An integral triangle is defined as a triangle whose sides are measurable in whole numbers. Find all integral triangles whose perimeter equals their area.
At first, I thought this would be one of the easier contest math problems but I have only been able to work out a couple of things so far. The area of a triangle and the perimeter can be related by the same variables only using herons formula.
So assuming p to be the semiperimeter we get $\sqrt{p(p-a)(p-b)(p-c)}$ = $2p$.
Now Assuming $p-a$ = $x$, $p-b$ = $y$, $p-c$ = $z$ we get,
$\sqrt{(x+y+z)(xyz)}$= $2(x+y+z)$.
After squaring and simplifying we get, $xyz$ = $4(x+y+z)$.
After this, however, I'm not sure how to go about solving this equation.
Any help would be much appreciated.