Plane Geometry - Bruce E. Shapiro

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254 SECTION 45. EUCLIDEAN CONSTRUCTIONSFigure 45.3: Construction 45.2. A copy of ∠EDF on ray −−→ DE is constructedsuch that ∠EDF ∼ = ∠BAC.5. Draw an arc of c 3 = C(B, r).6. Circles c 2 and c 3 intersect at two points: one of these points is A; callthe other point Q.7. Draw line n = ←→ P Q. Then n ‖ l.Construction 45.8 (Partition a Segment) Given a line segment AB anda positive integer n, divide the segment into n segments of equal length.1. Pick any point Q that is not on the line ←→ AB.2. Construct ray −→ AQ.3. Pick any point Q 1 ∈ −→ AQ.4. Define r = AQ 1 .5. Define points Q 2 , Q 3 , ..., Q n ∈ −→ AQ such that Q i Q i+1 = r.6. Construct line m n = ←−→ Q n B.7. Construct lines m 1 , m 2 , ..., m n−1 through points Q 1 , Q 2 , ..., Q n−1 thatare parallel to to m n .8. Define the points P 1 = m 1 ∩ −→ AB, P2 = m 2 ∩ −→ AB,..., Pn−1 = m n−1 ∩−→AB.9. Then the segments AP 1 , P 1 P 2 ,...,P n−2 P n−1 , P n−1 B are the desiredsegments of equal length.Construction 45.9 (Tangent to a Circle) Given a circle γ = C(O, r)« CC BY-NC-ND 3.0. Revised: 18 Nov 2012
SECTION 45. EUCLIDEAN CONSTRUCTIONS 255Figure 45.4: Illustration of construction 45.3. Ray −→ AG bisects angle ∠BAC.Figure 45.5: Construction 45.4. Line ←→ QP passes through A and is perpendicularto l.and a point P that is outside γ, construct a line through P that is tangentto γ.1. Let γ = C(O, r) and let P be any point oustide γ.2. Draw the line segment OP .3. Find the midpoint M of OP .4. Draw the circle δ = C(M, OM).5. Let Q and R be the two points where the two circles intersect. Lines←→P R and ←→ P Q are both tangent to γ.Constructible NumbersGiven a segment of length 1, we can ask the question of which other segmentswe can construct. For example, we know that by duplicating theRevised: 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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30 SECTION 7. EUCLID’S ELEMENTSEu
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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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192 SECTION 38. THE PYTHAGOREAN THE
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202 SECTION 39. CIRCLESFigure 39.1:
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Page 210 and 211: 206 SECTION 39. CIRCLESFigure 39.5:
Page 212 and 213: 208 SECTION 39. CIRCLESTheorem 39.1
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Page 256 and 257: 252 SECTION 45. EUCLIDEAN CONSTRUCT
Page 268 and 269: 264 SECTION 46. HYPERBOLIC GEOMETRY
Page 276 and 277: 272SECTION 47.PERPENDICULAR LINES I
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304 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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Plane Geometry - Bruce E. Shapiro

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