Minimum number of midpoints

Question

Given $n$ points on a plane, join all pairs of points and find the midpoint of the respective line segment. What is the minimum number of midpoints?

My attempt: this might be a standard (and trivial) question in combinatorics geometry, but I do not know the answer.

Clearly, the maximum number is $\binom{n}{2}$. For the minimum, it seems that one can put the $n$ points equally distanced on a line. However, how can I show that this is the minimum?

Answer 1

Choose coordinate axes so that all of the points have different $x$ coordinates, and let $P_1,P_2,\dots,P_n$ be a list of the points in order of increasing $x$-coordinate. If we let $M(P,Q)$ be the midpoint of $P$ and $Q$, it follows that this a list of $2n-3$ distinct midpoints, because their $x$-coordinates strictly increase from left to right: $$ M(P_1,P_2),M(P_1,P_3),M(P_1,P_4),\dots,M(P_1,P_n),M(P_2,P_n),M(P_3,P_n)\dots,M(P_{n-1},P_n) $$ Since every arrangement has at least $2n-3$ district midpoints, and the equally spaced collinear arrangement attains exactly $2n-3$ midpoints, this linear arrangment is optimal.

Answer 2

Just to quantify the proposed minimum, $n$ evenly spaced points on a line (with $n \ge 2$) have $2n-3$ distinct midpoints. I think the first case where $n$ non-collinear points must have more midpoints than $n$ collinear points is $n = 5$.