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 center similarly constructed equilateral triangles based on the other two sides of triangle ABC. Then the lines AA', BB' and CC' are concurrent and the point of concurrence is the 1st isogonal center. Its trilinear coordinates are Expressing these coordinates in terms of "a", "b" and "c", one can verify that they indeed satisfy the defining properties of the coordinates of a triangle center. Hence the 1st isogonic center is also a triangle center. Then "f" is bisymmetric and homogeneous so it is a triangle center function. Moreover the corresponding triangle center coincides with the obtuse angled vertex whenever any vertex