For convex quadrilaterals, the midpoint polygon is a parallelogram with sides parallel to the diagonals (see figure 2). The diagonals divide the original quadrilateral up into four triangles. If we consider one of these triangles, say AOB, it is clear that the portion of the midpoint polygon contained within the triangle (shaded red) has exactly half the area of the entire triangle (red and blue regions). Therefore the total area of the midpoint polygon is 1/2 the area of the original quadrilateral.
Quadrilaterals were discussed here, here, and here in the original discussion.
At this point we can go in two directions. We can ask about non-convex quadrilaterals, and we can go further to ask about self-intersecting quadrilaterals. Or we can move up to pentagons.
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Next: Non-Convex Quadrilaterals*
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Or skip to: Convex Pentagons
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