  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 the field of
complex numbers, or both. Here we show that Poncelet's theorem holds for any
two conics C and D in the projective plane over an algebraically closed field k
of 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 the
situation in characteristic two in the appendix.
