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If G is the centroid of the triangle ABC,prove that {\bar  {GA}}+{\bar  {GB}}+{\bar  {GC}}={\bar  {O}}\, where A,B,C\, are the vertices of the triangle ABC and {\bar  {O}}\, is the point vector

Let A={\bar  {a}},B={\bar  {b}},C={\bar  {c}}\, be the vertices of the given triangle.

Then the position vector of the centroid G={\frac  {1}{3}}({\bar  {a}}+{\bar  {b}}+{\bar  {c}})\,

{\bar  {GA}}={\bar  {A}}-{\bar  {G}}={\bar  {a}}-[{\frac  {1}{3}}({\bar  {a}}+{\bar  {b}}+{\bar  {c}})]={\frac  {1}{3}}(2{\bar  {a}}-{\bar  {b}}-{\bar  {c}})\,

Similarly {\bar  {GB}}={\frac  {1}{3}}(2{\bar  {b}}-{\bar  {a}}-{\bar  {c}}),{\bar  {GC}}={\frac  {1}{3}}(2{\bar  {c}}-{\bar  {a}}-{\bar  {b}})\,

{\bar  {GA}}+{\bar  {GB}}+{\bar  {GC}}={\frac  {1}{3}}(2{\bar  {a}}-{\bar  {b}}-{\bar  {c}})+{\frac  {1}{3}}(2{\bar  {b}}-{\bar  {a}}-{\bar  {c}})+{\frac  {1}{3}}(2{\bar  {c}}-{\bar  {a}}-{\bar  {b}})={\frac  {1}{3}}({\bar  {O}})={\bar  {O}}\,

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