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