Two Externally Tangent Circles, Line Does Not Intersect Them

Steps

0. Given two circles with centres C and E with outer tangent D and a line that has no intersection with these circles.
1. Start with a circle centered at D and with an arbitrary radius, e.g. |DE|. This will serve as the axis of the circular inversion.
2. Represent the objects from the assignment using circular inversion over the circle from the previous point. The circles shown from the assignment will be parallel lines c' and e', and the line will appear on the circle f' centered at S.
3. Draw the tangents m and l of the circle f' parallel to the lines c' and e'.
4. Draw the axis of symmetry n of the parallel lines c' and e' and the intersection B of the axis n and the perpendicular line drawn from point S to the line c'.
5. Draw a circle centered at point B with radius that is the sum of the radius of the circle f' and the distance |Bc'|.
6. The intersections of this circle with the line n are the centers of the circles o and p with radii as |Bc'|.
7. Draw a circle with center at B and radius that is the difference of the radius of the circle f' and the distance |Bc'|.
8. The intersections of this circle with the line n are the centers of the circles q and r with radii as |Bc'|.
9. Then, we back-plot the circles o, p, q,r using circular inversion over the same circle. These circles touch all inverted objects c', e', f'.
10. Next, we back-plot the lines m, l using circular inversion over the same circle. These lines touch all inverted objects c', e', f'.
11. We have all 6 solutions to the problem.

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. Construct a circle centered at E with radius BF. Name its intersections with the perpendicular line J and K.
3. At points H, I, J and K, construct parallel lines with the given line
4. 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 S₁, S₂, S₃, S₄, S₅ and S₆
5. 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.
6. Construct circles with centres at points S that intersect the point of tangency.
7. We have 6 solutions