MathDB

2

Part of 2021 German National Olympiad

Problems(1)

Interior points of the sides minimizing the perimeter iff cyclic

Source: Germany 2021, Problem 2

6/16/2021
Let PP on ABAB, QQ on BCBC, RR on CDCD and SS on ADAD be points on the sides of a convex quadrilateral ABCDABCD. Show that the following are equivalent:
(1) There is a choice of P,Q,R,SP,Q,R,S, for which all of them are interior points of their side, such that PQRSPQRS has minimal perimeter.
(2) ABCDABCD is a cyclic quadrilateral with circumcenter in its interior.
geometryperimetergeometry proposedcyclic quadrilateralinterior