Exploring Euclidean Geometry

Review for Introduction to Geometry (AoPS)

This is an adaptation of a review for Goodreads. Originally published May 21, 2020.

I didn't rate this book poorly [1] because it's a bad book, by any means. But this is probably one of the most overrated math textbooks out there. It's not bad, it's in fact far above average and I would definitely prefer this over any school textbook. It has competition problems too, which aligns with my own philosophy of "you can only remember hard stuff by doing hard problems." It's also a decent place to pick up the basics, but the book's structure doesn't lend itself to it. In fact, to use this book optimally, you probably have to actively go against the design of this book.

The pacing of the book is very slow and it's very hard to slog through. Unless you're a complete beginner to competition geometry, you're probably better off just picking stuff up from doing problems. That's not to say you can't pick up problems from doing them in the book - the book has a quite decent selection of them. But actually reading the text should be treated as a last resort. [2]

This might be due to personal preference, but the structuring of the book is terrible. Things that do not deserve to be chapters are artificially inflated in length, and chapter labeling tries to be clever at the expense of clarity. Specific complaint: "Special Parts of Triangles" (otherwise known as "Triangle Centers" or "Cevians," both of those would've been fine). And the book takes up far too much time and space with stuff that really needs to be stated succinctly: "A perpendicular bisector of a segment is the locus of points equidistant from its endpoints," and "An angle bisector of two lines is the locus of points equidistant from those two lines." The proofs are also one-liners - "Notice triangle X is congruent to triangle Y." And this kind of stuff needs to be made succinct to stand out. The use of one-way proofs instead of just a biconditional argument that encompasses the whole thing is also terribly inefficient and teaches bad habits. (There are some USAMO/IMO problems where proving one direction and proving the other are actually two separate non-trivial tasks, but this is rare and being able to biconditionally prove something with one sweeping argument is still good.) And worst of all, there's no way to tell what the most important facts are: The four triangle centers and the four cevians. (Perpendicular bisector isn't a cevian, but my point stands.) In a good manuscript I should be able to, without effort, tell what the triangle centers are, and the gist of their existence/property proofs. For this one it's very hard to tell.

The book also treats people like idiots, maybe because it's just "geometry standard" (even though nobody actually good at math cares what the "standard" is). The book insists on saying things like "SAS congruence" or "HL congruence" or whatever. Stuff that can be and should be one-liners aren't. What this book doesn't seem to understand is this: You don't need to spell out all of the trivial details. Just leave enough for the reader to be able to (somewhat) quickly pick it up on their own. Well-known theorems are presented as "problems," so the fact that they're well known and the fact that they aren't novel/interesting, just necessary to build upon, is not communicated at all to the reader.

Even though I harp on this book, I realize that the reason it's so praised is in part because it deserves to be. It's also very hard to write a good introduction to geometry - my attempts have also ended up being much more suited to people who already have some experience (though I do try to make my manuscript beginner friendly)! So I think the world is made a much better place with this book rather than without, because this book fills a void that really needs to be filled. But it's not the holy grail and it's not the only way to get good at Geometry.

What do I recommend instead? I recommend doing lots of hard 2D geometry problems. You'll notice a pattern - at least in the AMC/AIME, it's not proving some triangle is similar to some other triangle or some angle is congruent to some other angle. In fact that part is usually trivial. The hard part is actually noticing what you need to prove, and convincing yourself that it's true instead of convincing yourself of some other stupid things. Consider PUMAC 2016/G7 - it's a very hard problem (which is why it's the second-to-last problem, after all), but not because we use some very obscure theorem that you have to read 100 textbooks to memorize. It's because the similar triangle (sorry, spoilers!) is very well hidden. What's even more amazing is that problems like these are one-liners. The gist of the solution to really, really hard geometry problems can just be summarized in 4,5 or 6 lines, which is really impressive considering how long the solutions to some harder Algebra, Combinatorics, and Number Theory problems are.

You might also hear people say that geometry is the most theory-heavy subject. This is true, but it's in the context of the IMO. This is not the case with the material AoPS Geometry is covering. There is lots of theory in computational geometry, but AoPS Geometry doesn't get a pass here; the material it presents is artificially inflated to be longer than it needs to be, and the book doesn't include the real meat despite how easy it would be for them to include it. Take Radical Axes, for example - at least tell everyone that the common chord of two circles bisects their common external tangent!

In summary: This book is a good starting point for beginners, but it should not be treated as the holy grail and you should try to get away with using it as little as possible.


[1] I gave it a rating of 3/5.

[2] This is what my opinion is in general, but in this case I feel it much more strongly.