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

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.