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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 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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