Counting Isosceles Triangles Among Points

Given n points in the 2D coordinate plane, count the number of ways to choose three distinct points that form an isosceles triangle. A triangle is isosceles if at least two of its sides have equal length. In other words, if the three chosen points are P₁, P₂, and P₃ with pairwise squared distances \(d_{12}\), \(d_{13}\), and \(d_{23}\), then the triangle is isosceles if one of the following holds:

[ \begin{aligned} &d_{12} = d_{13} \quad \text{or}\ &d_{12} = d_{23} \quad \text{or}\ &d_{13} = d_{23} \end{aligned} ]

Note: An equilateral triangle (where all three sides are equal) is a special kind of isosceles triangle. However, when counting the triangles using a method that fixes a vertex as the apex (where the equal sides meet), an equilateral triangle would be counted three times (once for each vertex). To account for this, if we denote:

[ S = \text{sum over all points of } \sum_{d} \binom{\text{count}_d}{2}, ]

and let \(E\) be the number of equilateral triangles, then the correct number of distinct isosceles triangles is:

[ \text{Answer} = S - 2E. ]

Your task is to compute the number of different isosceles triangles that can be formed from the given points.

inputFormat

The first line contains an integer n (the number of points). Following that, there are n lines. Each of these lines contains two space-separated integers representing the x and y coordinates of a point.

outputFormat

Output a single integer which is the count of isosceles triangles that can be formed from the given points.

sample

3
0 0
2 0
1 1

1