First notice that if you have a triplet $(a,b,c)$, then
If the triplet is ordered increasingly, then, you may show that
With the new information, you get the triplet $(a,b,a+b)$. Therefore
which means that we need to have
Having $a<b$ and $b|2a$, show that $b=2a$.
The second part is to show that you cannot have more members in the set, for if you add a new integer $d$, to be the greatest in the set, $(a,b,d)$ will not make a triplet.