You may have a look at the book

"Combinatorial Geometry in the Plane" by Hadwiger, DeBrunner, and Klee.

For example, one of my favorites is Proposition #9:

"If an infinite set of points is such that all points are at integral distances from each other, then all of the points lie on a straight line."

I guess one could argue that there are significant hypotheses here, but the conclusion seems surprisingly strong.