The spheres tangent externally to the tritangent spheres of a triangle (Q2904609)

Given a triangle in the plane, it is well-known that the nine-point circle is simultaneously tangent to the incircle and the three excircles.NEWLINENEWLINEWe can consider these five circles as the equatorial circles of five spheres in the space. Obviously the sphere corresponding to the nine-point circle is tangent to the other four considered spheres. In this situation it naturally arises the question of whether there is any other sphere simultaneously tangent to those four spheres.NEWLINENEWLINEIn this paper, the previous question is answered in the affirmative explicitly constructing one such sphere.