Euclidean Geometry in Mathematical Olympiads

Euclidean Geometry in Mathematical Olympiads

With 248 Illustrations

Contents

Preface xi
Preliminaries xiii
0.1 The Structure of This Book. .xiii
0.2 Centers of a Triangle . .xiv
0.3 Other Notations and Conventions . .xv

I Fundamentals . .1

1 Angle Chasing 3
1.1 Triangles and Circles . .3
1.2 Cyclic Quadrilaterals . .6
1.3 The Orthic Triangle. .7
1.4 The Incenter/Excenter Lemma . .9
1.5 Directed Angles . .11
1.6 Tangents to Circles and Phantom Points . .15
1.7 Solving a Problem from the IMO Shortlist . .16
1.8 Problems . .18

2 Circles 23
2.1 Orientations of Similar Triangles. .23
2.2 Power of a Point . .24
2.3 The Radical Axis and Radical Center . .26
2.4 Coaxial Circles. .30
2.5 Revisiting Tangents: The Incenter . .31
2.6 The Excircles . .32
2.7 Example Problems . .34
2.8 Problems . .39

3 Lengths and Ratios 43
3.1 The Extended Law of Sines . .43
3.2 Ceva’s Theorem . .44
3.3 Directed Lengths and Menelaus’s Theorem . .46
3.4 The Centroid and the Medial Triangle . .48
3.5 Homothety and the Nine-Point Circle. .49
3.6 Example Problems . .51
3.7 Problems . .56

4 Assorted Configurations 59
4.1 Simson Lines Revisited . .59
4.2 Incircles and Excircles . .60
4.3 Midpoints of Altitudes . .62
4.4 Even More Incircle and Incenter Configurations . .63
4.5 Isogonal and Isotomic Conjugates. . .63
4.6 Symmedians . .64
4.7 Circles Inscribed in Segments . .66
4.8 Mixtilinear Incircles . .68
4.9 Problems . .70

II Analytic Techniques . .73

5 Computational Geometry 75
5.1 Cartesian Coordinates . .75
5.2 Areas . .77
5.3 Trigonometry . .79
5.4 Ptolemy’s Theorem . .81
5.5 Example Problems . .84
5.6 Problems . .91

6 Complex Numbers 95
6.1 What is a Complex Number? . .95
6.2 Adding and Multiplying Complex Numbers . .96
6.3 Collinearity and Perpendicularity . .99
6.4 The Unit Circle . .100
6.5 Useful Formulas . .103
6.6 Complex Incenter and Circumcenter . .106
6.7 Example Problems . .108
6.8 When (Not) to use Complex Numbers . .115
6.9 Problems . .115

7 Barycentric Coordinates 119
7.1 Definitions and First Theorems . .119
7.2 Centers of the Triangle . .122
7.3 Collinearity, Concurrence, and Points at Infinity . .123
7.4 Displacement Vectors . .126
7.5 A Demonstration from the IMO Shortlist . .129
7.6 Conway’s Notations . .132
7.7 Displacement Vectors, Continued . .133
7.8 More Examples . .135
7.9 When (Not) to Use Barycentric Coordinates. .142
7.10 Problems . .143