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

Abstract

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