Line EF bisects line AB at F, line EF is a perpendicular bisector of segmens AD at C and the interior bisector of right angle AED

Finding a perpendicular bisector of a line using a compass and ruler

In geometry, bisection is the division of something into two equal parts, usually by a line, which is then called a bisector. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position The most often considered types of bisectors are segment bisectors and angle bisectors.

A segment bisector passes through the midpoint of the segment. The midpoint (also known as class mark in relation to Histogram) is the middle point of a Line segment. Particularly important is the perpendicular bisector of a segment, which, according to its name, meets the segment at right angles. In Geometry, two lines or planes (or a line and a plane are considered perpendicular (or orthogonal) to each other if they form congruent In Geometry and Trigonometry, a right angle is an angle of 90 degrees corresponding to a quarter turn (that is a quarter of a full circle The perpendicular bisector of a segment also has the property that each of its points is equidistant from the segment's endpoints. Distance is a numerical description of how far apart objects are Therefore Voronoi diagram boundaries consist of segments of such lines or planes. In Mathematics, a Voronoi diagram, named after Georgy Voronoi, also called a Voronoi Tessellation, a Voronoi decomposition, or

An angle bisector divides the angle into two equal angles. In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called Equality is the paradigmatic example of the more general concept of Equivalence relations on a set those binary relations which are reflexive, symmetric 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. Pentagon constructgif|thumb|right|Construction of a regular pentagon]] Compass-and-straightedge or ruler-and-compass construction is the construction of lengths or Angles Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the 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.

Bisection of an angle using a compass and ruler

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 Pierre Wantzel). The problem of trisecting the angle is a classic problem of Compass and straightedge constructions of ancient Greek mathematics. Pierre Laurent Wantzel ( June 5, 1814 in Paris – May 21, 1848 in Paris was a French Mathematician who proved

The angle bisectors of the angles of a triangle are concurrent in a point called the incenter of the triangle. In Geometry, the incircle or inscribed circle of a triangle is the largest Circle contained in the triangle it touches (is Tangent

See also

• angle bisector theorem
• bisector plan

External links

• The Angle Bisector at cut-the-knot
• Angle Bisector definition. Math Open Reference With interactive applet
• Line Bisector definition. Math Open Reference With interactive applet
• Perpendicular Line Bisector. With interactive applet
• Animated instructions for bisecting an angle and bisecting a line Using a compass and straightedge

This article incorporates material from Angle bisector on PlanetMath, which is licensed under the GFDL. In Geometry, the angle bisector theorem relates the Length of the side opposite one Angle of a Triangle to the lengths of the other two sides Cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in Mathematics. PlanetMath is a free, collaborative online Mathematics Encyclopedia.