Two Circles With Inner Contact, Point Outside The Circle

Number of solutions: 1

We choose the reference circle for the inversion. We take the point of tangency of the two circles as its center.
We apply a circular inversion to the given objects. Since both circles pass through the center of the reference circle, they are mapped to lines.
The image of the solution circle is a line that is parallel to the images of the circles and passes through the image of the point.
We invert the obtained line back using the original inversion.
The problem has one solution.

We are given two touching circles and point A. Let's denote their point of tangency as T₁.
Draw a line f passing through the centers of the given circles.
Draw the bisector of segment |AT₁|.
The intersection of this bisector and line f is the center of our solution.
There is one solution.

We choose the reference circle for the inversion. We take the given point as its center.
We apply a circular inversion to the given objects. Since the given point is the center of the reference circle, it is mapped to infinity.
The image of the solution circle is the common tangent of the two transformed circles.
We invert the obtained line back using the original inversion.
The problem has one solution.

Circle Inversion