We have provided several demonstrations to help explain the solution to the problem, and to facilitate exploration of midpoint polygons. It should be noted that these demonstrations calculate the oriented area, which can be negative. If you are unfamiliar with the concept of oriented area, you might want to read the short introduction that we have prepared.
Area of midpoint triangles
Perimeter of midpoint polygons for rectangles
Area of midpoint quadrilaterals
Area maximum for midpoint pentagons
Infinite area ratios for non-convex pentagons
Another nice demonstration can be found on the NCTM standards site, though it calculates the absolute value of the oriented area. Although we are taught that "area is always positive", using the oriented area allows us to make a more powerful statement about midpoint polygons in the most general case.