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Geometry with polygons formed within drawn stars

  1. Jun 30, 2011 #1
    1. The problem statement, all variables and given/known data
    I'm working with the polygons on the interior of stars that have been drawn in a specific manner. For an example, I'm currently using a 16-gram. To construct the same one I have, you array sixteen vertices equidistant from one another around a central point (as in the construction of a regular hexadecagram). Starting at one point, connect it directly to the point that is five spaces from it clockwise (so there will be four points in between that have nothing through them). Continue with this pattern of fives until you eventually get back to the beginning.

    Inside this star, another hexadecagram should exist. My question is whether this polygon is regular, and if so, how do you prove it? My intuition tells me that it ought to be nothing more than a smaller version of the hexadecagram that would be created by directly connecting each of the exterior points in a circular fashion, but I've learned well enough by now that 1) intuition is not always right, and 2) intuition is very difficult to cite in a paper.


    2. Relevant equations
    Honestly, I'm not sure of any relevant equations here. That's sort of why I'm here, haha.


    3. The attempt at a solution
    Well... I drew it. And it looks mildly regular-ish. That hardly constitutes a proof in itself, though. I'm pretty decent at algebra and calculus, but I've never been the best problem-solver when it comes to geometry, haha.

    Mostly if someone could help me figure out an actual proof for whether or not the interior polygon is regular, I would be most appreciative. Any ideas? Thanks in advance!
     
  2. jcsd
  3. Jun 30, 2011 #2

    Dick

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    Think about angles. Clearly each new line you are drawing has the same minimal radius to the center point of the original 16-gon. If you can prove that there are 16 of them spaced at equal central angles (think about how multiples of 5 divide into 16) wouldn't that prove the interior 16-gon is also regular?
     
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