AMC 12 2004b Problem 24

In $\triangle ABC$, $AB = BC$, and $\overline{BD}$ is an altitude. Point $E$ is on the extension of $\overline{AC}$ such that $BE = 10$. The values of $\tan \angle CBE$, $\tan \angle DBE$, and $\tan \angle ABE$ form a geometric progression, and the values of $\cot \angle DBE,$ $\cot \angle CBE,$ $\cot \angle DBC$ form an arithmetic progression. What is the area of $\triangle ABC$?

size(120);
defaultpen(0.7);
pair A = (0,0), D = (5*2^.5/3,0), C = (10*2^.5/3,0), B = (5*2^.5/3,5*2^.5), E = (13*2^.5/3,0);
draw(A--D--C--E--B--C--D--B--cycle);
label("$A$",A,S);
label("$B$",B,N);
label("$C$",C,S);
label("$D$",D,S);
label("$E$",E,S);

$\mathrm{(A)}\ 16 \mathrm{(B)}\ \frac {50}3 \mathrm{(C)}\ 10\sqrt{3} \mathrm{(D)}\ 8\sqrt{5} \mathrm{(E)}\ 18$