Poncelet Parabola Pirouettes

TL;DR

This paper explores intriguing geometric phenomena involving parabolas related to Poncelet triangles, revealing how key parabola features often trace simple geometric loci, supported by numerical evidence and posing open challenges.

Contribution

It uncovers and documents numerous novel geometric behaviors of parabolas in Poncelet triangle configurations, many of which are unproven and invite further mathematical proof.

Findings

Parabolas' focus, vertex, and directrix often sweep elementary loci.

Numerical evidence supports many observed phenomena.

The paper proposes open problems and experimental challenges.

Abstract

We describe some three-dozen curious phenomena manifested by parabolas inscribed or circumscribed about certain Poncelet triangle families. Despite their pirouetting motion, parabolas' focus, vertex, directrix, etc., will often sweep or envelop rather elementary loci such as lines, circles, or points. Most phenomena are unproven though supported by solid numerical evidence (proofs are welcome). Some yet unrealized experiments are posed as "challenges" (results are welcome!).