The Point Lies On The Line

Steps

1. Draw a perpendicular to the line from the problem and point A. The centres of all solutions have to lie on this line.
2. Dilate point A by the radius of the given circle. Label the intersections with the line from step 1 with A' and A''.
3. Draw a line segment between point A' and the center of the given circle and another line segment between point A'' and the center of the given circle. The center of the solution always lies at the intersection of one of these axes and the line from step 1.
4. Draw the solution.

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. First, find the hyperbola on which all centers of circles tangent to the given circle and passing through the given point lie. This hyperbola has one focus at the given point and the other at the center of the given circle. To construct the hyperbola, we need at least one point on it. This point can be found on the line connecting the two foci: it is located halfway between the intersection of the line with the circle and the given point A.
2. Construct the hyperbola based on the previous step.
3. The centers of circles tangent to the given line at point A lie on the perpendicular to line p passing through point A.
4. The centers of the desired solution circles are located at the intersections of the hyperbola and the perpendicular line.
5. The problem has two solutions.