Bisectors

A bisector in geometry is any line, ray, or segment that divides a figure into two congruent parts. The term is used in several contexts, most commonly for angle bisectors, segment bisectors, and perpendicular bisectors. Bisectors are fundamental in compass-and-straightedge constructions and in defining notable triangle centers.

Angle bisector: An angle bisector is a line or ray that splits an angle into two equal angles. In a triangle, the three angle bisectors concur at the incenter, which is equidistant from the triangle’s three sides. The Angle Bisector Theorem states that an angle bisector inside a triangle divides the opposite side into segments proportional to the adjacent side lengths.

Segment and perpendicular bisectors: A segment bisector passes through the midpoint of a given segment, dividing it into two equal parts. If the bisector is perpendicular to the segment, it is the perpendicular bisector, and any point on it is equidistant from the segment’s endpoints. In a triangle, the perpendicular bisectors of the three sides meet at the circumcenter, the center of the circumscribed circle, which is equidistant from the triangle’s vertices.

In coordinate geometry, the equation of a perpendicular bisector can be obtained from the endpoints of the segment, and the midpoint formula gives the point of intersection for the bisector with the segment. Bisectors also appear in various geometric constructions and proofs, including center definitions and distance relations.

Notable related concepts include the incenter, circumcenter, midpoint, and triangle centers in general.