【摘要】 本文把欧氏平面，半球面和非欧面之中，不含给定边界，含有给定边界和含有边界而且在其上给定端点这样三种等周问题、给以初等、统一的证明。其要点在于把它们的存在性和唯一性简明扼要地归结到下述初等引理，即一个给定凹边边长的四边形的面积以四顶共圆时为其唯一的极大更多还原

【Abstract】 This note provides an elementary, unified proof of the isoperimetric problems in three kinds of geometries (namely, Euclidean, spherical and non-Euclidean) for the three cases of without boundary, with given boundary and with given boundary and initial points. The main idea of such a proof is to reduce the proof of both the existence and the uniqueness of isoperimetric solutions in all the above nine cases to the following elementary geometric lemma, namely Basic Lemma. The area function of quadrilaterals with given set of side-lengths (in Euclidean, spherical and non-Euclidean geometries) attains its unique maximal when its four vertices are cocircular.更多还原