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
Nonhypotenuse number Nonhypotenuse number In mathematics, a nonhypotenuse number is a natural number whose square "cannot" be written as the sum of two nonzero squares. The name stems from the fact that an edge of length equal to a nonhypotenuse number "cannot" form the hypotenuse of a right angle triangle with integer sides. The numbers 1, 2, 3 and 4 are all nonhypotenuse numbers. The number 5, however, is "not" a nonhypotenuse number as 5 equals 3 + 4. The first fifty nonhypotenuse numbers are: Although nonhypotenuse numbers are common among small integers, they become more-and-more sparse for larger numbers. Yet, there