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

TL;DR

This paper presents a fully visual dissection proof of the law of cosines that replaces rectangles with congruent triangles, simplifying the conceptual understanding and making it suitable for educational purposes.

Contribution

It introduces a novel dissection proof of the law of cosines using congruent triangles, avoiding trigonometry and enhancing visual clarity.

Findings

Provides the only dissection proof of its kind

Simplifies understanding of the law of cosines

Suitable for teaching geometry courses

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.