Right triangle two legs (the two sides that meet at a right angle). This can be stated in equation form as where "c" is the length of the hypotenuse, and "a" and "b" are the lengths of the remaining two sides. Pythagorean triples are integer values of "a, b, c" satisfying this equation. The radius of the incircle of a right triangle with legs "a" and "b" and hypotenuse "c" is The radius of the circumcircle is half the length of the hypotenuse, Thus the sum of the circumradius and the inradius is half the sum of the legs: One of the
Triangle inequality angles satisfy: Therefore, and so, in particular, That means side opposite angle is shorter than side opposite the larger angle . But . Hence: A similar construction shows , establishing the theorem. An alternative proof (also based upon the triangle postulate) proceeds by considering three positions for point : (i) as depicted (which is to be proven), or (ii) coincident with (which would mean the isosceles triangle had two right angles as base angles plus the vertex angle , which would violate the triangle postulate), or lastly, (iii) interior to the right triangle between points and (in which case angle