The Art and Craft of Problem Solving

8.3.36 Here is an example of the utility of trigonom­

etry. Use the !absin8 area fonnula to prove the fol­

lowing surprising fact: Any trapezoid is divided by its

diagonals into four triangles. Two of these are similar,

and the other two have equal area. For example, in the

picture below, the shaded triangles have equal area.

B B C

8.3.37 Ponder the figure below which depicts right tri­

angle ABC, with right angle at B and AB = 1. Let D be

a point on AC such that AD = 1. Let DE II CB.

c

A

(a) Recall that the length of an arc of a circle with

radius r subtended by a central angle of 8 is r8.

(Of course, 8 is measured in radians.) Use this

to prove that for acute angles 8,

sin8 < 8 < tan 8.

(b) Prove, without calculus, that lim

Sin

8

8

= 1.

(^6) -0
8.3.38 (Putnam 1999) Right triangle ABC has right
angle at C and LBAC = 8; the point D is chosen on AB
so that AC = AD = 1; the point E is chosen on BC so
that LCDE = 8. The perpendicular to BC at E meets
AB at F. Evaluate lim 6 _oEF. Hint: The answer is
not I, or 0, or 1/2.
8.3 SURVIVAL GEOMETRY II 281
A�--------��--�B
8.3.39 The figure below suggests another "dissection"
proof of the Pythagorean Theorem. Figure out (and
prove) why it works.
8.3.40 Escribed circles. The inscribed circle of a tri­
angle is tangent to the three sides, and the incenter, of
course, lies in the interior of the triangle. Yet there
are three more circles that are each tangent to all three
sides of the triangle-just allow the center to lie in the
exterior of the triangle, and extend sides as needed!
For example, here is one escribed circle, with radius
ra (since it is opposite vertex A).
In a similar fashion, we can define the other two es­
cribed circles-also called excircles-with radii rb
and rc.
(a) Prove that the center of each excircle is the in­
tersection of the three bisectors of the angles of
the triangle (one interior angle and two exterior
angles).