[Vectors] If M is the mid point of AB and N is the midpoint of CD, show that 2MN = AC + BD?

∵ M is midpoint of AB ∴ m↑ = ( a↑ + b↑ ) / 2 .......... (1)

∵ N is midpoint of CD ∴ n↑ = ( c↑ + d↑ ) / 2 ............ (2)

∴ (AC↑) + (BD↑)

= ( c↑ - a↑ ) + ( d↑ - b↑ )

= ( c↑ + d↑ ) - ( a↑ + b↑ )

= 2( n↑ ) - 2( m↑ ) ............... from (1), (2)

= 2( n↑ - m↑ )

= 2( MN↑ ) ................................... Q.E.D.
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