15211Re: The area of prime sided triangles
- Aug 8, 2004There 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
(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. #
--- In firstname.lastname@example.org, "cino hilliard"
> I was playing with prime sided triangles and area and deduced the
> 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
> cannot be squareirrational
> 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
> Fractions are allowed, ie., (3/2)^2 + (4/2)^2 = (5/2)^2
> are also true.
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