Two Non-Tangent Circles, Third Touches Both Externally

Number of solutions: 4

We begin by focusing on one 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 the two given circles. The foci of the hyperbola are the centers of the given circles, and the hyperbola also passes through their common point of tangency.
Similarly, we find the loci of centers of circles tangent to the second pair of tangent circles. Again, the centers 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 identified hyperbolas and lines. However, not every intersection is a valid center — only those for which the type of tangency (external or internal) 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 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