Concentric Circles, Line Intersects Both

Steps

1. Draw a line passing through the centers of the given circles and find its intersection points with both circles.
2. The intersection points are the points of tangency of a circle that is tangent to the given circles. The center of this circle lies at the midpoint of the segment defined by the two tangency points. The distance from the center to a tangency point gives the radius of the solution circles.
3. A circle concentric with the given circles and passing through the found center is the locus of all centers of circles that are internally tangent to the larger given circle and externally tangent to the smaller one.
4. Circles tangent to the given line and with the radius found earlier have their centers on lines parallel to the given line. The distance of these parallels equals the radius of the solution circles.
5. The intersections of the parallel lines and the concentric circle give the centers of the solution circles.
6. The problem has four solutions.