Two Tangent Circles Of Different Sizes, Point Lies On The Common Tangent Line Which Passes Through The Tangent Point Of The Circles

Steps

0. Two circles and a point that lies on a line perpendicular to the line joining the centres and passing through the point of contact of the circles are given.
1. We will solve the problem by circular inversion. The center of the control circle of the circular inversion is the given point.
2. We will represent the given circles in the circular inversion.
3. The given point is displayed to infinity. Therefore, the images of the solution circles are the common tangents of the circles displayed.
4. We display the found tangents in circular inversion. Their images are the searched solutions.
5. The problem has a total of two solutions.

Steps

1. This task will be solved using a set of points which can be used to draw a hyperbola. Lets two draw lines through point C and the points A and B respectively. Name the intersections of these lines.
2. Find the centerpoint between these intersections and the point C. These points are on the desired hyperbola.
3. Using one center of a circle and the point C as focus points and a point from step two which lies on the hyperbola construct a hyperbola. Repeat this for all four of the points constructed in step two - four hyperbolas should be constructed.
4. The intersections of these hyperbolas are the centers of the wanted circles. Call these points S₁ and S₂.
5. Draw the final circles with their center in points S₁ and S₂ respectively and passing through point C.
6. Continue using the process of solving the circle, circle, point solutions.