As some of you might have heard, I'm working on an olympiad geometry textbook which I hope will replace basically all the other existing resources in olympiad geometry.

I'm happy to announce that I've completed a zeroth draft; that is, I've said everything that I wanted to say and am going through with editing / polishing now. I've shown a draft briefly to a few people, with the most common reaction being "this is massive". But considering I've spent less than a year on it, I'm surprised no one beat me to doing this.

Better still, the zeroth draft is now official in the queue for the MAA Problem Books editorial board Towards the end of summer I'll hear what they think of it... I'm very excited!

In the meantime, here is the table of contents.

Preliminaries
Section 0.1 The Structure of This Book 1
Section 0.2 Notations and Centers 2

Part I Fundamentals 5
Chapter 1 Angle Chasing 7
Section 1.1 Vanilla Angle Chasing 7
Section 1.2 Cyclic Quadrilaterals 10
Section 1.3 The Orthic Triangle 12
Section 1.4 The Incenter/Excenter Lemma 15
Section 1.5 Directed Angles 16
Section 1.6 Tangents to Circles and Phantom Points 22
Section 1.7 Solving a Problem from the IMO Shortlist 24
Section 1.8 Problems 26

Chapter 2 Circles 31
Section 2.1 Orientations of Similar Triangles 31
Section 2.2 Power of a Point 32
Section 2.3 The Radical Axis and Radical Center 35
Section 2.4 Coaxal Circles 40
Section 2.5 Revisiting Tangents: The Incenter 41
Section 2.6 The Excircles 42
Section 2.7 Example Problems 44
Section 2.8 Problems 51

Chapter 3 Lengths and Ratios 55
Section 3.1 The Extended Law of Sines 55
Section 3.2 Ceva's Theorem 56
Section 3.3 Directed Ratios of Lengths and Menelaus' Theorem 59
Section 3.4 The Centroid and the Medial Triangle 61
Section 3.5 Homothety and The Nine-Point Circle 62
Section 3.6 Example Problems 65
Section 3.7 Problems 70

Chapter 4 Assorted Configurations 75
Section 4.1 Simson Lines Revisited 75
Section 4.2 Incircles and Excircles 76
Section 4.3 Midpoints of Altitudes 78
Section 4.4 Circles Inscribed in Segments 79
Section 4.5 More Incircle/Incenter Configurations 82
Section 4.6 Isogonal and Isotomic Conjugates 83
Section 4.7 Symmedians 84
Section 4.8 Problems 86


Part II Dark Arts 89
Chapter 5 Computational Geometry 91
Section 5.1 Cartesian Coordinates 91
Section 5.2 Areas 93
Section 5.3 Trigonometry 96
Section 5.4 Ptolemy's Theorem 98
Section 5.5 Example Problems 102
Section 5.6 Problems 109

Chapter 6 Complex Numbers 113
Section 6.1 What is a Complex Number? 113
Section 6.2 Adding and Multiplying Complex Numbers 115
Section 6.3 Collinearity and Perpendicularity 118
Section 6.4 The Unit Circle 120
Section 6.5 Useful Formulas 123
Section 6.6 Complex Incenter and Circumcenter 126
Section 6.7 Example Problems 129
Section 6.8 When to Not Use Complex Numbers 137
Section 6.9 Problems 138

Chapter 7 Barycentric Coordinates 141
Section 7.1 Definitions and First Theorems 141
Section 7.2 Centers of the triangle 145
Section 7.3 Collinearity, Concurrence, and Points at Infinity 146
Section 7.4 Displacement Vectors 149
Section 7.5 A Demonstration from the IMO Shortlist 152
Section 7.6 Conway's Notations 157
Section 7.7 Displacement Vectors, Continued 158
Section 7.8 More Examples 160
Section 7.9 When to Not Use Barycentric Coordinates 169
Section 7.10 Problems 170


Part III Farther from Kansas 175
Chapter 8 Inversion 177
Section 8.1 Circles are Lines 177
Section 8.2 Where Do Clines Go? 179
Section 8.3 An Example from the USAMO 183
Section 8.4 Overlays and Orthogonal Circles 186
Section 8.5 More Overlays 189
Section 8.6 The Inversion Distance Formula 191
Section 8.7 More Example Problems 191
Section 8.8 When to Invert 197
Section 8.9 Problems 198

Chapter 9 Projective Geometry 201
Section 9.1 Completing the Plane 201
Section 9.2 Cross Ratios 203
Section 9.3 Harmonic Bundles 206
Section 9.4 Apollonian Circles 210
Section 9.5 Poles/Polars and Brokard's Theorem 212
Section 9.6 Pascal's Theorem 216
Section 9.7 Projective Transformations 218
Section 9.8 Three Solutions to an Example Problem 221
Section 9.9 Problems 224

Chapter 10 Complete Quadrilaterals 229
Section 10.1 Spiral Similarity 230
Section 10.2 Miquel's Theorem 232
Section 10.3 The Gauss-Bodenmiller Theorem 233
Section 10.4 More Properties of General Miquel Points 235
Section 10.5 Miquel Points of Cyclic Quadrilaterals 236
Section 10.6 Example Problems 238
Section 10.7 Problems 241

Chapter 11 Personal Favorites 245
Section 11.1 Problems 245
Section 11.2 Solutions 249

Part IV Appendices 267
Chapter A An Ounce of Linear Algebra 269
Section A.1 Matrices and Determinants 269
Section A.2 Cramer's Rule 271
Section A.3 Vectors and the Dot Product 272
Chapter B Hints 277
Chapter C Lists 307
Section C.1 List of Contests and Abbreviations 307
Section C.2 List of Theorems and Configurations 309
Section C.3 List of Contest Problems 313