A Line Is Tangent To A Circle, A Point Independently

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 three solutions.

Steps

1. Create a line p₂ that will pass through the center of S₁ and the given point A. Name the intersection of the circle k₁ and the line p₂ closer to point A B and find the center C between points A and B.
2. Draw a perpendicular line p₃ that is perpendicular to the line p₁ and passes through the given point A.
3. Find points equidistant from the circle k₁ and point A. Thus, we are looking for a hyperbola d with foci at points A and S₁ that passes through point C.
4. We look for points equidistant from the line p₁ and point A. Thus, we are looking for a parabola c with a control line p₁ and foci at point A.
5. Name the intersections of the hyperbola e and the parabola g S.
6. The points S are the centres of the circles we are looking for. They must all pass through point A.