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#51 2013-05-30 05:27:52

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,257

Re: triangle inequality

Then I used a couple of more relationships and the triangle inequality to test the integers from 7 to 84. 83 was the biggest.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#52 2013-05-30 05:29:46

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,521

Re: triangle inequality

Well, the 83 comes directly from the upper bound.

anonimnystefy wrote:

And, what exactly are we doing now. We have the value, we have constructed the triangle. What else is there to do?

Last edited by anonimnystefy (2013-05-30 05:32:37)


“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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#53 2013-05-30 05:32:40

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,257

Re: triangle inequality

The upper bound I showed is 84. 83 works but 84 does not.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#54 2013-05-30 05:34:03

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,521

Re: triangle inequality

Well, of course it doesn't work. The triangle inequality states a+b>c, not >=c.


“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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#55 2013-05-30 05:44:59

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,257

Re: triangle inequality

It took a little M code to try from 7 to 84.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#56 2013-05-30 05:45:41

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,521

Re: triangle inequality

There was no need to try them, but whatever.


“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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#57 2013-05-30 05:48:19

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,257

Re: triangle inequality

With what I dug up, I had to try them all. What did you have to shorten that?


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#58 2013-05-30 06:32:01

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,521

Re: triangle inequality

Well, I had the fact that the lengths 1/12, 1/14 and 1/hc must form a triangle, where 1/hc is as small as possible. Because of the triangle inequality we have that 1/hc>1/12-1/14=1/84. So, hc<84. The maximum possible length is 83.


“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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#59 2013-05-30 06:38:33

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,257

Re: triangle inequality

Okay but what about this?

1/hc>1/12-1/14


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#60 2013-05-30 06:40:26

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,521

Re: triangle inequality

What about it?


“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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#61 2013-05-30 06:41:33

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,257

Re: triangle inequality

Why the minus and not a plus?


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#62 2013-05-30 06:44:31

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,521

Re: triangle inequality

The triangle inequality states that 1/hc+1/14>1/12.


“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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#63 2013-05-30 06:46:32

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,257

Re: triangle inequality

What triangle? That inequality is for the sides. You have altitudes there.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#64 2013-05-30 06:47:46

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,521

Re: triangle inequality

Yes, but, as I already said, 1/ha=1/12, 1/hb=1/14 and 1/hc must be length of some triangle in order to be a valid set of altitudes.


“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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#65 2013-05-30 06:54:38

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,257

Re: triangle inequality

I do not get it but it works so the problem is done.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#66 2013-05-30 06:58:27

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,521

Re: triangle inequality

Well, it is because a*ha=b*hb=c*hc. From this we can get a:b:c=(1/ha):(1/hb):(1/hc), which means that, if a, b and c can form a triangle, 1/ha, 1/hb and 1/hc must be able to form a triangle as well.

Last edited by anonimnystefy (2013-05-30 06:58:35)


“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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#67 2013-05-30 07:02:38

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,257

Re: triangle inequality

Hi;

Okay, see you later.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#68 2013-05-30 07:17:45

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,521

Re: triangle inequality

Okay, see you! smile


“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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#69 2013-06-01 00:19:55

ElainaVW
Member
Registered: 2013-04-29
Posts: 291

Re: triangle inequality

I have a different solution.

Last edited by ElainaVW (2013-06-01 00:28:27)

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#70 2013-06-01 00:20:44

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,257

Re: triangle inequality

Hi;

What is it, please post what you have.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#71 2013-06-01 00:23:32

ElainaVW
Member
Registered: 2013-04-29
Posts: 291

Re: triangle inequality

[Code fixed by admin]

Solve[{12 ==1/a Sqrt[2] \[Sqrt]((a + b + c) (1/2 (a + b + c) - a) (1/2 (a + b + c) - b) (1/2 (a + b + c) - c)), 14 == 1/b Sqrt[2] \[Sqrt]((a + b + c) (1/2 (a + b + c) - a) (1/2 (a + b + c) - b) (1/2 (a + b + c) - c)), 
   83 == 1/c Sqrt[2] \[Sqrt]((a + b + c) (1/2 (a + b + c) - a) (1/2 (a + b + c) - b) (1/2 (a + b + c) - c)), 12 == (b*c)/(2 R)}, {a, b, c, R}] // N

Only had to try 84 and 83.

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#72 2013-06-01 00:27:38

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,257

Re: triangle inequality

Hi;

That is very good. Nice work.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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