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).

Case 1.

Assume mod mod 4 mod 4.

Assume mod 3 mod 3 mod 3.

Case 2.

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