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 with proof experience 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 material#
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. Here is a freely available subset of the book:
- Table of contents.
- Chapter 2 (Circles) and Chapter 8 (Inversion) (available for free).
- Chapters 1-3 on Google Books preview.
- Parody version of entire book: check out Undergraduate Math 011: a firsT yeaR coursE in geometrY, made for April Fool’s Day 2019.
Errata (aka selected solutions to Problem 11.0)#
Here is a list of all mistakes that I am aware of. Most mistakes are harmless, but a few of the mistakes are significant; the important ones are reproduced here, too.
- On page 12, Theorem 1.22, the four points could also be collinear.
- On page 76, Theorem 5.1 is missing a factor of $\frac12$.
- 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 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 159, in Lemma 8.16, change “fixes $B$ and $C$” to “swaps $B$ and $C$”.
- On page 184, in Theorem 9.33, uniqueness is not true for (b) or (c). The last sentence is also wrong as written. The correct sentence is: if the circumcircle of a cyclic quadrilateral is preserved, then so is the cross ratio of the cyclic quadrilateral.
- On page 274, the solution 8.37 is wrong, it assumes $AB$ passes through the center of $\omega_2$.
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 places to look:
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).
There is a user-created forum which has a link to discussion for every problem.
I included sources for as many of the problems as possible, so looking up the corresponding thread on Art of Problem Solving’s Contest Index will generally yield a solution to any sourced problem. (The search function can work too, but is usually less helpful.)
If all of the above fail, try posting on the AoPS forums directly.
Of course there are other good geometry textbooks too. A few that I recommend are:
- 106, 107, and 110 Geometry Problems
- A Beautiful Journey Through Olympiad Geometry, which is pay-what-you-want!
- Lemmas in Olympiad Geometry
- UKMT Plane Euclidean Geometry, but consider starting from Chapter 3
See also advice from Geoff Smith on geometry.