Bisection

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Not to be confused with Dissection. Bisection_sentence_0

For the bisection theorem in measure theory, see Ham sandwich theorem. Bisection_sentence_1

For the root-finding method, see Bisection method. Bisection_sentence_2

For other uses, see Bisect (disambiguation). Bisection_sentence_3

In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a bisector. Bisection_sentence_4

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). Bisection_sentence_5

In three-dimensional space, bisection is usually done by a plane, also called the bisector or bisecting plane. Bisection_sentence_6

Line segment bisector Bisection_section_0

A line segment bisector passes through the midpoint of the segment. Bisection_sentence_7

Particularly important is the perpendicular bisector of a segment, which, according to its name, meets the segment at right angles. Bisection_sentence_8

The perpendicular bisector of a segment also has the property that each of its points is equidistant from the segment's endpoints. Bisection_sentence_9

Therefore, Voronoi diagram boundaries consist of segments of such lines or planes. Bisection_sentence_10

In classical geometry, the bisection is a simple compass and straightedge construction, whose possibility depends on the ability to draw circles of equal radii and different centers. Bisection_sentence_11

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. Bisection_sentence_12

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. Bisection_sentence_13

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. Bisection_sentence_14

Brahmagupta's theorem states that if a cyclic quadrilateral is 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. Bisection_sentence_15

Angle bisector Bisection_section_1

An angle bisector divides the angle into two angles with equal measures. Bisection_sentence_16

An angle only has one bisector. Bisection_sentence_17

Each point of an angle bisector is equidistant from the sides of the angle. Bisection_sentence_18

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. Bisection_sentence_19

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. Bisection_sentence_20

To bisect an angle with straightedge and compass, one draws a circle whose center is the vertex. Bisection_sentence_21

The circle meets the angle at two points: one on each leg. Bisection_sentence_22

Using each of these points as a center, draw two circles of the same size. Bisection_sentence_23

The intersection of the circles (two points) determines a line that is the angle bisector. Bisection_sentence_24

The proof of the correctness of this construction is fairly intuitive, relying on the symmetry of the problem. Bisection_sentence_25

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). Bisection_sentence_26

Triangle Bisection_section_2

Concurrencies and collinearities Bisection_section_3

The interior angle bisectors of a triangle are concurrent in a point called the incenter of the triangle, as seen in the diagram at right. Bisection_sentence_27

The bisectors of two exterior angles and the bisector of the other interior angle are concurrent. Bisection_sentence_28

Three intersection points, each of an external angle bisector with the opposite extended side, are collinear (fall on the same line as each other). Bisection_sentence_29

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. Bisection_sentence_30

Angle bisector theorem Bisection_section_4

Main article: Angle bisector theorem Bisection_sentence_31

The angle bisector theorem is concerned with the relative lengths of the two segments that a triangle's side is divided into by a line that bisects the opposite angle. Bisection_sentence_32

It equates their relative lengths to the relative lengths of the other two sides of the triangle. Bisection_sentence_33

Lengths Bisection_section_5

or in trigonometric terms, Bisection_sentence_34

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. Bisection_sentence_35

No two non-congruent triangles share the same set of three internal angle bisector lengths. Bisection_sentence_36

Integer triangles Bisection_section_6

There exist integer triangles with a rational angle bisector. Bisection_sentence_37

Quadrilateral Bisection_section_7

The internal angle bisectors of a convex quadrilateral either form a cyclic quadrilateral (that is, the four intersection points of adjacent angle bisectors are concyclic), or they are concurrent. Bisection_sentence_38

In the latter case the quadrilateral is a tangential quadrilateral. Bisection_sentence_39

Rhombus Bisection_section_8

Each diagonal of a rhombus bisects opposite angles. Bisection_sentence_40

Ex-tangential quadrilateral Bisection_section_9

The excenter of an ex-tangential quadrilateral lies at the intersection of six angle bisectors. Bisection_sentence_41

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. Bisection_sentence_42

Parabola Bisection_section_10

Main article: Parabola § Tangent bisection property Bisection_sentence_43

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. Bisection_sentence_44

Bisectors of the sides of a polygon Bisection_section_11

Triangle Bisection_section_12

Medians Bisection_section_13

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). Bisection_sentence_45

The three medians intersect each other at the 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. Bisection_sentence_46

The centroid is twice as close to the midpoint of any one side as it is to the opposite vertex. Bisection_sentence_47

Perpendicular bisectors Bisection_section_14

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. Bisection_sentence_48

The three perpendicular bisectors of a triangle's three sides intersect at the circumcenter (the center of the circle through the three vertices). Bisection_sentence_49

Thus any line through a triangle's circumcenter and perpendicular to a side bisects that side. Bisection_sentence_50

In an acute triangle the circumcenter divides the interior perpendicular bisectors of the two shortest sides in equal proportions. Bisection_sentence_51

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. Bisection_sentence_52

Quadrilateral Bisection_section_15

The two bimedians of a convex quadrilateral are the line segments that connect the midpoints of opposite sides, hence each bisecting two sides. Bisection_sentence_53

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. Bisection_sentence_54

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. Bisection_sentence_55

If the quadrilateral is cyclic (inscribed in a circle), these maltitudes are concurrent at (all meet at) a common point called the "anticenter". Bisection_sentence_56

Brahmagupta's theorem states that if a cyclic quadrilateral is 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. Bisection_sentence_57

The perpendicular bisector construction forms a quadrilateral from the perpendicular bisectors of the sides of another quadrilateral. Bisection_sentence_58

Area bisectors and perimeter bisectors Bisection_section_16

Triangle Bisection_section_17

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. Bisection_sentence_59

The three cleavers concur at (all pass through) the center of the Spieker circle, which is the incircle of the medial triangle. Bisection_sentence_60

The cleavers are parallel to the angle bisectors. Bisection_sentence_61

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. Bisection_sentence_62

The three splitters concur at the Nagel point of the triangle. Bisection_sentence_63

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). Bisection_sentence_64

There are either one, two, or three of these for any given triangle. Bisection_sentence_65

A line through the incenter bisects one of the area or perimeter if and only if it also bisects the other. Bisection_sentence_66

Parallelogram Bisection_section_18

Any line through the midpoint of a parallelogram bisects the area and the perimeter. Bisection_sentence_67

Circle and ellipse Bisection_section_19

All area bisectors and perimeter bisectors of a circle or other ellipse go through the center, and any chords through the center bisect the area and perimeter. Bisection_sentence_68

In the case of a circle they are the diameters of the circle. Bisection_sentence_69

Bisectors of diagonals Bisection_section_20

Parallelogram Bisection_section_21

The diagonals of a parallelogram bisect each other. Bisection_sentence_70

Quadrilateral Bisection_section_22

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. Bisection_sentence_71

Volume bisectors Bisection_section_23

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. Bisection_sentence_72

Thus any plane containing a bimedian (connector of opposite edges' midpoints) of a tetrahedron bisects the volume of the tetrahedron Bisection_sentence_73


Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/Bisection.