I was confused after finding out, by computing counterexamples, that the Cayley-Menger determinant for a tetrahedron being positive does not actually imply that the triangle inequality and Ptolemy's inequality holds for all points, since I assumed that being a representation of the volume meant that all positive values for any 3-dimensional CM determinant would lead to a set of distances for points that could be realised as a tetrahedron in Euclidean 3D space.
Therefore, I decided to try to find examples of the 8 different cases: one in which Ptolemy's inequality holds but the triangle inequality does not and the Cayley-Menger determinant is negative, one in which both Ptolemy's inequality and the triangle inequality holds but Cayley-Menger determinant is negative and so on.
I ended up finding examples of each type except for one case: The Cayley-Menger determinant is positive and the triangle inequality holds but Ptolemy's inequality does not hold. This leads me to believe that, together, the Cayley-Menger determinant being positive and the triangle inequality holding for all points implies that Ptolemy's inequality holds.
I have tried to find proofs of Ptolemy's inequality that rely both on the triangle inequality and the volume of a tetrahedron being positive but have not found any. I have also attempted to prove it to no avail. I feel like there would be an elegant proof of this using matrices, since Ptolemy's inequality holding is equivalent to the (1,1) minor of the 3-Dimensional CM determinant being negative, but I have failed to find one.
Disclaimer: this is a different question to the one I made a few days ago (one which was disproved by one of the counterexamples mentioned)