Two Tangent Circles, Different Sizes, Point Inside A Circle

Number of solutions: 1

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.
The image of the solution circle is a line parallel to the images of the given circles and passing through the image of the given point.
We invert the obtained line back using the original inversion.
The problem has one solution.

Draw a point B₁ where the circles touch eachother.
Draw a line segment AB₁.
Draw an axis of the line segment AB₁. The centre of the final circle belongs to this axis.
Draw a line h going through the centres of the given circles - points B and C.
The intersection point of the line segment AB₁ and the line h is named S₁ and it is the centre of the final circle.
Draw the final circle the centre of which is the point S₁ and which goes through the point A.
The final circle of the task "point, circle, circle"

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