Plane Geometry - Bruce E. Shapiro

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196 SECTION 38. THE PYTHAGOREAN THEOREMEuclid, of course, did not state the Pythagorean theorem in terms of thesum of the squares of the edges; to do so would have required algebra,which was not invented for another thousand years after Euclid. Instead,the theorem was expressed in terms of area.Theorem 38.9 (Euclid’s Version of the Pythagorean Theorem) Thearea of the square on the hypoteneuse of a right triangle is equal to the sumof areas of the squares on the legs.Proof. (Second Proof of the Pythago- rean Theorem, due to Euclid6 )Let △ABC be a right triangle with right angle at C. We construct thesquares on each edge and define the points as indicated in figure 38.5.Point J is the foot of the perpendicular to AB dropped from C, and pointK is the intersection of the same perpendicular with DE.From the Euclidean Area Postulate,α(□ABED) = α(□AJKD) + α(□JBEK) (38.1)Since ∠ACB is a right angle and □ACHI is a square, H ∈ ←→ BC and AI ‖←→BC.Thus, since triangles △IAC and △IAB each have base IA and height AC,they both have the same area:Furthermore, since IA = AC, AB = AD, andthen by SASα(△IAC) = α(△IAB) (38.2)∠IAB = 90 + ∠CAB = ∠CAD△IAB ∼ = △CADand since congruent triangles have equal areas (by the neutral area postulate),α(△IAB) = α(△CAD) (38.3)Since △CAD and △JAD each have base AD and height AJ, they haveequal area,α(△CAD) = α(△JAD) (38.4)Combining equations 38.2, 38.3 and 38.4 givesα(△IAC) = α(△JAD) (38.5)6 Version given in Book 1 of the Elements; another proof due to Euclid is in Book VI.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012
SECTION 38. THE PYTHAGOREAN THEOREM 197Figure 38.5: Euclid’s Proof of the Pythagorean Theorem.Butα(□ACHI) = 2α(△IAC)= 2α(△JAD)= α(□AJKD) (38.6)A similar argument is used to show thatα(□CBF G) = α(□JBEK) (38.7)Substituting equations 38.6 and 38.7 into 38.1 givesα(□ABED) = α(□ACHI) + α(□CBF G)Illustration of the proof the Guogu thoerem (third proof) from Zhoubi suanjing (c 100BCE). The illustrations were not in the original text but were added sometime beforethe 16th century, when this version was printed. [MAA Digitial Sciences Library]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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24 SECTION 5. LOGIC AND PROOF“I
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26 SECTION 6. REAL NUMBERSThe inter
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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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246 SECTION 44. ESTIMATING πTable
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248 SECTION 44. ESTIMATING πnums =
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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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276 SECTION 48. PARALLEL LINES IN H
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284 SECTION 49. TRIANGLES IN HYPERB
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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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Plane Geometry - Bruce E. Shapiro

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