Since is the midpoint of , the similarity implies that , from which the result follows. fourth proof. Let be the midpoint of the arc . , so is the angle bisector of . Let be the midpoint of , and It follows that is the Inverse of with respect to the circumcircle. From that, we know that the circumcircle is an Apollonian circle with foci . So is the bisector of angle , and we have achieved our wanted result. Tetrahedra The concept of a symmedian point extends to (irregular) tetrahedra. Given a tetrahedron two planes through are isogonal conjugates if they form equal angles with the planes and . Let be the midpoint of the side . The plane containing the side that is isogonal to the plane is called a symmedian plane of the tetrahedron. The symmedian planes can be shown to intersect at a point, the symmedian point. This is also the point that minimizes the squared distance from the faces of the tetrahedron. References External links Symmedian and Antiparallel at cut-the-knot Symmedian and 2 Antiparallels at cut-the-knot Symmedian and the Tangents at cut-the-knot An interactive Java applet for the symmedian point Isogons and Isogonic Symmetry Straight lines defined for a triangle