Right triangle and "G" and hypotenuse "A", then and where formula_40 is the golden ratio formula_41 Since the sides of this right triangle are in geometric progression, this is the Kepler triangle. Thales' theorem states that if "A" is any point of the circle with diameter "BC" (except "B" or "C" themselves) "ABC" is a right triangle where "A" is the right angle. The converse states that if a right triangle is inscribed in a circle then the hypotenuse will be a diameter of the circle. A corollary is that the length of the hypotenuse is twice the distance from the right
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