Problem 4

Is there an infinite set of points in the plane such that no three points are collinear, and the distance between any two points is rational?

Step 4 of 4: Verify distinctness, rational distances, and no three collinear

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P_mP_n=2|\sin((m-n)\theta)|\in\mathbb Q

Detailed analysis

The chord formula gives $P_mP_n=2|\sin((m-n)\theta)|$ , which is rational. If $P_m=P_n$ , then $(m-n)\theta$ is an integer multiple of $\pi$ , contradicting the preceding nonvanishing result unless $m=n$ . Thus the set is infinite and distinct. All points lie on one circle, and a line meets a circle in at most two points, so no three are collinear.