Suppose $\overline{u},\overline{v},\overline{w},\overline{x}$ are four binary vectors, pairwise distance d apart. Show that d must be even, there's exactly one vector which is a distance $d\over 2$ from each of $\overline{u},\overline{v},\overline{w}$ and there's atmost one vector which is at a distance $d\over 2$ from each of $\overline{u},\overline{v},\overline{w},\overline{x}$.
I know that i have to use the triangular inequality somewhere and perhaps i only need three of the vectors-and may be let them be equidistant...- but i got no idea how to apply it or proceed with the proof. Any suggestions, ideas are welcome. it could be text for reading. Thanks alot.