length , then the internal angle bisector from one of the two equal-angled vertices satisfies as well as and conversely, if the latter condition holds, an isosceles triangle parametrized by and exists. The Steiner–Lehmus theorem states that every triangle with two angle bisectors of equal lengths is isosceles. It was formulated in 1840 by C. L. Lehmus. Its other namesake, Jakob Steiner, was one of the first to provide a solution. Although originally formulated only for internal angle bisectors, it works for many (but not all) cases when, instead, two external angle bisectors are equal. The 30-30-120 isosceles triangle makes a boundary case for this variation of the theorem, as it has four equal angle bisectors (two internal, two external). Radii The inradius and circumradius formulas for an isosceles triangle may be derived from their formulas for arbitrary triangles. The radius of the inscribed circle of an isosceles triangle with side length , base , and height is: The center of the circle lies on the symmetry axis of the triangle, this distance above the base. An isosceles triangle has the largest possible inscribed circle among the triangles with the same base and apex angle, as well as also having the largest area and perimeter among the same class of triangles. The radius of the circumscribed circle is: The center of the circle lies on the symmetry axis of the triangle, this distance below the apex. Inscribed square For any isosceles triangle, there is a unique square with one side collinear with the base of the triangle and the opposite two corners on its sides. The Calabi triangle is 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 and height , the side length of the inscribed square on the base of the triangle is Isosceles subdivision of other shapes For any integer , any triangle can be partitioned into isosceles triangles. In a right triangle, the median from the hypotenuse (that is, the line segment from the midpoint of the hypotenuse to the right-angled vertex) divides the right triangle into two isosceles triangles. This is because the midpoint of the hypotenuse is the center of the circumcircle