310 CHAPTER 8 GEOMETRY FOR AMERICANS
Actually, our proof is incomplete, since our diagram only handles the case when
X is outside co. If X lies on r, but inside co, the image X, will be on the line AB, not on
the segment AB. The proof is similar; we leave it to you.
While we strongly encourage you to completely prove that inversion takes "cir
cles" to "circles," assuming this as a fact gives you tremendous new powers to simplify
problems. The basic strategy is to look for a point that involves several circles, and
invert about this point, striving to make a simpler image. Usually lines are simpler
than circles, so try to turn some of the circles into lines. If the inversion image is easier
to understand, it may shed light on the original figure. Here is a spectacular example;
we defy you to prove it without inversion!Example 8.5. 16 Let Cl, C 2 , q, q be circles tangent "cyclically;" i.e., C} is tangent to
C 2 , C 2 is tangent to C 3 , C 3 is tangent to q, and C 4 is tangent to Cl. Prove that the four
points of tangency are concyclic.Solution: We wish to show that A, B, C, Dare concyclic. Without loss of gener
ality, we invert about C, choosing an arbitrary radius. Denote the inversion circle by
co. Let < denote the image of q under this inversion.What happens to our diagram? By Example 8.5.15, c; is the common chord of C} and
co, extended to a line; likewise, c� is another line, parallel to c;. This makes sense,
since these image "circles" are still tangent-at oo!
What are the fates of C 2 and C 3? Since neither circle intersects C, their images are
ordinary circles, homothetic images of the originals about the center C. Furthermore,
c 2 is tangent to both c; and c;, and c; is tangent to c�. Again, this makes sense, since
inversion will not disturb tangency. Here is the result.