Two Externaly 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.

Steps

1. We first focus on pair of the given tangent circles. The centers of circles that touch both of them 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. Next, we find the loci of centers of circles that touch the second pair of circles. Again, the centers of circles that touch both either externally or internally lie on a hyperbola. Its foci are the centers of the given circles, and it passes through their intersection points.
3. 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 the centers of the given circles, and the ellipse passes through their intersection points.
4. The centers of the solution circles lie at the intersections of the constructed hyperbolas, the ellipse, and the line. However, not every intersection is a valid center of a solution circle — only those for which the type of tangency (external or internal) to the common circle agrees on both curves.
5. The problem has two solutions.