The Art and Craft of Problem Solving

258 CHAPTER 8 GEOMETRY FOR AMERICANS

8.1.2 The Angle Bisector Theorem. Let ABC be a triangle and let the angle bisector of

LA intersect side CB at D. Prove that

A

CD AC

DB AB·

B

8.1. 3 The Centroid Theorem. A median of a triangle is a line segment from a vertex

to the midpoint of the opposite side. Prove that the medians of a triangle meet in a

single point, and moreover, that this intersection point divides each median in a 2 : 1

ratio. For example, in the figure below,

BG/GD =AG/GF = CG/GE = (^2).
A
c B

8.2 Survival Geometry I

When you attempt Problems 8.1.1-8.1.3, you may wonder just what you may assume

and what you must prove. As you know, Euclidean geometry is based on a very small

set of undefined objects (including "points" and "lines") and postulates or axioms

(theorems that are assumed to be true, that lie "above" proof). In the interest of time,

we will play fast and loose with this, and start you out with a much larger collection

of "facts" that you can safely assume for now.

Together with these facts, we will introduce some simple lemmas and theorems

that are consequences of them. Some of these we will prove, to begin illustrating some

of the most important geometrical problem-solving techniques. The others you should

prove on your own, as essential exercises (not problems) to master the material. We

will label "facts" that you need not prove as such; any unlabeled problem is begging

for you to solve it.