Two Tangent Circles, Third Intersecting at the Point of Tangency

Steps

1. Choose the circle of inversion so that its center lies at the common point of all three given circles.
2. Apply circular inversion to the given circles. They are transformed into two parallel lines and one non-parallel line.
3. Now we seek circles tangent to all three lines. The centers of such circles lie at the intersections of the angle bisectors formed by the lines.
4. Construct the circles that are the images of the solutions under inversion. Their centers are located at the identified intersection points, and their radii are determined using perpendicular segments from the centers to the lines.
5. Apply the inverse transformation to the constructed circles.
6. The problem has two solutions.