To prove that a CAT(0) space is Ptolemaean is relatively easy, and the converse does not always hold (obviously, e.g. the Heisenberg group with the Koranyi is Ptolemaean). There must be some other equally intrinsic property, such that in addition to the Ptolemaean property turns a space to a CAT(0) space. Might this be as elementary as well? Any ideas are welcome.