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In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, bisection is the division of something into two equal or congruent parts, usually by a
line Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Art ...
, 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 In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...
, that divides it into two equal angles). In
three-dimensional space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informal ...
, bisection is usually done by a plane, also called the ''bisector'' or ''bisecting plane''.


Perpendicular line segment bisector


Definition

*The
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
bisector of a line segment is a line, which meets the segment at its
midpoint In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment. Formula The midpoint of a segment in ''n''-dimens ...
perpendicularly. The Horizontal intersector of a segment AB also has the property that each of its points X is
equidistant A point is said to be equidistant from a set of objects if the distances between that point and each object in the set are equal. In two-dimensional Euclidean geometry, the locus of points equidistant from two given (different) points is ...
from the segment's endpoints:
(D)\quad , XA, = , XB, . The proof follows from and
Pythagoras' theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...
: :, XA, ^2=, XM, ^2+, MA, ^2=, XM, ^2+, MB, ^2=, XB, ^2 \; . Property (D) is usually used for the construction of a perpendicular bisector:


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 arcs of equal radii and different centers: The segment AB is bisected by drawing intersecting circles of equal radius r>\tfrac 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 M, the construction is used for determining M 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'' g at a ''given point'' P: drawing a circle whose center is P such that it intersects the line g in two points A,B, and the perpendicular to be constructed is the one bisecting segment AB.


Equations

If \vec a,\vec b are the position vectors of two points A,B, then its midpoint is M: \vec m=\tfrac and vector \vec a -\vec b is a
normal vector In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve ...
of the perpendicular line segment bisector. Hence its vector equation is (\vec x-\vec m)\cdot(\vec a-\vec b)=0. Inserting \vec m =\cdots and expanding the equation leads to the vector equation (V) \quad \vec x\cdot(\vec a-\vec b)=\tfrac 1 2 (\vec a^2-\vec b^2) . With A=(a_1,a_2),B=(b_1,b_2) one gets the equation in coordinate form: (C) \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) \; . Or explicitly:
(E)\quad y = m(x - x_0) +y_0,
where \; m = - \tfrac, \;x_0 = \tfrac(a_1 + b_1)\;, and \;y_0 = \tfrac(a_2 + b_2)\;.


Applications

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


Perpendicular line segment bisectors in space

*The
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
bisector of a line segment is a ''plane'', which meets the segment at its
midpoint In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment. Formula The midpoint of a segment in ''n''-dimens ...
perpendicularly. Its vector equation is literally the same as in the plane case: (V) \quad \vec x\cdot(\vec a-\vec b)=\tfrac 1 2 (\vec a^2-\vec b^2) . With A=(a_1,a_2,a_3),B=(b_1,b_2,b_3) one gets the equation in coordinate form: (C3) \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) \; . Property (D) (see above) is literally true in space, too:
(D) The perpendicular bisector plane of a segment AB has for any point X the property: \;, XA, = , XB, .


Angle bisector

An
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...
bisector divides the angle into two angles with
equal Equal(s) may refer to: Mathematics * Equality (mathematics). * Equals sign (=), a mathematical symbol used to indicate equality. Arts and entertainment * ''Equals'' (film), a 2015 American science fiction film * ''Equals'' (game), a board game ...
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 the
supplementary angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles are ...
(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. To bisect an angle with
straightedge and compass In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an Idealiz ...
, 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 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 In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
. If the angle is formed by the two lines given algebraically as l_1x+m_1y+n_1=0 and l_2x+m_2y+n_2=0, then the internal and external bisectors are given by the two equations :\frac = \pm \frac.


Triangle


Concurrencies and collinearities

The bisectors of two
exterior angle In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or ) if ...
s and the bisector of the other interior angle are concurrent. Three intersection points, each of an external angle bisector with the opposite extended side, are
collinear In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned o ...
(fall on the same line as each other). 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.


Angle bisector theorem

The angle bisector theorem is concerned with the relative
length Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Inte ...
s of the two segments that a
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colli ...
'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 isJohnson, Roger A., ''Advanced Euclidean Geometry'', Dover Publ., 2007 (orig. 1929). : \frac, or in trigonometric terms, :\frac\cos \frac. 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 :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 :\fract_a^2+ \fract_b^2+\fract_c^2 = (a+b+c)^2. No two non-congruent triangles share the same set of three internal angle bisector lengths.


Integer triangles

There exist integer triangles with a rational angle bisector.


Quadrilateral

The internal angle bisectors of a
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...
quadrilateral In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...
either form a
cyclic quadrilateral In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the ''circumcircle'' or ''circumscribed circle'', and the vertices are said to be '' ...
(that is, the four intersection points of adjacent angle bisectors are concyclic), or they are concurrent. In the latter case the quadrilateral is a tangential quadrilateral.


Rhombus

Each diagonal of a
rhombus In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. Th ...
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 extensions of opposite sides intersect.


Parabola

The
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
to a
parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descri ...
at any point bisects the angle between the line joining the point to the focus and the line from the point and
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
to the directrix.


Bisectors of the sides of a polygon


Triangle


Medians

Each of the three medians of a triangle is a line segment going through one 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 In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ...
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 In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
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 An acute triangle (or acute-angled triangle) is a triangle with three acute angles (less than 90°). An obtuse triangle (or obtuse-angled triangle) is a triangle with one obtuse angle (greater than 90°) and two acute angles. Since a triangle's ang ...
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.Mitchell, Douglas W. (2013), "Perpendicular Bisectors of Triangle Sides", ''Forum Geometricorum'' 13, 53-59. http://forumgeom.fau.edu/FG2013volume13/FG201307.pdf For any triangle the interior perpendicular bisectors are given by p_a=\tfrac, p_b=\tfrac, and p_c=\tfrac, where the sides are a \ge b \ge c and the area is T.


Quadrilateral

The two bimedians of a
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...
quadrilateral In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...
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.Altshiller-Court, Nathan, ''College Geometry'', Dover Publ., 2007. 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 (inscribed in a circle), these maltitudes are concurrent at (all meet at) a common point called the "anticenter". Brahmagupta's theorem states that if a cyclic quadrilateral is
orthodiagonal In Euclidean geometry, an orthodiagonal quadrilateral is a quadrilateral in which the diagonals cross at right angles. In other words, it is a four-sided figure in which the line segments between non-adjacent vertices are orthogonal (perpendicu ...
(that is, has
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
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 Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while ''surface area'' refers to the area of an open su ...
of a
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colli ...
. Three of them are the medians of the triangle (which connect the sides' midpoints with the opposite vertices), and these are concurrent at the triangle's
centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ...
; 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+1:1. 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 An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter or card. Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a ...
of the infinitude of area bisectors is a
deltoid Deltoid (delta-shaped) can refer to: * The deltoid muscle, a muscle in the shoulder * Kite (geometry), also known as a deltoid, a type of quadrilateral * A deltoid curve, a three-cusped hypocycloid * A leaf shape * The deltoid tuberosity, a part of ...
(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). 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

The sides of the deltoid are arcs of
hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, c ...
s that are
asymptotic In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
to the extended sides of the triangle. The ratio of the area of the envelope of area bisectors to the area of the triangle is invariant for all triangles, and equals \tfrac \log_e(2) - \tfrac, i.e. 0.019860... or less than 2%. A 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 concur at (all pass through) the center of the Spieker circle, which is the incircle of the medial triangle. The cleavers are parallel to the angle bisectors. A 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.


Parallelogram

Any line through the midpoint of a
parallelogram In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of eq ...
bisects the area and the perimeter.


Circle and ellipse

All area bisectors and perimeter bisectors of a circle or other ellipse go through the center, and any
chord Chord may refer to: * Chord (music), an aggregate of musical pitches sounded simultaneously ** Guitar chord a chord played on a guitar, which has a particular tuning * Chord (geometry), a line segment joining two points on a curve * Chord ( ...
s through the center bisect the area and perimeter. In the case of a circle they are the
diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid fo ...
s of the circle.


Bisectors of diagonals


Parallelogram

The
diagonal In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Gree ...
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) is itself bisected by the 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 tetrahedronAltshiller-Court, N. "The tetrahedron." Ch. 4 in ''Modern Pure Solid Geometry'': Chelsea, 1979.


References


External links


The Angle Bisector
at cut-the-knot
Angle Bisector definition. Math Open Reference
With interactive applet

With interactive applet

With interactive applet

an

Using a compass and straightedge * {{PlanetMath attribution, id=3623, title=Angle bisector Elementary geometry