Do infinitely many points in a plane with integer distances lie on a line? Someone posted a question on the notice board at my University's library. I've been thinking about it for a while, but fail to see how it is possible. Could someone verify that this is a valid question and point me in the right direction?
'Given an infinite set of points in a plane, if the distance between any two points is an integer, prove that all these points lie on a straight line.'

Say $A$, $B$ and $C$ are three points with pairwise integer distances. Any other point $P$ with integer distances from those three will satisfy $|AP-PB| \leq AB$ and $|AP-PC| \leq AC$, by the triangle inequality. So $P$ lies on an intersection of a hyperbola $|AP-BP| = k$ (for some integer $k \leq AB$) and a hyperbola $|AP-PC| = m$ (for some integer $m \leq AC$). There are finitely many such points $P$ (at most $4\;AB\;AC$).
(I'd be surprised if the argument in the Anning and Erdős paper is very different from this one.)
