296 CHAPTER 8 GEOMETRY FOR AMERICANS
8.4.30 Ptolemy's Theorem is a lovely formula relat
ing the sides and diagonals of a cyclic quadrilateral:
The sum of the products of the lengths of the oppo
site sides is equal to the product of the lengths of the
diagonals. In other words, the diagram below satisfies
AD ·BC+DC·AB =AC ·DB.
There are many proofs of this remarkable equality.
Perhaps the shortest uses similar triangles, plus the
very clever auxiliary construction suggested below
(LDAE = LFAB). Don't be ashamed to get such a
good hint. Just finish the proof! When you are done,
prove the converse: that the equality above forces the
four points to be concyclic.
A
8.4.3 1 Let Ii and 12 be two non-intersecting circles
external to each other (i.e., one does not lie inside the
other). Let their centers, respectively. be P and Q.
Through P, draw line segments PA and PB that are
tangent to 12 (so A and B are on 12). Let C, D be the
intersection of PA and PB with Ii. Likewise, draw tan-
8.5 Transformations
Symmetry Revisited
gents QE, QF to Ii, intersecting 12 at G and H. Prove
that CD = GH.
8.4.32 (Bay Area Mathematical Olympiad 1999; orig
inally a proposal for the 1998 IMO) Let ABCD be
a cyclic quadrilateral (a quadrilateral that can be in
scribed in a circle). Let E and F be variable points on
the sides AB and CD, respectively, such that AE / EB =
CF /FD. Let P be the point on the segment EF such
that PE/PF =AB/CD. Prove that the ratio between
the areas of triangle AP D and BPC does not depend on
the choice of E and F.
8.4.33 (Leningrad Mathematical Olympiad 1987)
Given /::,ABC with B = 60°. Altitudes CE and AD in
tersect at point O. Prove that the circumcenter of ABC
lies on the common bisector of angles AOE and COD.
8.4.34 (Leningrad Mathematical Olympiad 1987)
Two circles intersect at points A and B, and tangent
lines to these circles are perpendicular at each of points
A and B (in other words, the circles "meet at right an
gles"). Let M be an arbitrary point chosen on one of
the circles so that it lies inside the other circle. Denote
the intersection points of lines AM and BM with the
latter circle by X and Y, respectively. Prove that XY is
a diameter of this circle.
8.4.35 (lMO 1990) Chords AB and CD of a circle in
tersect at a point E inside the circle. Let M be an in
terior point of the segment EB. The tangent line of E
to the circle through D, E, M intersects the lines BC
and AC at F and G, respectively. If AM / AB = t, find
EG/EF in terms oft.
We first discussed symmetry in Section 3.1, and by now you have seen how important
it is. Why is symmetry so ubiquitous? Many mathematical situations, when suit
ably massaged, reveal structures that are invariant under carefully chosen "transforma
tions." For example, the sum
1+2+3+···+ 100
is not symmetrical, but if we place the reverse sum underneath it, we create something
that is symmetric with respect to rotation about the center:
1 + 2
100 + 99
+ ... + 99
+ ... + 2
+
+
100,
1.
