Answer: E

Small Hint:

Parameterize points on $FB$ by their fraction of the distance from $F$ to $B$

Big Hint:

If $F=tD$ in affine coordinates based at $A,$ the parameters of $G$ and $E$ are $\frac{t-1}{t}$ and $\frac{t}{t+1}$

Solution:

Use affine coordinates $A=0,$ $B=u,$ $D=v,$ $C=u+v,$ and $F=tv.$ A point on $FB$ is $su+t(1-s)v.$ Intersecting $DC$ gives $s_G=\frac{t-1}{t},$ while intersecting $AC$ gives $s_E=\frac{t}{t+1}.$ Therefore $$\begin{aligned} \frac{GF}{EF} &=\frac{s_G}{s_E} =\frac{t^2-1}{t^2}\\ &=\frac{24}{32}=\frac34, \end{aligned}$$ so $t=2$ and $s_E=\frac{2}{3}.$ Thus $EF=(\frac{2}{3})FB=32,$ making $FB=48$ and $BE=16.$

Therefore, the correct answer is E.