Geometry of factorization identities for discriminants

Let $\Delta_n$ be the discriminant of a general polynomial of degree $n$ and $\mathcal{N}$ be the Newton polytope of $\Delta_n$. We give a geometric proof of the fact that the truncations of $\Delta_n$ to faces of $\mathcal{N}$ are equal to products of discriminants of lesser $n$ degrees. The proof is based on the blow-up property of the logarithmic Gauss map for the zero set of $\Delta_n$.