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This is a disambiguation page; that is, one
that points to other pages that might otherwise have the same name. If you followed a link here, you might want to go back and
fix that link to point to the appropriate specific page.
In numerical analysis, the bisection method is a simple root-finding algorithm.
In geometry, bisection refers to dividing an object exactly in
half, 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 midpoint of the segment. Particularly
important is the perpendicular bisector 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 equidistant from the segment's endpoints.
An angle bisector 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.
In classical geometry, the bisection is a simple ruler-and-compass construction, whose possibility depends on the ability to draw circles of
equivalent radius 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.
Alternately, if a line and a point on it are given, we can find a perpendicular bisector by drawing a single circle whose center
is that point. The circle intersects the line in two more points, and from here the problem reduces to 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) is somewhat more difficult, and cannot
be achieved with the ruler and compass alone.
(please add figures to this entry! Also, please provide the proof, as a nice one escapes me right now.)
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