AMC 12 2005a Problem 15

Let $\overline{AB}$ be a diameter of a circle and $C$ be a point on $\overline{AB}$ with $2 \cdot AC = BC$. Let $D$ and $E$ be points on the circle such that $\overline{DC} \perp \overline{AB}$ and $\overline{DE}$ is a second diameter. What is the ratio of the area of $\triangle DCE$ to the area of $\triangle ABD$?

unitsize(2.5cm);
defaultpen(fontsize(10pt)+linewidth(.8pt));
dotfactor=3;
pair O=(0,0), C=(-1/3.0), B=(1,0), A=(-1,0);
pair D=dir(aCos(C.x)), E=(-D.x,-D.y);
draw(A--B--D--cycle);
draw(D--E--C);
draw(unitcircle,white);
drawline(D,C);
dot(O);
clip(unitcircle);
draw(unitcircle);
label("$E$",E,SSE);
label("$B$",B,E);
label("$A$",A,W);
label("$D$",D,NNW);
label("$C$",C,SW);
draw(rightanglemark(D,C,B,2));

$\textbf{(A) } \frac{1}{6} \textbf{(B) } \frac{1}{4} \textbf{(C) } \frac{1}{3} \textbf{(D) } \frac{1}{2} \textbf{(E) } \frac{2}{3}$