Two Externally Tangent Circles, Line Intersects Both Of Them

Number of solutions: 6

The problem is two circles with centres C and E with outer tangent D and a line that has an intersection with both circles.
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.
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.
Draw the tangents m and l of the circle f' parallel to the lines c' and e'.
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'.
Draw a circle centered at point B with radius that is the sum of the radius of the circle f' and the distance |Bc'|.
The intersections of this circle with the line n are the centers of the circles o and p with radius as |Bc'|.
Then, we back-plot the circles o, p using circular inversion over the same circle. These circles touch all inverted objects c', e', f'.
Next, we will use circular inversion to map back over the same circle the lines m, l. These lines touch all inverted objects c', e', f'.
We have all 4 solutions to the problem.

The problem is two tangent circles and a line that passes through both circles.
Construct perpendiculars from the given line to the centres of the given circles.
Construct circles identical to the ones in the problem with the center as the foot of these perpendiculars (points I and J).
Draw all 4 tangents of these new circles that are parallel to the given line.
For each of the tangents just drawn, construct a parabola defined always by the center of the circle that the tangent touches (points I and J).
This will produce 6 intersections of hyperbolas that were not defined by the same point.
Finally, construct 6 circles centered at these points and with radii such that they touch the given line.
These are our resulting circles.

Circle inversion