If-D-E-and-F-are-midpoints-of-the-sides-BC-CA-and-AB-respectively-of-the-ABC-and-O-be-any-point-Prove-that-OA-OB-OC-OD-OE-OF-

Question Number 61823 by necx1 last updated on 09/Jun/19

I f D, E a n d F a r e m i d p o i n t s o f t h e s i d e s

B C, C A a n d A B r e s p e c t i v e l y o f t h e △ A B C

a n d O b e a n y p o i n t . P r o v e t h a t

O \vec{A} + O \vec{B} + O \vec{C} = O \vec{D} + O \vec{E} + O \vec{F}

Answered by Kunal12588 last updated on 09/Jun/19

In △ AOF

\vec{OA} + \vec{AF} + \vec{FO} = 0 (1)

In △ BOF

\vec{OB} + \vec{BF} + \vec{FO} = 0 (2)

adding (1) and (2)

\vec{OA} + \vec{OB} + (\vec{AF} + \vec{BF}) + 2 \cdot \vec{FO} = 0

\Rightarrow 2 \cdot \vec{OF} = \vec{OA} + \vec{OB} [∵ \vec{AF} + \vec{BF} = 0, \vec{OF} = - \vec{FO}]

similarly

2 \cdot \vec{OD} = \vec{OB} + \vec{OC}

2 \cdot \vec{OE} = \vec{OA} + \vec{OC}

adding blue ones

2 (\vec{OD} + \vec{OE} + \vec{OF}) = 2 (\vec{OA} + \vec{OB} + \vec{OC})

\Rightarrow \vec{OD} + \vec{OE} + \vec{OF} = \vec{OA} + \vec{OB} + \vec{OC} proved

Commented by Kunal12588 last updated on 09/Jun/19