Generalizations
The midpoint is actually an affine invariant. Hence, aforementioned formulas for Cartesian coordinates are feasible for any affine coordinate system.
The midpoint is not defined in projective geometry. Any point inside a projective range may be projectively mapped to any another point inside (the same or some else) projective range. Fixing one of such points as a midpoint actually defines an affine structure on the projective line containing that range. The projective harmonic conjugate of the two endpoints together with the midpoint is the point at infinity.
The definition of the midpoint of a segment may be extended to geodesic arcs on a Riemannian manifold. Note that, unlike affine case, the midpoint between two points may be defined not uniquely.
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