A synthetic proof of the spherical and hyperbolic Pythagorean theorem on models in Euclidean and Minkowski space

Abstract: There are multiple generalisations of the Pythagorean theorem to spherical and hyperbolic geometry. A natural one, involving areas of disks with radii equal to the sides of a proper triangle, was discovered in the hyperbolic case by Maria Teresa Calapso and generalised to the spherical case by Paolo Maraner. All known proofs are analytic, and Maraner posed the question of whether there is a synthetic proof. In this paper, we explain the statement of the theorem in a way that is accessible to a wide audience. Next, we give an elementary geometric proof of this theorem using the sphere in Euclidean space and the hyperboloid in Minkowski space as models for spherical and hyperbolic geometry.