Geometry with trigonometry “All things stand by proportion.” George Puttenham (1529–1590) “Mathematics possesses not only truth, but supreme beauty – a beauty cold and austere like that of sculpture, and capable of stern perfection, such as only great art can show.” Bertrand Russell in The Principles of Mathematics (1872–1970) About the author Paddy Barry was born in Co. Westmeath in 1934 and his family moved to Co. Cork five years later. After his secondary school education at Patrician Academy, Mallow he studied at University College, Cork from 1952 to 1957, obtaining a degree of BSc in Mathematics and Mathematical Physics in 1955 and the degree of MSc in 1957. He then did research in complex analysis under the supervision of Professor W. K. Hayman, FRS, at Imperial College of Science and Technology, London (1957–1959) for which he was awarded the degree of PhD in 1960. He took first place in the Entrance Scholarships Examination to University College, Cork in 1952 and was awarded a Travelling Studentship in Mathematical Science at the National University of Ireland in 1957. He was appointed Instructor in Mathematics at Stanford University, California in 1959–1960. Returning to University College, Cork in 1960 he made his career there, becoming Professor of Mathematics and Head of Department in 1964. He has participated in the general administration of the College, being a member of the Governing Body on a number of occasions, and was the first modern Vice-President of the College from 1975 to 1977. He was also a member of the Senate of the National University of Ireland from 1977 to 1982.University College, Cork was upgraded to National University of Ireland, Cork in 1997. His mathematical interests expanded in line with his extensive teaching experience. As examiner for matriculation for many years he had to keep in contact with the detail of secondary school mathematics and the present book arose from that context, as it seeks to give a thorough account of the geometry and trigonometry that is done, necessarily incompletely, at school. Dedication To my wife Fran, and Conor, Una and Brian Geometry with trigonometry Patrick D. Barry Oxford Cambridge Philadelphia New Delhi Published by Woodhead Publishing Limited, 80 High Street, Sawston, Cambridge CB22 3HJ, UK www.woodheadpublishing.com www.woodheadpublishingonline.com Woodhead Publishing, 1518 Walnut Street, Suite 1100, Philadelphia, PA 19102- 3406, USA Woodhead Publishing India Private Limited, 303, Vardaan House, 7/28 Ansari Road, Daryaganj, New Delhi – 110002, India www.woodheadpublishingindia.com First published by Horwood Publishing Limited, 2001 Reprinted by Woodhead Publishing Limited, 2014 © P. D. Barry, 2001, 2014. The author has asserted his moral rights. 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Printed by Lightning Source v Contents Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv 1 Preliminaries 1 1.1 Historical note . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Note on deductive reasoning . . . . . . . . . . . . . . . . . . . 3 1.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Euclid’s The Elements . . . . . . . . . . . . . . . . . . . . . . 4 1.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3.3 Postulates and common notions . . . . . . . . . . . . . . 6 1.3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.5 Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.6 Quantities or magnitudes . . . . . . . . . . . . . . . . . . 8 1.4 Our approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4.1 Type of course . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4.2 Need for preparation . . . . . . . . . . . . . . . . . . . . 9 1.5 Revision of geometrical concepts . . . . . . . . . . . . . . . . . 10 1.5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.5.2 The basic shapes . . . . . . . . . . . . . . . . . . . . . . . 10 1.5.3 Distance; degree-measure of an angle . . . . . . . . . . . 15 1.5.4 Our treatment of congruence . . . . . . . . . . . . . . . . 18 1.5.5 Parallel lines . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.6 Pre-requisites . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.6.1 Set notation . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.6.2 Classical algebra . . . . . . . . . . . . . . . . . . . . . . . 20 1.6.3 Other algebra . . . . . . . . . . . . . . . . . . . . . . . . 20 1.6.4 Distinctive property among fields of real numbers . . . . 20 2 Basic shapes of geometry 21 2.1 Lines, segments and half-lines . . . . . . . . . . . . . . . . . . 21 2.1.1 Plane, points, lines . . . . . . . . . . . . . . . . . . . . . 21 2.1.2 Natural order on a line . . . . . . . . . . . . . . . . . . . 22 2.1.3 Reciprocal orders . . . . . . . . . . . . . . . . . . . . . . 23 2.1.4 Segments . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.1.5 Half-lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2 Open and closed half-planes . . . . . . . . . . . . . . . . . . . 26 2.2.1 Convex sets . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.2.2 Open half-planes . . . . . . . . . . . . . . . . . . . . . . . 26 2.2.3 Closed half-planes . . . . . . . . . . . . . . . . . . . . . . 27 2.3 Angle-supports, interior and exterior regions, angles . . . . . . 29 Contents vi 2.3.1 Angle-supports, interior regions . . . . . . . . . . . . . . 29 2.3.2 Exterior regions . . . . . . . . . . . . . . . . . . . . . . . 30 2.3.3 Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.4 Triangles and convex quadrilaterals . . . . . . . . . . . . . . . 31 2.4.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.4.2 Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.4.3 Pasch’s property, 1882 . . . . . . . . . . . . . . . . . . . . 33 2.4.4 Convex quadrilaterals . . . . . . . . . . . . . . . . . . . . 33 3 Distance; degree-measure of an angle 37 3.1 Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.1.1 Axiom for distance . . . . . . . . . . . . . . . . . . . . . 37 3.1.2 Derived properties of distance . . . . . . . . . . . . . . . 38 3.2 Mid-points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.2.1 Existence and properties of mid-points . . . . . . . . . . 40 3.3 A ratio result . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.3.1 Distances of three collinear points . . . . . . . . . . . . . 42 3.4 The cross-bar theorem . . . . . . . . . . . . . . . . . . . . . . 42 3.4.1 Statement and proof of a key result . . . . . . . . . . . . 42 3.5 Degree-measure of angles . . . . . . . . . . . . . . . . . . . . . 43 3.5.1 Axiom for degree-measure . . . . . . . . . . . . . . . . . 43 3.5.2 Derived properties of degree-measure . . . . . . . . . . . 44 3.6 Mid-line of an angle-support . . . . . . . . . . . . . . . . . . . 46 3.6.1 Right-angles . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.6.2 Perpendicular lines . . . . . . . . . . . . . . . . . . . . . 46 3.6.3 Mid-lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.7 Degree-measure of reflex angles . . . . . . . . . . . . . . . . . 48 3.7.1 Extension of concept of degree-measure . . . . . . . . . . 48 4 Congruence of triangles; parallel lines 51 4.1 Principles of congruence . . . . . . . . . . . . . . . . . . . . . 51 4.1.1 Congruence of triangles . . . . . . . . . . . . . . . . . . . 51 4.2 Alternate angles, parallel lines . . . . . . . . . . . . . . . . . . 55 4.2.1 Alternate angles . . . . . . . . . . . . . . . . . . . . . . . 55 4.2.2 Parallel lines . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.3 Properties of triangles and half-planes . . . . . . . . . . . . . . 57 4.3.1 Side-angle relationships; the triangle inequality . . . . . . 57 4.3.2 Properties of parallelism . . . . . . . . . . . . . . . . . . 58 4.3.3 Dropping a perpendicular . . . . . . . . . . . . . . . . . . 58 4.3.4 Projection and axial symmetry . . . . . . . . . . . . . . . 59 vii Contents 5 The parallel axiom; Euclidean geometry 63 5.1 The parallel axiom . . . . . . . . . . . . . . . . . . . . . . . . 63 5.1.1 Uniqueness of a parallel line . . . . . . . . . . . . . . . . 63 5.2 Parallelograms . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.2.1 Parallelograms and rectangles . . . . . . . . . . . . . . . 65 5.2.2 Sum of measures of wedge-angles of a triangle . . . . . . 66 5.3 Ratio results for triangles . . . . . . . . . . . . . . . . . . . . . 67 5.3.1 Lines parallel to one side-line of a triangle . . . . . . . . 67 5.3.2 Similar triangles . . . . . . . . . . . . . . . . . . . . . . . 70 5.4 Pythagoras’ theorem, c. 550B.C. . . . . . . . . . . . . . . . . . 71 5.4.1 Proof of theorem . . . . . . . . . . . . . . . . . . . . . . . 71 5.4.2 Proof of converse . . . . . . . . . . . . . . . . . . . . . . 72 5.5 Mid-lines and triangles . . . . . . . . . . . . . . . . . . . . . . 72 5.5.1 Harmonic ranges . . . . . . . . . . . . . . . . . . . . . . . 72 5.6 Area of triangles, convex quadrilaterals etc. . . . . . . . . . . . 75 5.6.1 Area of a triangle . . . . . . . . . . . . . . . . . . . . . . 75 5.6.2 Area of a convex quadrilateral . . . . . . . . . . . . . . . 77 5.6.3 Area of a convex polygon . . . . . . . . . . . . . . . . . . 78 6 Cartesian coordinates; applications 83 6.1 Frame of reference, Cartesian coordinates . . . . . . . . . . . . 83 6.1.1 Development of rectangular Cartesian coordinates . . . . 83 6.2 Algebraic note on linear equations . . . . . . . . . . . . . . . . 87 6.2.1 Preparatory detail on linear algebraic equations . . . . . 87 6.3 Cartesian equation of a line. . . . . . . . . . . . . . . . . . . . 88 6.3.1 Development of Cartesian equation of a line . . . . . . . 88 6.4 Parametric equations of a line . . . . . . . . . . . . . . . . . . 91 6.4.1 Development of parametric equations of a line . . . . . . 91 6.5 Perpendicularity and parallelism of lines . . . . . . . . . . . . 94 6.5.1 Coordinate treatment of perpendicular/parallel lines . . . 94 6.6 Orthogonal projection to a line . . . . . . . . . . . . . . . . . . 96 6.6.1 Perpendicular from a point to a line . . . . . . . . . . . . 96 6.6.2 Formula for area of a triangle . . . . . . . . . . . . . . . . 97 6.6.3 Inequalities for closed half-planes. . . . . . . . . . . . . . 97 6.7 Coordinate treatment of harmonic ranges . . . . . . . . . . . . 98 6.7.1 New parametrization of a line . . . . . . . . . . . . . . . 98 6.7.2 Interchange of pairs of points . . . . . . . . . . . . . . . . 99 6.7.3 Distances from mid-point . . . . . . . . . . . . . . . . . . 100 6.7.4 Distances from end-point . . . . . . . . . . . . . . . . . . 101 6.7.5 Construction for a harmonic range . . . . . . . . . . . . . 102 Contents viii 7 Circles; their basic properties 105 7.1 Intersection of a line and a circle . . . . . . . . . . . . . . . . . 105 7.1.1 Terminology concerning a circle . . . . . . . . . . . . . . 105 7.2 Properties of circles . . . . . . . . . . . . . . . . . . . . . . . . 107 7.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 7.2.2 Equation of a circle . . . . . . . . . . . . . . . . . . . . . 108 7.2.3 Circle through three points . . . . . . . . . . . . . . . . . 109 7.3 Formula for mid-line of an angle-support . . . . . . . . . . . . 109 7.3.1 Equation of mid-line of angle-support . . . . . . . . . . . 109 7.4 Polar properties of a circle . . . . . . . . . . . . . . . . . . . . 110 7.4.1 Tangents from an exterior point . . . . . . . . . . . . . . 110 7.4.2 The power property of a circle . . . . . . . . . . . . . . . 111 7.4.3 A harmonic range . . . . . . . . . . . . . . . . . . . . . . 113 7.5 Angles standing on arcs of circles . . . . . . . . . . . . . . . . 115 7.5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 7.5.2 Minor and major arcs of a circle . . . . . . . . . . . . . . 117 7.6 Sensed distances . . . . . . . . . . . . . . . . . . . . . . . . . . 117 7.6.1 Sensed distance . . . . . . . . . . . . . . . . . . . . . . . 117 7.6.2 Sensed products and a circle . . . . . . . . . . . . . . . . 119 7.6.3 Radical axis and coaxal circles . . . . . . . . . . . . . . . 121 8 Translations; axial symmetries; isometries 125 8.1 Translations and axial symmetries . . . . . . . . . . . . . . . . 125 8.1.1 Development of translations . . . . . . . . . . . . . . . . 125 8.1.2 Development of axial symmetries . . . . . . . . . . . . . . 126 8.2 Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 8.2.1 Development of isometries . . . . . . . . . . . . . . . . . 127 8.2.2 Result for mid-lines . . . . . . . . . . . . . . . . . . . . . 131 8.3 Translation of frame of reference . . . . . . . . . . . . . . . . . 132 9 Trigonometry; cosine and sine; addition formulae 133 9.1 Indicator of an angle . . . . . . . . . . . . . . . . . . . . . . . 133 9.1.1 Terminology concerning angles . . . . . . . . . . . . . . . 133 9.2 Cosine and sine of an angle . . . . . . . . . . . . . . . . . . . . 134 9.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 9.2.2 Polar coordinates . . . . . . . . . . . . . . . . . . . . . . 137 9.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 9.3 Angles in standard position . . . . . . . . . . . . . . . . . . . . 138 9.3.1 Angles in standard position . . . . . . . . . . . . . . . . . 138 9.3.2 Addition of angles . . . . . . . . . . . . . . . . . . . . . . 139 9.3.3 Modified addition of angles . . . . . . . . . . . . . . . . . 140 9.3.4 Subtraction of angles . . . . . . . . . . . . . . . . . . . . 142 9.3.5 Integer multiples of an angle . . . . . . . . . . . . . . . . 143 9.3.6 Standard multiples of a right-angle . . . . . . . . . . . . 143 ix Contents 9.4 Half-angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 9.4.1 Half-angles, a tricky concept . . . . . . . . . . . . . . . . 143 9.5 The cosine and sine rules . . . . . . . . . . . . . . . . . . . . . 145 9.5.1 The cosine rule . . . . . . . . . . . . . . . . . . . . . . . . 145 9.5.2 The sine rule . . . . . . . . . . . . . . . . . . . . . . . . . 146 9.5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 9.5.4 The Steiner-Lehmus theorem, 1842 . . . . . . . . . . . . 147 9.6 Cosine and sine of angles equal in magnitude . . . . . . . . . . 148 9.6.1 Cosine and sine of equal angles . . . . . . . . . . . . . . . 148 10 Complex coordinates; rotations, duo-angles 151 10.1 Complex coordinates . . . . . . . . . . . . . . . . . . . . . . . 151 10.1.1 Introduction of complex numbers as coordinates . . . . . 151 10.2 Complex-valued distance . . . . . . . . . . . . . . . . . . . . . 154 10.2.1 Complex-valued distance . . . . . . . . . . . . . . . . . . 154 10.2.2 A complex-valued trigonometric function . . . . . . . . . 155 10.3 Rotations and axial symmetries . . . . . . . . . . . . . . . . . 156 10.3.1 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 10.3.2 Formula for an axial symmetry . . . . . . . . . . . . . . . 158 10.4 Sensed angles . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 10.4.1 Development of sensed angles . . . . . . . . . . . . . . . . 159 10.5 Sensed-area . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 10.5.1 Preparatory formula . . . . . . . . . . . . . . . . . . . . . 163 10.5.2 Sensed-area of a triangle . . . . . . . . . . . . . . . . . . 163 10.5.3 A basic feature of sensed-area . . . . . . . . . . . . . . . 164 10.5.4 An identity for sensed-area . . . . . . . . . . . . . . . . . 164 10.6 Isometries as compositions . . . . . . . . . . . . . . . . . . . . 165 10.6.1 General isometries expressed as compositions of special isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 10.7 Orientation of a triple of non-collinear points . . . . . . . . . . 167 10.7.1 Orientation and how it is affected by isometries . . . . . 167 10.8 Sensed angles of triangles, the sine rule . . . . . . . . . . . . . 169 10.8.1 Extension of the sine rule to sensed angles of a triangle . 169 10.9 Some results on circles . . . . . . . . . . . . . . . . . . . . . . 171 10.9.1 A necessary condition to lie on a circle . . . . . . . . . . 171 10.9.2 A sufficient condition to lie on a circle . . . . . . . . . . . 172 10.9.3 Complex cross-ratio . . . . . . . . . . . . . . . . . . . . . 173 10.9.4 Ptolemy’s theorem, c.200A.D. . . . . . . . . . . . . . . . 173 10.10 Angles between lines . . . . . . . . . . . . . . . . . . . . . . . 175 10.10.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 175 10.10.2 Duo-sectors . . . . . . . . . . . . . . . . . . . . . . . . . . 175 10.10.3 Duo-angles . . . . . . . . . . . . . . . . . . . . . . . . . . 176 10.10.4 Duo-angles in standard position . . . . . . . . . . . . . . 176 10.10.5 Addition of duo-angles in standard position . . . . . . . . 178