Two Internally Tangent Circles, Third Intersects Both, Tangency Point Lies Inside the Third Circle

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 circular inversion to the given circles. The circles that pass through the center of the reference circle are transformed into lines.
3. We find circles tangent to the images of the given circles. Their centers are found using sets of points with given properties. The four resulting circles are the images of four solution circles.
4. The remaining two solutions appear as tangents to the image of a circle that is parallel to the lines representing the images of the remaining two given circles.
5. We invert the obtained circles and tangents back using the same inversion.
6. The problem has six solutions.