Question 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 $\triangle ABC?$
