A Pascal's Theorem for rational normal curves

Pascal’s Theorem gives a synthetic geometric condition for six points $a,\dots ,f$ in ${\mathbb{P}}^2$ to lie on a conic. Namely, that the intersection points $\overline{ab}\cap \overline{de}$, $\overline{af}\cap \overline{dc}$, $\overline{ef}\cap \overline{bc}$ are aligned. One could ask an analogous question in higher dimension: is there a coordinate-free condition for $d+4$ points in ${\mathbb{P}}^d$ to lie on a degree $d$ rational normal curve? In this paper we find many of these conditions by writing in the Grassmann-Cayley algebra the defining equations of the parameter space of $d+4$ ordered points in ${\mathbb{P}}^d$ that lie on a rational normal curve. These equations were introduced and studied in a previous joint work of the authors with Giansiracusa and Moon. We conclude with an application in the case of seven points on a twisted cubic.