  Let $P$ be a set of $n$ points in the plane that determines at most $n/5$
distinct distances. We show that no line can contain more than $O(n^{43/52}{\rm
polylog}(n))$ points of $P$. We also show a similar result for rectangular
distances, equivalent to distances in the Minkowski plane, where the distance
between a pair of points is the area of the axis-parallel rectangle that they
span.
