Two Circles with External Tangency, Third Intersecting Both

Steps

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

Steps

1. We first focus on one pair of the given circles. The centers of circles that touch both given circles either externally or internally lie on a hyperbola. The foci of this hyperbola are the centers of the given circles, and the hyperbola passes through their common point of tangency.
2. The centers of circles that touch both given circles at their common tangency point lie on the line passing through their centers.
3. Next, we find the loci of centers of circles touching the second pair of circles. The centers of circles that touch both of these circles either externally or internally again lie on a hyperbola. The foci of this hyperbola are the centers of the given circles, and it passes through their points of intersection.
4. The centers of circles that touch one of the circles externally and the other internally lie on an ellipse. The foci of this ellipse are again the centers of the given circles, and the ellipse passes through their intersection points.
5. The centers of the solution circles lie at the intersections of the constructed hyperbolas, the line, and the ellipse. However, not every intersection corresponds to a valid solution — only those where the type of tangency (external or internal) to the shared circle matches across all involved curves.
6. The problem has six solutions.