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Poncelet’s theorem states that if there exists an  n -sided polygon which is inscribed in a given conic  C  and circumscribed about another conic  D , then there are infinitely many such  n -gons. Proofs of this theorem that we are aware of, including Poncelet’s original proof and the celebrated modern proof by Griffiths and Harris, assume the two conics to be in general position (that is, not tangent or at least not osculating), or be defined over, or both. Here we show that Poncelet’s theorem holds for  any  two conics  C  and  D  in the projective planeover an algebraically closed field  k  of  any  characteristic other than two. If  C  and  D  are osculating and, our result shows that there  always  exist infinitely many polygons ofsides that are inscribed in  C  and circumscribed about  D . We also describe the situation in characteristic two in the appendix.

Poncelet’s theorem for conics in any position and any characteristic