From the earth to the moon

If we let the points be $(0,0), (2,0), \ldots, (598,0)$ , then the midpoint of any pair of points will be a point of the form $(k,0)$ with $1 \leq k \leq 597$ . So the minimum possible size of $B$ is at most $597$ . We will show that we are unable to do any better.

Because we have $300$ points, there are at most $\binom{300}{2}$ different slopes which occur between pairs of points. Choose a slope $m$ such that no pair of points has slope $\frac{-1}{m}$ between them. We can rotate the plane such that the line $y = mx$ is mapped onto the x-axis. It is clear that this transformation does not affect the number of distinct midpoints. After this transformation, every point in $A$ will have a distinct $x$ -coordinate. We can label the points $A_1, A_2, \ldots, A_n$ in ascending order of x-coordinate. The x-coordinates of the midpoints of $A_i,A_{i+1}$ will all be pairwise distinct, so the midpoints will also be pairwise distinct. This is also true for the midpoints for $A_i,A_{i+2}$ . No point can occur as a midpoint in both of these instances, so we get a total of $299 + 298 = 597$ distinct midpoints from this. Thus, we always have at least $597$ distinct midpoints, and $597$ distinct midpoints is attainable, so it is our minimum.