I hope you can do a little modular arithmatic. You're idea is indeed the right one:
Let be such that
The area of the triangle is given by:
Observe that: mod 4. And mod 3. (you can easily check this by looking at quadratic residu-classes modulo 4, and modulo 3).
Assume mod mod 4 mod 4.
Assume mod 3 mod 3 mod 3.
Assume mod 4. This implies mod 8, or mod 8
Now we'll look at quadratic residues modulo 8 wich are
Since there is no quadratic residu that is 5 mod 8. It follows that mod 8 mod 8, mod 8. Hence mod 8.
Assume mod 3 mod 3, mod 3 mod 3
So in both cases we have and