3. In $\triangle A B C$, it is known that $A B=12, A C=9, B C$ $=13$. From $A$, perpendiculars are drawn to the angle bisectors of $\angle C$ and $\angle B$, with the feet of the perpendiculars being $M$ and $N$, respectively. Connect $M N$. Then $\frac{S_{\triangle A M N}}{S_{\triangle A B C}}=$ $\qquad$ .
3. $\frac{2}{13}$.

As shown in Figure 2, extend $A M$ to intersect $B C$ at point $D$, and extend $A N$ to intersect $B C$ at point $E$. Then, $\triangle A C D$ and $\triangle A B E$ are both isosceles triangles.
Thus, $B E=A B=12, C D=A C=9$. Since $B C=13$, we have $D E=12+9-13=8$. Therefore, $\frac{S_{\triangle A D E}}{S_{\triangle A B C}}=\frac{8}{13}$.
Since $M$ and $N$ are the midpoints of $A D$ and $A E$ respectively, we have
$$
S_{\triangle I M N}=\frac{1}{4} S_{\triangle A D E} \text {. }
$$

Therefore, $\frac{S_{\triangle A M N}}{S_{\triangle M B C}}=\frac{8}{52}=\frac{2}{13}$.