Thanks, Warren, for all those Heron triangle links. Your next to last mentioned paper, namely: Paul Yiu: Construction of indecomposable Heronian triangles especially mentions Gaussian Integers in connection with lattice point positioning of Heron triangles which cannot be decomposed into two integer sided Pythagorean triangles, which has been the main thrust of this whole thread. It also reminded me of an old problem I looked at in the early 1990's, namely find all Heron triangles whose area is a perfect square integer. It turned out that these correspond to the rational points on a certain family of elliptic curves (or elliptic surface), and the nicest way to do it was to use the tangents of the half-angles as variables, which Paul Yiu also makes use of in the paper I mentioned above. The main thing I discovered was that given any Heron triangle, say the tangents of its half-angles are t1, t2, t3, then there are three positive-rank elliptic curves over Q, one for each of t1, t2, t3 and the rational points on the t1-curve give infinitely many pairwise dissimilar "square Heron triangles" with a fixed angle theta such that tan(theta/2) = t1, etc for t2, t3. Since two new angles occur for each triangle one can play the same game with all the new triangles. What I never did figure out is whether the whole collection of "square Heron triangles" is generated in this way from some finite (or easily describable) list of start-triangles or whether there will always exist new ones not in the network of triangles arising from such a list. If anyone knows about this I would be interested. Jim Buddenhagen ------------------------------------ On Sat, Nov 19, 2011 at 8:13 AM, Warren Smith <warren.wds@gmail.com> wrote:
I googled Heronian triangles and a lot of what we've been saying was already known!! Duh!!
Paul Yiu: Heronian triangles are lattice triangles, Amer Math Monthly 108,3 (March 2001) 261-263 http://math.fau.edu/Yiu/AMM2001Heron.pdf Proves "Reid's" theorem. Indeed more strongly shows if the sides are sqrt(integer) and the area is rational, then embeddable on integer lattice. (But I think Reid's proof also shows this.) I think Yiu's proof is not as nice as Reid's and my stuff because he does not use the Gaussian Integers.
http://www.maa.org/mathdl/CMJ/methodoflastresort.pdf This 1998 paper reports how Brahmagupta found around 625AD that the consecutive-integer-sides Heronian triangles arise from Pell equation. The same result was rediscovered by Prof. Bill Richardson in 2010: http://www.math.wichita.edu/~richardson/heronian/heronian.html
http://sci-gems.math.bas.bg:8080/jspui/bitstream/10525/382/1/sjc058-vol2-num... http://www.wm.uni-bayreuth.de/fileadmin/Sascha/Publikationen/On_Heronian_Tri... Describe several algorithms for the generation of integer Heronian triangles with diameter at most n. Two of them have running time O(n^(2+epsilon)), which beats cubic time.
http://forumgeom.fau.edu/FG2007volume7/FG200718.pdf Heronian Triangles Whose Areas Are Integer Multiples of Their Perimeters
Paul Yiu: Construction of indecomposable Heronian triangles, Rocky Mountain Journal of Mathematics 28 (1998) 1189-1202. http://math.fau.edu/yiu/Southern080216.pdf
Heronian Triangles Wm. Fitch Cheney, Jr. The American Mathematical Monthly Vol. 36, No. 1 (Jan., 1929), pp. 22-28
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