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Rectangle and set of points

by Numeriprimi
Tags: perimeter, rectangle
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Numeriprimi
#1
Jan7-13, 08:36 AM
P: 138
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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tiny-tim
#2
Jan7-13, 10:40 AM
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Quote Quote by Numeriprimi View Post
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.
Vargo
#3
Jan7-13, 01:15 PM
P: 350
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.

Numeriprimi
#4
Jan7-13, 03:37 PM
P: 138
Rectangle and set of points

Sorry for my English... And Vargo, yes, I mean this, thanks for wrote it in better form :-)
tiny-tim
#5
Jan7-13, 05:51 PM
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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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