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
Fermat's right triangle theorem of a right triangle with square area. By dividing by any common factors, one can assume that this triangle is primitive and from the known form of all primitive Pythagorean triples, one can set formula_23, formula_24, and formula_25, by which the problem is transformed into finding relatively prime integers formula_26 and formula_27 (one of which is even) such that formula_28 is square. The four linear factors formula_26, formula_27, formula_31, and formula_32 are relatively prime and therefore must themselves be squares; let formula_33 and formula_34. Both formula_35 and formula_36 must be odd since exactly one of formula_26 or formula_27 is even