标题： 欧几里得空间和闵可夫斯基空间模型上的球面和双曲毕达哥拉斯定理的合成证明 显示英文标题

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

摘要： 在球面几何和双曲几何中，毕达哥拉斯定理有多种推广形式。 一个自然的推广形式涉及以适当三角形的边长为半径的圆盘的面积，该形式在双曲情况下由Maria Teresa Calapso发现，并由Paolo Maraner推广到球面情况。 所有已知的证明都是解析的，Maraner提出了是否存在一种综合证明的问题。 在本文中，我们以一种对广大读者易于理解的方式解释定理的陈述。 接下来，我们使用欧几里得空间中的球面和闵可夫斯基空间中的双曲面作为球面和双曲几何的模型，给出该定理的一个初等几何证明。 显示英文摘要

摘要： 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.