Poncelet Triangles: a Theory for Locus Ellipticity

TL;DR

This paper develops a predictive theory for the shape of triangle center loci in Poncelet families, identifying conditions for conic shapes and analyzing their geometric properties.

Contribution

It introduces a general theory for predicting when triangle center loci are conics in Poncelet families, extending beyond case-by-case analysis.

Findings

Locus' turning number is +/- 3

Conditions for locus degenerating to a segment or circle

Monotonicity of locus with vertex motion

Abstract

We present a theory which predicts if the locus of a triangle center over certain Poncelet triangle families is a conic or not. We consider families interscribed in (i) the confocal pair and (ii) an outer ellipse and an inner concentric circular caustic. Previously, determining if a locus was a conic was done on a case-by-case basis. In the confocal case, we also derive conditions under which a locus degenerates to a segment or a circle. We show the locus' turning number is +/- 3, while predicting its monotonicity with respect to the motion of a vertex of the triangle family.