13. As shown in Figure 5, in the circle $\odot O$ with radius $r$, $AB$ is the diameter, $C$ is the midpoint of $\overparen{AB}$, and $D$ is the one-third point of $\overparen{CB}$. Moreover, the length of $\overparen{DB}$ is twice the length of $\overparen{CD}$; connect $AD$ and extend it to intersect the tangent line $CE$ of $\odot O$ at point $E$ ($C$ is the point of tangency). Find the length of $AE$.
ミ，３，As shown in Figure 6, connect $B E$.
$\because A B$ is the diameter.
$$
\begin{array}{l}
\therefore \angle A D B=90^{\circ}, \\
\text { Also } \because \overparen{B D}=60^{\circ}, \\
\therefore \angle D A B=30^{\circ} .
\end{array}
$$

Thus, $B D=\frac{1}{2} A B=r$.
Connecting $O C$, we know that $O C \perp A B$ and $O C \perp C E$, so $A B \parallel$ CE. Therefore,
$$
\begin{array}{l}
S_{\angle A E B}=S_{\triangle A C B}=\frac{1}{2} \times 2 r \times r=r^{2} . \\
\because S_{\angle A E B}=\frac{1}{2} A E \cdot B D, \therefore r^{2}=\frac{1}{2} A E \cdot r .
\end{array}
$$

Thus, $A E=2 r$.