Two Circles With Contact, Third Circle Outside Both Of Them

Steps

1. We will solve the problem using circular inversion. The circle of inversion is chosen so that its center lies at the point of tangency of the given circles.
2. Apply the inversion to the given circles. The tangent circles are transformed into parallel lines.
3. Through the circular inversion, we have transformed the original problem into one involving two parallel lines and a circle. This new problem can be solved, for example, using loci of points with given properties. The images of the solution are four circles tangent to the two parallel lines and the image of the original circle. In addition, there are also two tangent lines to the inverted circle that are parallel to the pair of lines.
4. Apply the inverse transformation to the circles and tangents found in the inverted setting.
5. The original problem has six solutions.

Steps

1. Draw a line passing through the centers of the two given tangent circles. This line represents the locus of centers of all circles tangent to both given circles at their common point of tangency.
2. Construct a hyperbola with foci at the centers of the given circles and passing through their point of tangency. This hyperbola represents the locus of centers of all circles that are tangent to both given circles with either two external or two internal tangency points.
3. Select another pair of circles. Construct a line through their centers. Find the points where this line intersects the circles, and determine the midpoints of the segments defined by these intersection points. The hyperbolas passing through these midpoints represent the loci of centers of circles tangent to both selected circles.
4. Construct these hyperbolas. Their foci are the centers of the corresponding given circles.
5. Find the intersection points of the constructed hyperbolas and the initial line. Some of these intersections are centers of the solution circles. Others are not, specifically when they lie at the intersection of two loci—one corresponding to centers of circles internally tangent to a given circle, and the other corresponding to centers of circles externally tangent to it.
6. The problem has a total of six solutions.