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If A, B, C and D are four non-collinear ...

If A, B, C and D are four non-collinear points in the plane such that `bar(AD)+bar(BD)+bar(CD)=bar(0)`, then prove that the points D is the centroid of the triangle ABC.

Text Solution

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Let `bar(a),bar(b),bar(c),bar(d)` be the position vectors of A,B,C,D respectively.
`:.bar(AD)+bar(BD)=bar(CD)=bar(0)` gives
`(bar(d)-bar(a))+(bar(d)-bar(b))+(bar(d)-bar(c))=bar(0)`
`:.3bar(d)-(bar(a)+bar(b)+bar(c))=bar(0)`
`:.3bar(d)=bar(a)+bar(b)+bar(c)`
`:.bar(d)=(bar(a)+bar(b)+bar(c))/(3)`
Hence, D is the centroid of the `triangle ABC`.
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