tl;dr I parodied my own book, download the new version here.
People often complain to me about how olympiad geometry is just about knowing a bunch of configurations or theorems. But it recently occurred to me that when you actually get down to its core, the amount of specific knowledge that you need to do well in olympiad geometry is very little. In fact I’m going to come out and say: I think all the theory of mainstream IMO geometry would not last even a one-semester college course.
So to stake my claim, and celebrate April Fool’s Day, I decided to actually do it. What would olympiad geometry look like if it was taught at a typical college? To find out, I present to you the course notes for:
Undergrad Math 011: a firsT yeaR coursE in geometrY
It’s 36 pages long, title page, preface, and index included. So, there you go. It is also the kind of thing I would never want to read, and the exercises are awful, but what does that matter?
(I initially wanted to post this file as an April Fool’s gag, but became concerned that one would not have to be too gullible to believe these were actual course notes and then attempt to work through them.)
Power Overwhelming 
This is really hilarious, but raises an important pedagogical question: What would you suggest to go in the opposite direction ?
A run-of-the-mill undergrad textbook on a run-of-the-mill undergrad topic would be of course very structurally similar to this EGMO parody and don’t create any sort of deep understanding or whatever. Sadly there’s no books analogous to EGMO which helps in sharpening undergrad problem solving. I don’t think just learning theory helps and standard Putnam prep books like “Putnam and Beyond” are too easy/obvious/useless for someone with High School Contest experience.
As a former High School Olympian who was kinda successful (also an ardent fan of the Analytic technique section of EGMO ;), I don’t have too much of a trouble with the A1-A4/B1-B4 Putnam problems but when it comes to say contests like Miklós Schweitzer or even A5/A6/B5/B6 my naive problem solving skills don’t suffices. What should be done to improve problem solving skills at undergrad level, to say ace most of the problem of Miklós Schweitzer ?
Also, from my limited contest experience (this may be very wrong for the “more advanced” problems :) there are only a handful of “bag of tricks” for solving most of the ISL till say 5 or 6 (there are exceptions like IMO 2011 Q2 but they’re rare I guess) and once you’re well aware with the tricks and their triggers (eg “problems with a lot of dumb choices” => probabilistic method, “a degree of freedom on a point moving on a conic” => animate the point and look at projective map, “lots of clines through a point” => inversion imba etc) and once have some reasonable technical manipulation skills, they stop becoming “unapproachable”. Is the scenario similar for undergrad contests too (i.e are there really a handful of tricks/triggers) resurfacing in most of the problems under some disguises ? Is there some books (again, like EGMO/PfTB) which teaches you the tricks well ?
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I can’t think of any resources I recommend off-hand. As much as I lament the lack of materials for high school olympiads, the college competition situation is far more scarce, there is not really even a community. (This may not be a bad thing.)
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The sad thing is this actually very good compared to how geometry is taught in high schools; “Angles” is actually quite reasonable compared to the stuff we saw in our school.
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As in, this is something I could see actually being recommended at some level.
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There’s also more meaningful commentary than an average geometry textbook; for example, “We have a rare application of Cartesian coordinates” when discussing the proof for Radical Axes.
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What book did you first learn geometry from, if you don’t mind my asking?
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It was split between several books and lectures. The one I remember starting with was Geometry Revisited, but I later learned a lot from e.g. lectures at MOP as well.
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The books by Landau on basic analysis, which I think were just transcriptions of his “satz-beweis” style lectures, are exactly like this, but without the exercises. HAKMEM and Bourbaki come to mind (for different reasons) as well. Some of the historically important 20th century books on geometry, even textbooks, are deliberately very light (I don’t remember whether any were fully weightless) in diagrams so as to maintain rigor without appealing to a diagram. Hilbert foundations of geometry, Mumford intro to algebraic geometry, Dieudonne’s books on analysis and his anti-Euclid approach to geometry, Artin’s “geometric algebra”.
So you’re in good company despite the parody. This would be a respectable approach for teaching or documenting the material if used with an implied standing exercise for every statement, to draw the corresponding diagram(s). Plane geometry is suited to minimalist exposition because the proofs are all short and relatively independent of each other (shallow not deep), and depth of understanding is attained by solving problems and experiencing the connections more than by discursive explanations. It’s certainly possible to do worse than thoughfully arranged lists of theorem-proof pairs followed by lists of exercises.
Another thing is that when old textbooks indulged in this style it was sometimes literally to save paper, e.g., in the high school books for poor countries. Solutions in the book were unheard of. A lot of Soviet books didn’t have an index.
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