Two Circles With No Contact, Line Intersects None

Steps

1. The set of all centers of circles tangent to two non-touching circles is a pair of hyperbolas. The foci of the hyperbolas are the centers of the given circles. To construct them, we need at least one point on each hyperbola. We draw the line passing through the centers of the given circles and find its intersections with the circles. The midpoints of the segments defined by these intersection points are the vertices of the hyperbolas.
2. We construct hyperbolas with foci at the centers of the circles, passing through the found vertices.
3. The set of all centers of circles tangent to a given circle and a line is a pair of parabolas. The focus of both parabolas is the center of the circle. Their directrices are lines parallel to the given line. The distance between the directrices and the given line equals the radius of the circle.
4. The centers of the solution circles lie at the intersections of the constructed hyperbolas and parabolas.
5. The problem has eight solutions.

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.