Editing Bisection

{{Short description|Division of something into two equal or congruent parts}}
{{distinguish|Dissection}}
{{for-multi|the bisection theorem in measure theory|Ham sandwich theorem|the root-finding method|Bisection method|other uses|Bisect (disambiguation)}}
[[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 (having the same shape and size). Usually it involves a bisecting [[line (mathematics)|line]], also called a ''bisector''. The most often considered types of bisectors are the ''segment bisector'', a line that passes through the [[midpoint]] of a given [[line segment|segment]], and the ''angle bisector'', a line that passes through the [[Apex (geometry)|apex]] of an [[angle]] (that divides it into two equal angles).
In [[three-dimensional space]], bisection is usually done by a bisecting [[plane (geometry)|plane]], also called the ''bisector''.

== Perpendicular line segment bisector ==
{{anchor|Line segment bisector}}

=== Definition ===
[[File:Mittelsenkr-ab-e.svg|thumb|upright=0.8|Perpendicular bisector of a line segment]]
*The [[perpendicular]] bisector of a line segment is a line which meets the segment at its [[midpoint]] perpendicularly.
*The perpendicular bisector of a line segment <math>AB</math> also has the property that each of its points <math>X</math> is [[equidistant]] from segment AB's endpoints:
'''(D)'''<math>\quad |XA| = |XB|</math>.
 
The proof follows from <math>|MA|=|MB|</math> and [[Pythagoras' theorem]]:
:<math>|XA|^2=|XM|^2+|MA|^2=|XM|^2+|MB|^2=|XB|^2 \; .</math>

Property '''(D)''' is usually used for the construction of a perpendicular bisector:

=== Construction by straight edge and compass ===
[[File:Mittelsenkr-ab-konstr-e.svg|thumb|upright=0.8|Construction by straight edge and compass]]
In classical geometry, the bisection is a simple [[compass and straightedge construction]], whose possibility depends on the ability to draw [[arc (geometry)|arc]]s of equal radii and different centers:

The segment <math>AB</math> is bisected by drawing intersecting circles of equal radius <math>r>\tfrac 1 2 |AB|</math>, whose centers are the endpoints of the segment. The line determined by the points of intersection of the two circles is the perpendicular bisector of the segment.<br> 
Because the construction of the bisector is done without the knowledge of the segment's midpoint <math>M</math>, the construction is used for determining <math>M</math> as the intersection of the bisector and the line segment.

This construction is in fact used when constructing a ''line perpendicular to a given line'' <math>g</math> at a ''given point'' <math>P</math>: drawing a circle whose center is <math>P</math> such that it intersects the line <math>g</math> in two points <math>A,B</math>, and the perpendicular to be constructed is the one bisecting segment <math>AB</math>.

=== Equations ===
If <math>\vec a,\vec b</math> are the position vectors of two points <math>A,B</math>, then its midpoint is <math>M: \vec m=\tfrac{\vec a+\vec b}{2}</math> and vector <math>\vec a -\vec b</math> is a [[normal vector]] of the perpendicular line segment bisector. Hence its vector equation is <math>(\vec x-\vec m)\cdot(\vec a-\vec b)=0</math>. Inserting  <math>\vec m =\cdots</math> and expanding the equation leads to the vector equation

'''(V)''' <math>\quad \vec x\cdot(\vec a-\vec b)=\tfrac 1 2 (\vec a^2-\vec b^2) .</math>

With <math>A=(a_1,a_2),B=(b_1,b_2)</math> one gets the equation in coordinate form:

'''(C)''' <math>\quad (a_1-b_1)x+(a_2-b_2)y=\tfrac 1 2 (a_1^2-b_1^2+a_2^2-b_2^2) \; .</math>

Or explicitly:<br>
'''(E)'''<math>\quad y = m(x - x_0)  +y_0</math>, <br>
where <math>\; m = - \tfrac{b_1 - a_1}{b_2 - a_2}</math>, <math>\;x_0 = \tfrac{1}{2}(a_1 + b_1)\;</math>, and <math>\;y_0 = \tfrac{1}{2}(a_2 + b_2)\;</math>.

=== Applications ===
Perpendicular line segment bisectors were used solving various geometric problems:
#Construction of the center of a [[Thales's theorem|Thales' circle]],
#Construction of the center of the [[Excircle]] of a triangle,
#[[Voronoi diagram]] boundaries consist of segments of such lines or planes.

[[File:Mittelloteb-ab-3d-e.svg|thumb|upright=0.8|Bisector plane]]

=== Perpendicular line segment bisectors in space ===
*The [[perpendicular]] bisector of a line segment is a ''plane'', which meets the segment at its [[midpoint]] perpendicularly. 
Its vector equation is literally the same as in the plane case:

'''(V)''' <math>\quad \vec x\cdot(\vec a-\vec b)=\tfrac 1 2 (\vec a^2-\vec b^2) .</math>

With <math>A=(a_1,a_2,a_3),B=(b_1,b_2,b_3)</math> one gets the equation in coordinate form:

'''(C3)''' <math>\quad (a_1-b_1)x+(a_2-b_2)y+(a_3-b_3)z=\tfrac 1 2 (a_1^2-b_1^2+a_2^2-b_2^2+a_3^2-b_3^2) \; .</math>

Property '''(D)''' (see above) is literally true in space, too:<br>
'''(D)''' The perpendicular bisector plane of a segment <math>AB</math> has for any point <math>X</math> the property: <math>\;|XA| = |XB|</math>.

==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' or 'internal bisector' of an angle is the line, [[Ray (geometry)|half-line]], or line segment that divides an angle of less than 180° into two equal angles.  The 'exterior' or 'external 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 this construction is fairly intuitive, relying on the symmetry of the problem. The [[Angle trisection|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]]).

The internal and external bisectors of an angle are [[perpendicular]]. If the angle is formed by the two lines given algebraically as  <math>l_1x+m_1y+n_1=0</math> and <math>l_2x+m_2y+n_2=0,</math> then the internal and external bisectors are given by the two equations<ref>Spain, Barry. ''Analytical Conics'', Dover Publications, 2007 (orig. 1957).</ref>{{rp|p.15}}

:<math>\frac{l_1x+m_1y+n_1}{\sqrt{l_1^2+m_1^2}} = \pm \frac{l_2x+m_2y+n_2}{\sqrt{l_2^2+m_2^2}}.</math>

==Triangle ==
===Concurrencies and collinearities===
[[File:Incircle.svg|The interior angle bisectors of a triangle are [[Concurrent lines|concurrent]] in a point called the [[incenter]] of the triangle, as seen in the diagram.|alt=The interior angle bisectors of a triangle are concurrent in a point called the incenter of the triangle, as seen in the diagram.|thumb]]

The bisectors of two [[exterior angle]]s and the bisector of the other [[interior angle]] are concurrent.<ref name="Johnson" />{{rp|p.149}}

Three intersection points, each of an external angle bisector with the opposite [[extended side]], are [[collinearity|collinear]] (fall on the same line as each other).<ref name=Johnson/>{{rp|p. 149}}

Three intersection points, two of them between an interior angle bisector and the opposite side, and the third between the other exterior angle bisector and the opposite side extended, are collinear.<ref name=Johnson/>{{rp|p. 149}}

==== Angle bisector theorem ====
{{main|Angle bisector theorem}}
[[Image:Triangle ABC with bisector AD.svg|thumb|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 <math>a,b,c</math>, the semiperimeter <math>s=(a+b+c)/2,</math> and A is the angle opposite side <math>a</math>, 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).</ref>{{rp|p. 70}}

:<math> \frac{2 \sqrt{bcs(s-a)}}{b+c},</math>

or in trigonometric terms,<ref>Oxman, Victor. "On the existence of triangles with given lengths of one side and two adjacent angle bisectors", ''Forum Geometricorum'' 4, 2004, 215–218. http://forumgeom.fau.edu/FG2004volume4/FG200425.pdf</ref>

:<math>\frac{2bc}{b+c}\cos \frac{A}{2}.  </math>

If the internal bisector of angle A in triangle ABC has length <math>t_a</math> and if this bisector divides the side opposite A into segments of lengths ''m'' and ''n'', then<ref name=Johnson/>{{rp|p.70}}

:<math>t_a^2+mn = bc</math>

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 <math>t_a, t_b,</math> and <math>t_c</math>, then<ref>Simons, Stuart.  ''Mathematical Gazette'' 93, March 2009, 115-116.</ref>

:<math>\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.</math>

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]] (that is, the four intersection points of adjacent angle bisectors are [[concyclic points|concyclic]]),<ref>

Weisstein, Eric W. "Quadrilateral." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Quadrilateral.html</ref>  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.

====Ex-tangential quadrilateral====

The excenter of an [[ex-tangential quadrilateral]] lies at the intersection of six angle bisectors. These are the internal angle bisectors at two opposite vertex angles, the external angle bisectors (supplementary angle bisectors) at the other two vertex angles, and the external angle bisectors at the angles formed where the [[Extended side|extensions of opposite sides]] intersect.

===Parabola===
{{Main|Parabola#Tangent bisection property}}
[[File:Parabel 2.svg|thumb|right|''BE'' bisects ∠''FEC'']]

The [[tangent]] to a [[parabola]] at any point bisects the angle between the line joining the point to the focus and the line from the point and [[perpendicular]] to the directrix.

==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 a point which is called 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====
{{main|Circumcircle}}
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. http://forumgeom.fau.edu/FG2013volume13/FG201307.pdf</ref>{{rp|Corollaries 5 and 6}}

For any triangle the interior perpendicular bisectors are given by <math>p_a=\tfrac{2aT}{a^2+b^2-c^2},</math> <math>p_b=\tfrac{2bT}{a^2+b^2-c^2},</math> and <math>p_c=\tfrac{2cT}{a^2-b^2+c^2},</math> where the sides are <math>a \ge b \ge c</math> and the area is <math>T.</math><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 of a quadrilateral|perpendicular bisector construction]] forms a quadrilateral from the perpendicular bisectors of the sides of another quadrilateral.

==Area bisectors and perimeter bisectors==

===Triangle===

There is 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 <math>\sqrt{2}+1:1</math>.<ref name=Dunn>{{cite journal | last1 = Dunn | first1 = Jas. A. | last2 = Pretty | first2 = Jas. E. | date = May 1972 | doi = 10.2307/3615256 | issue = 396 | journal = The Mathematical Gazette | jstor = 3615256 | pages = 105–108 | title = Halving a triangle | volume = 56}}</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 [[asymptotic]] to the extended sides of the triangle.<ref name=Dunn/> The ratio of the area of the envelope of area bisectors to the area of the triangle is invariant for all triangles, and equals <math>\tfrac{3}{4} \log_e(2) - \tfrac{1}{2},</math> i.e. 0.019860... or less than 2%.

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 name=Dunn/> 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.

===Quadrilateral===

If a line segment connecting the diagonals of a quadrilateral bisects both diagonals, then this line segment (the [[Newton line|Newton Line]]) is itself bisected by the [[Quadrilateral#Remarkable points and lines in a convex quadrilateral|vertex centroid.]]

==Volume bisectors==

A plane that divides two opposite edges of a tetrahedron in a given ratio also divides the volume of the tetrahedron in the same ratio. Thus any plane containing a bimedian (connector of opposite edges' midpoints) of a tetrahedron bisects the volume of the tetrahedron<ref>Weisstein, Eric W. "Tetrahedron." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Tetrahedron.html</ref><ref>Altshiller-Court, N. "The tetrahedron." Ch. 4 in ''Modern Pure Solid Geometry'': Chelsea, 1979.</ref>{{rp|pp.89–90}}

==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]]