Pythagorean theorem three sides in place of squares (squares are a special case, of course). The upper figure shows that for a scalene triangle, the area of the parallelogram on the longest side is the sum of the areas of the parallelograms on the other two sides, provided the parallelogram on the long side is constructed as indicated (the dimensions labeled with arrows are the same, and determine the sides of the bottom parallelogram). This replacement of squares with parallelograms bears a clear resemblance to the original Pythagoras's theorem, and was considered a generalization by Pappus of Alexandria in 4 AD The
Fermat's right triangle theorem a congruent number. Equivalently, it is impossible for a square (the geometric shape) and a right triangle to have both equal areas and all sides commensurate with each other. Yet another equivalent form of Fermat's theorem involves the elliptic curve consisting of the points whose Cartesian coordinates formula_18 satisfy the equation This equation has the obvious pairs of solutions (0,0), (1,0), and (−1,0). Fermat's theorem is equivalent to the statement that these are the only points on the curve for which both "x" and "y" are rational. During his lifetime, Fermat challenged several other mathematicians to prove the non-existence of