Bisector

A bisector is a line, ray, or plane that divides a geometric object into two parts of equal measure. Common uses are with segments, angles, and dihedral angles in three-dimensional figures.

Segment bisector: A line, ray, or plane that intersects a segment AB at its midpoint. If M is the intersection point, then AM = MB. A perpendicular bisector is a segment, line, or plane that is perpendicular to AB at its midpoint; any point on the perpendicular bisector is equidistant from A and B.

Angle bisector: A ray that starts at the vertex of an angle and divides the angle into two congruent angles. Each interior angle bisector lies inside the angle, while an exterior angle bisector divides the corresponding exterior angle. In triangles, the angle bisectors meet at the incenter, the center of the incircle. The angle bisector theorem describes a relation between side lengths: if a triangle has an angle bisected by a line intersecting the opposite side at D, then BD/DC = AB/AC.

Loci and symmetry: The set of points equidistant from A and B forms the perpendicular bisector of segment AB. The set of points inside an angle that are equidistant from the sides lies along the angle bisector.

In summary, a bisector is a fundamental construct that splits a figure into two equal parts, with specific variants for segments and angles that yield important geometric results and centers.