Internal Tangency of Circles, Line Passes Through Point of Tangency and Intersects Both Circles

Steps

1. We choose the reference circle for the inversion. Its center is selected as the point of tangency of the two circles, which also lies on the given line.
2. We apply circular inversion to the given objects. Both circles are mapped to lines. The given line is invariant under the inversion.
3. The problem is thus transformed into finding circles tangent to three lines, two of which are parallel. The centers of such circles lie at the intersections of angle bisectors.
4. The resulting circles are mapped back via circular inversion.
5. The problem has two solutions.

Steps

1. The set of centers of all circles that are externally tangent to the smaller circle and internally tangent to the larger circle lies on an ellipse. The foci of the ellipse are the centers of the given circles. The ellipse passes through their point of tangency.
2. The centers of all circles tangent to a circle and its secant lie on a pair of parabolas. The focus of both parabolas is the center of the circle. The directrices are parallel to the given line. The distance between these parallels and the given line equals the radius of the circle.
3. The centers of the solution circles lie at the intersections of the ellipse with the parabolas.
4. The problem has two solutions.