Secant Line and Point Outside 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 centers of all circles tangent to a given circle and passing through a given point lie on a hyperbola. The foci of this hyperbola are the given point and the center of the given circle. To construct the hyperbola, we need one more point on it. Locate its vertex on the line defined by the two foci. A point located halfway between the intersection of this line with the circle and point A lies on the hyperbola.
2. Construct the hyperbola using the point found in the previous step.
3. The centers of all circles tangent to a given line and passing through the given point lie on a parabola, with the given point as the focus and the given line as the directrix.
4. The centers of the desired solution circles lie at the intersections of the hyperbola and the parabola.
5. The problem has two solutions.