Midpoint

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.

The midpoint of a segment in n-dimensional space whose endpoints are ${\displaystyle A=(a_{1},a_{2},\dots ,a_{n})}$ and ${\displaystyle B=(b_{1},b_{2},\dots ,b_{n})}$ is given by

That is, the i^th 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). 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.

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