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Prove that the sum of the three vectors determined by the medians of a triangle directed from the vertices is zero.

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Let \(\vec{a}, \vec{b}\) and \(\vec {c} \) are the position vectors of the vertices A, B and C respectively. Then we know that the position vector of the centroid O of the triangle is \(\frac{\vec{a} + \vec{b} + \vec{c}}{3}\)

Therefore sum of the three vectors \(\vec {OA}, \vec {OB}\) and \(\vec {OC},\) is

Hence, Sum os the three vectors determined by the medians of a triangle directed from the vertices is zero.

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