Perigal's Proof of the Pythagorean Theorem

We begin with a video approach to Perigal's proof (courtesy of YouTube).
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And this approach utilizes an applet.



Explanation:

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Assume b > a, so that the lines parallel and perpendicular to the hypotenuse of ABC are drawn in the second largest square ACBcBa through its midpoint, say 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.

A triangle will be formed with the sides parallel to those of ABC, but just half as big.

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. Since

(a + b)/2 - (b - a)/2  =  a 

we have the the emergence of a square of side a after rearrangement of the pieces on the hypotenuse.
 
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