3. As shown in Figure 1, the incircle of $\triangle A B C$ is $\odot O$. A line parallel to $B C$ is drawn through $O$, intersecting $A B$ and $A C$ at points $D$ and $E$, respectively. If the lengths of the three sides of $\triangle A B C$, $B C$, $C A$, and $A B$, are 8, 7, and 5, respectively, then the length of $D E$ is $\qquad$
3. $\frac{24}{5}$.

From the properties of the incenter and parallel lines, we get
$$
B D=O D, C E=O E \text {. }
$$

From $D E / / B C$ and the properties of proportional segments, we get
$$
\begin{array}{l}
\frac{B D}{5}=\frac{C E}{7}=\frac{D E}{12}, \frac{A D}{A B}=1-\frac{B D}{5}=\frac{D E}{8} \\
\Rightarrow 1-\frac{D E}{12}=\frac{D E}{8} \\
\Rightarrow D E=\frac{24}{5} .
\end{array}
$$