Circles with Internal Tangency, Third Passes Through the Point of Tangency

Steps

1. We choose the reference circle for the inversion. Its center is chosen at the intersection point of all three circles.
2. We apply the circular inversion to the given circles. Since all of them pass through the center of the reference circle, they are transformed into lines.
3. We find circles tangent to the images of the given circles. Their centers are found using angle bisectors.
4. We invert the found circles back using the same inversion.
5. The problem has two 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 the ellipse are the centers of the given circles, and the ellipse also passes through their common point of tangency.
2. We now find the loci of centers of circles tangent to the second pair of circles. The centers of circles that are externally tangent to both or internally tangent to both lie on a hyperbola. The foci of the hyperbola are the centers of the given circles, and the hyperbola passes through their intersection points.
3. 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 again the centers of the given circles, and it passes through their points of intersection.
4. The centers of the solution circles lie at the intersections of the found ellipses and 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.
5. The problem has two solutions.