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in Mathematics by (54.5k points)

If ABCDEF is a regular hexagon, and G is its centre, then prove that vector{AB+ AC+ AD+ AE + AF} = 3 vector AD = 6 vector AG .

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In the hexagon ABCDEF, the centre G bisects AD, BE, CF

vector AD = 2 vector AG

vector{AB + AC + AD + AC + AF} 

= vector{AB + AD + DC + AD  + DE + AF}

= vector{(AB + DE) + (AF + DC) + 3AD} 

= vector{(AB + BA) + (AF+ FA) + 3AD}

= 3AD 

= 6AG

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