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After taking a modern art class, Farmer John has become interested in finding geometric patterns in everything around his farm. He carefully plots the locations of his N cows (2 <= N <= 1000), each one occupying a distinct point in the 2D plane, and he wonders how many different lines of symmetry exist for this set of points. A line of symmetry, of course, is a line across which the points on both sides are mirror images of each-other.
Please help FJ answer this most pressing geometric question.
4 0 0 0 1 1 0 1 1
The 4 cows form the corners of a square.
There are 4 lines of symmetry -- one vertical, one horizontal, and two diagonal.