Pappus' Generalization of Euclid I.47
The applet below illustrates a theorem (ca 300 A.D.) of Pappus of Alexandria that generalizes Pythagorean theorem and its proof in Euclid I.47 in two ways: the triangle ABC is not required to be right-angled and the shapes built on its sides are arbitrary parallelograms instead of squares. (The latter feature brings it closer to Euclid VI.31 than I.47.)
The proof proceeds in two steps, each using nothing but parallelogram shearing transformation as in several other proofs of the Pythagorean theorem. Use the "Move" button to observe the steps. (The applet actually shows four steps: two forward and two backward transformations.) Use the small blue circles (A, B, C, D, F) to modify the configuration.
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