Two Concentric Circles, Third Lies in the Annular Region

Steps

1. First, we will look for all centers of circles tangent to both concentric circles. We find the centers of two such circles. We draw a line passing through the common center of the given circles. The centers of the desired circles lie at the midpoints of the segments defined by the intersections of this line with the given circles.
2. We draw circles concentric with the given ones and passing through the previously found points. All centers of circles tangent to the two given concentric circles lie on these new circles. The distance of these circles from the original concentric circles also determines the radii of the solution circles.
3. We draw a line through the center of the third given circle and find its intersection with that circle. We then draw circles centered at this intersection point, using the radii determined in the previous step. We find the intersections of these new circles with the line through the center.
4. We draw circles concentric with the third given circle, passing through the points found in the previous step. The centers of the solution circles will lie on these circles.
5. The intersections of the previously constructed circles determine the centers of the solution circles.
6. The problem has eight solutions.