Perigal's Proof of the Pythagorean Theorem

Explanation

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Assume b > a, so that the lines parallel and perpendicular to the hypotenuse of ΔABC are drawn in the square ACB_cB_a through its midpoint M. The length of both segments of these lines inside the square equal the length of the hypotenuse c. Each segment is divided by M into equal parts of length c/2.

Drop a perpendicular from M onto CB_c. Its foot will land in the middle of CB_c. Thus a triangle will be formed with the sides parallel to those of ΔABC, but just half as big.

From here we conclude that the endpoints on the add-on lines in square ACB_cB_a divide its sides (each of length b) into two parts (a + b)/2 and (b - a)/2. The difference of the two

(a + b)/2 - (b - a)/2 = a,

which explains the emergence of a square of side a after rearrangement of the pieces on the hypotenuse.

Equidecomposition by Dissection