Plane Geometry - Bruce E. Shapiro

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242 SECTION 44. ESTIMATING πso thatand so forth.H 2 = √ 2»H 3 = 2 − √ 2√ »H 4 = 2 − 2 + √ 2… √ »H 5 = 2 − 2 + 2 + √ 2… √ »H 6 = 2 − 2 + 2 + 2 + √ 2One could, for example, calculate the first i values of π i with the followingcode in Mathematica:f[x_]:= Sqrt[2*(1-Sqrt[1-(x/2)^2])];pi[i_] := NestList[f, Sqrt[2], i]*Table[2^n, {n, 1, i + 1}];To print the first 25 estimates to 20 significant figures, we enterN[pi[25],20]The resulting guesses would be{2.8284271247461900976, 3.0614674589207181738,3.1214451522580522856, 3.1365484905459392638,3.1403311569547529123, 3.1412772509327728681,3.1415138011443010763, 3.1415729403670913841,3.1415877252771597006, 3.1415914215111999740,3.1415923455701177423, 3.1415925765848726657,3.1415926343385629891, 3.1415926487769856695,3.1415926523865913458, 3.1415926532889927653,3.1415926535145931202, 3.1415926535709932089,3.1415926535850932311, 3.1415926535886182366,3.1415926535894994880, 3.1415926535897198008,3.1415926535897748791, 3.1415926535897886486,3.1415926535897920910, 3.1415926535897929516}The correct value of π to 20 figures is 3.1415926535897932385.Method of ArchimedesArchimedes estimated the value of π by assuming that the perimeter I n ofany inscribed regular n-sided polygon was less than the circumference ofa circle, and the perimeter C n of any regular-sided circumscribed polygonwas greater than the circumference of the same circle:I n < C < C n« CC BY-NC-ND 3.0. Revised: 18 Nov 2012
SECTION 44. ESTIMATING π 243As the number of sides of the corresponding polygons gets larger and larger,these two numbers approach one another.If we let Γ = C(O, r) be any circle with radius 1/2, then C = 2πr = π.HenceI n < π < C nArchimedes obtained the following recurrence relations: 1C 2n =2C nI nC n + I n(circumscribed polygon)I 2n = √ C 2n I n (inscribed polygon)where C 2n and I 2n are the circumferences of the 2n-sided regular circumbscribedand inscribed polygons.Archimedes obtained the bounds22371 < π < 227by setting n = 6·2 k for k = 0, 1, .., 4 (i.e., triangle, hexagon, 12-gon, 24-gon,48-gon, 96-gon). His limits were rational numbers rather than real numbersbecause he approximated all of his square roots with real numbers by somemethod that he did not describe. 2Taylor SeriesUsing the Taylor Series expansion for the arctangent,tan −1 x = x − 1 3 x3 + 1 5 x5 − 1 7 x7 + ...gives (using tan π/4 = 1),π = 4Å1 − 1 3 + 1 5 − 1 ã7 + ...(44.1)Unfortunately this converges very slowly - here are the results after up toa million iterations:1 If you are interested, you can find a derivation of these formulas using trigonometry athttp://mathworld.wolfram.com/ArchimedesRecurrenceFormula.html.2 A likely method that was known by the ancients to estimate square roots was therecursion relation x n+1 = (x n + a/x n)/2 whic converges to √ a as n → ∞. A perfectlygood starting guess is x 0 = a.Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.
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Foundations of GeometryLecture Note
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ivCONTENTS30 Triangles in Neutral G
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2 SECTION 1. PREFACEare enrolled in
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4 SECTION 1. PREFACEre-learning it
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6 SECTION 2. NCTM3. To guide in the
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8 SECTION 2. NCTMTechnology. Techno
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10 SECTION 2. NCTM« CC BY-NC-ND 3.
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12 SECTION 3. CA STANDARDIn 2005 Ca
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14 SECTION 3. CA STANDARDThe Califo
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16 SECTION 3. CA STANDARDCalifornia
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18 SECTION 4. COMMON COREAt all lev
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20 SECTION 5. LOGIC AND PROOFour th
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22 SECTION 5. LOGIC AND PROOFRene D
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24 SECTION 5. LOGIC AND PROOF“I
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26 SECTION 6. REAL NUMBERSThe inter
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28 SECTION 6. REAL NUMBERS“The Su
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32 SECTION 7. EUCLID’S ELEMENTSFi
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34 SECTION 8. HILBERT’S AXIOMSiom
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36 SECTION 8. HILBERT’S AXIOMS«
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38 SECTION 9. BIRKHOFF/MACLANE AXIO
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40 SECTION 9. BIRKHOFF/MACLANE AXIO
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42 SECTION 10. THE SMSG AXIOMSP is
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44 SECTION 10. THE SMSG AXIOMS« CC
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46 SECTION 11. THE UCSMP AXIOMSin s
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48 SECTION 11. THE UCSMP AXIOMS« C
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50 SECTION 12. VENEMA’S AXIOMS3.
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52 SECTION 12. VENEMA’S AXIOMS«
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54 SECTION 13. INCIDENCE GEOMETRYFi
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56 SECTION 13. INCIDENCE GEOMETRYEx
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58 SECTION 13. INCIDENCE GEOMETRYFi
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60 SECTION 13. INCIDENCE GEOMETRY«
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62 SECTION 14. BETWEENNESSFigure 14
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64 SECTION 14. BETWEENNESSTheorem 1
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66 SECTION 14. BETWEENNESS(a) To sh
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68 SECTION 14. BETWEENNESSThe follo
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70 SECTION 14. BETWEENNESSExample 1
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72 SECTION 14. BETWEENNESSorf(A) >
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74 SECTION 14. BETWEENNESS« CC BY-
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76 SECTION 15. THE PLANE SEPARATION
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78 SECTION 15. THE PLANE SEPARATION
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80 SECTION 16. ANGLESFigure 16.1: T
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82 SECTION 16. ANGLESCorollary 16.6
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84 SECTION 16. ANGLESFigure 16.6: I
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86 SECTION 16. ANGLESD cannot be in
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88 SECTION 17. THE CROSSBAR THEOREM
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90 SECTION 17. THE CROSSBAR THEOREM
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92 SECTION 18. LINEAR PAIRSFigure 1
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94 SECTION 18. LINEAR PAIRSγ + ∠
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96 SECTION 19. ANGLE BISECTORSwith
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98 SECTION 19. ANGLE BISECTORSAngle
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100 SECTION 20. THE CONTINUITY AXIO
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102 SECTION 20. THE CONTINUITY AXIO
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104 SECTION 21. SIDE-ANGLE-SIDEBirk
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106 SECTION 21. SIDE-ANGLE-SIDEFigu
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108 SECTION 22. NEUTRAL GEOMETRYEll
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110 SECTION 22. NEUTRAL GEOMETRY«
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112 SECTION 23. ANGLE-SIDE-ANGLEFig
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114 SECTION 23. ANGLE-SIDE-ANGLE«
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116 SECTION 24. EXTERIOR ANGLESTheo
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118 SECTION 24. EXTERIOR ANGLES« C
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120 SECTION 25. ANGLE-ANGLE-SIDEBy
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122 SECTION 26. SIDE-SIDE-SIDECase
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124 SECTION 26. SIDE-SIDE-SIDEBy si
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126 SECTION 27. SCALENE AND TRIANGL
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128 SECTION 27. SCALENE AND TRIANGL
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130 SECTION 28. CHARACTERIZATION OF
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132 SECTION 28. CHARACTERIZATION OF
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134 SECTION 29. TRANSVERSALSFigure
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136 SECTION 29. TRANSVERSALSCorolla
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138 SECTION 30. TRIANGLES IN NEUTRA
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146 SECTION 31. QUADRILATERALS IN N
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156 SECTION 32. THE EUCLIDEAN PARAL
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168 SECTION 33. RECTANGLESLemma 33.
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170 SECTION 33. RECTANGLESFigure 33
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172 SECTION 33. RECTANGLES« CC BY-
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174 SECTION 34. THE PARALLEL PROJEC
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176 SECTION 34. THE PARALLEL PROJEC
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178 SECTION 35. SIMILAR TRIANGLESBy
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180 SECTION 35. SIMILAR TRIANGLESCo
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182 SECTION 36. TRIANGLE CENTERSFig
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184 SECTION 36. TRIANGLE CENTERSThe
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186 SECTION 36. TRIANGLE CENTERSCen
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188 SECTION 37. AREADefinition 37.5
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190 SECTION 37. AREAToday’s Lesso
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292 SECTION 50. AREA IN HYPERBOLIC
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298 SECTION 51. THE POINCARE DISK M
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306 SECTION 52. ARC LENGTHAs we see
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308 SECTION 52. ARC LENGTHProof. Su
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310 SECTION 52. ARC LENGTHThus the
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312 SECTION 52. ARC LENGTHMeasuring
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314 SECTION 53. SPHERICAL GEOMETRYW
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316 SECTION 53. SPHERICAL GEOMETRYF
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318 SECTION 53. SPHERICAL GEOMETRYP
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320 APPENDIX A. SYMBOLS USEDAppendi
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322 APPENDIX A. SYMBOLS USEDTechnol
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