Two Circles With Inner Contact, Point On The Larger Circle

Steps

1. Imagine a circle that is a solution to the problem. If we enlarge this circle by the radius of the smaller of the two given circles, the smaller given circle shrinks to a point, and the larger one increases its radius by the same amount. The given point is adjusted so that it remains on the enlarged larger circle and stays tangent to the enlarged solution circle.
2. The center of the solution circle must lie on the perpendicular bisector of the segment A'S₂.
3. At the same time, it must lie on the line defined by the center of the larger circle and the given point. Therefore, we find the center of the solution circle at the intersection of this line and the bisector from the previous step.
4. The problem has one solution.

Steps

0. Given are the circles a and b such that a is internally tangent to b. Also given is the point A, which lies on b and is not the tangent point of the two circles.
1. Draw any circle c with A as its origin.
2. Invert this circle. Point A' is now at infinity. The inverted forms of a and b shall be named a' and b' respectively.
3. Now, construct the line d that is tangent to b' and parallel to a'. This line must touch a' and go through A' in infinity.
4. Invert d around c. The resulting circle is the solution.