One Circle Inside Another, Third Has Internal Tangency with the Outer and External Tangency with the Inner

Steps

1. We choose the reference circle for the inversion. Its center is chosen at the point of tangency of the given circles.
2. We apply the circular inversion to the given objects. The circle passing through the point of tangency is transformed into a line.
3. The first three solution circles are transformed into circles tangent to the images of the given circles. We find them using sets of points with given properties.
4. The last solution circle is transformed into tangent to a circle parallel to the images of the given circles.
5. We invert the images of the solution circles back using the same inversion.
6. The problem has four solutions.