Touching Circles, Line Intersects Both at the Point of Tangency

Steps

1. Choose the circle of inversion such that its center is the point of tangency between the two given circles.
2. Apply circular inversion to the given objects. The two circles are mapped to parallel lines, and the given line remains unchanged (self-inverse).
3. Solve the problem for three lines—two of which are parallel. This configuration yields two solutions. The centers of the solution circles lie at the intersections of the angle bisectors.
4. We have found two images of the solution circles.
5. Invert these images back using the circular inversion to obtain the original solutions.
6. The problem has two solutions.

Steps

1. All centers of circles tangent to the given circles outside their common point of tangency lie on a hyperbola. The foci of the hyperbola are the centers of the given circles, and the hyperbola passes through their point of tangency.
2. All centers of circles tangent to the given circle and the line lie on a pair of parabolas. The focus of both parabolas is the center of the given circle. The directrices are lines parallel to the given line. The distance between the directrices and the given line equals the radius of the given circle.
3. The centers of the solution circles lie at the intersections of the hyperbola and the parabolas.
4. The problem has two solutions.