Pythagoras twisted squares: Why did they not teach you any of this in school?

I never saw it anywhere ever, not in 55 years...

I saw the second (algebraic) proof in hight school

the area of the shaded region is root 5 divided by 4

The solution to the final puzzle is 1/5.
Perhaps the most visually pleasing way is to rearrange the 4 small triangles so they each fit onto one of the trapeziums to form 4 identical squares.
We know that these are also identical to the middle square with unknown area because the ratio of original lengths create similar triangles.
The result is the large square of area 1 is separated into 5 smaller squares of equal area, one of which is the middle square.

you made rectangles and I had to turn it off

Maybe you've hinted at it briefly but what I find super-duper-interesting is the fact that in any Pythagorean triple, the B length can be understood as the geometric mean of C+A and C-A.
What's more, if I swap A and B, it's still true, even though the factor is different!
Example #1:
75 + 45 = 120, 75 - 45 = 30. The factor in question is either 2 or 1/2.
Example #2:
75 + 60 = 135, 75 - 60 = 15. The factor in question is either 3 or 1/3.
This fact seems to often be omitted, with a notable exception being Eleanor Robson's paper about Mesopotamian mathematics.
[Side note: Not sure if any similar property exists for the Eisenstein triples.]
I think it would actually be interesting to watch the ratios of the corresponding factors. In the example I gave above, the ratio would either be 3/2 or 2/3.
Now since the ratio is getting closer and closer to 1 when A and B are getting closer and closer to each other, I gather it must have something to do with tangents or cotangents. But I'm not sure what exactly.

I had a math teacher in junior high who has shown us this proof :)

Both methods, in Algebra. 7th grade S. California public school 1969.

I think these twisted squares are a great design for a tri-oriented architectural design. Each story being smaller than the one below it, the inner squares bring rooms, with the left over triangles being balconies.

I first (?) saw the second proof on a poster in the Paris metro. The French are different from us anglophones. You also see poetry, philosophy, all kinds of non-commercial stuff.

Hmm I never learned any proof for Pythagorean theorem, but I did come up with the square one on my own a couple years ago, though only algebraically.

Imagine having to pay for school! That's fucked up

However I use much geometry in carpentry.

Graduated HS in 1977.
We had pretty good education system.
Grew up in a small town on Texas coast with one HS
Went up to Trig. We had a math teacher who could teach in any college. Great man
But, Never seen this.
But I understannd.

(Sorry for my bad english skills, Im from Germany)I made the first proof up myself while I was sitting bored in our classroom. +We didnt evenlearn the pythagoras theorem at that time. I basically had the same as the 2nd proof already wrote so much but then i clicked play and noticed that the same proof was in the video already. But the first and the second are basically the same, just visualized differently

Amazing. Thanks for sharing knowledge

This twisted square proof I was never ever exposed to and me, a graduate of the 29th grade of school! (It was a long haul to a PhD in Geophysics.)

Lol, every time you laugh i have this feeling it's live and you can see me and you can tell how stupid i am. Anyway, keep going.

As a mathematician, I am rather disappointed by the lack of an immense generalization of Pythagorean theorem to higher dimensions in standard undergraduate or graduate linear algebra textbooks. You project the "c"-vector (side) of the abc triangle onto the x and y axes to get the a and b vectors (sides) and then c^2 = a^2 + b^2. In higher dimensions, you define the length of a vector in a similar way where the square of the length of a vector-x is the sum of the squares of the lengths of the 1-projections of x. Now, what happens is that if you consider not just a vector x but a multitude of vectors x_1, ..., x_k in a n-dimensional space with k < n and look at the parallelepiped P formed by these vectors, then the square of the volume of P is equal to the sum of the squares of volumes of all k-projections of P. The key notion in this equality is the fact that the relevant volumes are equal to the determinant of relevant square matrices so that an algebraic definition of a determinant of a square matrix gets a geometric meaning.

I came across the algebraic proof in serge langs "Basic mathematics" text. I find it a lot more elegant than the first proof, simply because you don't have to do any visual manipulation

Bothe

Wow, using no calculator and taking 5 times as much as just doing the math by hand

If you already "knew" that 2 to the 10th power is 1024 as implied by the video, juat multiply 1024 by 1024 and subtract 1.

It does not always need to be about notable.products, you know

I actually figured out that 2nd proof of Pythagoras on my own.

In my retirement I have been working occasionally as a substitute public school teacher. (Math, Science, Spanish, only subjects I know well). I like to show these proofs to the students and jokingly quote your "get your money back" comment.
One day I was talking with a math faculty member and told him of your channel and some of the things I like to share with the students to demonstrate that math is fun. He had never seen a proof of the Pythagorean Theorem , and in fact had believed it was an axiom.
So thanks for your channel, you do more good than you know.

i'm farmiliar with the second proof
and I only learned it in my senior year of highschool in my multivariable calculus class. (i probably
have learned it before but just don't remember it)

I was made to derive the proof myself as homework in highschool - and came up with proof B. I've met other kids whose teachers did the same, so it is (or was) probably part of curriculum/recommended homework activity.

I first came across the second proof in my early 20s after graduating college. I became interested in being able to prove many of the tings I had a "learned" in school and the Pythagorean theorem was one of these. I forget where I learned about it, perhaps some math website. I was blown away by how beautiful and simple this proof is. I have no idea why this wasn't taught in school. Makes me kind of angry.

Jeje, one step away from Vortex math and all the occultry that comes along with it. LFG! 😂

At 55 seconds what the diagram is the same way the ancient mathematician of India named bodhayan described the pythagorus theorem many centuries before pythagorus. I am now convinced that pythagorus definitely visited India and learned here.

FUN FACT: this geometry is applicable to musical scales as well. I see music as geometry as strange and weird as that must sound lol

I live in a very secluded place in the Adirondack mountains and it is a place with a lot of woods, critters and nature and you can SEE the harmony not just HEAR it in your life, If you take the time to see cymatics in life. It is all frequencies energies and vibrations. Sound CREATES GEOMETRY! Om So Hum! I Find Geometry as soothing as the sound that produces it! It's all-encompassing harmony.

I have that shirt.

Well, not THAT shirt, but one that looks just like it.

so cool.

Never either heard them, iirc. Private school too, ie expensive!

sir i would be car full to what you do to me square because the cube is watching you :(

''Why did they not teach you any of this in school?''
Because school is intended to teach you subservience to the upper classes, and just sufficient so that you can carry out the work they want from you.
Anything more is superfluous to their requirements.

Both. In Grammar School 1972

You say the Chinese proof may be nicer than the first proof because you only have to move two triangles. Not so. In the first proof you can move just two triangles by sliding without rotating, and without them overlapping at any time. Furthermore, the diagram ends up with a nicer symmetry than the Chinese version.

Wonderful!

2000 years back this theorem prove by BODHYAN in India you can google its actually not pythagorus theorem

Root of A ^2 + Root B^2 = A +B

I folow you long time ago.
Tanks.

I think you are wrong in time 27min and 25 séc this vídeos

The equilateral triangle logo would make a nice Tshirt: Caution: We're about to do math!

It's not "a scandal" that it's named the Pythagorean theorem. We're European, we live in a society built by Europeans. That's the word for it in our part of the world. Words have meanings. We call spaghetti "spaghetti" even though the Chinese invented noodles. Should we stop using "spaghetti" and suddenly start calling it "mian tiao"? Words have meanings based on long, complicated histories. We don't just change the name of a thing just because we didn't have it first. That's moronic. It stems from the weird self-hate that so many whites have now. We want to destroy our own culture just to be "inclusive" in some round about way.

Any correlation between the Pyramids at Giza and this 60 degree Trithagorean theorem?