And where's the proof that the area increases as square of length? ....
Try this, and by the way try to remember that "square" didn't exist in the days of Pythagoras: 'proof by rearrangement' on the wikipedia article on Pythagorean theorem.
Correction; I should have written, 'However the assumption that similar shapes increase their area in proportion to THE SQUARE OF a corresponding length is an elementary yet non-trivial proposition. I blame the 500 character buffer limit myself . . that and inate stupidity, lol.
However the assumption that similar shapes increase their area in proportion to a corresponding length is an elementary yet non-trivial proposition which requires some argument and also lengthens the proof! At elementary level it is hardly more reasonable to assume this than to assume the Pythagoras theorem itself! You can always shorten an argument but in so doing you also shorten the comprehending audience.
The proof is correct because all the normalised triangles are the same right angled triangle, scaled up first by ‘a’ to achieve an arbitrary length then by ’b’ to match distinct adjacent sides then by ‘c’ to fit the remaining sides to the hypotenuse! Because it is just a matter of scaling, all triangles are similar! If they were either not similar or not right angled there would be no such fit. The trick is to make the details implicit to shorten the proof.
i dont even understand the preliminaries, if u let that area = k and hypo be 1, then i assume either breadth or length is 2k isn't it? cuz 0.5x2kx1= k.
then you change to a bigger triangle, we duno how much 2k changes, given hypo is now a, how did u just get a^2k ?
I have an idea. Since you can't answer the objections to your 'proof' which show that it is in fact invalid, why don't you remove it from youtube, go back to geometry class, and learn how to properly do proofs. Amazing how you think you can gloss over the fact that you used the same proportionality constant of K for all 3 triangles without stating how you can know they are all congruent.
Also, the fact that area is proportional to length squared is proven after this theorem so it's circular.
@luvphysics2009 Well you can say they're all congruent becuase they all have a right angle and one of the other angles that the big triangle has (obviously the big triangle itself has both of them). Since they share two angles with the big triangle they obviously share the third and are similar triangles. For a given shape of triangle area is proportional to the square of a given side because you increase both height and width. You do not need pythagoras theorem to prove this.
@newperve Ah, so you're assuming more than what is properly pointed out in the picture and requiring the viewer to fill in all those pesky details about SSA congruence and such. With those details and a few others made clear, it would be a proof.
What's strange is, pythagorean theorem is absolutely dependent on the parallel postulate, since it is not true in non-euclidean geometry. And yet your theorem uses little more than proposition 26 book one which doesn't require parallel postulate.
@luvphysics2009 I'm not assuming anything. The triangles A and B are both right angle triangles becuse they both have a corner where the perpendicular was dropped. If you acxtually look at the picture you will notice that one of the angles of C is also an angle of B. The other (non-right) angle is an angle of A. So each has an non-right angle equal to one of the angles of C and a right angle 2 eqaul angles therefore congruent.
They would not fit together if they were not similar triangles rotated at 90 degrees making their angles complementary. He makes it short by not going through the trouble of stating that they are similar right triangles and then showing that the angles are complementary, so this proof is based on a real proof but cheats to make it short.
@pckyt This proof is actually correct. The video did not state that all three original triangles are the exact same triangle, and hence they have the same area. (The fact that this works for any right angled triangle is implicit.)
@medeerbeer If the triangle wasn't a right-angled triangle the orange and purple triangles wouldn't fit together to form a straight line on the c-side of the combined triangle.
The best thing about this proof is that it's wrong because you claim that this is true for all triangles when it's only true for right triangles, and you don't even make use of that.
This proof is very wrong. You assume that the "normalized" versions of the yellow, violet and red areas all have size K. Obviously, that cannot be guaranteed.
Pythagoras works only for right-angled triangles. Your proof doesn't make use of that property anywhere, so if it were correct, the Pythagoras theorem would apply to any triangle.
I agree... why go to all the effort of putting up something that is totally wrong... Why not make a video about how 1 = 2... here is the proof.... well X = 1 and X = 2... X = X so by substitution 2 = 1...that is genuinely what you are doing here.
@pckyt the proof is correct, just read the NUAGESA comment, the proof starts with three equal triangles with area K, and then it just multiplies each of them by a, b and c respectively, in a way that they fit to each other as it shows on the picture. And they only fit that way if they are right-angled and similar.
@TheInfiniteQuestions Yes I see how it can be made to work now. Hell of a confusing way to write it down though. And it still needs a least two further lemmas on the way home (scaling and sum of angles = 180 deg) . Making "details implicit", as NUAGESA calls it, is one of the worst habits in mathematics.
Unluckly i about failed my real analysis class. The professor would have never take a picture has a proof. But be sure of this. I will retake the course until I passed it.
There are lively proofs and lifeless proofs. Lively proofs are those which get the reaction from the student: "what else can it be?" about the theorem statement.
waste of time makes no sense
Jamieduk999 2 weeks ago
Wait, so the son of the hide of the hippopotamus is greater than the sons of the squaws other two hides ?
Postie218 3 weeks ago
SHOULDN'T IT BE a+b = csq
kirkleblanc 3 weeks ago
And where's the proof that the area increases as square of length? ....
Try this, and by the way try to remember that "square" didn't exist in the days of Pythagoras: 'proof by rearrangement' on the wikipedia article on Pythagorean theorem.
Pay attention to the 90-degree angles.
Your proof is not a proof.
icl151 1 month ago
Correction; I should have written, 'However the assumption that similar shapes increase their area in proportion to THE SQUARE OF a corresponding length is an elementary yet non-trivial proposition. I blame the 500 character buffer limit myself . . that and inate stupidity, lol.
NUAGESA 1 month ago
However the assumption that similar shapes increase their area in proportion to a corresponding length is an elementary yet non-trivial proposition which requires some argument and also lengthens the proof! At elementary level it is hardly more reasonable to assume this than to assume the Pythagoras theorem itself! You can always shorten an argument but in so doing you also shorten the comprehending audience.
NUAGESA 1 month ago
The proof is correct because all the normalised triangles are the same right angled triangle, scaled up first by ‘a’ to achieve an arbitrary length then by ’b’ to match distinct adjacent sides then by ‘c’ to fit the remaining sides to the hypotenuse! Because it is just a matter of scaling, all triangles are similar! If they were either not similar or not right angled there would be no such fit. The trick is to make the details implicit to shorten the proof.
NUAGESA 1 month ago
@NUAGESA totally agreed, the proof is correct
TheInfiniteQuestions 1 month ago
The fact that this actually needs prooving, still now in the 21st century, says something highly alarming about people.
DrMadolite 1 month ago
brilliant video
shellybirdy 1 month ago
Lol, what a trolling video. Very dumb. Mathematicians would send you to fry potatoes.
proxsor 2 months ago
Shouldnt you mention that the 3 triangles are similar?
betamale3 3 months ago
labeled badly
circleseverywhere 3 months ago
i dont even understand the preliminaries, if u let that area = k and hypo be 1, then i assume either breadth or length is 2k isn't it? cuz 0.5x2kx1= k.
then you change to a bigger triangle, we duno how much 2k changes, given hypo is now a, how did u just get a^2k ?
sry im noob lol
DiabloMercy 4 months ago
Easier to understand and better proof:
watch?v=pVo6szYE13Y&list=FLrSpkSGdnA1sBsiponc7mwA&index=7
mmmimi50 5 months ago
The proof is wrong, K is the same only for same triangles, and not all right-angled triangles are same.
lucinos19 5 months ago
what a noob
chrisl2002 5 months ago
I have an idea. Since you can't answer the objections to your 'proof' which show that it is in fact invalid, why don't you remove it from youtube, go back to geometry class, and learn how to properly do proofs. Amazing how you think you can gloss over the fact that you used the same proportionality constant of K for all 3 triangles without stating how you can know they are all congruent.
Also, the fact that area is proportional to length squared is proven after this theorem so it's circular.
luvphysics2009 6 months ago
@luvphysics2009 Well you can say they're all congruent becuase they all have a right angle and one of the other angles that the big triangle has (obviously the big triangle itself has both of them). Since they share two angles with the big triangle they obviously share the third and are similar triangles. For a given shape of triangle area is proportional to the square of a given side because you increase both height and width. You do not need pythagoras theorem to prove this.
newperve 5 months ago
@newperve Ah, so you're assuming more than what is properly pointed out in the picture and requiring the viewer to fill in all those pesky details about SSA congruence and such. With those details and a few others made clear, it would be a proof.
What's strange is, pythagorean theorem is absolutely dependent on the parallel postulate, since it is not true in non-euclidean geometry. And yet your theorem uses little more than proposition 26 book one which doesn't require parallel postulate.
luvphysics2009 5 months ago
@luvphysics2009 I'm not assuming anything. The triangles A and B are both right angle triangles becuse they both have a corner where the perpendicular was dropped. If you acxtually look at the picture you will notice that one of the angles of C is also an angle of B. The other (non-right) angle is an angle of A. So each has an non-right angle equal to one of the angles of C and a right angle 2 eqaul angles therefore congruent.
newperve 5 months ago
They would not fit together if they were not similar triangles rotated at 90 degrees making their angles complementary. He makes it short by not going through the trouble of stating that they are similar right triangles and then showing that the angles are complementary, so this proof is based on a real proof but cheats to make it short.
michalchik 8 months ago
Comment removed
wkrepelin 6 months ago
@pckyt This proof is actually correct. The video did not state that all three original triangles are the exact same triangle, and hence they have the same area. (The fact that this works for any right angled triangle is implicit.)
@medeerbeer If the triangle wasn't a right-angled triangle the orange and purple triangles wouldn't fit together to form a straight line on the c-side of the combined triangle.
nfwu 10 months ago 2
The best thing about this proof is that it's wrong because you claim that this is true for all triangles when it's only true for right triangles, and you don't even make use of that.
medeerbeer 1 year ago 9
This proof is very wrong. You assume that the "normalized" versions of the yellow, violet and red areas all have size K. Obviously, that cannot be guaranteed.
Pythagoras works only for right-angled triangles. Your proof doesn't make use of that property anywhere, so if it were correct, the Pythagoras theorem would apply to any triangle.
pckyt 1 year ago 25
@pckyt
I agree... why go to all the effort of putting up something that is totally wrong... Why not make a video about how 1 = 2... here is the proof.... well X = 1 and X = 2... X = X so by substitution 2 = 1...that is genuinely what you are doing here.
theoriginalwasa 10 months ago
@pckyt the proof is correct, just read the NUAGESA comment, the proof starts with three equal triangles with area K, and then it just multiplies each of them by a, b and c respectively, in a way that they fit to each other as it shows on the picture. And they only fit that way if they are right-angled and similar.
TheInfiniteQuestions 1 month ago
@TheInfiniteQuestions Yes I see how it can be made to work now. Hell of a confusing way to write it down though. And it still needs a least two further lemmas on the way home (scaling and sum of angles = 180 deg) . Making "details implicit", as NUAGESA calls it, is one of the worst habits in mathematics.
pckyt 1 month ago
!!!!!!! NICe!!!!
Unluckly i about failed my real analysis class. The professor would have never take a picture has a proof. But be sure of this. I will retake the course until I passed it.
turser222 2 years ago
There are lively proofs and lifeless proofs. Lively proofs are those which get the reaction from the student: "what else can it be?" about the theorem statement.
kautilya33 2 years ago
This is my favourite proof as well, but a bit of animation would have been nice.
Sunduc 2 years ago
Unfortunately, I am a bit weak in animation:-)
kautilya33 2 years ago