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

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#### Solution

Let `bar a , bar b , bar c , bar d` be the position vectors of points A, B, C, D respectively

`bar(AD)+bar(BD)+bar(CD)=barO`

`(bard-bara)+(bard-barb)+(bard-barc)=barO`

`3bard-(bara+barb+barc)=barO`

`3bard=bara+barb+barc`

`bard=(bara+barb+barc)/3`

`bard` represents centroid of the triangle.

Point D is the centroid of the ΔABC.

Concept: Vector and Cartesian Equations of a Line - Centroid Formula for Vector

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