Two Concentric Circles, Third Has Internal Tangency with Both

Steps

1. Draw the line passing through the centers of the given circles and also through their common tangency points. The solution circles will pass through the tangency points between the circles and will touch the concentric circles at the intersection points of the line with those circles. The centers of the solution circles lie at the midpoints of the segments defined by one tangency point and the intersection point of the line with the third given circle.
2. The problem has two solutions.