Bisection

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In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, 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, that divides it into two equal angles).

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

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Line segment bisector

A line 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. Therefore Voronoi diagram boundaries consist of segments of such lines or planes.

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 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.

Angle bisector

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

To bisect an angle with straightedge 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 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 Pierre Wantzel).

The angle bisectors of the angles of a triangle are concurrent in a point called the incenter of the triangle.

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