Two Circles With No Contact, Line Intersects One

GeoGebra construction

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Steps

0. The input is the circles a and b and the line c that intersects the circle a.
1. Construct a random circle d centered at point A, which is one of the intersections of circle a and line c.
2. Make a circular inversion of the circle a and b according to the circle d.
3. Construct a circle e with radius the size of the radius of circle b' centered at point B, which is the intersection of lines c and a'.
4. Construct the perpendiculars f and g to the lines c and a' that pass through the point B.
5. Construct the parallel lines h, i, j, and k of the lines c and a' that pass through the intersections C, D, E, and F of the lines f and g and the circle e.
6. Construct the axis of the angle of the lines h and i.
7. Construct a random perpendicular line m to the line h.
8. Construct the circles n and o centered at the intersection G of the lines l and m that touch the lines k and h.
9. Construct the lines p and q that pass through the center H of the circle b' and through the intersections I and J of the lines h and i and j and k.
10. From the intersections K, L, M, and N of the lines p and q and the circles n and o, construct the lines r, s, t, and u that pass through the point G.
11. Construct the parallel lines v, w, x and y of the lines r, s, t and u that pass through the point H.
12. Construct the circles k_1' and k_2' which have their centers at the intersections O and R of the lines v and y with the line l and have an exterior contact with the circle b', and the circles k_3' and k_4' with their centers at the intersections P and Q of the lines w and x with the line l and have an interior contact with the circle b'.
13. Perform a circular inversion of the circles k_1', k_2', k_3' and k_4' according to the circle d.
14. The resulting circles k₁, k₂, k₃ and k₄ are the solutions.