Plane Geometry - Bruce E. Shapiro

More documents

Recommendations

Info

$NAME: Math 103L: Exercise on Functions ... - Bruce E. Shapiro$

$Introducing Latex - Bruce E. Shapiro$

$Applied Differential Equations, Math 280 at CSUN - Bruce E. Shapiro$

$Math 103 Section 1.2: Linear Equations and ... - Bruce E. Shapiro$

208 SECTION 39. CIRCLESTheorem 39.16 Let Γ = C(A, r) be a circle. Choose any point B ∈ Γ andany point C not in Γ. Define the functionbyf(α) : (0, 180) ↦→ [0, ∞)f(α) = CD(x)where D(x) ∈ H C,←→ ABis chosen so that µ(∠DAB) = α (see figure 39.7).Then f(x) is continuous.Figure 39.7: The function f(α) : (0, 180) ↦→ [0, ∞) is continuous.Proof. Let α ∈ (0, 180).By the triangle inequality,HenceCD(x) < CD(a) + D(a)D(x)CD(a) < CD(x) + D(a)D(x)By the definition of f(x),Taking limits,CD(a) − D(a)D(x) < CD(x)< CD(a) + D(a)D(x)f(a) − D(a)D(x) < f(x) < f(a) + D(a)D(x)lim (f(a) − D(a)D(x)) < lim f(x)x→α x→α< lim (f(a) + D(a)D(x))x→α« CC BY-NC-ND 3.0. Revised: 18 Nov 2012
SECTION 39. CIRCLES 209Since D(a)D(x) → 0 as x → a, we getlim f(x) = f(α)x→αTheorem 39.17 (Strong Circular Continuity) Let Γ = C(O, r) andΓ ′ = C(O ′ , r ′ ) be circles, let A be a point on Γ that is outside Γ ′ , and let Bbe a point on Γ that is inside Γ ′ . Then Γ ∩ Γ ′ contains precisely two points.Proof. Assume that A and B are not antipodal. If they are, by the continuityof f described in the previous theorem we can replace A with anotherpoint A ′ close to A ′ so that A is still outside Γ ′ .Chose a point C ∈ Γ such that A, B are on the same side of ←→ OC. Use thispoint to define a function f(x) like the one in theorem 39.16:f(x) : [0, µ(∠BOC)) ↦→ [0, ∞)byf(α) = O ′ Q(α)where Q(α) is the point on Γ such that α = µ(∠COQ(α)) (see figure 39.8).There exist numbers a, b ∈ (0, 180) such that Q(a) = A and Q(b) = B. ByFigure 39.8: Strong circular continuity.the previous theorem, f is continuous, with f(a) > r ′ and f(b) < r ′ .Hence by the intermediate value theorem there is a number x such thatf(x) = r ′ .Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.
Page 1:
Foundations of GeometryLecture Note
Page 4 and 5:
ivCONTENTS30 Triangles in Neutral G
Page 6 and 7:
2 SECTION 1. PREFACEare enrolled in
Page 8 and 9:
4 SECTION 1. PREFACEre-learning it
Page 10 and 11:
6 SECTION 2. NCTM3. To guide in the
Page 12 and 13:
8 SECTION 2. NCTMTechnology. Techno
Page 14 and 15:
10 SECTION 2. NCTM« CC BY-NC-ND 3.
Page 16 and 17:
12 SECTION 3. CA STANDARDIn 2005 Ca
Page 18 and 19:
14 SECTION 3. CA STANDARDThe Califo
Page 20 and 21:
16 SECTION 3. CA STANDARDCalifornia
Page 22 and 23:
18 SECTION 4. COMMON COREAt all lev
Page 24 and 25:
20 SECTION 5. LOGIC AND PROOFour th
Page 26 and 27:
22 SECTION 5. LOGIC AND PROOFRene D
Page 28 and 29:
24 SECTION 5. LOGIC AND PROOF“I
Page 30 and 31:
26 SECTION 6. REAL NUMBERSThe inter
Page 32 and 33:
28 SECTION 6. REAL NUMBERS“The Su
Page 34 and 35:
30 SECTION 7. EUCLID’S ELEMENTSEu
Page 36 and 37:
32 SECTION 7. EUCLID’S ELEMENTSFi
Page 38 and 39:
34 SECTION 8. HILBERT’S AXIOMSiom
Page 40 and 41:
36 SECTION 8. HILBERT’S AXIOMS«
Page 42 and 43:
38 SECTION 9. BIRKHOFF/MACLANE AXIO
Page 44 and 45:
40 SECTION 9. BIRKHOFF/MACLANE AXIO
Page 46 and 47:
42 SECTION 10. THE SMSG AXIOMSP is
Page 48 and 49:
44 SECTION 10. THE SMSG AXIOMS« CC
Page 50 and 51:
46 SECTION 11. THE UCSMP AXIOMSin s
Page 52 and 53:
48 SECTION 11. THE UCSMP AXIOMS« C
Page 54 and 55:
50 SECTION 12. VENEMA’S AXIOMS3.
Page 56 and 57:
52 SECTION 12. VENEMA’S AXIOMS«
Page 58 and 59:
54 SECTION 13. INCIDENCE GEOMETRYFi
Page 60 and 61:
56 SECTION 13. INCIDENCE GEOMETRYEx
Page 62 and 63:
58 SECTION 13. INCIDENCE GEOMETRYFi
Page 64 and 65:
60 SECTION 13. INCIDENCE GEOMETRY«
Page 66 and 67:
62 SECTION 14. BETWEENNESSFigure 14
Page 68 and 69:
64 SECTION 14. BETWEENNESSTheorem 1
Page 70 and 71:
66 SECTION 14. BETWEENNESS(a) To sh
Page 72 and 73:
68 SECTION 14. BETWEENNESSThe follo
Page 74 and 75:
70 SECTION 14. BETWEENNESSExample 1
Page 76 and 77:
72 SECTION 14. BETWEENNESSorf(A) >
Page 78 and 79:
74 SECTION 14. BETWEENNESS« CC BY-
Page 80 and 81:
76 SECTION 15. THE PLANE SEPARATION
Page 82 and 83:
78 SECTION 15. THE PLANE SEPARATION
Page 84 and 85:
80 SECTION 16. ANGLESFigure 16.1: T
Page 86 and 87:
82 SECTION 16. ANGLESCorollary 16.6
Page 88 and 89:
84 SECTION 16. ANGLESFigure 16.6: I
Page 90 and 91:
86 SECTION 16. ANGLESD cannot be in
Page 92 and 93:
88 SECTION 17. THE CROSSBAR THEOREM
Page 94 and 95:
90 SECTION 17. THE CROSSBAR THEOREM
Page 96 and 97:
92 SECTION 18. LINEAR PAIRSFigure 1
Page 98 and 99:
94 SECTION 18. LINEAR PAIRSγ + ∠
Page 100 and 101:
96 SECTION 19. ANGLE BISECTORSwith
Page 102 and 103:
98 SECTION 19. ANGLE BISECTORSAngle
Page 104 and 105:
100 SECTION 20. THE CONTINUITY AXIO
Page 106 and 107:
102 SECTION 20. THE CONTINUITY AXIO
Page 108 and 109:
104 SECTION 21. SIDE-ANGLE-SIDEBirk
Page 110 and 111:
106 SECTION 21. SIDE-ANGLE-SIDEFigu
Page 112 and 113:
108 SECTION 22. NEUTRAL GEOMETRYEll
Page 114 and 115:
110 SECTION 22. NEUTRAL GEOMETRY«
Page 116 and 117:
112 SECTION 23. ANGLE-SIDE-ANGLEFig
Page 118 and 119:
114 SECTION 23. ANGLE-SIDE-ANGLE«
Page 120 and 121:
116 SECTION 24. EXTERIOR ANGLESTheo
Page 122 and 123:
118 SECTION 24. EXTERIOR ANGLES« C
Page 124 and 125:
120 SECTION 25. ANGLE-ANGLE-SIDEBy
Page 126 and 127:
122 SECTION 26. SIDE-SIDE-SIDECase
Page 128 and 129:
124 SECTION 26. SIDE-SIDE-SIDEBy si
Page 130 and 131:
126 SECTION 27. SCALENE AND TRIANGL
Page 132 and 133:
128 SECTION 27. SCALENE AND TRIANGL
Page 134 and 135:
130 SECTION 28. CHARACTERIZATION OF
Page 136 and 137:
132 SECTION 28. CHARACTERIZATION OF
Page 138 and 139:
134 SECTION 29. TRANSVERSALSFigure
Page 140 and 141:
136 SECTION 29. TRANSVERSALSCorolla
Page 142 and 143:
138 SECTION 30. TRIANGLES IN NEUTRA
Page 144 and 145:
Page 146 and 147:
Page 148 and 149:
Page 150 and 151:
146 SECTION 31. QUADRILATERALS IN N
Page 152 and 153:
Page 154 and 155:
Page 156 and 157:
Page 158 and 159:
Page 160 and 161:
156 SECTION 32. THE EUCLIDEAN PARAL
Page 162 and 163: 158 SECTION 32. THE EUCLIDEAN PARAL
Page 172 and 173: 168 SECTION 33. RECTANGLESLemma 33.
Page 174 and 175: 170 SECTION 33. RECTANGLESFigure 33
Page 176 and 177: 172 SECTION 33. RECTANGLES« CC BY-
Page 178 and 179: 174 SECTION 34. THE PARALLEL PROJEC
Page 180 and 181: 176 SECTION 34. THE PARALLEL PROJEC
Page 182 and 183: 178 SECTION 35. SIMILAR TRIANGLESBy
Page 184 and 185: 180 SECTION 35. SIMILAR TRIANGLESCo
Page 186 and 187: 182 SECTION 36. TRIANGLE CENTERSFig
Page 188 and 189: 184 SECTION 36. TRIANGLE CENTERSThe
Page 190 and 191: 186 SECTION 36. TRIANGLE CENTERSCen
Page 192 and 193: 188 SECTION 37. AREADefinition 37.5
Page 194 and 195: 190 SECTION 37. AREAToday’s Lesso
Page 196 and 197: 192 SECTION 38. THE PYTHAGOREAN THE
Page 206 and 207: 202 SECTION 39. CIRCLESFigure 39.1:
Page 214 and 215: 210 SECTION 39. CIRCLESLet D be the
Page 216 and 217: 212 SECTION 39. CIRCLES« CC BY-NC-
Page 218 and 219: 214 SECTION 40. CIRCLES AND TRIANGL
Page 224 and 225: 220 SECTION 41. EUCLIDEAN CIRCLESFi
Page 232 and 233: 228 SECTION 41. EUCLIDEAN CIRCLESBy
Page 234 and 235: 230 SECTION 42. AREA AND CIRCUMFERE
Page 242 and 243: 238 SECTION 43. INDIANA BILL 246the
Page 244 and 245: 240 SECTION 44. ESTIMATING πFigure
Page 246 and 247: 242 SECTION 44. ESTIMATING πso tha
Page 248 and 249: 244 SECTION 44. ESTIMATING πn π n
Page 250 and 251: 246 SECTION 44. ESTIMATING πTable
Page 252 and 253: 248 SECTION 44. ESTIMATING πnums =
Page 254 and 255: 250 SECTION 44. ESTIMATING π« CC
Page 256 and 257: 252 SECTION 45. EUCLIDEAN CONSTRUCT
Page 262 and 263:
258 SECTION 45. EUCLIDEAN CONSTRUCT
Page 264 and 265:
Page 266 and 267:
Page 268 and 269:
264 SECTION 46. HYPERBOLIC GEOMETRY
Page 270 and 271:
Page 272 and 273:
Page 274 and 275:
Page 276 and 277:
272SECTION 47.PERPENDICULAR LINES I
Page 278 and 279:
274SECTION 47.PERPENDICULAR LINES I
Page 280 and 281:
276 SECTION 48. PARALLEL LINES IN H
Page 282 and 283:
Page 284 and 285:
Page 286 and 287:
Page 288 and 289:
284 SECTION 49. TRIANGLES IN HYPERB
Page 290 and 291:
Page 292 and 293:
Page 294 and 295:
290 SECTION 50. AREA IN HYPERBOLIC
Page 296 and 297:
Page 298 and 299:
Page 300 and 301:
Page 302 and 303:
298 SECTION 51. THE POINCARE DISK M
Page 304 and 305:
Page 306 and 307:
Page 308 and 309:
Page 310 and 311:
306 SECTION 52. ARC LENGTHAs we see
Page 312 and 313:
308 SECTION 52. ARC LENGTHProof. Su
Page 314 and 315:
310 SECTION 52. ARC LENGTHThus the
Page 316 and 317:
312 SECTION 52. ARC LENGTHMeasuring
Page 318 and 319:
314 SECTION 53. SPHERICAL GEOMETRYW
Page 320 and 321:
316 SECTION 53. SPHERICAL GEOMETRYF
Page 322 and 323:
318 SECTION 53. SPHERICAL GEOMETRYP
Page 324 and 325:
320 APPENDIX A. SYMBOLS USEDAppendi
Page 326 and 327:
322 APPENDIX A. SYMBOLS USEDTechnol
show all

Plane Geometry - Bruce E. Shapiro

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?