KöMaL 2010 B4311

$P$ is a given point in the interior of the acute-angled triangle ABC . The lines $AP$, $BP$ and $CP$ intersect the opposite sides at the points $A_1$, $B_1$ and $C_1$, respectively. Given that $PA_1$=$PB_1$=$PC_1$=3 and $AP$+$BP$+$CP$=43, prove that $AP^. BP^. CP$=441.