minimal number of extra points that should be added to create at least one square

You are given an array of points on the plane. What is the minimal number of extra points that should be added so that you could create at least one square using four of the points from the resulting set as its vertices? All given points are different. Square sides need not be parallel to the axes.

NOTE: This is a challenge from CodeFights, original here:

https://codefights.com/feed/uQWGzFRpNyTrWubWB