280 CHAPTER 8 GEOMETRY FOR AMERICANS
of the slope of ax + by + c = 0), then use the distance
fonnula. But this method is horribly ugly.(0,0)Instead, use similar triangles for a neat and elegant
two-line proof. Here's the proof; you supply the jus
tification. (Don't forget to explain why absolute value
is needed.)- d/lhl = Ibl/v'a^2 +b^2.
- Ibhl = lau+bv+cI-
8.3.28 Prove the converse to the angle bisector the
orem ( 8 .1.2): If ABC is a triangle, and D is a point
placed on on side BC so that CD/DB = AC/AB, then
AD bisects L.CAB.
8.3.29 Prove the converse to the Power of a Point the
orem ( 8 .1.1): If P is the intersection of lines XY and
X'Y' and PX. PY = PX'. PY', then X, Y, X', y' are
concyclic.
8.3.30 Similar Subtriangles. Suppose DABC �
b.DEF, and let X and Y be points on AB and DE, re
spectively, cutting these sides in the same ratios. In
other words, AX/X B = DY /Y E. Prove that b.AXC �
MY F. We call these "similar subtriangles" since
ACX is contained within ABC and DY F is contained
with DEF in "similar" ways. It seems completely
obvious-"proportional" subsets of similar objects
should be similar-but you'll need to do a little work
to prove it rigorously.
Similar subtriangles crop up often. See, for ex
ample, Problem 8.4.32 and Example 8.4.4.
8.3.3 1 Compass-and-Ruler Constructions. Find the
following constructions (make sure that you can prove
why they work).
(a) Given a line segment of unit length, construct a
segment of length
I. k, where k is an arbitrary positive integer. - alb, where a, b are arbitrary positive inte
gers.
(b) Given line segments of length I and x, construct
a segment with length y'X.
(c) Given the midpoints of the sides of a triangle,
construct the triangle.
8.3.32 Trigonometry Review. Some geometry purists
look down on trig as "impure," but that's just silly. Cer
tainly, there are many elegant geometric methods that
bypass messy trigonometric manipulations, but some
times trigonometry saves time and effort. After all, the
trig functions essentially encode ratios of similar tri
angles. Consequently, whenever you deal with similar
triangles, you should at least be aware that trigonom
etry may be applicable. At this point, you should
review, if you don't know already, the definitions of
sin, cos, tan and the basic angle-addition and angle
subtraction fonnulas. And then (without looking them
up!) do the following exercises/problems.
(a) If the lengths of two sides of a triangle are a and
b, and 8 is the angle between them, prove that
the area of the triangle is �ab sin 8.
(b) Use this fonnula to get a third proof of the angle
bisector theorem (8.1 .2).
(c) Prove the law of sines: Let ABC be a triangle,
with a = BC, b = AC, c = AB. Then
abc
--=--=--
sinA sinB sinC
(d) Prove the extended law of sines, which says
that the ratios above are equal to 2R, where R
denotes the circumradius.
(e) Prove the law of cosines, which says (using the
triangle notation in (c) above)
c^2 = a^2 + b^2 - 2abcosC.8.3.33 Use the law of cosines to prove the useful
Stewart's theorem: Let ABC be a triangle with point
X on side BC. If AB = c, AC = b, AX = p, BX = m,
and XC = n, then
ap^2 +amn = b^2 m+c^2 n.8.3.34 Use 8.3. 10 to find yet another proof of the
Pythagorean Theorem, using similar triangles.
8.3.35 Let a, b, c be the sides of a triangle with area
K, and let R denote the circumradius. Prove that
K = abc/4R.