Putnam 1989 A5

Let $m$ be a positive integer and let $\mathcal{G}$ be a regular $(2m+1)$-gon inscribed in the unit circle. Show that there is a positive constant $A$, independent of $m$, with the following property. For any points $p$ inside $\cal G$ there are two distinct vertices $v_1$ and $v_2$ of $\cal G$ such that $$\left|\,|p-v_1| - |p-v_2|\,\right| < \frac1{m} - \frac{A}{m^3}.$$ Here $|s-t|$ denotes the distance between the points $s$ and $t$.