On 6/30/12, Henry Baker <hbaker1@pipeline.com> wrote:
... Is there a name for triangles whose sides are integers, and _which can also be embedded in the plane with integer coordinates for the vertices_ ?? ]
"Heronian". If the vertices are rational, then the area is rational via the standard determinant giving area in terms of Cartesian coordinates [which ought to have a name, though I know of none]. If the sides are integers and the area rational, then the triangle is Heronian by definition [and furthermore its area 6x integer]. Conversely, if a triangle is Heronian then it may be embedded with integer vertices via Yiu's theorem --- see the remarkable complex GCD proof by Michael Reid which was discussed in math-fun last year. An analogous argument applies in 3 dimensions. Fred Lunnon