In triangle $A B C$, on the midline $D E$, parallel to $A B$, a circle is constructed with $D E$ as its diameter, intersecting sides $A C$ and $B C$ at points $M$ and $N$.

Find $M N$, if $B C=a, A C=b, A B=c$.
$D N$ and $E M$ are the altitudes of triangle $C D E$, which means that triangles $C M N$ and $C E D$ are similar with a coefficient of $\cos \angle C$.

Therefore, $M N=D E \cos \angle C=\frac{c}{2} \cdot \frac{a^{2}+b^{2}-c^{2}}{2 a b}$.

## Answer

$\frac{c\left(a^{2}+b^{2}-c^{2}\right)}{4 a b}$