Loci of Poncelet Triangles in the General Closure Case

TL;DR

This paper investigates the loci of triangle centers in generalized Poncelet porisms, revealing that despite complex geometry, these loci often remain conics or circles.

Contribution

It extends classical Poncelet porisms to a more general setting with separate caustics, showing invariance of certain loci as conics or circles.

Findings

Loci of triangle centers are conics or circles in the general case.

Despite complex geometry, certain loci remain invariant.

The work generalizes classical Poncelet porisms to new configurations.

Abstract

We analyze loci of triangle centers over variants of two-well known triangle porisms: the bicentric and confocal families. Specifically, we evoke the general version of Poncelet's closure theorem whereby individual sides can be made tangent to separate in-pencil caustics. We show that despite the more complicated dynamic geometry, the locus of certain triangle centers and associated points remain conics and/or circles.