Do you mean Euclid's postulates? You'd need to first define what lines are in your space. A distance function isn't taken as fundamental in Euclid's geometry. The fundamental notions are "point", "line", "equal lengths" and "equal angles", and in modern formulations there is also a concept of "between" or something analogous, which is used implicitly in Euclid. So you'd need to define these concepts.

If what you mean is a proof that your metric space isn't isometric to R\^2 with the Euclidean distance, I think you'll find many examples where distinct "circles" have more than two points of intersection. (I'm assuming that your underlying set is R\^2. If all points have integer coordinates, then it's much simpler.)