S is a finite set of points in the plane. Show that there is at most one point P in the plane such that if A is any point of S, then there is a point A' in S with P the midpoint of AA'.
Solution
The points must form disjoint pairs (A,A') with P the midpoint of each pair. But P is the centroid of each pair and hence the centroid of all the points, but that is a uniquely defined point.
© John Scholes
jscholes@kalva.demon.co.uk
1 Nov 2003
Last corrected/updated 1 Nov 03