Two Concentric Circles, Third Has One Internal and One External Tangency

Steps

1. We draw a line passing through the centers of the given circles and their tangency points. Two of the solution circles will pass through the tangency points between the circles and will touch the concentric circles at the intersections of the line with these circles. The centers of the solution circles lie at the midpoints of the segments defined by a tangency point and the intersection of the line with the third given circle.
2. We have found the first two solutions to the problem.
3. The centers of the remaining two solution circles lie on a circle concentric with the two concentric given circles, passing through the center of the third given circle.
4. The centers of the solution circles are at a distance from the third given circle equal to their radius, which is the same as the radius of that circle. We therefore draw a circle sharing the same center with the third given circle and having twice its radius. The centers of the solution circles are the intersection points of this circle with the one from the previous step.
5. The problem has a total of four solutions.