  The $\widetilde{A}_n$ Coxeter groups are known to not be systolic or
cocompactly cubulated for $n\geq 3$. We prove that these groups act
geometrically on weakly modular graphs, a weak notion of nonpositive curvature
generalizing the 1-skeleta of $\mathrm{CAT}(0)$ cube complexes and systolic
complexes. To prove weak modularity we describe the canonical emeddings of the
1-skeleta of $\widetilde{A}_n$ Coxeter complexes into the Euclidean spaces
$\mathbb{R}^{n+1}$. We also prove weak modularity for buildings of type
$\widetilde{A}_3$.
