Circle Inversion

Circle inversion entails taking a point in the plane and specifying it as the center of a circle C of radius a. A point is transformed to the point such that [1]

It is relatively simple to prove a range of theorems involving this transformation.

1. A circle that is completely outside C is transformed to a circle wholly inside C, but not passing through the (the center of inversion), and vice versa.

2. A circle that intersects C and passes through is transformed to a straight line that passes through the points of intersection of the two circles, and vice versa.

3. The inverse of a line that does not pass through is a circle inside C that passes through O, and so forth [1].