The Art and Craft of Problem Solving

284 CHAPTER 8 GEOMETRY FOR AMERICANS MN; let's call thisX. Then we will need to show that QX ·XP=MX ·XN. 5. Use the converse of ...

8.4 THE POWER OF ELEMENTARY GEOMETRY 285 AB' = AB cosA. Substituting, we have AM^2 =AC'·AB = AC·ABcosA and AP^2 =AB'· AC =AB ·AC ...

286 CHAPTER 8 GEOMETRY FOR AMERICANS the second solution, line AA' was key. This was no accident, either: since the problem was ...

8.4 THE POWER OF ELEMENTARY GEOMETRY 287 But there are other area ratios known to us. Since [CAF]: [CFB] = I: 2, and [CFB] = 6y ...

288 CHAPTER 8 GEOMETRY FOR AMERICANS It should be possible, with the techniques we have acquired, to determine when three cevian ...

8.4 THE POWE:R OF ELEMENTARY GEOMETRY 289 Similar Triangles and Collinear Points You have noticed, no doubt, that similar triang ...

290 CHAPTER 8 GEOMETRY FOR AMERICANS B c Now BHC is a right triangle with midline DE and median BE. And sure enough, it is obvio ...

8.4 THE POWER OF ELEMENTARY GEOMETRY 291 Our strategy for proving that H, M, and K are collinear is to show that H M and M K hav ...

292 CHAPTER 8 GEOMETRY FOR AMERICANS Phantom Points and Concurrent Lines You have seen the method of phantom points before, wher ...

8.4 THE POWER OF ELEMENTARY GEOMETRY 293 Another way to draw a false "line" is to start at F, intersect X (so as to use a point ...

294 CHAPTER (^8) GEOMETRY FOR AMERICANS cut AB in D. Prove that the length of CD is 2ab cos (C /2) a+b 8.4. 10 (Canada 1986) A c ...

8.4 THE POWER OF ELEMENTARY GEOMETRY 295 Our weight assignment appears to be consistent, since the interior balancing point I re ...

296 CHAPTER 8 GEOMETRY FOR AMERICANS 8.4.30 Ptolemy's Theorem is a lovely formula relat­ ing the sides and diagonals of a cyclic ...

8.5 TRANSFORMATIONS 297 And of course, what makes this particular symmetry useful is the fact that the sums of each column are i ...

298 CHAPTER 8 GEOMETRY FOR AMERICANS symmetry is important. The great German mathematician Felix Klein introduced this "transfor ...

� -B .. - _^4 - -- p\ � If , , o "11--- c �- - - -�-- B B 8.5 TRANSFORMATIONS 299 /:"' F (; To indicate the motion along a s ...

300 CHAPTER 8 GEOMETRY FOR AMERICANS We can write AB = B -A, etc., and everything will cancel, or we can think dynamically: The ...

5 TRANSFORMATIONS 301 S that is symmetric to A k with respect to P. (Note that Bk = Aj for some j.) By the definition of symme ...

302 CHAPTER 8 GEOMETRY FOR AMERICANS Rotations A rotation with center C and angle 8 does just what you'd expect: rotates every p ...

8.5 TRANSFORMATIONS 303 is transformed and the rest is left alone. Here is a dramatic example of a problem that is quite thorny ...