Suppose $P$ is an interior point of a triangle $ABC$ and $[AP]$, $[BP]$, $[CP]$ meet the opposite sides $[BC]$, $[CA]$, $[AB]$ in $D$, $E$, $F$ respectively. Find the set of all possible values of the following quantities can take:
$$ \frac{|AP|}{|PD|}+\frac{|BP|}{|PE|}+\frac{|CP|}{|PF|}\\ \frac{|AP|\cdot|BP|\cdot|CP|}{|PD|\cdot|PE|\cdot|PF|} $$
If you drag $P$ to one of the corners, say $A$, then both expressions diverge to infinity. Since, while $\frac{|AP|}{|DP|}$ converges to zero, both $\frac{|BP|}{|EP|}$ and $\frac{|CP|}{|FP|}$ goes to infinity roughly same speed. Therefore there is no upper bound to these expressions.
As for the lower bound, i can only say that this happens when $P$ is on the centroid; giving a lower bound of $6$ and $8$ respectively. However, i cannot prove this fact, even though clearly there is no counterexample.