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![[Post New]](/templates/default/images/icon_minipost_new.gif) 23 Feb 2007 14:11:59 IST
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Hi,
If  then prove that the difference of the arguments of atleast two of the complex numbers is greater than  . Salute promised!
thanks.
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Guide to latex:
http://www.goiit.com/posts/list/community-shelf-a-guide-to-latex-48056.htm
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18 Jan 2008 19:34:00 IST
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hey if arg of all d cn differ by 2 /3 and the mod of each cn is same then wat u r asking is wrong
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z1 , z2 and z3 are the sides of a triangle. geometrically, u can see that the difference of the arguments is the external angle of each vertex of the triangle .
since the sum of the exterior angles = 2pi , there should be atleast one angle which is greater than or equal to 2pi/3
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6 Feb 2008 07:00:58 IST
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Just to elaborate on nadeem's post: If z1+z2+z3 = 0 then we can find complex numbers a, b and c such that z1=a-b; z2 = b-c and z3 = c-a. On the Argand Plane, let these be represented by the points A, B and C. Now, it is easy to see that the angle between any two zi is the exterior angle at the vertices A, B and C. Let the interior angles be  1, 2 and 3. Now let's assume that none of the external angles are greater than 2 /3 Summing up, 1+ 2+ 3 < which contradicts 1+ 2+ 3 = . Hence atleast one exterior angle is greater than 2 /3.
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Time wounds all heels |
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6 Feb 2008 11:12:23 IST
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hey z1 z2 z3 represent compl. no. of a triangle which has one of its vertices at origin right.............. mein ne bohat koshish kiya par finally came to d conc. that this quest. may b incorrect.............. what others have replied i dont clearly understand.............
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6 Feb 2008 21:09:37 IST
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maybe we should make it protocol that the poser of a problem must close the thread with his remarks
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6 Feb 2008 21:24:58 IST
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 yes sorry for the late reply. I didnt notice this thread had got replies. This was a question I asked last year! But no one had seemed to notice that a counter example exists when all three complex numbers are 0. They have to be distinct.
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Guide to latex:
http://www.goiit.com/posts/list/community-shelf-a-guide-to-latex-48056.htm
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6 Feb 2008 21:32:00 IST
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I hadnt noticed how ancient the problem was till u mentioned it. our frnd bhuwana has enthusiastically dug up the problem to see it doesn't get corroded.
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Time wounds all heels |
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Moderation Team
![[Up]](/templates/default/images/icon_up.gif "[Up]"){kind=icon}
. Use cosine rule to prove that the angle between segments 
and 
is at least 
. 
