The Art and Craft of Problem Solving

282 CHAPTER 8 GEOMETRY FOR AMERICANS

(b) Following the notation of Problem 8.3.25, prove

that

K = (S-a)ra = (s- bh = (s- c)rc.

(c) Prove the remarkable formula

I I I I

• r + -+-=-.
  a r
  b

rc r

8.3.4 1 (Bay Area Mathematical Olympiad 2006) In

triangle ABC, choose point Al on side BC, point BI

on side CA, and point CI on side AB in such a way

that the three segments AA I, BB I, and CC I intersect in

one point P. Prove that if P is the centroid of triangle

ABC if and only if P is the centroid of triangle A I B I C I.

(This is an "if and only if' statement; one part is much

easier than the other.)

8.3.42 In the figure below, D, E, and F are midpoints,

respectively, of AF, BD, and CEo If [ABC] = I, find

8.4 The Power of Elementary Geometry

[DEF].

A

8.3.43 (Putnam 1998) Let s be any arc of the unit cir­

cle lying entirely in the first quadrant. Let A be the

area of the region lying below s and above the x-axis

and let B be the area of the region lying to the right of

the y-axis and to the left of s. Prove that A + B depends

only on the arc length, and not on the position, of s.

Let's regroup from the high-speed review of the last section. After covering many

topics, we will now revisit the most important ones. This section develops several new

theorems and tools, but mostly, its focus is geometric problem-solving strategies and

tactics, by using simple ideas in creative ways.

The previous section introduced many problem-solving concepts, at all three lev­

els (strategies, tactics, tools). Let's formalize some of them into a checklist. This

list is by no means complete, but it is a good start, and will help the beginner to get

organized.

1. Draw a careful diagram.

2. Draw auxiliary objects, but sparingly.

3. Start with angle chasing. Don't rely on it for everything!

4. Seek out your best friends: right triangles, parallel lines, and concyclic points.

5. Compare areas.

6. Relentlessly exploit similar triangles.

7. Look for symmetry and "distinguished" points or lines.

8. Create a "phantom" point with a desired property, and show that it coincides

with an existing point.

Let's look at some of these in more detail, with some problem examples and some

new ideas. We have little to say about items 1-3 besides restating the obvious: with­

out a careful diagram, your time will be wasted; auxiliary objects are miraculous, but

sometimes require real artistry to find; angle chasing is fun, but will not solve all prob­

lems. Starting with item 4, there is much to learn by looking at some good examples.