Universal measure for Poncelet-type theorems

TL;DR

This paper introduces a new invariant measure on the circle to provide a simple proof of the Emch closing theorem, unifying and extending results related to Poncelet's and Zigzag theorems.

Contribution

The paper presents a novel invariant measure on the circle that simplifies proofs and generalizes classical Poncelet-type theorems.

Findings

A new invariant measure on the circle is introduced.

The measure simplifies proofs of classical theorems.

Generalizations of Poncelet-type theorems are achieved.

Abstract

We give a simple proof of the Emch closing theorem by introducing a new invariant measure on the circle. Special cases of that measures are well-known and have been used in the literature to prove Poncelet's and Zigzag theorems. Some further generalizations are also obtained by applying the new measure.