Cool Proof of Pythagorean Theorem

Cool Proof of Pythagorean Theorem
Flannel Jesus
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Posted Jan 30, 2013 - 3:10 AM:
Subject: Cool Proof of Pythagorean Theorem

I was asked to prove it myself once a long time ago in a galaxy far away, and I did it, but it wasn't nearly this pretty:


Anybody know of any other clever proofs of this or other interesting theorems? Pi? e? Anything at all.

On Feb 2, 2013 - 7:35 PM, andrewk responded: I know 3 other proofs but that's the best one I've seen, and the easiest to remember. I love it!
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Posted Jan 31, 2013 - 4:19 AM:

You can find several neat proofs made by ancient greeks over the internet. They generaly involve geometric considerations, but one of my favorites of all time is in Euclids' Elements, on the infinity of prime numbers. There is another one there about the irrationality of sqrt(2) I think,

I found one of Fermat on the irrationality of sqrt(3) that is so clever and simple and I think you'd like to see it. There is a general proof of the irrationality of sqrt(n) when n is not a perfect square, so it generalizes all the precedent.



Too bad I cannot use LATEX here.



take care.
Numan
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Posted Feb 1, 2013 - 12:43 PM:
Subject: Proof by Pappus
'
The pleasingly simple proof you have so clearly presented here was not in Euclid, but did appear in the Zhou Bei Suan Jing, a Chinese mathematical text which quite possibly antedated Euclid -- as Joseph Needham pointed out in the mathematics section of his magisterial work, Science and Civilisation in China

The proof presented in Euclid, is rather compicated in comparison, which has puzzled many people.

Proposition 47 of Book I

The reason, I think, is that it is meant to be understood in connection with a very similarly constructed proof concerning all possible triangles and the parallelograms that can be constructed on the sides of any triangle. This proof appears in Pappus, and it shows that the Pythagorean Theorem is merely a special case of a more general proof concerning the Platonic nature of all triangles.

Pappus' Proof

I suspect that this is evidence that in Euclid's time (~300 BC) geometry stll had something of the flavour of being a secret knowledge, known only to initiates. The similarity of Euclid's proof and that of Pappus seems too close to be accidental. Euclid seems to present the "baby" version of the proof, which would -- under instruction by a master geometer, or, as they say, "as a problem to be worked out by the student" -- lead to the deeper proof preserved in Pappus.
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Posted Feb 2, 2013 - 5:56 AM:


Einstein, at age 11, came up with a proof that is quite elegant.

The height divides the large triangle into two smaller and similar triangles. (The angles must be the same.)

According to friend Euclid the area ratio of two similar closed figures equals the square of the ratio of the linear dimensions.

Ea,Eb and Ec are the areas of the triangles and their hypoteneuses are related by

Ea = m*a2

Eb = m*b2

Ec = m*c2

where m is a dimensionless multiplier such that m # 0 and m is the same in all three expressions.

Observing the figure above we see that:

Ea + Eb = Ec

i.e. the area of the larger triangle is the sum of the areas of the two smaller ones.

Now, using the equations in m we get:

m*a2 + m*b2 = m*c2

and, of course, after dividing by m we see Pythagoras rear his head:

a2 + b2 = c2

johnson.mafoko
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Posted Feb 5, 2013 - 1:52 AM:

@Flannel Jesus
facinating proof, simple

@Ezekiel
hmmm.

My skeptical mind once looked a different pythagorus theorem proofs(i hear there are more than 200 known proofs) as waste of time. Its proved, why prove it again. Mathematical beauty yes, but philosophers are not interested in mathematics for beauty sake, they do math because math says something about the world. My last thoughts on the utility of finding different proofs of the same theorem was that different proofs can facilitate cohesiveness of mathematics.

But frankly, I am not interested in "blind mathematics"(automatic theorem generation, theorem proving programs can do the same I hear) where theorem after theorems is proved without seeking "philosophical". Look at the way Plato, Leibniz or Descartes studied mathematics. For them mathematics was a means to an end. Descartes had insight of his scientic method just by going through say a proof that -1x-1=+1.

Math proofs when done blindly really are dull and are not different from a chess game, but when mixed with philosophical insight, they are a source rich of analogies(read isomorphisms) that can be used in a philosophical discourse. That is why I like Ezekiel's proof above, though it is ephemeral, but is insightful, it doesn't end in itself,it helps you see beyond just triangles and areas, it links the concept of area, line, ratios in an interesting way. There is a source of contemplation on what area, line, point etc is. In summary, the proofs must lead to interesting philosophy. My opinion.
Numan
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Posted Feb 5, 2013 - 9:54 AM:
Subject: Are triangles circles?
'
Beauty is Truth, Truth, Beauty -- that is all
Ye know on Earth, and all ye need to know.

----Keats, Ode on a Grecian Urn

I think you are right; the true joy of mathematics is when many disparate facts fll together in a new pattern -- a Thing of Beauty appears which heretofore had never been dreamed of.

In Book III, Proposition 31 of Euclid, it is first proved that an inscribed triangle, constructed upon the diameter of a circle and touching the circumference at the third point of the triangle, is a right-triangle; and as you move this third point around the circumference you generate all possible right-triangles.

aleph0.clarku.edu/~djoyce/j...nts/bookIII/propIII31.html

Next is a generalization from this and deals with triangles constructed on all possible chords of a circle -- thereby generating all possible triangles. This demonstrates that circles and triangles are identical -- there is no difference between them in their Platonic essence.
Together, they form two aspects of a single geometrical nature, which could never be guessed by any amount of visual inspection of specific triangles and circles.
.
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Posted Feb 5, 2013 - 11:15 AM:

I like that approach, but I don't see how it's connected to the golden ratio...
I mean, other than you arbitrarily making A/B equal the golden ratio, I don't think there's any connection.

Is there?
johnson.mafoko
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Posted Feb 6, 2013 - 6:09 AM:

Numan wrote:
'


Next is a generalization from this and deals with triangles constructed on all possible chords of a circle -- thereby generating all possible triangles. This demonstrates that circles and triangles are identical -- there is no difference between them in their Platonic essence.

.




Do you mean by this that for any theorem of a triangle there is a correposponding similar theorem for a cycle like isomorphic groups in abstract algebra. Interesting.

On another note, I know Descartes gave an equation that shows connection between triangle centers (centroid,incenter,orthocenter,circumcenter,etc) but my thoughts at the time was "why the fuss?" Do you care to share your views on philosophical import of such a exercises.

Mine is this: like Descartes believed, numbers are simple and distinct ideas, and therefore must be true(materially true). But "the true" or Being is related or connected which entails if we keep playing with elementary mathematical objects(finding myriad mathematical connections) we will end up with complex mathematical objects which may then mirror the complex world.

Math then should proceed through asking questions(solving problems even if they seem useless, so called "for it on sake") till it is ready enough for applications. Mathematical problems therefore are good as far as they unite mathematics in their solutions, and they help us generate more theorems. So thats why I am for general solutions in number theory. Particular solutions should lead us to general solutions which will lead to general mathematics(which leads to true forms as plato believed).

My 2cent.
johnson.mafoko
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Posted Feb 6, 2013 - 9:13 AM:

Numan wrote:
'
Beauty is Truth, Truth, Beauty -- that is all
Ye know on Earth, and all ye need to know.

----Keats, Ode on a Grecian Urn

I think you are right; the true joy of mathematics is when many disparate facts fll together in a new pattern -- a Thing of Beauty appears which heretofore had never been dreamed of.

In Book III, Proposition 31 of Euclid, it is first proved that an inscribed triangle, constructed upon the diameter of a circle and touching the circumference at the third point of the triangle, is a right-triangle; and as you move this third point around the circumference you generate all possible right-triangles.

aleph0.clarku.edu/~djoyce/j...nts/bookIII/propIII31.html

Next is a generalization from this and deals with triangles constructed on all possible chords of a circle -- thereby generating all possible triangles. This demonstrates that circles and triangles are identical -- there is no difference between them in their Platonic essence.
Together, they form two aspects of a single geometrical nature, which could never be guessed by any amount of visual inspection of specific triangles and circles.
.


That relation between a cycle and triangle is amusing considering that classic geometry had to be done using ruler and compass. On a plane, the simple shapes suscribed by the 2 instruments are a cycle and triangle. I may be accused of going into mathematical mystism(numerology etc) but I if plato was right about that the heavenly bodies moving in cycles while the earthly objects move in straight lines then your theorem can be interpreted as cycle being a Form over and above particular triangles. It is the One Form over Many possible particular triangles it circumscribes. For more information on problem of One over many see Plato's Parmenides
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Posted Feb 6, 2013 - 9:19 AM:

Numan wrote:
(...) This demonstrates that circles and triangles are identical -- there is no difference between them in their Platonic essence. (...)




Hello Numan, I don't know what you mean by "platonic essence", but your claim as a whole seems philosophically challenging. Basically, you argue that two isomorphic structures are identical. What you mean?

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