Tangent Spheres and Triangle Centers

1. TANGENT SPHERES. Any four mutually tangent spheres determine six points of tangency. We say that a pair of tangencies {ti , t j } is opposite if the two spheres determining ti are distinct from the two spheres determining t j . Thus the six tangencies are naturally grouped into three opposite pairs, corresponding to the three ways of partitioning the four spheres into two pairs. Altshiller-Court [1, §630, p. 231] proved the following result about these opposite pairs, which we use to define two new triangle centers.