Three Intersecting Circles with a Common Interior Region

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. Construct three hyperbolas whose foccuses are at the centres of the circles and which pass through their intersection.
2. Construct two ellipses whose foccuses are at the centers of the given circles and which pass through their intersection (can be done with any circle from the given set).
3. Two intersections of three hyperbolas are two solutions, different intersections of hyperbolas and ellipses hide 6 more solutions.
4. We havae 8 solutions