Dissecting Circles to Prove a Square: A Novel Geometric Proof of the Pythagorean Theorem Using Circular Segments and Area Decomposition

TL;DR

This paper presents a new geometric proof of the Pythagorean Theorem using three circles, circular segments, and area decomposition, avoiding coordinate geometry and relying solely on classical Euclidean constructions and trigonometry.

Contribution

It introduces a novel approach combining circular symmetry and area analysis that has not been documented in prior literature, offering fresh insights into a classical theorem.

Findings

New proof based on three circumscribing circles

Area decomposition using circular segments and triangles

Elementary and rigorous geometric derivation

Abstract

The Pythagorean Theorem has been proved in hundreds of ways, yet it inspires fresh insights through geometry and trigonometry. In this paper, we offer a new proof based on three circles that circumscribe the sides of a right triangle. Rather than invoke coordinate geometry, the argument relies purely on classical Euclidean constructions, trigonometric identities independent of the theorem itself, and a careful analysis of the areas of circular segments. The key idea is to evaluate the area of the semicircle built on the hypotenuse in two distinct ways: directly and as a combination of areas formed by overlapping circular segments and triangles constructed on the legs of the triangle, as shown in Figure 10. Thales' Theorem, inscribed angle theorem, basic trigonometric identities, and segment area formulas all play a role in a derivation that is both elementary and rigorous. To the author's knowledge, this specific approach, which combines circular symmetry, angle decomposition, and area comparison, has not appeared in the prior literature, including Loomis' comprehensive catalog [3] and the extensive database at Cut-the-Knot [4]. As such, it provides both a new perspective on an ancient theorem and an example of how classical tools can still yield original insights.