Two Tangent Circles Of Different Sizes, Point Is Colinear With Both Centres

Steps

1. The center of a circle passing through point A and touching both given circles at their common tangential point lies on line p.
2. The locus of all centers of circles passing through point A and touching circle k₁ is a hyperbola. The foci of this hyperbola are the center of circle S₁ and point A. One point of the hyperbola lies at the midpoint of segment AD.
3. The locus of all centers of circles passing through point A and touching circle k₂ is another hyperbola. This time, the foci are the center of circle S₂ and point A. One point of the hyperbola lies at the midpoint of segment AF.
4. The centers of the desired circles are the intersections of the two hyperbolas, as well as line p.
5. The desired circles have their centers at the found intersection points and pass through point A.

Steps

1. Construct circle e with center at point C, passing through the intersection of circles a and b (a different circle would also work).
2. Perform a circular inversion using circle e.
3. Construct the common tangents to the images of circles a and b.
4. Invert the new tangents back with respect to the original guiding circle e and construct a circle k with center at point E passing through B.
5. k1, k2 and k3 are the solution.