Point $F$ is taken on the extension of side $AD$ of parallelogram $ABCD$.  $BF$ intersects diagonal $AC$ at $E$ and side $DC$ at $G$. If $EF = 32$ and $GF = 24$, then $BE$ equals:


[asy]
size(7cm);
pair A = (0, 0), B = (7, 0), C = (10, 5), D = (3, 5), F = (5.7, 9.5);
pair G = intersectionpoints(B--F, D--C)[0];
pair E = intersectionpoints(A--C, B--F)[0];
draw(A--D--C--B--cycle);
draw(A--C);
draw(D--F--B);
label("$A$", A, SW);
label("$B$", B, SE);
label("$C$", C, NE);
label("$D$", D, NW);
label("$F$", F, N);
label("$G$", G, NE);
label("$E$", E, SE);
//Credit to MSTang for the asymptote
[/asy]



$\textbf{(A)}\ 4 \qquad
\textbf{(B)}\ 8\qquad
\textbf{(C)}\ 10 \qquad
\textbf{(D)}\ 12 \qquad
\textbf{(E)}\ 16$
1. **Identify the given information and relationships:**
   - Parallelogram \(ABCD\) with \(F\) on the extension of side \(AD\).
   - \(BF\) intersects diagonal \(AC\) at \(E\) and side \(DC\) at \(G\).
   - \(EF = 32\) and \(GF = 24\).
   - We need to find \(BE\).

2. **Calculate \(GE\):**
   \[
   GE = EF - GF = 32 - 24 = 8
   \]

3. **Use similar triangles:**
   - Since \( \triangle{FAE} \sim \triangle{BCE} \) (by AA similarity, as they share angle \( \angle AEC \) and both have a right angle at \( \angle AFE \) and \( \angle BGE \)), we can write the ratio of corresponding sides:
   \[
   \frac{BE}{EF} = \frac{BC}{FA}
   \]
   Let \( BE = x \) and \( BC = AD = a \). Since \( F \) is on the extension of \( AD \), let \( FA = b \).

4. **Express \(FA\) in terms of \(x\):**
   \[
   \frac{x}{32} = \frac{a}{a + b}
   \]
   Solving for \( b \):
   \[
   x(a + b) = 32a \implies xa + xb = 32a \implies xb = 32a - xa \implies b = \frac{32a - xa}{x}
   \]

5. **Use the ratio of segments on \(DC\):**
   - Since \( \frac{FD}{DA} = \frac{FG}{GB} \) (by the properties of similar triangles and parallel lines):
   \[
   \frac{\frac{32a - xa}{x}}{a} = \frac{24}{x + 8}
   \]
   Simplify the left-hand side:
   \[
   \frac{32a - xa}{ax} = \frac{32 - x}{x}
   \]
   Equate the ratios:
   \[
   \frac{32 - x}{x} = \frac{24}{x + 8}
   \]

6. **Solve the equation:**
   \[
   (32 - x)(x + 8) = 24x
   \]
   Expand and simplify:
   \[
   32x + 256 - x^2 - 8x = 24x
   \]
   \[
   -x^2 + 24x + 256 = 24x
   \]
   \[
   -x^2 + 256 = 0
   \]
   \[
   x^2 = 256
   \]
   \[
   x = 16
   \]

The final answer is \( \boxed{16} \).