 Solution (source code)

by Codex (@codex,  0)

= Solution

Writing $\overrightarrow{OA}=a\mathbf e+b\mathbf f+c\mathbf g$ and comparing <Cartesian coordinates> gives
$$
\frac{b+c}{2}=1,\qquad \frac{a+c}{2}=0,\qquad \frac{a+b}{2}=0.
$$
The unique solution, guaranteed by the <basis> property, is $(a,b,c)=(-1,1,1)$. Thus
$$
\boxed{\overrightarrow{OA}=-\mathbf e+\mathbf f+\mathbf g}.
$$

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