Pythagorean Theorem & Its Inverse (my favorite proof)
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- Опубліковано 22 жов 2024
- I'll be sharing my favorite proof of the Pythagorean Theorem and its inverse, using similar triangles. The Pythagorean Theorem is a fundamental concept in geometry that states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. But why does this theorem hold true? In this video, I'll walk you through a simple and elegant proof that uses similar triangles to demonstrate the Pythagorean Theorem and its inverse. By understanding how the ratios of the sides of similar triangles relate to each other, we can prove this theorem and its inverse with ease. Whether you're a math enthusiast or just curious about the Pythagorean Theorem, this video is for you.
Pythagorean triple generator 👉 • finding ALL pythagorea...
This question was asked by Toby from my Chinese channel:
中文版 • 畢式定理有兩種: a^2+b^2=c^2 &...
@blackpenredpen
![Pythagoras Would Be Proud: High School Students' New Proof of the Pythagorean Theorem [TRIGONOMETRY]](http://i.ytimg.com/vi/p6j2nZKwf20/mqdefault.jpg)
Omg spiderman is supporting your channel!! You're going to be at 400k subscribers by the end of the year for sure
René Gado yup, it's very nice of him!!!
And someone special too
Sir please make videos related to all types of figure counting. Need for competitional exams 😑
It's stuck at 301K
Now it's 6.81lakh🥳
This is far clearer and more straight-forward than the traditional proofs.
Nathan Greene 😊
Possibly not as intuitively motivating, but the logic is as pristine as a crisp winter morning. There, who's not a poet?
Oh dude search president Garfield s proof
The inverse pythagorean theorem is some serious beautiful shit.
Brand new Maths for me. BTW, my 71st B. D. was this year 2019.
@@inyobill happy birthday, old one!
"Better old than dead", kiddo! And thanks for the kind wish. :-D
Ah lads don't worry about it, we all DIE!(Maniacal Laughter)
@@MrCoffeypaul What? What? Wait, what did he say?
I love math so much! Thank you so much for this video. For Pythagoras, as soon as you drew the vertical height, h, I decided to try to do the proof and I did it. I also just happen to label the parts of c as c1 and c2 also lol. For the inverse, I knew what was going to happen as you were going and it was beautiful! The simplicity of math can be so great; I just wish schools showed it the right way.
FACH2004 hahaha very nice. Thanks
The Pythagoras Lu! 😄
Pytha Lu? ;-)
Wise man once say triangle is strongest shape.
L H agree!!
Wow, I've never seen the inverse Pythagoras Theorem. Love your proof videos!
David Morris thanks!!!!
Awh the nostalgia. This video hit me in the heart. It's amazing how far we've come from counting numbers to algebra and to functions and calculus and to complex. I remember not understanding the Pythagorean theorem and back then trigonometry was like the real deal.
Loved the inverse proof, and also this is now my favourite proof as it uses similar triangles which is neat!
DavidAde yay!!!!
Indeed...totally agree considering that this proof is about a right triangle's feature and the fact it only relies on similar triangles (not squares or other polygons here) makes it my favourite demonstration of the PT as well. Beyond that I always keep it in mind as a special case of the cosine law which applies to all triangles.
My favorite proof of the Pythagorean theorem is Garfield's (the president, not the cat) proof, just because of how clever it is.
DerivativeOfSenpi
Oh man!! I just searched it. That is amazing!!!
Yeah, I love that one too. Basically the same as the square + four triangles = bigger square one, just cut in half to get a trapezoid made of three right triangles. Write the area of the three triangles, write the area for the whole trapezoid, equate them, do some math, and you get the theorem!
@@blackpenredpen There's a video that I made in my channel that I would love for you to check out, about sec^2x. I promise you will like it. Thanks !
Yes, it's nice, but it's just another of those "draw a square and populate the corners with right-angled triangles and do some algebra" proofs. He just ignored half of the square.
On the other hand, it's always nice to know politicians haven't always been entirely useless (I speak as a Brit).
@@davidgould9431 James Abram Garfield, 20th U.S. President, and the 2nd President to be assassinated, had been an Ohio schoolteacher before entering politics.
So his non-political contribution to society went well beyond his PT proof.
Incidentally, as I read it, some decades ago, Garfield's proof *did* include the whole square, complete with 4 right triangles.
Which makes the algebra a bit more direct, as there are no ½'s to be canceled from both sides. There's only:
(a + b)² = 4·½ab + c²
a² + 2ab + b² = 2ab + c²
a² + b² = c²
Fred
Cool! Thank you!
This proof doesn't ask for special math knowledge, just fundamental ones.
I have done it, but in different way:
I use sin and cos in my proof. After I finished, I have understood that sina^2 + cosa^2 = 1 is also Pythagorean theorem and I used Pythagorean theorem in my proof of Pythagorean theorem 👌
The Inverse Pythagorean theorem is really useful, I think
omg, I love that inverse to get the height from the hypot. Very nice, thank you!
Keith Masumoto yay!!!
Love how you tackle really hard and also simpler problems with the same enthusiasm!
I wish they taught this proof in schools to teach kids things don't just come out of the blue.
Really pleased to see the inverse proof. And who, I wonder, are the four people who don't like this video?
this concept has already helped me in solving question thanks for covering it
beautiful. the proof based on triangles similarity is really elegant.
I remember getting a 5 (best grade) in high school partially thanks to the inverse of the Pythagorean theorem (didn't know it was called that way though). In fact, using a bit of manipulation it can be further expanded to work for a regular three-sided pyramid: 1/H^2 = 1/a^2 + 1/b^2 + 1/c^2, where H is the height. This formula is super useful to find the height if you have the sides of the pyramid, otherwise you'll spend over an hour digging around.
I like your videos because I can see that you're genuinely excited about the maths. There's a million people out there that understand it well enough to explain it, but it's a good video because you love it.
Very easy to understand. Love the teacher's enthusiasm!
This is the first proof I loved... Like we use it every day and it's fun
Proof by area, really straightforward.
With more than 300 Pythag Theorem proofs, there's so very much to enjoy! My favorite again uses the 3 similar triangles that You use: let's call them T_a, T_b, T_c where the subscript tells which side is the hypotenuse of which similar triangle. Of course, Area(T_a) + Area(T_b) = Area(T_c) because T_a U T_b = T_c. Consider a fourth similar triangle, T_1, whose hypotenuse is of length 1, having area A_1. By principles of 2-dimensional scaling we know that Area(T_a) = a^2*A_1, Area(T_b) = b^2*A_1, and Area(T_c) = c^2*A_1. Thus a^2*A_1 + b^2*A_1 = c^2*A_1, and we just divide through by A_1.
Your mention of the "inverse Pythag Theorem" involving the height h reminds me of a heights-base version of Heron's area-formula for an arbitrary triangle, utilizing all three heights h_a, h_b, & h_c of the triangle:
1/Area = SqRt{ (1/h_a + 1/h_b + 1/h_c)*(1/h_a + 1/h_b - 1/h_c)*(1/h_a - 1/h_b + 1/h_c)*( - 1/h_a + 1/h_b + 1/h_c) }.
Some might enjoy this challenge: starting with three heights, h_a, h_b, h_c, how can one geometrically construct the unique triangle that they specify (without resorting to using the above area formula).
The inverse is too beautiful, I remember that elegance after deriving it myself :)
I NEVER KNEW THE INVERSE ONE!!!!!
I remember have to find a proof for the Pythagorean theorem for a pre-calc project and I found on that used the area of a square and area of a triangle formulas
Very simple and easy to teach demonstration of the famous theorem. Thanks a lot for providing brand smart different points of view!!!
The most amazing proof of the Pythagoras I've ever seen. Thanks Steve!
What a simple proof,you Chinese, have turned the world upside down.you have unearthed everything.you are amazing.
I hadn't seen the inverse pythagorean theorem before, very nice! My favorite proof is just the square of side length a+b with the square of side length c touching it with the corners 'a' away from each outer corner : this gives you 4 abc triangles inside the square of size a+b, The whole square is area (a+b)^2, the inner square is area c^2 and there are 4 triangles area (a+b)/2 which add up area 2ab for the triangles.... So the whole outer square is a^2 + 2ab + b^2, but it it is also c^2 + 4(ab)/2 = c^2 +2ab, so a^2+2ab+b^2 = c^2 + 2ab, and you subtract 2ab from both sides.
I like the one in which you draw a square with four right triangles on the inside. If done in a way such that the side of the square is a+b, being a and b the sides of the triangles, then the shape in the middle results to be another square of side c.
The area of the big square is (a+b)(a+b)=aa+2ab+bb, the area of the four triangles together is 4ab/2=2ab.
Now, the area of the square of side c is the area of the big square minus the area of the four triangles, but it's also just cc.
So, if we compute the above:
[area of the sq of side c] = [area of the sq of side a+b] - [area of the four tr]
cc = (aa+2ab+bb) - (2ab) => cc = aa + bb
c^2 = a^2 + b^2
I know it might be one of the most elementary proofs there are out there, but the reason I like it is because I came up with it by myself.
It's wonderful result!
I've used inverse Pythagorean theorem to find distances between point and line in cube and other polyhedra without using a single coordinate/vector at all, it was such refreshing and satisfying way to find it that way without linear algebra.
DAMN I LOVE THIS GUY HE IS AMAZING
Derivatives&integrations
Thank you!!
blackpenredpen YOU ARE AMAZING 😂 MATH IS MY FAVORITE LESSON AND U MADE ME TO LOVE IT EVEN MORE
@@snorlass oh same!!
Nice one ! I like to the Full Square Area proof. :: Build a square with side s = a + b such as the right triangle is used 4 times. The middle area of that construction is a square of side c. Trivial proof : by construction. So, Surface S = s^2 = c^2 + 4T where T = ab/2 ( area of the triangle )Thus,S = s^2 = (a+b)^2 = c^2 + 4*(ab/2) a^2 + 2ab + b^2 = c^2 + 2ab a^2 + b^2 = c^2 [DONE]
Make three copies of the triangle and scale each in order by a, b, and c so that the lengths are -
1) aa ab ac
2)ab bb bc
3)ac bc cc
Lay 3 down so that side cc is the base. Fit the ac side of 3 with the ac side from 1. Fit the bc side of 3 with the bc side of 2. Flip the smaller two so that you end up with a rectangle. The base will be cc and the top will be the sum of aa and bb.
I don't know what it's called but this geometric proof is my favorite, it's the most computationally simple one I know and isn't concerned about areas.
I really like this proof as well! So much easy to visualize somehow.
I really like this, as it gives a better explanation of proving the pythagorean theorem, and I love him showing the inverse. This will be a good way for me to help the students I work with, and will help with any geometry students I get working on similar triangles.
I too like this proof the most. It's so simple and only relies on similarity of triangles. In fact, since the whole Pythagorean theorem is a special case of the "Law of Cosines: C^2 = A^2 + B^2 - 2AB cos(C)", in a right triangle, cos(c) is cos(90) which is just zero, this then simply reduces to C^2 = A^2 + B^2 (Pythagorean Theorem). Perhaps you can make a video about the proof of laws of cosines and sines, and even Heron's formula. Would be interesting for sure.
Mohammad Khaled yay!!!
No. Law of cosines is derived using pythagorean theorem.
@@SanjeevKumar-js4mu you can derive the law of cosines with just the definition of cosine though.
@@SanjeevKumar-js4mu What I meant here was that law of cosines is the general form here. I wasn't talking about deriving which from which. There are definitely many many ways for derivations.
Dr Sanjeev Kumar No, the law of cosines is not derived using the Pythagorean theorem.
Nice proofs. You could actually get the relation ab=ch from the similar triangles too (from the first and the third triangle you have a/h=c/b).
Thank you!!! This was a beautiful proof!!! :-D
Thats a beatiful way to proof pythagorean thanks.
Mind blowing proofs, i am speechless. Keep up the good work 👍👍👍
Nicholas Yap thanks!!
I love to using James Grifeld formula,and your are excellent too for proof for Pythagorean theorem
Great! Simple, plain, non-obfuscated. Spot on!😎
Per Appelgren thank you!!
I like the square proof you said you didn't like. I find getting the same thing two different ways is a cool method. Like, deriving the Sine and Cosine rules. Never knew about the inverse theorem and I'm a maths teacher myself. Very elegant and a great way to get the height quickly.
Well explained. I wasn't aware of this cool proof.
Very nice proof of Pythagorean theorem ! It seems to me I didn't see this proof elsewhere. Thank you !!
Jordan Jordan yay! You're welcome!
I believe Einstein came up with it when he was young
Jordan Saenz interesting!! I actually learned about this when I was in college. And this kinda similar triangles in a right triangle problem happen a lot on standardized tests such as the SAT
@@blackpenredpen Yeah I saw one or two problems about triangle ratios in the PSAT
This proof is part of the Indian 10th grade similarity syllabus :)
Hello!
This is clearly sorted out!
Thank you for sharing!
Keep up the good work!
Last year on this day i subscribed this channel.... It has been an amazing journey with blackpenredpen #yay ...🖤
My favorite PT proof has for a long time, been President Garfield's proof.
My most unfavorite is the one from 9th grade geometry class. It was long, involved, and arcane. I don't even recall how it went any more.
I think it might have been right out of Euclid's _Elements._
Another one I rather like, sets the RT on c as its base, actually draws the squares on all 3 sides, then drops a perpendicular from the apex all the way down through the c square, and proceeds to equate each resulting rectangle with the square on the corresponding leg.
I do really like the one you've given, along with the inverse-squares version.
BTW, speaking of Pythagorean Theorem "cousins," the most remarkable one I've run across so far, involves areas of faces of a tri-right triangular pyramid (where 3 right angles meet at one vertex, so that those 3 faces are all right triangles). This then, is a 3D analog to the 2D right triangle. It can be made by a planar slice through any 3 vertices of a rectangular prism, no 2 of which share an edge - so that the 3 lines connecting pairs of them, are all face diagonals.
I'm having some trouble recalling the actual theorem at the moment; I must look it up; but I think it was just that the squares of the areas of those 3 RT's equals the square of the area of the 4th face.
I also seem to recall that it doesn't generalize into dimensions > 3.
In any case, have you seen that?
Have you perchance already done a video on it, that I missed?
Fred
I use the method of taking a square of side length a+b and drawing lines to create four triangles with base and height a and b respectively, with a square section of side length c in the middle. Comparison of areas shows the required identity
Great explanation as always. As a request would you please do a video explaining the Pythagorean Theorem proof that used Trigonometry and Calculus on a "Waffle cone" diagram.
Such a simple proof! I absolutely love it!
This is how I have been taught to demonstrate the theorem at school when I was 10. It is the easiest, most intuitive way in my opinion. It is a natural consequence of Euclid's theorems on triangles, so it fully makes sense.
Immediately liked the contrasting colors and title of click maths page.
@blackpenredpen
Correct me if I'm wrong but as far as I know, similar triangle is not the same as equal triangle. Therefore you cannot equate them.
You can only equate the ratio of sides of similar triangles (just like what you did in the pythagorean proof) but you cannot equate the area of similar triangles.
We can even see that the area of triangle with sides abc is bigger than the area of the triangle with sides bch.
this is the proof i got taught in school...always liked it more than the other ones
Around 1:20, you assumed which were the smallest sides in order to use triangle similarities. If you want to make more formal, just use the angles of the original triangles and verify which adjacent sides have the same angle.
This method is also better because, if the original triangle is isosceles, the smaller ones have this property too
Nice one! 👌🏻 Didn't know about the inverse version before. 😉
Btw @blackpenredpen, talking about inverse stuff here I have another little problem and I would be curious how you would solve it. I got the solution by now but I spent quite a lot of time thinking about it before I finally found the right approach... 🙈🤓 So:
Let f: IR -> IR, x -> x+e^x. (i) Show the existence of the inverse function g(x) on whole IR. (ii) Compute: g(1), g'(1), g''(1). 🧐😉
(By "IR" I mean the symbol for the real numbers.)
What do you mean by x-> x+e^x ?
@@schrysafis f(x)=x+e**x
This was the proof I learned as a kid 45yr ago... lol
My favourite one is the the areas of concentric circles (Short side is radius of inner circle [a]- outer radius is hypotenuse[c] - then use calculus to show area of outer ring is PI*b^2 - and hence pythagoras etc..) - love it cos it's just a crazy way to do it!
Just realised - there's a complex number proof!
a and b short sides, c hypotenuse. Pick one acute angle, label as theta.
we know e^(itheta)=cos theta+i sin theta.
Cos is an even function, sin is odd, thus also know
e^(-itheta)=cos theta-i sin theta. multiply both of these together...
the exponential side is e^(i theta - i theta) = e^0 = 1
the 2 complex trig sides are conjugates so becomes... cos^2 theta+sin^2 theta
which must equal 1 proving cos squared + sin squared = 1.
so (a/c)^2 + (b/c) ^2 =1 and multiply by c^2 gives Pythagoras. Complex arithmetic rocks haha
This is a very interesting video. I subscribed!
Davi Nonnenmacher yay!! Thanks!
No offense man, but I like the visual proof more. It's so cool.
Wow, very excellent reciprocal Pythagoras theorem.
Beautiful Proof.
Mandeep Singh thanks
They say the pythagorean theorem have 50 proof,but from now on ur first proof is my favorit.
There are more than 300 proofs...
Beautiful proof!
Hey I have a different proof for inverse Pythagorean theorem by using similar triangles 5:53
a^-2 + b^-2 = 1/a^2 +1/b^2
= (a^2+b^2)/a^2*b^2
= c^2 /(c*c1*c*c2)
(from Pythagorean theorem
a^2+b^2=c^2
And from diagram at given time stamp a^2=c*c1 and b^2=c*c2)
=1/(c1*c2)
=1/h^2
(from the diagram at the given time stamp in the triangle c1, a, h
And triangle h, b, c2
h/c1=c2/h
c1*c2 =h^2)
Interesting! I didn't really know there was an inverse PT and I pondered what it was going to be throughout the video because I was thinking in terms of inverse functions, but I often forget that inverse has the occasional meaning related to "reciprocal".
Good job! 😇👍👍
Very elegant proof...it blows me away. Is inverse Pythagorean theorem same as harmonic mean?
Blackredpen I admire your mathematical abilities!
This guy makes me happy for no reason
Damn wow I've never seen the inverse formula before! That is definitely gonna be useful
xMinix cool!!!!
Enjoyed! Brilliant!
Bravo! As always, super presentation!
Beautifull proof and formula
What a delightfully elegant proof
Elliott Manley thank you!!
my proof
Of the three similar triangles you identified, the hypotenuses are a and b for the smaller triangles and c for the complete triangle. since area scales with linear scale squared, the corresponding areas of those triangles are
ka^2, kb^2 and kc^2 where k is the constant of proportion. but from the sum of areas in the triangle
ka^2 +kb^2 =kc^2 hence a^2+b^2=c^2 .(eqn 1)
calculating the area of the whole triangle in two ways, we have that ab/2 = ch/2, doubling and squaring
gives a^2.b^2 =c^2.h^2 (eqn 2)
dividing eqn(1) by eqn 2 gives
(a^2 +b^2)/(a^2b^2) = c^2/(c^2h^2) ,
distributing the division on the LHS gives
a^2/(a^2b^2) + b^2/(a^2b^2) = c^2/(c^2h^2)
cancelling like terms gives
1/b^2 + 1/a^2 =1/h^2
My favourite proof of the Pythagorean theorem isn't a formal one for sure but I really like it nonetheless. You need to assume some physics for it. You build a right triangle, pin it down on the vertex with the right angle, fill it with air and put it in a vacuum. It can't start spinning due to conservation of energy. But it has torques acting on it- from the pressure inside of it. If you calculate the sum of these torques, them equalising is equivalent to the Pythagorean theorem.
definitly gonna do this on next math test
Like the most beautiful equation(Euler's law) It's the most beautiful proof 👌👌👍👍👍👏👏👏Fantastic
这些视频都太好了。 多谢！
ab/2=ch/2 h=ab/c
The Jacob Bronowski's proof shown in the episode 'Music of the spheres' of the series 'The Ascent of Man'. It only needs a simple sum. No need of multiplication (neither division) of triangle's sides.
What is so beautiful about the inverse Pythagorean theorem is that it can be rewritten as h^2 = HarmonicMean(a^2, b^2)
Yea! That is very cool!!!
The inverse Pythagorian theorem says the hypotnusr is forming as diameter of a semi circle conveving towards a single point making a perperndicular h towards hypotnuse forming the new formula as area 1/2 ab = 1/2 ch ab = ch .a^2+b^2 =c^2 inversed as h^2 =(c1+c2 )^2 with similar triangle many formulas may be generated.Now the question of pi connected as sweeping area of semi circle making many 90 degrees cornered triangles and some how will be connected with pi^2.
Inverse of cosinus theorem:
(h^-2)*((sinC)^2)=(a^-2)+(b^-2)-2*(a^-1)*(b^-1)*cosC, where C - angle between a and b
Blackpenredpen you can do it in a simpler manner
Construct h
1/2ab=1/2ch
ab=ch
c= ab/h
a= ch/b
b = ch/a
c^2=a^2+b^2
(ab/h)^2=(ch/b)^2+(ch/a)^2
ab^2/h^2=c^2h^2/b^2+c^2h^2/a^2
=1/h^2=1/b^2+a^2
After striking out the common terms
My favourite used to be the one I learned in linear algebra which involved vectors and inner products. However, I gotta say after watching this, I prefer this proof.
Two equation Pythagoras and inverse are symmetrical.very cute.keep it up.
Very cool proof. I am a big fan.
This was majestic to watch 😢
Thank yyou for proove this theory to me at last
The inverse is very useful.... lovely video
That is an Awesome proof!
Bro, I am so glad I subscribed to your channel. Cheers.
My favorite proof of the pithagorean theorem is using Thales' theorems
(1) pythagoras theorem Einstein’s method ... the area of each similar triangle is proportional to the square of the hypotenuse( rules of similarity) by summing areas of triangles we get directly the area of the two smaller triangles ka^2 +kb^2 = the area of the total triangle kc^2 dividing by the constant of proportionality we get a^2+b^2=c^2
Wow! Thank you!
The proof of Chinese mathematician'趙爽'is my favorite proof,just brcause there is no algebra in it