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
Isosceles triangle a special isosceles triangle with the property that the other two inscribed squares, with sides collinear with the sides of the triangle, are of the same size as the base square. A much older theorem, preserved in the works of Hero of Alexandria, states that, for an isosceles triangle with base formula_3 and height formula_1, the side length of the inscribed square on the base of the triangle is For any integer formula_35, any triangle can be partitioned into formula_36 isosceles triangles. In a right triangle, the median from the hypotenuse (that is, the line segment from the midpoint of