In general, for an $n$-gon, we have $\dfrac {n} {n-1} \leq \sum_{i=1}^n \dfrac {a_i} {s - a_i}< 2$. The left inequality comes from Jensen's, with equality for an equilateral polygon; the right inequality is proven as above, with equality only reached for a degenerated polygon (but we can come as close to $2$ as wanted).

the left inequality can be also proved by Cauchy directly. for the right side,by $a_{i}\le\ s-a_{i}$,we have that $\sum_{i=1}^{n}\frac{a_{i}}{s-a_{i}}\le\sum_{i=1}^{n}\frac{2a_{i}}{s}=2$

let the polygon be $n$ sided and the sides be $a_1,a_2,...,a_n$ let $t_{i}=\frac{a_{i}}{a_{1}+....+a_{i-1}+a_{i+1}+....+a_{n}}$ note that , the sum of any (n-1) side lengths is greater than the remaining side length. so , $t_{i} <\frac{2a_{i}}{a_{1}+....+a_{n}}$ and the rest is trivial