Pythagoras Would Be Proud: High School Students' New Proof of the Pythagorean Theorem [TRIGONOMETRY]

Calcea Johnson and Ne’Kiya Jackson are two high school students at St. Mary's Academy in New Orleans who recently presented a new proof of the Pythagorean theorem at the Spring Southeastern Sectional Meeting of the American Mathematical Society. There are, of course, many proofs of the Pythagorean theorem, but what sets this one apart is that it's (mostly) trigonometric, but does not circularly rely on the Pythagorean identity. You can read the AMS abstract here (meetings.ams.org/math/spring2..., and you can read more about these remarkable young women here (www.theguardian.com/us-news/2....
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There's not a lot of information online yet about how their proof proceeds, but from a few of the slides you can pull out of news coverage (you can see de-skewed versions of three of them here: www.dropbox.com/s/h99ezl8l360..., it seems the proof relies roughly on the definition of the sine ratio itself and the Law of Sines, along with a very clever "waffle cone" triangular shape.
If you'd like to play around with the waffle cone shape, I built a little illustration in Desmos that you van view here: www.desmos.com/calculator/tyw.... You can see the interplay of the infinitely many right triangles going down to the right from the isosceles triangle, and it also includes a folder with the proof.
Finally, some news coverage has gone a little overboard on stating the accomplishment here. It is true that a collection of proofs of the Pythagorean theorem from roughly a hundred years ago stated that a trigonometric proof was impossible (Elisha Loomis wrote "no trigonometric proof is possible" here: files.eric.ed.gov/fulltext/ED.... But in 2009, Jason Zimba published just such a result in Forum Geometricorum: forumgeom.fau.edu/FG2009volum.... You can see that proof and other related proofs at Cut the Knot:
+ Jason Zimba's proof: www.cut-the-knot.org/pythagor...
+ More Trigonometric Proofs: www.cut-the-knot.org/pythagor...
+ Nuno Luzia's Half-Angle Proof: www.cut-the-knot.org/pythagor...
+ John Arioni's Convergent Geometric Series Proof: www.cut-the-knot.org/pythagor...
#pythagoreantheorem #pythagoras #impossibleproof #highschool
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[@polymathematic]

Appreciate all the thoughtful comments! If you're interested in pursuing more mathematics, I recommend checking out Brilliant.org. Brilliant's got an offer for my viewers right now. Head to brilliant.org/polymathematic/ to try everything Brilliant has to offer-free-for a full 30 days. Thanks for watching!

[@HoSza1]

It's most simple to use the law of cosines (LoC) to prove the Pythagorean theorem, if someone wants a purely trigonometric way. LoC itself has a trigonometric proof itself, check the Wikipedia article for it: the proof does not use the sin²x+cos²x=1 identity anywhere, so it is not circular. Why they thought then trigonometric proofs were impossible? Or am I missing something? Someone please enlighten me!

[@ilhamisgndrov6180]

kzbin.info/www/bejne/n6nKcoSdor6ifqssi=O3wU7qznu0AXP9Kl

[@ilhamisgndrov6180]

Посмотрите тоже на это видео ролик

[@YoungNino2017]

@@HoSza1 It's just a bullshit strawman claim to further pretend these two high school frauds did something new; which they didn't

[@jimhabegger3712]

The real value that I see in what those students did is that it's a proof that's creative, resourceful, and possibly never thought of before; and that they tried to do something that they thought (mistakenly) mathematicians have considered impossible for 2000 years, and never gave up until they succeeded. Unfortunately, the reporting and public discussion has been almost exclusively around that false claim, instead of the real value of what they did.

[@polymathematic]

Totally agree. The news coverage definitely got out of hand. One story I read said mathematicians thought it was true, but it had never been proven!

[@jimhabegger3712]

@@polymathematic I would want those students to know that a person doesn't have to believe any false stories about what's new in their proof, to see value in it and in what they did.

[@leif1075]

@@polymathematic I'm going to try to come up with the same proof they did before watching this video or consuming any other media about them to see if I.can independently replicate their work..I mean if they did it, I should be able to without much difficulty right? Thanks for sharing .

[@marcomoreno6748]

​@@leif1075"...i should be able to do it without MUCH difficulty right?" How did you come to that assertion?

[@AlFredo-sx2yy]

@@marcomoreno6748 how did the news come to the assertion that mathematicians have thought it was impossible for 2k years when there's a paper from the early 2000s by a mathematician named Zimba that back then was already considered to be the first fully trigonometric proof of pythagoras? A lot of people reach some assertions that make no sense. It is good to question them. The problem is when you only question half of the nonsense.

[@kingarth0r]

I love that while all the math here is high school level, it's still creative.

[@flyingraccoon5262]

Bro i always double take seeing you in yt comments and not on disc 💀💀

[@haideri0313]

hi mr arthur

[@tasse0599]

We didn't do series in highschool

[@mitchratka3661]

The infinite sum is typically taught in Calc II, which is a college-level math, especially with the geometric series convergence. But if your high-school offers AP Calc II then I guess it counts.

[@Random131_]

@@mitchratka3661 I mean I took it extensively starting from 10th grade, but I do believe they got help, with the “waffle cone” idea I presume

[@loganreidy7055]

Really cool proof. There's 2 general takeaways from this, 1) as scientists we should keep open minds to old problems and tackle them with ingenuity and 2) draw triangles

[@thatweakpowerlifter2515]

Scientists? Math is not science.

[@loganreidy7055]

@@thatweakpowerlifter2515 I guess I will tell my mathematician friends that their B.S. and M.S. degrees are no longer "Bachelors of Science" or "Masters of Science"

[@stuartdparnell]

Haha, how do we do this to come up with quantum gravity that fits with current observations of the universe?

[@slothbearanonymous]

@@thatweakpowerlifter2515 I literally have a Bachelor Degree of Science in Mathematics

[@thatweakpowerlifter2515]

@@slothbearanonymous when we say science we are referring to natural science. Mathematics is formal science, like computer science. It doesn't follow the scientific methods.

[@chrislankford7939]

I'm guessing this "waffle cone" shape can be used to derive the Taylor series expansion for sin(x) or cos(x) using only geometry, too, which is neat. Props to Johnson & Jackson.

[@dotatwo5413]

Was thinking just that. This infinite right triangle cascade can be a seed for series expansions of many trigonometry expression.

[@griof]

A series os even and odd powers speaks sin and cos!! I was thinking the same

[@PrismariLaura]

Props to Calcea and Ne'Kiya. Use their first names, in order to hopefully show to young girls that they, too, can make history in maths.

[@johnrickard8512]

Well...using geometry and a limit anyway. Calculus is such a handy thing.😊

[@bobweiram6321]

I wonder if their proof can also lead to another proof of Fermat's theorem. It might reveal new insights to help find a proof.

[@squorsh]

I can totally see why nobody had thought to prove it this way yet. This is really complicated, but still elegant. Major props to the students who discovered this.

[@iyziejane]

Nobody has thought to use a sledgehammer to kill a fly either, as effective as it would be

[@iyziejane]

@@telanis9 The idea that proofs should use weak techniques to prove strong results has been part of mathematics since Euclid, at least. So it's less of my opinion, and more of explaining what is valued in mathematical proofs over the past 2000 years.

[@JeremyOuelletteNH]

@@iyziejane first its a "sledgehammer", now it's "weak". Sounds like someone's just feeling a bit jelly 🥹🤣

[@iyziejane]

@@JeremyOuelletteNH I don't care about the girls, I just hate the media and the way they lie about everything.

[@randomnobody660]

@@JeremyOuelletteNH no comment about iyzie's claims in general, but you are misreading. Sledgehammer is presumably talking using calculus-ish tools to prove trig identities. Weak techniques is what SHOULD have been used, which the sledgehammer isn't.

[@dkuhlmann]

Greαt insight by those students. Technical point: This proof doesn't work is α=β or a=b since those long lines added will be parallel. Not to worry though since 2α=90 degrees the we have sin(β)=a/c from original triangle and sin(β)=c/(2a) from larger right triangle. So 2a^2=c^2.

[@polymathematic]

Very nice evaluation of the degenerate case! I dealt with it a little lazily: I just programmed it into Desmos to make that impossible :) Not sure how the girls dealt with it, but I'm eager to see once they release their final paper.

[@ericerpelding2348]

Right angles with side a less than side b appear to be the preferred case in the proof(s).

[@jaimeabs]

Great observation, but probably the two stunning girls must have seen that.

[@dkuhlmann]

@@jaimeabs I assume so, also.

[@grant1390]

@@polymathematic I don't think they did deal with it. It's not a proof.

[@michaellarson2184]

Brilliant! I’m amazed that two high school students came up with something so creative! It’s a wonderful achievement!

[@ianjohnston379]

Yeah, I don't buy it.

[@tinsalopek7740]

​@@ianjohnston379 why, not that hard probably if you are actually trying to solve it, most of the people just did not bother to since there was a good reason to believe that its not possible. proof itself is not that hard to come up with tho

[@francisluglio6611]

@@ianjohnston379 there’s always someone pathetic like you commenting

[@SayAhh]

​@@ianjohnston379 We need ppl like you (cynics), who are impossible to convince no matter how much proof or evidence are presented. Unfortunately, while their distrust might serve a great purpose to the entire population (or subset thereof), they sometimes also tend to fall for the most common (and least sophisticated) scams at the same time. Quite puzzling.

[@Random131_]

@@tinsalopek7740 I believe the “waffle cone” was an addition of a helping teacher

[@secondchancecookies]

As a math teacher in the New Orleans area, super proud of these girls.

[@signumcrucis71]

Surprisingly, it was easier to solve the Pythagorean theorem problem than to prove that a woman with a penis is not a woman!!

[@billyscenic5610]

@@signumcrucis71 How do women come from Rib bones? explain it mathematically.

[@signumcrucis71]

@@billyscenic5610 Don't ignore the riddle here, is a woman with a penis a woman???.....since you're so smart with mathematical equations, why don't you solve this riddle?

[@japphan]

What intrigues me the most is the waffle cone, becuase it is a general idea. You can use it to prove the pythagorean theorem, but it might be useful for other things. It is a concept. You can generate waffle cones out of any straight line. Wherever there is a straight line, there is a waffle cone. But it is not just the waffle cone itself. It highlights how geometric problems can be solved using infinite converging series. Any shape that can generate an infinite number of shrinking (and possibly growing) copies of itself, can provide some insight to that shape. And you can do this for any symmetrical 2D object, using the actual waffle cone, i.e. use the side of a dodecagon to create a waffle cone, use the side of a smaller triangle to draw a new dodecagon. We can use the same method for some 3D objects, what about higher dimensions? Can we distinguish between waffleconeable shapes and non-waffleconeable objects, and is that distinction somehow useful?

[@signumcrucis71]

Surprisingly, it was easier to solve the Pythagorean theorem problem than to prove that a woman with a penis is not a woman!!

[@japphan]

@@signumcrucis71 Please take your transphobia back to nazi Germany.

[@Jim-be8sj]

This is fantastic. I love any proofs that bring in a geometric series and these students did it twice. Very inspiring. I hope math instructors far and wide share this accomplishment with their students. Having great role models like these two could inspire many people to keep pushing themselves to achieve more than they had thought possible.

[@shimrrashai-rc8fq]

Yeah, but how many of those will achieve enough to make the peer of this in whatever field they go at, even with the exact same diligence?

[@ashb2483]

I’m gonna show my students today

[@swampwiz]

How about all the role models that have also reproven this or any other proof? Something that must be kept in mind with anyone doing a reproof is that xe knows what the outcome will be beforehand.

[@signumcrucis71]

Surprisingly, it was easier to solve the Pythagorean theorem problem than to prove that a woman with a penis is not a woman!!

[@permanentmigraine]

This honestly might be one of my favorite proofs ever; it's just so clever and thoughtful. What a wonderful thing.

[@Mossbeehave]

Brilliant & humble young women from St. Mary’s high school in New Orleans! More blessings 🙏

[@jcortese3300]

I love it when complex things are done with high school level math. That's one of the main reasons I love special relativity so much -- a kid can get a grip on it as a sophomore in high school. Congratulations to those kids!

[@heisenberg4996]

Definitely not high school level math. The calculus used is primarily taught in Calculus II at most colleges.

[@swaroopkamble2387]

@@heisenberg4996 Calculus is taught at many high schools, albeit, as an AP class (Calc AB/BC).

[@heisenberg4996]

@@swaroopkamble2387 Calc I maybe, but not Calc II.

[@swampwiz]

President Grover Cleveland came up with a reproof, and I'm fairly certain that he only had a basic high-school mathematics background.

[@signumcrucis71]

Surprisingly, it was easier to solve the Pythagorean theorem problem than to prove that a woman with a penis is not a woman!!

[@michaelaristidou2605]

Finally, some accurate description of the issue. Because the media grossly misinformed the people of what the students actually achieved and did. They were like "they solved a problem that mathematician couldn't do for 2000 years", etc. Or "First proof of Pythagoras Theorem", etc. Crazy!

[@saurelius5217]

More likely it was solved long ago but the information was lost.

[@NathanHedglin]

​@@saurelius5217 like the fast Fourier transform

[@MrVirus9898]

Its frustrating, but what do you expect? Most folks have a hard enough time with sales tax.

[@peelysl]

@@saurelius5217 What was solved exactly?

[@profd65]

They still did more than you'll ever do. Maybe the media will get that part right.

[@yellowlynx]

The key is that the mathematics involved are not difficult at all, even for me who left college decades ago and barely touched mathematics for years. I could see the ingenuity and elegance in the proof they discovered. The girls did a marvelous job.

[@forthelulz8085]

I thought this would be fun, but my headache says otherwise. I am glad there are smarter people on this planet than myself, because this breaks my mind.

[@polymathematic]

ha! fair enough :)

[@brucewayne1777]

The biggest thing for looking at proofs, I've realized, is to slow down. Every step in this case is coming at a mile a minute. The girls who wrote the proof didn't do it this fast, and polymathematic didn't understand it and derive it as fast as he did in the video. If you slow each individual step down and don't move on until you full understand it, it makes a _lot_ more sense!

[@regroff]

@@polymathematic Nice video! I was wondering what tablet and software you are using for writing in this video?

[@jaimeabs]

​@@regroff I'm also looking forward to hearing the answer to this important question of yours for those who teach.

[@simpleman283]

I think all of us know about that headache. I worked on a problem for 7 days, about a year ago & I was exhausted for 2 months.

[@YashvardhanVerma]

That's such a creative and beautiful proof. And thank you for the video. I was looking for the proof since the news came out, and looking at it, is certainly a great achievement.

[@ckq]

4:00 Consider extending the line with length b That creates a new right triangle with sides c, ca/b, and hypotenuse b+a/b = (a²+b²)/b Note that the hypotenuse also equals c²/b And we're done

[@barbrcuejoe]

The last step, "Note that the hypotenuse also equals c²/b"" comes from the similarity of the original triangle to the big triangle. Although the law of sines can be used, it is not essential; everything results from similar triangles. Nice job, ckq!

[@AlWaller-zp9rg]

Excellent alternate proof!! See my other reply to Polymathmatic asking him to create video of your proof. And you inspired me to make a similar alternate proof (interior construction).

[@charlesstimler9276]

WOW, Fantastic Awesome Proof!!

[@apriljones1381]

I’m not really skilled at math, but I understood the challenge was to prove the theorem using trigonometry, which had not been done for 2000 years until someone did in 2009 I think? And then these two young women each came up with a trigonometric solution, so now there are 3 after 2000 years of none. Yours seems like a geometric solution to me. Is that not the case?

[@serdnasocram]

I've been waiting for a video like this to drop. Every news outlet that I read had their brains frying over some high school math. Thanks for making the video

[@graysilver007]

Those students are doing really well! It's not until about Calc II that you start doing limits and integrals and only certain highschools will have lectures for that. Kudos to them for thinking of taking things to the limit to breathe life into the sciences.

[@theobserver314]

Limits are taught in Calculus 1. Infinite series on the other hand is taught in Calculus 2. But I suppose it depends on the curriculum of the school.

[@graysilver007]

@@theobserver314 yes Infinite series is Calculus II. You'd have to do algebra I/II/Trig by middle school so by about 10th grade you could complete Calc I/Calc II. Then 11th grade you could start applying the stuff from Calc that you've learned toward things that have not been proven.

[@teikokugirl]

Thank you for the desmos link :) this was very nice to watch!!! Love the energy and love the math :)

[@polymathematic]

glad you enjoyed it! thanks for watching :)

[@amethystklintberg7436]

This is gorgeous and creatively motivating! Thanks for explaining, and well done Johnson and Jackson!

[@psingh63]

A brilliant piece of mathematical thinking. Well done to those high school students. Just one minor thought. The proof depends on the convergence of the series for a/b < 1. ( If b < a can easily redraw the triangle ). When a = b and alpha = beta the u and v values the series sum to are divergent, i.e. there are singularities in the formulae for u and v at a = b. But in that later case you don't need the series part of the proof.

[@ankan1627]

Thanks for a great explanation. The waffle cone was definitely extremely clever and an inspired path to take. Great job ladies, and congratulations !

[@PragMaliceTV]

This is a wonderful argument for MOST right triangles where a is not equal to b. The technique of extending the triangle infinitely using a scale factor of a/b technically only works if b > a, as this ensures that the size of subsequent triangles are getting smaller and smaller. Thankfully, if a>b you would just extend in a different direction and use a scale factor of b/a, and you will end up with the same result, so therefore assuming b>a is fine in this regard. However, this technique falls apart if a=b, as a/b (or b/a) would be equal to 1, and the lines formed by the hypotenuses would in fact be parallel, never intersecting to form the larger triangle in question. This is further evidenced later in the proof where the construction has to deal with b^2 - a^2 in a denominator, except if a=b, then we've just divided by 0 and therefore the resulting expression is undefined. Hopefully when the full proof is released, we will see how they have (or haven't) accounted for this special case (specifically by way of the trigonometric ratios). Regardless, as a high school math teacher myself I still have to applaud these students' rare display of ingenuity in developing this approach to at least a partial proof.

[@JoelRosenfeld]

That case can also be handled by using the Law of Sines. I show how to do it here in my video on the topic. kzbin.info/www/bejne/rabcp5eBrb2qhZo

[@winnetou9706]

Good point, I would suggest that the case a=b is limiting. Using a sequence of triangles with fixed "a", and "bk" sequence such that "bk ->a" from above. This way the resulting sequence of hypothenuses "ck ->c" from above as well. Since for each term in the sequence we have "a^2 + bk^2 = ck^2" with "bk -> a" and "ck ->c", then by continuity of multiplication and addition. we must have "a^2 + b^2 = c^2" as well for the case a=b.

[@gvc76]

for the case a=b, see the answer from @dkuhlmann above.

[@ishwar8119]

Might be a classic case of "if it doesn't hold at a single point (or a countable set of isolated points) then since everything's continuous it still holds for this case". Useful sometimes in physics, where you divide by a parameter that changes with time which might be zero at particular instants.

[@danfretwell2214]

Another way to interpret this proof is that it gives a geometric interpretation of the construction of new pythagorean triples from old ones. The usual formula to generate Pythagorean triples over the integers is (k(m^2-n^2), 2kmn, k(m^2+n^2)), for arbitrary 0

[@Grunchy005]

The infinite series part is an elegant expansion, excellent thinking there. It's one step further than the typical "double angle" construction, well, I guess it's an infinite number of steps further!

[@victoralvarez1610]

Awesome! I’m currently taking a Calculus 2 class, and found this proof fascinating. However, I wonder if this works for 45-45-90 triangles: because the infinite series couldn’t be assumed to be convergent if b=a, correct?

[@polymathematic]

that's exactly right. you couldn't use the convergent series for any a greater than or equal to b. the a is greater than b case is easy enough to deal with, because you can always just flip the triangle around. but you have to deal with the a = b case differently. i'm not sure what the new orleans teens did to deal with that case, but i'm looking forward to finding out!

[@gvc76]

There is a comment from Dkuhlmann above, answering that exact question.

[@Rajeev_Walia]

Great work by the students! The part of the proof that you were saying was not purely trigonometric (infinite serries) can actually be proved by trigonometry. Consider the triangle made of all of the infinitely many triangles except the first two and apply Sine Law on this triangle. You will be able to find u and (v - c) in terms of a,b,c. The rest is the same.

[@polymathematic]

very cool!

[@garyknight8966]

A most remarkable tour de force of patience and imagination by Calcea and Ne'Kiya (I would say patience was the larger part). I note that strictly speaking the steps leading to the two long sides as ratio'd over a^2-b^2 fails when a=b , still less of final cancellations of a^2-b^2, is strictly not justified when a=b , the 45 degree special case of the original triangle, for which the 'cone' goes to its u=v = \infty limit with parallel sides. Nevertheless, this is a 'trivial' lacuna because then it is certain that a^2+b^2=2a^2 and the equation on the right already says (for this triangle) that 2a^2=c^2 . Without having seen their proof, I will just assume they set aside this trivial case at the start. As a high-school math teacher myself, I say Kudos to these ladies !

[@karlove9360]

That was awesome, thank you so much for your explanation, so much better than just reading the news article

[@JustChives]

1: Why did I click on this? 2: Why did I watch the whole thing? 3: Why was it so cool?

[@NitroniumGaming]

This proof is certainly a beautiful and unique one, though the significance of it being one of the first "Trig" proofs is more up to debate. The same proof can essentially be completed without the need for trig (and keeping the same logic) by replacing all trig identities with side length ratios instead. After all, this is the original proof of the sin law. It's a few lines of algebra separated from a completely algebraic proof, so I'm not fully convinced it should be considered a trig proof. Jason Zimba's proof does use deeper trig realizations, but at the end of the day, the lines of what "method" a proof uses can become blurry.

[@asaejapan7143]

Thank you! A swarm of praising voices often makes it hard to express a cool headed angle even shroud the brain's excellent intuitions.

[@topdog5252]

Fantastic proof! Congrats to the two girls. I wonder if Pythagoras himself would have actually liked this proof if you could present it to him? I know Archimedes used techniques that look a lot like summing infinite series over 2000 years ago, but even he came centuries after Pythagoras. I don't know if Pythagoras ever did anything like that, but considering the Pythagoreans were meant to have been horrified by discovering the irrationality of √2, they might have struggled with the idea of infinity and summing infinite series. Would he actually have been proud? He might have a lot of thinking to do before he became satisfied.

[@polymathematic]

i presume the infinite series would definitely have been looked down on :)

[@imanplays89]

The square root of two shows itself exactly when a=b which is a degenerate case. It is interesting.

[@caspermadlener4191]

Only if Pythagoras did exist. That is not really a given.

[@topdog5252]

@@caspermadlener4191 true

[@JordanWeitz]

@@imanplays89 Indeed. And the proof breaks down because of division by zero.

[@trueriver1950]

You don't actually have to sum the series, just show that it converges for all ab by swapping the letters around, or by using the incantation "wlog". We still have to do a special case for a=b, unless we are physicists when cancelling by infinity is called renormalization (which makes it OK -- see for example the renormalization of the mass of the electron in Quantum Electro Dynamics). For some reason mathematicians are doubtful of this procedure...

[@Pyrokan]

It's always neat to see how cool intersections of math branches can be - trigonometry and calculus in this case. Brilliant!

[@NiceMicroTV]

I like it when it is math that is elegant, creative, and I can follow the whole thing. Thank you!

[@ReneGrothmann]

I admire the persistence of these students. I wish I'd have had students like this more often when I was active. However, this is just a very involved way of getting the theorem. It had to come out in the end, of course, simply because it is true. The problem with the approach is the following: If you use the definition of sin(\alpha) as a/c in any other triangle than the unit one (c=1), you are using similarity. I.e., triangles with equal angles have equal proportions between their sides. But there are very easy proofs of the theorem using similarity only. So the computation shown here is based on facts which yield the theorem in a much easier way.

[@douggarfinkel2415]

When I teach trig, we regularly use the theorem to derive a missing ratio, and we also use it in defining the law of cosines. So it is intriguing to see a student use a measure that is based on the theorem to prove the theorem.

[@austin6876]

awesome proof! great accomplishment for the two high school students

[@polymathematic]

yep! i hope they both had a great experience, and go on to more math conferences in the future :)

[@brian8507]

​@polymathematic I don't think those girls did this. I think someone else did and is letting the girls have the credit for woke reasons. This is why u ain't seeing these girls present. Also this proof is cool... but it ain't ground breaking lol. Yes I am a real mathematician lol. And real mathematicians secretly think something funny is going on

[@polymathematic]

@Zigest whatever you have to tell yourself!

[@salmanel-farsi3744]

Great explanation of the proof. This proof is more about ratios and algebra. The trig notation, I admit is very useful to declare those ratios. Great job by the students.- very creative solution!

[@Galileosays]

I agree, the trig notation is obsolete. It would be more in line with Pythagorean approach not to use it.

[@realman1936]

The proof has nothing to do with trigonometry. The trig. functions were just shorthand notions for ratios. That these young ladies did was to find yet another proof of the P.T.

[@l.h.308]

My favourite proof is where you take 4 equal triangles with hypotenuse c (and sides a, b, a > b) and place them so that the 4 hypotenuses make a square with area c^2 with all triangles inside it, getting a smaller square in the middle, with sides a - b. Putting the area of the big square equal to the 4 triangles plus the small square you get immediately a^2 + b^2 = c^.

[@refrainrestrainresist-3rs49]

Great proof! I'd also like to add that if we are using the infinite series sum formula "(first term)/1-(common ratio)" then that means (common ratio)

[@henrik6151]

How about when a=b?

[@nakhleasmar9175]

Very nice proof. I am very impressed that high school students persisted with this multi-step proof. It's a level of maturity that you don't see in most college students these days. Congratulations to CJ and NJ!

[@arjankroonen4319]

"these days"... nice. So what did your generation ever do for us?

[@your_-_mom]

@@arjankroonen4319 computers, quantum physics, etc

[@profd65]

@@arjankroonen4319 A lot more than your sorry TikTok generation.

[@1mol831]

@@profd65 What about the ones born recently?

[@arjankroonen4319]

@@profd65 "Your sorry TikTok generation" Not sure how old you think I am but I guess you are a few years off...

[@aspieguy9133]

Very impressive to view things in such a dynamic way (never mind still in high school) that I despite my love of maths still wouldn't have came up with that with pointed questions. Didn't realise that there was a belief that trig couldn't simply be used to prove pythagora's theorem. I believe I have one (which I think is fairly simple) and sure avoided inherit cyclic proof.

[@ProfessorBeautiful]

Beautiful proof, beautiful and clear presentation!

[@bigolbearthejammydodger6527]

congrats to those 2 students! Love to see original thinking by kids, Id also like to say its great to see a teacher that RECOGNIZES this, actually bothered to understand the presented work and then promoted their students work. That I think is sadly the truly rare thing here. When I was at uni, my self and my lab partner came up with an algorithm for hit detection in 3d space that was an order faster (n squared + n) than the expected result presented in the course (n cubed). We thankfully had one professor who understood what we had done and pushed it out there, the professor who gave out the assignment simply told us we had 'done it wrong' - and we had to fight it because we knew it worked. So kudos to the young ladies! BUT also kudos to the teacher!

[@MottiShneor]

This is a beautiful proof! mostly because it only uses what high-school students know and can play with. No "Higher math mind set" needed. It is direct and as far as I can see doesn't use any "dirty tricks". The building of infinite set of triangles in imaginative and ingenious, The use of sum of infinite series is great. Coming to think about it... when you PROVE the sum of those infinite series, don't you, somewhere, rely on Pythagoras theorem? that needs to be checked - or the proof is circular.

[@codtelly1124]

Thanks for breaking this down into clear steps.

[@polymathematic]

glad you liked it!

[@ChaineYTXF]

Sir, you gained a subscriber. Congrats to Mmes. Johnson and Jackson. A superb feat. The Greeks of Old would have adored this. Very geometric. Although Zeno might have had something to object For those who don't know: look up Zeno's paradox.

[@brian8507]

I don't think these girls came up with this. I think it's a media stunt

[@rodschmidt8952]

Mlles., I think you mean

[@nigelmansfield3011]

Absolutely brilliant work by the two students. I'm so impressed and excited that I can barely think.

[@SureshSuresh-qg3xr]

I saw a similar explanation of the proof in another site. You have b^2 -a^2 in the denominator which will not work for an isosceles triangle. In that website, he made a separate case for this and proved it trivially using sin 45 degrees = 1/sqrt(2), which of course depends on Pythagoras in a circular fashion

[@mbfanz]

Wow, very interesting proof. I didn't follow the infinite series part because I forgot most of what I learned at school lol. But everything else surely makes sense. Good job to the students! 👏

[@niwaka273]

Thank you for the video. Even though I love Math, my ADHD makes it real hard to focus enough to follow proofs. But your explanation was a great treat! :D Loved the delivery

[@johnp1]

This was on the news. I didn't understand how they did it until now. Thanks for the explanation.

[@stigmontgomery7901]

Wonderful. Congratulations to those two ladies. You have to wonder why this approach was never thought of or attempted before? Congratulations again.

[@LizhouSha]

This is an awesome proof, but I don’t think you need to use the law of sines at all to prove it. You can draw the altitude from the vertex of the angle beta to one of the sides of length c in the reflected triangle, which by definition is equal to c * sin 2alpha, which for brevity I will call h. But the area of that triangle can also be expressed two ways: either as (2ab)/2, or as hc/2. So h = 2ab/c as expected, and no law of sines required. Similar argument applies to the big triangle: its area is either cu/2, or hv/2, so h = cu/v. Of course one could argue that this is exactly how you would prove the law of sines in the first place, but I really think that the crux of this proof is in the use of the sum of the geometric series, not in the law of sines! The a=b case is trivial because it can be proved by arranging four of the same right triangles in a big square, with 2a being the diagonal. The big square’s are is either c^2, or 4(ab/2) = a^2 + b^2. So no infinite series needed.

[@Bietente]

Thank you for the detailed explanation. Based on some of the news articles you would think they revolutionized maths as a whole and proven all mathematicians of the last 2000 years wrong so I was a bit sceptical. It's great to hear that maths still stand but they managed to do a really cool thing especially for their age, that deserves to be celebrated without the need for exaggeration😊!

[@gafam7251]

You got scared didn’t you? The relief that breathes through your typed words are telling.

[@johnnyragadoo2414]

Wow! In hindsight, it really makes sense. A right triangle's area is always greater or equal to 2ab, so dividing that by a fraction sort of fits the topology of things. May I offer the observation of a dummy who just didn't get math in high school? My geometry and trig was taught via experimental education without any mention of the unit circle. When I found that on my own, I thought it was a beautiful thing. Identities like sin^2+cos^2 strike me as a little disembodied. I recently worked out a straightedge and compass proof of the law of cosines, and one thing that helped was to remember that identity is incomplete in the real world. I know, it's just me, but I always remember (sin(ϴ)c)^2 + (cos(ϴ)c)^2 = c^2. Only for the special case of c=1 is sin^2+cos^2=1, and at that the units of "1" don't show it's a squared quantity. I know, I'm not the brightest bulb. It's ok.

[@elielc2431]

Awesome explanation! Thanks a lot for taking the time to do this! I was wondering, what software are you using to write? it seems really easy to manipulate the texts and everything else. TIA

[@galaxyquestminute7490]

Wouldn't this proof technique fail for a 45-45-90 right triangle? For any other right triangle you can assign your alpha and beta, without loss of generality, such that (2 * alpha) is less than 90 degrees, leading to the convergent waffle cone shape. But in a 45-45-90 right triangle, (2 * alpha) will itself be 90 degrees, as well as the measure of (alpha + beta), so instead of a waffle cone, you'd get an infinitely-long rectangle

[@polymathematic]

yes, you'd have to demonstrate the a = b case separately. i'm told they used four different proofs in their paper, so it's possible one of those other proofs addressed that case, but i don't know.

[@EuropaE]

The amount of creativity and intelligence needed to find a NEW proof for a theorem as old and well known as the Pythagorean theorem is immense. I hope those two young women get a full ride through college from this!

[@en2-joserivera896]

Hi.

[@rodschmidt8952]

I wonder if they will go to the same college and collaborate

[@kitsurubami]

You explained it so well that even i was able to follow! Subscribed

[@Amb3rjack]

Well, you know what, I could actually follow all of that! I am well made up right now! Fantastic stuff! Thanks for an amazing vid upload!

[@safdghjklyftdrseawehhjk]

Thanks for breaking it down! These girls are awesome and their proof is so creative! I haven't really gotten their story yet, not that it's very important, but does anyone know if they developed the solution independently on the same bonus question or was it a pair-work situation wherein they created this new proof?

[@vivianomelime4008]

actually the other student created another independent proof. watch 60 minutes on their theorems

[@sesurin]

Happy to see an explanation. When the story broke awhile back, they never went into the math.

[@ofir.ll2004]

hey i think i found a problem in this proof. in 8:20 you use the formula for infinite series, that only works when the common ratio's value is lower than 1. if this is proof for the pythagorean theorem then who says a^2/b^2 is less than 1? this then excludes the posibility of what happens when both sides are equal (if they are then the sum is undefined).

[@azureskybox3024]

Actually, the limit part can be avoided. The shape (triangle) with side lengths u, 2a and v-c is similar to the one with side lengths v-c, 2a²/b and u-(2ac/b). Taking ratios, we get simultaneous linear equations involving u and v which can be easily solved. This was a really great proof!

[@mychaelsmith6874]

You can also use the law of sines.

[@nicolangelolavermicocca8093]

I have a question, the convergence of the geometric series is only possible if a^2/b^2 < 1, what happens when that ratio is equal to 1, that would be the case of alpha and beta equal to 45 degrees

[@polymathematic]

yes, you have to presume that a

[@nicolangelolavermicocca8093]

@@polymathematic thanks I had not seen that comment

[@rickthompson3231]

Amazing job!

[@damon22441]

Wow, this makes me think I should go back and look at how I found another way to solve matrix translations. It didn't work in specific circumstances (my hs precalc teacher made sure to put those circumstances on the test when I showed her my method) but maybe I could refine my rule on the exceptions and do something with it?

[@6884]

chapeau to the two young students, this was a beautiful journey. Isn't it mindblowing that thousands of years later we can still get interesting stuff from one of the most known cornerstones of elementary mathematics?

[@johnchessant3012]

very creative, bravo to them!

[@celestialnubian]

I think the most important thing that Calcea and Ne’Kiya proved is that our high schools kids are capable of so much more that they are currently expecting of themselves.

[@solodiety996]

so their another solution for this form, and that is u can add the value of tan(a)+tan(b)=a/b+b/a as tan(a)=a/b,sin(a)=a/c,cos(a)=b/c similarly to tan(b) if u can find that next is we can add and convert the lhs to sin cos function sina.cob+sinb.cosa(whole divided by)cosa.cosb=a^2+b^2(whole divided by)ab here u have remember some situation related formulae is angle is lower than 90 degrees i.e sin(a+b)=sina.cosb+sinb.cosa and that angles a+b+90 degree=180 degree so a+b=90 and sin(a+b)=sin90=1 so u can hence write like 1/cosa.cosb=a^2+b^2(whole divided by)ab here u can do some mathematical operations, funny right we will multiply 2 to numerator and denominator respectively to lhs hers 2cosa.cosb can be written as 2sina.cosa why? cause u can find that cosb is equivalent ot sina so we have another formula that is sin2a=2sina.cosa we can replace it and we get 2/sin2a=a^2+b^2(whole divided by)ab here u can see 2ab/sin2a can be founded easy which is equivalent to a^2+b^2=c^2 thank u if u get it.

[@idolgin776]

I thoroughly enjoyed this proof. It is pretty complicated, but I think that looking for new connections between mathematical concepts is the best thing a Math student (high school, or even college) can do, stepping outside the prescribed curriculum. Kudos to these young ladies!

[@ericerpelding2348]

I am going to venture to say that the last page of the screenshot document shows a construction that will be used to determine the value of sin(beta - alpha), and possibly also sin(2*alpha).

[@polymathematic]

i was a little curious about that slide myself, since I didn't end up needing that altitude. Their approach could turn out to be quite a bit different depending on what they were doing with that.

[@ericerpelding2348]

@@polymathematic One news story mentioned that the students gave four proofs in their presentation.

[@kirkhodges1946]

I like to think that despite involving an infinite series it’s still trig (you do get to see why Greece never arrived at this, which is cool. Newton would have been proud).

[@guillaumehuguet3243]

For the sake of completeness, it should be mentioned somewhere that b>a, aka that by definition b is choosen to be the longest side length. This is necessary afterwards so that the series converge. From this side remark one can notice that there is a small issue with the case a=b (the waffle cone becomes an infinite strip). Of course it is quite trivial to complement with a proof for this special case.

[@litovillar6027]

Ooohhh I love how the girls used different mathematical tools to find another proof of Pythagorean’s theorem❤❤❤❤❤ So much math skills😊😊😊😊

[@StarOnCheek]

The fact that highschoolers found this makes me wonder how many times someone did this without realizing that it was so special

[@graememorrison333]

Sadly, that's my take on it also.

[@TheDeltaboss]

Absolutely amazing! Kudos to the Ladies.

[@aaronbraskcapital7155]

Elegant! My company logo is actually an implicit proof of the Pythagorean theorem :)

[@frederickd.provoncha8671]

You know, the math involved isn't even all that complicated. Basic trigonometry, basic algebra, basic calculus. I was able to follow the process pretty well. The idea of the "waffle cone" is the truly creative, original part. I'm a bit jealous that I didn't think of that. My hat is off to these two ladies. Great job, brilliant idea!

[@realman1936]

One can't use 'basic' or normal trigonometry to prove the P.T. All of trig. is dependent on the P.T. being true. They used basic algebra and calculus.

[@user-wk3bw9zg9b]

Hello Polymathematic, my question might be dumb but how to demonstrate properly that a^2/b^2< 1 in your geometric serie. It seems obvious since it is a right triangle, but how to demonstrate that in a right triangle the hypothenuse is the longer side without actually using pythagorean theorem.

[@polymathematic]

not dumb at all! so, this proof will only work if the geometric series converges, which as you say, requires that a² is less than b² (or simply that a is less than b). If a is greater than b, of course, you could just switch them around, so that case is no problem. where you do run into the problem is if a equals b. but in that case, the original right triangle can be turned into a self-similar isosceles right triangle, and you can get to 2a² = c² in the way that another commenter did below. hope that helps!

[@aman-qj5sx]

It is possible to prove that for any triangle, the side opposite the largest angle is the largest side. This can be done without pythagoras. Since the sum of angles in a triangle is 180 degrees (provable from the parallel postulate), this is all we need.

[@johnschmidt1262]

Thank you for the explanation.

[@jimspalding1614]

A somewhat less tedious proof results from finding the total area of an infinite series of triangles obtained by dividing the original triangle into similar triangles using lines perpendicular to the original hypotense or base.

[@Green0Photon]

Yo this is crazy. The waffle cone is a brilliant idea, and what seems to be the biggest insight here -- though there's still quite a bit of work even outside of that. Fucking brilliant work by those two on this. Wow!

[@lavendec]

Epic! Now I don’t have to learn the Pythagorean formula 😮 I’ll just derive it myself using this method whenever I see a triangle in my exams 😏

[@njo6480]

They have done it! Great job Calcea and Ne'Kiya

[@Odonianos]

Awesome proof. I haven't checked it, but don't you need to have b>a for this to work? It means you need to make the initial "doubling" of the right triangle along the longest of the two vertical sides and also that it makes the equilateral right triangle case interesting (u and v are parallel lines). Does it still work?

[@nivington]

As a former high schooler I can confirm this is true

[@nonnnth]

Somebody shared an article about this on twitter and I can’t wrap my head around what I just read, so I came here to try to understand what a noble thing they contributed to humanity. I still don’t understand how any of that make sense, but great job to the two girls who discovered it.

[@brownie3454]

it doesn’t actually contribute to society, just more propaganda from Big Girl

[@michaelbauers8800]

It should be noted that mathematics often contributes nothing to society. It's great to see such things written in the news, because I think it's encouraging. Of course whenever such things are reported in the news, some percentage of the population blames the wrong people. The news is often full of hyperbole. I read a whole book on this Irish student who found a few way to encrypt, and the news was full of hyperbole. She herself, seemed very aware the news media wanted to make too much of it. I loved the book, it was down to earth, and a good story. Doesn't have to be earth shattering to be a good story.

[@harrys2331]

@@michaelbauers8800 uh mathematics is the foundation of all engineering, physics, chemistry, and really everything at all relating to the construction of society at all. If you mean it doesn’t get spotlight, you would be correct. To mean it didn’t contribute anything to society? Look at your phone. Every pixel, code, and even the construction of the phone is all math.

[@brownie3454]

@@harrys2331 and we had every single thing you just listed years and years and years before this “new proof”, supporting that it actually contributes nothing

[@harrys2331]

@@brownie3454 I never even mentioned this new proof. I responded to Michael who stated that mathematics contributes nothing to society and I’m sorry but that’s completely not true.

[@st3althyone]

It’s amazing seeing it all come together!

[@Good13man]

Appreciate your work

[@426F6F]

Wow, this really clears things up. I thought I was confused about Pythagorean Theorem before, but now I totally understand.

[@oystercatcher943]

Thanks for explaining. I was so happy for those two girls and really wanted this proof to work out. It's a beautiful proof. I was worried about when a=b and u and v become infinite but it seems it can be dealt with.

[@rupertopasillas]

This is awesome, thanks for the explanation - also I'd love to know what hardware and whiteboard software you use to make your videos?

[@polymathematic]

thank you! i record myself on an iphone at the same time that i screen-record my ipad using goodnotes. then i composite the video together in final cut pro.

[@JayMaverick]

Kudos to the teacher for recognising what was going on and not just dismiss it out of hand.