Pythagorean theorem straightedge and compass. Pythagoras's theorem enables construction of incommensurable lengths because the hypotenuse of a triangle is related to the sides by the square root operation. The figure on the right shows how to construct line segments whose lengths are in the ratio of the square root of any positive integer. Each triangle has a side (labeled "1") that is the chosen unit for measurement. In each right triangle, Pythagoras's theorem establishes the length of the hypotenuse in terms of this unit. If a hypotenuse is related to the unit by the square root of a positive integer that is
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