Say we have a convex quadrilateral $ABCD$ and a set of points $S$ (no three of which are collinear) such that all $P \in S$ are in the closed set of $ABCD$, and that the points $E$, $F$, $G$, and $H$ lie on $AB$, $BC$, $CD$, and $DA$, which form the convex quadrilateral $EFGH$. For all $P \in S$, $P$ is not in the closed set of $EFGH$. Prove (or disprove) that there exist four points $P_1, P_2, P_3, P_4$ such that the convex quadrilateral $P_1P_2P_3P_4$ can be formed such that no points in $S$ are in the closed set of $P_1P_2P_3P_4$.
My intuition tells me that yes, a quadrilateral does exist, and it would be points that minimize the Euclidean distance from a point $P$ to one of the sides of $EF$. But I'm not sure how to prove this rigorously. I was also thinking about convex hulls, but I don't have too much knowledge on those.
