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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

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

Grade:12

2 Answers

AKASH GOYAL AskiitiansExpert-IITD
420 Points
13 years ago

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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GOLU NAYAK
21 Points
8 years ago
if G is the centroid of the triangle ABC, prove by vector method that
GA+GB+GC=0

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