Triangle containment

Three distinct points are plotted at random on a Cartesian plane, for which $-1000$ ≤ x, y ≤ $1000$, 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 $27$K 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.