2004 AMC 12B 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 $\triangle ABC$?

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

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.