The Art and Craft of Problem Solving

264 CHAPTER 8 GEOMETRY FOR AMERICANS Circles and Angles Circles are the most important entities in Euclidean geometry. You canno ...

8.2 SURVIVAL GEOMETRY I 265 The following lemma is perhaps the first interesting and unexpected result proven in high-school geo ...

266 CHAPTER 8 GEOMETRY FOR AMERICANS 8.2. 17 Inscribed Right Triangles. Let AB be the diameter of a circle, and let C be an arbi ...

8.2 SURVIVAL GEOMETRY I 267 It is not immediately obvious that an arbitrary triangle even has an inscribed or circumscribed circ ...

268 CHAPTER 8 GEOMETRY FOR AMERICANS and BD intersect outside the triangle (remember, they are lines, not line segments). A �c / ...

Problems and Exercises 8.2 SURVIVAL GEOMETRY I 269 The problems below are elementary, in that no geometric facts other than thos ...

270 CHAPTER 8 GEOMETRY FOR AMERICANS I. Start with a long strip of paper, and visualize a crease for folding (thin line). The cr ...

8.3 SURVIVAL GEOMETRY II 271 version of the Principle of Inclusion-Exclusion (Section 6.3). But how do we compute area? We need ...

272 CHAPTER 8 GEOMETRY FOR AMERICANS ~ B D C We should point out that 8.3.1 and 8.3.2 are intuitively obvious by visualizing a p ...

8.3 SURVIVAL GEOMETRY II 273 The beautiful crux idea is an auxiliary line m drawn through the right angle (B), per­ pendicular t ...

274 CHAPTER 8 GEOMETRY FOR AMERICANS Similar Triangles Contemplating area gives our geometric investigations a powerful "multipl ...

8.3 SURVIVAL GEOMETRY II 275 8.3.9 Let ABC be a triangle and let m be a line parallel to BC, intersecting AB and AC respectively ...

276 CHAPTER 8 GEOMETRY FOR AMERICANS We need to include two obvious auxiliary lines (XY' and X'Y) in order to find similar trian ...

A 8.3 SURVIVAL GEOMETRY II 277 B This gives us two similar triangles: ACE rov ABD. What do we do next? Investigate the diagram f ...

278 CHAPTER 8 GEOMETRY FOR AMERICANS triangles CAF and DAG are similar. (If two right triangles have a single pair of equal angl ...

Problems and Exercises 8.3 SURVIVAL GEOMETRY II 279 Here is a wide selection of elementary problems and exercises. No geometric ...

280 CHAPTER 8 GEOMETRY FOR AMERICANS of the slope of ax + by + c = 0), then use the distance fonnula. But this method is horribl ...

8.3.36 Here is an example of the utility of trigonom­ etry. Use the !absin8 area fonnula to prove the fol­ lowing surprising fac ...

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. ( ...

Concyclic Points 8.4 THE POWER OF ELEMENTARY GEOMETRY 283 Let's begin with a problem that explicitly asks us to find concyclic p ...