  A Moebius structure (on a set X) is a class of metrics having the same
cross-ratios. A Moebius structure is ptolemaic if it is invariant under
inversion operations. The boundary at infinity of a CAT(-1) space is in a
natural way a Moebius space, which is ptolemaic. We give a free of
classification proof of the following result that characterizes the rank one
symmetric spaces of noncompact type purely in terms of their Moebius geometry:
Let X be a compact Ptolemy space which contains a Ptolemy circle and allows
many space inversions. Then X is Moebius equivalent to the boundary at infinity
of a rank one symmetric space.
