Is it possible to find an infinite set of point belonging to the circle of radius r such that $\sqrt{r} \notin \mathbb{Q}$ and where the distance between any pair is rational?
For example, can we find such set for $r=\frac{4}{3}$
This is related to the following question: nD SPACE RATIONAL DISTANCE
The difference is the new condition on the radius (in the link, the argument is given for the unit circle).
My idea would be then to construct infinite sets in n-dimensional spaces considering the unions of such circles in 2D subspaces (and adding points on an axis when n is odd)
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You could adapt the method documented here. In that method, the circle's radius is 1, and the method to determine each point from the previous was to use the right-angled triangle with legs $3/5$ and $4/5$ and hypotenuse 1.
Now scale things up by a factor of 5. The radius is now $r=5$. $r$ is rational but $\sqrt r$ is not. We now use copies of the Pythagorean triangle with legs $3$ and $4$ and hypotenuse $5$. Choose each point to be at distance 3 from the previous. (Or distance 4; it doesn't matter which leg is used as chords of the circle.)
