A finite set in which every point is a midpoint

Idea. An extreme point can never be the midpoint of two other points.

Extreme point. Choose the point $P=(p,q)$ of $S$ with the largest abscissa $p$, and, among the points with abscissa $p$, the largest ordinate $q$.

Contradiction. Suppose $P$ is the midpoint of two other points $A=(a_1,a_2)$ and $B=(b_1,b_2)$ of $S$. Then $\frac{a_1+b_1}2=p$ with $a_1\le p$ and $b_1\le p$, so $a_1=b_1=p$. Now $A$ and $B$ both have abscissa $p$, hence $a_2\le q$ and $b_2\le q$, and $\frac{a_2+b_2}2=q$ forces $a_2=b_2=q$. Then $A=B=P$, but $A$ and $B$ were supposed to differ from $P$. Contradiction.

Remark. For an infinite set the statement is false: $\mathbb{Z}^2$ has this property.

Lesson. Extremal principle: look at a point that maximises or minimises some quantity.