Two Circles With No Contact, Line Intersects One

Steps

1. The centers of circles tangent to both given circles lie on a pair of ellipses. The foci of the ellipses are the centers of the given circles. To construct them, we need at least one point on each ellipse. We draw a line through the centers of the given circles and find the midpoints of the segments whose endpoints are the intersections of this line with the given circles. These points are the centers of circles tangent to both given circles.
2. We construct ellipses with foci at the centers of the given circles that pass through these points.
3. The centers of circles tangent to the given line and the inner circle lie on a pair of parabolas. The focus of both parabolas is the center of the circle; the directrices are parallel to the given line, and their distance from the given line equals the radius of the circle.
4. The centers of the solution circles lie at the intersections of the ellipses and the parabolas.
5. The problem has four solutions.

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 k₁ ' and k₂ ' 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 k₃ ' and k₄ ' 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 k₁ ', k₂ ', k₃ ' and k₄ ' according to the circle d.
14. The resulting circles k₁, k₂, k₃ and k₄ are the solutions.