Perhaps the most famous proof of all times is Euclid's geometric proof (Tropfke 1921ab; Tietze 1965, p. 19), although it is neither the simplest nor the most obvious. Euclid's proof used the figure below, which is sometimes known variously as the bride's chair, peacock tail, or windmill. The philosopher Schopenhauer has described this proof as a "brilliant piece of perversity" (Schopenhauer 1977; Gardner 1984, p. 153).
Shearing these triangles gives two more equivalent triangles
Therefore,
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