Lin4.1.1

From Exampleproblems

Jump to: navigation, search

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}\,

Main Page:Linear Algebra

Retrieved from "http://www.exampleproblems.com/wiki/index.php/Lin4.1.1"
Argan Oil
Natural Skin Care
Organic Skin Care
visitor stats