Two Circles with External Tangency, Third Intersecting One of Them

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 a circle concentric with the transformed circle, whose radius is larger 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 four solutions.