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In the figure below G is the centroid of triangle ABC prove that triangles AGB,AGC,BGC have the same area?


Mid points of sides BC; AC and AB of  ΔABC are respectively D,E and F. So lines AD, BE and CF are medians of the triangle.
      G is the point of intersection of these medians.  So G is the centroid of Δ ABC. The centroid divides each median of a triangle in the ratio 2 : 1.
      ∴ AG : GD = 2 : 1
      Perpendicular from  A to opposite side BC  cuts BC at M  and the parallel line to BC through G at N.
      So AN : NM = 2 :1
      ∴ AM = AN + NM = 2 NM + NM = 3NM.
      Area of ΔABC = 1/2 BC  x AM.
                              = 1/2 BC x 3NM  = 3 x 1/2  BC x NM
      Area of ΔABC = 1/2 BC x NM
      ∴ Area of   ΔABC = 3 x area of  ΔBGC.
      ie  Area of  ΔBGC = 1/3 area of   ΔABC.
      Like  this we can show that areas of  ΔAGC and ΔAGB are equal to 1/3 are of ΔABC. So areas of   ΔAGB,  ΔAGC and   ΔBGC are equal.


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