# Perigal's Proof of the Pythagorean Theorem

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A few words and an explanation.

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 What if applet does not run?

Assume b > a, so that the lines parallel and perpendicular to the hypotenuse of ΔABC are drawn in the square ACBcBa 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 CBc. Its foot will land in the middle of CBc. 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 ACBcBa 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.