  Let $W$ be a $2$-dimensional Coxeter group, that is, a one with
$\frac{1}{m_{st}}+\frac{1}{m_{sr}}+\frac{1}{m_{tr}}\leq 1$ for all triples of
distinct $s,t,r\in S$. We prove that $W$ is biautomatic. We do it by showing
that a natural geodesic language is regular (for arbitrary $W$), and satisfies
the fellow traveller property. As a consequence, by the work of Jacek
\'{S}wi\k{a}tkowski, groups acting properly and cocompactly on buildings of
type $W$ are also biautomatic. We also show that the fellow traveller property
for the natural language fails for $W=\widetilde{A}_3$.
