Two Circles Inside Each Other, Line Tangent To The Outer Circle

Steps

2. Since the line is tangent to the outer circle k_1, the circles we are looking for must touch at their tangent point.
3. We find this by forming a perpendicular to the given line p_1, which also passes through the centre C of the circle k_1.
4. Let us call this tangent point F. The centres of the circles we are looking for must lie somewhere on this perpendicular line.
5. Prepare a circle for circular inversion.
6. It can have any radius, the important thing is that it has a center at point F. Let's call it circle e.
7. Make a circular inversion for the inner given circle k_2.
8. Since the circle over which we do the circular inversion has its center at point F, we send point F to infinity.
9. But since we are looking for something to touch point F and the inverted circle, this something must pass through infinity and also be tangent to the inverted circle.
10. We also want this something to touch the given line p₁ at only one point, point F.
11. But this means that even in the inverted image, this something must touch the inverted line p₁ at one point.
12. Since this one location is supposed to be point F, which is at infinity, this something must be a line parallel to the image of the line p_1.
13. Construct two parallel lines to the line p₁ that are also tangents to the inverted circle k_2.
14. We invert the parallel lines through the circle e back. This gives the two resulting circles.