In $ \triangle ABC$, $ AB \equal{} BC$, and $ BD$ is an altitude. Point $ E$ is on the extension of $ \overline{AC}$ such that $ BE \equal{} 10$. The values of $ \tan CBE$, $ \tan DBE$, and $ \tan ABE$ form a geometric progression, and the values of $ \cot DBE$, $ \cot CBE$, $ \cot DBC$ form an arithmetic progression. What is the area of $ \triangle ABC$?
[asy]unitsize(3mm);
defaultpen(linewidth(.8pt)+fontsize(10pt));

pair D=(0,0), C=(3,0), A=(-3,0), B=(0, 8), Ep=(6,0);

draw(A--B--Ep--cycle);
draw(D--B--C);

label("$A$",A,S);
label("$D$",D,S);
label("$C$",C,S);
label("$E$",Ep,S);
label("$B$",B,N);[/asy]$ \textbf{(A)}\ 16 \qquad \textbf{(B)}\ \frac {50}{3} \qquad \textbf{(C)}\ 10\sqrt3 \qquad \textbf{(D)}\ 8\sqrt5 \qquad \textbf{(E)}\ 18$