Apollonius Meets Computer Algebra The circles of Apollonius problem is a classic geometry problem dating from Greek antiquity. Although there are various generalizations and special cases, the main problem was to find the circle(s) tangent to three given circles in the plane. Here we are concerned with higher dimensional generalizations. First, to find equations for a sphere tangent to four given spheres in three-space. Secondly, to similarly find equations for a hypersphere tangent to n+1 given hyperspheres in n-dimensional space. Thirdly, to replace some of these spheres with ellipsoids. The three-dimensional problem is motivated by work in trying to compute the medial axis or surface (a sort of skeleton) of the space around molecules, with eventual medical research applications. We will discuss computer algebra solutions based on Groebner bases and on Dixon resultants. Robert H. Lewis Department of Mathematics Fordham University New York Stephen Bridgett Queen's University Belfast