triangle

TL;DR

This paper provides a synthetic geometric solution to an open problem involving radical axes and circumcircles, introducing two generalizations and leveraging projective geometry concepts like isogonal conjugation and polarity.

Contribution

It offers a novel synthetic approach to a longstanding open problem, incorporating generalizations and advanced projective geometry techniques.

Findings

Solved an open problem in geometry.

Introduced two new generalizations related to radical axes.

Connected isogonal conjugation and polarity with circumconics.

Abstract

In this paper, we present a synthetic solution to a geometric open problem involving the radical axis of two strangely defined circumcircles. The solution encapsulates two generalizations, one of which uses a powerful projective result relating isogonal conjugation and polarity with respect to circumconics.