Triangle inequality is an exterior angle of a right triangle and therefore larger than , meaning the other base angle of the isosceles triangle also is greater than and their sum exceeds in violation of the triangle postulate). This theorem establishing inequalities is sharpened by Pythagoras' theorem to the equality that the square of the length of the hypotenuse equals the sum of the squares of the other two sides. Consider a triangle whose sides are in an arithmetic progression and let the sides be , , . Then the triangle inequality requires that The first of these quadratic inequalities requires to
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