Two Circles with External Tangency, Third Intersecting One of Them

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 points of intersection of the two circles.
Next, we find the loci of centers of circles tangent to the pair of tangent circles. These 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 constructed hyperbolas, the ellipse, and the line. However, not every intersection of these curves is a 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.

Choose the circle of inversion, placing its center at the tangency point of the two circles.
Apply circular inversion to the given circles. The circles passing through the center of the inversion circle transforms into two parallel lines.
The centers of the images of the solution circles will lie on the axis (midline) of the strip between the parallel lines.
The centers lie at the intersections of this axis and a circle concentric with the transformed circle, whose radius is larger by half the distance between the parallel lines.
The remaining two solutions appear as tangents to the transformed circle, parallel to both parallel lines.
Map the found images of the solutions back using circular inversion.
The problem has four solutions.

Loci of Points Satisfying Certain Properties