View source for Midpoint

{{Short description|Point on a line segment which is equidistant from both endpoints}}
{{other uses}}
[[Image:Midpoint.svg|282px|thumb|The midpoint of the segment ({{mvar|x}}<sub>1</sub>, {{mvar|y}}<sub>1</sub>) to ({{mvar|x}}<sub>2</sub>, {{mvar|y}}<sub>2</sub>)]]

In [[geometry]], the '''midpoint''' is the middle [[point (geometry)|point]] of a [[line segment]]. It is [[Distance|equidistant]] from both endpoints, and it is the [[centroid]] both of the segment and of the endpoints. It [[Bisection|bisects]] the segment.

==Formula==

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

:<math>\frac{A+B}{2}.</math>

That is, the ''i''<sup>th</sup> coordinate of the midpoint (''i'' = 1, 2, ..., ''n'') is

:<math>\frac{a_i+b_i} 2.</math>

==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 (geometry)|plane]], can be located by first constructing a [[Lens (geometry)|lens]] using [[circular arc]]s of equal (and large enough) [[radius|radii]] centered at the two endpoints, then connecting the [[cusp (singularity)|cusp]]s 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]].<ref>{{cite web|url=http://mathworld.wolfram.com/Midpoint.html|title=Wolfram mathworld|date=29 September 2010}}</ref>

==Geometric properties involving midpoints==
===Circle===

* The midpoint of any [[diameter]] of a [[circle]] is the center of the circle.

*Any line [[perpendicular]] to any [[chord (geometry)|chord]] of a circle and passing through its midpoint also passes through the circle's center.

*The [[butterfly theorem]] states that, if {{math|''M''}} is the [[midpoint]] of a [[Chord (geometry)|chord]] {{math|''PQ''}} of a [[circle]], through which two other chords {{math|''AB''}} and {{math|''CD''}} are drawn; {{math|''AD''}} and {{math|''BC''}} intersect chord {{math|''PQ''}} at {{math|''X''}} and {{math|''Y''}} correspondingly, then {{math|''M''}} is the midpoint of {{math|''XY''}}.

===Ellipse===

*The midpoint of any segment which is an [[area]] [[bisection|bisector]] or [[perimeter]] bisector of an [[ellipse]] is the ellipse's center.

*The ellipse's center is also the midpoint of a segment connecting the two [[focus (geometry)|foci]] of the ellipse.

===Hyperbola===

*The midpoint of a segment connecting a [[hyperbola]]'s vertices is the center of the hyperbola.

===Triangle===

*The [[bisection#Triangle|perpendicular bisector of a side]] of a [[triangle]] is the line that is perpendicular to that side and passes through its midpoint. The three perpendicular bisectors of a triangle's three sides intersect at the [[circumcenter]] (the center of the circle through the three vertices). 

*The [[median (geometry)|median]] of a triangle's side passes through both the side's midpoint and the triangle's opposite [[vertex (geometry)|vertex]]. The three medians of a triangle intersect at the triangle's [[centroid]] (the point on which the triangle would balance if it were made of a thin sheet of uniform-density metal).

*The [[nine-point center]] of a triangle lies at the midpoint between the circumcenter and the [[orthocenter]]. These points are all on the [[Euler line]].

*A ''midsegment'' (or ''midline'') of a triangle is a line segment that joins the midpoints of two sides of the triangle. It is parallel to the third side and has a length equal to one half of that third side.

*The [[medial triangle]] of a given triangle has vertices at the midpoints of the given triangle's sides, therefore its sides are the three midsegments of the given triangle. It shares the same centroid and medians with the given triangle. The [[perimeter]] of the medial triangle equals the [[semiperimeter]] (half the perimeter) of the original triangle, and its area is one quarter of the area of the original triangle. The [[orthocenter]] (intersection of the [[altitude]]s) of the medial triangle coincides with the [[circumcenter]] (center of the circle through the vertices) of the original triangle.

*Every triangle has an [[inscribed figure|inscribed]] [[ellipse]], called its [[Steiner inellipse]], that is internally tangent to the triangle at the midpoints of all its sides. This ellipse is centered at the triangle's centroid, and it has the largest area of any ellipse inscribed in the triangle.

*In a [[right triangle]], the circumcenter is the midpoint of the [[hypotenuse]].

*In an [[isosceles triangle]], the median, [[Altitude (triangle)|altitude]], and perpendicular bisector from the [[base (geometry)|base]] side and the [[angle bisector]] of the [[Apex (geometry)|apex]] coincide with the Euler line and the [[axis of symmetry]], and these coinciding lines go through the midpoint of the base side.

===Quadrilateral===

*The two [[Quadrilateral#Bimedians|bimedians]] of a [[Convex polygon|convex]] [[quadrilateral]] are the line segments that connect the midpoints of opposite sides, hence each bisecting two sides. The two bimedians and the line segment joining the midpoints of the diagonals are [[Concurrent lines|concurrent]] at (all intersect at)a point called the "vertex centroid", which is the midpoint of all three of these segments.<ref name=Altshiller-Court>Altshiller-Court, Nathan, ''College Geometry'', Dover Publ., 2007.</ref>{{rp|p.125}}

*The four "maltitudes" of a convex quadrilateral are the perpendiculars to a side through the midpoint of the opposite side, hence bisecting the latter side. If the quadrilateral is [[Cyclic quadrilateral|cyclic]] (inscribed in a circle), these maltitudes all meet at a common point called the "anticenter".

*[[Brahmagupta's theorem]] states that if a cyclic quadrilateral is [[Orthodiagonal quadrilateral|orthodiagonal]] (that is, has [[perpendicular]] [[diagonals]]), then the perpendicular to a side from the point of intersection of the diagonals always goes through the midpoint of the opposite side.

*[[Varignon's theorem]] states that the midpoints of the sides of an arbitrary quadrilateral form the vertices of a [[parallelogram]], and if the quadrilateral is not self-intersecting then the area of the parallelogram is half  the area of the quadrilateral.

*The [[Newton line]] is the line that connects the midpoints of the two diagonals in a convex quadrilateral that is not a parallelogram. The line segments connecting the midpoints of opposite sides of a convex quadrilateral intersect in a point that lies on the Newton line.

===General polygons===

*A [[regular polygon]] has an [[inscribed circle]] which is [[tangent]] to each side of the polygon at its midpoint.

*In a regular polygon with an even number of sides, the midpoint of a [[diagonal]] between opposite vertices is the polygon's center.

*The [[midpoint-stretching polygon]] of a [[cyclic polygon]] {{mvar|P}} (a [[polygon]] whose vertices all fall on the same circle) is another cyclic polygon inscribed in the same circle, the polygon whose vertices are the midpoints of the [[circular arc]]s between the vertices of {{mvar|P}}.<ref name="dhz">{{Citation |last1=Ding |first1=Jiu |last2=Hitt |first2=L. Richard |last3=Zhang |first3=Xin-Min |date=1 July 2003 |title=Markov chains and dynamic geometry of polygons |journal=Linear Algebra and Its Applications |volume=367 |pages=255–270 |doi=10.1016/S0024-3795(02)00634-1 |url=http://www.rhitt.com/research/markov.pdf |access-date=19 October 2011}}.</ref> Iterating the midpoint-stretching operation on an arbitrary initial polygon results in a sequence of polygons whose shapes converge to that of a [[regular polygon]].<ref name="dhz"/><ref>{{Citation |first1=Francisco |last1=Gomez-Martin |first2=Perouz |last2=Taslakian |first3=Godfried T. |last3=Toussaint|author3-link=Godfried Toussaint |year=2008 |contribution=Convergence of the shadow sequence of inscribed polygons|title=18th Fall Workshop on Computational Geometry |isbn=978-84-8181-227-5 |url=http://oa.upm.es/4442/}}</ref>

==Generalizations==

The [[#Formulas|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,<ref>{{citation|last=Fishback|first=W.T.|title=Projective and Euclidean Geometry|edition=2nd|publisher=John Wiley & Sons|year=1969|page=214|isbn=0-471-26053-3}}</ref> the midpoint can still be defined since it is an affine [[invariant (mathematics)|invariant]]. The [[Synthetic geometry|synthetic]] affine definition of the midpoint {{mvar|M}} of a segment {{mvar|AB}} is the [[projective harmonic conjugate]] of the [[point at infinity]], {{mvar|P}}, of the line {{mvar|AB}}. That is, the point {{mvar|M}} such that {{math|H[''A'',''B''; ''P'',''M'']}}.<ref>{{citation|last=Meserve|first=Bruce E.|title=Fundamental Concepts of Geometry|year=1983|orig-year=1955|publisher=Dover|page=156|isbn=0-486-63415-9}}</ref> When coordinates can be introduced in an affine geometry, the two definitions of midpoint will coincide.<ref>{{citation|last=Young|first=John Wesley|title=Projective Geometry|year=1930|publisher=Mathematical Association of America|series=Carus Mathematical Monographs #4|pages= 84–85}}</ref>

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 segment]]s, such as [[geodesic]] [[arc (geometry)|arcs]] on a [[Riemannian manifold]]. Note that, unlike in the affine case, the ''midpoint'' between two points may not be uniquely determined.

==See also==
* {{section link|Center (geometry)|Projective conics}}
* [[Midpoint polygon]]
* {{multi-section link|Bisection|Line segment bisector}}
* {{multi-section link|Numerical integration|Quadrature rules based on interpolating functions}}

==References==
{{Reflist}}

==External links==
*[https://web.archive.org/web/20161212001049/http://www.mathopenref.com/midpoint.html Animation] – showing the characteristics of the midpoint of a line segment

[[Category:Elementary geometry]]
[[Category:Affine geometry]]
[[Category:Analytic geometry]]
[[Category:Point (geometry)]]