毕达哥拉斯证明他的定理Pythagoras's proof of his theorem

45° Pythagoras's proof of his theorem. Tradition is unanimous in ascribing to Pythagoras the independent discovery of the theorem on the right triangle that now universally bears his name, that the square on the hypotenuse of a right triangle is equal to the sum of the squares on the two legs. We have seen, in Item 17°, that this theorem was known to the Babylonians of Hammurabi's time, more than a thousand years earlier, but the first general proof of the theorem may well have been given by Pythagoras. There has been much conjecture as to the proof Pythagoras might have offered, and it is generally felt that it probably was a dissection type of proof* like the following, illustrated in Figure 5. Let a, b, c denote the legs and hypotenuse of the given right triangle, and consider the two squares in the figure, each having a + b as side. The first square is dissected into six pieces, namely the two squares on the legs and four right triangles congruent to the given triangle. The second square is dissected into five pieces, namely the square on the hypotenuse and again four right triangles congruent to the given triangle. By subtracting equals from equals, it now follows that the square on the hypotenuse is equal to the sum of the squares on the legs.

To prove that the central piece of the second dissection is actually a square of side c we need to employ the fact that the sum of the angles of a right triangle is equal to two right angles. But the Greek commentator Proclus attributes this theorem for the general triangle to the Pythagoreans. Since a proof of this theorem requires, in turn, a knowledge of some properties of parallels, the early Pythagoreans are also credited with the development of that theory.

See, however, Daniel Shanks, Solved and Unsolved Problems in Number Theory*, Vol. 1 (Washington, D.C.: Spartan Books, 1962), pp. 124, 125.