If G is the centroid of the triangle ABC and O is any other point in the plane then prove that (OA)² + (OB)² + (OC)²=(GA)² + (GB)² + (GC)² + 3(GO)²

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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