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If G is the centroid of Delta ABC and G'...

If G is the centroid of `Delta ABC and G' ` is the centroid of `Delta A' B' C' " then " vec(A A)' + vec(B B)' + vec(C C)' = `

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Here,
G is centroid of `DeltaABC` and G' is centroid of `DeltaA'B'C'`, shown as in figure.

Clearly, `A A'=AG+G G'+G'A'` (polygon law)
`B B'=BG+G G'+G'B'`
`C C'=CG+CG'+G'C'`
On adding these
`A A'+B B'+C C'=3G G'+(AG+BG+CG)+(G'A'+G'B'+G'C')`
`=3G G'+(AG+2DG)+(G'A'+2G'D')`
(usingg AD and A'D' as the medians of `DeltaABC` and `DeltaA'B'C'`, respectively).
`=3G G'+(AG+GA)+G'A'+A'G'`
`=3G G'+O+O`
`therefore A A'+B B'+C C'=3G G'`
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