AND THE APOLLONIUS PROBLEM

**BORUT JURCIC ZLOBEC AND
NEZA MRAMOR KOSTA**

Given spheres and planes of dimension in , the
Apollonius problem is to find a common tangent sphere or plane,
and the generalized Apollonius problem is to find a sphere or plane
intersecting them under prescribed angles. In Lie geometry, an
Apollonius problem is given by an -frame of points on the
Lie quadric
. The solutions are described as the
intersections of the projective line determined by the orthogonal
complement to this frame with respect to the Lie product in
and the quadric. Two special points span this line, and
the connection between the position of these two points and the
existence and geometric properties of the solutions of the
Apollonius problem are described.