- 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
marvins-underground-k-12
(Marvins-Underground-K-12)
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