6. In $\triangle A B C$, $D$ is the midpoint of side $B C$. When a moving point $E$ is on the line segment $A D$, if $\overrightarrow{B E}=\lambda \overrightarrow{B A}+\mu \overrightarrow{B C}$, then the minimum value of $s=\lambda^{2}+\mu$ is $\qquad$.
Solve:
$\overrightarrow{B E}=\lambda \overrightarrow{B A}+2 \mu \overrightarrow{B D}$, then $A, E, D$ are collinear $\Rightarrow \lambda+2 \mu=1$, and $\lambda, \mu \geqslant 0$. Thus $\lambda^{2}+\mu=\lambda^{2}+\frac{1-\lambda}{2}=\left(\lambda-\frac{1}{4}\right)^{2}+\frac{7}{16} \geqslant \frac{7}{16}$, equality holds when $\lambda=\frac{1}{4}, \mu=\frac{3}{8}$.