Pythagorean Theorem & Its Inverse (my favorite proof) 

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This is far clearer and more straight-forward than the traditional proofs.

The inverse pythagorean theorem is some serious beautiful shit.

The Pythagoras Lu! 😄

Wow, I've never seen the inverse Pythagoras Theorem. Love your proof videos!

My favorite proof of the Pythagorean theorem is Garfield's (the president, not the cat) proof, just because of how clever it is.

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!

I wish they taught this proof in schools to teach kids things don't just come out of the blue.

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

Really pleased to see the inverse proof. And who, I wonder, are the four people who don't like this video?

omg, I love that inverse to get the height from the hypot. Very nice, thank you!

What a simple proof,you Chinese, have turned the world upside down.you have unearthed everything.you are amazing.

This is the first proof I loved... Like we use it every day and it's fun

I NEVER KNEW THE INVERSE ONE!!!!!

The inverse is too beautiful, I remember that elegance after deriving it myself :)

Proof by area, really straightforward.

DAMN I LOVE THIS GUY HE IS AMAZING

beautiful. the proof based on triangles similarity is really elegant.

Love how you tackle really hard and also simpler problems with the same enthusiasm!

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]

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.

They say the pythagorean theorem have 50 proof,but from now on ur first proof is my favorit.

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.

Thank you!!! This was a beautiful proof!!! :-D

Very nice proof of Pythagorean theorem ! It seems to me I didn't see this proof elsewhere. Thank you !!

The most amazing proof of the Pythagoras I've ever seen. Thanks Steve!

Last year on this day i subscribed this channel.... It has been an amazing journey with blackpenredpen #yay ...🖤

Thats a beatiful way to proof pythagorean thanks.

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

I love to using James Grifeld formula,and your are excellent too for proof for Pythagorean theorem

It's wonderful result!

This is a very interesting video. I subscribed!

Great! Simple, plain, non-obfuscated. Spot on!😎

this concept has already helped me in solving question thanks for covering it

Mind blowing proofs, i am speechless. Keep up the good work 👍👍👍

this is the proof i got taught in school...always liked it more than the other ones

Enjoyed! Brilliant!

Very simple and easy to teach demonstration of the famous theorem. Thanks a lot for providing brand smart different points of view!!!

No offense man, but I like the visual proof more. It's so cool.

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.

This guy makes me happy for no reason

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.

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.

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.

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.

Bravo! As always, super presentation!

What a delightfully elegant proof

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!

Beautiful proof!

Beautiful Proof.

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).

What is so beautiful about the inverse Pythagorean theorem is that it can be rewritten as h^2 = HarmonicMean(a^2, b^2)

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.

Like the most beautiful equation(Euler's law) It's the most beautiful proof 👌👌👍👍👍👏👏👏Fantastic

Bro, I am so glad I subscribed to your channel. Cheers.

Damn wow I've never seen the inverse formula before! That is definitely gonna be useful

Blackredpen I admire your mathematical abilities!

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.

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.

Beautifull proof and formula

My favorite proof of Pythagoras theorem is the one I made few months ago

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.

definitly gonna do this on next math test

Two equation Pythagoras and inverse are symmetrical.very cute.keep it up.

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

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).

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.

Immediately liked the contrasting colors and title of click maths page.

The proof of Chinese mathematician'趙爽'is my favorite proof,just brcause there is no algebra in it

Well explained. I wasn't aware of this cool proof.

This was majestic to watch 😢

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)

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

The inverse is very useful.... lovely video

My favorite proof of the pithagorean theorem is using Thales' theorems

I also like this demostración the most, but it’s risky: people will start wondering if the similar triangles theorems use the pitagorean (they don’t need it ) the areas stuff is more “clean” un that sense.

ab/2=ch/2 h=ab/c

Thank yyou for proove this theory to me at last

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

Very elegant proof...it blows me away. Is inverse Pythagorean theorem same as harmonic mean?

My favorite proof is one by Einstein where he also uses the similarity of the triangles you used, but he uses their areas to prove Pythagorean theorem

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.)

@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.

That is an Awesome proof!

When π comes out from ζ(2)

where da heck we use this inverse pytho. theo. concepts at?? havent learn in my school

My favourite proof is the algebraic proof by contradiction because this one is just unusual.

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.

(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

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.

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".

What an awesome proof

I love the President Garfield proof

I didnot knew about inverse pythagoras theoram but thanks bprp class 9

Good job! 😇👍👍

My favorite proof is the euclidean one. It is very cool

这些视频都太好了。 多谢！

MAGNIFIQUE !!! MERCI INFINIMENT ...