Two concentric circles, the third intersects both

Steps

1. The diameter of the solution circles is determined by the difference between the radii of the given concentric circles. The centers of the solution circles lie on a circle concentric with the given pair, with radius equal to the average of their radii.
2. The centers of the solution circles lie on circles concentric with the third given circle. Their radii differ from the radius of the given circle by the radius of the solution circles.
3. The centers of the solution circles lie at the intersections of the constructed circles.
4. The problem has four solutions.