  We provide a new proof of the elementary geometric theorem on the existence
and uniqueness of cyclic polygons with prescribed side lengths. The proof is
based on a variational principle involving the central angles of the polygon as
variables. The uniqueness follows from the concavity of the target function.
The existence proof relies on a fundamental inequality of information theory.
  We also provide proofs for the corresponding theorems of spherical and
hyperbolic geometry (and, as a byproduct, in $1+1$ spacetime). The spherical
theorem is reduced to the euclidean one. The proof of the hyperbolic theorem
treats three cases separately: Only the case of polygons inscribed in compact
circles can be reduced to the euclidean theorem. For the other two cases,
polygons inscribed in horocycles and hypercycles, we provide separate
arguments. The hypercycle case also proves the theorem for "cyclic" polygons in
$1+1$ spacetime.
