Is there a rational point in a given open set such that the distancse from given rational points to it are all rational numbers on $R^2$?

Question

On $R^2$ there is a nonempty open set $A$ and n rational points $a_1,...,a_n$. Is there a rational point $a$ in $A$ such that $\forall n\in\{1,..,n\},~|a-a_i|$ is a rational number. My idea is to find a measuring method, in which the "length" of all rational points in unit circle is $0$ with universe set being $Q$ in unit circle, and then we can solve this question in a way similar to measure theory.