Two Concentric Circles, Third Circle Intersecting Inner Circle

Steps

1. The centers of circles tangent to both concentric circles lie on circles that are also concentric with the given circles. It is therefore sufficient to find just one point on such a circle. We construct a line passing through the center of the concentric circles and find the intersections of this line with the concentric circles.
2. The circles on which the centers of circles tangent to both concentric circles lie pass through the midpoint of segment AB and the midpoint of segment AC. On one circle lie the centers of circles that are externally tangent to circle k₂, on the other lie those that are internally tangent to it.
3. Circles that are externally tangent to circle k₂ have a radius equal to the distance AS'. Their centers therefore lie at this distance from circle k₁.
4. Circles that are internally tangent to circle k₂ have a radius equal to the distance AS''. Their centers therefore lie on a circle concentric with circle k₁. Its radius is the difference between the distance AS'' and the radius of k₁.
5. The problem has four solutions.