## Task 5 - 070935

For a triangle $A B C$, let the side lengths be $a, b$, and $c$.

Calculate the length $s_{c}$ of the median to side $A B$!
Let $S$ be the midpoint of the segment $A B$. By applying the cosine rule in triangles $A B C$ and $A S C$, we obtain:

$$
\frac{b^{2}+c^{2}-a^{2}}{2 b c}=\cos (\alpha)=\frac{b^{2}+\left(\frac{c}{2}\right)^{2}-s_{c}^{2}}{2 b \frac{c}{2}}
$$

and thus

$$
\frac{1}{2}\left(b^{2}+c^{2}-a^{2}\right)=b^{2}+\frac{1}{4} c^{2}-s_{c}^{2} \Longleftrightarrow s_{c}^{2}=\frac{1}{4}\left(2 a^{2}+2 b^{2}-c^{2}\right)
$$