Non-Tangent Circles, Line Tangent to One of Them

Steps

1. We choose the reference circle for the inversion. Its center is selected as the point of tangency between the line and the circle.
2. We apply circular inversion to the given objects. A circle passing through the center of the reference circle is mapped to a line. The given line is invariant under the inversion.
3. We look for circles tangent to the transformed objects: two parallel lines and a circle. The problem is solved using loci of points with specific properties. The four circles found are the images of the solution circles to the original problem.
4. The images of the remaining two solutions are the tangents to the transformed circle. These tangents are parallel to the given line.
5. All images of the found solutions are mapped back using circular inversion.
6. The problem has six solutions.

Steps

1. The centers of circles tangent to a given circle and its tangent line at their point of tangency lie on a line perpendicular to the tangent, passing through the point of tangency.
2. The centers of circles tangent to a given circle and its tangent line, but not at the point of tangency, lie on a parabola. The focus of the parabola is the center of the circle. The directrix is a line parallel to the given line. The distance between these parallel lines equals the radius of the given circle.
3. The centers of all circles tangent to both given circles lie on a pair of hyperbolas. Their foci are the centers of the given circles. To construct the hyperbolas, we need at least one point on each. We draw a line connecting the centers of the two circles. On this line lie the centers of circles whose points of tangency with the given circles lie on a single line. These are the intersections of the drawn line with the given circles. Points on the hyperbolas are then the midpoints of segments defined by these intersection points.
4. We construct hyperbolas with foci at the centers of the given circles passing through the found points.
5. The centers of circles tangent to all three given objects lie at the intersections of the parabola, the perpendicular line, and both hyperbolas. However, some of these intersection points are not centers of solution circles. This occurs when one curve contains centers of circles internally tangent to the given circle, and another contains those externally tangent to it (or vice versa).
6. The problem has six solutions.