A Poncelet theorem for lines

TL;DR

This paper proves a Poncelet type theorem for line configurations in the complex projective plane, establishing conditions for the existence of infinitely many polygons inscribed in lines and circumscribed around a conic.

Contribution

It introduces a new Poncelet theorem for line configurations, providing an elementary proof using Frégier's involution and connecting classical theorems like Pascal and Brianchon.

Findings

Existence of infinitely many polygons inscribed in line configurations

Connection between involutions and classical projective geometry theorems

Elementary proof based on Frégier's involution

Abstract

Our aim is to prove a Poncelet type theorem for a line configuration on the complex projective. More precisely, we say that a polygon with 2n sides joining 2n vertices A1, A2,..., A2n is well inscribed in a configuration Ln of n lines if each line of the configuration contains exactly two points among A1, A2, ..., A2n. Then we prove : "Let Ln be a configuration of n lines and D a smooth conic in the complex projective plane. If it exists one polygon with 2n sides well inscribed in Ln and circumscribed around D then there are infinitely many such polygons. In particular a general point in Ln is a vertex of such a polygon." We propose an elementary proof based on Fr\'egier's involution. We begin by recalling some facts about these involutions. Then we explore the following question : When does the product of involutions correspond to an involution? It leads to Pascal theorem, to its dual version proved by Brianchon, and to its generalization proved by M\"obius.