Extension Problems for Marion Walter's Theorem

The number three is central to the original problem in two ways: the number of sides in a triangle and the number of parts each side is divided into.  If you replace "three" with another integer like "four" or "five" can you pose similar questions?  Hint:  In some cases the questions might not make sense.

There are six other points of intersection between the lines in Figure 1.  Is the area of the hexagon that they determine related to the area of the triangle?

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