Given a vertex $v$ of a polytope P defined by the intersection of $n$ linearly independent hyperplanes with normals $v_1, \ldots, v_n$ and a supporting hyperplane $c^Tx \leq d$ that passes through $v$, is it true $c \in Cone(v_1, \ldots, v_n)$? This seems like it should be obvious, but I cannot find a proof of it anywhere. Thanks for the help.
Intuition why it should be true: if the normal is outside of this cone, the supporting hyperplane should cut into P, contradiction that it is a supporting hyperplane. Unfortunately I have no clue how to formalize this.
My attempt so far: Farkas lemma says that if $c$ is not in the cone then there exists $a \in \mathbb{R}^n$ such that $c^Ta<0$ and $v_i^Ta \geq 0$. Maybe there is a way to scale $a$ so that it satisfies the hyperplanes defining the vertex, but falsifies the supporting hyperplane, thus contradicting that this hyerplane is supporting.