Here is my commentary for the 2019 International Math Olympiad, consisting of pictures and some political statements about the problem. Summary This year’s USA delegation consisted of leader Po-Shen Loh and deputy leader Yang Liu. The USA scored 227 points, … Continue reading
While making preparations for this year’s MOP, I imagined to myself what I would say on orientation night if I was director of the camp, and came up with the following speech. I thought it might be nice to share on this blog. Of course, it represents my own views, not the actual views of MOP or MAA. And since I am not actually director of MOP, the speech was never given.
People sometimes ask me, why do we have international students at MOP? Doesn’t that mean we’re training teams from other countries? So I want to make this clear now: the purpose of MOP is not to train and select future IMO teams.
I know it might seem that way, because we invite by score and grade. But I really think the purpose of MOP is to give each one of you the experience of working hard and meeting new people, among other things. Learn math, face challenges, make friends, the usual good stuff, right? And that’s something you can get no matter what your final rank is, or whether you make IMO or EGMO or even next year’s MOP. The MOP community is an extended family, and you are all part of it now.
What I mean to say is, the camp is designed with all 80 of you in mind. It made me sad back in 2012 when one of my friends realized he had little chance of making it back next year, and told me that MAA shouldn’t have invited him to begin with. Even if I can only take six students to the IMO each year, I never forget the other 74 of you are part of MOP too.
This means one important thing: everyone who puts in their best shot deserves to be here. (And unfortunately this also means there are many other people who deserve to be here tonight too, and are not. Maybe they solved one or two fewer problems than you did; or maybe they even solved the same number of problems, but they are in 11th grade and you are in 10th grade.)
Therefore, I hope to see all of you put in your best effort. And I should say this is not easy to do, because MOP is brutal in many ways. The classes are mandatory, we have a 4.5-hour test every two days, and you will be constantly graded. You will likely miss problems that others claim are easy. You might find out you know less than you thought you did, and this can be discouraging. Especially in the last week, when we run the TSTST, many of you will suddenly realize just how strong Team USA is.
So I want to tell you now, stay determined in the face of adversity. This struggle is your own, and we promise it’s worth it, no matter the outcome. We are rooting for you, and your friends sitting around you are too. (And if the people around you aren’t your friends yet, change that asap.)
In yet another contest-based post, I want to distinguish between two types of thinking: things that could help you solve a problem, and things that could help you understand the problem better. Then I’ll talk a little about how you can use the latter. (I’ve talked about this in my own classes for a while by now, but only recently realized I’ve never gotten the whole thing in writing. So here goes.)
1. More silly terminology
As usual, to make these things easier to talk about, I’m going to introduce some words to describe these two. Taking a page from martial arts, I’m going to run with hard and soft techniques.
A hard technique is something you try in the hopes it will prove something — ideally, solve the problem, but at least give you some intermediate lemma. Perhaps a better definition is “things that will end up in the actual proof”. Examples include:
- Angle chasing in geometry, or proving quadrilaterals are cyclic.
- Throwing complex numbers at a geometry problem.
- Plugging in some values into a functional equation (which gives more equations to work with).
- Taking a given Diophantine equation modulo to get some information, or taking -adic evaluations.
- Trying to perform an induction, for example by deleting an element.
- Trying to write down an inequality that when summed cyclically gives the desired conclusion.
- Reducing the problem to one or more equivalent claims.
and so on. I’m sure you can come up with more examples.
In contrast, a soft technique is something you might try to help you understand the problem better — even if it might not prove anything. Perhaps a better definition is “things not written up”. Examples include:
- Examining particular small cases of the problem.
- Looking at the equality cases of a min/max problem.
- Considering variants of the problem (for example, adding or deleting conditions).
- Coming up with lots of concrete examples and playing with them.
- Trying to come with a counterexample to the problem’s assertion and seeing what the obstructions are.
- Drawing pictures, even on non-geometry problems (see JMO2 and JMO5 in my 2019 notes for example).
- Deciding whether or not a geometry problem is “purely projective”.
- Counting the algebraic degrees of freedom in a geometry problem.
- Checking all the linear/polynomial solutions to a functional equation, in order to get a guess what the answer might be.
- Blindly trying to guess solutions to an algebraic equation.
- Making up an artificial unnatural function in a functional equation, and then trying to see why it doesn’t work (or occasionally being surprised that it does work).
- Thinking about why a certain hard technique you tried failed, or even better convincing yourself it cannot work (for example, this Diophantine equation has a solution modulo every prime, so stop trying to one-shot by mods).
- Giving a heuristic argument that some claim should be true or false (“probably is odd infinitely often”), or even easy/hard to prove.
and so on. There is some grey area between these two, some of the examples above might be argued to be in the other category (especially in context of specific problems), but hopefully this gives you a sense of what I’m talking about.
If you look at things I wrote back when I was in high school, you’ll see this referred to as “attacking” and “scouting” instead. This is too silly for me now even by my standards, but back then it was because I played a lot of StarCraft: Brood War (I’ve since switched to StarCraft II). The analogy there is pretty self-explanatory: knowing what your opponent is doing is important because your army composition and gameplay decisions should change in reaction to more information.
2. Using soft techniques: an example
Now after all that blabber, here’s the action item for you all: you should try soft techniques when stuck.
When you first start doing a problem, you will often have some good ideas for what to try. (For example: a wild geometry appeared, let’s scout for cyclic quadrilaterals.) Sometimes if you are lucky enough (especially if the problem is easier) this will be enough to topple the problem, and you can move on. But more often what happens is that eventually you run out of steam, and the problem is still standing. When that happens, my advice is to try doing some soft techniques if you haven’t already done so.
Here’s an example that I like to give.
A common first thing that people will try to do is add the first two equations, since that will cause the terms to cancel. This gives a factor of in the left and an in the right, so then maybe you try to submit that into the in the last equation, so you get , cool, there’s no more linear terms. Then. . .
Usually this doesn’t end well. You add this and subtract that and in the end all you see is equation after equation, and after a while you realize you’re not getting anywhere.
So we’re stuck now. What to do? I’ll now bring in two of the soft techniques I mentioned earlier:
- Let’s imagine the problem had replaced with . In this new problem, you can imagine solving for in terms of using the first equation, then in terms of , and then finally putting everything into the last equation to find a degree polynomial in . I say “imagine” because wow would that be ugly.
But here’s the kicker: it’s a polynomial. It should have exactly complex roots, with multiplicity. That’s a lot. Really?
So here’s a hint you might take: there’s a good reason this is over but not . Often these kind of things end up being because there’s an inequality going on somewhere, so there will only be a few real solutions even though there might be tons of complex ones.
- Okay, but there’s an even more blatant thing we don’t know yet: what is the answer, anyways?
This was more than a little bit embarrassing. We’re half an hour in to the problem and thoroughly stuck, and we don’t even have a single that works? Maybe it’d be a good idea to fix that, like, right now. In the simplest way possible: guess and check.
It’s much easier than it sounds, since if you pick a value of , say, then you get from the third equation, from the first, then check whether it fits the second. If we restrict our search to integer values of , then there aren’t so many that are reasonable.
I won’t spoil what the answer is, other than saying there is an integer triple and it’s not hard to find it as I described. Once you have these two meta-considerations, you suddenly have a much better foothold, and it’s not too hard to solve the problem from here (for a USA TST problem anyways).
I pick this example because it really illustrates how hopeless repeatedly using hard techniques can be if you miss the right foothold (and also because in this problem it’s unusually tempting to just think that more manipulation is enough). It’s not impossible to solve the problem without first realizing what the answer is, but it is certainly way more difficult.
3. Improving at soft techniques
What this also means is that, in the after-math of a problem (when you’ve solved/given up on a problem and are reading and reflecting on the solution), you should also add soft techniques into the list of possible answers to “how might I have thought of that?”. An example of this is at the end of my earlier post On Reading Solutions, in which I describe how you can come up with solutions to two Putnam problems by thinking carefully about what should be the equality case.
Doing this is harder than it sounds, because the soft techniques are the ones that by definition won’t appear in most written solutions, and many people don’t explicitly even recognize them. But soft techniques are the things that tell you which hard techniques to use, which is why they’re so valuable to learn well.
In writing this post, I’m hoping to make the math contest world more aware that these sorts of non-formalizable ideas are things that can (and should) be acknowledged and discussed, the same way that the hard techniques are. In particular, just as there are a plethora of handouts on every hard technique in the olympiad literature, it should also be possible to design handouts aimed at practicing one or more particular soft techniques.
At MOP every year, I’m starting to see more and more classes to this effect (alongside the usual mix of classes called “inversion” or “graph theory” or “induction” or whatnot). I would love to see more! End speech.
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 … Continue reading
Pictures, thoughts, and other festives from the 2019 Romania Masters in Math. See also the MAA press release. Summary Po-Shen Loh and I spent the last week in Bucharest with the United States team for the 11th RMM. The USA … Continue reading
I think it would be nice if every few years I updated my generic answer to “how do I get better at math contests?”. So here is the 2019 version. Unlike previous instances, I’m going to be a little less olympiad-focused than I usually am, since these days I get a lot of people asking for help on the AMC and AIME too.
(Historical notes: you can see the version from right after I graduated and the version from when I was still in high school. I admit both of them make me cringe slightly when I read them today. I still think everything written there is right, but the style and focus seems off to me now.)
0. Stop looking for the “right” training (or: be yourself)
These days many of the questions I get are clearly most focused on trying to find a perfect plan — questions like “what did YOU do to get to X” or “how EXACTLY do I practice for Y”. (Often these words are in all-caps in the email, too!) When I see these I always feel very hesitant to answer. The reason is that I always feel like there’s some implicit hope that I can give you some recipe that, if you follow it, will guarantee reaching your goals.
I’m sorry, math contests don’t work that way (and can’t work that way). I actually think that if I gave you a list of which chapters of which books I read in 2009-2010 over which weeks, and which problems I did on each day, and you followed it to the letter, it would go horribly.
Why? It’s not just a talent thing, I think. Solving math problems is actually a deeply personal art: despite being what might appear to be a cold and logical discipline, learning math and getting better at it actually requires being human. Different people find different things natural or unnatural, easy or hard, et cetera. If you try to squeeze yourself into some mold or timeline then the results will probably be counterproductive.
On the flip side, this means that you can worry a lot less. I actually think that surprisingly often, you can get a first-order approximation of what’s the “best” thing to do by simply doing whatever feels the most engaging or rewarding (assuming you like math, of course). Of course there are some places where this is not correct (e.g., you might hate geometry, but cannot just ignore it). But the first-order approximation is actually quite decent.
That’s why in the introduction to my geometry book, I explicitly have the line:
Readers are encouraged to not be bureaucratic in their learning and move around as they see fit, e.g., skipping complicated sections and returning to them later, or moving quickly through familiar material.
Put another way: as learning math is quite personal, the advice “be yourself” is well-taken.
1. Some brief recommendations (anyways)
With all that said, probably no serious harm will come from me listing a little bit of references I think are reasonable — so that you have somewhere to start, and can oscillate from there.
For learning theory and fundamentals:
For sources of additional practice problems (other than the particular test you’re preparing for):
- The collegiate contests HMMT November, PUMaC, CMIMC will typically have decent short-answer problems.
- HMMT February is by far the hardest short-answer contest I know of.
- At the olympiad level, there are so many national olympiads and team selection tests that you will never finish. (My website has an archive of USA problems and solutions if you’re interested in those in particular.)
The IMO Shortlist is also good place to work as it contains proposals of varying difficulty from many countries — and thus is the most culturally diverse. As for other nations, as a rule of thumb, any country that often finishes in the top 20 at the IMO (say) will probably have a good questions on their national olympiad or TST.
For every subject that’s not olympiad geometry, there are actually surprisingly few named theorems.
2. Premature optimization is the root of all evil (so just get your hands dirty)
For some people, the easiest first step to getting better is to double the amount of time you spend practicing. (Unless that amount is zero, in which case, you should just start.)
There is a time and place for spending time thinking about how to practice — one example is if you’ve been working a while and feel like nothing has changed, or you’ve been working on some book and it just doesn’t feel fun, etc. Another common example is if you notice you keep missing all the functional equations on the USAMO: then, maybe it’s time to search up some handouts on functional equations. Put another way, if you feel stuck, then you can start thinking about whether you’re not doing something right.
On the other extreme, if you’re wondering whether you are ready to read book X or do problems from Y contest, my advice is to just try it and see if you like it. There is no commitment: just read Chapter 1, see how you feel. If it works, keep doing it, if not, try something else.
(I can draw an analogy from my own life. Whenever I am learning a new board game or card game, like Catan or Splendor or whatever, I always overthink it. I spend all this time thinking and theorizing and trying to come up with this brilliant strategy — which never works, because it’s my first game, for crying out loud. It turns out that until you start grappling at close range and getting your hands dirty, your internal model of something you’ve never done is probably not that good.)
3. Doing problems just above your level (and a bit on reflecting on them)
There is one pitfall that I do see sometimes, common enough I will point it out. If you mostly (only?) do old practice tests or past problems, then you’re liable to be spending too much time on easy problems. That was the topic of another old post of mine, but the short story is that if you find yourself constantly getting 130ish on AMC10 practice tests, then maybe you should spend most of your time working on problems 21-25 rather than repeatedly grinding 1-20 over and over. (See 28:30-29:00 here to hear Zuming make fun of them.)
The common wisdom is that you should consistently do problems just above your level so that you gradually increase the difficulty of problems you are able to solve. The situation is a little more nuanced at the AMC/AIME level, since for those short-answer contests it’s also important to be able to do routine problems quickly and accurately. However, I think for most people, you really should be spending at least 70% of your time getting smarter, rather than just faster.
I think in this case, I want to give concrete descriptions. Here’s some examples of what can happen after a problem.
- You looked at the problem and immediately (already?) knew how to do it. Then you probably didn’t learn much from it. (But at least you’ll get faster, if not smarter.)
- You looked at the problem and didn’t know right away how to start, but after a little while figured it out. That’s a little better.
- You struggled with the problem and eventually figured out a solution, but maybe not the most elegant one. I think that’s a great situation to be in. You came up with some solution to the problem, so you understand it fairly well, but there’s still more for you to update your instincts on. What can you do in the future to get solutions more like the elegant one?
- You struggled with the problem and eventually gave up, then when you read the solution you realize quickly what you were missing. I think that’s a great situation to be in, too. You now want to update your instincts by a little bit — how could you make sure you don’t miss something like that again in the future?
- The official solution quoted some theorem you don’t know. If this was among a batch of problems where the other problems felt about the right level to you, then I think often this is a pretty good time to see if you can learn the statement (better, proof) of the theorem. You have just spent some time working on a situation in which the theorem was useful, so that data is fresh in your mind. And pleasantly often, you will find that ideas you came up with during your attempt on the problem correspond to ideas in the statement or proof of the theorem, which is great!
- You didn’t solve the problem, and the solution makes sense, but you don’t see how you would have come up with it. It’s possible that this is the fault of the solutions author (many people are actually quite bad at making solutions read naturally). If you have a teacher, this is the right time to ask them about it. But it’s also possible that the problem was too hard. In general, I think it’s better to miss problems “by a little”, whatever that means, so that you can update your intuition correctly.
- You can’t even understand the solution. Okay, too hard.
You’ll notice how much emphasis I place on the post-problem reflection process. This is actually important — after all the time you spent working on the problem itself, you want to update your instincts as much as possible to get the payoff. In particular, I think it’s usually worth it to read the solutions to problems you worked on, whether or not you solve them. In general, after reading a solution, I think you should be able to state in a couple sentences all the main ideas of the solution, and basically know how to solve the problem from there.
For the olympiad level, I have a whole different post dedicated to reading solutions, and interested readers can read more there. (One point from that post I do want to emphasize since it wasn’t covered explicitly in any of the above examples: by USA(J)MO level it becomes important to begin building intuition that you can’t explicitly formalize. You may start having vague feelings and notions that you can’t quite put your finger on, but you can feel it. These non-formalizable feelings are valuable, take note of them.)
4. Leave your ego out (e.g. be willing to give up on problems)
This is easy advice to give, but it’s hard advice to follow. For concreteness, here are examples of things I think can be explained this way.
Sometimes people will ask me whether they need to solve every problem in each chapter of EGMO, or do every past practice test, or so on. The answer is: of course not, and why would you even think that? There’s nothing magical about doing 80% of the problems versus 100% of them. (If there was, then EGMO is secretly a terrible book, because I commented out some problems, and so OH NO YOU SKIPPED SOME AAAHHHHH.) And so it’s okay to start Chapter 5 even though you didn’t finish that last challenge problem at the end. Otherwise you let one problem prevent you from working on the next several.
Or, sometimes I learn about people who, if they do not solve an olympiad problem, will refuse to look at the solution; instead they will mark it in a spreadsheet and to come back to later. In short, they never give up on a problem: which I think is a bad idea, since reflecting on missed problems is so important. (It is not as if you can realistically run out of olympiad problems to do.) And while this is still better than giving up too early, I mean, all things in moderation, right?
I think if somehow people were able to completely leave your ego out, and not worry at all about how good you are and rather just maximize learning, then mistakes like these two would be a lot rarer. Of course, this is impossible to do in practice (we’re all human), but it’s good to keep in mind at least that this is an ideal we can strive for.
5. Enjoy it
Which leads me to the one bit that everyone already knows, but that no platitude-filled post would be complete without: to do well at math contests (or anything hard) you probably have to enjoy the process of getting better. Not just the end result. You have to enjoy the work itself.
Which is not to say you have to do it all the time or for hours a day. Doing math is hard, so you get tired eventually, and beyond that forcing yourself to work is not productive. Thus when I see people talk about how they plan to do every shortlist problem, or they will work N hours per day over M time, I always feel a little uneasy, because it always seems too results-oriented.
In particular, I actually think it’s quite hard to spend more than two or three good hours per day on a regular basis. I certainly never did — back in high school (and even now), if I solved one problem that took me more than an hour, that was considered a good day. (But I should also note that the work ethic of my best students consistently amazes me; it far surpasses mine.) In that sense, the learning process can’t be forced or rushed.
There is one sense in which you can get more hours a day, that I am on record saying quite often: if you think about math in the shower, then you know you’re doing it right.
There’s a recent working paper by economists Ruchir Agarwal and Patrick Gaule which I think would be of much interest to this readership: a systematic study of IMO performance versus success as a mathematician later on.
Despite the click-baity title and dreamy introduction about the Millenium Prizes, the rest of the paper is fascinating, and the figures section is a gold mine. Here are two that stood out to me:
There’s also one really nice idea they had, which was to investigate the effect of getting one point less than a gold medal, versus getting exactly a gold medal. This is a pretty clever way to account for the effect of the prestige of the IMO, since “IMO gold” sounds so much better on a CV than “IMO silver” even though in any given year they may not differ so much. To my surprise, the authors found that “being awarded a better medal appears to have no additional impact on becoming a professional mathematician or future knowledge production”. I included the relevant graph below here.
The data used in the paper spans from IMO 1981 to IMO 2000. This is before the rise of Art of Problem Solving and the Internet (and the IMO was smaller back then, anyways), so I imagine these graphs might look different if we did them in 2040 using IMO 2000 – IMO 2020 data, although I’m not even sure whether I expect the effects to be larger or smaller.
(As usual: I do not mean to suggest that non-IMO participants cannot do well in math later. This is so that I do not get flooded with angry messages like last time.)
Some of the stuff covered in this handout:
- Advice for constructing the triangle centers (hint: circumcenter goes first)
- An example of how to rearrange the conditions of a problem and draw a diagram out-of-order
- Some mechanical suggestions such as dealing with phantom points
- Some examples of computer-generated figures
Some thoughts about some modern trends in mathematical olympiads that may be concerning.
I. The story of the barycentric coordinates
I worry about my geometry book. To explain why, let me tell you a story.
When I was in high school about six years ago, barycentric coordinates were nearly unknown as an olympiad technique. I only heard about it from whispers in the wind from friends who had heard of the technique and thought it might be usable. But at the time, there were nowhere where everything was written down explicitly. I had a handful of formulas online, a few helpful friends I can reach out to, and a couple example posts littered across some forums.
Seduced by the possibility of arcane power, I didn’t let this stop me. Over the spring of 2012, spring break settled in, and I spent that entire week developing the entire theory of barycentric coordinates from scratch. There were no proofs I could find online, so I had to personally reconstruct all of them. In addition, I set out to finding as many example problems as I could, but since no one had written barycentric solutions yet, I had to not only identify which problems like they might be good examples but also solve them myself to see if my guesses were correct. I even managed to prove a “new” theorem about perpendicular displacement vectors (which I did not get to name after myself).
I continued working all the way up through the summer, adding several new problems that came my way from MOP 2012. Finally, I posted a rough article with all my notes, examples, and proofs, which you can still find online. I still remember this as a sort of magnus opus from the first half of high school; it was an immensely rewarding learning experience.
Today, all this and much more can be yours for just $60, with any major credit or debit card.
Alas, my geometry book is just one example of ways in which the math contest scene is looking more and more like an industry. Over the years, more and more programs dedicated to training for competitions are springing up, and these programs can be quite costly. I myself run a training program now, which is even more expensive (in my defense, it’s one-on-one teaching, rather than a residential camp or group lesson).
It’s possible to imagine a situation in which the contest problems become more and more routine. In that world, math contests become an arms race. It becomes mandatory to have training in increasingly obscure techniques: everything from Popoviciu to Vieta jumping to rectangular circumhyperbolas. Students from less well-off families, or even countries without access to competition resources, become unable to compete, and are pushed to the bottom of the IMO scoreboard.
(Fortunately for me, I found out at the 2017 IMO that my geometry book actually helped level the international playing field, contrary to my initial expectations. It’s unfortunate that it’s not free, but it turned out that many students in other countries had until then found it nearly impossible to find suitable geometry materials. So now many more people have access to a reasonable geometry reference, rather than just the top countries with well-established training.)
II. Another dark future
The first approximation you might have now is that training is bad. But I think that’s the wrong conclusion, since, well, I have an entire previous post dedicated to explaining what I perceive as the benefits of the math contest experience. So I think the conclusion is not that training is intrinsically bad, but rather than training must be meaningful. That is, the students have to gain something from the experience that’s not just a +7 bonus on their next olympiad contest.
I think the message “training is bad” might be even more dangerous.
Imagine that the fashion swings the other way. The IMO jury become alarmed at the trend of train-able problems, and in response, the problems become designed specifically to antagonize trained students. The entire Geometry section of the IMO shortlist ceases to exist, because some Asian kid wrote this book that gives you too much of an advantage if you’ve read it, and besides who does geometry after high school anyways? The IMO 2014 used to be notable for having three combinatorics problems, but by 2040 the norm is to have four or five, because everyone knows combinatorics is harder to train for.
Gradually, the IMO is redesigned to become an IQ test.
The changes then begin to permeate down. The USAMO committee is overthrown, and USAMO 2050 features six linguistics questions “so that we can find out who can actually think”. Math contests as a whole become a system for identifying the best genetic talent, explicitly aimed at weeding out the students who have “just been trained”. It doesn’t matter how hard you’ve worked; we want “creativity”.
This might be great at identifying the best mathematicians each generation, but I think an IMO of this shape would be actively destructive towards the contestants and community as well. You thought math contests were bad because they’re discouraging to the kids who don’t win? What if they become redesigned to make sure that you can’t improve your score no matter how hard you work?
What this means is that we have a balancing act to maintain. We do not want to eliminate the role of training entirely, because the whole point of math contests is to have a learning experience that lasts longer than the two-day contest every year. But at the same time, we need to ensure the training is interesting, that it is deep and teaches skills like the ones I described before.
Paying $60 to buy a 300-page PDF is not meaningful. But spending many hours to work through the problems in that PDF might be.
In many ways this is not a novel idea. If I am trying to teach a student, and I give them a problem which is too easy, they will not learn anything from it. Conversely, if I give them a problem which is too difficult, they will get discouraged and are unlikely to learn much from their trouble. The situation with olympiad training feels the same.
This applies to the way I think about my teaching as well. I am always upset when I hear (as I have) things like “X only did well on USAMO because of Evan Chen’s class”. If that is true, then all I am doing is taking money as input and changing the results of a zero-sum game as output, which is in my opinion rather pointless (and maybe unethical).
But I really think that’s not what’s happening. Maybe I’m a good teacher, but at the end of the day I am just a guide. If my students do well, or even if they don’t do well, it is because they spent many hours on the challenges that I designed, and have learned a lot from the whole experience. The credit for any success thus lies solely through the student’s effort. And that experience, I think, is certainly not zero-sum.
One of the major headaches of using complex numbers in olympiad geometry problems is dealing with square roots. In particular, it is nontrivial to express the incenter of a triangle inscribed in the unit circle in terms of its vertices.
The following lemma is the standard way to set up the arc midpoints of a triangle. It appears for example as part (a) of Lemma 6.23.
Theorem 1 is often used in combination with the following lemma, which lets one assign the incenter the coordinates in the above notation.
Unfortunately, the proof of Theorem 1 in my textbook is wrong, and I cannot find a proof online (though I hear that Lemmas in Olympiad Geometry has a proof). So in this post I will give a correct proof of Theorem 1, which will hopefully also explain the mysterious introduction of the minus signs in the theorem statement. In addition I will give a version of the theorem valid for quadrilaterals.
2. A Word of Warning
I should at once warn the reader that Theorem 1 is an existence result, and thus must be applied carefully.
To see why this matters, consider the following problem, which appeared as problem 1 of the 2016 JMO.
By experimenting with the diagram, it is not hard to guess that the correct fixed point is the midpoint of arc , as seen in the figure below. One might be tempted to write , , , and assert the two incenters are and , and that the fixed point is .
This is a mistake! If one applies Theorem 1 twice, then the choices of “square roots” of the common vertices and may not be compatible. In fact, they cannot be compatible, because the arc midpoint of opposite is different from the arc midpoint of opposite .
In fact, I claim this is not a minor issue that one can work around. This is because the claim that the circumcircle of passes through the midpoint of arc is false if lies on the arc on the same side as ! In that case it actually passes through instead. Thus the truth of the problem really depends on the fact that the quadrilateral is convex, and any attempt with complex numbers must take this into account to have a chance of working.
3. Proof of the theorem for triangles
Fix now, so we require , , . There are choices of square roots , , we can take (differing by a sign); we wish to show one of them works.
We pick an arbitrary choice for first. Then, of the two choices of , we pick the one such that . Similarly, for the two choices of , we pick the one such that . Our goal is to show that under these conditions, we have again.
The main trick is to now consider the arc midpoint , which we denote by . It is easy to see that:
Thus, we have
From this we can see why the minus signs are necessary.
4. A version for quadrilaterals
We now return to the setting of a convex quadrilateral that we encountered in Example 3. Suppose we preserve the variables , , that we were given from Theorem 1, but now add a fourth complex number with . How are the new arc midpoints determined? The following theorem answers this question.
This setup is summarized in the following figure.
Note that unlike Theorem 1, the four arcs cut out by the sides of do not all have the same sign (I chose to have coordinates ). This asymmetry is inevitable (see if you can understand why from the proof below).
Proof: We select , , with Theorem 1. Now, pick a choice of such that is the arc midpoint of not containing and . Then the arc midpoint of not containing or is given by
On the other hand, the calculation of for the midpoint of follows by applying Lemma 4 again. (applied to triangle ). The midpoint of is computed similarly.
In other problems, the four vertices of the quadrilateral may play more symmetric roles and in that case it may be desirable to pick a setup in which the four vertices are labeled in order. By relabeling the letters in Theorem 6 one can prove the following alternate formulation.
To test the newfound theorem, here is a cute easy application.