Line ED bisects line AB at D, line EF is a perpendicular bisector of segment 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. The most often considered types of bisectors are segment bisectors and angle bisectors. The Ham sandwich theorem, also known as the Stone-Tukey theorem in topology in mathematics, states that given n objects in n-dimensional space, it is possible to divide each one in half with a single (n − 1)-dimensional hyperplane. ... Bisect may mean: bisection, a mathematical construction bisect (philately), the use of postage stamp halves This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... Image File history File links Bisectors. ... Image File history File links Bisectors. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Calabi-Yau manifold Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ... “Line” redirects here. ...

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. Therefore Voronoi diagram boundaries consist of segments of such lines or planes. Segment may mean: Segment (biology), a repeated part of the metazoan body. ... In mathematics, a line segment is a part of a line that is bounded by two end points. ... Fig. ... This article is about angles in geometry. ... personal space, proxemics. ... An endpoint or end point is a mark of termination or completion. ... This is the Voronoi diagram of a random set of points in the plane (all points lie within the image). ...

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. 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. ∠, the angle symbol. ... In mathematics, two mathematical objects are considered equal if they are precisely the same in every way. ...

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. 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. Creating a regular hexagon with a ruler and compass Construction of a regular pentagon Compass and straightedge or ruler-and-compass construction is the construction of lengths or angles using only an idealized ruler and compass. ... Circle illustration This article is about the shape and mathematical concept of circle. ...

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. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ...

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 (Pierre Wantzel). A number of ancient problems in geometry involve the construction of lengths or angles using only an idealized ruler and compass, or more properly a straightedge and compass. ... Pierre Wantzel (1814–1848) was a French mathematician. ...

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 to) the three sides. ...

See also

• angle bisector theorem

In this diagram, BD:DC = AB:AC. 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 of the triangle. ...

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. 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. ...