Special right triangle : , which follows immediately from the Pythagorean theorem. Of all right triangles, the 45°–45°–90° degree triangle has the smallest ratio of the hypotenuse to the sum of the legs, namely . Of all right triangles, the 45°–45°–90° degree triangle has the greatest ratio of the altitude from the hypotenuse to the sum of the legs, namely . Triangles with these angles are the only possible right triangles that are also isosceles triangles in Euclidean geometry. However, in spherical geometry and hyperbolic geometry, there are infinitely many different shapes of right isosceles triangles. This is a triangle whose three angles
Automedian triangle satisfying the formula "a" = "b" + "c". That is, in order for the three numbers "a", "b", and "c" to be the sides of an automedian triangle, the sequence of three squared side lengths "b", "a", and "c" should form an arithmetic progression. If "x", "y", and "z" are the three sides of a right triangle, sorted in increasing order by size, and if 2"x" < "z", then "z", "x" + "y", and "y" − "x" are the three sides of an automedian triangle. For instance, the right triangle with side lengths 5, 12, and 13 can be used