ABCD is a parallelogram, with P,Q,R and S the midpoints of AB, BC, CD and DA, respectively.Use Vector methods to prove that PQRS is a parallelogram

You should come up with the following notations: =
vector of position of the point A,   = vector of position of the point B,  
= vector of position of the point C,   = vector of position of the point AD,
  = vector of position of the point P,   = vector of position of the point
Q,  = vector of position of the point R,   = vector of position of the point
S.

You should use the formula of the mid-point to denote
the vectors   such that:

(bara + bard)/2 ; barq = (bara + barb)/2 ; barr = (barb+barc)/2 ; bars = (barc +
bard)/2

Join the points P and S and express the vector
.

bard)/2

(barc - bara)/2

Join the points Q and R. If you prove that
the vector QR is parallel and equal to the vector PS, then PQRS is
parallelogram.

(barb+barc)/2 - (bara + barb)/2

 
bara)/2

Since the vector QR is a scalar multiple of PS,
then .

Hence, since
, also , then PQRS
parallelogram.