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We are dealing here with a 'circle inversion', though it may not be easy to infer this. Points inside the given circle are mapped onto points outside the circle and vice versa. For any point P and its image P', OPxOP' is a constant.

More specifically, OAxOA' = OBxOB' = OCxOC = 4x4 = 16 (as our circle has a radius of 4 units).

A circle inversion (or 'plane inversion') has the property that all circles are mapped onto circles (if one includes straight lines as circles, ie circles of infinite radius). This result is fairly obvious for circles with centre O.

We start by considering the line segment BD (which we can think of as part of an infinitely large circle).
We then consider the image of circles that pass through O,
and then the image of circles that don't pass through O.

These pages provide
• two JAVA worksheets that allow one to get a feel for what happens
• various movies that demonstrate what happens
• two diagrams that hint towards proofs of what happens 