= Solution
Fix a positive rational radius $r$. Distinct disjoint circles of radius $r$ have centers more than $2r$ apart, so their open interior disks are pairwise disjoint. Each such disk contains a point of the countable dense set $\mathbb Q^2$. Fix an enumeration of $\mathbb Q^2$ and assign to each disk the first rational point it contains. Disjointness makes this assignment injective, so there are only countably many circles of radius $r$.
The positive rational numbers are countable. Taking the <countable union of countable sets> over all rational radii proves that the whole collection is countable.
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