Prove that the quadrilateral formed by joining the mid-points of the pairs of consecutive sides of a quadrilateral is a parallelogram.

Explanation

Let a,B,c and D be the position vector Of A,b,,c and D.
Hence , position vector of mid -points P,Q, R and s will be
`=(b+c)/(2),(b+c)/(2),(c+d)/(2) and (d+a)/(2)`
We have ,
PQ= position vector of Q-position vector of P
`=(b+c)/(2)-(a+b)/(2)=(c-a)/(2)`
and SR= postion vector of R- position vector of S
`=(c+d)/(2)-(d-a)/(2)=(c-a)/(2)`
From Eq.(i)and (ii),We get
PQ=SR
QR=PS
`thereforePQRS` is a parallelogram.