Let $ P,Q $ be the midpoints of the diagonals $ BD, $ respectively, $ AC, $ of the quadrilateral $ ABCD, $ and points $ M,N,R,S $ on the segments $ BC,CD,PQ, $ respectively $ AC, $ except their extremities, such that $$ \frac{BM}{MC}=\frac{DN}{NC}=\frac{PR}{RQ}=\frac{AS}{SC} . $$Show that the center of mass of the triangle $ AMN $ is situated on the segment $ RS. $
Problem
Source: Romanian National Olympiad 2014, Grade IX, Problem 3
Tags: geometry, center of mass, analytic geometry