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

$$IfD,EandFaremidpointsofthesides$$$$BC,CAandABrespectivelyofthe△ABC$$$$andObeanypoint.Provethat$$$$O\overrightarrow{A}+O\overrightarrow{B}+O\overrightarrow{C}=O\overrightarrow{D}+O\overrightarrow{E}+O\overrightarrow{F}$$

Answered by Kunal12588 last updated on 09/Jun/19

$$\mathrm{In}△\mathrm{AOF}$$$$\overrightarrow{\mathrm{OA}}+\overrightarrow{\mathrm{AF}}+\overrightarrow{\mathrm{FO}}=0\left(1\right)$$$$\mathrm{In}△\mathrm{BOF}$$$$\overrightarrow{\mathrm{OB}}+\overrightarrow{\mathrm{BF}}+\overrightarrow{\mathrm{FO}}=0\left(2\right)$$$$\mathrm{adding}\left(1\right)\mathrm{and}\left(2\right)$$$$\overrightarrow{\mathrm{OA}}+\overrightarrow{\mathrm{OB}}+(\overrightarrow{\mathrm{AF}}+\overrightarrow{\mathrm{BF}})+2\cdot \overrightarrow{\mathrm{FO}}=0$$$$\Rightarrow 2\cdot \overrightarrow{\mathrm{OF}}=\overrightarrow{\mathrm{OA}}+\overrightarrow{\mathrm{OB}}[\because \overrightarrow{\mathrm{AF}}+\overrightarrow{\mathrm{BF}}=0,\overrightarrow{\mathrm{OF}}=-\overrightarrow{\mathrm{FO}}]$$$$\mathrm{similarly}$$$$2\cdot \overrightarrow{\mathrm{OD}}=\overrightarrow{\mathrm{OB}}+\overrightarrow{\mathrm{OC}}$$$$2\cdot \overrightarrow{\mathrm{OE}}=\overrightarrow{\mathrm{OA}}+\overrightarrow{\mathrm{OC}}$$$$\mathrm{adding}\mathrm{blue}\mathrm{ones}$$$$2(\overrightarrow{\mathrm{OD}}+\overrightarrow{\mathrm{OE}}+\overrightarrow{\mathrm{OF}})=2(\overrightarrow{\mathrm{OA}}+\overrightarrow{\mathrm{OB}}+\overrightarrow{\mathrm{OC}})$$$$\Rightarrow \overrightarrow{\mathrm{OD}}+\overrightarrow{\mathrm{OE}}+\overrightarrow{\mathrm{OF}}=\overrightarrow{\mathrm{OA}}+\overrightarrow{\mathrm{OB}}+\overrightarrow{\mathrm{OC}}\mathrm{proved}$$

Commented by Kunal12588 last updated on 09/Jun/19

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