Intersecting Circles, Line Tangent to One of Them

Steps

1. Choose the circle of inversion with its center at the point of tangency between the given line and circle.
2. Apply circular inversion to the given objects. The given line remains fixed under inversion (self-inverse), and the point of tangency maps to infinity. A circle passing through the tangency point becomes a line.
3. The first two images of the solution circles are circles tangent to the parallel lines and the image of the given circle. Their centers can be found using loci of points with specific properties.
4. The solution circles that pass through the point of tangency are mapped to tangents of the inverted circle that are parallel to the given line.
5. Map the found solution circles back using circular inversion.
6. The problem has four solutions.

Steps

1. The centers of all circles that are tangent to a given circle and its tangent line at their point of tangency lie on the perpendicular to the given line passing through that point.
2. The centers of all circles tangent to the given circle and its tangent line at a point different from their point of tangency lie on a parabola. The focus of this parabola is the center of the circle, and the directrix is a line parallel to the given tangent. The distance between the circle and the directrix equals the radius of the given circle.
3. The centers of all circles tangent to two intersecting circles, with one internal and one external tangency, lie on an ellipse. The foci of the ellipse are the centers of the two given circles, and the ellipse passes through their intersection points.
4. The centers of all circles that are tangent to both given circles, with either two internal or two external tangencies, lie on a hyperbola. The foci of the hyperbola are the centers of the given circles, and it also passes through their points of intersection.
5. One center of the solution circles lies at the intersection of the perpendicular through the tangency point and the ellipse. Among the two possible intersection points, only one corresponds to a valid solution. The other represents a center of a circle that is tangent to both given circles and simultaneously a center of a circle tangent to the given circle and the line at their point of contact — but these are not the same circles. For this reason, not all intersections in the following steps will yield valid solutions either.
6. Another center of a solution circle lies at the intersection of the perpendicular and the hyperbola.
7. The remaining two centers of the solution circles lie at the intersections of the hyperbola and the parabola.
8. The problem has four solutions.