`A B C D`are fourpoints in a plane and `Q`is the pointof 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.

`A B C D`are fourpoints in a plane and `Q`is the pointof intersection

1.	`A B C D`are fourpoints in a plane and `Q`is the pointof 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.
Answer» 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` `vec(PA)+vec(PB)+vec(PC)+vec(PD)=4vec(PQ)` `veca+vecb+vecc+vecd=4[(veca+veccb+vecc+vecd)/4]` Hence proved.

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`A B C D`are fourpoints in a plane and `Q`is the pointof 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.

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