Special right triangle than 256: Isosceles right-angled triangles cannot have sides with integer values, because the ratio of the hypotenuse to either other side is , but cannot be expressed as a ratio of two integers. However, infinitely many "almost-isosceles" right triangles do exist. These are right-angled triangles with integral sides for which the lengths of the non-hypotenuse edges differ by one. Such almost-isosceles right-angled triangles can be obtained recursively, "a" is length of hypotenuse, "n" = 1, 2, 3, ... Equivalently, where {"x", "y"} are the solutions to the Pell equation , with the hypotenuse "y" being the odd terms of the
Isosceles triangle formula_10. When the isoperimetric inequality becomes an equality, there is only one such triangle, which is equilateral. If the two equal sides have length formula_2 and the other side has length formula_3, then the internal angle bisector formula_22 from one of the two equal-angled vertices satisfies as well as and conversely, if the latter condition holds, an isosceles triangle parametrized by formula_2 and formula_22 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