The Solution

	Triangles: The Base Case
	Convex Quadrilaterals     Non-Convex Quadrilaterals*     Self-Intersecting Quadrilaterals*
	Convex Pentagons
	Convex Hexagons And Up     The General Case*

Pedagogy

	Math 104     Original Discussion
	Oriented Area Part 1
	Oriented Area Part 2
	Vector Notation

Java Demonstrations

	Area of Midpoint Triangles
	Area of Midpoint Quadrilaterals
	Area Max for Convex Pentagons
	Infinite Area Ratios
	Star Pentagons
	Non-Midpoint Polygons

Bibliography

Summary

• The midpoint polygon for convex quadrilaterals is a parallelogram.
• The area ratio is always 1/2.

Break It Up Into Four Sections

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.

Figure 2. Finding the area of a midpoint 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.