I'm not sure that the hint is very helpful, and I don't think that stereographic projection is needed.

If such an isometry exists then the points form an equilateral triangle with side 1, in some 2-dimensional subspace of . So do the points . Thus the images of the four points lie in some 3-dimensional subspace of , so we may as well assume that n=3.

The points and must both lie at a distance from the midpoint of the line segment joining and . So the distance from to is at most , contradicting the condition that is an isometry.