KöMaL 2019 B5058

Let $P$ be an arbitrary point in the interior of triangle $ABC$. Lines $AP$, $BP$ and $CP$ intersect sides $BC$, $AC$, and $AB$ at points $A_1$, $B_1$ and $C_1$, respectively. Prove that $\frac{AP}{A_1P}\cdot\frac{BP}{B_1P}\cdot\frac{CP}{C_1P}\ge 8.$