Prove using vectors: The diagonals of a quadrilateral bisect each other iff it is a parallelogram.

Explanation

Let `ABCD` is a parallelogram and the position vectors of `A,B,C,D` be `veca,vecb,vecc,vecd` respectively.

`vec{AB}=vec{DC}`

`implies vecb-veca=vecc-vecd`

`implies vecb+vecd=veca+vecc`

`implies 1/2(vecb+vecd)=1/2(veca+vecc)`

`implies` Diagonals of a parallelogram bisects each other.

Conversely,

Let diagonals of a quadrilateral `ABCD` with position vectors of `A,B,C,D` be `veca,vecb,vecc,vecd` respectively, bisects each other.

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