Secant Line, Point on Line Inside Circle

Steps

1. Choose the circle of inversion so that the given point is its center.
2. Apply circular inversion to the given circle. The given line remains invariant under this inversion, and the given point maps to infinity.
3. The images of the solution circles are tangents to the image of the circle that are parallel to the given line.
4. Invert the found tangents back to obtain the solution circles.
5. The problem has two solutions.

Steps

1. The set of centers of all circles passing through a given point and tangent to a given circle forms an ellipse. Its foci are the given point and the center of the given circle. To construct the ellipse, we need one more point on it. Draw a line through the given point and the center of the circle. The point located halfway between the given point and the intersection of this line with the circle is a point on the ellipse.
2. Construct the ellipse using the identified point.
3. The centers of all circles tangent to the given line at the given point lie on the perpendicular line passing through that point.
4. The centers of the desired solution circles are located at the intersections of the ellipse and the perpendicular line.
5. The problem has two solutions.