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.

Formula

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

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.

Geometric properties involving midpoints

Circle

M is the midpoint of a chord PQ of a circle, through which two other chords AB and CD are drawn; AD and BC intersect chord PQ at X and Y correspondingly, then M is the midpoint of XY .

Ellipse

Hyperbola

Triangle

Quadrilateral

General polygons

Generalizations

The abovementioned formulas for the midpoint of a segment implicitly use the lengths of segments. However, in the generalization to affine geometry, where segment lengths are not defined, the midpoint can still be defined since it is an affine invariant. 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. That is, the point M such that H[A,B; P,M] . When coordinates can be introduced in an affine geometry, the two definitions of midpoint will coincide. The midpoint is not naturally defined in projective geometry since there is no distinguished point to play the role of the point at infinity (any point in a projective range may be projectively mapped to any other point in (the same or some other) projective range). However, fixing a point at infinity defines an affine structure on the projective line in question and the above definition can be applied. The definition of the midpoint of a segment may be extended to curve segments, such as geodesic arcs on a Riemannian manifold. Note that, unlike in the affine case, the midpoint between two points may not be uniquely determined.