Rational distances from given rational points in the plane

In this paper we consider sets of points in the plane with rational distances from a prescribed finite set of n rational points. We show that for n 3, the points are dense in the real topology. On the other hand, for n 4, we show that they correspond to rational points in a surface of general type, hence conjecturally degenerate. However, at the present, we lack methods to prove this, given the fact that the surface is simply-connected, as we shall show. We give explicit proofs as well as describe in detail the geometry of the surfaces involved. In addition we discuss certain cases of density of points with distances in certain ring of integers.