Midpoint Polygons

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

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

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

    • Bibliography

    Java Demonstrations

    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.

    1. Area of midpoint triangles
    2. Area of midpoint quadrilaterals
    3. Area maximum for midpoint pentagons
    4. Infinite area ratios for non-convex pentagons
    5. Star Pentagons
    6. Changing the side-length ratio: non-midpoint polygons

    Another nice demonstration can be found on the NCTM standards site, though it calculates the absolute value of the oriented area, and areas covered an even number of times are drawn empty. 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.

    Solution to the problem
    Full text of the online discussion
    Pedagogical Notes
    Problem -- Solution -- Pedagogy -- Demos -- Bibliography