@warren: that is pretty cool if you have n,m,k, but is there a way to make it constructive if you are just given the sides a,b,c? For example, suppose [a,b,c] = [471, 518, 611] which are the sides of a primitive Heron triangle. How can I find appropriate n,m,k and then G, so as to actually compute the vertices A,B,C of this triangle? On Fri, Nov 18, 2011 at 10:03 AM, Warren Smith <warren.wds@gmail.com> wrote:
I also point out that (assuming Reid proof valid) by combining my Buchholz-->nonprimitive coordinates formula, and the Gaussian GCD, we can actually write just a single formula in terms of n,m,k, rational operations, and Gaussian GCD, giving coordinates for every PRIMITIVE Heronian triangle's vertices A,B,C. That is quite cool.
Here are the nonprimitive coordinate formulas: xC = m*(n-k)*(n+k); yC = 2*k*m*n; xA = n*(m-k)*(m+k) + xC; yA = 0; xB = 0; yB = 0; and now the primitivization as Gaussian integers is 0, xA/G, and (xC+yC*i)/G where G=GaussianGCD(xA, xC+yC*i).
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