Intersecting Circles with External Line

Steps

We choose the reference circle for the circular inversion. Its center is placed at the intersection point of the given circles. 1. We choose the reference circle for the circular inversion. Its center is placed at the intersection point of the given circles. 2. We apply circular inversion to the given objects. The circles are transformed into intersecting lines, and the given line is transformed into a circle. 3. Now we look for circles tangent to the transformed objects – a circle and two lines. To find them, we use homothety, specifically four homotheties in which the given circle is the image of each solution circle. The centers of these homotheties are the points of tangency between the given circle and the solution circles. The given lines are transformed into parallel tangents to the circle. 4. The intersection point of the original lines is transformed into the intersection points of the tangents. We draw lines connecting these points. The intersections of these lines with the circle are the centers of the homotheties – i.e., the tangency points between the given circle and the solution circles. 5. We draw lines passing through the found points of tangency and the center of the given circle. These lines represent, in the homotheties, the images of the center of the given circle mapped to the centers of the solution circles. 6. Since the solution circles must be tangent to the given lines, their centers lie on the angle bisector of the two lines. We find the centers of the solution circles as intersections of this bisector with the previously constructed lines. 7. We have found four solutions to the problem in the inverted image. 8. These are the images of the original problem’s solutions under circular inversion. 9. The problem has four solutions.

Steps

1. The centers of all circles that are either externally tangent to both given circles or internally tangent to both lie on a hyperbola. The foci of this hyperbola are the centers of the two given circles. The hyperbola passes through the intersection points of the two circles.
2. The centers of all circles that are tangent to the given line and externally tangent to the given circle lie on a parabola. The focus of the parabola is the center of the circle, and the directrix is a line parallel to the given line, shifted by the radius of the circle away from the line.
3. The centers of all circles that are tangent to the given line and internally tangent to the given circle lie on a parabola. The focus is again the center of the circle, and the directrix is a parallel line shifted toward the circle by the value of its radius.
4. The centers of the circles tangent to all three given objects lie at the intersections of the hyperbola and the parabolas. Some of these intersections do not correspond to valid solution circles—this happens when one curve contains centers of circles with internal tangency, while the other contains those with external tangency (or vice versa).
5. The problem has four solutions.