Concave Quadrilaterals

The file attached with this problem contains a list having \(100\) lines. Eight numbers \(x_1, y_1, x_2, y_2, x_3, y_3, x_4, y_4\) are written in each line, which correspond to a quadrilateral whose vertices have coordinates \[A_1=(x_1, y_1),\ A_2=(x_2, y_2),\ A_3=(x_3, y_3),\ A_4=(x_4, y_4)\] (in Cartesian plane). There are \(N\) quadrilaterals in this list which are concave.

Find \(N+6.\)

Details and Assumptions:

  • As an explicit example, suppose the list had \(2\) lines, and the following numbers were written on those two lines: \[\begin{array}{lrrrrrrrrl} &1, &1, &2, &2, &3, &3, &4, &4 \\ &1, &-1, &2, &-2, &3, &-3, &4, &-4 &.\end{array} \] We have two quadrilaterals, the first one having its vertices at points \((1, 1), (2, 2), (3, 3), (4, 4),\) and the second one having its vertices at points \((1, -1), (2, -2), (3, -3), (4, -4).\) Your job is to find out how many of them are concave.

  • Link to the file:

  • If one of the vertices lies on the line joining two other vertices, consider the resulting figure as convex.

  • This is a computer science problem, and is inspired from ProjectEuler (I don't remember the problem number at the moment).


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