Midpoint

In geometry, the midpoint is the middle point of a line segment.

is given by That is, the ith coordinate of the midpoint (i = 1, 2, ..., n) is Given two points of interest, finding the midpoint of the line segment they determine can be accomplished by a compass and straightedge construction.

The midpoint of a line segment, embedded in a plane, can be located by first constructing a lens using circular arcs of equal (and large enough) radii centered at the two endpoints, then connecting the cusps of the lens (the two points where the arcs intersect).

It is more challenging to locate the midpoint using only a compass, but it is still possible according to the Mohr-Mascheroni theorem.

The synthetic affine definition of the midpoint M of a segment AB is the projective harmonic conjugate of the point at infinity, P, of the line AB.

[6] When coordinates can be introduced in an affine geometry, the two definitions of midpoint will coincide.

However, fixing a point at infinity defines an affine structure on the projective line in question and the above definition can be applied.

Note that, unlike in the affine case, the midpoint between two points may not be uniquely determined.