Loading ...
Sorry, an error occurred while loading the content.

15211Re: The area of prime sided triangles

Expand Messages
  • Adam
    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. #

      Adam



      --- In primenumbers@yahoogroups.com, "cino hilliard"
      <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
    • Show all 4 messages in this topic