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Geometry with Trigonometry

(Marvins-Underground-K-12) #1

  • 1 Preliminaries Glossary

    • 1.1 Historicalnote.............................

    • 1.2 Noteondeductivereasoning......................

    • 1.2.1 Definitions...............................

    • 1.2.2 Proof..................................

    • 1.3 Euclid’sThe Elements.........................

    • 1.3.1

    • 1.3.2 Definitions...............................

    • 1.3.3 Postulatesandcommonnotions....................

    • 1.3.4

    • 1.3.5 Congruence

    • 1.3.6 Quantities or magnitudes

    • 1.4 Ourapproach

    • 1.4.1 Typeofcourse.............................

    • 1.4.2 Needforpreparation..........................

    • 1.5 Revisionofgeometricalconcepts

    • 1.5.1

    • 1.5.2 Thebasicshapes............................

    • 1.5.3 Distance; degree-measure of an angle

    • 1.5.4 Our treatment of congruence

    • 1.5.5 Parallellines..............................

    • 1.6 Pre-requisites

    • 1.6.1 Setnotation

    • 1.6.2 Classical algebra

    • 1.6.3 Otheralgebra

    • 1.6.4 Distinctivepropertyofrealnumbersamongfields



  • 2 Basic shapes of geometry

    • 2.1 Lines, segments and half-lines.....................

    • 2.1.1 Plane,points,lines...........................

    • 2.1.2 Naturalorderonaline.........................

    • 2.1.3 Reciprocalorders

    • 2.1.4 Segments................................

    • 2.1.5 Half-lines

    • 2.2 Open and closed half-planes

    • 2.2.1 Convexsets

    • 2.2.2 Open half-planes

    • 2.2.3 Closed half-planes

    • 2.3 Angle-supports, interior and exterior regions, angles

    • 2.3.1 Angle-supports, interior regions.................... viii Contents

    • 2.3.2 Exteriorregions

    • 2.3.3 Angles

    • 2.4 Triangles and convex quadrilaterals

    • 2.4.1 Terminology..............................

    • 2.4.2 Triangles................................

    • 2.4.3 Pasch’s property,

    • 2.4.4 Convex quadrilaterals



  • 3 Distance; degree-measure of an angle

    • 3.1 Distance

    • 3.1.1 Axiomfordistance

    • 3.1.2 Derived properties of distance.....................

    • 3.2 Mid-points...............................

    • 3.2.1

    • 3.3 Aratioresult..............................

    • 3.3.1

    • 3.4 Thecross-bartheorem.........................

    • 3.4.1

    • 3.5 Degree-measure of angles

    • 3.5.1 Axiom for degree-measure

    • 3.5.2 Derived properties of degree-measure

    • 3.6 Mid-line of an angle-support

    • 3.6.1 Right-angles

    • 3.6.2 Perpendicularlines

    • 3.6.3 Mid-lines................................

    • 3.7 Degree-measure of reflex angles....................

    • 3.7.1



  • 4 Congruence of triangles; parallel lines

    • 4.1 Principles of congruence

    • 4.1.1 Congruence of triangles

    • 4.2 Alternateangles,parallellines.....................

    • 4.2.1 Alternateangles

    • 4.2.2 Parallellines..............................

    • 4.3 Properties of triangles and half-planes

    • 4.3.1 Side-angle relationships; the triangle inequality

    • 4.3.2 Properties of parallelism

    • 4.3.3 Dropping a perpendicular

    • 4.3.4 Projectionandaxialsymmetry



  • 5 The parallel axiom; Euclidean geometry Contents

    • 5.1 Theparallelaxiom...........................

    • 5.1.1 Uniquenessofaparallelline

    • 5.2 Parallelograms.............................

    • 5.2.1 Parallelogramsandrectangles.....................

    • 5.2.2 Sum of measures of wedge-angles of a triangle

    • 5.3 Ratio results for triangles

    • 5.3.1 Lines parallel to one side-line of a triangle

    • 5.3.2 Similar triangles

    • 5.4 Pythagoras’ theorem, c. 550B.C.

    • 5.4.1

    • 5.5 Mid-lines and triangles

    • 5.5.1 Harmonicranges............................

    • 5.6 Area of triangles, and convex quadrilaterals and polygons

    • 5.6.1 Area of a triangle

    • 5.6.2 Area of a convex quadrilateral.....................

    • 5.6.3 Area of a convex polygon



  • 6 Cartesian coordinates; applications

    • 6.1 Frameofreference,Cartesiancoordinates

    • 6.1.1

    • 6.2 Algebraicnoteonlinearequations

    • 6.2.1

    • 6.3 Cartesianequationofaline

    • 6.3.1

    • 6.4 Parametric equations of a line

    • 6.4.1

    • 6.5 Perpendicularity and parallelism of lines

    • 6.5.1

    • 6.6 Projectionandaxialsymmetry

    • 6.6.1

    • 6.6.2 Formula for area of a triangle

    • 6.6.3 Inequalities for closed half-planes...................

    • 6.7 Coordinatetreatmentofharmonicranges...............

    • 6.7.1 Newparametrisationofaline

    • 6.7.2 Interchangeofpairsofpoints

    • 6.7.3 Distancesfrommid-point

    • 6.7.4 Distancesfromend-point

    • 6.7.5 Constructionforaharmonicrange



  • 7 Circles; their basic properties

    • 7.1 Intersectionofalineandacircle ...................

    • 7.1.1 .....................................

    • 7.2 Properties of circles ..........................

    • 7.2.1 ..................................... Contents

    • 7.2.2 Equationofacircle ..........................

    • 7.2.3 Circle through three points ......................

    • 7.3 Formula for mid-line of an angle-support ...............

    • 7.3.1 .....................................

    • 7.4 Polar properties of a circle .......................

    • 7.4.1 Tangentsfromanexteriorpoint ....................

    • 7.4.2 Thepowerpropertyofacircle.....................

    • 7.4.3 Aharmonicrange ...........................

    • 7.5 Anglesstandingonarcsofcircles...................

    • 7.5.1 .....................................

    • 7.5.2 Minorandmajorarcsofacircle....................

    • 7.6 Senseddistances............................

    • 7.6.1 Senseddistance ............................

    • 7.6.2 Sensed products and a circle......................

    • 7.6.3 Radicalaxisandcoaxalcircles ....................



  • 8 Translations; axial symmetries; isometries

    • 8.1 Translations and axial symmetries .................

    • 8.1.1 .....................................

    • 8.2 Isometries ............................

    • 8.2.1 .....................................

    • 8.2.2 .....................................

    • 8.3 Translationofframeofreference ...................



  • 9 Trigonometry; cosine and sine; addition formulae

    • 9.1 Indicatorofanangle..........................

    • 9.1.1 .....................................

    • 9.2 Cosineandsineofanangle ......................

    • 9.2.1 .....................................

    • 9.2.2 Polarcoordinates ...........................

    • 9.2.3 .....................................

    • 9.3 Angles in standard position ......................

    • 9.3.1 Angles in standard position ......................

    • 9.3.2 Addition of angles ...........................

    • 9.3.3 Modified addition of angles ......................

    • 9.3.4 Subtractionofangles .........................

    • 9.3.5 Integer multiples of an angle .....................

    • 9.3.6 Standard multiples of a right-angle .................

    • 9.4 Halfangles...............................

    • 9.4.1 .....................................

    • 9.5 Thecosineandsinerules .......................

    • 9.5.1 Thecosinerule.............................

    • 9.5.2 Thesinerule..............................

    • 9.5.3 ..................................... Contents

    • 9.5.4 The Steiner-Lehmus theorem, 1842 ...............

    • 9.6 Cosineandsineofanglesequalinmagnitude.............

    • 9.6.1 .....................................



  • 10 Complex coordinates; sensed angles; angles between lines

    • 10.1 Complexcoordinates .........................

    • 10.1.1 .....................................

    • 10.2 Complex-valueddistance .......................

    • 10.2.1Complex-valueddistance .......................

    • 10.2.2 A complex-valued trigonometric function...............

    • 10.3 Rotations and axial symmetries ....................

    • 10.3.1Rotations................................

    • 10.3.2Formulaforanaxialsymmetry ....................

    • 10.4 Sensedangles .............................

    • 10.4.1 .....................................

    • 10.5 Sensed-area ..............................

    • 10.5.1 .....................................

    • 10.5.2 Sensed-area of a triangle .....................

    • 10.5.3Abasicfeatureofsensed-area.....................

    • 10.5.4 An identity for sensed-area .....................

    • 10.6 Isometries as compositions .....................

    • 10.6.1 .....................................

    • 10.7 Orientation of a triple of non-collinear points .....

    • 10.7.1 .....................................

    • 10.8 Sensed angles of triangles, the sine rule ...............

    • 10.8.1 .....................................

    • 10.9 Someresultsoncircles ........................

    • 10.9.1 A necessary condition to lie on a circle ...............

    • 10.9.2 A sufficient condition to lie on a circle ...............

    • 10.9.3Complexcross-ratio ..........................

    • 10.9.4 Ptolemy’s theorem, c. 200A.D. ....................

    • 10.10 Anglesbetweenlines .........................

    • 10.10.1Motivation ..............................

    • 10.10.2 Duo-sectors ............................

    • 10.10.3Duo-angles ..............................

    • 10.10.4 Duo-angles in standard position ...................

    • 10.10.5 Addition of duo-angles in standard position .....

    • 10.10.6 Addition formulae for tangents of duo-angles .....

    • 10.10.7 Associativity of addition of duo-angles ...............

    • 10.10.8 Group properties of duo-angles; sensed duo-angles .........

    • 10.10.9Anapplication ............................

    • 10.11 A case of Pascal’s theorem, 1640 ...................

    • 10.11.1 ....................................

    • 10.11.2 .................................... Contents

    • 10.11.3 ....................................



  • 11 Vector and complex-number methods

    • 11.1 Equipollence..............................

    • 11.1.1 .....................................

    • 11.2 Sum of couples, multiplication of a couple by a scalar .....

    • 11.2.1 .....................................

    • 11.2.2 Vector space overR ..........................

    • 11.3 Scalar or dot products .......................

    • 11.3.1 .....................................

    • 11.4 Components of a vector .......................

    • 11.4.1 Components ............................

    • 11.4.2Arealcoordinates ...........................

    • 11.4.3Cartesiancoordinatesfromarealcoordinates .............

    • 11.4.4 .....................................

    • 11.4.5 .....................................

    • 11.5 Vector methods in geometry......................

    • 11.5.1 Menelaus’ theorem, c. 100A.D. ....................

    • 11.5.2 Ceva’s theorem and converse, 1678 A.D. ...............

    • 11.5.3 Desargues’ perspective theorem, 1648 A.D. ...........

    • 11.5.4 Pappus’ theorem, c. 300A.D. .....................

    • 11.5.5 Centroid of a triangle .......................

    • 11.5.6 Orthocentre of a triangle .......................

    • 11.5.7 Incentre of a triangle..........................

    • 11.6 Mobilecoordinates ..........................

    • 11.6.1 Grassmann’s supplement of a vector .................

    • 11.6.2 .....................................

    • 11.6.3 Handling a triangle ..........................

    • 11.6.4 Circumcentre of a triangle .......................

    • 11.6.5 Other distinguished points for a triangle ...............

    • 11.6.6 Euler line of a triangle .......................

    • 11.6.7 Similar triangles ............................

    • 11.6.8 Centroids of similar triangles erected on the sides of a triangle....

    • 11.6.9 Circumcentres of similar triangles on sides of triangle .....

    • 11.6.10 Triangle with vertices the mid-points of sides of given triangle

    • 11.6.11Thenine-pointcircle.........................

    • 11.6.12 Parametric equations of lines. ....................

    • 11.7 Somewell-knowntheorems......................

    • 11.7.1 Feuerbach’s theorem, 1822 ......................

    • 11.7.2 The Wallace-Simson line, 1797 ....................

    • 11.7.3 The incentre on the Euler line of a triangle ...........

    • 11.7.4 Miquel’s theorem, 1838 .......................

    • 11.8 Isogonal conjugates ..........................

    • 11.8.1 Isogonal conjugates .......................... Contents xiii

    • 11.8.2Concurrency..............................

    • 11.8.3Symmedians..............................



  • 12 Trigonometric functions in calculus

    • 12.1 Repeatedbisectionofanangle ....................

    • 12.1.1 .....................................

    • 12.2 Circularfunctions ...........................

    • 12.2.1 .....................................

    • 12.2.2 Definition ofπ.............................

    • 12.3 Derivativesofcosineandsinefunctions................

    • 12.3.1 .....................................

    • 12.4 Parametric equations for a circle ...................

    • 12.4.1Areaofadisk .............................

    • 12.4.2Lengthofanarcofacircle ......................

    • 12.4.3 Radian measure ...........................

    • 12.5 Extensionofdomainsofcosineandsine ...............

    • 12.5.1 .....................................

    • 12.5.2 .....................................



  • Appendix; List of axioms

  • Bibliography

  • Index

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