8.2 SURVIVAL GEOMETRY I 265The following lemma is perhaps the first interesting and unexpected result proven
in high-school geometry.
Example 8.2.16 The Inscribed Angle Theorem. The measure of an arc is twice the
measure of the inscribed angle that subtends it. (In other words, for any arc, the angle
subtended at the center is twice the angle subtended at the circle.)
This is truly remarkable. It says that no matter where we place the point A on the
circumference of the circle r below, the measure of the angle CAB will be fixed (and
moreover, equal to half of the central angle COB).
Solution: Let (J = LCOB. We wish to show that LCAB = (J /2. In order to do
this, we need to gather as much angular information as we can. It certainly makessense to draw in one auxiliary object, namely, line segment AO, since this adds two
new triangles that have a vertex at A. And moreover, these triangles are isosceles, since
AO, BO, and CO are radii of circle r.
r�c
BLet f3 = LAOB. Our strategy is simple: we will compute LCAB by finding LCAO and
LBAO. And these are easy to find, since by Fact 8.2.1 1, we have
and
f3
LBAO=90 --
2 '
LCAO = 90 _
360
- f3
(J
= Ii + �90.
2 2 2
Adding these, we conclude that
LCAB = LCAO + LBAO = �. •
That wasn't too hard! A single, fairly obvious auxiliary line, followed by as much
angular information as possible, led almost inexorably to a solution. This method is
called angle chasing. The idea is to compute as many angles as possible, keeping the
number of variables down. In the above example, we used two variables, one of which
cancelled at the end.
Angle chasing is easy, fun, and powerful. Unfortunately, many students use it
all the time, perhaps with trigonometry and algebra, to the exclusion of other, more
elegant tactics. A key goal of this chapter is to teach you how to get the most from
angle chasing, and when you should transcend it.
Here are a few simple and useful corollaries of the inscribed angle theorem.