# Perigal's Proof of the Pythagorean Theorem

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A few words and an explanation.

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Copyright © 1996-2018 Alexander Bogomolny### 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_{c}B_{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_{c}B_{a} divide its sides (each of length b) into two parts

(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

- Carpet With a Hole
- Equidecomposition of a Rectangle and a Square
- Equidecomposition of Two Parallelograms
- Equidecomposition of Two Rectangles
- Equidecomposition of a Triangle and a Rectangle
- Equidecomposition of a Triangle and a Rectangle II
- Perigal's Proof of the Pythagorean Theorem
- Two Symmetric Triangles Are Directly Equidecomposable
- Wallace-Bolyai-Gerwien Theorem

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Copyright © 1996-2018 Alexander Bogomolny63969414 |