Plane Geometry - Bruce E. Shapiro

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38 SECTION 9. BIRKHOFF/MACLANE AXIOMSBirkhoff’s AxiomsUndefined Terms: point, line, distance, angle.1. Axiom of Line Measure. The points A, B, .. of any line l can be put into (1, 1)correspondence with the real numbers x so that |x B − x A | = d(A, B) for all pointsA, B. Here d(A, B) denotes the distance between the points A and B. In otherwords, you are allowed to use a rule to measure the length of a line.2. The point-line postulate. One and only one straight line l contains two given pointsP, Q (P ≠ Q).3. The Axiom of angle measure. The half lines l, m, .. through any point O can be putinto (1,1) correspondence with teh real numbers a mod 2π so that, if A ≠ 0 andB ≠ O are points of l and m respectively, the difference a m − a l mod 2π is ∠A0B.4. The Postulate of triangle similarity. If in two triangles, △ABC and △A ′ B ′ C ′ ,of for some constant k > 0, d(A ′ , B ′ ) = kd(A, B), d(A ′ , C ′ ) = kd(A, C), andalso ∠B ′ A ′ C ′ = ±∠BAC then also d(B ′ , C ′ ) = kd(B, C), ∠C ′ B ′ A ′ = ±∠CBA,∠A ′ C ′ B ′ = ±∠ACB.1. There are at least two points.MacLane’s Axioms on Distance2. If A and B are points, d(AB) is a nonnegative number (that gives the distancebetween the points).3. For points A and B, d(AB) = 0 if and only if A = B.4. If A and B are points then d(AB) = d(BA).MacLane’s Axioms on Lines1. A l ine is a set of points containing more than one points.2. Through two distinct points there is one and only one l ine.3. Three distinct points on a line if and only if one of them is between the other two.4. On each ray from a point O and to each positive real number b there is a point Bwith d(OB) = bMacLane’s Axioms on Angles1. If r and s are rays from the same point, then ∠rs is a real number (mod 360).2. If r, s, t are three rays from the same point, the ∠rs + ∠st = ∠rt.3. If r is a ray from O and c is a real number, then there is a ray s from O such that∠rs = c ◦ .« CC BY-NC-ND 3.0. Revised: 18 Nov 2012
SECTION 9. BIRKHOFF/MACLANE AXIOMS 39McLane’s Axiom on SimilarityIf two triangles △ABC and △A ′ B ′ C ′ have ∠ABC = ±∠A ′ B ′ C ′ , d(AB) = kd(A ′ B ′ ), andd(BC) = kd(B ′ C ′ ) for some postive number k then they are similar.MacLane’s Axiom of ContinuityLet ∠AOB be proper. If D is between A adn B then 0 < ∠AOD < ∠AOB.Here a proper angle is an angle x such that 0 < x < 180.The 37th View of Mt Fuji by Katsushika Hokusai (1760-1849) never madeit to wood block.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
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Page 80 and 81: 76 SECTION 15. THE PLANE SEPARATION
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Page 84 and 85: 80 SECTION 16. ANGLESFigure 16.1: T
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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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192 SECTION 38. THE PYTHAGOREAN THE
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202 SECTION 39. CIRCLESFigure 39.1:
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208 SECTION 39. CIRCLESTheorem 39.1
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210 SECTION 39. CIRCLESLet D be the
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212 SECTION 39. CIRCLES« CC BY-NC-
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214 SECTION 40. CIRCLES AND TRIANGL
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238 SECTION 43. INDIANA BILL 246the
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240 SECTION 44. ESTIMATING πFigure
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242 SECTION 44. ESTIMATING πso tha
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244 SECTION 44. ESTIMATING πn π n
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246 SECTION 44. ESTIMATING πTable
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248 SECTION 44. ESTIMATING πnums =
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250 SECTION 44. ESTIMATING π« CC
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252 SECTION 45. EUCLIDEAN CONSTRUCT
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264 SECTION 46. HYPERBOLIC GEOMETRY
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272SECTION 47.PERPENDICULAR LINES I
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274SECTION 47.PERPENDICULAR LINES I
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306 SECTION 52. ARC LENGTHAs we see
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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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320 APPENDIX A. SYMBOLS USEDAppendi
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322 APPENDIX A. SYMBOLS USEDTechnol
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Plane Geometry - Bruce E. Shapiro

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