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
Special right triangle Pell numbers 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378... .. The smallest Pythagorean triples resulting are: Alternatively, the same triangles can be derived from the square triangular numbers. The Kepler triangle is a right triangle whose sides are in a geometric progression. If the sides are formed from the geometric progression "a", "ar", "ar" then its common ratio "r" is given by "r" = where "φ" is the golden ratio. Its sides are therefore in the ratio . Thus, the shape of the Kepler triangle is uniquely determined (up to a scale factor) by the requirement that