Two Circles With No Contact, Line Intersects None

Steps

0. The problem is two circles and a line that do not touch each other. Use circular inversion four times to find two of the resulting circles. With each inversion we will have a differently dilated assignment. We'll call the smaller circle c and the larger circle d (with the same radius, we can interchange)
1. The first dilated assignment is the circle d with radius c smaller and
2. the specified line of radius c is "closer" (it is parallel to the specified line and the distance between them is c).
3. Perform a circular inversion of the dilated line and the dilated circle d over the circle c. This creates two circles.
4. Construct 4 tangent images of the circular inversion.
5. Plot the tangents back over the circular inversion - again over the circle c.
6. Dilate the tangent (circle) images back (try increasing or decreasing back by the radius c, in steps 13-22 by the radius d). Just two of these solutions touch all the given objects.
7. For steps 8-12, the dilated input is a circle d smaller by radius c and the input line "further" by radius c (cf. step 2). Again, solve by circular inversion over the circle c.
8. For steps 13-18, the dilated input is a circle c larger by radius d and the input line "closer" by radius d (cf. step 2). We solve by circular inversion, but over the circle d.
9. For steps 19-22, the dilated input is a circle c larger by radius d and the input line "farther" by radius d (cf. step 2). We solve by circular inversion, but over the circle d.