Tangent Circles, Line Is Tangent to One and Secant to the Other

Steps

1. We choose the reference circle for the circular inversion. The center of the circle is chosen as the point of tangency between the given circle and the line.
2. We apply circular inversion to the given objects. A circle passing through the point of tangency is mapped to a line. The given line is invariant under the inversion.
3. In the image, we solve the problem of finding circles tangent to two parallel lines and a circle. We solve it using loci of points with given properties. The three found solutions are the images, under inversion, of the solutions to the original problem.
4. The image of the solution that passes through the point of tangency of the given circle and the line is a tangent to the transformed circle. This tangent is parallel to the given line.
5. The found images of the solutions are mapped back using circular inversion.
6. The problem has four solutions.

Steps

1. The centers of circles tangent to the given circle and its tangent line at their common tangency point lie on a line perpendicular to the given line and passing through the point of tangency.
2. The centers of circles tangent to the given circle and its tangent line at a point other than their point of tangency lie on a parabola. The focus of the parabola is the center of the given circle, and the directrix is a line parallel to the given line. The distance between these two lines equals the radius of the given circle.
3. The centers of circles tangent to both given circles at their common tangency point lie on the line connecting the centers of the two circles.
4. The centers of circles that are either externally tangent to both given circles or internally tangent to both lie on a hyperbola. The foci of the hyperbola are the centers of the given circles, and the hyperbola passes through the point of tangency of the circles.
5. The centers of the solution circles are located at the intersections of the identified lines, the parabola, and the hyperbola. Some of these intersections do not correspond to valid solution circles—this happens when one curve contains centers of circles with internal tangency, while the other contains those with external tangency (or vice versa).
6. The problem has four solutions.