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#26 2012-09-30 08:16:47

bobbym
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Re: Triangle Problem

Supposing a was large and b,c were small? I do not know if that step is rigorous enough or requires more.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#27 2012-09-30 08:45:40

anonimnystefy
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Re: Triangle Problem

You are forgeting one thing which might be crucial. a, b and c are sides of a triangle. There is a great chance that has some other purpose than just stating that a, b and c are positive.


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#28 2012-09-30 08:56:54

bobbym
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Re: Triangle Problem

One property is

a + b > c
a + c > b
b + c > a


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#29 2012-09-30 09:00:59

anonimnystefy
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Re: Triangle Problem

Exactly what I had in mind. I will try to do the problem.


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#30 2012-09-30 19:35:12

bob bundy
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Re: Triangle Problem

hi



But a, b, c all > 0



Similarly





Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

#31 2012-09-30 19:49:38

zetafunc.
Guest

Re: Triangle Problem

bob bundy wrote:

hi



But a, b, c all > 0



Similarly





Bob

Thanks for this, I didn't think about setting c(b+a) > 0... so, would all my reasoning be mathematically sound? I agree with what you have written above -- I'm wondering if a geometric solution is also possible however, since it appears that this solution doesn't take advantage of any triangle properties...

#32 2012-09-30 19:59:27

bob bundy
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Re: Triangle Problem

hi zetafunc

Your post with (a+1)(b+1)(c+1) = 4 + (a-1)(b-1)(c-1) is the way to go with this.

But  you just needed to justify a,b,c all < 1

I experimented using Sketchpad with a number of values for a b and c and found

it doesn't hold if a b c are not the sides of a triangle and the expression = 4 when any of a b or c = 1. (and is > 4 if over 1)

So the two constraints (triangle and ab + bc + ca = 1) are necessary.

Therefore you have to use a property of triangles.

My contribution uses b < a + c

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

#33 2012-09-30 20:05:10

anonimnystefy
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Re: Triangle Problem

bob bundy wrote:

How'd you get 2b<b(a+c)? It would imply that a+c>2, so one of them has to be greater than 1...


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#34 2012-09-30 20:24:08

bob bundy
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Re: Triangle Problem

Arhh!  Once more you have spotted my error.  Curses. (not aimed at you of course!)

My brain did this.

triangle property

b < a + c   and b(a+c) < 2 .... => 2b < 2.

But it was wishful thinking.

I should have written b^2 < 2 which is not any help.  Sorry zetafunc.  Back to the drawing board.

Bob

ps. Nevertheless, some triangle property seems essential here.


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

#35 2012-09-30 20:48:56

anonimnystefy
Real Member

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Re: Triangle Problem

It's okay. For I second there I thought we finally had proof!


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#36 2012-10-01 04:27:20

bobbym
Administrator

Online

Re: Triangle Problem

Hi zetafunc.;

zetafunc wrote:

a(b + c) + bc = 1
b(a + c) + ca = 1
c(a + b) + ab = 1

You were on the right track when you posted that.

You need to prove that a,b,c <1.

Let's assume WLOG that a>1 then



By the triangle inequality



If a>1 then (b+c) > 1 and a(b+c) >1 but bc cannot be less than or equal to 0 ( see equation 2 ) so we have a contradiction. Therefore a,b,c<1

Now put your proof all together and present it.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#37 2012-10-02 01:29:59

zetafunc.
Guest

Re: Triangle Problem

I see now. Thank you.

I suppose the proof would look like this:















#38 2012-10-02 04:06:20

bobbym
Administrator

Online

Re: Triangle Problem

Hi;

Yes, that is what I would do. If it is wrong then at least you have company.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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