  Given a polytope $\mathcal{P}$ in $\mathbb{R}^d$ and a subset $U$ of its
vertices, is there a triangulation of $\mathcal{P}$ using $d$-simplices that
all contain $U$? We answer this question by proving an equivalent and
easy-to-check combinatorial criterion for the facets of $\mathcal{P}$. Our
proof relates triangulations of $\mathcal{P}$ to triangulations of its
"shadow", a projection to a lower-dimensional space determined by $U$. In
particular, we obtain a formula relating the volume of $\mathcal{P}$ with the
volume of its shadow. This leads to an exact formula for the volume of a
polytope arising in the theory of unit equations.
