Internal Tangency, Third Tangent to Inner and Crossing Outer Circle

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 circle passing through the center of the inversion circle transforms into two parallel lines.
3. The center of the first solution in the inverted setting is found at the intersection of the axis of the strip between the parallel lines and the perpendicular to the parallels passing through the tangency point and the center of the transformed circle.
4. The centers of the next two inverted solution circles lie on the same axis as the previous circle and also on a circle whose radius equals the sum of the radius of the transformed circle and half the distance between the parallel lines.
5. The fourth solution appears as a tangent to the circle, parallel to both parallel lines.
6. Map the four found solutions back using circular inversion.
7. The problem has four solutions.