Harmonious loci of Poncelet triangles about the incircle and their degeneracies

TL;DR

This paper explores special properties and loci of Poncelet triangles inscribed in an ellipse and tangent to the incircle, highlighting how degeneracies occur with equilateral triangles.

Contribution

It investigates the geometric loci and degeneracies of Poncelet triangles related to the incircle, revealing new insights into their Euclidean properties.

Findings

Loci of triangle centers are characterized.

Envelope structures of key geometric objects are described.

Degenerate cases are linked to equilateral triangles.

Abstract

We tour several Euclidean properties of Poncelet triangles inscribed in an ellipse and circumscribing the incircle, including loci of triangle centers and envelopes of key objects. We also show that a number of degenerate behaviors are triggered by the presence of an equilateral triangle in the family.