# Euclidean Geometry in Mathematical Olympiads (EGMO)

You can get a hard copy from Amazon or the AMS. You can also purchase 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.

I wrote this textbook while serving time as a high school clerk. You can read about the publication process.

### 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.

### On Solutions to Problems

I included solutions to about a quarter of the practice problems in the back of the textbook (it was impossible to include more for space reasons). If you are stuck on a problem for which no solution is provided, here are some possible hints:

1. Each problem in the text has one or more hints, the intention is that the union of all the hints should form an outline of the complete solution (even if the details are omitted).

2. I included sources for as many of the problems as possible, so looking up the corresponding thread on the AoPS Contest Index will generally yield a solution to any sourced problem. (The search function can work too, but is often less helpful.)

3. If both of the above fail, try posting on the AoPS forums directly.

### 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 12, Proposition 1.24, the isosceles triangle condition holds only when $A$, $B$, $C$ are not collinear.
• 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 20, in problem 1.47, change "Let $ABC$ be triangle" to "Let $ABC$ be a triangle".
• 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 33, in Lemma 2.20, swap the definitions of $X$ and $D$. (The problem is technically correct as stated, but it should be consistent with Figure 2.6A.)
• 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 37, Example 2.23, change 2010 to 2011 in the source.
• On page 40, Problem 2.29, change "six points" to "the six points".
• On page 40, Problem 2.30, the lines may also be pairwise parallel.
• On page 48, Theorem 3.8, the lines may also be pairwise parallel.
• On page 49, third to last line, lengths are multiplied by $|k|$.
• On page 51, before Lemma, in "this circles is called the nine-point circle", change "circles" to "circle".
• On page 54, third paragraph, change "pick let" to "let".
• 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 105, in Problem 6.20 change "Theorem 6.16" to "Theorem 6.15".
• 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 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 121, "his idea" to "this idea".
• On page 132, in the proof of Example 7.20, change $C = (0, 0, 1)$ to $D = (0, 0, 1)$.
• 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 144, second line of 7.42, change "tangency points" to "tangency point".
• On page 149, second paragraph of 8.1, change "three ordinary points" to "three noncollinear ordinary points".
• On page 150, in Lemma 8.1, rplace "tangents from $A^\ast$" with "tangency points from $A^\ast". • On page 151, at start of 8.2, in "simplest example is a just a line", delete the extra "a". • On page 151, immediately before figure, add a period after 8.2A. • On page 155, first bullet, delete the extra comma. • On page 159, second line of Example 8.15, change "tangent to$\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$. • On page 179, top of page, first sentence,$\ell$is allowed to pass through$O$. • On page 181, in Lemma 9.27, change "pole" to "polar" (two instances). • On page 184, the last sentence of Theorem 9.33 is wrong as written. The correct statement is: if the circumcircle of a cyclic quadrilateral is preserved, then so is the cross ratio of the cyclic quadrilateral. • On page 184, in Example 9.34, swap the definitions of$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)$. • On page 206, problem 10.23, change "IMO 2005/2" to "IMO 2005/5", and "lie of the sides" to "lie on the sides". • On page 209, problem 11.6, change "circumcenter" to "circumcircle". • On page 210, problem 11.8, assume$AB \neq AC$. • On page 210, problem 11.10, change "$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$". • On page 223, hint 96 is wrong (see page 274 correction below). • On page 224, hint 112 is wrong (see page 274 correction below). • On page 225, hint 164 should be: let$X = \overline{EF} \cap \overline{BC}$and$Y = \overline{AD} \cap \overline{EF}$, show$(X,Y;E,F) = -1$. • On page 226, hint 183, replace "$ABCD$" with "$ABXY$". • On page 226, hint 185, replace "$\angle WXY = 40^{\circ}$" with "$\angle XZY = 40^{\circ}$". • On page 226, hint 193, replace "major arc$BC$" with "arc$\widehat{BAC}$". • On page 226, hint 197 is wrong (see correction to Solution 6.45 below). • On page 226, hint 218, change$AC$and$BD$to$AB$and$XY$. • On page 227, hint 231, replace$P,C,D$with$P,A,B$; replace$ABCD$with$AXBY$. • On page 227, hint 252, replace$O$with$O^\ast$. • On page 227, hint 255, replace$C$with$C^\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$. • On page 229, hint 316, change "the circle is" to "the circle is centered at". • On page 229, hint 321, delete "applies". • On page 230, hint 355, change$AC$to$AB$. • On page 230, hint 340, replace$P$with$\overline{A_1A_2} \cap \overline{C_1C_2}$. • On page 230, hint 362,$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 380, replace "$\sin 30^{\circ}$" with "$i \sin 30^{\circ}$". • On page 231, hint 383, change "trigonemetric" to "trigonometric". • On page 231, hint 389, change$L^\ast$to$A^\ast$. • 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 243, 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 249, Solution 3.29, add a remark that$(AMN)$is tangent to$(ABC)$. • On page 250, in Solution 4.50, second-to-last sentence, change "circumcircle" to "circumcenter". • On page 252, in Solution 4.52, last display, change$\measuredangle BED + \measuredangle BDE$to$\measuredangle BED + \measuredangle DBE$. • On page 252, 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 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 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 268, at the end of Solution 7.42, the end of the display should be$-2S_A+2bc$. • On page 268, Solution 7.44,$A_0$should be$(0:a-uc/a:uc/a)$in the second display. • On page 271, the solution 7.52, second-to-last display, the expressions actually equal$-\gamma$, not$\gamma$. • On pages 273-274, 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 277, in Solution 9.50, change$\overline{CG} \cap \overline{BE}$to$\overline{CG'} \cap \overline{BE}$. • 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$X' = \left( a^2vw : S_A(c^2S_C+t)((ab)^2+2t) : S_A(b^2S_B+t)((ac)^2+2t) \right)$. 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!
• On page 305, reference [10], change "Geogebral" to "Geogebra".

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

## 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

Last modified: Wed Mar 28 10:40:29 2018