Two Tangent Circles, Third Circle Internally Tangent to One

Steps

1. Choose the circle of inversion, with its center at one of the points of tangency of the given circles.
2. Apply circular inversion to the given circles. The result is two parallel lines and a circle that is tangent to one of them.
3. The images of the solutions in the inversion are a circle tangent to all three objects and a tangent line to the inverted circle that is parallel to both lines.
4. Invert the found tangent and circle back to the original setting.
5. The problem has two solutions.