Two Intersecting Circles, Third Passes Through One Intersection and Intersects the Common Region

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 four solutions.

Steps

1. We first focus on one pair of the given circles. The centers of circles that touch both given circles either externally or internally lie on a hyperbola. The foci of this hyperbola are the centers of the given circles, and the hyperbola passes through their intersection points.
2. The centers of circles that touch one of the given circles externally and the other internally lie on an ellipse. The foci of this ellipse are also the centers of the given circles, and the ellipse passes through their intersection points.
3. Similarly, we find the loci of centers of circles touching the second pair of circles. Centers of circles that touch both circles either externally or internally lie on another hyperbola. Its foci are the centers of the given circles, and it passes through their points of intersection.
4. The centers of circles that touch one circle externally and the other internally lie on an ellipse. The foci of this ellipse are again the centers of the given circles, and it also passes through their intersection points.
5. The centers of the solution circles lie at the intersections of the constructed hyperbolas and ellipses. However, not every intersection point corresponds to a solution circle — only those where the type of tangency (external or internal) to the shared circle is the same for both curves.
6. The problem has four solutions.