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#26 2012-09-29 10:16:47

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,394

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.

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#27 2012-09-29 10:45:40

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,809

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.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#28 2012-09-29 10:56:54

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,394

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.

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#29 2012-09-29 11:00:59

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,809

Re: Triangle Problem

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


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#30 2012-09-29 21:35:12

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,092

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

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#31 2012-09-29 21:49:38

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 2012-09-29 21:59:27

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,092

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

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#33 2012-09-29 22:05:10

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,809

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...


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#34 2012-09-29 22:24:08

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,092

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

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#35 2012-09-29 22:48:56

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,809

Re: Triangle Problem

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


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#36 2012-09-30 06:27:20

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,394

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.

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#37 2012-10-01 03:29:59

zetafunc.
Guest

Re: Triangle Problem

I see now. Thank you.

I suppose the proof would look like this:

#38 2012-10-01 06:06:20

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,394

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.

Offline

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