Question

By vector method, show that in a parallelogram, the sum of the squares of the diagonals is twice the sum of the squares of its adjacent sides.

Answer 1

Let ABCD be a parallelogram

vector{AC2 + BD2} = vector{(AB + BC)2 + (BC + CD)2}

= vector{AB2 + BC2 + 2AB.BC + BC2 + CD2 + 2BC.CD}

= vector{AB2 + DC2 + 2BC2 + 2AB.BC + 2BC.CD}

= vector{2AB2 + 2BC2 +2BC . (AB - DC)}

= 2 vector AB2 + 2 vector BC2