Two Circles with External Tangency, Third Intersecting Both

Steps

1. Choose the circle of inversion, placing its center at the tangency point of the two circles.
2. Apply circular inversion to the given circles. The circles passing through the center of the inversion circle transforms into two parallel lines.
3. The centers of the images of the solution circles will lie on the axis (midline) of the strip between the parallel lines.
4. The centers lie at the intersections of this axis and circles concentric with the transformed circle, whose radii differ from the radius of the transformed circle by half the distance between the parallel lines.
5. The remaining two solutions appear as tangents to the transformed circle, parallel to both parallel lines.
6. Map the found images of the solutions back using circular inversion.
7. The problem has six solutions.