Right triangle and "b" are the legs of the triangle. If the incircle is tangent to the hypotenuse AB at point P, then denoting the semi-perimeter as "s", we have and , and the area is given by This formula only applies to right triangles. If an altitude is drawn from the vertex with the right angle to the hypotenuse then the triangle is divided into two smaller triangles which are both similar to the original and therefore similar to each other. From this: In equations, where "a", "b", "c", "d", "e", "f" are as shown in the diagram. Thus Moreover, the
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