reflexivity, i.e. aRa
symmetry, i.e. aRb <--> bRa
transitivity, i.e. (aRb,bRc) --> aRc
I don't see that restricting your points to Q makes any difference.
given that your relation is just the square of the distance between the points (x1,y1) and (x2,y2) the geometric interpretation of the equivalence classes should be pretty clear.