Topic title says it all, I think. Does there exist a circle with EXACTLY two rational points, no more, no less? I know that as soon as it has three rational points, its center must be rational too, therefore an isomorphism with the unit circle yields infintely many (dense) rational points. I have examples with only one and zero rational points, but I'm wondering about the case 2.
I already know that the center must be irrational, because else one rational point on the circle would suffice to yield infinitely many points.
Best Answer
From cut-the-knot.org:
So, choose two rational points $P,Q$ and an irrational point $C$ on their perpendicular bisector. The circle with center $C$ that goes through $P,Q$ answers your question.