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
Special right triangle Ancient Egypt; that their surveyors did use ropes for measurement; that Plutarch recorded in "Isis and Osiris" (around 100 AD) that the Egyptians admired the 3 : 4 : 5 triangle; and that the Berlin Papyrus 6619 from the Middle Kingdom of Egypt (before 1700 BC) stated that "the area of a square of 100 is equal to that of two smaller squares. The side of one is ½ + ¼ the side of the other." The historian of mathematics Roger L. Cooke observes that "It is hard to imagine anyone being interested in such conditions without knowing the Pythagorean