Newsgroups: sci.math
From: pmontgom@cwi.nl (Peter L. Montgomery)
Subject: Re: Heron's Formula
Date: Thu, 16 Mar 1995 05:30:27 GMT

In article <3k8572$2cv@uwm.edu> radcliff@alpha2.csd.uwm.edu (David G Radcliffe) writes:
>Heron's Formula is as follows.  Let a,b,c be the sides of a triangle, 
>and let A be the area of the triangle.  Then A^2 = s(s-a)(s-b)(s-c), 
>where s = (a+b+c)/2.
>
>I can derive this formula using analytic geometry and some tricky
>factorization, but there should be an easier way.  

    Let C be the angle separating the sides of lengths a and b.  Then

         2 ab cos C = a^2 + b^2 - c^2     (law of cosines)
                  A = (ab/2) sin C        (draw altitude on side of length b)
              16A^2 = (2ab sin C)^2 = (2ab)^2 - (2ab cos C)^2
                    = (2ab)^2 - (a^2 + b^2 - c^2)^2
                    = (2ab + a^2 + b^2 - c^2) (2ab - a^2 - b^2 + c^2) 
                    = (a + b + c)(a + b - c)(c + a - b)(c - a + b) 
-- 
        Peter L. Montgomery    pmontgom@cwi.nl    San Rafael, California
Mathematically gifted, unemployed, U.S. citizen.  Interested in computer
architecture, program optimization, computer arithmetic, cryptography, 
compilers, computational mathematics.  17 years industrial experience.