Preservation of the Inverse Points Property by Extended Mobius Transformation

Mobius transformations map generalized circles (circles and lines) to generalized circles, but they do not map the centres of circles to the centres of circles. They do however, preserve the inverse property of points, so that ifandare inverse points with respect to a circlethen after being transformed by an extended mobius transformationandare inverse points with respect to

Proof

Ifandare inverse points with respect to a generalized circlethen there must be an extended mobius transformationthat mapsto 0,toandto the unit circle. Thusmapsto 0,toandto the unit circle. It follows thatandare inverse points with respect to

On the other hand ifinthenso once againandare inverse points with respect to