Angle bisector

Angle bisector

Angle Bisector

The angle bisector of an angle in geometry is the line or line segment that divides the angle into two equal parts. Angle bisectors are a fundamental concept in geometry, playing a crucial role in various constructions, theorems, and problems.

Definition

An angle bisector is a line that passes through the vertex of an angle and divides it into two angles of equal measure. For an angle ∠ABC, the angle bisector of ∠ABC would be a line from vertex B that splits ∠ABC into two angles with equal measure.

Properties

Angle bisectors have several important properties:

• The angle bisector of an angle in a triangle is the path from the angle's vertex to the opposite side, dividing the angle into two equal parts.
• In a triangle, the angle bisectors intersect at a point called the incenter, which is the center of the triangle's incircle.
• The angle bisector theorem states that the angle bisector of a triangle divides the opposite side into two segments that are proportional to the adjacent sides.

Applications

Angle bisectors are used in various geometric constructions and proofs. They are essential in constructing the incenter of a triangle, solving problems related to angle bisector theorem, and designing with compass and straightedge.

Angle Bisector Theorem

The Angle Bisector Theorem is a fundamental theorem in geometry that relates the lengths of the sides of a triangle to the lengths of segments created by bisecting one of the triangle's angles. It states that the angle bisector of a triangle divides the opposite side into two segments that are proportional to the lengths of the other two sides.

Construction

Constructing an angle bisector involves using a compass and straightedge. The process typically involves: 1. Drawing an arc centered at the vertex of the angle that intersects both sides of the angle. 2. From the points of intersection, drawing two arcs of equal radius inside the angle that intersect each other. 3. Drawing a line from the vertex through the point of intersection of the arcs inside the angle. This line is the angle bisector.

See Also

• Incenter
• Bisector
• Compass and straightedge constructions
• Triangle