Two Intersecting Circles Inside a Third One

Number of solutions: 4

We first focus on a pair of intersecting circles. The centers of the solution circles lie on a hyperbola. The foci of the hyperbola are the centers of the given circles, and the hyperbola also passes through their intersection points.
Next, we similarly find the loci of centers of circles tangent to another pair of circles. The centers of the solution circles lie on a pair of ellipses. The foci of the ellipses are the centers of the given circles. To construct each ellipse, we need at least one point on it. We draw a line through the centers of the given circles and find its intersections with the circles. The midpoints of the segments defined by these intersection points are the vertices of the ellipses.
We construct ellipses with foci at the centers of the given circles that pass through the identified vertices.
The centers of the solution circles lie at the intersections of the constructed ellipses and hyperbola. However, not every such intersection point corresponds to a solution circle — only those for which the type of tangency (internal or external) to the shared circle agrees on both curves.
The problem has four solutions.

We choose the reference circle for circular inversion. Its center is selected as the intersection point of the given circles.
We apply the circular inversion to the given objects. Circles passing through the center of the reference circle are mapped to lines.
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.
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.
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.
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.
We have found four solutions of the problem in the image.
We map the obtained circles back via circular inversion.
The problem has four solutions.

Loci of Points Satisfying Certain Properties