Theorem: (Coinciding Circles)

If circles K and C coincide, the image of C when inverted through K is C again.

Proof:

Let P be a point on circle C.

Let P' be the inversion of P with respect to circle K.

Let T be the point on circle C that defines the radius of circle C or segment ST.

By the definition of an Inverse Image, SP * SP' = ST².

Since ST and SP are radii of circle C, ST ²= SP².

SP * SP' = SP²

SP' = SP

Therefore, the inverse image of circle C inverted through circle K is again C.