Albrecht Hess, Daniel Perrin, and Mehdi Trense,
A group theoretic interpretation of Poncelet's theorem -- the real case,
Forum Geometricorum, 16 (2016) 381--395.

Abstract. Poncelet's theorem about polygons that are inscribed in a conic and at the same time circumscribe another one has a greater companion, in which the second conic is substituted by possibly different conics for different sides of the polygon, while all conics belong to a fixed pencil. Here, a construction is presented that gives a visual group theoretic interpretation of both theorems and, eventually, leads to a generalization exposing the role of commutativity in Poncelet's theorem. There is no new thing about the ingredients but we hope that a dynamical view sheds new light on them. Finally, the occurrence of conics in a Poncelet grid [14] of lines constructed on a pencil of circles is explained with a simple proof.

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