Two Non-Tangent Circles, Third Touches Both Internally

Number of solutions: 4

We first focus on one pair of tangent circles. The centers of the solution circles lie on a straight line and an ellipse. The line passes through the centers of both circles. The centers of the given circles are also the foci of the ellipse, which passes through their point of tangency.
Next, we find the loci of centers of circles tangent to the second pair of tangent circles. Again, the centers of the solution circles lie on a straight line and an ellipse. The line passes through the centers of both circles. The centers of the given circles serve as the foci of the ellipse, which passes through their point of tangency.
The centers of the solution circles lie at the intersections of the identified lines and ellipses. However, not every intersection corresponds to a valid solution — only those points for which the type of tangency (internal or external) to the common circle is consistent 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 three 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 last solution circle is transformed into tangent 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