# rectangle and set of points

by Numeriprimi
Tags: perimeter, rectangle
 P: 94 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?
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P: 25,521
 Quote by Numeriprimi Where is the set of points when their (for each point) sum of distance lines AB, BC, CD, DA is 2/3o ?
Sorry, Numeriprimi, I don't understand.
 P: 350 I think the problem is to identify the set of points X such that $d(X,AB)+d(X,BC)+d(X,CD)+d(X,DA)= \frac{2}{3}*\text{Perimeter of Rectangle ABCD}.$ 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.
P: 94

## rectangle and set of points

Sorry for my English... And Vargo, yes, I mean this, thanks for wrote it in better form :-)
 PF Patron HW Helper Sci Advisor Thanks P: 25,521 Thanks, Vargo! 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. (and yes, it's an octagon!)

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