A Poncelet Theorem for Lines

Our aim is to prove a Poncelet type theorem for a line configuration on the complex projective plane P. 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 : Theorem Let Ln be a configuration of n lines and D a smooth conic in P. 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. This result was probably known by Poncelet or Darboux but we did not find a similar statement in their publications. Anyway, even if it was, we would like to propose an elementary proof based on Frégier’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? We give a partial answer in proposition 2.6. This question leads also to Pascal theorem, to its dual version proved by Brianchon, and to its generalization proved by Möbius (see [1], thm.1 and [6], page 219). In the last section, using the Frégier’s involutions and the projective duality we prove the main theorem quoted above.