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:
$$\begin{gather} A(-340,495), B(-153,-910), C(835,-947)\\ X(-175,41), Y(-421,-714), Z(574,-645) \end{gather}$$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.
Solution
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