## 15211Re: The area of prime sided triangles

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• Aug 8, 2004
There is no triangle with rational area and all odd sides.

Proof: Let the three sides be a=2k+1,b=2m+1,c=2n+1. Use Heron's
formula area=sqrt[s*(s-a)*(s-b)*(s-c)]=sqrt[(a+b+c)(-a+b+c)(a-b+c)
(a+b-c)/16] where s=(a+b+c)/2 is the semiperimeter. The numerator is
a mess, but it is also 3 mod 8 (once you distribute all the k's to
l's to n's, etc.) and so not a perfect square. #

<hillcino368@h...> wrote:
> Hi,
> I was playing with prime sided triangles and area and deduced the
following.
>
> Theorem 1: The area of a prime sided triangle is irrational.
>
> The area of a triangle of sides a < b < c is easily derived as
>
> A = 1/4*sqrt((-a+b+c)*(a-c+b)*(a+c-b)*(a+c+b))
>
> for all real a,b, c <=a+b
>
> Is it sufficient to prove Theorem 1 by proving if a+b+c is square
then a+c-b
> cannot be square
> and if a+c-b is square then a+c+b cannot be square?
>
> If so, maybe someone can fill in the details.
>
> I would guess
>
> Theorem 2: The area of an odd sided triangle is irrational.
>
> Theorem 3: The area of a square sided triangle is irrational
>
> Theorem 4: The area of a non-pythagorean sided triangle is
irrational
>
> Fractions are allowed, ie., (3/2)^2 + (4/2)^2 = (5/2)^2
>
> are also true.
>
> CLH
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