Concave Quadrilaterals

The file attached with this problem contains a list having 100100 lines. Eight numbers x1,y1,x2,y2,x3,y3,x4,y4x_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 A1=(x1,y1), A2=(x2,y2), A3=(x3,y3), A4=(x4,y4)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 NN quadrilaterals in this list which are concave.

Find N+6.N+6.

Details and Assumptions:

  • As an explicit example, suppose the list had 22 lines, and the following numbers were written on those two lines: 1,1,2,2,3,3,4,41,1,2,2,3,3,4,4.\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),(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).(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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