Does every circle in $\mathbb{R^2}$ contain a point with rational coordinates?

Question

Is it true that any circle in $\mathbb{R^2}$ contains a point with rational coordinates? what about any simple closed curve?

If it is, could you please help me with the proof?

Answer 1

No, consider $$x^2+y^2=r^2$$ there are continuum many $r$ giving disjoint circles, but only countably many rational points.

Answer 2

For circles, no. Pick any number $r$ with $r^2$ irrational. Then the circle $$x^2+y^2=r^2$$ does not have any rational solution, or else $r^2$ is rational.

For arbitrary closed curves, even more counterexamples exist. Pick a rectangle such that for each straight line part, say a horizontal line of the rectangle, the $y$-coordinate is irrational, while for vertical line, the $x$-coordinate is irrational.