IZhO 2008 Problem 1

Points $K,L,M,N$ are repectively the midpoints of sides $AB,BC,CD,DA$ in a convex quadrliateral $ABCD$ .Line $KM$ meets dioganals $AC$ and $BD$ at points $P$ and $Q$ ,respectively.Line $LN$ meets dioganals $AC$ and $BD$ at points $R$ and $S$ ,respectively. Prove that if $AP\cdot PC=BQ\cdot QD$ ,then $AR\cdot RC=BS\cdot SD$ .