# Bisection

{{Confuse|Dissection}} {{two other uses||the bisection theorem in measure theory|Ham sandwich theorem||Bisect}} [[Image:Bisectors.svg|right|thumb|Line DE bisects line AB at D, line EF is a perpendicular bisector of segment AD at C, and line EF is the interior bisector of right angle AED]] In [[geometry]], '''bisection''' is the division of something into two equal or [[congruence (geometry)|congruent]] parts, usually by a [[line (mathematics)|line]], which is then called a ''bisector''. The most often considered types of bisectors are the ''segment bisector'' (a line that passes through the midpoint of a given segment) and the ''angle bisector'' (a line that passes through the apex of an angle, that divides it into two equal angles). In [[three-dimensional space]], bisection is usually done by a plane, also called the ''bisector'' or ''bisecting plane''. ==Line segment bisector== [[File:PerpendicularBisector.svg|right|700px|Bisection of a line segment using a compass and ruler]] A [[line segment]] bisector passes through the [[midpoint]] of the segment. Particularly important is the [[perpendicular]] bisector of a segment, which, according to its name, meets the segment at [[right angle]]s. The perpendicular bisector of a segment also has the property that each of its points is [[equidistant]] from the segment's endpoints. Therefore [[Voronoi diagram]] boundaries consist of segments of such lines or planes. In classical geometry, the bisection is a simple [[compass and straightedge]], whose possibility depends on the ability to draw [[circle]]s of equal radii and different centers. The segment is bisected by drawing intersecting circles of equal radius, whose centers are the endpoints of the segment and such that each circle goes through one endpoint. The line determined by the points of intersection of the two circles is the perpendicular bisector of the segment, since it crosses the segment at its center. This construction is in fact used when constructing a line perpendicular to a given line at a given point: drawing an arbitrary circle whose center is that point, it intersects the line in two more points, and the perpendicular to be constructed is the one bisecting the segment defined by these two points. [[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 bisects the opposite side. ==Angle bisector== [[File:Bisection construction.gif|right|thumb|Bisection of an angle using a compass and straightedge]] An [[angle]] bisector divides the angle into two angles with [[equality (mathematics)|equal]] measures. An angle only has one bisector. Each point of an angle bisector is equidistant from the sides of the angle. The interior bisector of an angle is the line, [[half-line]], or line segment that divides an angle of less than 180° into two equal angles. The exterior bisector is the line that divides the [[supplementary angle]] (of 180° minus the original angle), formed by one side forming the original angle and the extension of the other side, into two equal angles.<ref>[http://mathworld.wolfram.com/ExteriorAngleBisector.html Weisstein, Eric W. "Exterior Angle Bisector." From MathWorld--A Wolfram Web Resource.]</ref> To bisect an angle with [[straightedge and compass]], one draws a circle whose center is the vertex. The circle meets the angle at two points: one on each leg. Using each of these points as a center, draw two circles of the same size. The intersection of the circles (two points) determines a line that is the angle bisector. The proof of the correctness of these two constructions is fairly intuitive, relying on the symmetry of the problem. It is interesting to note that the [[trisecting the angle|trisection of an angle]] (dividing it into three equal parts) cannot be achieved with the compass and ruler alone (this was first proved by [[Pierre Wantzel]]). ===Triangle=== The interior angle bisectors of a [[triangle]] are concurrent in a point called the [[incenter]] of the triangle. ==== Angle bisector theorem ==== {{main|Angle bisector theorem}} [[Image:Triangle ABC with bisector AD.svg|thumb|240px|right|In this diagram, BD:DC = AB:AC.]] The angle bisector theorem is concerned with the relative [[length]]s of the two segments that a [[triangle]]'s side is divided into by a line that bisects the opposite angle. It equates their relative lengths to the relative lengths of the other two sides of the triangle. ====Lengths==== If the side lengths of a triangle are $a,b,c$, the semiperimeter $s=(a+b+c)/2,$ and A is the angle opposite side $a$, then the length of the internal bisector of angle A is<ref name=Johnson>Johnson, Roger A., ''Advanced Euclidean Geometry'', Dover Publ., 2007 (orig. 1929), p. 70.</ref> :$\frac{2 \sqrt{bcs(s-a)}}{b+c}.$ If the internal bisector of angle A in triangle ABC has length $t_a$ and if this bisector divides the side opposite A into segments of lengths ''m'' and ''n'', then<ref name=Johnson/> :$t_a^2+mn = bc$ where ''b'' and ''c'' are the side lengths opposite vertices B and C; and the side opposite A is divided in the proportion ''b'':''c''. If the internal bisectors of angles A, B, and C have lengths $t_a, t_b,$ and $t_c$, then<ref>Simons, Stuart. ''Mathematical Gazette'' 93, March 2009, 115-116.</ref> :$\frac{(b+c)^2}{bc}t_a^2+ \frac{(c+a)^2}{ca}t_b^2+\frac{(a+b)^2}{ab}t_c^2 = (a+b+c)^2.$ No two non-congruent triangles share the same set of three internal angle bisector lengths.<ref>Mironescu, P., and Panaitopol, L., "The existence of a triangle with prescribed angle bisector lengths", ''[[American Mathematical Monthly]]'' 101 (1994): 58–60.</ref><ref>[http://forumgeom.fau.edu/FG2008volume8/FG200828.pdf Oxman, Victor, "A purely geometric proof of the uniqueness of a triangle with prescribed angle bisectors", ''Forum Geometricorum'' 8 (2008): 197–200.]</ref> ====Integer triangles==== There exist [[Integer triangle#Integer triangles with a rational angle bisector|integer triangles with a rational angle bisector]]. ===Quadrilateral=== The internal angle bisectors of a [[Convex polygon|convex]] [[quadrilateral]] either form a [[cyclic quadrilateral]] or they are [[Concurrent lines|concurrent]]. In the latter case the quadrilateral is a [[tangential quadrilateral]]. ====Rhombus==== Each diagonal of a [[rhombus]] bisects opposite angles. ==Bisectors of the sides of a polygon== ===Triangle=== ====Medians==== Each of the three [[Median (geometry)|medians]] of a triangle is a line segment going through one [[Vertex (geometry)#Of a polytope|vertex]] and the midpoint of the opposite side, so it bisects that side (though not in general perpendicularly). The three medians intersect each other at the [[Centroid#Of triangle and tetrahedron|centroid]] of the triangle, which is its [[center of mass]] if it has uniform density; thus any line through a triangle's centroid and one of its vertices bisects the opposite side. The centroid is twice as close to the midpoint of any one side as it is to the opposite vertex. ====Perpendicular bisectors==== The interior [[perpendicular]] bisector of a side of a triangle is the segment, falling entirely on and inside the triangle, of the line that perpendicularly bisects that side. The three perpendicular bisectors of a triangle's three sides intersect at the [[circumcenter]] (the center of the circle through the three vertices). Thus any line through a triangle's circumcenter and perpendicular to a side bisects that side. In an [[acute triangle]] the circumcenter divides the interior perpendicular bisectors of the two shortest sides in equal proportions. In an [[obtuse triangle]] the two shortest sides' perpendicular bisectors (extended beyond their opposite triangle sides to the circumcenter) are divided by their respective intersecting triangle sides in equal proportions.<ref name=Mitchell>Mitchell, Douglas W. (2013), "Perpendicular Bisectors of Triangle Sides", ''Forum Geometricorum'' 13, 53-59.</ref>{{rp|Corollaries 5 and 6}} For any triangle the interior perpendicular bisectors are given by $p_a=\tfrac{2aT}{a^2+b^2-c^2},$ $p_b=\tfrac{2bT}{a^2+b^2-c^2},$ and $p_c=\tfrac{2cT}{a^2-b^2+c^2},$ where the sides are $a \ge b \ge c$ and the area is $T.$<ref name=Mitchell/>{{rp|Thm 2}} ===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 at a point called the "vertex centroid" and are all bisected by this point.<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 are [[Concurrent lines|concurrent]] at (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 bisects the opposite side. The [[perpendicular bisector construction]] forms a quadrilateral from the perpendicular bisectors of the sides of another quadrilateral. ==Area bisectors and perimeter bisectors== ===Triangle=== There are an infinitude of lines that bisect the [[area]] of a [[triangle]]. Three of them are the [[Median (geometry)|medians]] of the triangle (which connect the sides' midpoints with the opposite vertices), and these are [[Concurrent lines|concurrent]] at the triangle's [[centroid]]; indeed, they are the only area bisectors that go through the centroid. Three other area bisectors are parallel to the triangle's sides; each of these intersects the other two sides so as to divide them into segments with the proportions $\sqrt{2}+1:1$.<ref name=Dunn>Dunn, J. A., and Pretty, J. E., "Halving a triangle," ''[[Mathematical Gazette]]'' 56, May 1972, 105-108.</ref> These six lines are concurrent three at a time: in addition to the three medians being concurrent, any one median is concurrent with two of the side-parallel area bisectors. The [[Envelope (mathematics)|envelope]] of the infinitude of area bisectors is a [[Deltoid curve|deltoid]] (broadly defined as a figure with three vertices connected by curves that are concave to the exterior of the deltoid, making the interior points a non-convex set).<ref name=Dunn/> The vertices of the deltoid are at the midpoints of the medians; all points inside the deltoid are on three different area bisectors, while all points outside it are on just one. [http://www.se16.info/js/halfarea.htm] The sides of the deltoid are arcs of [[hyperbola]]s that are [[Asymptote|asymptotic]] to the extended sides of the triangle.<ref name=Dunn/> A [[Cleaver (geometry)|cleaver]] of a triangle is a line segment that bisects the [[perimeter]] of the triangle and has one endpoint at the midpoint of one of the three sides. The three cleavers [[Concurrent lines|concur]] at (all pass through) the [[Spieker center|center of the Spieker circle]], which is the [[incircle]] of the [[medial triangle]]. The cleavers are parallel to the angle bisectors. 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 (the center of its [[incircle]]). There are either one, two, or three of these for any given triangle. A line through the incenter bisects one of the area or perimeter if and only if it also bisects the other.<ref>Kodokostas, Dimitrios, "Triangle Equalizers," ''[[Mathematics Magazine]]'' 83, April 2010, pp. 141-146.</ref> ===Parallelogram=== Any line through the midpoint of a [[parallelogram]] bisects the area<ref>Dunn, J. A., and J. E. Pretty, "Halving a triangle", ''Mathematical Gazette'' 56, May 1972, p. 105.</ref> and the perimeter. ===Circle and ellipse=== All area bisectors and perimeter bisectors of a circle or other ellipse go through the [[Center (geometry)|center]], and any [[Chord (geometry)|chord]]s through the center bisect the area and perimeter. In the case of a circle they are the [[diameter]]s of the circle. ==Bisectors of diagonals== ===Parallelogram=== The [[diagonal]]s of a parallelogram bisect each other. ===Cyclic quadrilateral=== If a line segment connecting the diagonals of a [[cyclic quadrilateral]] bisects both diagonals, then this line segment is itself bisected by the [[Cyclic quadrilateral#Anticenter and collinearities|anitcenter]] (the intersection of the four maltitudes, each of which is perpendicular to one side and bisects the other). ==References== <references/> ==External links== * [http://www.cut-the-knot.org/triangle/ABisector.shtml The Angle Bisector] at [[cut-the-knot]] * [http://www.mathopenref.com/bisectorangle.html Angle Bisector definition. Math Open Reference] With interactive applet * [http://www.mathopenref.com/bisectorline.html Line Bisector definition. Math Open Reference] With interactive applet * [http://www.mathopenref.com/bisectorperpendicular.html Perpendicular Line Bisector.] With interactive applet * [http://www.mathopenref.com/constbisectangle.html Animated instructions for bisecting an angle] and [http://www.mathopenref.com/constbisectline.html bisecting a line] Using a compass and straightedge * {{MathWorld|title=Line Bisector|urlname=LineBisector}} {{PlanetMath attribution|id=3623|title=Angle bisector}} [[Category:Elementary geometry]]