Triangle inequality ≥ b ≥ c" and "formula_8" is the golden ratio, as In the case of right triangles, the triangle inequality specializes to the statement that the hypotenuse is greater than either of the two sides, and less than their sum. The second part of this theorem is already established above for any side of any triangle. The first part is established using the lower figure. In the figure, consider the right triangle . An isosceles triangle is constructed with equal sides . From the triangle postulate, the angles in the right triangle satisfy: Likewise, in the isosceles triangle , the
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