Pythagorean theorem four elementary lemmata: Next, each top square is related to a triangle congruent with another triangle related in turn to one of two rectangles making up the lower square. The proof is as follows: This proof, which appears in Euclid's "Elements" as that of Proposition 47 in Book 1, demonstrates that the area of the square on the hypotenuse is the sum of the areas of the other two squares. This is quite distinct from the proof by similarity of triangles, which is conjectured to be the proof that Pythagoras used. We have already discussed the Pythagorean proof, which was
Square root of 2 as shown in Figure 1. The square overlap region in the middle () must equal the sum of the two uncovered squares (). But these squares on the diagonal have positive integer sides that are smaller than the original squares. Repeating this process we can find arbitrarily small squares one twice the area of the other, yet both having positive integer sides, which is impossible since positive integers cannot be less than 1. Another geometric reductio ad absurdum argument showing that is irrational appeared in 2000 in the American Mathematical Monthly. It is also an example of proof by infinite