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Poncelet's theorem states that if there exists an n-sided polygon which isinscribed in a given conic C and circumscribed about another conic D, thenthere are infinitely many such n-gons. Proofs of this theorem that we are awareof, including Poncelet's original proof and the celebrated modern proof byGriffiths and Harris, assume the two conics to be in general position (that is,not tangent or at least not osculating), or be defined over the field ofcomplex numbers, or both. Here we show that Poncelet's theorem holds for anytwo conics C and D in the projective plane over an algebraically closed field kof any characteristic other than two. If C and D are osculating and char(k)>2,our result shows that there always exist infinitely many polygons of char(k)sides that are inscribed in C and circumscribed about D. We also describe thesituation in characteristic two in the appendix.

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