Kasner meets Poncelet

TL;DR

This paper extends Kasner's theorem, which states the commutativity of certain pentagon constructions, to Poncelet polygons, revealing new geometric properties involving inscribed and circumscribed conics.

Contribution

It generalizes Kasner's theorem from pentagons to Poncelet polygons, broadening the understanding of geometric transformations involving conics.

Findings

Kasner's theorem is extended to Poncelet polygons.

The operations on Poncelet polygons commute under the extended theorem.

New geometric properties of polygons inscribed in and circumscribed about conics are established.

Abstract

Given a planar pentagon, construct two new pentagons: the vertices of the first one are the intersection points of the diagonals of the original pentagon, and the vertices of the second one are the tangency points of the conic inscribed in the original pentagon. E. Kasner theorem, published in 1928, asserts that these two operations on pentagons commute. We extend Kasner's result to Poncelet polygons, that is, the polygons inscribed into a conic and circumscribed about a conic.