Siberian Mathematical Journal

, Volume 45, Issue 4, pp 740-762

First online:

Around the Proof of the Legendre–Cauchy Lemma on Convex Polygons

  • I. Kh. SabitovAffiliated withMoscow State University

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We briefly describe the history of the proofs of the well-known Cauchy lemma on comparison of the distances between the endpoints of two convex open polygons on a plane or sphere, present a rather analytical proof, and explain why the traditional constructions lead in general to inevitable appearance of nonstrictly convex open polygons. We also consider bendings one to the other of two isometric open or closed convex isometric polygons.

convex polygon isometry of polygons distance between endpoints of an open polygon isometric deformation of a polygon