The Child and Polygon

题意翻译

给出$n$个点，求这个多边形的三角剖分的方案数 mod 1e9+7

题目描述

This time our child has a simple polygon. He has to find the number of ways to split the polygon into non-degenerate triangles, each way must satisfy the following requirements: - each vertex of each triangle is one of the polygon vertex; - each side of the polygon must be the side of exactly one triangle; - the area of intersection of every two triangles equals to zero, and the sum of all areas of triangles equals to the area of the polygon; - each triangle must be completely inside the polygon; - each side of each triangle must contain exactly two vertices of the polygon. The picture below depicts an example of a correct splitting. Please, help the child. Calculate the described number of ways modulo $ 1000000007 $ $ (10^{9}+7) $ for him.

输入输出格式

输入格式

The first line contains one integer $ n $ $ (3<=n<=200) $ — the number of vertices of the polygon. Then follow $ n $ lines, each line containing two integers. The $ i $ -th line contains $ x_{i},y_{i} $ $ (|x_{i}|,|y_{i}|<=10^{7}) $ — the $ i $ -th vertex of the polygon in clockwise or counterclockwise order. It's guaranteed that the polygon is simple.

输出格式

Output the number of ways modulo $ 1000000007 $ $ (10^{9}+7) $ .

输入输出样例

输入样例 #1

4
0 0
0 1
1 1
1 0

输出样例 #1

2

输入样例 #2

4
0 0
1 0
0 1
-1 0

输出样例 #2

1

输入样例 #3

5
0 0
1 0
1 1
0 1
-2 -1

输出样例 #3

3

说明

In the first sample, there are two possible splittings: In the second sample, there are only one possible splitting: