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#1 2008-08-25 07:20:51

atreyyu
Member
Registered: 2008-08-01
Posts: 5

Line passing through a square

Straight line

passes through the center of a square
, whose area is
. Let
denote the shortest distances between line
and points
respectively. How to prove that
?

Last edited by atreyyu (2008-08-25 07:21:14)

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#2 2008-08-25 08:59:48

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,905

Re: Line passing through a square

Look at the picture. It is sufficient to prove that

.
To show this, notice AO=DO=a half of the diagonal. The angle AOD is 90degs, so we end up with 2 congruent triangles, namely APO and ODQ. That means that OQ = AP, and by Pythagorean theorem,

Last edited by krassi_holmz (2008-08-25 09:00:12)


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#3 2008-08-25 09:32:41

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Line passing through a square

Very nicely done!
I had a go at this but just ended up fumbling about with quadratic equations and trigonometry and didn't really get anywhere.


Why did the vector cross the road?
It wanted to be normal.

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