314 CHAPTER 8 GEOMETRY FOR AMERICANS
8.5.47 Using compass and ruler, draw a circle that is
externally tangent to three given mutually external cir
cles.
8.5.48 (USAMO 19 93) Let ABCD be a convex
quadrilateral such that diagonals AC and BD intersect
at right angles, and let E be their intersection. Prove
that the reflections of E across AB, BC, CD, DA are
concyclic. (There is also a non-inversive solution, us
ing homothety.)
8.5.49 Ptolemy's Theorem via Inversion. Prob
lem 8.4.30 suggested a way to prove Ptolemy's the-orem using similar triangles. Now find an inversive
proof. Hint: Invert about one of the vertices, sending
the other three onto a line. Now the image distances of
are more easily handled; then use the image-distance
formula of 8.5. 14 (b) to relate these with the original
side lengths.
8.5.50 (USAMO 2000) Let AIA 2 A 3 be a triangle and
let COl be a circle in its plane passing through A I and
A 2. Suppose there exist circles CO2, CO] , ••• , CI>7 such
that for k = 2,3, ... ,7, circle COk is externally tan
gent to COk- 1 and passes through Ak and Ak+ I, where
An+ 3 = An for all n � I. Prove that CI>7 = COl.