Circle Inversion

Definition of Circular Inversion

Circular inversion is a transformation of the plane that maps points relative to a given circle (called the inversion circle) as follows:

• Let k be the inversion circle with center S and radius R.
• For any point A in the plane (except the center S), we denote its image as A′.
• Points A and A′ lie on the same ray emanating from S, and the following holds: ∣SA∣⋅∣SA′∣=R2.

Points inside the circle k are mapped outside the circle and vice versa. The center S has no image or is mapped to infinity. This property is useful because a circle passing through S is mapped to a line, precisely because S "is sent to infinity."

Properties of Circular Inversion

• Lines and circles:
  • A line passing through the center S is mapped onto itself.
  • A line not passing through S is mapped onto a circle passing through S.
  • A circle passing through S is mapped onto a line.
  • A circle not passing through S is mapped onto another circle.
• Angles: Circular inversion preserves angles between curves (it is a conformal transformation).
• Tangency: If two curves are tangent, their images will also be tangent.

Application of Circular Inversion in Apollonius Problems

Apollonius problems often involve finding circles that satisfy certain conditions relative to given objects. Circular inversion simplifies these problems by transforming some configurations of circles and lines into simpler ones (e.g., converting circles into lines or concentric circles).

Example: A Circle Tangent to Three Given Circles

1. Choosing the Inversion Circle: Select a circle k that simplifies the configuration. Usually, we choose a circle whose center is at the center of the problem (e.g., at the center of one of the given circles) and a radius such that one of the circles is transformed into a line.
2. Transforming the Problem: Perform the inversion of all objects relative to the circle k. This can result in:
   1. Transforming one of the circles into a line.
   2. Simplifying the tangency conditions between objects.
3. Solving in the Inverted Plane: In the new plane, solve the simpler problem (e.g., finding a circle tangent to lines and circles).
4. Inverse Transformation of the Result: Convert the found circle back using circular inversion, obtaining the solution to the original problem.

Practical Construction

For manual inversion construction, use the relationship ∣SA∣⋅∣SA′∣=R2 to find the image of a point.

Here, we can utilize the Euclidean theorem on the altitude, which states that a2=c⋅cA. If we set a = R, we obtain a tool for mapping points via circular inversion. This method can be used in all three cases of mapping A to A':

• |SA| > R: Here, we obtain a right triangle satisfying our assumptions by constructing the Thales circle t over SA. One of the intersections of t and k is marked as T, and we draw a perpendicular p to SA passing through T. The point A' is at the intersection of p and SA.

• |SA| = R: In this case, the point A' maps to A. We can prove this either using the formula ∣SA∣⋅∣SA′∣=R2 or by following the construction steps for |SA| > R and |SA| < R. All methods confirm that A = A'.

• |SA| < R: Here, we already know the foot of the altitude of our triangle, which is A. We find T by drawing a perpendicular p to SA at A—T is any intersection of p and k. Then, we draw a perpendicular e to ST passing through T. The intersection of e and the ray SA gives the mapped point A'.

For more complex problems, we often use computer tools or dynamic geometry software (e.g., GeoGebra).

Tips for using circular inversion:

Circular inversion is particularly useful for finding a circle that is tangent to given circles or lines. It helps transform problems involving circles into problems involving lines, which are generally easier to solve.

If the problem is symmetrical, circular inversion can not only simplify the problem but also reveal additional important properties, such as the existence of multiple solutions.