BG-MO 1973 Problem 5

Let the line $\ell$ intersects the sides $AC,BC$ of the triangle $ABC$ respectively at the points $E$ and $F$ . Prove that the line $\ell$ is passing through the incenter of the triangle $ABC$ if and only if the following equality is true: $$BC\cdot\frac{AE}{CE}+AC\cdot\frac{BF}{CF}=AB.$$ H. Lesov