Exploring Euclidean Geometry


Originally published on May 19, 2020.

One of the most common mistakes beginners to the math competition scene make is thinking they solved a problem and then realizing they didn't actually solve it correctly. This is especially prevalent in beginning olympiads, such as BAMO, USAMTS, and other similar competitions, but this also happens quite commonly with computational contests. [1]

So how do you prevent this from happening? First and foremost, by getting more experience by solving more problems. You're much less likely to think you solved a problem while not making any significant progress if you get better at noticing things and actually making progress.

Geometry Specifics

Geometry is one of the subjects most prone to fakesolving. [2] My rule of thumb for checking if you've actually solved a problem: Have you actually found anything meaningful? For most olympiad geometry problems, you can't just randomly chase some angles and hope to finish. You actually have to notice that certain angles are equal, certain triangles are similar, certain homotheties can be taken, or whatever. [3] Here's a proper subset of the questions you should be asking yourself whenever you feel like your solution to a problem might have been sketchy:

You don't have to explicitly ask yourself each of these questions. At some point in time you'll be able to internalize it, and instead you'll just get a gut feeling that something went wrong if you haven't really done anything. [4]


[1] To clarify: This isn't because they sillied, missed an edge case, or whatever - it's because they actually didn't make any significant progress in the problem (but they think they did).

[2] For Algebra, Combo, and NT, the advice is really similar - "finding a point" just becomes "finding a bijection," "finding the equality case," "finding a contradiction," et cetera. It's just much easier to formalize with geometry, which is one of the reasons I used it as an example.

[3] Actually, this is just the list of what to look for/how to tell you're making progress in the first place. You'll find that the list of "have I actually solved this problem" is very similar to "how should I solve this problem?"

[4] This is also why it's super important to solve problems and actually check your answers/compare your solutions to other people's - so you can establish a quick and accurate feedback loop that iterates many times.