Unizor - Geometry2D - Inversion

Unizor - Creative Minds through Art of Mathematics - Math4Teens

Notes to a video lecture on http://www.unizor.com

Inversion

Before addressing construction problems of Apollonius, where a circle is one of the given elements, to which a constructed circle must be tangential, we will define a special transformation on a plane that allows to convert these types of problems into problems that involve only lines and points.
In other words, we will try to simplify our task using this transformation and to reduce its solution to a known solution with only points and lines given.
This is a rather clever method which is not easy to come up with by yourself. Yet, it is relatively simple and allows easily to solve Apollonius problems that involve given circles.

The transformation we would like to address is called inversion or symmetry relatively to a circle.
Any transformation of the points on a plane prescribes how each point is transformed. For instance, symmetry relatively to a straight line transforms all points lying on one side of a plane (let's call it here "left of the symmetry line") to points on another side ("right of the symmetry line") and all points from the "right" side to corresponding points on the "left". This transformation has a rule that the source and the image points lie on a perpendicular to a symmetry line and on the same distance from it. Points on the symmetry line are transformed into themselves.

The definition of inversion is somewhat similar, but slightly more complex. We divide the plane by a circle of certain radius R with a center O (inversion circle) into two parts - inside and outside of this circle.
All points of the inside area (except a center of an inversion circle) are transformed into points outside and all outside points are transformed to inside. The rule of transformation is as follows.
For any point P inside of an inversion circle we connect a center of an inversion circle with this point by a ray and find on this ray on the outside of a circle such a point P' that
OP·OP' = R²

Similarly, for any point P' outside of an inversion circle we find point P inside that satisfies the same condition.
Obviously, if point P inside an inversion circle is transformed into point P', then this point P' is transformed by an inversion to point P. Also obvious is that all points on the inversion circle are transformed into themselves. These properties qualify the transformation of inversion to be called "symmetry relative to a circle".

The closer point P lies to a center - the farther from a center will be P' to preserve the main rule of inversion.
The center of an inversion circle does not participate in the transformation, though sometimes one might say that a center is transformed into "infinity", which only means that, as we move point P infinitely closer to a center, its image P' moves infinitely far from a center.

Inversion of a Straight Line

Theorem
Any straight line lying outside of an inversion circle is transformed into a circle inside the inversion circle that passes through a center of inversion.

Inversion of a Circle

Lets consider a circle of radius r concentric with an inversion circle and lying inside it. For any point P on it the segment OP will have a length r. Therefore, the distance from a center of an inversion circle to an image of this point P' must be R²/r for the main rule of inversion to hold. As point P goes around a circle of radius r, its image P' should always be on a distance R²/r from point O - that is, all image points must lie on a circle of radius R²/r concentric with an inversion circle.
We see now that a circle, concentric with an inversion circle is transformed into another concentric circle.

The interesting theorem we are going to prove is that the fact of transformation of a circle into a circle is much broader and expands to circles that are not concentric with an inversion circle. This is the main subject of this lecture.

Theorem
Any circle lying inside an inversion circle and not passing through its center is transformed into a circle outside the inversion circle.