3. In a regular triangular prism $A B C-A_{1} B_{1} C_{1}$, $E$ is the midpoint of $B C$, and $D$ is a moving point on $A A_{1}$, with $\frac{A D}{D A_{1}}=m$. If $A E / /$ plane $D B_{1} C$, then the value of $m$ is ( ).
(A) $\frac{1}{3}$
(B) $\frac{1}{2}$
(C) $\frac{2}{3}$
(D) 1
3.D.

Pass through $A A_{1}$ and $A E$ to make plane $A_{1} A E F$ intersect with plane $B C C_{1} B_{1}$ at $E F$. Let $E F$ intersect $B_{1} C$ at point $O$, and connect $D O$. Since $A A_{1} / /$ plane $B C C_{1} B_{1}, A E / /$ plane $D B_{1} C$, therefore,
$$
A A_{1} / / E F, A E / / D O \text {. }
$$

Thus, quadrilateral $A E O D$ is a parallelogram, which means $A D=E O$. Since $E$ is the midpoint of $B C$, therefore, $O$ is the midpoint of $B_{1} C$, and also the midpoint of $E F$, i.e., $A D=E O=\frac{1}{2} E F=\frac{1}{2} A A_{1}$. Therefore, $\frac{A D}{D A_{1}}=m=1$.