Plane Geometry - Bruce E. Shapiro

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128 SECTION 27. SCALENE AND TRIANGLE INEQUALITYFigure 27.4: Hinge theorem (theorem 27.3) when B ∗ H ∗ G.Definition 27.4 The distance from a point P to a line l, denoted byd(P, l) is the distance from P to the foot of the perpendicular dropped fromP to l. (See figure 27.5.)Theorem 27.5 (Shortest Distance from a Point to a Line) Let l bea line, let P ∉ l a point, and let F be the foot of the perpendicular from Pto l. If R ∈ l such that R ≠ F , then P R > P F . (See figure 27.5.)Proof. (Exercise)Figure 27.5: The shortest distance from a point P to a line l is the lengthof the perpendicular segment dropped from P to l (theorem 27.5). Hence(F P < F R)(∀R ∈ l, R ≠ F ).« CC BY-NC-ND 3.0. Revised: 18 Nov 2012
Section 28Characterization ofBisectorsTheorem 28.1 (Pointwise Characterization of Angle Bisector) LetA, B, C be non-collinear points and let P be in the interior of ∠BAC. ThenP lies on the angle bisector of ∠BAC if and only if d(P, ←→ AB) = d(P, ←→ AC).Proof. ( ⇒) Assume P is a point on the bisector of ∠BAC (see figure 28.1).Drop perpendiculars from P to ←→ AB and ←→ AC to their respective feet D ∈ ←→ ABand E ∈ ←→ AC.∠P AE = ∠P AD by definition of angle bisection, ∠P EA = ∠P DA becauseboth are right angles, and P A = P A. Hence by AAS, △ADP ∼ = △AEP .Hence DP = P E because they are corresponding sides of congruent trian-Figure 28.1: Theorem 28.1.129
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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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34 SECTION 8. HILBERT’S AXIOMSiom
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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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46 SECTION 11. THE UCSMP AXIOMSin s
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50 SECTION 12. VENEMA’S AXIOMS3.
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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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76 SECTION 15. THE PLANE SEPARATION
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188 SECTION 37. AREADefinition 37.5
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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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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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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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