FOURPTS - Editorial

The answer will be impossible if the polygon is concave.

There are the special cases, rectangle and parallelogram, and when points are collinear, and the fourth point is inside the triangle.

In other cases, the problem can be solved by finding the intersections of the lines. The idea is to construct convex hull of points, let it be A, B, C, D in the order. From C make a line parallel to BD, and intersect this line with AB and AD to get the points E, F on line BD. Then AEF is the desired triangle.