Concentric Circles, Third Touches the Inner One Externally

Steps

1. We draw a line passing through the centers of the given circles. The common tangent point of two given circles lies on this line. The two solution circles will also be tangent to these circles at this point. These two solution circles will be tangent to the third given circle at its two intersection points with the line. The centers of the solution circles lie at the midpoints of the segments defined by the corresponding intersections of the line with the circles.
2. We have found the first two solutions.
3. The centers of the remaining solution circles will lie on circles concentric with the given concentric circles. These new circles pass through the previously found two centers.
4. The radii of the solution circles are determined by the distance between the found concentric circles and the given circles, and they are the same as the radii of the two solution circles already found. Therefore, the centers of the remaining solution circles will lie on circles concentric with the third given circle, whose radii equal the sum of the radius of the given circle and the radii of the solution circles.
5. We have found the remaining four solutions.
6. The problem has a total of six solutions.