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
Proofs of Fermat's theorem on sums of two squares formula_196. The area of the fundamental parallelogram (in this case a rectangle) of the lattice is formula_82. The area of the open disk, formula_198, of radius formula_199 centered around the origin is formula_200. Furthermore, formula_198 is convex and symmetrical about the origin. Therefore, by Minkowski's theorem there exists a nonzero vector formula_196 such that formula_203. Both formula_204 and formula_205 so formula_206. Hence formula_82 is the sum of the squares of the components of formula_208. Let formula_209 be prime, let formula_210 denote the natural numbers (with or without zero), and consider the finite set formula_211 of triples of numbers. Then formula_212