Three Intersecting Circles, the Third Intersects the Common Region of Two

Steps

1. We choose the reference circle for the inversion. Its center is chosen as one of the intersection points of the given circles.
2. We apply circular inversion to the given objects. The circles that pass through the center of the reference circle are transformed into lines.
3. We look for circles tangent to the images of the given circles. Their centers lie on the angle bisectors defined by the lines.
4. To find the centers of these circles, we use homotheties in which the desired circles are mapped onto the circle they are supposed to touch. The centers of these homotheties are the points of tangency between circles. The given lines are transformed in these homotheties into parallel tangents of the circle.
5. The intersections of lines are mapped to the intersections of tangents. The line connecting them passes through the centers of the homotheties, which are also the points of tangency between circles.
6. The centers of the desired circles lie at the intersections of lines passing through the center of the given circle and the points of tangency, along with the previously found angle bisectors.
7. We have found eight circles that are the images of the solution circles.
8. We invert the found circles back using the same circular inversion.
9. The problem has eight 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 eight solutions.