Table of Contents
1 Basic Euclidean Geometry 1
1.1 Triangles 1
1.2 Similar Triangles 2
1.3 Congruent Triangles 3
1.4 Quadrilaterals 4
1.5 Polygons 5
1.6 Circles and Angles 6
1.7 Cyclic Quadrilaterals 11
1.8 Intersecting Chords 14
1.9 Inversion 16
1.9.1 Reflection in a line 16
1.9.2 Inversion in a circle 16
2 Trigonometry 21
2.1 Basic Definitions 21
2.2 Complementary Angles 22
2.3 Supplementary Angles 23
2.4 Extensions 24
2.5 Addition Formulae 26
2.6 Double Angle Formulae 27
2.7 Half Angle Formulae 28
2.8 The Sine Rule 30
2.9 The Cosine Rule 32
2.10 The Area of a Triangle 34
2.11 Inverse Trigonometric Functions 35
2.11.1 The Sine and Cosine Functions 35
2.11.2 The Tangent Function 38
2.12 De Moivre's Theorem 39
2.12.1 Roots of Unity and Euler's Formula 40
2.13 Hyperbolic Functions 44
2.14 Inverse Hyperbolic Functions 45
3 Concurrency and Collinearity 49
3.1 Menelaus' Theorem 49
3.2 Ceva's Theorem 52
3.3 Particular Concurrency Results 53
3.3.1 Medians 53
3.3.2 Altitudes 54
3.3.3 Perpendicular Bisectors of the Sides 55
3.3.4 Interior Angle Bisectors 56
3.3.5 Exterior Angle Bisectors 57
3.3.6 The Gergonne Point 58
3.4 The Wallace-Simson Line 59
3.5 The Nine Point Circle 60
3.6 The Euler Line 63
3.7 The Miquel Point 65
3.8 Joshua Klehr's Theorem and its Extensions 66
3.8.1 First extension 67
3.8.2 Second extension 74
4 Circumcircle, Inradius and Semiperimeter Formulae 77
4.1 Semi-perimeter Formulae 77
4.2 Half Angle Formulae 79
4.3 In-radius Formulae 81
4.4 Escribed Circles 82
5 Conic Sections 85
5.1 Review 85
5.2 Basic Definitions 85
5.3 The Parabola 86
5.4 The Ellipse 87
5.5 The Hyperbola 88
5.6 Parametric Equations 92
5.6.1 Parabolas 92
5.6.2 Ellipses 92
5.6.3 Hyperbolas 93
5.7 General Second Degree Equations 93
6 Constructions 95
6.1 Elementary Constructions 95
6.1.1 Bisection of a line segment 95
6.1.2 Bisection of an angle 97
6.1.3 Drawing a perpendicular to a line 98
6.1.4 Drawing a parallel line 99
6.1.5 Trisection of a line segment 99
6.2 Regular Polygons 100
6.2.1 Three Sides - equilateral triangle 101
6.2.2 Four Sides - square 101
6.2.3 Five Sides - regular pentagon 102
6.2.4 Seventeen Sides - regular 17-gon 104
6.2.5 Other polygons 105
6.2.6 The Carpenter's "Theorem" 106
6.2.7 The theory of the Carpenter's "Theorem" 107
6.2.8 The Draughtsman's "Theorem" 107
6.2.9 Theory of the Draughtsman's "Theorem" 108
6.2.10 The Master Carpenter's "Theorem" 109
6.2.11 Comparison of the three constructions 110
6.3 Real Numbers 111
6.3.1 Integral Multiples 111
6.3.2 Rational Multiples 111
6.3.3 Irrational Multiples 111
7 The Broken Chord Theorem 115
8 Cleavers and Splitters 119
8.1 Cleavers 119
8.2 Splitters 121
8.3 Collinearity 122
9 Remarkable Bisections 125
10 Miscellaneous Problems 129
10.1 Problems from Crux Mathematicorum with Mathematical Mayhem 152
11 Problems with Solutions 157
11.1 Problems from Crux Mathematicorum with Mathematical Mayhem 290
Index 305
