KöMaL 2024 C1812

Let $a$, $b$ and $c$ denote the lengths of the sides of a triangle satisfying $a+b=3c$. Let $\alpha$ and $\beta$ denote the angles opposite to sides $a$ and $b$, respectively. Prove that $\cot{\frac{\alpha}{2}}\cdot{\cot{\frac{\beta}{2}}}=2$.