The problem relates to four points on the unit circle with centroid coincident with the centre of the circle.Four points satisfy the conditions

Show that the points lie either at the vertices of a square inscribed in the unit circle or else coincide in pairs.

As such, what you are asked to show is false. Let the points be the vertices of any rectangle, but not a square, inscribed in the circle. Then the specified properties hold for this configuration, but it is not a square nor do any of the points coincide.

RonL