Let $x, y \in \mathbb{R}$ such that $x^2+y^2 \in \mathbb{Q}$. Do there exist $a, b \in \mathbb{Q}$ such that $a^2 + b^2 = x^2 + y^2$?
Interpreted geometrically: Consider the circle in $\mathbb{C}$ with center $0$ and radius $r := |x+iy|$. Does it contain a rational point, i.e. do there exist $a, b \in \mathbb{Q}$ such that $a+ib$ lies on that circle?
My first step to approach this was the following: There exist coprime $p, q \in \mathbb{Z}$ such that $r^2 = p/q$. I think, that we can assume that $p$ and $q$ are both square-free. For if $p=k^2p_0$ for some $k, p_0$, then we maybe can replace $r$ with $r/k$ and $p$ with $p_0$. If this works, then it works similarly for $q$.
But then I'm not sure how to proceed.
