`A B C D` are four points in a plane and `Q` is the point of intersection of the lines joining the mid-points of `A B` and `C D ; B C` and `A Ddot` Show that ` vec P A+ vec P B+ vec P C+ vec P D=4 vec P Q ,` where `P` is any point.

Explanation

Let`vec(PA)=veca,vec(PB)=vecb,vec(PC)=vecc,(PD)=vecd`

`vec(PH)=(veca+vecd)/2`

`vec(PF)=(vecb+vecc)/2`

`vec(PE)=(veca+vecb)/2`

`vec(PG)=(vecc+vecd)/2`

Q is the mid point of HF anf EG

`vec(PQ)=(vec(PH)+vec(PF))/2 or (vec(PE)+vec(PG))/2`

`=(veca+vecb+vecc+vecd)/4 or (veca+vecb+vecc+vecb)/4`

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