Two Internally Tangent Circles, Third Intersects Both, Tangency Point Lies Outside 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.

Steps

1. We first focus on a pair of tangent given circles. The centers of circles that are externally tangent to one and internally tangent to the other lie on an ellipse. The foci of this ellipse are the centers of the given circles, and the ellipse passes through their common point of tangency.
2. The centers of circles that touch both given circles at their common point of tangency lie on the line passing through their centers.
3. Next, we find the loci of centers of circles tangent to the second pair of circles. The centers of circles that touch both either externally or both internally lie on a hyperbola. The foci of the hyperbola are the centers of the given circles, and it passes through their points of intersection.
4. The centers of circles that are externally tangent to one and internally tangent to the other lie on an ellipse. The foci of the ellipse are the centers of the given circles, and it passes through their points of intersection.
5. The centers of the solution circles lie at the intersections of the found ellipses, the line, and the hyperbola. However, not every such intersection is the center of a solution circle — only those for which the type of tangency (external or internal) to the common circle matches on both curves.
6. The problem has six solutions.