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2016-05-03T20:53:37+05:30

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In a triangle there are many centers,   Incenter, circum center, center of gravity - centroid,    and  so  on.... The centroid is not at the same distance from the three vertices.. so is incenter.. incenter is equidistant from the sides...

I suppose the given triangle is an equilateral triangle ABC with mid points of the sides AC, AB and BC  being D, E and F respectively.  The medians BD, CE and AF intersect at G.

Since that is an equilateral triangle, the center is the centroid G.

The three medians have equal length.  so  GA = GB = GC.  So the magnitude of the electric fields at the center due to charges at A, B and C are equal..

Now  we prove by vector method that  vectors
           \vec{AG} + \vec(BG)+ \vec{CG} =0

the following are all vectors..

  AG = 2/3 (AB + BC/2)
  BG = 2/3 (BC+CA/2)  
  CG = 2/3(CB+BA/2)
adding :   AG + BG + CG = 2/3 (AB+BC+CB) + 1/3(BC+CA+BA)
                                        = 0
since  resultant field at G =  K * Q / |AG|^2 *  [ AG + BG + CG ] 
                      that is 0.

I think that this is only good for an equilateral triangle.

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