Points at rational distances from the vertices of certain geometric objects

We consider various problems related to finding points in Q2Q2 and in Q3Q3 which lie at rational distance from the vertices of some specified geometric object, for example, a square or rectangle in Q2Q2, and a cube or tetrahedron in Q3Q3. In particular, as one of several results, we prove that the set of positive rational numbers a   such that there exist infinitely many rational points in the plane which lie at rational distance from the four vertices of the rectangle with vertices (0,0)(0,0), (0,1)(0,1), (a,0)(a,0), and (a,1)(a,1), is dense in R+R+.