Problem

Source: Chinese National Olympiad 2009 P1

Tags: trigonometry, geometry, trapezoid, Gauss, cyclic quadrilateral, congruent triangles, angle bisector



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.