A shear is a transformation of a rectangle into a parallelogram which preserves one base and the corresponding height.
Shears can be used in proving Pythagoras' Theorem because shears preserve area. However, there are several different ways to use them.
Jim Morey's first Java demonstration
Byrnes' title page
Logically, these are among the most elementary proofs. Euclid's version (his Proposition I.47) is contained already in his first book.
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