D, then the center of E is not the inverse of the center of C inverted in D. The center of E is determined, however, by three of the points on E, and three points on E can by found by taking any three points on C and inverting them in D. Thus, we'll need the construction C to find the center of a circle given three points on its circumference.
Let P,Q, and R be three given points. We are to construct the center O of the circle passing through them. We'll find O by inverting everything in the circle C with center P and radius PQ.
First, invert the point R in C to get the point R' = RC.
Next draw two circles, one with center Q and radius QP, the other with center R' and radius R'P. Besides at P, these two circles will intersect at another point O'. It is evident that this point O' is the reflection of P in the line QR', that is, O' = PQR'. Finally, invert O' in C to get O.
Next, we'll show O is the center of the circle PQR. Now, when the circle PQR is inverted in the circle C, then the result is the straight line QR', since P will be sent to the point 
at infinity, Q is fixed, and R is sent to R'. Algebraically, that says PQRC = QR'. To show that O is the center of the circle PQR, we only need to show that O =
PQR. That will follow from a property of , namely, distributes over itself on the right, as shown in the next section.
PQR =
(PC) (QR'C) =
(PQR')C =
O'C = O.
Thus, we have constructed O, the center of the circle passing through the three points P, Q, and R.
March, 2002.
David E. Joyce
Department of Mathematics and Computer Science
Clark University
Worcester, MA 01610
Email: djoyce@clarku.edu
My Homepage