KMO 2007 Problem 5

For the vertex $A$ of a triangle $ABC$ , let $l_a$ be the distance between the projections on $AB$ and $AC$ of the intersection of the angle bisector of ∠ $A$ with side $BC$ . Define $l_b$ and $l_c$ analogously. If $l$ is the perimeter of triangle $ABC$ , prove that $\frac{l_a l_b l_c}{l^3}\le\frac{1}{64}$ .