Let $Γ$ denote a distance-regular graph with diameter $D \geq 2$. Let $E$ denote a primitive idempotent of $Γ$ with respect to which $Γ$ is $Q$-polynomial. Assume that there exists a $3$-clique $\{x,y,z\}$ such that $E\hat{x},E\hat{y},E\hat{z}$ are linearly dependent. In this paper, we classify all the $Q$-polynomial distance-regular graphs $Γ$ with the above property. We describe these graphs from multiple points of view.