Concentric Circles, Line Tangent to Inner Circle

Steps

1. The centers of the desired circles lie on the line passing through the tangency point of the given line and the inner circle, and the common center of the two concentric circles. This line is perpendicular to the given line.
2. Find the intersection points of this line with the larger of the two concentric circles.
3. The centers of the first two solution circles lie at the midpoints of the segments whose endpoints are the tangency point and each of the found intersections.
4. These are the first two solutions.
5. The remaining two solution circles are externally tangent to the smaller circle and internally tangent to the larger one. Their centers lie on a circle located between the two given concentric circles. This intermediate circle is also concentric with the given ones and passes through the center of one of the previously constructed solution circles.
6. Since the desired circles are also tangent to the given line, their centers lie on a line parallel to the given line. The distance between these parallel lines corresponds to the radius of the solution circles, which equals the distance between the intermediate circle and either of the given concentric circles.
7. This yields the remaining two solutions.
8. The problem has four solutions.