Generalized Chapple-Euler Relation

TL;DR

This paper introduces a generalized relation extending the classical Chapple-Euler relation, providing new proofs and properties for triangles inscribed in a circle and circumscribed about a central conic, with implications for Poncelet triangles.

Contribution

It offers a new proof of the existence condition for such triangles and explores invariance properties of Poncelet triangles related to conic centers and foci.

Findings

The generalized relation reduces to the classical Chapple-Euler relation in special cases.

The sum of squared sides of Poncelet triangles is invariant under specific conditions.

Several new properties of triangles inscribed in a circle and circumscribed about a conic are derived.

Abstract

We present a new proof of the necessary and sufficient condition for the existence of a triangle that is simultaneously inscribed in a circle and circumscribed about a central conic (an ellipse or a hyperbola). In the limiting case where the foci of the conic coincide, the condition reduces to the classical Chapple-Euler relation. We also prove that the sum of the squared sides lengths of a Poncelet triangle is invariant over a family of Poncelet triangles inscribed in a circle and circumscribed about a central conic if and only if the circle is centered either at the center of the conic or at one of the foci of the conic, among several other properties of such triangles that we derive.