The midpoint of a segment in n-dimensional space whose endpoints are and is given by
That is, the ith coordinate of the midpoint (i=1, 2, ..., n) is
Given two points, finding the midpoint is one of the compass and straightedge constructions. The midpoint of a line segment can be located by first constructing a lens using circular arcs, then connecting the cusps of the lens. The point where the cusp-connecting line intersects the segment is then the midpoint. It is more challenging to locate the midpoint using only a compass, but it is still possible.
The midpoint is not defined in projective geometry. Any point inside a projective range may be projectively mapped to any other 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.
- Animation – showing the characteristics of the midpoint of a line segment