Ksusha and Square

题意翻译

求在一给定凸边形内随机取不同两整点并以此作为正方形的对角线的正方形面积期望

题目描述

Ksusha is a vigorous mathematician. She is keen on absolutely incredible mathematical riddles. Today Ksusha came across a convex polygon of non-zero area. She is now wondering: if she chooses a pair of distinct points uniformly among all integer points (points with integer coordinates) inside or on the border of the polygon and then draws a square with two opposite vertices lying in the chosen points, what will the expectation of this square's area be? A pair of distinct points is chosen uniformly among all pairs of distinct points, located inside or on the border of the polygon. Pairs of points $ p,q $ $ (p≠q) $ and $ q,p $ are considered the same. Help Ksusha! Count the required expectation.

输入输出格式

输入格式

The first line contains integer $ n $ $ (3<=n<=10^{5}) $ — the number of vertices of Ksusha's convex polygon. Next $ n $ lines contain the coordinates of the polygon vertices in clockwise or counterclockwise order. The $ i $ -th line contains integers $ x_{i},y_{i} $ $ (|x_{i}|,|y_{i}|<=10^{6}) $ — the coordinates of the vertex that goes $ i $ -th in that order.

输出格式

Print a single real number — the required expected area. The answer will be considered correct if its absolute and relative error doesn't exceed $ 10^{-6} $ .

输入输出样例

输入样例 #1

3
0 0
5 5
5 0

输出样例 #1

4.6666666667

输入样例 #2

4
-1 3
4 5
6 2
3 -5

输出样例 #2

8.1583333333

输入样例 #3

3
17 136
859 937
16 641

输出样例 #3

66811.3704155169