DE-MO 2021 Problem 2

Let $P$ on $AB$ , $Q$ on $BC$ , $R$ on $CD$ and $S$ on $AD$ be points on the sides of a convex quadrilateral $ABCD$ . Show that the following are equivalent:

(1) There is a choice of $P,Q,R,S$ , for which all of them are interior points of their side, such that $PQRS$ has minimal perimeter.

(2) $ABCD$ is a cyclic quadrilateral with circumcenter in its interior.