18 year old students just discovered a proof of Pythagoras that mathematicians said was impossible 

It took these 18 year olds 2000 years to solve this math problem. That’s what I call dedication

Very clever high school students. This would have made a great undergraduate senior project in mathematics.

Good for them. Even if this proof may not be commonly used in everyday-life solving the Pythagoras theorem, it is still a spectacular discovery.

Thanks so much for the shout out! You didn't have to be so forthright with your citation but you were and i greatly appreciate it.

The most fascinating thing that no one is talking about is that we human beings uses shared knowledges across time periods and always figure out a way to make things better, and discoveries aren't always made by people that society deems genius, but people that are curious enough to put in the time and effort.

To be fair, trigonometry is based on the parallel postulate, which is equivalent to the Pythagoras theorem. The students theorem is based on the assumption the extended lines must intersect.

These two are geniuses for figuring this out. One thing that I was interested in hearing is your opinions on Animation vs. Math.

No mathematician ever said it was 'impossible'. Nor has any professional mathematician been trying to do a trig proof in the past 2000 years. There is so much nonsense in this story that it makes my brain hurt.

You lost me but i trust you

I can see how all this mathematics is accessible to high schools (albeit advanced high schoolers), but it also took the creativity and grit of two young mathematicians to put it all together. Bravo!

There is another trigonometric proof I have been taught at school. The only thing you need is to come up with a shape of an outer square with the side a+b, inner square with side c and the rest of space filled with 4 equal right triangles rotated 4 ways (0, 90, 180, 270 degrees). Essentially, you put those in the corners of a big square. Then you write the area of the big square two ways and voila - very easy;)

I'm an old PhD in mathematics & I never heard the assertion anywhere at any time that it was "impossible to prove the Pythagorean theorem using trigonometry alone". Lol! Perhaps you can send me a link to someone making that assertion? ;)

I would assume this proof was overlooked until now because it relied on taking the sum of an inifinite series, which is a relatively modern concept. By the time that idea had been well established, the dogma that there were no trigonometric proofs was well established, so no one went looking until now. Bravo!

It depends a lot on what we mean by *only* trigonometry. In particular, it means that there is something used in the common proofs of the theorem that has to be excluded. If not one could just apply a thin varnish of trigonometry on another proof. An obvious candidate would be to exclude geometry (which would then exclude this new proof) but then how do we define the trigonometric functions? We should do it axiomatically. If we give differential equations as axioms, it might be possible to prove the theorem using analysis but would not really fit the notion of what trigonometry *is*. If we want to use some of the common trigonometric formulas as axioms, we would need to not include sin²x+cos²x=1 (as that is essentially what we want to prove) but even from other formulas (like the formulas for sum of angles), we can quite trivially 'prove' the sum of squares (and so decide that we kind of put the result in the axioms). Nevertheless it is an interesting proof with non-trivial trigonometry, geometry and limits in the long tradition of alternative proves of this most famous theorem. Congrats to them!

I don't know... To me Pyth and Trig were always different formulations of the same qualities of triangles, expressed differently, but NOT from different principles. It always felt like they both started from the way that right triangles can be inscribed into circles. But I never devoted much thought into this. So it's interesting but ultimately I don't think a proof from Trig is independent at all.

Man, his voice by the end of the second proof sounds so happy and excited, it shows genuine love to mathematics and the topic in hand.

I think the most beautiful proof of all is the following. You have a right triangle. Divide the triangle into two triangles by a line that is perpendicular to the hypotenuse and connects to the vertex of the right angle. Now you have three similar triangles: the original one and the two pieces. Because of the fact that the triangles are similar, each occupies the same fraction 'x' of the square on its hypothenuse. If the hypotenuses and surface areas of the three triangles are a, b, c and A, B, C, respectively, you therefore have A = x*a^2, B = x*b^2 and C = x*c^2. Because of the fact that A + B = C, then x*a^2 + x*b^2 = x*c^2 and therefore a^2 + b^2 = c^2.

Media blowing up with nonsense: Zimba already did the "impossible" (itself a questionable statement by Elisha Loomis) nearly 15 years ago.

My go at it; The angle addition formula can be prooven without pythagoras theorem cos(a+b) = cos(a)cos(b) - sin(a)sin(b) let b = -a cos(a-a) = cos(a)cos(-a) - sin(a)sin(-a) = cos(a)cos(a) + sin(a)sin(a) = cos²(a) + sin²(a) We also have cos(a-a) = cos(0) = 1 cos²(a) + sin²(a) = cos(a-a) = 1 q.e.d.

Well... Actually this proof uses not only trigonometry. It uses also limits and similarity. And you can easily prove the theorem using similarity only. :-)

There are well over 371 Pythagorean Theorem proofs, originally collected and put into a book in 1927, which includes those by a 12-year-old Einstein (who uses the theorem two decades later for something about relatively), Leonardo da Vinci and President of the United States James A. Garfield.

I like the history you are providing in recent days. is this related to your earlier poll and will you discuss more mathematical history? It's definitely interesting to know, especially considering how despite different cultures and different times, mathematics connects us all. I remember seeing depictions of 'Pascal's Triangle' from a Chinese mathematician and Ancient Indian Mathematician. Even without knowing the language, I can understand it as Pascal's triangle.

It's called a geometric series and scaling. It's infinitely repetitive. It says a^2 + b^2 = C^2 because a^2 + b^2 = c^2 over and over and over again for infinity. It's was also already done in a previous proof: John Arioni "Pythagorean Theorem via Geometric Progression"

The thing that is still missing is that it is BY DEFINITION what sin and cos mean, and they are based on right triangles. There is nothing new in the “proof” here, except to take as defined one thing vs another, neither of which are independent (of course not, otherwise Pyth theorem would not be true).

the sine rule is based on right angle triangle drawn by taking one of the heights. so even though law of sine extends to any triangle the law inherently is based on basic definition of sin in relation to a right angled triangle. so I don't think the paragraph is wrong, these students found nothing new but just a longer proof of proving something which can be itself written as sin²a + cos²a=1

Put a small square inside a larger square with their centers coinciding. Rotate the smaller square until its 4 corners touch the sides of the large square. The large square will be comprised of 4 identical right triangles and the small square. Write an equation stating that the area of the large square is equal to the area of the small square + the area of the 4 identical triangles. Rearrange the terms in the equation and voila!

In that last proof, after drawing in the rest of the chord that made up sin(theta), you can use the intersecting chord theorem to get sin^2+cos^2=1

Can someone explain why exactly the other proofs told at the end of the video are not considered trigonometric proofs for the Pythagoras theorem, and why does this new proof special?

The funny thing is that neither of them wanted to pursue mathematics in college but instead, if my memory is correct, environmental engineering and biochemistry. Of course both are math heavy but only math. Also, I believe it originated out of school competition and not just a paragraph in a textbook.

Congrats to these high students. I would not say they used trigonometry to prove the theorem. Instead, they used calculus plus a clever geometry construction to prove it, which is a great effort. BTW, I never heard such comments from mathematicians that a particular tool is IMPOSSIBLE to solve a specific problem.

There's an infinite limit, so that's a calculus proof disguised as a trigonometry proof.

7:46 If alpha + beta is 90 degrees then you're assuming the euclidian parallel postulate, which is equivalent to the pythagorean theorem. Seems to me that the students are implicitly assuming that pythagoras is true in this proof, but I might be wrong.

At 12:18 you have expressions for all three sides of the big triangle, which is a right triangle. You can show that the sum of the squares of the two shorter sides is equal to the square of the hypotenuse. That proves the pythagorean theorem without needing to go back to the original triangle.

When it comes to lost and/or missing proofs, I want you to reflect on how much knowledge has been lost to time because our own destructive tendencies.

The last proof involving just the circle is (at 16:21s) the best and elegant proof of fundamental trig identity w/o using "pythagoras theorem"

the need for a < b was for the similarity between the triangles ?

1:27 I wonder, why they came up with this rather strange statement. The sine theorems in (right) triangles can easily be deducted purely from the definition of the sine, just like it's done in the video. Also it seems quite obvious that sin alpha = a/c will hold if and only if gamma=90° . So anyway I just googled this book... it's from 1940. Not that people couldn't have known better back then, but it is quite imaginable that nowadays such books would be checked much more thouroughly by many more people. So that might explain it in conjection with someone being a bit too excited about the topic.

This strikes me as more of a geometric proof than a trigonometric one. Most of the steps in Case 2 are using trig functions to add extra steps to simply taking ratios of similar triangles. Nonetheless, it is a very nice and creative proof.

I don't know if I missed it in the video; but I was a tad unconvinced, at first, that the similarity factor of all of the waffle-triangles would be the same. They ARE definitely similar, but I think it just needed a short explanation as to how you can tell the similarity factor for all of the triangles would be the same. Your old side "a" becomes your new side "b" (ie, your short side length from the larger triangle is now your shorter side length of the smaller triangle). Based on the similar triangles, the ratio of the short side to the long side is a/b. That's basically it. Every time you want to calculate the next unknown perpendicular length, you would use this ratio (its also the tangent ratio!), so your similarity scaling factor is a/b. After typing this out, I think this was touched on, but I hope this extra explanation is helpful for anyone who might have been initially unconvinced.

Wow! When I first heard about this proof from my mom, who hardly knows any math, by the way, and she told me that the students used trig to prove PT, naturally I was quite skeptical, but after watching this video, I now see that this is indeed a remarkable proof! Congrats to Ne'Kiya Jackson and Calcea Johnson! I wish I could have done something half this great when I was in high school!

By the way, the last proof that you showed is also one of the fundamental equalities in basic trigonometry, taken nearly straight from the definition of cosinus.

That’s innovative. Never would have thought of this. Very nice

The infinite geometric series part is a pre-existing proof. They just slapped a couple of triangles on top of it and it just happened to work. Makes me wonder if you could do the same with other known proofs.

This is a clever construction, but I would not be surprised at all if some previous mathematics student (undergrad or grad) made a similar observation that such an infinite series could be constructed to provide a demonstration of this famous theorem. I'm happy to see that a HS student had the patience and determination to explore this approach because that is how mathematics is done in the professional level, just with higher complexity of concepts/techniques and typically a much longer period of time.

It's possible to prove the law of cosines without using the Pythagorean theorem. The proof can be found on Wikipedia for example. Now using the law of cosines on a right triangle just gives the Pythagorean theorem as a special case.

I love the new proof using infinite series … I think i recall learning the first alternate/other proof in high school, - a new proof - genius! these students are legends in high school. Thanks everyone for shining light on this!

Using trig on a triangle with side a, b, c and angle A. a = c*cosA. b = c*sinA, a^2 + b^2 = c^2*(cosA)^2 + c^2*(sinA)^2 = c^2*((cosA)^2 + (sinA)^2) = c^2. I used only trignometry.

There are literally hundreds of trigonometric proofs of the Pythagorean Theorem. Every trigonometric function is only a ratio of 2 geometric knowns. EVERY geometric proof to the pythagorean theorem can be expressed in terms of trigonometric functions.

Whoa.. that is quite a work! Thank you for making the video so we could understand it easier👍

i've could have derived that, but I screwed up my knee

I learned a long time ago that Einstein proved PT at age 12 by partitioning an arbitrary right triangle into two similar ones using the altitude of the hypotenuse, and using the fact that the areas of these two partitions, which are proportional to a^2 and b^2, respectively, is equal to the area of the original triangle, which is proportional to c^2 with the same constant of proportionality.

This proof is implicitly using the flatness of euclidean geometry in that current form of the law of sines (there is a modified form in the other geometries), and is using this flatness in saying sin(alpha)=a/c etc. since this is directly derived from the law of sines. In general, this just simply is not the case. Therefor, this proof is using implicitly the assumption of a flat euclidean geometry, which is the geometry which assumes the parallel postulate, and the parallel postulate is equivalent to the pythagorean theorem. Therefor, the proof is starting with the assumption that the pythagorean theorem is true, albeit subtly and implicitly. It is a smart proof, but it is circular.

Perhaps even more amazing than these students coming up with this never-before-discovered proof, is that the proof they came up with is relatively understandable by lay people. Great job!!

Did you know that the Pythagorean Theorem or the Equation to the Unit Circle is embedded within the simplest of all arithmetic equations: 1+1=2? And with that, so are all the definitions, identities and properties of Trigonometry. If you can construct a Unit Circle you already have the Pythagorean Theorem and from that Unit Circle you can construct a Right Triangle and this is why the Trigonometric Functions have a Pythagorean Identity. The equation 1+1 = 2 is the unit circle located with its center (h,k) located at (1,0). Each unit value of 1 is a radii and their sum of 2 is the unit circle's diameter. In order to understand this proof, you can not think of 1 and 2 as being a scalar value. You have to think of them as being unit vectors with the operation of addition (+) as being a linear transformation of those unit vectors. When you look at them within this context, then it becomes clear as to why the cosine also has a direct relationship to the dot product. I could even go one step deeper than this. All of these properties are actually embedded not just within this arithmetic or algebraic expression but they are also embedded within the identity properties of both addition and multiplication a+0 = a, a*1 = a, as well as in the equality or assignment operator as in a = a. If you don't think so, then consider the following equation: y = x. This is a specific case of the slope-intercept form of y = mx+b where b is the y-intercept and m is the slope of the line defined as rise over run which can be solved by m = (y2-y1)/(x2-x1) where (x1,y1) and (x2,y2) are any two points on that line. This can be simplified to dy/dx. With the case of y = x, the intercept is 0 as this line crosses through the origin (0,0) and the slope here is 1. The line y = x bisects the first and third quadrants at 45 and 225 degrees or PI/4 and 5*PI/4 radians. At 45 degrees or PI/4 radians, both the sine and cosine of those functions are equal, sqrt(2)/2 . This is also tan(45) or tan(PI/4). Within the linear equation in the slope-intercept form we can substitute m = dy/dx with sin(t)/cos(t) = tan(t) where t, theta is the angle between the line y=mx+b and the +x-axis. And this can be derived or constructed simply from the line y = x which is equivalent to x = x which is equivalent to x+0 = x and/or x*1 = x. x | y ==== .. | ... -3 | -3 -2 | -2 -1 | -1 0 | 0 1 | 1 2 | 2 3 | 3 ... | ... And this is just an simple expression or equation which doesn't even involve function composition. The reason this works is because the Pythagorean Theorem A^2 + B^2 = C^2 for all tense and purposes is equivalent to or a simplified form the equation to a circle (x-h)^2 + (y-k)^2 = r^2 where (x,y) is any point on the circumference (h,k) is the center of the circle and r is it's radius. So going back to the expression 1+1 = 2 (full circle) the tail of the first vector of (1,0) starts at (0,0) and it's head ends at (1,0). The addition operator (+) translates this vector in the same direction (+) by convention to the right by its magnitude of (1) since the second operand of the equation is also a unit vector. The result of this linear transformation (translation) now has the 2nd vector starting at (1,0) and its head is at (2,0). The center of the unit circle is (1,0) and both points (0,0) and (2,0) are on the circumference of the circle. We can substitute these in to the equation of the circle (x-h)^2 + (y-k)^2 = r^2 == (2-1)^2 + (0-0)^2 = 1^2 which == 1^2 = 1^2 = 1 which is the radius of the unit circle and is also the unit vector. The diameter of the unit circle is 2r which is 2*1. Therefore when you see any term in any mathematics of 2x or 2n, you actually have a solution to a given circle where x or n is its radius.This is also why the distance formula between any two points is directly related to the equations of a line as well as the Pythagorean Theorem. You don't need a "Triangle" to have a "Pythagorean Theorem". You only need a Line Segment. With that, everything within mathematics is related from basic arithmetics to algebra to geometry to trigonometry to linear algebra to calculus, and so on... These are all possible simply just because we can Count or Enumerate. This is why I appreciate numbers and mathematics.

I love when a proof that seems to be spiraling away into unintelligible complexity suddenly is resolved into simplicity itself, seemingly out of the blue. Bravo!

I read the story in the Guardian and am glad to have watched your explanation of their clever approach

I'm an engineering student and I don't know who said no proof existed before. I've proven it with a basic method that takes 2 minutes to explain with 1 square

I saw this proof when it first came out last April. A very elegant proof and good on the students.

Here's another way to prove it with trigonometry: d/dx (cosx) = -sinx and d/dx (sinx) = cosx is proven without pythagoras. We say that: (sinx)^2 + (cosx)^2 = f(x), for some function f DIFFERENTIATE BY X TO GET: 2sinx(cosx) + 2cosx(-sinx) = f'(x) 0 = f'(x) --> f(x) = C, where C is a constant (sin0)^2 + (cos0)^2 = 1 --> C = 1 --> (sinx)^2 + (cosx)^2 = 1

É incrível como na matematica existem coisas que ainda não descobrimos, e ainda não provamos A matemática sempre me fascina pois ela é um alfabeto completo que podemos descobrir coisas com puro raciocínio e apenas lendo de forma alfabética, é legal resolver problemas nessa elegância de ler matematicamente, e por isso apoio totalmente olimpíadas de matemática, são incríveis A matemática é cheia de incompletudes e apesar de parecer estressante, só me deixa mais fascinado com essa idéia A matemática tem muita coisa ainda a ser tratada. Mesmo que demore 300 anos pra provar algum problema proposto por algum matematico maldoso. Kkkkk quem sabe mexer de verdade com a matemática, pode usufruir de tudo dela e dos problemas, mas também pode ser um cara chato que aterroriza todos do ensino médio com sua fórmula que ele cria para provar alguma coisa bem doida. Amo

I think you are implicitly assuming the parallel postulate here. The sum angles in a triangle don’t have to be 180 degrees, but if they are, then you basically assuming thing that make the Pythagorean theorem true

15:17 in our maths textbook similar proof is given only difference is instead of sin it relies on rules of similar triangles

While proofs like the one at 15:10 use trigonometry, it doesn’t really require it - you could do almost the same proof just using similar triangles and not mentioning sine. The student’s proof is arguably “more trigonometric”, since it uses the law of sines, while still not requiring anything beyond trigonometry.

Muito bom! Parabéns e obrigado!

15:06 This is such an elegant, easy to understand proof, I love it.

I really wish math instructors would focus more on the details. I get that this is youtube, and there is a certain standard for the length of videos, but I would have loved to learn all the reasoning for example visually of all the connecting ideas within the equations. (and can literally be a second-long, you can throw a picture in with text that explains how you were able to move forward). I loved that I remembered how to simplify complex formulae, but even for youth, I can see it being extremely useful for reviewing material. Also, how much do I bet that length and rate of 'pause' in a video is the single most defining factor in how interested people are in your videos.

Very interesting! - You can use same approach in generating triangles right inside base one triangle, but in innerside direction, by making first triangle by line going from 90⁰ to hypotenuse. So it will bring same area you have already. For both of starting triangles. Another question - is how to prove that square area is a². And proving anything you use something you need to prove, always. It leads you to axioms. - It can be stop, if it has no changeable base.

"With the few screenshots we got, we can predict what they did to do this" Proceeds to give a detailed explanation for how they did it

I was taught in grad school the Greeks did not have algebra. The Greeks had a straight edge and a compass, so using trigonometry with algebra seems slightly anachronistic. I would be really blown away to see the proof using just a straight edge and a compass! On the other hand, my hat goes off to the two high school students. Great work.

Right on. Thanks for sharing.

Great work by the students! The part of the proof that involves 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 the unknown sides of the big right triangle (the waffle cone) in terms of a,b,c. The rest is the same.

This proof is awesome. My biggest question is how did these two students specifically collaborate on such a unique proof?

...................I'm gonna go watch Animation vs Math

I really appreciated this presentation, thank you.

1) It's not the first trigonometric proof of the Pythagorean Theorem. 2) It's not pure trigonometry. 3) There is an exception: It doesn't work for isosceles triangles.

I am surprised no one has thought of this before for 2000 years. It was so satisfying when I realized how they were going to attack the proof. Good job!

I once sat a test and I wrote "therefore by Baudhayan a^2 + b^2 = c^2" and I got 0/10 The teacher commented, "It's the Pythagoras theorem, what is wrong with you?"

Okay, that was really impressive!👍🏻

Just draw in any right angled triangle the vertical to the hypotenuse c going through the opposite corner. Name the 2 segments a and b. With similarity follows a^2 + b^2 = c^2

very sceptical this hasn't been done before. isn't sin() and Pyth connected?

What makes me fascinated in mathematics is that even an 18 year old can make a groundbreaking discovery.

All of the theorems about not solving the ancient theorem cannot be solved by trigonometry had been shattered by these two girls.

I think PT is only needed to prove sin²+cos² if they're defined geometrically. It's easy to prove it if they're defined as sin x = (e**ix - e**-ix)/2i, cos x = (e**ix + e**-ix)/2. Just substitute and simplify: ((e**ix - e**-ix)/2i)**2 + ((e**ix + e**-ix)/2)**2 = (e**ix - e**-ix)**2/-4 + (e**ix + e**-ix)**2/4 = ((e**ix + e**-ix)**2 - (e**ix - e**-ix)**2)/4 = (e**2x*2*e**-2x*2)/4 = e**0 = 1

Presh said “Pythagorus”! I’ve been waiting years for this day.

Trying to cancel Pythagoras for 18 minutes straight

Since when infinite series is "trigonometry alone"? There is another branch of mathematics that gives as the sum of it, it's called calculus. That was NOT a proof using "trigonometry alone".

Even with everything it's really really hard to follow but hell I had a hell of a time with plain geometry so this was practically impossible for me to follow

These 18 year olds took 2000 years to solve this question. That’s some mad dedication

I'm confused... you say the students found the impossible: A trigonometric proof of the Pythagorean Theorem. Then you proceed to show it, and it uses infinite iterations of smaller triangles. That's fine. But then in the "More Proofs" section of this video you show a different, much simpler trigonometric proof. Wasn't that simpler proof already known?

Excellent work by these two young scholars!!! Now lets get the proof added to Algebra, Geometry, Trigonometry, & Calculus texts. This is good for young people to see they too can add to our math base. Most classes at the HS level are boring, uninspired, and cover nothing new or interesting. Students need to see they can connect to advanced mathematics and that math when connected to actual practical problems is useful, exciting and very much connected to their lives.

The simplest is to take the limit as spherical trigonometry approaches plane trignometry. Use the formula cos A = cos B cos C + sin B sin C cos(a - b). A B and C are the sides and a, b, c the vertex angles of a spherical triangle - all dimensionless. In the limit A, B, and C are very small and cos A -> 1 - 1/2 A^2 and sin A -> A to highest order. Let the angle a-b be pi/2, then we get (1 - 1/2 A^2) = (1 - 1/2 B^2) (1 - 1/2 C^2) - working out to highest order, A^2 = B^2 + C^2. This proof does not presuppose the theorem in any way, just the power series expansions for sin and cos and the angular geometry of a sphere. It works in any number of dimensions as well.

The true master equation of planar geometry is Norman J. Wildberger's Cross Law, which is the cosine law squared (=1-sin²), out of which Pythagora's theorem is a limit case when the spread ("sine squared") between two sides of a triangle is 1. The other limit case is when a spread of a (degenerate) triangle is 0, hence about 3 collinear points, as strong and as fundamental as Pythagora's theorem. This video is much wind for nothing, NJWildberger cracked geometry properly, without the sine qua none need for infinite processes 😊 Nevertheless, I enjoy the channel and content very much!! Keep up the good work 😁🙏🙋

@MindYourDecisions I appreciate that you try to be historically sensitive. I really do. But there is a lot of value in naming conventions. If a large portion of people know the thing under the same name, the Pythagorean theorem, then that's an achievement all on its own. And it's not worth giving that up for the chase of "Who did it first".

1) It's not the first trigonometric proof of the Pythagorean Theorem. 2) It's not pure trigonometry. 3) The same proof could be done without any trigonometric equations at all. It's only superficially trigonometric.

In this "proof" sin(alpha) is just the ratio a/c where a is the side opposite alpha and c is the hypotenuse. So we don't need to mention the sin function and in that sense, it's not "trigonometric". The "proof" assumes the ratio properties of similar triangles, which themselves are proved from the Euclidean axioms. Euclid gives a far simpler ratio proof. He also gives an area proof, but this depends on further axioms to do with areas.

They used trigonometry to prove that x = 2ac/b (see ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-juFdo2bijic.html) , but they could have proven it by the proportionality of sides in similar triangles (in fact, that's what Pythagoras used). In other words, it is really just one more proof of the theorem. Anyway, it is great for high school students.

Bravo, students. Brilliant!!!

This makes me want to revisit the idea I had about measuring the one way speed of light

I like how the proof has that iconic wedge shape. I wonder if the high-school students will get any offers from colleges.

truly amazing!