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.
Problem
Source: Germany 2021, Problem 2
Tags: geometry, perimeter, geometry proposed, cyclic quadrilateral, interior
21.06.2021 03:43
I'm sorry I can't understand the firse statement. I guess you mean P,Q,R,S is on the line of the four sides oringinly, the first statement means the minimal perimeter holds when they are all on the segments of the sides. Is that right?
21.06.2021 21:42
No no. They are always on the segments (or as the problem says "on the sides") but the condition (1) is about whether for the choice with the minimal parameters among all such $PQRS$, these four points are interior points (as opposed to being one of the endpoints of the sides/segments).
30.03.2022 18:03
Okay if I get the problem statement right, then we actually only consider the sides an not the lines on which the sides lie, so with "interior point", we mean that it lies on one of the sides and it not equal to $A$, $B$, $C$ or $D$. However, then $(1)\Longrightarrow (2)$ does not make sense. $(2) \Longrightarrow (1)$: Wlog, (or OBdA, as we germans say), let $P\in \overline{AB}$, $Q\in \overline{BC}$, $R\in \overline{CD}$ and $S\in \overline{DA}$, so that non of the points lie on and assume that $A$, $B$, $C$ or $D$. Let $PQRS$ now be any such quadrilateral. By the triangle inequality, we get $PB+BQ>PQ$. We get similar results for the other points. Adding these inequalities up gives $$PB+BQ+CQ+CR+DR+DS+AS+AP=AB+BC+CD+DA>PQ+QR+RS+SP$$Therefore, the perimeter of $ABCD$ is larger than any quadrilateral inscribed on the sides, so it can be the minimal one.
30.03.2022 18:13
Joel.Gerlach wrote: However, then $(1)\Longrightarrow (2)$ does not make sense. I don't understand. Why does it not make sense?