$\mathrm{PQR}$ ত্রিভূজের $Q R, R P$ \& $P Q$ বাহুগুলোর মধ্যবিন্দু যথাক্রমে L, M ও N । $\overrightarrow{\mathrm{PL}}+\overrightarrow{\mathrm{QM}}+\overrightarrow{\mathrm{RN}}=?$

প্রমাণ: QR এর মধ্যবিন্দু L বলে,$\dot{\mathrm{PL}}=\frac{1}{2}(\overrightarrow{\mathrm{PQ}}+\overrightarrow{\mathrm{PR}})$

অনুরূপভাবে,

$\begin{aligned} \text { L.H.S. } & =\overrightarrow{\mathrm{PL}}+\overrightarrow{\mathrm{QM}}+\overrightarrow{\mathrm{RN}} \mathrm{Q} \\ & =\frac{1}{2}(\overrightarrow{\mathrm{PQ}}+\overrightarrow{\mathrm{PR}}+\overrightarrow{\mathrm{QP}}+\overrightarrow{\mathrm{QR}}+\overrightarrow{\mathrm{RP}}+\overrightarrow{\mathrm{RQ}}) \\ & \left.=\frac{1}{2}\{(\overrightarrow{\mathrm{PQ}}+\overrightarrow{\mathrm{QP}})+\overrightarrow{(\mathrm{RQ}}+\overrightarrow{\mathrm{QR}})+(\overrightarrow{\mathrm{RP}}+\overrightarrow{\mathrm{PR}})\right\} \\ & =\frac{1}{2}(\underline{0}+\underline{0}+\underline{0})=\underline{0}=\text { R.H.S. (Proved) } \end{aligned}$