One Circle Inside Another, Third Intersects the Inner and Is Tangent to the Outer

Number of solutions: 4

We first focus on the pair of intersecting circles. The centers of the solution circles lie on an ellipse and a hyperbola. The foci of both the ellipse and the hyperbola are the centers of the given circles, and both curves pass through the intersection points of the circles.
Next, we determine the loci of centers of circles tangent to the pair of tangent circles. The centers of the solution circles lie on a straight line and a hyperbola. The line passes through the centers of both circles. The foci of the hyperbola are the centers of the given circles, and the hyperbola passes through the point of tangency of the two circles.
The centers of the solution circles lie at the intersections of the identified ellipses, hyperbola, and line. However, not every intersection of these curves is a center of a solution circle — 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