Midpoint

For other uses, see Midpoint (disambiguation).

The midpoint of the segment (

x

₁,

y

₁) to (

x

₂,

y

₂)

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.

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 i^th coordinate of the midpoint (i=1, 2, ..., n) is

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

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Construction

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.^[1]

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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.^[2]

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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