  We show that for $m$ points and $n$ lines in the real plane, the number of
distinct distances between the points and the lines is
$\Omega(m^{1/5}n^{3/5})$, as long as $m^{1/2}\le n\le m^2$. We also prove that
for any $m$ points in the plane, not all on a line, the number of distances
between these points and the lines that they span is $\Omega(m^{4/3})$. The
problem of bounding the number of distinct point-line distances can be reduced
to the problem of bounding the number of tangent pairs among a finite set of
lines and a finite set of circles in the plane, and we believe that this latter
question is of independent interest. In the same vein, we show that $n$ circles
in the plane determine at most $O(n^{3/2})$ points where two or more circles
are tangent, improving the previously best known bound of $O(n^{3/2}\log n)$.
Finally, we study three-dimensional versions of the distinct point-line
distances problem, namely, distinct point-line distances and distinct
point-plane distances. The problems studied in this paper are all new, and the
bounds that we derive for them, albeit most likely not tight, are non-trivial
to prove. We hope that our work will motivate further studies of these and
related problems.
