# 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

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### 1 Answer

You should come up with the following notations: `bara` = vector of position of the point A, `barb` = vector of position of the point B, `barc` = vector of position of the point C, `bard` = vector of position of the point AD, `barp` = vector of position of the point P, `barq` = vector of position of the point Q,`barr` = vector of position of the point R, `bars` = vector of position of the point S.

You should use the formula of the mid-point to denote the vectors `barp, barq, barr, bars` such that:

`barp = (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 `barPS = bars - barp` .

`barPS = (barc + bard)/2 - (bara + bard)/2`

`barPS = (barc + bard - bara- bard)/2 =gt barPS = (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.

`barQR = barr - barq =gt barQR = (barb+barc)/2 - (bara + barb)/2`

`` `barQR =(barc - bara)/2`

Since the vector QR is a scalar multiple of PS, then `barQR||barPS` .

**Hence, since `barQR||barPS` , also `barQR=barPS` , then PQRS parallelogram.**