Dear Sohan
You can do it by coordinates and vectors both methods
vector method is simple and involves less variables
assume o as origin
Position vectors of A,B,C as a,b,c respectively
OA=a, OB=b,OC=c
G=a+b+c/3
GA=a-(a+b+c)/3=2a-b-c/3
similarly you can write for GB,GC,GO
GA2=(4a2+b2+c2-4a.b+b.c-4a.c)/9
similarly write for others and just calculate the expression
similarly using coordinate 
you can take o as (0,0)and A,B,C as (x1,y1),(x2,y2)and (x3,y3)
G will be ((x1+x2+x3)/3 , (y1+y2+y3)/3)
and using distance formula solve the expressions on LHS and RHS
 
 All the best.                                                          
AKASH GOYAL
AskiitiansExpert-IIT Delhi
 
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if G is the centroid of the triangle ABC, prove by vector method that
GA+GB+GC=0