# If A, B, C, D are four non-collinear points in the plane such that AD+ BD+CD=O then prove that point D is the centroid of the ΔABC. - Mathematics and Statistics

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.

#### 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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