Two Intersecting Circles, Third Lies Inside the Common Region and Is Tangent to One of Them

Number of solutions: 4

We begin by focusing on a pair of intersecting circles. The centers of the solution circles lie on both an ellipse and a hyperbola. The foci of the ellipse and the hyperbola are the centers of the given circles, and both curves pass through the points where the circles intersect.
Next, we determine the loci of centers of circles tangent to a pair of tangent circles. The centers of the solution circles lie on a line and a hyperbola. The line passes through the centers of the two given circles. The foci of the hyperbola are the centers of the circles, and the hyperbola is constructed to pass through their common point of tangency.
The centers of the solution circles lie at the intersections of the constructed ellipses, hyperbola, and line. However, not every such intersection corresponds to a valid solution – only those for which the type of tangency (internal or external) to the shared circle matches on both curves.
The problem has four solutions.

We choose the reference circle for the inversion. Its center is chosen at the point of tangency of the given circles.
We apply the circular inversion to the given objects. The circle passing through the point of tangency is transformed into a line.
The first two solution circles are transformed into circles tangent to the images of the given circles. We find them using sets of points with given properties.
The remaining two solution circles are transformed into tangents to a circle parallel to the images of the given circles.
We invert the images of the solution circles back using the same inversion.
The problem has four solutions.

Loci of Points Satisfying Certain Properties