A New Lemoine-Type Circle

TL;DR

This paper introduces a new geometric circle related to six-point configurations, proves its properties, and compares it to existing Lemoine and Tucker circles.

Contribution

It defines a novel Lemoine-type circle, establishes its properties, and clarifies its relationship with known geometric circles.

Findings

The new circle is defined by a specific six-point configuration.

A converse theorem for the new circle is proved.

The new circle is shown not to be a Tucker circle.

Abstract

This paper presents a new Lemoine-type circle defined by a six-point configuration satisfying a cocyclicity criterion. We prove the result, establish a converse theorem, and relate the new circle to previously known Lemoine circles, in particular the one introduced by Q.T. Bui. We show that the new circle does not belong to the family of Tucker circles.