A dissection proof of the law of cosines, replacing Cuoco-McConnell's\n rectangles with congruent triangles

2018/06/01 by Martin Celli, Celli, Martin
Engineering · Mathematics · #Advanced Theoretical and Applied Studies in Material Sciences and Geometry #FOS: Mathematics #History and Overview (math.HO) #History and Theory of Mathematics #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.1806.00207

openalex publication_date 2018/06/01 · openalex created_date 2022/08/31 · openalex updated_date 2026/07/28

Abstract

Taking up the challenge McConnell laid down at the end of his proof of the\nlaw of cosines, we give a completely visual dissection proof of this theorem,\nwhich applies to any triangle. In order to avoid the trigonometric expressions\nof Cuoco-McConnell's proof, we replaced the equal-area rectangles with\ncongruent triangles. As a matter of fact, trigonometric expressions are\nimplicitly based on the similarity of two right triangles with a common\nnon-right angle. So they are conceptually less simple than our congruent\ntriangles which are, moreover, easy to visualize. This makes our proof the only\ndissection proof and the simplest proof of its family, and thus one of the best\noptions for a course of geometry.\n

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