I found Fred's cool paper on arxiv. Perhaps some of my ideas can
make a 2D version of the embedding theorem a little more elegant.
BTW,
my determinant III of complex numbers a,b,c:
|0 a b c |
|a 0 c' b'|
|b c' 0 a'|
|c b' a' 0 |
when c=-a-b can be simplified with row & column operations to
|a a' 0 c'|
|b b' 0 0 |
|0 0 a b |
|0 0 a' b'|
thus making obvious the factorization (ab'-a'b)^2.
(We note again that ab'-a'b is pure imaginary, thus
the square is negative real.)
At 03:52 AM 6/30/2012, Fred lunnon wrote:
>On 6/30/12, Henry Baker <hbaker1(a)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
