If `A B C D` is quadrilateral and `Ea n dF` are the mid-points of `A Ca n dB D` respectively, prove that ` vec A B+ vec A D` +` vec C B` +` vec C D` =4 ` vec E Fdot`

Explanation

F is the middle point of BD. Therefore,
`" " vec(AB) + vec(AD) = 2 vec(AF)" "` (i)
Similarly, `vec(CB) + vec(CD) = 2 vec(CE)" "` (ii)
Adding (i) and (ii), we get
`vec(AB) + vec(AD) + vec(CB) + vec(CD)`
`" " = 2(vec(AF) + vec(CF) ) = -2( vec(FA) + vec(FC))`
`= -2(vec(FE))` (because E is the midpoint of AC )
`" " = 4 vec(EF)`