Concentric Circles, Line is a Secant of the Outer Circle

Steps

1. Draw a line passing through the centers of the given circles, and find its intersection points with each circle.
2. There exist circles that are tangent to both given circles at those intersection points. The centers of these circles lie at the midpoints of the segments between the tangency points. The distances between the centers and the tangency points determine the radii of the solution circles.
3. All circles tangent to the given circles have their centers lying on circles concentric with the original ones, passing through the previously found midpoints.
4. Circles that are tangent to the given line and have the same radii as found earlier must have their centers located on lines parallel to the given line at a distance equal to the respective radius.
5. The centers of the solution circles lie at the intersections of the concentric circles and the parallel lines.
6. The problem has four solutions.