Stewart's Theorem, Proof and AMC 12 Example

[@randomacc6151]

Love this, helped a lot!! ❤

[@TheBeautyofMath]

Thank you for the feedback! Good luck with your prep!

[@buxeessingh2571]

Stewart's theorem and mass points are two of the top geometry topics for math contests.

[@TheBeautyofMath]

Agreed. I just felt on the AMC 10 and 12 that it essentially serves as a potential avenue for time savings, while not a "if you don't know this, then you can't solve the problem" kind of topic. However it may very well be that it is a requirement for certain AIME questions or questions from other competitions that I am less familiar with. In my opinion though, it is a must-know concept simply for it's utility. Thanks for the comment Buxees, good to see you around. 🙂

[@EpicMethGaming]

picks theorem. just cheese with it

[@bobjoe5524]

You mix in humor so well when you teach lol. Yeah its one of those "helps a lot on the later geo problems" because the MAA masterminds who made the problem probably have a very subtle solution which is not attainable for the standard test taker in 75 minutes.

[@ericqiang5578]

I wanted a refresher on some theorems as a part of my quick review seeing that the test is next week. So happy to see that your channel alone has everything I need. Really appreciate the work and thanks for giving these top notch explanations that are so hard to find not to mention free as well!

[@TheBeautyofMath]

Thank you for the comment! I am honestly grateful to know that they are helping. Every word of encouragement helps me in return. :)

[@altacc3653]

Hey, beauty of math, I'm hoping you respond to this comment because you've been really great about providing detailed answers to questions that I'm sure everybody appreciates. Before watching this video I had no idea the test date had been moved to November. This is pretty bad news for me - I've hadn't done competition math since two years ago and I'm basically a newbie. I've been banking on the amc 10 as a fresh start for competition math and I and I've recently cracked open my only aops textbook - the into one - and I've done up to chapter 20. As anyone might, I want to do well on this test, qualifying for AIME and whatnot, but with a little over eight weeks, it seems near impossible, and I'm hoping my chances aren't totally null. I know I should start by doing some past papers, but I have no idea how to approach this vast and new world, let alone in so little time. I have no idea which key concepts or topics I should learn, which strategies for learning I should use, how to approach a problem and basically anything I don't think I can learn from experience in 8 weeks. I'm aware that this is a lengthier than usual youtube comment and one of the more demanding questions you might have been asked, but I hope you can help me. I thank you in advance and for all the free math tutorials you've made for hopeful mathematicians like me.

[@TheBeautyofMath]

A little tired at the moment. Just saw this was caught in KZbin comment filter. I will reply hopefully tomorrow when I am able to think clearly. :]

[@TheBeautyofMath]

Ok just got off of work a little before midnight Pacific Time. First, I cannot tell you your likelihood of qualifying without a recent benchmark. Take something around say 2015 A and B tests. Tell me your scores. You can send this to me by email if you don't want to put it on KZbin comments. If you are below 70 the chance of getting to over 100 in 8 weeks is virtually zero. If you are above 85.5, then you have a reasonable shot to do so with dedicated practice time inside of 8 weeks. Reasonable meaning...above 50% chance. These are rough estimates of percent chances based on an averages person's progress in learning and information recall and knowledge acquisition. I would start with those two tests and see whereabouts your score is. Time them both. The honest reality is most students who qualify prepare diligently for more than 1 year with small breaks of maybe a few weeks. You must understand 2 things then. First...you can do absolutely nothing to change the past. Nothing. You can think about the past and become sad about the things that were not done. Then you will have a new situation. That is...the past cannot be changed and furthermore your present mood and happiness have deteriorated as you dwell upon the regret of not doing different things in the past, however what all of that sadness cannot do...is change the past. So...stop feeling sad. Stop feeling discouraged. It does NOTHING to help you. Look in the metaphorical mirror. Tell yourself to snap out of it and do the best you can. Second, all you can do is try. Let's say you don't qualify this year. Ok. What did it cost you? Time spent studying and improving yourself and increasing your skill level while solving enjoyable math problems. You will have a better interconnected understanding of Mathematics based on your study time. Further...you will now have the foundation on which to build a springboard to catapult yourself to a 2022-2023 School Year AIME qualification be it on AMC 10 or 12 depending on your grade/year in school. Now you will have an early start on next year's prep. When the tests go by this year assuming you didn't qualify, which to be fair, knowing nothing about you is at least a possibility...then you hit the ground running in prep for next year. Start with the intro book you mentioned. AoPS actually has 5 intro books. I don't know which of those you have. It would be like Intro to Algebra, or Intro to Geo, or Intro to Counting and Probability, or Intro to Number Theory, etc. I would finish that book before this years tests. Do every problem in it. Next target the book called Volume 1. This has the most content that has the greatest overall impact. It is like light coverage of the majority of the major concepts on the test. It will give you by itself a much firmer foundation than any one of the books by itself. Next between now and November, do all tests from 2010 to present. About 21 tests. You will need to do about 2 to 3 a week. If that is not feasible...start from 2013 or so. After this years tests...work at a more gradual pace. Pick a different Intro book that represents one of your weaknesses. Then go back all the way to the year 2000 and do all the tests from before the starting point you chose(2010, 2013 etc.) for the last 8 weeks of prep. Use the free resource Alcumus too. It's an AoPS problem resource. The next thing is...you have to change your current outlook. This is the hardest part. Paradigm shift. From a defeatist and pessimistic thinking to one focused on your next step. The next step is to start. Immediately. Do the suggested tests. Don't worry if you are going about your preparation wrong. Solving problems, past tests, and AoPS books in any order is a great plan. There is no perfect plan. I can tell you what doesn't work. Quitting before you start. Telling yourself you cannot do it. Think about how you feel now knowing you didn't study the previous 8 months as you weren't aware of the test date being moved. Sadness? Regret? Etc. Now look into the future at yourself in mid November after you sat out this year's tests because you felt you wouldn't have enough time. How will you feel? Likely more regret. So care about that future self enough to do the best you can. Not qualifying is not the end of the world. It might be the start you need to get the qualification next year. My final advice then is to pick yourself up...get running...and don't look back. What's done is done. All that now exists is what lies before you...and what you are going to do about it. Best of luck.

[@altacc3653]

@@TheBeautyofMath Dear TheBeautyofMath, The amount of effort you give to help students you don't know at all is stunning...I'm sure everybody on your channel is incredibly grateful for the help you give. If you were worried that your comment fell on deaf ears, I have in fact read it, but only just now. As my account name suggests, this is an alt account and after you hadn't responded to my comment for a while, I was worried you were never going to as this was a huge request of someone, and thus I hadn't checked it since. Despite my heavy school work load, I have been able to a test everyday, covering all amc's from 2002-2010, and I have just finished 10b yesterday. Miraculously, I am getting 110-120ish on my tests, and for my most recent one, if I hadn't made any sillies (calculation errors and such), I would have gotten upward of 125. Just looking at the amc historical results, since the aime threshold gets lower and lower, I can only assume the tests get progressively harder, so im not sure how well these scores will reflect. As for your question about which into textbook I have, I actually have both Volume 1 and Volume 2, and I have finished the entirety of Volume 1. Because geometry is the most content heavy subject, I was initially my greatest weakness, but as I did more tests, I started to notice that all of the problems were quite similar, although I haven't memorised a lot of the advanced theorems, such as Stewart's and ceva's. I am pretty much ok with combinatorics, but I very often make silly errors in my casework or misunderstanding the set-up. For number theory, I am yet to fully feel comfortable with diophantine equations and I am yet to learn some of the major theorems. My main weakness is actually the earlier problems, because this is where I tend to make sillies and I just spend too much time on them. My next step is to probably move on to the volume 2, since I have it at hand, but I'm not too sure which chapters are actually helpful for the amc 10. Once again, I thank you for your tremendous help...you have motivated me to push myself harder and you are an inspiration to all.

[@altacc3653]

@@TheBeautyofMath One quick question if and when you have the time, I have access to the Volume 2 aops book, through my local library, as well as the intermediate counting one. My question is which chapters in both books are beneficial to the amc 10, because some chapters seem too advanced for the topics covered and there is only a limited time before the test date in November. Thanks a lot.

[@TheBeautyofMath]

Hmmm good question. I think all chapters in the Intermediate Counting one with the exception of the first have content that can benefit, but it's not every question for sure. I would have to go thru on case by case basis to determine that. But the thinking processes can certainly help even if the problems themselves are slightly more advanced than current AMC 10 content(keep in mind the tests get seemingly more difficult almost every year). As for Volume 2 I would do Chapters: 3,4,5,6,7 12-rest of the book. But you can do them in almost any order. Pick one first that you have high confidence of increased likelihood of making an appearance on the AMC 10. Chapters 1,2, and 9 can be done when you have moved on to the AMC 12. In addition I think if you grind hard this next month on tests and practice problems...with the scores you mention above you have a great shot in spite of losing prep time. Don't give up. Finish Strong. Also start trying to analyze why the silly mistakes are occurring and try to avoid them by being hyper-vigilant of previous mistake types from past tests.

[@jndd7373]

Tysm! First time learning Stewart's Theorem :)

[@TheBeautyofMath]

Right? it's kind of fun. :)

[@ishaanpanigrahi8258]

that little caesars joke was pretty funny :P, also thank you for this video. Are you planning to do anymore concept videos before the AMC? I struggle with forming bijections, and was wondering if you could a video on that. Thanks again.

[@TheBeautyofMath]

Ishaan, thank you for the joke appreciation. lol I will likely do more concept videos before the test, but... Sadly one-to-one correspondences are not a strength of mine either. Two people who could probably give a great explanation if they have time are @Sohil Rathi (kzbin.info/door/DIW29U4zATB2m1lVFu4cyQ) and @Evan Chen (kzbin.info) . Maybe some day I will improve myself in this area enough to explain it a level I am satisfied with, but my strong suspicion is that will not be before this year's tests.

[@rahulsanklecha5612]

Thanks :)

[@TheBeautyofMath]

My pleasure.

[@zephyr3790]

Could you do a video on a number 17+ AMC 10 problem? I'm a 7th grader really cramming studying because of the upcoming test and was wondering if you could do a couple problems on that. Thanks!

[@TheBeautyofMath]

I actually have a lot of questions from beyond problem number 17 from the AMC 10 on my channel. For example here is the complete 2017 AMC 10A test all 25 problems: kzbin.info/aero/PL-27w0UNlunwrSKk51Rl-APHLDAux7rre but if you go through my content, there is probably at least 60 problems from beyond number 17 currently on the channel. Just use the videos tab or the playlist tab to locate them. I also have 1-24 from the 2020 AMC 10B for instance. Does this help? If not, let me know which problems you had in mind. Thanks :)

[@zephyr3790]

My main struggling points are those counting and probability questions that generally appear after the number 17(that is an approximation)

[@zephyr3790]

Also, I have a question about timings. I am consistently getting around 100-105. When do you think in these 8 weeks should I introduce the 'time' in the practice tests I take? Im kind of afraid that once I introduce time, I score will go from around 100-105 to a 85-90. Along with me, I think a lot of people know what happens if you dont practice with time on MATHCOUNTS and AMC problems before hand(I generally dont finish) based on what I understood from friends as well.

[@TheBeautyofMath]

On the time question...immediately. I only recommemd untimed to students for about the first 4 tests they take in their life. After that...ALL of them should be timed. But that doesn't mean that when time runs out you stop working. Keep trying every problem the test for as long as you can. But as you track your scores all questions you answer after time has run out count as skips whether you you got them right or wrong. The purpose is you need to practice being time pressured. As much as possible. There is literally no point in doing untimed tests if one's purpose is AIME qualification. A timed test can give all the same benefits as untimed but it also gives additional benefits.

[@TheBeautyofMath]

For Counting and Probability...here is a 14 video playlist with every problem coming after number 17: kzbin.info/aero/PL-27w0UNlunxisWf_AuSLJrW0aax5fzLC There is a 2 question AIME C and P playlist too if you look in the playlist.

[@God_Of_Sniper]

thx for the fantastic vid =

[@sophiayao7041]

Nice!

[@TheBeautyofMath]

Thanks!

[@robomania5346]

Will you have an AMC strat video this year?

[@TheBeautyofMath]

Hmmm I could? Did you mean one essentially like this linked one or something different? kzbin.info/www/bejne/fHfTc4ChgNOCoZo I am not sure that I would say much that is that different from what is said here...but if you have any ideas or you just think it would be good to make a similar video like a pre game speech...then I probably will. Maybe I can make it a tradition.

[@robomania5346]

@@TheBeautyofMathI was thinking something like that if you have the time for it. Kind of almost motivation or something for those who need it.

[@TheBeautyofMath]

You know...based on this comment I think I will make a video this year and each year. Thank you for the suggestion. 😀

[@TheBeautyofMath]

Update...I have something in mind to come out soon. This week hopefully.

[@SandunSuranga]

Did you shoot this video using your Logitech Brio 4k webcam, please let me know. and accept my heartiest congratulations for your great effort.

[@TheBeautyofMath]

Thanks Sandun, I did indeed yes. I think it's a great webcam. A tripod is a must for it though in my opinion. It comes with a tiny one...but not enough stability unless it's just sitting on a desk.

[@SandunSuranga]

@@TheBeautyofMath definitely 🖐️, it's as clear as crystal But , I noticed that the lights automatically change when you move. How can you fix it?

[@TheBeautyofMath]

The problem is what's called exposure. I think? I have had to learn a lot about filming and such since I started. If you look at my older videos they had terrible lighting and sound. I did some research on that. In the logitech controls software you have to tell it to focus on either exposure or frame rate. But I can't manually control exposure with their software. So the solution is...I need to use different software called OBS Studio. Sadly though, there are literally hundreds of tutorial videos to teach how to use it, and until I watch those videos I can't figure some aspects of it out. I don't know where I want the exposure and why once I can manage it's controls. So it takes a lot of playing around with it and self teaching how to use it. But I don't have the time right now. I work 7 days a week. No days off. 8 to 10 hours a day...and then have to manage my life(dentist, doctor, grocery, etc.) Outside of that. But once the AMC 10/12 is over in November I plan to cut my schedule back to 6 days or 5 day a week. Maybe then I will have time to teach myself these things. In the meantime if I select exposure focus then the framerate sometimes isn't accurate and like...my lips will move but the sound isn't at the same point. If I put it on framerate focus you have this lighting effect. So my recommendation is if you buy it, you will also need to learn OBS Studio to properly use it.

[@SandunSuranga]

@@TheBeautyofMath It's really kind of you 💝 Thank you very much for your long response. May all your dreams come true 🖐️

[@SandunSuranga]

I'm your Sri Lankan subscriber 🖐️💝

[@happyproblemsolver944]

I remember using this problem lol. I remember myself going too fast on this problem and mistakingly reading the problem and sillying this. I thought that the problem was saying AM = CM = BM when it was actually AM = BC. So I thought the triangle was a right triangle. FOr some reason, I thought angle B looked right so I assumed it. I remember saying then BC^2 + 1^2 = 2^2 so BC = sqrt(3). And I put answer E...

[@TheBeautyofMath]

Man...I have done very similar things in my learning process as well. It feels embarrassing at the moment, but that's ok...if you learn from it. I hope it taught you humility, and being careful about reading problems carefully. If it accomplished that, then that's all that matters...the mistake is an opportunity to improve and nothing more. Onward and Forward :)

[@abhyuditatripathi5949]

video on menaulus theorem and ceva theorem if pssbl those are kinda tough for me :(

[@TheBeautyofMath]

I will see what I can do. Might not be before the test this year. But I will try and come back around to it at some point. Add it to the Problem Request Form on the About page if you can so I don't forget.

[@Russell-zk4lb]

ngl, i think this is the only vid u made dad jokes on

[@TheBeautyofMath]

Oh no Russell, you need to watch more videos. They are quite common lol. You probably skip around in the videos a lot amd miss them. A lot of times they are subtle so, people can miss them even when watching that part.

[@Russell-zk4lb]

@@TheBeautyofMath lol ok will look for tem

[@anvithkakkera7196]

Lol, man + dad

[@TheBeautyofMath]

right? lol it's funny...but it's effective for information recall. :)

[@SivamuruganVelmurugan]

Who else looked up the wooga looga theorem as soon as he mentioned it?

[@TheBeautyofMath]

😂

[@randomacc6151]

@@TheBeautyofMathLove your haircut 😊😊