One Circle Inside Another, Line Tangent to Outer Circle, Centers and Point of Tangency Collinear

Steps

1. The centers of all circles tangent to both the given line and circle at their common point of tangency lie on the perpendicular to the line passing through that point.
2. The centers of the solution circles lie at the midpoints of the segments whose endpoints are the intersections of the perpendicular with the smaller circle and the tangency point of the line and the circle.
3. The problem has two solutions.