bisects

Bisect is a geometric term meaning to divide something into two equal parts. In mathematics, the verb bisect is used for segments, angles, or other figures, while the noun form bisector refers to the line, ray, or plane that performs the division. Common types in two-dimensional geometry are segment bisectors, angle bisectors, and perpendicular bisectors.

A segment bisector is a line, ray, or plane that intersects a given segment at its midpoint, thereby dividing the segment into two congruent parts. The perpendicular bisector is a notable special case: it intersects the segment at right angles and, every point on it, is equidistant from the segment’s endpoints.

An angle bisector is a ray from the vertex that divides the angle into two equal angles. A key property is that any point on the angle bisector is equidistant from the two sides of the angle. In triangles, the angle bisector theorem states that the bisector of an angle at a vertex divides the opposite side into segments proportional to the adjacent sides.

Construction and applications: with compass and straightedge, a segment can be bisected by drawing two circles centered at the endpoints and connecting their intersection points; the resulting line is the perpendicular bisector. An angle can be bisected by marking equal arcs on its sides and drawing the line from the vertex to the arc’s intersection. In triangle geometry, the perpendicular bisectors of the sides meet at the circumcenter, while angle bisectors meet at the incenter.

In coordinate and analytic geometry, bisectors are described by equations that encode equidistance or equal angles, and they play a central role in loci, centers, and triangle constructions.