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If G and Gprime are the centroids of the...

If `G and Gprime` are the centroids of the triangle `ABC and AprimeBprimeCprime,` then the value of `bar(A Aprime)+bar(B Bprime)+bar(C Cprime)` equals

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Let `bar(a),bar(b),bar(c),bar(a'),bar(b'),bar(c'),bar(g)` and `bar(g')` be the position vectors of the points A,B,C,A',B',C,G and G' respectively. G and G' are the centroids of `triangleABC` and `triangleA'B'C'` respectively`.
`:.` by the centroid formula,
`bar(g)=(bar(a)+bar(b)+bar(c))/(3) andbar(g')=(bar(a')+bar(b')+bar(c'))/(3)`
`:.bar(a)+bar(b)+bar(c)=3bar(g)andbar(a')+bar(b')+bar(c')=3bar(g')` . . . (1)
Now `bar(A A')=bar(a')=bar(a'),bar(BB')=bar(b')-bar(b),bar(C C')=bar(c')=bar(c)andbar(GG')=bar(g')=bar(g)`
`:.bar(A A')+bar(BB')+bar(C C')=(bar(a')-bar(a))+(bar(b')-bar(b))+(bar(c')-bar(c))`
`=(bar(a')+bar(b')+bar(c'))-(bar(a)+bar(b)+bar(c))`
`=3bar(g')-3bar(g)`
`=3(bar(g')-bar(g))` . . . [By (1)]
`=3bar(GG')`.
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