A note on inversion through a circle:

Let C be a circle, with center o. Then for any point a, at distance s from o, the inverse a-1 is the point that lies on the line ao at distance 1/s from o.

If a is inside C, then a-1 will be outside C

If a is outside C, then a-1 will be inside C

If a is on C, then a-1 = a

And if a=o, then a-1 is undefined (or if you prefer, is the point at infinity)

Inversion has many beautiful properties. For example, circular arcs, when inverted, are still circular arcs. The two shapes above are inverses of one another. You can download an interactive version of this image!

(To use the downloaded interactive image, you must have Geometer's Sketchpad. A demo version is available here)