Given median of a triangle divides the third side into two equal parts. So E and F are the mid points of side AC and AB respectively.
Const: JOIN EF
Proof ; As we know that the line joining the mid points of 2 sides of a triangle is // to the third side
Triangles on the same base and between same // lines are equal in area.
therefore, ar(BCF)= ar(BCE)
implies ar(BCG)+ ar(CEG) = ar(BCG) + ar(BFG)
implies ar(CEG)= ar(BFG) 1)
Now the median of triangle divides the triangle into 2 equal areas
BE is the median of triangle ABC
implies ar(BCG) + ar(CEG)= ar(BFG) + ar(AFGE)
implies ar(BCG) + ar(CEG)= ar(CEG) + ar(AFGE) from 1
hence ar(BCG) = ar(AFGE)