The Art and Craft of Problem Solving

Chapter 8

Geometry for Americans

8.1 Three "Easy" Problems

256

We call this chapter "Geometry for Americans" instead of "Geometry for Dummies"

so as not to offend. The sad truth is that most mathematically inclined Americans know

very little geometry, in contrast to their luckier peers in Eastern Europe and Asia. But

it is never too late to learn. Geometry is a particularly fun topic to study, because you

are compelled to draw lots and lots of pictures.

Indeed, before you begin seriously studying this chapter, we recommend that you

procure the following drawing supp lies:

• Compass (good quality),


• Protractor/ruler,


• Mechanical pencil with good eraser,


• Colored pencils,


• Pencil sharpener,

• Plenty of unlined paper.

When you put your drawing materials into a pencil case, you may feel as though you

are back in elementary school. That's fine; you may as well start with few assumptions,

and little ego about your skill. There's nothing like a few nicely sharpened, brightly

colored pencils to reduce inhibitions, which we all know is good for creative problem

solving.

We highly recommend that you supplement your low-tech drawing tools with a

computer program such as Geometer's Sketchpad, Geometer, or Geogebra. Programs

of this kind allow you to quickly draw flexible, accurate pictures that can be easily

modified, which greatly facilitates investigation. I

We shall begin with a review of the basic facts of plane geometry: definitions and

theorems about points, lines, circles, etc. Few of these facts will be new to you, but

most likely, it has been a while-if ever-since you proved them. Once we are done

I Geometer's Sketchpad is commercially available from Key Curriculum Press (http: //www • keypress.

coml sketchpad!) and Geometer and Geogebra are both currently free (http: //www. geometer. org I

geometer I index. html and http: //www.geogebra.at/. respectively).