triangle inequality

bobbym · 2013-05-30 05:27:52

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

anonimnystefy · 2013-05-30 05:29:46

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)

bobbym · 2013-05-30 05:32:40

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

anonimnystefy · 2013-05-30 05:34:03

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

bobbym · 2013-05-30 05:44:59

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

anonimnystefy · 2013-05-30 05:45:41

There was no need to try them, but whatever.

bobbym · 2013-05-30 05:48:19

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

anonimnystefy · 2013-05-30 06:32:01

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.

bobbym · 2013-05-30 06:38:33

Okay but what about this?

1/hc>1/12-1/14

anonimnystefy · 2013-05-30 06:40:26

What about it?

bobbym · 2013-05-30 06:41:33

Why the minus and not a plus?

anonimnystefy · 2013-05-30 06:44:31

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

bobbym · 2013-05-30 06:46:32

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

anonimnystefy · 2013-05-30 06:47:46

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.

bobbym · 2013-05-30 06:54:38

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

anonimnystefy · 2013-05-30 06:58:27

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)

bobbym · 2013-05-30 07:02:38

Hi;

Okay, see you later.

anonimnystefy · 2013-05-30 07:17:45

Okay, see you!

ElainaVW · 2013-06-01 00:19:55

I have a different solution.

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

bobbym · 2013-06-01 00:20:44

Hi;

What is it, please post what you have.

ElainaVW · 2013-06-01 00:23:32

[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.

bobbym · 2013-06-01 00:27:38

Hi;

That is very good. Nice work.

Math Is Fun Forum

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

Re: triangle inequality

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

Re: triangle inequality

#53 2013-05-30 05:32:40

Re: triangle inequality

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

Re: triangle inequality

#55 2013-05-30 05:44:59

Re: triangle inequality

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

Re: triangle inequality

#57 2013-05-30 05:48:19

Re: triangle inequality

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

Re: triangle inequality

#59 2013-05-30 06:38:33

Re: triangle inequality

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

Re: triangle inequality

#61 2013-05-30 06:41:33

Re: triangle inequality

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

Re: triangle inequality

#63 2013-05-30 06:46:32

Re: triangle inequality

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

Re: triangle inequality

#65 2013-05-30 06:54:38

Re: triangle inequality

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

Re: triangle inequality

#67 2013-05-30 07:02:38

Re: triangle inequality

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

Re: triangle inequality

#69 2013-06-01 00:19:55

Re: triangle inequality

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

Re: triangle inequality

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

Re: triangle inequality

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

Re: triangle inequality

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