A different proof of the Pythagorean Theorem

Really many proofs are known for the Pythagorean Theorem. A huge collection you can find at Alexander Bogomolny - Cut the Knot. Of course, in fact one proof is sufficient. The nice thing about having various proofs is that you see that there are a great many ways to see the theorem. In this proof we derive the general theorem from one of its special cases. On the Cut the Knot site, this is proof number 64.

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Isosceles right triangle

We show that in an isosceles right triangle de hypothenusa has length √2 times the length of the right angle legs. This is a special case of the Pythagorean Theorem.

Let s be the number such that the hypothenusa has length s times the right angle legs. We get the following figure.

In this figure we see on one hand that the area of the complete triangle equals a², as the green and blue triangle together are a square with side a. On the other hand the area equals ½ · sa · sa = ½ s² a². So s² = 2 (and s = √2).

The general case

The general case we get from the figure above. The segment with length s·c divides the square in congruent parts. In one part the two pink right triangles both have area ½ ab, together ab. In the other part the area of the purple right triangle equals ½ · sa · sb = ab.

In both parts an equal area is left when we remove these right triangles. So ½ c² = ½ a² + ½ b². And the Pythagorean theorem is proved.

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