Do infinitely many points in a plane with integer distances lie on a line?

Solution 1:

MR0013511 (7,164a) Anning, Norman H.; Erdős, Paul Integral distances. Bull. Amer. Math. Soc. 51, (1945). 598–600.

The authors show that for any n there exist noncollinear points $P_1,\dots,P_n$ in the plane such that all distances $P_iP_j$ are integers; but there does not exist an infinite set of non-collinear points with this property. Reviewed by I. Kaplansky

I can add that the first result mentioned requires lots of points to be on a circle. I believe the current record for points in the plane with all distances integers, no three on a line, no 4 on a circle, is 8.

Solution 2:

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.)