Two Tangent Circles Inside Another Circle

Number of solutions: 6

We first find the centers of circles tangent to one pair of the given circles. The centers of the solution circles lie on a pair of ellipses. The foci of the ellipses are the centers of the given circles, and to construct them, we need to know at least one point on each ellipse. We draw the line passing through the centers of the given circles and find its intersections with the circles. The midpoints of the segments defined by the resulting intersection points are the vertices of the ellipses.
We draw the ellipses with foci at the centers of the given circles, passing through the identified vertices.
Next, we find all centers of circles tangent to another pair of given circles. This time, the centers of the solution circles lie on a line and a hyperbola. The line passes through the centers of the two circles. The foci of the hyperbola are the centers of the given circles, and the hyperbola passes through their common tangent point.
The centers of the solution circles lie at the intersections of the found ellipses, the hyperbola, and the line. However, not every intersection is the center of a solution circle — only those where the type of tangency (internal or external) to each of the two given circles matches on both curves.
The problem has six solutions.

We choose the reference circle for the inversion. Its center is chosen at the point of tangency of the given circles.
We apply the circular inversion to the given objects. The circle passing through the point of tangency is transformed into a line.
The first four solution circles are transformed into circles tangent to the images of the given circles. We find them using sets of points with given properties.
The remaining two solution circles are transformed into tangents to a circle parallel to the images of the given circles.
We invert the images of the solution circles back using the same inversion.
The problem has six solutions.

Loci of Points Satisfying Certain Properties