Problem 2: Symmetry [Brian Dean, 2012]

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.

PROBLEM NAME: symmetry

INPUT FORMAT:

• Line 1: The single integer N.

• Lines 2..1+N: Line i+1 contains two space-separated integers
  representing the x and y coordinates of the ith cow (-10,000
  <= x,y <= 10,000).

SAMPLE INPUT:

4
0 0
0 1
1 0
1 1

INPUT DETAILS:

The 4 cows form the corners of a square.

OUTPUT FORMAT:

• Line 1: The number of different lines of symmetry of the point set.

SAMPLE OUTPUT:

4

OUTPUT DETAILS:

There are 4 lines of symmetry -- one vertical, one horizontal, and two
diagonal.

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