Fine proprieties of Lemoine point and the inscribed conic

TL;DR

This paper explores the properties of the Lemoine point and inscribed conics, providing new insights into their geometric relationships and offering a synthetic proof of a specific property involving the conic center and isotomic conjugate.

Contribution

It introduces a synthetic proof of a property relating the Lemoine point, the center of an inscribed conic, and its isotomic conjugate, expanding understanding of these geometric concepts.

Findings

The center of an inscribed conic with a given perspector is the complement of its isotomic conjugate.

A synthetic proof of this property is provided based on properties of the Lemoine point.

The paper clarifies the relationship between the Lemoine point and inscribed conics.

Abstract

The center of an inscribed conic which have a given perspector is the complement of its isotomic conjugate. We provide a synthetic proof, based on fine proprieties of Lemoine point.