Yet another elementary program glitch caused misprinted vertex coordinates --- for [a, b, c] = [5, 29, 30] the lattice vertices should have read [0, 0], [3, 4], [21, -20] . The data list sent out subsequently was similarly garbled, and a corrected version will follow [embarrassed groan]. WFL On 11/17/11, Fred lunnon <fred.lunnon@gmail.com> wrote:
As far as I can see, splitting a Heronian triangle (by its altitude) into two Pythagorean (right-angled, with rational edge-lengths, hence areas) constructs merely a pose with rational coordinates.
For example, suppose one edge joins [0, 0] and [a, 0]; then the third vertex lies at [(a^2+b^2-c^2) / 2 a, 2 d / a], where d denotes the area. This will not be an integer unless some edge-length a happens to divide 2d .
For triangles such as [a, b, c] = [5, 29, 30], where s, d = 32, 72, there is no such edge. However, the rotated pose with vertices [0, 0], [-4, 3], [-20, 21] still lies on the lattice. A similar apparently happy accident overtakes every case with s <= 200.
Fred Lunnon