Let $S$ be an infinite set of points in the plane. The distance between two points of $S$ is integral. Prove that $S$ is a subset of a straight line.
Here is my attempt: Suppose not, then for each line passing through two points $a$ and $b$, you can find a third point $c$ such that $c$ does not lie on the line. Hence $a,b,c$ form a triangle. So pick a triangle $\triangle_s$ constructed like this of smallest area, then all other points lie outside of the triangle $\triangle_o$ whose midpoints of its edges are the vertices of $\triangle_s$. Then I'm stuck.