Let $A C$ and $C E$ be perpendicular line segments, each of length $18 .$ Suppose $B$ and $D$ are the midpoints of $A C$ and $C E$ respectively. If $F$ be the point of intersection of $E B$ and $A D,$ then the area of $\triangle B D F$ is?

[list=1]
[*] $27\sqrt{2}$
[*] $18\sqrt{2}$
[*] 13.5
[*] 18
[/list]
1. **Define the coordinates of the points:**
   - Let $C$ be at the origin $(0, 0)$.
   - Let $A$ be at $(18, 0)$ since $AC = 18$.
   - Let $E$ be at $(0, 18)$ since $CE = 18$ and $AC \perp CE$.

2. **Find the midpoints $B$ and $D$:**
   - $B$ is the midpoint of $AC$, so $B = \left(\frac{18+0}{2}, \frac{0+0}{2}\right) = (9, 0)$.
   - $D$ is the midpoint of $CE$, so $D = \left(\frac{0+0}{2}, \frac{0+18}{2}\right) = (0, 9)$.

3. **Find the equations of lines $EB$ and $AD$:**
   - Line $EB$ passes through $E(0, 18)$ and $B(9, 0)$. The slope of $EB$ is $\frac{0-18}{9-0} = -2$. Thus, the equation of $EB$ is $y = -2x + 18$.
   - Line $AD$ passes through $A(18, 0)$ and $D(0, 9)$. The slope of $AD$ is $\frac{9-0}{0-18} = -\frac{1}{2}$. Thus, the equation of $AD$ is $y = -\frac{1}{2}x + 9$.

4. **Find the intersection point $F$ of $EB$ and $AD$:**
   - Set the equations equal to each other: $-2x + 18 = -\frac{1}{2}x + 9$.
   - Solve for $x$:
     \[
     -2x + 18 = -\frac{1}{2}x + 9 \implies -2x + \frac{1}{2}x = 9 - 18 \implies -\frac{3}{2}x = -9 \implies x = 6
     \]
   - Substitute $x = 6$ back into $y = -2x + 18$ to find $y$:
     \[
     y = -2(6) + 18 = -12 + 18 = 6
     \]
   - Thus, $F$ is at $(6, 6)$.

5. **Calculate the area of $\triangle BDF$:**
   - Use the vertices $B(9, 0)$, $D(0, 9)$, and $F(6, 6)$.
   - The area of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$ is given by:
     \[
     \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
     \]
   - Substituting the coordinates:
     \[
     \text{Area} = \frac{1}{2} \left| 9(9 - 6) + 0(6 - 0) + 6(0 - 9) \right| = \frac{1}{2} \left| 9 \cdot 3 + 0 + 6 \cdot (-9) \right| = \frac{1}{2} \left| 27 - 54 \right| = \frac{1}{2} \left| -27 \right| = \frac{27}{2} = 13.5
     \]

The final answer is $\boxed{13.5}$