  Let $\mathbb{K}$ be an algebraically closed field. In this paper, we consider
the class of smooth plane curves of degree $n+1>3$ over $\mathbb{K}$,
containing three points, $P_1,P_2,$ and $P_3$, such that $nP_1+P_2$,
$nP_2+P_3$, and $nP_3+P_1$ are divisors cut out by three distinct lines. For
such curves, we determine the dimension of certain special divisors supported
on $\{P_1,P_2,P_3\}$, as well as an explicit description of all pure gaps at
any subset of $\{P_1,P_2,P_3\}$. When $\mathbb{K}=\overline{\mathbb{F}}_q$,
this class of curves, which includes the Hermitian curve, is used to construct
algebraic geometry codes having minimum distance better than the Goppa bound.
