Let $\mathrm{D}$ and E be the midpoints of the sides $\mathrm{AC}$ and $\mathrm{BC}$ of a triangle $\mathrm{ABC}$ respectively. If $\mathrm{O}$ is an interior point of the triangle $\mathrm{ABC}$ such that $\overrightarrow{\mathrm{OA}}+\overrightarrow{2\mathrm{OB}}+\overrightarrow{3\mathrm{OC}}=\overrightarrow{0},$ then the area (in sq.units) of the triangle $\mathrm{ODE}$ is 
 

$$\begin{gathered} \overline{\mathrm{OA}}=\overline{\mathrm{a}} \overline{\mathrm{OB}}=\overline{\mathrm{b}} \overline{\mathrm{OC}}=\overline{\mathrm{c}} \\ \overline{\mathrm{OD}}=\frac{\overline{\mathrm{a}}+\overline{\mathrm{c}}}{2} \overline{\mathrm{OE}}=\frac{\overline{\mathrm{b}}+\overline{\mathrm{c}}}{2} \\ \overline{\mathrm{a}}+2\overline{\mathrm{b}}+3\overline{\mathrm{c}}=0 \Rightarrow 2(\overline{\mathrm{a}}\times \overline{\mathrm{b}})=3(\overline{\mathrm{c}}\times \overline{\mathrm{a}}) \\ \overline{\mathrm{b}}\times \overline{\mathrm{a}}+0+3(\overline{\mathrm{b}}\times \overline{\mathrm{c}})=0 \Rightarrow 3(\overline{\mathrm{b}}\times \overline{\mathrm{c}})=\overline{\mathrm{a}}\times \overline{\mathrm{b}} \\ \Rightarrow \overline{\mathrm{a}}\times \overline{\mathrm{b}}=3(\overline{\mathrm{b}}\times \overline{\mathrm{c}})=\frac{3}{2}(\overline{\mathrm{c}}\times \overline{\mathrm{a}}) \end{gathered}$$

$$\begin{gathered} \mathrm{area} ∆ \mathrm{ODE}=\frac{1}{2}\left|\overline{\mathrm{OD}}\times \overline{\mathrm{OE}}\right| \\ =\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\left|\left(\frac{\overline{\mathrm{a}}+\overline{\mathrm{c}}}{2}\right)\times \left(\frac{\overline{\mathrm{b}}\times \overline{\mathrm{c}}}{2}\right)\right| \\ =\frac{1}{8}\left|\overline{\mathrm{a}}\times \overline{\mathrm{b}}+\overline{\mathrm{a}}\times \overline{\mathrm{c}}+\overline{\mathrm{c}}\times \overline{\mathrm{b}}\right| \\ =\frac{1}{8}\left|\overline{\mathrm{a}}\times \overline{\mathrm{b}}-\frac{2}{3}(\overline{\mathrm{a}}\times \overline{\mathrm{b}})-\frac{1}{3}(\overline{\mathrm{a}}\times \overline{\mathrm{b}})\right|=0 \end{gathered}$$