Secant Line and Point Inside Circle

Steps

1. First, choose the circle of inversion so that the given point is its center.
2. Apply the inversion to the given line and the given circle.
3. Since the given point, being the center of the inverting circle, is mapped to infinity, the solution circles appear as common tangents to the images of the inverted objects.
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 lies on an ellipse, whose foci are the given point and the center of the given circle. To construct the ellipse, we need at least one point on it. Such a point can be found on the line passing through the foci: it is the midpoint of the segment whose endpoints are the given point and the intersection of the line with the given circle.
2. Construct the ellipse passing through the identified point.
3. The set of centers of all circles passing through the given point and tangent to a given line is a parabola, with the given point as the focus and the given line as the directrix.
4. The centers of the desired solution circles are located at the intersections of the ellipse and the parabola.
5. The problem has two solutions.