Editing Bisection (section)

==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.