Concentric Circles, Third Innternaly Tangent to Inner and Intersecting Outer

Steps

1. Draw a line passing through the centers of the given circles. The common tangency point of the two given circles lies on this line. The two solution circles will also be tangent to these circles at this point. These solution circles will also be tangent to the third given circle at its two intersections with the line. The centers of the solution circles lie at the midpoints of the segments defined by the respective intersections of the line with the circles.
2. We have found the first two solutions.
3. The centers of the remaining solution circles lie on a circle concentric with the given concentric circles. This circle passes through the center of one of the previously found solution circles.
4. The radii of the solution circles are given by the distance from this concentric circle to the given circles and are equal to the radius of one of the already constructed solution circles. The centers of the remaining solution circles therefore lie on a circle concentric with the third given circle. Its radius is equal to the difference of the radius of the given circle and the radius of the solution circles.
5. We have found the remaining two solutions.
6. The problem has a total of four solutions.