# Midpoint

For other uses, see Midpoint (disambiguation).
The midpoint of the segment (x1, y1) to (x2, y2)

In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment.

## Formulas

The midpoint of a segment in n-dimensional space whose endpoints are $A = (a_1, a_2, \dots , a_n)$ and $B = (b_1, b_2, \dots , b_n)$ is given by

$\frac{A+B}{2}.$

That is, the ith coordinate of the midpoint (i=1, 2, ..., n) is

$\frac{a_i+b_i}{2}.$

## Construction

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). The point where the line connecting the cusps intersects the segment is then the midpoint of the segment. It is more challenging to locate the midpoint using only a compass, but it is still possible according to the Mohr-Mascheroni theorem.[1]

## Generalizations

The midpoint is actually an affine invariant. Hence, the abovementioned formulas for Cartesian coordinates are applicable in any affine coordinate system.

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 other) projective range. Fixing one such point as a midpoint actually defines an affine structure on the projective line containing that range. The projective harmonic conjugate of such a "midpoint" with respect to the two endpoints is the point at infinity.[2]

The definition of the midpoint of a segment may be extended to geodesic arcs on a Riemannian manifold. Note that, unlike in the affine case, the midpoint between two points may not be uniquely determined.