USACO 2012 February Contest, Gold Division

Problem 2. Symmetry

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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 (file 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 (file symmetry.out): 4 OUTPUT DETAILS: There are 4 lines of symmetry -- one vertical, one horizontal, and two diagonal.
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