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 of which is a sum of two squares, is itself a sum of two squares." 2. "If a number which is a sum of two squares is divisible by a prime which is a sum of two squares, then the quotient is a sum of two squares." (This is Euler's first Proposition). 3. "If a number which can be written as a sum of two squares is divisible by a number which is not a sum of two squares, then the quotient has a factor which is not a sum of two squares." (This is Euler's second Proposition). 4. "If