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Rectangle and set of points

  1. Jan 7, 2013 #1
    I found one interesting example...

    We have a rectangle ABCD with his perimeter o. Where is the set of points when their (for each point) sum of distance lines AB, BC, CD, DA is 2/3o ?

    I tried it, but geometric problems is quite hard for me and I don't know how do it. So, have you got any idea?
  2. jcsd
  3. Jan 7, 2013 #2


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    Sorry, Numeriprimi, I don't understand. :confused:
  4. Jan 7, 2013 #3
    I think the problem is to identify the set of points X such that
    [itex] d(X,AB)+d(X,BC)+d(X,CD)+d(X,DA)= \frac{2}{3}*\text{Perimeter of Rectangle ABCD}.[/itex]
    The distances on the left hand side are lengths of perpendiculars dropped from the point X to the corresponding line. For example, if X lies inside the interior, then the left hand side would equal (1/2) the perimeter. So the locus cannot include points inside the rectangle. Without doing an explicit calculation, I bet the answer is an octogon.
  5. Jan 7, 2013 #4
    Sorry for my English... And Vargo, yes, I mean this, thanks for wrote it in better form :-)
  6. Jan 7, 2013 #5


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    Thanks, Vargo! :smile:

    Numeriprimi, divide the exterior of the rectangle into 8 sections (by extending the sides),

    and consider each section separately …

    you should be able to find an easy equation for each section. :wink:

    (and yes, it's an octagon!)
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