Non-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 two found solutions are the images, under inversion, of the solutions to the original problem.
4. The images of the solutions that passes through the point of tangency of the given circle and the line are tangents to the transformed circle. These tangents are 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 set of all centers of circles tangent to two non-touching circles is a pair of hyperbolas. The foci of the hyperbolas are the centers of the given circles. To construct them, we need at least one point on each hyperbola. We draw the line passing through the centers of the given circles and find its intersections with the circles. The midpoints of the segments defined by these intersection points are the vertices of the hyperbolas.
2. We construct hyperbolas with foci at the centers of the circles, passing through the found vertices.
3. The set of all centers of circles tangent to a given circle and a line is a pair of parabolas. The focus of both parabolas is the center of the circle. Their directrices are lines parallel to the given line. The distance between the directrices and the given line equals the radius of the circle.
4. The centers of the solution circles lie at the intersections of the constructed hyperbolas and parabolas.
5. The problem has four solutions.