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CTK Exchange
Bui Quang Tuan
Member since Jun-23-07
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Aug-16-10, 02:39 PM (EST) |
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1. "RE: 3 trapezoids proof of the Pythagorean Theorem"
In response to message #0
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Dear John Molokach, I think it is not neccessary to cut or rearrange to "3 trapezoids" configuration. Immediately from your first drawing we have:
(2b + a)*(2a + b) = a^2 + b^2 + c^2 + 4*a*b/2 + 3*a*b and from it: a^2 + b^2 = c^2 Best regards,
Bui Quang Tuan
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John Molokach
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Aug-16-10, 11:20 PM (EST) |
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2. "RE: 3 trapezoids proof of the Pythagorean Theorem"
In response to message #1
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Your algebra suggests 4 triangles and 3 rectangles which I suppose I could have drawn in the original figure... John... |
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John Molokach
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Aug-17-10, 01:24 PM (EST) |
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4. "RE: 3 trapezoids proof of the Pythagorean Theorem"
In response to message #3
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Would the picture alone suffice as proof or would I need the algebra to accompany it? I was thinking that the second image could show geometrically the terms 2a^2, 5ab, and 2b^2... |
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Bui Quang Tuan
Member since Jun-23-07
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Aug-18-10, 02:06 AM (EST) |
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13. "RE: 3 trapezoids proof of the Pythagorean Theorem"
In response to message #9
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>>>Your algebra suggests 4 triangles and 3 rectangles which I
>>>suppose I could have drawn in the original figure...
>>>
>>>John...
>>
>>Yes, of course! They are very easy to draw and I think it is
>>interesting original proof of Pythagorean theorem!
>>Best regards,
>>Bui Quang Tuan
>
>Thank you so much. I am glad you think the proof is
>interesting and original. I wonder if "original" means it
>varies significantly from proof 4 and 9. Dear John,
For me privately, it is original proof of you! It is interesting because it proves one algebra:
(2*a + b)*(2*b + a) = a^2 + b^2 + (a + b)^2 + 3*a*b
and I like this configuration symmetric with A, B and special with C. I think from this we can "see" some more interesting, not only Pythagorean theorem. Please note that inventions often come to us when we finding another things.
If you see my attached image then color filling part is Pythagorean proof 4 in CTK.
Some times, when fall down, we get the most successfull results!
Best regards,
Bui Quang Tuan |
Attachments
https://www.cut-the-knot.org/htdocs/dcforum/User_files/4c6c805a69c90ebd.jpg
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jmolokach
Member since Aug-17-10
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Aug-23-10, 03:01 PM (EST) |
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17. "RE: 3 trapezoids proof of the Pythagorean Theorem"
In response to message #0
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I was reading the comment you wrote at the end of proof #39: "Instead of using either of the identities directly, Loomis adds the two:
2(a² + b²) = 2c², which appears as both graphical and algebraic overkill." I wonder if the trapezoid variant I have is much the same idea? Graphical and algebraic overkill, or do you think it is useful for students, etc... to "see" the original a^2, b^2, and c^2 in the diagram? Success is getting up one more
time than you fall down... |
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alexb
Charter Member
2591 posts |
Sep-03-10, 06:03 AM (EST) |
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29. "RE: 3 trapezoids proof of the Pythagorean Theorem"
In response to message #28
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They judge a quality of a proof by elegance and amount of new learning that comes along. Elegance in mathematics is often associated with the minimalistic principle known as Occam's razor. Proofs 3, 4, 5, 6, 9, 24, 51, 64, 69, 81 all can be treated as having embedded squares and triangles and offering at least two ways of evaluating the same quantity (area), which leads to the Pythagorean identity. MAA editors probably judge that your proof, while doing its job, adds much in the way of novelty to the better known ones. |
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jmolokach
Member since Aug-17-10
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Sep-03-10, 11:18 AM (EST) |
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30. "RE: 3 trapezoids proof of the Pythagorean Theorem"
In response to message #29
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>They judge a quality of a proof by elegance and amount of
>new learning that comes along. Elegance in mathematics is
>often associated with the minimalistic principle known as
>Occam's razor. Proofs 3, 4, 5, 6, 9, 24, 51, 64, 69, 81 all
>can be treated as having embedded squares and triangles and
>offering at least two ways of evaluating the same quantity
>(area), which leads to the Pythagorean identity. MAA editors
>probably judge that your proof, while doing its job, adds
>much in the way of novelty to the better known ones. rightfully so... I just received the following response from MAA: "Dear Mr. Molokach, I agree with Bogomolny that there is some virtue in your proof. But the Monthly receives so many submissions that we cannot publish every submission that has some virtue; we must choose those that we think will be most interesting to our readers. I continue to believe that because of the similarity to known proofs, your proofs will not be of sufficient interest to our readers to merit publication in the Monthly." Success is getting up one more
time than you fall down... |
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