geometry, bisection is the division of something into two equal or congruentparts, usually by a line, which is then called a "bisector". The most often considered types of bisectors are "segment bisectors" and "angle bisectors".
A segment bisector passes through the
midpointof the segment.Particularly important is the perpendicularbisector of a segment, which, according to its name, meets the segment at right angles.The perpendicular bisector of a segment also has the property that each of its points is equidistantfrom the segment's endpoints. Therefore Voronoi diagramboundaries consist of segments of such lines or planes.
anglebisector divides the angle into two equal angles.An angle only has one bisector.Each point of an angle bisector is equidistant from the sides of the angle.The interior bisector of an angle is the line or line segment that divides it into two equal angles on the same side as the angle.The exterior bisector of an angle is the line or line segment that divides it into two equal angles on the opposite side as the angle.
In classical geometry, the bisection is a simple
compass and straightedge, whose possibility depends on the ability to draw circles of equal radii and different centers. The segment is bisected by drawing intersecting circles of equal radius, whose centers are the endpoints of the segment. The line determined by the points of intersection is the perpendicular bisector, and crosses our original segment at its center. This construction is in fact used when constructing a line perpendicular to a given line at a given point: drawing an arbitray 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.
To bisect an angle, 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 these two constructions is fairly intuitive, relying on the symmetry of the problem. It is interesting to note that the trisection of an angle (dividing it into three equal parts) cannot be achieved with the ruler and compass alone (this was first proved by
The angle bisectors of the angles of a triangle are concurrent in a point called the
incenterof the triangle.
angle bisector theorem
* [http://www.cut-the-knot.org/triangle/ABisector.shtml The Angle Bisector] at
* [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
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Look at other dictionaries:
Bisection — Bi*sec tion, n. [Cf. F. bissection.] Division into two parts, esp. two equal parts. [1913 Webster] … The Collaborative International Dictionary of English
Bisection — (v. lat.), das Zerschneiden in 2 Theile … Pierer's Universal-Lexikon
Bisection — Bisection, lat., das Zerschneiden in 2 Theile … Herders Conversations-Lexikon
bisection — index dichotomy, split, subdivision Burton s Legal Thesaurus. William C. Burton. 2006 … Law dictionary
bisection — division in two, 1650s, noun of state from BISECT (Cf. bisect). Related: Bisectional … Etymology dictionary
bisection — (Roget s Thesaurus) >Division into two parts. < N PARAG:bisection >N GRP: N 1 Sgm: N 1 bisection bisection bipartition Sgm: N 1 dichotomy dichotomy subdichotomy Sgm: N 1 halving halving &c. >V. Sgm: N 1 dimidiation dimidiation … English dictionary for students
bisection — bisect ► VERB ▪ divide into two parts. DERIVATIVES bisection noun bisector noun. ORIGIN from Latin secare to cut … English terms dictionary
bisection — noun see bisect … New Collegiate Dictionary
bisection — See bisect. * * * … Universalium
bisection — noun A division into two parts, especially into two equal parts See Also: bisect … Wiktionary