Area of Integer Heronian Triangle is Multiple of 6

Theorem
Let $\triangle {ABC}$ be an integer Heronian triangle.

Then the area of $\triangle {ABC}$ is a multiple of $6$.

Proof
Heron's Formula gives us that:


 * $\AA = \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} }$

where:
 * $\AA$ denotes the area of the triangle
 * $a$, $b$ and $c$ denote the lengths of the sides of the triangle
 * $s = \dfrac {a + b + c} 2$ denotes the semiperimeter of the triangle.

We set out to eliminate $s$ and simplify as best possible:

This is now in the form $p^2 + q^2 = r^2$.

From Solutions of Pythagorean Equation, $\tuple {p, q, r}$ has the parametric solution:
 * $\tuple {m^2 - n^2, 2 m n, m^2 + n^2}$