See how euclid defines everything before explaining properly and correctly the properties of the figures considered without telling things like " with the basic assumption or truth that.....". Though The Elements may be a difficult book considering the amount of interest and attention needed to learn the proofs for many people, the book gives u a reason of what geometry and math is and its role in day to life..no human can ever write a book as clearly as Euclid has about Geometry...
Looking at Euclid's Proofs really gives you an appreciation of how much simpler things became with the invention of algebra. My favorite proof of the Pythagorean theorem is James Garfield's but this is definitely more eloquent.
He keeps asserting that a square is congruent to a parallelogram, when what he means is, they have equal areas. They are certainly not congruent. I noticed the overlays (replacing the congruent symbols with equal signs) when the statements were written on screen; however, the video ought to be redone with the verbal corrections, to eliminate the confusion. After all, this is supposed to be a formal proof. Also, can we not simply state that an angle IS a right angle instead of saying that it's congruent to a right angle? I don't understand the point of asserting the congruence. Later he says all right angles are congruent, which is fine.
+Ken Haley The efforts are appreciated except the word congruent is used too often and sometimes incorrectly. Congruency in 2D means two shapes are exactly same includes all sides and angles.
Thanks so much for describing the proof. With seeing the videos of the needed postulates, I could easily understand and can remember the proof. First I was confused, because we define congruent in german as the same form of objects, nit the same area. But I found it out. Thanls so much what a great channel.
You made my life man! Up until now i used to believe that there was no mathematical proof of the Pythagorean theorem except for the visual square cutting method and that a major chunk of mathematics was based on a questionable assumption that A square + B square equals C square. You saved mathematics for me. Thanks a LOT man! Keep it up!
very good proof. However, I think there is a mistake. in 2:34, you stated that the definition of the square, and highlighted def.6. In Euclid's "the Element", he defines that the square in his book I, definition 22, but not definition 6.
Go to wikipedia "Pythagoras Theorem". There's a pretty cute proof there too. Ctrl+F this: Geometric proof of the Pythagorean theorem from the Zhou Bi Suan Jing.
It's a good idea to hand out this information to them. So they can replicate the steps themselves, with typing paper and a compass. If they don't have a compass, they can use string. (Many kids will want to do the exercises themselves if they forget. It's also wise to show them "adult level" applications...building things, etc. On my first construction jobs, I had to build jigs based on these rules for duplicating struts...)
@@JakeWitmer thank you for your reply, but I'm not a teacher, I'm a student :D My homework was to explain Euclid's proof with a pair (a person who sits beside me). The teacher gave everyone someone's proof, and I got Euclid's proof. Tbh, I got unlucky because Euclid's proof was harder to explain than everyone else's proof and my pair didn't do anything. But that was a year ago, so it doesn't matter anymore
this proof is everything i remember ,loads of letters and Confusing because you start in the middle. to proof the rectangle has the same area as the square, it is sufficient to prove that the triangles in each formed by the diagonals FA ( in the square) and DJ in the rectangle are equal. starting with the DJL shear it along LA to form ABD. rotate it to FBC and shear it to FBA. Since shear and rotate are area-preserving the areas of the halVES and hence the wholes of the rectangle and the square are equal. Not rigorous but shear-rotate-shear is easy to remember.
c=a+b c^2 = (a+b)^2 = [a^2 + b^2] + [2ab] (binomial expansion) c^2 a^2+b^2 The "proof" in the video is only valid in the imagination. (Pythagoras was also confused).
Euclid was a total basket-case. There is a *much* easier way of proving the Pythagoras theorem. Place a small square of side c inside a larger square and turn it so that its corners touch the outer square's edges, creating four right angle triangles of hypotenuse c and opposite sides a and b. The total area is then c^2+4*1/2*ab=c^2+2ab. Now the outer square has side a+b so its area must also be a^2+2ab+b^2. Therefore c^2+2ab=a^2+2ab+b^2. Subtract 2ab from both sides and you have your result a^2+b^2=c^2..
Genial! Pero hay algunos conceptos que merecen un trato especial, la CONGRUENCIA , la IGUALDAD, y la EQUIVALENCIA. La congruencia se da cuando 2 figuras geométricas tienen la misma FORMA y el mismo TAMAÑO, teniendo en cuanta que todos sus elementos homólogos tengan la misma medida. La igualdad necesita lo anterior mencionado y además necesita que las figuras se encuentren exactamente en la misma posición, o sea, como superpuestas, por lo que no tiene sentido trabajar con este concepto, salvo cuando se habla de NÚMEROS o cantidades. Y la equivalencia, que se da entre figuras geométricas de la misma clase dimensional, o que estén contenidas en la misma dimensión, cumpliéndose una igualdad (ahora sí) entre por ejemplo áreas o volúmenes, aunque aquellas no tengan la misma forma.
Why do you keep calling them parallelograms when they are rectangles? I know rectangles are a special case of parallelogram, but why use the more generalized term, when we know they are rectangles?
Because the proposition that talks/speaks about triangles and paralelograms areas uses the term "paralelogram" then, he is using the the term palalelogram just like the preposition says. We know theyre rectangles but the proposition says paralelograms, no only rectangles.
Why do those who explain this proof call it a parallelogram and not a rectangle. I understand both terms are correct. But I think most folks would call that a rectangle, and all the times I've seen this proof explained (ok, about 4 times?), it is called a parallelogram. Why? Thanks.
You don't *know* for that it's a rectangle - that would have to be proven. (But actually it's unnecessary to prove that it's a rectangle, since we don't need to use the fact that it's a rectangle in the proof of Pythagoras' Theorem).
Thanks for the explanation it's really helpful to me . I have to make an assiments of Euclid proofs and then I found your video. ,Now I can easily make assiments.... 👍 for explaining 😊😊😊
Great work. Just incorrect terminology, congruence just means the same shape. Other than that these videos are very good. keep going, you'll make mistakes, we all do, but just keep these coming they are great.
Good Grief !! - how to make things over complicated !!!! I honestly don't think for one moment that Pythagoras thought about triangles and squares drawn on each side of a triangle ? Why would anyone think that ? - No !!! What I believe Pythagoras ACTUALLY thought about was SQUARES - and quite simply.- I reckon he was thinking thinking about a tiled floor and how you could make patterns using square tiles of different sizes That is .. IF you take two square tiles of different sizes - and put one " inside " the other - then the DIFFERENCE between the two areas MUST be the difference in area of the two squares (tiles) . - Obvious or what ??? - of COURSE it is !!! . So NOW we have on the Agenda SOMETHING to do with TWO squares and their areas and the patterns they make when one square is inside the other. so ……. … NOW … if one square is "slightly " smaller that the other .. then you "can" "position" the smaller one centrally inside the larger one so it "locks" in position inside the larger square forming 4 EQUAL right angled triangles - one triangle at each corner of the larger square .. and so ...… we have …. The LARGE square area LESS the SMALLER square area MUST BE equal to the area of FOUR right angle triangles that ARE IDENTICAL .... ( Got to be !!! ) since they WERE formed by rotating ONE square about its centre inside the other square. "centre" .. And so --- it is now soooo easy to see that Large square - small square = 4 * area of rt angled triangle ( the hypotenuse of which is equal in length to the length of the side of the smaller square ... … and .. a bit of simple algebra - and ….. voila … you have .. A^2 + B^2 = C^2 !!! No need to go to all that grief as per video !!
Too much complex, therefore boring. I started yawning at aproximately half of the video (might add to it, that I watched it in the evening). Yes, it might be educational, but there's too much information and lots of stuff going on. Why not simply explaining, that this square and this square are equal to that square and this triangle shifthed, or pushed in a way is equal to another triangle and so on? There are angles and another angles and stuff, that people actually don't concern about. Stuff, that everybody, who went to the school, already knows. This angle equals to that angle and the sum of those angles is 180° and so on. Not really interesting stuff, so to say.
Some people learn this stuff in private schools, and it is helpful, so if you think it was boring don't watch it because then it won't help you. Try doing almost the entire book one of his propositions. This one of the longer proofs he does in book one
![Pythagoras Would Be Proud: High School Students' New Proof of the Pythagorean Theorem [TRIGONOMETRY]](http://i.ytimg.com/vi/p6j2nZKwf20/mqdefault.jpg)
Just came to say "THANK YOU SO MUCH" I....literally can't find any sources explaining it as well as you do! Thank you
props to euclid for finding the long way around
Notice! There are some corrections to this video, please turn on annotations to see them.
See how euclid defines everything before explaining properly and correctly the properties of the figures considered without telling things like " with the basic assumption or truth that.....". Though The Elements may be a difficult book considering the amount of interest and attention needed to learn the proofs for many people, the book gives u a reason of what geometry and math is and its role in day to life..no human can ever write a book as clearly as Euclid has about Geometry...
Euclid is THE FATHER OF GEOMETRY.
Thank you, you are one of the only Euclidean proof videos I have found.
Looking at Euclid's Proofs really gives you an appreciation of how much simpler things became with the invention of algebra.
My favorite proof of the Pythagorean theorem is James Garfield's but this is definitely more eloquent.
Euclid’s proof is definitely more eloquent, but Garfield’s is superbly concise.
have you thought about putting Q.E.D. at the end?
Q.E.D, quod erat demonstrandum, the Latin phrase for "which was to be demonstrated."
He keeps asserting that a square is congruent to a parallelogram, when what he means is, they have equal areas. They are certainly not congruent. I noticed the overlays (replacing the congruent symbols with equal signs) when the statements were written on screen; however, the video ought to be redone with the verbal corrections, to eliminate the confusion. After all, this is supposed to be a formal proof.
Also, can we not simply state that an angle IS a right angle instead of saying that it's congruent to a right angle? I don't understand the point of asserting the congruence. Later he says all right angles are congruent, which is fine.
+Ken Haley The efforts are appreciated except the word congruent is used too often and sometimes incorrectly. Congruency in 2D means two shapes are exactly same includes all sides and angles.
+Ken Haley
True- And this is a crucial mistake too.
Actually the proof breaks down when he claims that area equality is the same as congruency.
Thanks so much for describing the proof. With seeing the videos of the needed postulates, I could easily understand and can remember the proof.
First I was confused, because we define congruent in german as the same form of objects, nit the same area. But I found it out. Thanls so much what a great channel.
this video saved my grade in maths
You made my life man! Up until now i used to believe that there was no mathematical proof of the Pythagorean theorem except for the visual square cutting method and that a major chunk of mathematics was based on a questionable assumption that A square + B square equals C square. You saved mathematics for me. Thanks a LOT man! Keep it up!
What do they do? karate-chop boards that add to the same area?
Pythagoras studied at Alexandria, Egypt. The Egyptians were already practicing this.
very good proof. However, I think there is a mistake. in 2:34, you stated that the definition of the square, and highlighted def.6. In Euclid's "the Element", he defines that the square in his book I, definition 22, but not definition 6.
Go to wikipedia "Pythagoras Theorem". There's a pretty cute proof there too.
Ctrl+F this:
Geometric proof of the Pythagorean theorem from the Zhou Bi Suan Jing.
Are you aware of any type of way in solving any side lengths of ABC, without using the Pythagorean Theorem?
Even though it's been 8 years, thank you very much. Tomorrow at school I have to explain Euclid's proof and your video helped me
Best of luck!
It's a good idea to hand out this information to them. So they can replicate the steps themselves, with typing paper and a compass. If they don't have a compass, they can use string. (Many kids will want to do the exercises themselves if they forget. It's also wise to show them "adult level" applications...building things, etc. On my first construction jobs, I had to build jigs based on these rules for duplicating struts...)
@@JakeWitmer thank you for your reply, but I'm not a teacher, I'm a student :D
My homework was to explain Euclid's proof with a pair (a person who sits beside me). The teacher gave everyone someone's proof, and I got Euclid's proof. Tbh, I got unlucky because Euclid's proof was harder to explain than everyone else's proof and my pair didn't do anything. But that was a year ago, so it doesn't matter anymore
Blown away by the fact that the ratio of resulting bottom orange and red rectangles is related to cos of corresponding angles
Or otherwise the projection line of A cutting through the BC square
Thank you so much for this gem)
this proof is everything i remember ,loads of letters and Confusing because you start in the middle.
to proof the rectangle has the same area as the square, it is sufficient to prove that the triangles in each formed by the diagonals FA ( in the square) and DJ in the rectangle are equal.
starting with the DJL shear it along LA to form ABD. rotate it to FBC and shear it to FBA.
Since shear and rotate are area-preserving the areas of the halVES and hence the wholes of the rectangle and the square are equal. Not rigorous but shear-rotate-shear is easy to remember.
Nice proof! Well done
Do we really need a proof that squares can have any side length?
yes
Great job and thanks for sharing.
ah... but we must remove all doubt, hence it must be proven!
c=a+b
c^2 = (a+b)^2 = [a^2 + b^2] + [2ab] (binomial expansion)
c^2 a^2+b^2
The "proof" in the video is only valid in the imagination.
(Pythagoras was also confused).
almost 2 years early in subscribed u
but i never recieved any notification from u
This was very thorough. Awesome.
Euclid was a total basket-case. There is a *much* easier way of proving the Pythagoras theorem. Place a small square of side c inside a larger square and turn it so that its corners touch the outer square's edges, creating four right angle triangles of hypotenuse c and opposite sides a and b. The total area is then c^2+4*1/2*ab=c^2+2ab. Now the outer square has side a+b so its area must also be a^2+2ab+b^2. Therefore c^2+2ab=a^2+2ab+b^2. Subtract 2ab from both sides and you have your result a^2+b^2=c^2..
You are using algebra, which the Ancient Greeks hadn't invented yet.
amazing explanation. thank you!!
why does your equals sign have 3 lines?
in my geometry class we dont really care so we say theyre equal, it doesnt really matter
Nice-and thorough! Thanks!
Genial! Pero hay algunos conceptos que merecen un trato especial, la CONGRUENCIA , la IGUALDAD, y la EQUIVALENCIA. La congruencia se da cuando 2 figuras geométricas tienen la misma FORMA y el mismo TAMAÑO, teniendo en cuanta que todos sus elementos homólogos tengan la misma medida. La igualdad necesita lo anterior mencionado y además necesita que las figuras se encuentren exactamente en la misma posición, o sea, como superpuestas, por lo que no tiene sentido trabajar con este concepto, salvo cuando se habla de NÚMEROS o cantidades. Y la equivalencia, que se da entre figuras geométricas de la misma clase dimensional, o que estén contenidas en la misma dimensión, cumpliéndose una igualdad (ahora sí) entre por ejemplo áreas o volúmenes, aunque aquellas no tengan la misma forma.
Why do you keep calling them parallelograms when they are rectangles? I know rectangles are a special case of parallelogram, but why use the more generalized term, when we know they are rectangles?
Because the proposition that talks/speaks about triangles and paralelograms areas uses the term "paralelogram" then, he is using the the term palalelogram just like the preposition says. We know theyre rectangles but the proposition says paralelograms, no only rectangles.
bc they have to be parallelagrams to use I.41
Thank you so much! This is sooo helpful!!
Simply love mathematicsonline!Really helpful!👍
Why do those who explain this proof call it a parallelogram and not a rectangle. I understand both terms are correct. But I think most folks would call that a rectangle, and all the times I've seen this proof explained (ok, about 4 times?), it is called a parallelogram. Why? Thanks.
You don't *know* for that it's a rectangle - that would have to be proven. (But actually it's unnecessary to prove that it's a rectangle, since we don't need to use the fact that it's a rectangle in the proof of Pythagoras' Theorem).
For some reason I got lost at the part where the triangles were half the area of the square.
look at it this way A=B+C, B+C=D+E, D+E=F. F=B+C, which =A ergo A=F
Thanks for the explanation it's really helpful to me . I have to make an assiments of Euclid proofs and then I found your video. ,Now I can easily make assiments.... 👍 for explaining 😊😊😊
Awesome, you made it really easy to understand,,
Thank you
Hi Krypton from Krypton ;)
Thank you !
Great work. Just incorrect terminology, congruence just means the same shape. Other than that these videos are very good. keep going, you'll make mistakes, we all do, but just keep these coming they are great.
thanks!
The Ptah-Horus Theorem 👍
Great idea that you sir
good one
this is so interesting omfg
Wow, so that's how it's proven.
its all about thinking different it is genius
Brother you smart
Awesomeeee
Great!
Fantastic thx
Guarda mio video geometrico
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fastest i subscribed
nice
türkiyeden selamlar
Good Grief !! - how to make things over complicated !!!!
I honestly don't think for one moment that Pythagoras thought about triangles and squares drawn on each side of a triangle ? Why would anyone think that ? - No !!!
What I believe Pythagoras ACTUALLY thought about was SQUARES - and quite simply.- I reckon he was thinking thinking about a tiled floor and how you could make patterns using square tiles of different sizes
That is .. IF you take two square tiles of different sizes - and put one " inside " the other - then the DIFFERENCE between the two areas MUST be the difference in area of the two squares (tiles) . -
Obvious or what ??? - of COURSE it is !!! .
So NOW we have on the Agenda SOMETHING to do with TWO squares and their areas and the patterns they make when one square is inside the other. so …….
… NOW … if one square is "slightly " smaller that the other .. then you "can" "position" the smaller one centrally inside the larger one so it "locks" in position inside the larger square forming 4 EQUAL right angled triangles - one triangle at each corner of the larger square ..
and so ...… we have …. The LARGE square area LESS the SMALLER square area MUST BE equal to the area of FOUR right angle triangles that ARE IDENTICAL .... ( Got to be !!! ) since they WERE formed by rotating ONE square about its centre inside the other square. "centre" ..
And so --- it is now soooo easy to see that Large square - small square = 4 * area of rt angled triangle ( the hypotenuse of which is equal in length to the length of the side of the smaller square ... … and .. a bit of simple algebra
- and ….. voila … you have .. A^2 + B^2 = C^2 !!!
No need to go to all that grief as per video !!
thats too much information
Lol id rather just fail than remember this uterly useless shit
Too much complex, therefore boring. I started yawning at aproximately half of the video (might add to it, that I watched it in the evening). Yes, it might be educational, but there's too much information and lots of stuff going on. Why not simply explaining, that this square and this square are equal to that square and this triangle shifthed, or pushed in a way is equal to another triangle and so on? There are angles and another angles and stuff, that people actually don't concern about. Stuff, that everybody, who went to the school, already knows. This angle equals to that angle and the sum of those angles is 180° and so on. Not really interesting stuff, so to say.
Alexandra Hefnerová yea, stuff
Some people learn this stuff in private schools, and it is helpful, so if you think it was boring don't watch it because then it won't help you. Try doing almost the entire book one of his propositions. This one of the longer proofs he does in book one
Guarda mio video geometrico
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