Let K be a circle with center S and radius k. Let C₁ be another circle not
    passing through S. Then the inverse of the points on C₁ form another circle,
    C₂, which is the image of C₁ under the dialation about point S with factor
    k²/S(C₁).

Proof:

    We must show that the image of C₁ is another circle, C₂, such that C₂'s radius is equal to
    (k²/S(C₁))(the radius of  C₁.

    We first construct a line through S that intersects the circle  C₁ at arbitrary points P and Q
    such that P does not equal Q.

    We know by the Inverse Image that the inverse or P, P', is found on the line SP such that (SP)(SP') =  k².

    Similarily, we can find the inverse of Q, Q'.

     (1)   We know that: (SP)(SP') = k^{2
     (2)}   We also know that (SQ)(SQ') = k².

     (3)   We also know that S(C₁) = (SP)(SQ).

    By dividing equations (1) and (2) by equation (3) we get:

    (SP')/(SQ) = k²/S(C₁)
    (SQ')/(SP) = k²/S(C₁)

    Since k²/S(C₁) is just a constant determined by the size of K and the position of C₁ we can call it a.

    Now we have that (SP')/(SQ) = c = (SQ')/(SP).

    From this we can see that in the inverse of C₁ P' corresponds to Q and Q' corresponds to P.

    We also know that dialation preserves circles (as well as other shapes).

    Therefore, the inverse of C₁ is another circle obtained by a dialation of k²/S(C₁).

Q.E.D.