Question 125
In $\triangle ABC$, $D$ is the midpoint of $BC$, $E$ is the foot of the perpendicular from $A$ to $BC$, and $F$ is the foot of the perpendicular from $D$ to $AC$. Given that $BE = 5$, $EC = 9$, and the area of triangle $ABC$ is $84$, compute $|EF|$.