Problem

Source: Germany 2021, Problem 2

Tags: geometry, perimeter, geometry proposed, cyclic quadrilateral, interior



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.