jmc

Let the first triangle have side lengths $a$, $a$, $14c$, and the second triangle have side lengths $b$, $b$, $16c$, where $a,b,2c\in \mathbb{Z}$. Equal perimeter: Equal Area: Since $a$ and $b$ are integer, the minimum occurs when $a=233$, $b=218$, and $c=15$. Hence, the perimeter is $2a+14c=2(233)+14(15)=\boxed{676}$.