Problem 1262

Theorem Let $A_{1}, A_{2}, \cdots, A_{n}$ be a plane convex $n$-sided polygon, with area $\Delta$, and side lengths $A_{i} A_{i+1}=a_{i}$ $\left(i=1,2, \cdots, n, n \geqslant 3\right.$, and assume $A_{n+1}=A_{1}$, the same below), and $\theta_{i} \in(0, \pi)(i=1,2, \cdots, n$, $n \geqslant 3)$, and $\sum_{i=1}^{n} \theta_{i}=\pi, n \geqslant 3$, then
$$\sum_{i=1}^{n} \cot \theta_{i} a_{i}^{2} \geqslant 4 \Delta$$

Equality holds in (1) if and only if the convex $n$-sided polygon is inscribed in a circle, and
$$\frac{a_{1}}{\sin \theta_{1}}=\frac{a_{2}}{\sin \theta_{2}}=\cdots=\frac{a_{n}}{\sin \theta_{n}}=2 R$$
( $R$ is the radius of the circle).