Two Touching Circles Inside Another, Both Tangent to It

Steps

1. We choose the reference circle for the inversion. Its center is chosen as the point of tangency of the given circles.
2. We apply the circular inversion to the given objects. The circles passing through the point of tangency, which is the center of the reference circle, are transformed into lines.
3. We find circles tangent to the images of the given circles. These are the images of the solution circles. To find their centers, we use sets of points with given properties.
4. We invert the found circles back using the circular inversion.
5. The problem has two solutions.