Tangent Circles with a Common Tangent Line

Steps

1. Choose the circle of inversion with its center at the tangency point of the given circle and the line.
2. Apply circular inversion to the given objects. The circle passing through the center of the inversion transforms into a straight line. The given line is self-inverse.
3. Find the circles tangent to the transformed objects. To locate their centers, use loci of points defined by specific geometric properties.
4. Invert the found images of the solution circles back to obtain the original solutions.
5. The problem has two solutions.

Steps

1. The solution circles will be externally tangent to the given circles. The centers of these circles lie on a hyperbola. The foci of the hyperbola are the centers of the given circles. The hyperbola passes through their point of tangency.
2. The centers of circles tangent to a given line and a circle, other than at their common 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 the given line and the directrix equals the radius of the given circle.
3. The centers of the solution circles lie at the intersections of the parabola and the hyperbola.
4. The problem has two solutions.