SHARYGIN 2008 Problem 23

(V.Protasov, 10--11) In the space, given two intersecting spheres of different radii and a point $A$ belonging to both spheres. Prove that there is a point $B$ in the space with the following property: if an arbitrary circle passes through points $A$ and $B$ then the second points of its meet with the given spheres are equidistant from $B$ .