Ingeometry,bisection is the division of something into two equal orcongruent parts (having the same shape and size). Usually it involves a bisectingline, also called abisector. The most often considered types of bisectors are thesegment bisector, a line that passes through themidpoint of a givensegment, and theangle bisector, a line that passes through theapex of anangle (that divides it into two equal angles).Inthree-dimensional space, bisection is usually done by a bisectingplane, also called thebisector.

• Theperpendicular bisector of a line segment is a line which meets the segment at itsmidpoint perpendicularly.
• The perpendicular bisector of a line segment${\displaystyle AB}$  also has the property that each of its points${\displaystyle X}$  isequidistant from segment AB's endpoints:

(D)${\displaystyle |XA|=|XB|}$ .

The proof follows from${\displaystyle |MA|=|MB|}$  andPythagoras' theorem:

${\displaystyle |XA{|}^2=|XM{|}^2+|MA{|}^2=|XM{|}^2+|MB{|}^2=|XB{|}^2.}$ 

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

In classical geometry, the bisection is a simplecompass and straightedge construction, whose possibility depends on the ability to drawarcs of equal radii and different centers:

The segment${\displaystyle AB}$  is bisected by drawing intersecting circles of equal radius${\displaystyle r>\frac{1}{2}|AB|}$ , 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.
Because the construction of the bisector is done without the knowledge of the segment's midpoint${\displaystyle M}$ , the construction is used for determining${\displaystyle M}$  as the intersection of the bisector and the line segment.

This construction is in fact used when constructing aline perpendicular to a given line${\displaystyle g}$  at agiven point${\displaystyle P}$ : drawing a circle whose center is${\displaystyle P}$  such that it intersects the line${\displaystyle g}$  in two points${\displaystyle A,B}$ , and the perpendicular to be constructed is the one bisecting segment${\displaystyle AB}$ .

If${\displaystyle \overrightarrow{a},\overrightarrow{b}}$  are the position vectors of two points${\displaystyle A,B}$ , then its midpoint is${\displaystyle M:\overrightarrow{m}=\frac{\overrightarrow{a}+\overrightarrow{b}}{2}}$  and vector${\displaystyle \overrightarrow{a}-\overrightarrow{b}}$  is anormal vector of the perpendicular line segment bisector. Hence its vector equation is${\displaystyle (\overrightarrow{x}-\overrightarrow{m})\cdot (\overrightarrow{a}-\overrightarrow{b})=0}$ . Inserting${\displaystyle \overrightarrow{m}=\cdots }$  and expanding the equation leads to the vector equation

(V)${\displaystyle \overrightarrow{x}\cdot (\overrightarrow{a}-\overrightarrow{b})=\frac{1}{2}({\overrightarrow{a}}^2-{\overrightarrow{b}}^2).}$ 

With${\displaystyle A=(a_1,a_2),B=(b_1,b_2)}$  one gets the equation in coordinate form:

(C)${\displaystyle (a_1-b_1)x+(a_2-b_2)y=\frac{1}{2}(a_1^2-b_1^2+a_2^2-b_2^2).}$ 

Or explicitly:
(E)${\displaystyle y=m(x-x_0)+y_0}$ ,
where${\displaystyle m=-\frac{b_1-a_1}{b_2-a_2}}$ ,${\displaystyle x_0=\frac{1}{2}(a_1+b_1)}$ , and${\displaystyle y_0=\frac{1}{2}(a_2+b_2)}$ .

Perpendicular line segment bisectors were used solving various geometric problems:

1. Construction of the center of aThales' circle,
2. Construction of the center of theExcircle of a triangle,
3. Voronoi diagram boundaries consist of segments of such lines or planes.

• Theperpendicular bisector of a line segment is aplane, which meets the segment at itsmidpoint perpendicularly.

Its vector equation is literally the same as in the plane case:

(V)${\displaystyle \overrightarrow{x}\cdot (\overrightarrow{a}-\overrightarrow{b})=\frac{1}{2}({\overrightarrow{a}}^2-{\overrightarrow{b}}^2).}$ 

With${\displaystyle A=(a_1,a_2,a_3),B=(b_1,b_2,b_3)}$  one gets the equation in coordinate form:

(C3)${\displaystyle (a_1-b_1)x+(a_2-b_2)y+(a_3-b_3)z=\frac{1}{2}(a_1^2-b_1^2+a_2^2-b_2^2+a_3^2-b_3^2).}$ 

Property(D) (see above) is literally true in space, too:
(D) The perpendicular bisector plane of a segment${\displaystyle AB}$  has for any point${\displaystyle X}$  the property:${\displaystyle |XA|=|XB|}$ .

Anangle bisector divides theangle into two angles withequal 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,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 thesupplementary 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.^[1]

To bisect an angle withstraightedge 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. Thetrisection of an angle (dividing it into three equal parts) cannot be achieved with the compass and ruler alone (this was first proved byPierre Wantzel).

The internal and external bisectors of an angle areperpendicular. If the angle is formed by the two lines given algebraically as${\displaystyle l_{1}x+m_{1}y+n_1=0}$  and${\displaystyle l_{2}x+m_{2}y+n_2=0,}$  then the internal and external bisectors are given by the two equations^[2]: p.15

${\displaystyle \frac{l_{1}x+m_{1}y+n_1}{\sqrt{l_1^2+m_1^2}}=\pm \frac{l_{2}x+m_{2}y+n_2}{\sqrt{l_2^2+m_2^2}}.}$ 

The bisectors of twoexterior angles and the bisector of the otherinterior angle are concurrent.^[3]: p.149

Three intersection points, each of an external angle bisector with the oppositeextended side, arecollinear (fall on the same line as each other).^{[3]: 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.^{[3]: p. 149}

The angle bisector theorem is concerned with the relativelengths of the two segments that atriangle'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.

If the side lengths of a triangle are${\displaystyle a,b,c}$ , the semiperimeter${\displaystyle s=(a+b+c)/2,}$  and A is the angle opposite side${\displaystyle a}$ , then the length of the internal bisector of angle A is^{[3]: p. 70}

${\displaystyle \frac{2\sqrt{bcs(s-a)}}{b+c},}$ 

or in trigonometric terms,^[4]

${\displaystyle \frac{2bc}{b+c}\cos \frac{A}{2}.}$ 

If the internal bisector of angle A in triangle ABC has length${\displaystyle t_a}$  and if this bisector divides the side opposite A into segments of lengthsm andn, then^[3]: p.70

${\displaystyle t_a^2+mn=bc}$ 

whereb andc are the side lengths opposite vertices B and C; and the side opposite A is divided in the proportionb:c.

If the internal bisectors of angles A, B, and C have lengths${\displaystyle t_a,t_b,}$  and${\displaystyle t_c}$ , then^[5]

${\displaystyle \frac{(b+c{)}^2}{bc}{t}_a^2+\frac{(c+a{)}^2}{ca}{t}_b^2+\frac{(a+b{)}^2}{ab}{t}_c^2=(a+b+c{)}^2.}$ 

No two non-congruent triangles share the same set of three internal angle bisector lengths.^[6][7]

There existinteger triangles with a rational angle bisector.

The internal angle bisectors of aconvexquadrilateral either form acyclic quadrilateral (that is, the four intersection points of adjacent angle bisectors areconcyclic),^[8] or they areconcurrent. In the latter case the quadrilateral is atangential quadrilateral.

Each diagonal of arhombus bisects opposite angles.

The excenter of anex-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 theextensions of opposite sides intersect.

Thetangent to aparabola at any point bisects the angle between the line joining the point to the focus and the line from the point andperpendicular to the directrix.

Each of the threemedians of a triangle is a line segment going through onevertex 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 thecentroid of the triangle, which is itscenter 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.

The interiorperpendicular 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 thecircumcenter (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 anacute triangle the circumcenter divides the interior perpendicular bisectors of the two shortest sides in equal proportions. In anobtuse 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.^{[9]: Corollaries 5 and 6}

For any triangle the interior perpendicular bisectors are given by${\displaystyle p_a=\frac{2aT}{a^2+b^2-c^2},}$ ${\displaystyle p_b=\frac{2bT}{a^2+b^2-c^2},}$  and${\displaystyle p_c=\frac{2cT}{a^2-b^2+c^2},}$  where the sides are${\displaystyle a\ge b\ge c}$  and the area is${\displaystyle T.}$ ^{[9]: Thm 2}

The twobimedians of aconvexquadrilateral 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.^{[10]: 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 iscyclic (inscribed in a circle), these maltitudes areconcurrent at (all meet at) a common point called the "anticenter".

Brahmagupta's theorem states that if a cyclic quadrilateral isorthodiagonal (that is, hasperpendiculardiagonals), then the perpendicular to a side from the point of intersection of the diagonals always bisects the opposite side.

Theperpendicular bisector construction forms a quadrilateral from the perpendicular bisectors of the sides of another quadrilateral.

There is an infinitude of lines that bisect thearea of atriangle. Three of them are themedians of the triangle (which connect the sides' midpoints with the opposite vertices), and these areconcurrent at the triangle'scentroid; 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${\displaystyle \sqrt{2}+1:1}$ .^[11] 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.

Theenvelope of the infinitude of area bisectors is adeltoid (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).^[11] 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.[1]The sides of the deltoid are arcs ofhyperbolas that areasymptotic to the extended sides of the triangle.^[11] The ratio of the area of the envelope of area bisectors to the area of the triangle is invariant for all triangles, and equals${\displaystyle \frac{3}{4}{\log }_e(2)-\frac{1}{2},}$  i.e. 0.019860... or less than 2%.

Acleaver of a triangle is a line segment that bisects theperimeter of the triangle and has one endpoint at the midpoint of one of the three sides. The three cleaversconcur at (all pass through) thecenter of the Spieker circle, which is theincircle of themedial triangle. The cleavers are parallel to the angle bisectors.

Asplitter 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 theNagel 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 itsincircle). 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.^[12]

Any line through the midpoint of aparallelogram bisects the area^[11] and the perimeter.

All area bisectors and perimeter bisectors of a circle or other ellipse go through thecenter, and anychords through the center bisect the area and perimeter. In the case of a circle they are thediameters of the circle.

Thediagonals of a parallelogram bisect each other.

If a line segment connecting the diagonals of a quadrilateral bisects both diagonals, then this line segment (theNewton Line) is itself bisected by thevertex centroid.

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^{[13][14]: pp.89–90}

• The Angle Bisector atcut-the-knot
• Angle Bisector definition. Math Open Reference With interactive applet
• Line Bisector definition. Math Open Reference With interactive applet
• Perpendicular Line Bisector. With interactive applet
• Animated instructions for bisecting an angle andbisecting a line Using a compass and straightedge
• Weisstein, Eric W."Line Bisector".MathWorld.

This article incorporates material from Angle bisector onPlanetMath, which is licensed under theCreative Commons Attribution/Share-Alike License.