CMO 2009 Problem 1

Given an acute triangle $PBC$ with $PB\neq PC.$ Points $A,D$ lie on $PB,PC,$ respectively. $AC$ intersects $BD$ at point $O.$ Let $E,F$ be the feet of perpendiculars from $O$ to $AB,CD,$ respectively. Denote by $M,N$ the midpoints of $BC,AD.$ $(1)$ : If four points $A,B,C,D$ lie on one circle, then $EM\cdot FN = EN\cdot FM.$ $(2)$ : Determine whether the converse of $(1)$ is true or not, justify your answer.