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
Heronian triangle triangles with these sidelengths, and joins them together along the sides of length "a", to obtain a triangle with integer side lengths "c", "e", and "b" + "d", and with area If "a" is even then the area "A" is an integer. Less obviously, if "a" is odd then "A" is still an integer, as "b" and "d" must both be even, making "b"+"d" even too. Some Heronian triangles cannot be obtained by joining together two right-angled triangles with integer sides as described above. For example, a 5, 29, 30 Heronian triangle with area 72 cannot be constructed from two