Two Concentric Circles, Third Circle Intersecting One Of Them

Steps

1. The centers of circles tangent to both given concentric circles lie on circles that are also concentric with the given ones. Therefore, it is sufficient to find one point on each such circle. To do this, draw a line passing through the center of the concentric circles and find its intersections with the two given circles.
2. The circles on which the centers of the solution circles lie pass through the midpoints of segments AB and AC. On one circle lie the centers of circles externally tangent to circle k₂, and on the other, the centers of circles internally tangent to k₂.
3. Circles that are externally tangent to k₂ have a radius equal to the distance AS′. Their centers are therefore located at this distance from circle k₁.
4. Circles that are internally tangent to k₂ have a radius equal to the distance AS″. Their centers are thus located at this distance from circle k₁.
5. The problem has four solutions.