Cyclic polygons in classical geometry

Abstract

Formulas about the side lengths, diagonal lengths or radius of the circumcircle of a cyclic polygon in Euclidean geometry, hyperbolic geometry or spherical geometry can be unified.

Volumes of Polytopes in Spaces of Constant Curvature

“…Its side and diagonal lengths are related by the equation Together with the Ptolemy's theorem this equation allows to express the diagonal lengths of a cyclic quadrilateral by its side lengths. It was noted in [22] that the above mentioned relationships between sides and diagonals of a cyclic quadrilateral are valid also in the hyperbolic geometry. To make them true, one can change the length side a by the quantity s(a) = sinh a 2 .…”

“…In these cases, instead of function s(a) one should take the functions s(a) = a and s(a) = sin a 2 respectively. See [22] and [17] for the arguments in the spherical case.…”