View source for Concurrent lines

{{short description|Lines which intersect at a single point}}
[[File:Concurrent Lines.svg|thumb|Lines A, B and C are concurrent in Y.]]
In [[Geometry|geometry]], [[Line (geometry)|lines]] in a [[Plane (geometry)|plane]] or higher-dimensional [[Space|space]] are '''concurrent''' if they intersect at a single [[Point (geometry)|point]].

The set of all lines through a point is called a ''[[Pencil (geometry)|pencil]]'', and their common intersection is called the ''[[Vertex (geometry)|vertex]]'' of the pencil. In any [[Affine space|affine space]] (including a [[Euclidean space]]) the set of lines parallel to a given line (sharing the same [[Orientation (geometry)|orientation]]) is also called a ''pencil'', and the vertex of each pencil of parallel lines is a distinct [[Point at infinity|point at infinity]]; including these points results in a [[Projective space|projective space]] in which every pair of lines has an intersection.

==Examples==

===Triangles===

In a [[Triangle|triangle]], four basic types of sets of concurrent lines are [[Altitude (triangle)|altitudes]], [[Bisection#Angle bisector|angle bisectors]], [[Median (geometry)|medians]], and [[Bisection#Perpendicular bisectors|perpendicular bisectors]]:

* A triangle's altitudes run from each [[Vertex (geometry)|vertex]] and meet the opposite side at a [[Right angle|right angle]]. The point where the three altitudes meet is the orthocenter.
* Angle bisectors are rays running from each vertex of the triangle and bisecting the associated [[Angle|angle]]. They all meet at the incenter.
* Medians connect each vertex of a triangle to the midpoint of the opposite side. The three medians meet at the [[Centroid|centroid]].
* Perpendicular bisectors are lines running out of the midpoints of each side of a triangle at 90 degree angles. The three perpendicular bisectors meet at the circumcenter.

Other sets of lines associated with a triangle are concurrent as well. For example:

* Any median (which is necessarily a [[Bisection#Area bisectors and perimeter bisectors|bisector of the triangle's area]]) is concurrent with two other area bisectors each of which is parallel to a side.<ref>Dunn, J. A., and Pretty, J. E., "Halving a triangle," ''Mathematical Gazette'' 56, May 1972, 105-108.</ref>
* A [[Cleaver (geometry)|cleaver]] of a triangle is a line segment that [[Bisection#Area bisectors and perimeter bisectors|bisects the perimeter]] of the triangle and has one endpoint at the midpoint of one of the three sides. The three cleavers concur at the center of the [[Spieker circle]], which is the incircle of the [[Medial triangle|medial triangle]].
* A [[Splitter (geometry)|splitter]] of a triangle is a line segment having one endpoint at one of the three vertices of the triangle and bisecting the perimeter. The three splitters concur at the Nagel point of the triangle.
* Any line through a triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter, and each triangle has one, two, or three of these lines.<ref>Kodokostas, Dimitrios, "Triangle Equalizers," ''Mathematics Magazine'' 83, April 2010, pp. 141-146.</ref> Thus if there are three of them, they concur at the incenter.
* The Tarry point of a triangle is the point of concurrency of the lines through the vertices of the triangle perpendicular to the corresponding sides of the triangle's first [[Brocard triangle]].
* The Schiffler point of a triangle is the point of concurrence of the [[Euler line]]s of four triangles: the triangle in question, and the three triangles that each share two vertices with it and have its incenter as the other vertex.
* The Napoleon points and generalizations of them are points of concurrency. For example, the first Napoleon point is the point of concurrency of the three lines each from a vertex to the centroid of the equilateral triangle drawn on the exterior of the opposite side from the vertex. A generalization of this notion is the Jacobi point.
* The de Longchamps point is the point of concurrence of several lines with the [[Euler line]].
* Three lines, each formed by drawing an external equilateral triangle on one of the sides of a given triangle and connecting the new vertex to the original triangle's opposite vertex, are concurrent at a point called the [[Triangle center#1st isogonic center|first isogonal center]]. In the case in which the original triangle has no angle greater than 120°, this point is also the [[Fermat point]].
* The Apollonius point is the point of concurrence of three lines, each of which connects a point of tangency of the circle to which the triangle's excircles are internally tangent, to the opposite vertex of the triangle.

===Quadrilaterals===

*The two [[Quadrilateral#Special line segments|bimedians]] of a [[Quadrilateral|quadrilateral]] (segments joining midpoints of opposite sides) and the line segment joining the midpoints of the diagonals are concurrent and are all bisected by their point of intersection.<ref name=Altshiller-Court/>{{rp|p.125}}
*In a [[Tangential quadrilateral|tangential quadrilateral]], the four angle bisectors concur at the center of the incircle.<ref>Andreescu, Titu and Enescu, Bogdan, ''Mathematical Olympiad Treasures'', Birkhäuser, 2006, pp. 64–68.</ref>
*Other concurrencies of a tangential quadrilateral are given [[Tangential quadrilateral#Concurrent and perpendicular lines|here]].
*In a [[Cyclic quadrilateral|cyclic quadrilateral]], four line segments, each [[Perpendicular|perpendicular]] to one side and passing through the opposite side's [[Midpoint|midpoint]], are concurrent.<ref name=Altshiller-Court>{{citation |first=Nathan |last=Altshiller-Court |title=College Geometry: An Introduction to the Modern Geometry of the Triangle and the Circle |year=2007 |publisher=Courier Dover |isbn=978-0-486-45805-2 |edition=2nd |orig-year=1952 |oclc=78063045 |pages=131, 137–8}}</ref>{{rp|p.131;}}<ref>{{citation |first=Ross |last=Honsberger |title=Episodes in Nineteenth and Twentieth Century Euclidean Geometry |chapter-url=https://books.google.com/books?id=6oduPgvOAhwC&pg=PA35 |year=1995 |publisher=Cambridge University Press |isbn=978-0-88385-639-0 |pages=35–39 |chapter=4.2 Cyclic quadrilaterals |series=New Mathematical Library |volume=37}}</ref> These line segments are called the ''maltitudes'',<ref>{{mathworld|title=Maltitude|urlname=Maltitude}}</ref> which is an abbreviation for midpoint altitude. Their common point is called the ''anticenter''.
*A convex quadrilateral is [[Ex-tangential quadrilateral|ex-tangential]] if and only if there are six concurrent angles bisectors: the internal [[Bisection#Angle bisector|angle bisector]]s at two opposite vertex angles, the external angle bisectors at the other two vertex angles, and the external angle bisectors at the angles formed where the extensions of opposite sides intersect.

===Hexagons===

*If the successive sides of a cyclic [[Hexagon|hexagon]] are ''a'', ''b'', ''c'', ''d'', ''e'', ''f'', then the three main diagonals concur at a single point if and only if {{nowrap|''ace'' {{=}} ''bdf''}}.<ref>Cartensen, Jens, "About hexagons", ''Mathematical Spectrum'' 33(2) (2000-2001), 37-40.</ref>
*If a hexagon has an [[Inscribed figure|inscribed]] conic, then by [[Brianchon's theorem]] its principal [[Diagonal|diagonal]]s are concurrent (as in the above image).
*Concurrent lines arise in the dual of [[Pappus's hexagon theorem]].
*For each side of a cyclic hexagon, extend the adjacent sides to their intersection, forming a triangle exterior to the given side. Then the segments connecting the circumcenters of opposite triangles are concurrent.<ref>Nikolaos Dergiades, "Dao's theorem on six circumcenters associated with a cyclic hexagon", ''Forum Geometricorum'' 14, 2014, 243--246.  http://forumgeom.fau.edu/FG2014volume14/FG201424index.html</ref>

===Regular polygons===

*If a regular polygon has an even number of sides, the [[Diagonal|diagonal]]s connecting opposite vertices are concurrent at the center of the polygon.

===Circles===

*The [[Bisection#Line segment bisector|perpendicular bisectors]] of all [[Circle#Chord|chords]] of a [[Circle|circle]] are concurrent at the center of the circle.
*The lines perpendicular to the tangents to a circle at the points of tangency are concurrent at the center.
*All [[Area|area]] [[Bisection|bisectors]] and [[Perimeter|perimeter]] bisectors of a circle are [[Diameter|diameter]]s, and they are concurrent at the circle's center.

===Ellipses===

*All area bisectors and perimeter bisectors of an [[Ellipse|ellipse]] are concurrent at the center of the ellipse.

===Hyperbolas===

*In a [[Hyperbola|hyperbola]] the following are concurrent: (1) a circle passing through the hyperbola's [[Focus (geometry)|foci]] and centered at the hyperbola's center; (2) either of the lines that are tangent to the hyperbola at the vertices; and (3) either of the asymptotes of the hyperbola.
*The following are also concurrent: (1) the circle that is centered at the hyperbola's center and that passes through the hyperbola's vertices; (2) either directrix; and (3) either of the asymptotes.

===Tetrahedrons===

*In a [[Tetrahedron|tetrahedron]], the four medians and three bimedians are all concurrent at a point called the ''centroid'' of the tetrahedron.<ref>Leung, Kam-tim; and Suen, Suk-nam; "Vectors, matrices and geometry", Hong Kong University Press, 1994, pp. 53-54</ref>
*An [[Tetrahedron#Other special cases|isodynamic tetrahedron]] is one in which the [[Cevian|cevian]]s that join the vertices to the [[Incircle and excircles of a triangle|incenter]]s of the opposite faces are concurrent, and an [[Tetrahedron#Other special cases|isogonic tetrahedron]] has concurrent cevians that join the vertices to the points of contact of the opposite faces with the [[Inscribed sphere|inscribed sphere]] of the tetrahedron.
*In an [[Orthocentric tetrahedron|orthocentric tetrahedron]] the four altitudes are concurrent.

==Algebra==

According to the [[Rouché–Capelli theorem]], a system of equations is consistent if and only if the [[Rank (linear algebra)|rank]] of the [[Coefficient matrix|coefficient matrix]] is equal to the rank of the [[Augmented matrix|augmented matrix]] (the coefficient matrix augmented with a column of intercept terms), and the system has a ''unique'' solution if and only if that common rank equals the number of variables. Thus with two variables the ''k'' lines in the plane, associated with a set of ''k'' equations, are concurrent if and only if the rank of the ''k'' × 2  coefficient matrix and the rank of the ''k'' × 3 augmented matrix are both 2. In that case only two of the ''k'' equations are [[Independent equation|independent]], and the point of concurrency can be found by solving any two mutually independent equations simultaneously for the two variables.

==Projective geometry==

In [[Projective geometry|projective geometry]], in two dimensions concurrency is the [[Duality (projective geometry)|dual]] of [[Collinearity|collinearity]]; in three dimensions, concurrency is the  dual of [[Coplanarity|coplanarity]].

==References==

{{reflist}}

==External links==
* [http://mathworld.wolfram.com/Concurrent.html Wolfram MathWorld Concurrent], 2010.

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[[Category:Elementary geometry]]

{{Sourceattribution|Concurrent lines}}