Two Tangent Circles, Different Sizes, Point Outside A Circle

Number of solutions: 3

We choose the reference circle for the inversion. We take the point of tangency of the two given circles as its center.
We perform a circular inversion of the given objects. Since the point of tangency lies on both given circles, they are transformed into lines.
We find circles tangent to the images of the original circles and passing through the image of the given point. For finding their centers, we can use, for example, loci of points with given properties. We have found two such circles. These are the images of the solution circles.
The image of the third solution circle is a line parallel to the images of the original circles and passing through the image of the given point.
We invert the found images of the solution circles back using the original inversion.
The problem has three solutions.

We will solve this problem using a set of points of given properties, which for a pair of a point and a circle represents a hyperbola. Draw the lines through point C and the centers of the given circles from the problem, i.e., points B and A. Label the intersections of the lines and circles from the problem.
Find the midpoints between the intersections and point C, these points will lie on the hyperbolas we are looking for.
Using the two foci (the center of the given circle and point C from the assignment) and the point lying on the search hyperbola, one of the centers from the previous step, draw hyperbolas for the two circles from the assignment and point C.
The intersections of these hyperbolas are the centers of the circles we are looking for, denote them S₁, S₂ and S₃.
Draw the resulting circles centered at S₁, S₂, and S₃, which pass through point C from the problem.
Resulting circles of the circle, circle, point problem.

Circle Inversion