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 and the other is odd. Therefore, both formula_39 and formula_40 are even, one of which is divisible by 4. From these two numbers Fermat derives two more numbers formula_41 and formula_42, one of which is even by the previous sentence. Because formula_43 is a square, formula_44 and formula_45 are the legs of another primitive Pythagorean triangle whose area formula_46. Since formula_27 is itself a square and since formula_48 is even, formula_49 is a square. Thus, any Pythagorean triangle with square area leads to a smaller Pythagorean triangle with square area, completing the proof. Fermat's right triangle theorem Fermat's right triangle