Lin4.1.1

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If AB,BE,CF\, are the medians of a triangle,then prove that \quad {\bar  {AB}}+{\bar  {BE}}+{\bar  {CF}}={\bar  {O}}\,

Let A={\bar  {a}},B={\bar  {b}},C={\bar  {c}}\, be the vertices of a tiangle ABC.Let,D,E,F be the midpoints of the sides BC,CA,AB respectively,since AD,BE and CF are given as medians.

D={\frac  {1}{2}}({\bar  {b}}+{\bar  {c}}),E={\frac  {1}{2}}({\bar  {c}}+{\bar  {a}}),F={\frac  {1}{2}}({\bar  {a}}+{\bar  {b}})\,

{\bar  {AD}}={\bar  {D}}-{\bar  {A}}={\frac  {1}{2}}({\bar  {b}}+{\bar  {c}}-2{\bar  {a}})\,

Similarly {\bar  {BE}}={\frac  {1}{2}}({\bar  {c}}+{\bar  {a}}-2{\bar  {b}}),{\bar  {CF}}={\frac  {1}{2}}({\bar  {a}}+{\bar  {b}}-2{\bar  {c}})\,

Therefore {\bar  {AB}}+{\bar  {BE}}+{\bar  {CF}}={\frac  {1}{2}}({\bar  {b}}+{\bar  {c}}-2{\bar  {a}}+{\bar  {c}}+{\bar  {a}}-2{\bar  {b}}+{\bar  {a}}+{\bar  {b}}-2{\bar  {c}})={\frac  {1}{2}}({\bar  {O}})={\bar  {O}}\,

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