$$\begin{gathered} \text{ If }\overline{a},\overline{b},\overline{c}\text{ are non coplanar vectors and }\overline{d}\text{ is such that }\overline{d}=\frac{1}{x}(\overline{a}+\overline{b}+\overline{c})\text{ and }\overline{a}=\frac{1}{y}(\overline{b}+\overline{c}+\overline{d})\text{ where }\mathbf{x}\text{ and }y\text{ are } \\ \text{ non zero real numbers, then }\frac{1}{xy}(\overline{a}+\overline{b}+\overline{c}+\overline{d})= \end{gathered}$$

$$\begin{gathered} \overline{a}+\overline{b}+\overline{c}=x\overline{d}\to 1 \\ \overline{b}+\overline{c}+\overline{d}=y\overline{a}\to 2 \\ \overline{d}=y\overline{a}-(\overline{b}+\overline{c}) \\ \overline{a}+\overline{b}+\overline{c}=xy\overline{a}-x\overline{b}-x\overline{c} \\ (1-xy)\overline{a}+(1+x)\overline{b}+\overline{c}(1+x)\overline{c}=0 \\ x=-1\text{, sub in }1.\overline{a}+\overline{b}+\overline{c}=-\overline{d}\Rightarrow \overline{a}+\overline{b}+\overline{c}+\overline{d}=0 \\ \frac{1}{xy}(\overline{a}+\overline{b}+\overline{c}+\overline{d})=\frac{1}{xy}\left(0\right)=\overline{0} \\ \frac{1}{xy}(\overline{a}+\overline{b}+\overline{c}+\overline{d})=\frac{1}{xy}\left(0\right)=\overline{0} \end{gathered}$$