KMO 2001 Problem 2

Let $P$ be a given point inside a convex quadrilateral $O_1O_2O_3O_4$ . For each $i = 1,2,3,4$ , consider the lines $l$ that pass through $P$ and meet the rays $O_iO_{i-1}$ and $O_iO_{i+1}$ (where $O_0 = O_4$ and $O_5 = O_1$ ) at distinct points $A_i(l)$ and $B_i(l)$ , respectively. Denote $f_i(l) = PA_i(l) \cdot PB_i(l)$ . Among all such lines $l$ , let $l_i$ be the one that minimizes $f_i$ . Show that if $l_1 = l_3$ and $l_2 = l_4$ , then the quadrilateral $O_1O_2O_3O_4$ is a parallelogram.