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What is the most elegant proof of the Pythagorean theorem?

How do we prove that the Pythagorean theorem holds for a right angled isoceles triangle with sides, $a,b,a$.

For a right angled triangle with sides $a,b,c$, where $\angle C = 90^{\circ}$, we have $a^2+b^2=c^2$

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    you say that "Pythagoras stated $a^2+b^2=c^2$, assuming a≠b≠c. ", however, I think a more correct statement is "Pythagoras stated $a^2+b^2=c^2$, including cases where a≠b≠c. "2012-08-20

1 Answers 1

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Make a paper square with sides $b$. Divide it into $4$ triangles by drawing the two diagonals. Cut along the diagonals.

We get $4$ congruent isosceles right-angled triangles. Let the right-angled triangles we get have legs $a$. They each have hypotenuse $b$.

We can put these triangles together in pairs along their hypotenuses to form two $a\times a$ squares. So we have cut a $b\times b$ square into four pieces and reassembled the pieces to make two $a\times a$ squares. Since area is conserved, it follows that $a^2+a^2=b^2.$

Remark: This is a simple dissection proof of a very special case of the Pythagorean Theorem. There are several general dissection proof of the Theorem.