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 | | From: | Virgil | | Subject: | Equilateral Triangle Redux. | | Date: | Sat, 22 Jan 2005 21:29:21 -0700 |
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 | A while back someone posted the following:
Given an equilateral triangle and a fourth point not coincident with
any vertex of the triangle to find the smallest possible triangle with
integer sides so that each of the distances from a vertex to the fourth
point is also integral.
In trying to analyse the problem generalized to find integer solutions
not necessarily for the smallest triangle, I ran into a different
problem towards which I am not making much headway.
The problem is to find necessary and sufficient conditions on integers
a and b, with a < b and coprime, so that a^2 + a b + b^2 is the square
of an integer.
If anyone can point me towards a solution, I should be grateful.
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| | From: | N. Silver | | Subject: | Re: Equilateral Triangle Redux. | | Date: | Sun, 23 Jan 2005 06:17:20 GMT |
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| Virgil wrote:
> The problem is to find necessary and sufficient conditions on integers
> a and b, with a < b and coprime, so that a^2 + a b + b^2 is the square
> of an integer.
> If anyone can point me towards a solution, I should be grateful.
This problem is in Dickson's History of the Theory of Numbers.
In the College Math Journal in the middle 80's, it was analyzed
as part of getting integer solutions for a calculus problem: Take
a rectangular piece of cardboard and fold it into the bottom of
a box. Find the max volume of such a box. Another version of
the same problem was published in the Mathematics Magazine
around the same time.
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| | From: | JEMebius | | Subject: | Re: Equilateral Triangle Redux.- | | Date: | Sun, 23 Jan 2005 15:32:37 +0100 |
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| Please take a look at http://www.geocities.com/fredlb37/triples8.pdf -
mentioned in a previous posting in news:sci.math (see below) - this will
solve your problem at least in part.
Johan E. Mebius
Virgil wrote:
>A while back someone posted the following:
>
> Given an equilateral triangle and a fourth point not coincident with
>any vertex of the triangle to find the smallest possible triangle with
>integer sides so that each of the distances from a vertex to the fourth
>point is also integral.
>
>In trying to analyse the problem generalized to find integer solutions
>not necessarily for the smallest triangle, I ran into a different
>problem towards which I am not making much headway.
>
>The problem is to find necessary and sufficient conditions on integers
>a and b, with a < b and coprime, so that a^2 + a b + b^2 is the square
>of an integer.
>
>If anyone can point me towards a solution, I should be grateful.
>
>
(Raw copy of newsgroup post:)
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From: "heck"
Newsgroups: sci.math
Subject: A 120 degree triangle preserving matrix
Date: 15 Jan 2005 06:43:52 -0800
Organization: http://groups.google.com
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Xref: news.xs4all.nl sci.math:732660
Perhaps someone will find this interesting.
Let M be the matrix
4 3 6 .
3 4 6
4 4 7
If a^2+b^2+ab = c^2 then (a b c)M = (u v w) where
u^2+v^2+uv = w^2, and u-v = a-b.
http://www.geocities.com/fredlb37/triples8.pdf
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