Circles with Internal Tangency, Line Tangent to Inner Circle

Steps

1. Construct the circle of inversion. Choose the point of tangency T as its center.
2. Apply the inversion to the given objects. In the chosen inversion, the given line remains invariant. One of the given circles is transformed into a line parallel to it.
3. Identify the images of the solutions. The first three are circles located in the strip between the two parallel lines. The fourth is a tangent to the image of the circle, which is parallel to both lines. The image of the point of tangency with the remaining two objects lies at infinity.
4. Apply the inverse transformation to the found circles and line.
5. The problem has four solutions.

Steps

1. All centers of circles that are tangent to both given circles at their common point of tangency lie on the line passing through the centers of the two given circles and their common tangent point.
2. The solution circle that is tangent to both given circles at their common point also shares the tangent line passing through this point.
3. The center of the desired circle lies on the angle bisector defined by the common tangent and the given line.
4. The first center of a solution circle lies at the intersection of the line through the centers of the given circles and the angle bisector.
5. The centers of all circles tangent to both given circles outside their common point lie on an ellipse. The foci of the ellipse are the centers of the given circles. The ellipse passes through their common tangent point.
6. The centers of all circles tangent to the given line and one of the circles lie on the perpendicular to the given line passing through the point of tangency and on a parabola. The focus of the parabola is the center of the circle. The directrix is parallel to the given line and at a distance equal to the radius of the given circle.
7. The centers of the remaining three solution circles lie at the intersections of the ellipse, the perpendicular, and the parabola.
8. The problem has a total of four solutions.