# Euclidean Geometry in Mathematical Olympiads (EGMO)

**You can get a hard copy from Amazon or MAA.
You can also get a PDF.**

*Euclidean Geometry in Mathematical Olympiads* (often abbreviated *EGMO*,
despite an olympiad having the same name)
is a comprehensive problem-solving book in Euclidean geometry.
It was written for competitive students training for national
or international mathematical olympiads.
However it has no prerequisites other than a good deal of courage:
any student who is interested in the subject matter
should be able to follow the exposition.

The book contains a selection of about 300 problems from around the world and is accompanied by about 250 figures.

### Prerequisites and sample chapters

There are essentially no geometry prerequisites;
EGMO is **entirely self-contained**.
(This was one of the design goals.)
The main limiting factor is instead the ability to **read proofs**;
as long as you can follow mathematical arguments,
then you should be able to follow the exposition
even if you don't know any geometrical theorems.

- Chapter 2: Circles. Power of a point, radical axis.
- Chapter 8: Inversion. Inversion around a circle.
- Chapters 1-3 on Google Books preview.

### Errata (aka selected solutions to Problem 11.0)

Here are all the mistakes that I am aware of. If you find another one, please let me know by email.

- On
**page xi**, append a comma after "lectures at MOP". - On
**page xiv**, second bullet, the phrase "intersection the medians" is missing "of". - On
**page 7**, beneath Figure 1.3A, change "orthocenter of $H$" to "orthocenter of $ABC$". - On
**page 16**, in problem 1.33, change "$\angle KC=90^{\circ}$" to "$\angle KCB=90^{\circ}$". - On
**page 18**, in problem 1.37, delete the word "again" in the definition of $Q$. - On
**page 19**, in problem 1.45, change "ray $BI$" to "the $\angle B$-bisector". - On
**page 24**, in Problem 2.2, change $\measuredangle BCA = \measuredangle YZX$ to $\measuredangle ABC = \measuredangle XYZ$. - On
**page 29**, in Theorem 2.9, excise "of Intersecting Circles" from the theorem name. - On
**page 29**, in proof of Theorem 2.9, change both $>0$'s to $<0$'s. - On
**page 30**, bottom of page, "coaxal" should be "coaxial". - On
**page 31**, in Lemma 2.13, the circles can also be tangent to one another at $X$ (i.e. the intersection is counted with multiplicity). - On
**page 34**, in solution to Example 2.21, in "we already know that that lines $PQ$, $RS$, and $XY$ concur at a point $X$", the extraneous "that" and "$X$" should both be deleted. - On
**page 34**, end of third paragraph after Example 2.21, change $O_1O_3$ to $\overline{O_2O_3}$. - On
**page 35**, in the second set of aligned equations, change $O_2X^2$ to $OO_2^2$. - On
**page 40**, the lines my also be pairwise parallel. - On
**page 51**, before Lemma, in "this circles is called the nine-point circle", change "circles" to "circle". - On
**page 65**, Problem 4.25, change $\frac{BM}{MC}$ to $\frac{CM}{MB}$. - On
**page 67**, in Lemma 4.33, change the second $\omega$ to $\Omega$. - On
**page 76**, Theorem 5.1 is missing a factor of $\frac12$. - On
**page 92**, in Problem 5.23, when defining point $G$, line $HE$ should intersect $\Gamma_1$, not $\Gamma_2$. Also, "interest" should be "intersect" in the first line. - On
**page 101**, in the proof of Example 6.10, Lemma 6.3 should be Lemma 6.5. Also, $\frac{xa}{bc}$ should be $\frac{bc}{xa}$ (two changes). - On
**page 107**, the proof of part (a) of the theorem has several issues, and is probably best to just ignore. (The result is still true, and the proof of part (b) is correct.) - On
**page 105**, in Problem 6.20 change "Theorem 6.16" to "Theorem 6.15". - On
**page 111**, the last expression should actually be negated. - On
**page 112**, in the second displayed line, change $y^2+x^2z/y$ to $-y^2+x^2z/y$. In the fourth, change the second $y^2/z^2$ to $z^2/y^2$. In the ninth, change the second $y^2/z^2$ to $z^2/y^2$. - On
**page 113**, in the definition of $M_2$, change $DH_A$ to $AH_A$. - On
**page 114**, Solution 6.27, the $a$ in the numerator of $a'$ should be $\bar a$. Follow through with the rest of the solution. - On
**page 132**, in the proof of Example 7.20, change $D = (0, 0, 1)$ to $C = (0, 0, 1)$. - On
**page 133**, Proposition 7.21, last display, change $S_a$ to $S_A$. - On
**page 136**, very top, $c=AE$ should be $c=AC$. - On
**page 138**, eighth line from top, change $AD:AC$ to $AD:CD$. - On
**page 139**, Solution 7.29, change the first display to $0 = c^2(t-1) + (a^2-b^2) \implies t = \frac{c^2+b^2-a^2}{c^2}$. - On
**page 140**, in the second display, $x+y$ should be $x-y$. - On
**page 142**, very top, in $a^2=S_{AB}+S_{AC}$, change LHS to $a^2S_A$. Also, the rest of the solution is wrong, since a factor of two is dropped in the first display. - On
**page 145**, Problem 7.44, change the angle condition to $\measuredangle BC_1A_1 = \measuredangle ACB = \measuredangle B_1C_1A$. - On
**page 146**, Problem 7.52, change $\angle PCB$ to $\angle PBC$. - On
**page 149**, second paragraph of 8.1, change "three ordinary points" to "three noncollinear ordinary points". - On
**page 151**, at start of 8.2, in "simplest example is a just a line", delete the extra "a". - On
**page 151**, immediately before figure, add a period after 8.2A. - On
**page 155**, first bullet, delete the extra comma. - On
**page 159**, second line of Example 8.15, change "tangent to $\omega$ at $T$" to "tangent to $\Omega$ at $K$". Also, in the second paragraph of the proof, change the last $\Gamma$ to $\Omega$. - On
**page 159**, in Lemma 8.16, change "fixes $B$ and $C$" to "swaps $B$ and $C$". - On
**page 162**, item 5 of list, change $G_1$ to $G_1^\ast$. Also in the first paragraph of the solution. - On
**page 163**, switch $C^\ast$ and $D^\ast$ in the diagram. - On
**page 163**, step 3, change $BS^\ast$ to $BC^\ast$. - On
**page 164**, switch $S^\ast$ and $R^\ast$ in the diagram. - On
**page 171**, Theorem 9.2, change "$\overline{AB}$ and $\overline{XY}$" to "segments $AB$ and $XY$". - On
**page 173**, in Problem 9.4, add $k \neq 0$. - On
**page 174-175**, in names of Lemmas 9.11-9.12, change "Induces" to "Induce". - On
**page 175**, "directed form*of*Ceva's theorem". - On
**page 176**, in Problem 9.14, delete "and Lemma 9.18" (and "proofs" to "proof"). - On
**page 178**, in the proof of Theorem 9.19, change $\angle CAY = \angle YBC$ to $\angle ACY = \angle YCB$. - On
**page 181**, in Lemma 9.27, change "pole" to "polar" (two instances). - On
**page 184**, the last sentence of Theorem 9.33 is wrong as written. The correct statement is: if the*circumcircle*of a cyclic quadrilateral is preserved, then so is the cross ratio of the cyclic quadrilateral. - On
**page 184**, in Example 9.34, swap the definitions of $P$, $Q$. - On
**page 187**, in solution to 9.38, $I_A$ is the $A$-excenter (of course). - On
**page 189**, in Solution 1, $T$ should be $\overline{AA} \cap \overline{CR}$. - On
**page 190**, immediately before problems, $P$ is the point at infinity along $\overline{AC}$ instead. - On
**page 193**, Problem 9.58, delete the last "again" in definitions of $P$, $Q$. - On
**page 200**, in third sentence of proof, add "of" after "radical center". - On
**page 202**, part (a), the latter four circles should be $(PAB)$, $(PCD)$, $(QAD)$, $(QBC)$. - On
**page 206**, problem 10.23, change "IMO 2005/2" to "IMO 2005/5". - On
**page 209**, problem 11.6, change "circumcenter" to "circumcircle". - On
**page 210**, problem 11.10, change "$PA$, $PB$, $PC$" to "$AP$, $BP$, $CP$". - On
**page 216**, in phrase "third column from the first column", change "first" to "second". - On
**page 223**, hint 77, replace "$\measuredangle CMN = \measuredangle BMN$" with "$\measuredangle CNM = \measuredangle BNM$". - On
**page 223**, hint 96 is wrong (see page 274 correction below). - On
**page 223**, hint 101, change "Lemma 1.17" to "Lemma 1.18". - On
**page 224**, hint 112 is wrong (see page 274 correction below). - On
**page 225**, hint 164 should be: let $X = \overline{EF} \cap \overline{BC}$ and $Y = \overline{AD} \cap \overline{EF}$, show $(X,Y;E,F) = -1$. - On
**page 226**, hint 185, replace "$\angle WXY = 40^{\circ}$" with "$\angle XZY = 40^{\circ}$". - On
**page 226**, hint 197 is wrong (see correction to Solution 6.45 below). - On
**page 227**, hint 252, replace $O$ with $O^\ast$. - On
**page 228**, hint 257, change "as do $C$ and $F$" to "as do $B$ and $D$". - On
**page 229**, hint 303, change first two instances of $de$ to $bc$. - On
**page 229**, hint 316, change "the circle is" to "the circle is centered at". - On
**page 230**, hint 355, change $AC$ to $AB$. - On
**page 230**, hint 362, $X$ should instead be the second intersection of $QI$ with the circumcircle. - On
**page 231**, hint 373, replace "$90^{\circ}-A$" with "$90^{\circ}-B$". - On
**page 231**, hint 383, change "trigonemetric" to "trigonometric". - On
**page 231**, hint 393, delete "$M=(0:1:1)$" and change the later "$L$" to "$M$". - On
**page 238**, hint 649 is wrong (see correction to Solution 6.45 below). - On
**page 238**, hint 653 is missing a period. - On
**page 238**, hint 666, change $\frac{XB}{XA}$ to $\frac{XA}{XB}$ - On
**page 246**, in Solution 1.50, one also needs to check $A$, $P$, $W$ are collinear. Thus, add the remark $\measuredangle NPA = \measuredangle NMA = \measuredangle NMC = \measuredangle BNC = \measuredangle BNW = \measuredangle NPW$. - On
**page 246**, in Solution 3.17, change $X_1Y_1$ to $X'Y'$ at the bottom (twice). - On
**page 259**, in Solution 6.36, change $ab-1$ to $ab+1$ and follow through. - On
**page 262**, there is a minus sign missing on fifth display. Carrying through, we actually get $x = h - \frac{bc(a+b+c)}{b^2+bc+c^2}$ instead, id est we want $x/h = 1 - \frac{bc}{b^2+bc+c^2}$ to be real. - On
**page 265**, in Solution 4.52, last display, chaneg $\measuredangle BED + \measuredangle BDE$ to $\measuredangle BED + \measuredangle DBE$. - On
**page 265**, in Solution 5.21, change $-BI \cdot CI \cdot \sqrt2$ to $+ BI \cdot CI \cdot \sqrt2$. Thus, in last display change $\sqrt2$ to $-\sqrt2$. - On
**page 265**, in Solution 6.45, the solution proves $|(a-b)(c-e)(d-f)|=|(d-e)(f-b)(a-c)|$. It should instead prove $|(b-c)(a-e)(f-d)|=|(c-a)(e-f)(d-b)|$, which is the same up to permutation of point labels. - On
**page 271**, the solution 7.52, second-to-last display, the expressions actually equal $-\gamma$, not $\gamma$. - On
**page 273**, Solution 8.31, swap $A$ and $C$ everywhere. - On
**page 274**, the solution 8.37 is wrong, it assumes $AB$ passes through the center of $\omega_2$. - On
**page 276**, in Solution 9.47, change $(A,X;B,C)$ to $(A,X;C,B)$. - On
**page 281**, in Solution 10.26, the last line, change $\measuredangle HMN$ to $\measuredangle HNM$. - On
**page 282**, in Solution 10.29, change $(P,E;X,Y)$ to $(F,E;X,Y)$ in last paragraph. - On
**page 296**, in Solution 11.18, very very end, change $2t$ to $(b^2+c^2)t$. - On
**page 297**, in Solution 11.18, the first display should read $-a^2v + b^2w + c^2v = (b^2+c^2)t + (abc)^2 - (ab)^2S_B - a^2t = S_A ((ab)^2 + 2t)$. The next display should be $X' = \left( a^2vw : S_A(c^2S_C+t)((ab)^2+2t) : S_A(b^2S_B+t)((ac)^+2t) \right)$. Similarly for $Y'$ and $Z'$. - On
**page 297**, in Solution 11.19, start of last paragraph, change $DBC_1$ to $DB_1C_1$. - On
**page 305**, the link to reference [3] is broken. Thankfully, it's on this website!

### Table of Contents

Here is the most recent table of contents.

**Preliminaries**

Section 0.1 The Structure of This Book

Section 0.2 Notations and Centers

## Part I Fundamentals

**Chapter 1 Angle Chasing**

Section 1.1 Triangles and Circles

Section 1.2 Cyclic Quadrilaterals

Section 1.3 The Orthic Triangle

Section 1.4 The Incenter/Excenter Lemma

Section 1.5 Directed Angles

Section 1.6 Tangents to Circles and Phantom Points

Section 1.7 Solving a Problem from the IMO Shortlist

Section 1.8 Problems

**Chapter 2 Circles**

Section 2.1 Orientations of Similar Triangles

Section 2.2 Power of a Point

Section 2.3 The Radical Axis and Radical Center

Section 2.4 Coaxal Circles

Section 2.5 Revisiting Tangents: The Incenter

Section 2.6 The Excircles

Section 2.7 Example Problems

Section 2.8 Problems

**Chapter 3 Lengths and Ratios**

Section 3.1 The Extended Law of Sines

Section 3.2 Ceva's Theorem

Section 3.3 Directed Ratios of Lengths and Menelaus' Theorem

Section 3.4 The Centroid and the Medial Triangle

Section 3.5 Homothety and The Nine

Point Circle

Section 3.6 Example Problems

Section 3.7 Problems

**Chapter 4 Assorted Configurations**

Section 4.1 Simson Lines Revisited

Section 4.2 Incircles and Excircles

Section 4.3 Midpoints of Altitudes

Section 4.4 Circles Inscribed in Segments

Section 4.5 More Incircle/Incenter Configurations

Section 4.6 Isogonal and Isotomic Conjugates

Section 4.7 Symmedians

Section 4.8 More on Mixtilinear Incircles

Section 4.9 Problems

## Part II Analytic Techniques

**Chapter 5 Computational Geometry**

Section 5.1 Cartesian Coordinates

Section 5.2 Areas

Section 5.3 Trigonometry

Section 5.4 Ptolemy's Theorem

Section 5.5 Example Problems

Section 5.6 Problems

**Chapter 6 Complex Numbers**

Section 6.1 What is a Complex Number?

Section 6.2 Adding and Multiplying Complex Numbers

Section 6.3 Collinearity and Perpendicularity

Section 6.4 The Unit Circle

Section 6.5 Useful Formulas

Section 6.6 Complex Incenter and Circumcenter

Section 6.7 Example Problems

Section 6.8 When to Not Use Complex Numbers

Section 6.9 Problems

**Chapter 7 Barycentric Coordinates**

Section 7.1 Definitions and First Theorems

Section 7.2 Centers of the triangle

Section 7.3 Collinearity, Concurrence, and Points at Infinity

Section 7.4 Displacement Vectors

Section 7.5 A Demonstration from the IMO Shortlist

Section 7.6 Conway's Notations

Section 7.7 Displacement Vectors, Continued

Section 7.8 More Examples

Section 7.9 When to Not Use Barycentric Coordinates

Section 7.10 Problems

## Part III Farther from Kansas

**Chapter 8 Inversion**

Section 8.1 Circles are Lines

Section 8.2 Where Do Clines Go?

Section 8.3 An Example from the USAMO

Section 8.4 Overlays and Orthogonal Circles

Section 8.5 More Overlays

Section 8.6 The Inversion Distance Formula

Section 8.7 More Example Problems

Section 8.8 When to Invert

Section 8.9 Problems

**Chapter 9 Projective Geometry**

Section 9.1 Completing the Plane

Section 9.2 Cross Ratios

Section 9.3 Harmonic Bundles

Section 9.4 Apollonian Circles

Section 9.5 Poles/Polars and Brokard's Theorem

Section 9.6 Pascal's Theorem

Section 9.7 Projective Transformations

Section 9.8 Three Solutions to an Example Problem

Section 9.9 Problems

**Chapter 10 Complete Quadrilaterals**

Section 10.1 Spiral Similarity

Section 10.2 Miquel's Theorem

Section 10.3 The Gauss-Bodenmiller Theorem

Section 10.4 More Properties of General Miquel Points

Section 10.5 Miquel Points of Cyclic Quadrilaterals

Section 10.6 Example Problems

Section 10.7 Problems

**Chapter 11 Personal Favorites**

Section 11.1 Problems

Section 11.2 Solutions

## Part IV Appendices

**Chapter A An Ounce of Linear Algebra**

Section A.1 Matrices and Determinants

Section A.2 Cramer's Rule

Section A.3 Vectors and the Dot Product
**Chapter B Hints**
**Chapter C Selected Solutions**
**Chapter D List of Contests and Abbreviations**