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Reframing the Pythagorean Theorem

Given the right triangle in Figure 1 with leg lengths a and b and hypotenuse length c, the modern formulation of the golden Theorem of Pythagoras is that a2 + b2 = c2. There are scores of different proofs of this theorem. At the website Cut the Knot [2], Bogolmony details 118 diverse proofs of the theorem. In The Pythagorean Proposition [7], Loomis provides about 370 different proofs. In this article, we explore a proof technique that uses four congruent framing right triangles (each triangle as pictured in Figure 1) to create a diagram illustrating a derivation of the theorem. There are two standard framing configurations, namely the large frame model (Figure 2) and small frame model (Figure 3). At this point, we encourage you to attempt derivations of the Pythagorean theorem from these two diagrams. For each, there is a wordless geometric proof that results