Yes. Exists a partition of the plane $\mathbb{R}\times\mathbb{R}$ in $4$ subsets such that any circle in the plane contains inside points from each subset: $A=\{(x,y)|\;x\in\mathbb{Q}, y\in\mathbb{Q}\}$; $B=\{(x,y)|\;x\in\mathbb{Q}, y\in\mathbb{R}\setminus\mathbb{Q}\}$; $C=\{(x,y)|\;x\in\mathbb{R}\setminus\mathbb{Q}, y\in\mathbb{Q}\}$; $D=\{(x,y)|\;x\in\mathbb{R}\setminus\mathbb{Q}, y\in\mathbb{R}\setminus\mathbb{Q}\}$.

Basically divide the real number line into two dense sets. Do that to each line on the plane.