Concentric Circles, Third Touches the Inner One Internally

Steps

1. We draw a line passing through the centers of the given circles. The common tangent point of two of the given circles lies on this line. The solution circles will also be tangent to these two circles at that point. They will be tangent to the third given circle at the two intersection points of the line with that circle. The centers of the solution circles lie at the midpoints of the segments defined by the respective intersection points of the line and the circles.
2. The problem has two solutions.