Pythagorean Triangle whose Area is Half Perimeter

Theorem
The $3-4-5$ triangle is the only Pythagorean triangle whose area is half its perimeter.

Proof
Let $a, b, c$ be the lengths of the sides of a Pythagorean triangle $T$.

Thus $a, b, c$ form a Pythagorean triple

By definition of Pythagorean triple, $a, b, c$ are in the form:
 * $2 m n, m^2 - n^2, m^2 + n^2$

We have that $m^2 + n^2$ is always the hypotenuse.

Thus the area of $T$ is given by:
 * $\mathcal A = m n \left({m^2 - n^2}\right)$

The perimeter of $T$ is given by:
 * $\mathcal P = m^2 - n^2 + 2 m n + m^2 + n^2 = 2 m^2 + 2 m n$

We need to find all $m$ and $n$ such that $\mathcal P = 2 \mathcal A$.

Thus:

As $m$ and $n$ are both (strictly) positive integers, it follows immediately that:
 * $n = 1$
 * $m - n = 1$

and so:
 * $m = 2, n = 1$

and the result follows.