Plane Geometry - Bruce E. Shapiro

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284 SECTION 49. TRIANGLES IN HYPERBOLIC GEOMETRYFigure 49.1: Construction of an isoceles triangle with measure less that ɛ.Proof. Let ɛ > 0 be given.Construct a right triangle △ABC with one interior angle 90 − ɛ as follows:Let A and B be points, and define Q so that ∠ABQ = 90 − ɛ. The dropa perpendicular from A to BQ and call its foot C. Then △ABC is a righttriangle with one interior angle of 90 − ɛ.Thenandσ(△ABC) = 90 + 90 − ɛ + µ(∠ACB)> 180 − ɛδ(△ABC) = 180 − σ(△ABC)< 180 − (180 − ɛ) = ɛTheorem 49.3 For every ɛ > 0 there is a number d > 0 such that if leg oftriangle △ABC has length less than d then δ(△ABC) < ɛ.Proof. Let ɛ > 0 be given.By theorem 49.2 there is a right triangle △EF G with right angle at E suchthat δ(△EF G) < ɛ/2.Let d be the length of the shortest side of △EF G.Let △ABC be any triangle with sides shorter d. Let D be the foot ofthe perpendicdular from C to AB. Then △ADC and △BDC are righttriangles with right angles at D (if D does not satisfy A ∗ D ∗ B relabel thevertices and choose another edge as AB).(See figure 49.2.)By construction we chose AC < d, BC < d, and AB < d.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012
SECTION 49. TRIANGLES IN HYPERBOLIC GEOMETRY 285Figure 49.2: Left:Any triangle can be divided into two right triangles. IfD is not between A and B then relabel the vertices to use a different edge.Right: △A ′ EC ′ ∼ = △ADC if A ′ E = AC and AD = C ′ E. This is possiblebecause the shortest side of △EF G has length d and all the sides of △ABChave length less than d.Since AB = AD + BD then BD < d and AD < d.By the scalene inequality CD < AC < d.As shown in figure 49.2, we can embed a triangle that is congruent to eachof △BDC and △ADC in △EF G.By additivity of defecthenceδ(△BDC) < δ(△EF G) < ɛ/2δ(△ADC) < δ(△EF G) < ɛ/2δ(△ABC) = δ(△BDC) + δ(△ADC)< ɛ 2 + ɛ 2 = ɛTheorem 49.4 For every pair of points A and B and for every ɛ > 0 thereexists a d > 0 such that if C ∉ ←→ AB and AC < d then δ(△ABC) < ɛ.Proof. Let ɛ > 0 be given.Choose a point P such that P ∗ A ∗ B, and choose D such thatµ(∠BP D) = 90 − ɛRevised: 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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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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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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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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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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230 SECTION 42. AREA AND CIRCUMFERE
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232 SECTION 42. AREA AND CIRCUMFERE
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Page 256 and 257: 252 SECTION 45. EUCLIDEAN CONSTRUCT
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Plane Geometry - Bruce E. Shapiro

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