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
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