Given a surface $S$ in a contact $(2n+1)$-manifold $(M,\xi)$ one can define the characteristic foliation of $S$ via looking at where the tangent bundle to $S$ coincides with $\xi$ (the singular part of the foliation), and otherwise where it intersects.
It turns out that locally, if $\theta$ is a volume form on $S$ and $\beta$ is a 1-form induced by $\xi$, the characteristic foliation is defined by the vector field $X$ such that: $$\iota(X)\theta = \beta\wedge(d\beta)^{n-1}.$$ This is equivalent definition for the characteristic foliation stated in both Giroux and Geiges, however the proof the Geiges gives I do not quite follow.
Geiges begins by showing that $\beta\wedge(d\beta)^{n-1}$ is non-zero apart from where $\beta$ itself is zero. Then he shows that if we are given such a vector field $X$, $X\in\ker\beta$. Finally, he shows that if $\beta_p$ is non-zero, then for all $v\in T_pS\cap\xi_p$, $d\beta(X_p,v)=0$. This finishes his proof.
However, I don't see how this proves the equivalence. Can anyone shed insight on this proof, or perhaps refer me to an alternative proof of this equivalence (or help me figure it out directly with hints)? Thanks!