defined,I46
Fibonacci, 216
monotonic, 318
symmetry, 67
series
arithmetic, 157
geometric, 51,105,131,133, 158
hannonic, see hannonic series
Taylor, 315, 344-346
sets
Cartesian product, 145
complex numbers (C), 144
integers (2:), 144
natural numbers (!Ii), 144
rational numbers (1Qi), 144
real numbers (IR), 144
Seven Sisters, 22
shearing, 27 1
shearing tool, 272
a-function, see functions, number theo
retic
similar triangles, 274-275, 289 -29 1
and parallel lines, 275
and right triangles, 275
as crux, 289
conditions for, 274
definition, 274
subtriangles, 280
similarity, 305
direct, 306
opposite, 306
simplification, see algebraic methods,
simplification
Siobodnik, S., 86
Soifer, A., 86
Spivak, Michael, 357
squares
algebraic manipulation, 149
and consecutive numbers, 4
and number of divisors, 30
and pigeonhole principle, 97
difference of two, 149
extracting, 150, 154, 173
sum of, 159
SSS condition, 260
stealing ideas, 18, 22
stick your butt out, 61
Stirling numbers, 22 1
straight angle, 259
strategies
angle chasing, 265 , 282
limitations of, 289
change point of view, 58
defined, 3
draw a picture, 53
drawing an auxiliary object, 262, 276
drawing anauxiliary object, 282
generalize, 90
geometric (various), 282
get your hands dirty, 26
is there a similar problem?, 37
make it easier, 16
opportunistic, 43
optimistic, 15
orientation, 6, 25
penultimate step, 5, 262, 289
peripheral vision, 54
phantom point, 263, 278, 282, 292-
293
produce a contradiction, 43
psychological, 14-25
recasting, 54
wishful thinking, 32
Stuyvesant High School, xi
subconscious, 15
substitution, 69, 150- 155, 346
sudoku, 23
sum, see series
of divisors, 235
of squares, 159
supplementary angles, 261
symmetry, 62-73, 282
applied to calculus, 338 -339
applied to probabilty, 73, 350 -353
center of, 300
importance of, 298
imposing it on a problem, 297
symmetry-product principle, 177, 179tactics, see separate entries for each item
complex numbers, 120- 132
defined, 3
extreme principle, 73-83
factoring, 148-14 9
generating functions, 132-141
graph theory, 109- 1 20
invariants, 92- 106
modular arithmetic, 100
modulo m filter, 230, 242
monotonize, 75
monovariant, 102-106
parity, 94-99
pigeonhole principle, 84-92
symmetry, 62-73
Tai Chi, 23
tangent (to a circle), 264
tangent line, 264, 316 , 328, 330
tangent line (to a circle)
and center, 264
and radius, 264INDEX 365Taylor series, 315, 344-346
telescope tool, 158-160, 162, 181
theorem
angle bisector, 258
converse, 278, 280
proof using area, 277
proof using trigonometry, 280
proof with auxiliary line, 276
centroid, 258
proof, 278
Ceva's, 288, 294
converse, 288, 294
inscribed angle, 265 , 279
Menalaus's
converse, 295
Menelaus's, 295
power of a point
application, 292
converse, 283, 285
power of a point (POP), 257
converse, 280
proof, 275
Ptolemy's
converse, 284
proof using auxiliary construction,
296
proof using complex numbers, 132
proof using inversion, 314
Pythagorean, 272
proof using dissection, 273, 28 1
proof using shearing, 272
proof using similar triangles, 280
Stewart's, 280
tiling, 54, 98, 101, 215
tools
add zero creatively, 149, 228
catalyst, 160
completing the square, 149
define a function, 90
defined, 3
extracting squares, 150
factorization, 5
Gaussian pairing, 67-69, 250
geometric senes, 133, 346
identity principle, 172
invent a font, 21 1
monic polynomial, 171
partial fractions, 135
reflection, 64, 66
roots of unity filter, 141
telescope, 158- 160, 162, 181
trigonometric, 57
undetennined coefficients, 6
weights, 294
and Ceva's theorem, 295