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 Fermat's right triangle theorem Fermat's right triangle theorem is a non-existence proof in number theory, the only complete proof left by Pierre de Fermat. It has several equivalent formulations: An immediate consequence of the last of these formulations is that Fermat's last theorem is true for the exponent formula_3 and hence for any multiple of 4. In 1225, Fibonacci was challenged to find a construction for triples of square numbers that are equally spaced from each other, forming an arithmetic progression, and for the spacing between these numbers, which he called a congruum. One way of describing Fibonacci's solution is