3012: The Child and Polygon
Description
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 (109+7) for him.
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 xi,yi (|xi|,|yi|≤107) − the i-th vertex of the polygon in clockwise or counterclockwise order.
It's guaranteed that the polygon is simple.
Output the number of ways modulo 1000000007 (109+7).
4
0 0
0 1
1 1
1 0
2
4
0 0
1 0
0 1
-1 0
1
5
0 0
1 0
1 1
0 1
-2 -1
3
In the first sample, there are two possible splittings:
In the second sample, there are only one possible splitting:
