KöMaL 2011 B4380

$m$ points are chosen on the sides of a convex $k$-sided polygon, such that the resulting point set $\mathcal{P}$ of $n$=$k$+$m$ elements is centrally symmetric. Prove that there may be at most 2$n$-3 line segments of equal length among the $\binom{n}{2}$ line segments determined by the points of the set $\mathcal{P}$ . For what point sets $\mathcal{P}$ are there 2$n$-3 line segments of equal length?