According to < http://en.wikipedia.org/wiki/Erdős–Anning_theorem >, infinitely many points in the plane having only integer distances must be collinear. So I guess the hypotenuse method must always result in only finitely many points and/or collinear points. The same Wikipedia article mentions what may be equivalent to Michael's hypotenuse idea: Consider all the points on the unit circle at the angle theta such that tan(theta/4) is rational. Then these must have a rational (straight-line) distance between any two of them. Hence, any finite set of them can be uniformly scaled to have integer distances. But this can't be done for any infinite set of them. --Dan On 2013-04-04, at 11:44 AM, Michael Kleber wrote:
It's easy to get infinitely many points in the plane with all pairwise distances integers. First of all, if you don't mind them being collinear, just take (n,0). Second, Ptolemy's Theorem implies that if two Pythagorean triangles share a hypotenuse, then their right-angled vertices are a rational distance from one another. This is a recipe for infinitely many points at pairwise rational distances (or any finite number at integer pairwise distances, after scaling), all on a circle.