Q. 64.3( 6 Votes )

If G be the centroid of a triangle ABC, prove that:

AB2 + BC2 + CA2 = 3 (GA2 + GB2 + GC2)

Answer :

We know that centroid of a triangle for (x1 , y1), (x2 , y2) and (x3 , y3) is


G(x, y) = ( , )


We assume centroid of ∆ABC at origin.


For x=0 and y=0


= 0 and = 0


= 0 and =0


Squaring on both sides, we get


x12 + x22 + x32 + 2x1x2 + 2x2x3 + 2x3x1 = 0 and y12 + y22 + y32 + 2y1y2 + 2y2y3 + 2y3y1 = 0 … (1)


AB2 + BC2 + CA2


= [(x2 – x1)2 + (y2 – y1)2] + [(x3 – x2)2 + (y3 – y2)2] + [(x1 – x3)2 + (y1 – y3)2]


= (x12 + x22 – 2x1x2 + y12 + y22 – 2y1y2)+(x22 + x32 – 2x2x3 + y22 + y32 – 2y2y3)+(x12 + x32 – 2x1x3 + y12 + y32 - 2y1y3)


= (2x12 + 2x22 + 2x32 – 2x1x2 – 2x2x3 – 2x1x3) + (2y12 + 2y22 + 2y32 – 2y1y2 – 2y2y3 – 2y1y3)


= (3x12 + 3x22 + 3x32) + (3y12 + 3y22 + 3y32) …from 1


= 3(x12 + x22 + x32) + 3(y12 + y22 + y32) … (2)


3(GA2 + GB2 + GC2)


= 3 [(x1 – 0)2 + (y1 – 0)2 + (x2 – 0)2 + (y2 – 0)2 + (x3 – 0)2 + (y3 – 0)2]


= 3 (x12 + y12 + x22 + y22 + x32 + y32)


= 3 (x12 + x22 + x32) + 3(y12 + y22 + y32) … (3)


From (2) and (3), we get


AB2 + BC2 + CA2 = 3(GA2 + GB2 + GC2)


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