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#101 2012-12-11 04:28:49

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 84,454

Re: A few questions

Hi;

Not necessary Agnishom found the link below.


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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#102 2012-12-11 04:31:59

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
Posts: 15,856
Website

Re: A few questions

Hello Bobbym,
Would you mind if I go to bed now?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Humanity is still kept intact. It remains within.' -Alokananda

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#103 2012-12-11 04:35:30

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 84,454

Re: A few questions

Hi;

Of course you can get some sleep see you later.


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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#104 2012-12-11 04:35:32

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 15,856
Website

Re: A few questions

Well, are you seeing this link?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Humanity is still kept intact. It remains within.' -Alokananda

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#105 2012-12-11 04:39:32

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 84,454

Re: A few questions

Hi;

Oh you found it, thanks. I could not remember where I copied it from. I cleaned it up a little in my notes.


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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#106 2012-12-11 08:53:38

noelevans
Member
Registered: 2012-07-20
Posts: 236

Re: A few questions

Hi Agnishom!

There are other triangles other than right triangles that work.  I was just pointing out that there are
only two that are right triangles.  The proof of that is probably much easier than the proof for triangles
in general.  It will be interesting to see the general proof. smile

P.S.  I tried your link and found the proof.  Thanks!

Last edited by noelevans (2012-12-11 09:26:34)


Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).
LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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#107 2012-12-11 15:07:26

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
Posts: 15,856
Website

Re: A few questions

Hi noelevans!
Will you explain the proof?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Humanity is still kept intact. It remains within.' -Alokananda

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#108 2012-12-12 19:28:10

noelevans
Member
Registered: 2012-07-20
Posts: 236

Re: A few questions

Hi!  This is Dan's proof of 4 years ago from your link.

Let a,b,c denote the sides of triangle ∆ABC and P and A its perimeter and area respectively.
Note that Heron’s formula states that the area of ∆ABC is: A = √[s(s-a)(s-b)(s-c)]
Where s denotes the semi-perimeter, s = (a+b+c)/2.
To find all such triangles such that P=A we must have
√(a+b+c)(b+c-a)(a+c-b)(a+b-c) = 4(a+b+c).
Put x=b+c-a, y=a+c-b and z=a+b-c, where x,y,z are positive integers (true by the triangle
inequality). We see that Heron reduces to the Diophantine equation 16(x+y+z)=xyz.

We see from above that x,y,z must all be even integers. So, we may put x=2m, y=2n, z=2k to get,

  mnk=4(m+n+k)
  ⇒ mnk-4m = 4(n+k)
  ⇒ m = 4(n+k)/(nk-4)

Without loss of generality assume m≥n≥k, then we have 2m≥2n≥n+k ⇒ nk≤12.
Since 4<nk≤12 we may test integral values of n and k to find all such triangles.
Finally, we see that there is only 5 such triangles:
(a,b,c) = (5,12,13), (6,8,10), (6,25,29), (7,15,20), (9,10,17)
..................................................................................................................................

I followed pretty easily to the  three little lines starting with mnk=4(m+n+k) but had a little
difficulty figuring out why 4<nk≤12.  So I went back to the line mnk=4(m+n+k) and got it
from there.  First rewrite mnk=4(m+n+k) as  nk = 4(m+n+k)/m. 


Since m≥n≥k then 3m = (m+m+m) ≥ (m+n+k).  Therefore
                              4(3m)          4(m+m+m)          4(m+n+k)
            12 = 3*4 = _____    =     __________    ≥   ________   =  nk  because m≥n≥k.
                             
                                 m                    m                        m
implies  (m+m+m) ≥ (m+n+k).

Also rewriting mnk=4(m+n+k) as  4 = nk*[m/(m+n+k)]  we see that nk is being multiplied
by the quantity [m/(m+n+k)] which is less than one.  Hence nk>4.

So now we have the 4<nk≤12.

Testing all n≥k being bigger than four but less than or equal to 12 gives us the combinations
(n,k) ∈ {(12,1),(6,2),(4,3),(3,2),(4,2),(5,2),(3,3),(6,1),(8,1),(5,1),(7,1), (9,1),(10,1),(11,1)}
to test (14 of them).

Testing means substitute the n and k into mnk=4(m+n+k) and solve for m.  If the result is a
positive integer then double the m, n, and k to get the x, y, and z.  Then put these values
into the equations x=-a+b+c
                           y= a-b+c
                           z= a+b-c    and solve this system for a, b and c (the sides of the triangles).

Example:  n=4 and k=3 gives  m*4*3=4(m+4+3); that is, 12m=4m+28  so  m=28/8 = 7/2
                so this case doesn't give an integral value of m.  Case fails.

Example:  n=6 and k=1 gives  m*6*1=4(m+6+1); that is, 6m=4m+28  so  m=14
                This case gives an integral value for m so x=2m=28, y=2n=12 and z=2k=2.

Substituting these in to the three equations involving a, b, and c we obtain

                           28 = -a + b + c
                           12 =  a -  b  + c
                            2  =  a + b - c

Solving this system by addition/subtraction we obtain a=7, b=15, c=20  which is one of the
triangles that works.  (Adding each pair of equations eliminates two variables leaving the
third which is easy to solve.)

A nice solution to the problem, but Dan leaves out a few steps that one must scratch the
head about to fill in the gaps. up  wave  gotta sleep


Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).
LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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#109 2012-12-15 01:01:23

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 15,856
Website

Re: A few questions

Sorry for my late reply.
In the first place, how do you know that if 16(x+y+z) = xyz then x, y and z are even?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Humanity is still kept intact. It remains within.' -Alokananda

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#110 2012-12-15 01:13:07

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 84,454

Re: A few questions

Hi Agnishom;

The statement 16(x+y+z)=xyz is not generally true. Just look at x = 6, y = 7 and z =8.

He says above, there might something else that implies that they are all even.


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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#111 2012-12-15 01:44:43

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 15,856
Website

Re: A few questions

Should I post my next question?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Humanity is still kept intact. It remains within.' -Alokananda

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#112 2012-12-15 01:50:38

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 84,454

Re: A few questions

Hi Agnishom;

You can but since Dan did not explain you have to consider the possibility that his proof is flawed. I am not saying that it is, just that it is possible.


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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#113 2012-12-15 03:32:38

noelevans
Member
Registered: 2012-07-20
Posts: 236

Re: A few questions

Hi!

I believe you all are right.  The x, y an z do not have to be even, only the product xyz.  So only
one of the three x, y, z has to be even; that is, one or more of x, y, z are even.  But I think his
proof essentially holds if his evenness assumption is ignored.

Starting with the 16(x+y+z)=xyz and reasoning in a similar fashion as before for the
4<nk≤12 we can arrive at 16<yz≤48 (assuming z≤y≤x without loss of generality).

Then for y=6 and z=4, 16<yz≤48  and substituting these into 16(x+y+z)=xyz we obtain x=20.
Putting these values for x, y and z into the system of equations  involving a, b, and c
we obtain a=5, b=12 and c=13 which is one of the triangles that works.

I'll leave it to you to check out the other possibilities for yz to see if the other 4 triangles are
obtained. smile

Edit:  P. S.  Actually all three of x, y and z must be odd or all three must be even to make a, b
and c come out to be integers when solving for a, b and c.

If Dan had just left out the sentence: "We see from above that x,y,z must all be even integers."
then he could have still made the substitutions x=2m, y=2n and z=2k but readers might wonder
why he did so.  It appears that all that does is change the limits from '16 to 48' to '4 to 12' which
makes for different pairs to try for solving for x and m, respectively.

Last edited by noelevans (2012-12-15 06:29:47)


Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).
LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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#114 2012-12-15 17:07:05

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 15,856
Website

Re: A few questions

Do you know Dan?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Humanity is still kept intact. It remains within.' -Alokananda

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#115 2012-12-15 17:44:48

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 84,454

Re: A few questions

Hi Agnishom;

I do not know him.


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

#116 2012-12-15 18:17:44

noelevans
Member
Registered: 2012-07-20
Posts: 236

Re: A few questions

I don't know him either. smile


Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).
LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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#117 2012-12-15 23:39:16

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 15,856
Website

Re: A few questions

Anyone saw this: http://jwilson.coe.uga.edu/emt668/EMAT6 … ngles.html?

BTW, Someone please explain me the x+y, y+z, x+z theorem about the sides of triangles?

Last edited by Agnishom (2012-12-15 23:43:13)


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Humanity is still kept intact. It remains within.' -Alokananda

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#118 2012-12-16 00:05:09

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 84,454

Re: A few questions

Hi Agnishom;

Thanks for the link! I did not know that one.


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

#119 2012-12-16 00:11:28

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 15,856
Website

Re: A few questions

I think I understand this proof


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Humanity is still kept intact. It remains within.' -Alokananda

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#120 2012-12-16 11:00:08

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 84,454

Re: A few questions

Hi Agnishom;

I did a lot of searching for a solution to that. How did you find 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.

Offline

#121 2012-12-17 00:46:49

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 15,856
Website

Re: A few questions

Just googling!
Hope it is correct
Please explain me this thing: Why do they assume the sides are in the form of (x+y), (y+z) and (x+z)?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Humanity is still kept intact. It remains within.' -Alokananda

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#122 2012-12-17 00:59:50

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 84,454

Re: A few questions

Hi Agnishom;

Looks like a clever trick. But the sides can be labelled anything as long as the triangle inequality holds. And it does hold,


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

#123 2012-12-17 01:03:09

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 15,856
Website

Re: A few questions

That means every triangle can be put into that form.
How can you say?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Humanity is still kept intact. It remains within.' -Alokananda

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#124 2012-12-17 01:07:07

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 84,454

Re: A few questions

Hi;

Yes, as long as x, y and z are greater than 0.


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

#125 2012-12-17 01:27:23

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 15,856
Website

Re: A few questions

Question 9:  A triangle has sides of length at most 2, 3 & 4. What is the maximum area the triangle can have?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Humanity is still kept intact. It remains within.' -Alokananda

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