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Problem 102

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Triangle containment

Three distinct points are plotted at random on a Cartesian plane, for which $-1000 \le x, y \le 1000$, such that a triangle is formed.

Consider the following two triangles:

$$A(-340,495), B(-153,-910), C(835,-947)$$
$$X(-175,41), Y(-421,-714), Z(574,-645)$$

It can be verified that triangle $ABC$ contains the origin, whereas triangle $XYZ$ does not.

Using triangles.txt (right click and ‘Save Link/Target As…’), a 27K text file containing the co-ordinates of one thousand “random” triangles, find the number of triangles for which the interior contains the origin.

NOTE: The first two examples in the file represent the triangles in the example given above.

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包含原点的三角形

在平面直角坐标系中选择三个不同的点，其坐标均满足$-1000 \le x, y \le 1000$，并以这三个点为顶点构成三角形。

考虑如下两个三角形：

$$A(-340,495), B(-153,-910), C(835,-947)$$
$$X(-175,41), Y(-421,-714), Z(574,-645)$$

可以验证三角形$ABC$包含原点，而三角形$XYZ$不包含原点。

在文本文件triangles.txt中包含了一千个“随机”三角形的坐标，求其中包含原点在其内部的三角形的数目。

注意：文件中的前两个三角形就是上述样例。

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