KöMaL 2008 B4117

Three lines passing through a point $P$ are cut by two other lines at the points $A$, $B$, $C$and $A$', $B$', $C$', respectively. Prove that $\frac{AA'}{A'P}\overrightarrow{BC} +\frac{BB'}{B'P}\overrightarrow{CA} +\frac{CC'}{C'P}\overrightarrow{AB}=\mathbf{0}$, where the ratios are considered directed.