Two subsets A, B of the plane are betweenness isomorphic if there is a bijection \(f:A\rightarrow B\) such that, for every \(x,y,z\in A\) , the point f(z) lies on the line segment connecting f(x) and f(y) if and only if z lies on the line segment connecting x and y. In general, it is quite difficult to tell whether two given subsets of the plane are betweenness isomorphic. We concentrate on the case when the sets A, B belong to the family \( {\mathcal {A}}_c\) of unions of pairs of concentric circles in the plane. We prove that \(A, B \in {\mathcal {A}}_c\) are betweenness isomorphic if and only if they are similar. In particular, there are continuum many betweenness isomorphism classes in \({\mathcal {A}}_c\) , and each of these classes consists exactly of all scaled translations of an arbitrary representative of the class. Furthermore, we show that every betweenness isomorphism between sets \(A,B\in {\mathcal {A}}_c\) is exactly the restriction of a scaled isometry of the plane.
