  We consider problems involving rich homotheties in a set S of n points in the
plane (that is, homotheties that map many points of S to other points of S). By
reducing these problems to incidence problems involving points and lines in
R^3, we are able to obtain refined and new bounds for the number of rich
homotheties, and for the number of distinct equivalence classes, under
homotheties, of k-element subsets of S, for any k >= 3. We also discuss the
extensions of these problems to three and higher dimensions.
