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An equilibrium point exists at the center of the cube ONLY if all the
charges at the vertices have the same magnitude and the same sign.
Field lines from (or to) each charge lie symmetrically around each body
diagonal.
As they approach the center, they veer away symmetrically on
either side so that the center is devoid of lines.
Does this violate the (textbook) rule that field lines start and end on
charges? If it does, are there any other comparable violations?
No. All the lines end on charges an infinite distance away whose sign is
opposite to the sign of the vertex charges.