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
Fermat's right triangle theorem single Diophantine equation which can be simplified to Conversely, any solution to this equation could be factored to give a square congruum. (In particular, the squares formula_13, formula_14, and formula_15 form an arithmetic progression with congruum formula_16, which is a square itself.) Thus, the solvability of this equation is equivalent to the existence of a square congruum. But, if Fermat's last theorem were false for the exponent formula_17, then squaring one of the three numbers in any counterexample would also give three numbers that solve this equation. Therefore, Fermat's proof that no Pythagorean triangle has a square area implies that