Two Intersecting Circles Inside a Third One

Steps

1. We choose the reference circle for circular inversion. Its center is selected as the intersection point of the given circles.
2. We apply the circular inversion to the given objects. Circles passing through the center of the reference circle are mapped to lines.
3. In the image, we look for circles tangent to two lines and a circle. The solution circles of this configuration are the images of the original problem's solutions. The centers of these circles lie on the angle bisector defined by the two lines.
4. We use four homotheties in which the given circle is mapped onto the solution circles. In these homotheties, the given lines are mapped to parallel tangents of the given circle.
5. The intersections of the given lines are mapped to the intersections of the tangents. We connect them with lines. The intersections of these lines with the given circle are the homothety centers and the tangency points between the given circle and the solution circles.
6. The center of the given circle is mapped, under homothety, to the centers of the solution circles. Therefore, these centers lie on lines passing through the center of the given circle and the corresponding tangency points. The centers of the solution circles are the intersections of these lines with the angle bisector.
7. We have found four solutions of the problem in the image.
8. We map the obtained circles back via circular inversion.
9. The problem has four solutions.