Problem

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 $\triangle ABC$ ?

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

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

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

Show/Hide Problem Tags

Problem Tags: 2-d Geometry Trigonometry

Want to contribute problems and receive full credit? Click here to add your problem!
Please report any issues to us in our Discord server
Go to previous contest problem (SHIFT + Left Arrow) Go to next contest problem (SHIFT + Right Arrow)

Problem feedback

Difficulty —

Category: AMC 12B
Points: 4
Back to practice
Train more like this