Three points related to the incenter and excenters of a triangle. (Q960582)

Summary: The incenter and the three excenters of a triangle \(\triangle\) are the centers of those four circles touching every side of \(\triangle\). From these points three other points can be derived. The resulting triangle \(\triangle_S\) is produced from the triangle \(\triangle_A\) of the excenters by a \(180^\circ\) rotation around the Feuerbach point of \(\triangle\). The triangles \(\triangle_A\) and \(\triangle_S\) share the Euler line and the Feuerbach nine-point circle. The author uses elementary means to prove these and other results connected with them.