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 appears in article 182 of the Disquisitiones Arithmeticae. An (integral binary) quadratic form is an expression of the form formula_105 with formula_106 integers. A number formula_107 is said to be "represented by the form" if there exist integers formula_108 such that formula_109. Fermat's theorem on sums of two squares is then equivalent to the statement that a prime formula_82 is represented by the form formula_111 (i.e., formula_112, formula_113) exactly when formula_82 is congruent to formula_91 modulo formula_116. The discriminant of the quadratic form is defined to be formula_117. The discriminant of formula_111 is then equal to formula_119. Two forms formula_120