Two Externally Tangent Circles, Line Does Not Intersect Them

Steps

1. Draw perpendiculars to the given line passing through the centres of the given circles and name their intersections with the given line D and E and their intersections with the given circles F and G.
2. Construct a circle centered at G with radius AG. We name its intersections with the perpendicular line H and I.
3. Construct a circle centered at E with radius BF. Name its intersections with the perpendicular line J and K.
4. At points H, I, J and K, construct parallel lines with the given line
5. Construct a parabola with a focus at point A whose control line is the parallel passing through H. Construct a parabola with a focus at point A whose control line is the parallel passing through I. Construct a parabola with a focus at point B whose control line is the parallel passing through J. Construct a parabola with a focus at point B whose control line is the parallel passing through K. Label their intersections S_1, S_2, S_3, S_4, S₅ and S₆
6. From these found centers of the solution circles, run a perpendicular line to the given line. The intersection of each with the given line forms the points of contact of the circle, which has a center at S, through which the perpendicular also passes.
7. Construct circles with centres at points S that intersect the point of tangency.
8. We have 6 solutions