The Invariance Theorem

Inversion in a circle preserves the size of any angle, but reverses the sense.

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