[ Extreme properties of a triangle (miscellaneous).]

Consider all acute-angled triangles with a given side $a$ and angle $\alpha$.

What is the maximum value of the sum of the squares of the lengths of sides $b$ and $c$?
According to the cosine theorem, $b^{2}+c^{2}=a^{2}+2 b c \cos \alpha$. Since $2 b c \leq b^{2}+c^{2}$ and $\cos \alpha>0$, then $b^{2}+c^{2} \leq a^{2}+\left(b^{2}+c^{2}\right) \cos \alpha$, that is,

$b^{2}+c^{2} \leq \frac{a^{2}}{1-\cos \alpha}=\frac{a^{2}}{2 \sin ^{2} \frac{\alpha}{2}}$. Equality is achieved when $b=c=\frac{a}{2 \sin \frac{\alpha}{2}}$.