, 2011

Without loss of generality, we may assume that the $n$ points form a regular $n$-gon. We will partition all the possible chords of the $n$-gon into $n$ different sets such that all the chords in each set are parallel. We distinguish two cases:

If $n$ is odd, then each chord is parallel to exactly one of the sides of the $n$-gon, so define $A_i$ to be the set of chords parallel to side $i$.

If $n = 2k$ is even, let the $n$-gon be $P_1P_2\dots P_{2k}$. Then each chord is parallel to a pair of sides, or parallel to a chord of the form $P_iP_{i+2}$ for some $i$. For $1 \le i \le k$, let $A_i$ be the set of chords parallel to side $P_iP_{i+1}$, and for $k < i \le 2k$, let $A_i$ be the set of chords parallel to $P_{i-2}P_i$.

Since we have $nk + 1$ chords, at least $k + 1$ of these chords must lie in one of the $n$ sets $A_i$. Since the chords in each set $A_i$ are parallel, no two of them intersect.