[ Signs of similarity ] [ Theorem of cosines ]

In triangle $A B C$, on the midline $D E$, parallel to $A B$, a circle is constructed, intersecting the extensions of sides $A C$ and $B C$ at points $M$ and $N$. Find $M N$, if $B C=a, A C=b, A B=c$.
It is clear from the condition that angle $C$ is obtuse. Triangles $C M N$ and $C E D$ are similar with a coefficient of $-\cos \angle C$.

Therefore,

$M N=-D E \cos \angle C=\frac{c}{2} \cdot \frac{c^{2}-a^{2}-b^{2}}{2 a b}=\frac{c\left(c^{2}-a^{2}-b^{2}\right)}{4 a b}$.

## Answer

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