The Invariance Theorem Inversion in a circle preserves the size of any angle, but reverses the sense. Proof In the sketch, O is the centre of inversion, PQ and PR are arcs, meeting at P. Using P' to denote the inverse of P (and so on), The arcs invert to the arcs P'Q' and P'R'. By the Corollary to the Equal Angles Theorem, <Q'P'R' = -<QPR. As Q moves towards P along its arc, so does R, and Q' and R' move toward P'. In the limit, they give the tangent rays. Since <Q'P'R' is always the reverse of <QPR, the result follows.

The Inversion Theorem

Main inversive page