Line Tangent to Circle, Point Inside

Steps

1. The centers of all circles tangent to a given line and a given circle at their common point of tangency lie on the line perpendicular to the given line and passing through the center of the given circle.
2. The centers of all circles passing through the given point and the point of tangency lie on the perpendicular bisector of the segment connecting these two points.
3. The center of the desired solution circle is located at the intersection of the two constructed lines.
4. The problem has one solution.

Steps

1. To find the solution, we use a homothety centered at the point of tangency, in which the solution circle is mapped onto the given circle. Therefore, the centers of both circles and the center of the homothety must lie on the same straight line.
2. The given point A and its image under the homothety must also be collinear with the center of the homothety.
3. Since a homothety maps lines to parallel lines, the line connecting the given point A and the center of the solution circle must be parallel to the line connecting the image of point A and the center of the given circle.
4. The center of the solution circle lies at the intersection of the constructed lines.
5. The problem has a unique solution.