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Geometry Level 3

Let ABCDEF be any hexagon , and let $A_{1}B_{1}C_{1}D_{1}E_{1}F_{1}$ be the hexagon of the centroid of the triangles ABC ,BCD ,CDE ,DEF ,EFA ,FAB . Then the $A_{1}B_{1}C_{1}D_{1}E_{1}F_{1}$ has parallel and equal opposite sides. What is true about this statement???

The statement is true for some special cases!!! The statement seems to be true but actually it is false always!!! Depends on the type of hexagon taken!!! The hexagon is regular!!! No doubt it's Always true!!!

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Adarsh Adi
Mar 7, 2018

Let's take it as a question of vector geometry then we have to prove that vector $A_{1}B_{1}$ = vector $E_{1}D_{1}$ or $B_{1}$ - $A_{1}$ = $D_{1}$ - $E_{1}$ indeed we have,. $A_{1}$ = $\frac{A+B+C}{3}$ , $B_{1}$ = $\frac{B+C+D}{3}$ , $D_{1}$ = $\frac{D+E+F}{3}$ , $E_{1}$ = $\frac{E+F+A}{3}$ apply this and get the proof !!![Remember the properties of vector are applied to centroid of triangles]

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