Non-Tangent Circles, Line Is Tangent to One, the Other Circle Lies in the Opposite Half-Plane

Steps

1. We choose the reference circle of the circular inversion. We select its center to be the tangency point of the given circle and the line.
2. We apply circular inversion to the given objects. The circle passing through the center of the reference circle is mapped to a line. The given line is invariant.
3. The solution circles are mapped under the inversion to tangents of the image of the circle that is parallel to the given line.
4. The obtained tangents are mapped back through the circular inversion.
5. The problem has two solutions.

Steps

1. The solution circles will be tangent to the given circle and the line, which is their common tangent, at their shared point of tangency. The centers of the solution circles will lie on the perpendicular to the given line passing through this tangency point.
2. All centers of circles tangent to the given line and the circle that does not touch it lie on a pair of parabolas. The common focus of both parabolas is the center of the given circle. The directrices are parallel to the given line, and their distance from the line equals the radius of the given circle.
3. The centers of the solution circles lie at the intersections of the parabolas and the perpendicular.
4. The problem has two solutions.

Steps

1. The point of tangency between the given circle and the given line will also be a point of tangency with the solution circles. Therefore, the centers of the solution circles will lie on the perpendicular to the line passing through the point of tangency.
2. To find the solution circles, we use two homotheties in which the solution circles are mapped onto the given circle. The centers of these homotheties are the remaining two points of tangency between the circles. In these homotheties, the given line is mapped to tangents of the given circle that are parallel to each other.
3. Points of tangency map to points of tangency. The lines passing through them also pass through the centers of the homotheties. These are the intersection points of the lines with the given circle. These points are also points of tangency between the given circle and the solution circles.
4. The centers of the solution circles lie on the perpendicular bisectors of segments defined by the points of tangency.
5. The centers of the solution circles lie at the intersections of the perpendicular bisectors and the perpendicular line.
6. The problem has two solutions.