13. In triangle $A B C$, a circle is constructed on the midline $D E$, parallel to $A B$, as its diameter, intersecting sides $A C$ and $B C$ at points $M$ and $N$. Find $M N$, if $|B C|=a,|A C|=b,|A B|=c$.
13. If $\hat{C}=\alpha$, then $|C N|=\frac{b}{2} \cos \alpha,|C M|=\frac{a}{2} \cos \alpha\left(\alpha<90^{\circ}\right)$; therefore, $\triangle C M N$ is similar to $\triangle C A B$ with a similarity coefficient of $\frac{\cos \alpha}{2}$, so $|M N|=|A B| \frac{\cos \alpha}{2}=\frac{c}{2} \frac{a^{2}+b^{2}-c^{2}}{2 a b}=\frac{\left(a^{2}+b^{2}-c^{2}\right) c}{4 a b}$.