What simple proofs of the Pythagorean theorem do you know? Which of them could Pythagoras know? Why did he come up with his own, more complex proof (placed in Kiselev’s textbook)?
The simplest proof in the framework of visual geometry is as follows: a square with sides of length a + b can be divided into two small squares with sides a and b and four more right-angled triangles with legs a, b. In the same square, you can inscribe a square with side c so that a triangle with a hypotenuse c and legs a, b is adjacent to each side of this square. Comparing the areas of the figures participating in these two partitions, we get the equality c ^ 2 = a ^ 2 + b ^ 2.
Another proof is algebraic: the Pythagorean formula is derived from the fact that the length of each leg is equal to the square root of the product of the hypotenuse and the projection of this leg onto the hypotenuse.
Pythagoras considered both of these proofs to be unsuccessful, since the first is conducted in the language of areas, and the second in the language of equations. Correct proof, according to Pythagoras, should be carried out in the same visual geometric language in which the theorem was formulated.
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