Angle Bisector

Dependencies: Angle, Ray. A derived definition: draw a ray from the vertex of an angle and divide the angle into two equal parts.

Definition

Let $\angle AOB$ be an angle.

If a ray $OD$ starts at vertex $O$, lies inside the angle, and divides $\angle AOB$ into two equal angles, so that $\angle AOD=\angle DOB$, then $OD$ is called the bisector of $\angle AOB$.

The angle bisector is the angle's axis of symmetry: folding the angle along it makes the two sides coincide. Because it starts at the vertex and extends only into the angle, it is a ray rather than a complete line.

Remarks

• Existence and uniqueness. Every angle has exactly one bisector. Once the Protractor axiom axiom assigns a real-number measure to the angle, the direction with half that measure is uniquely determined (Angle bisector ⇔ equidistant from sides).
• Locus of points equidistant from the sides. Every point on an angle bisector is the same distance from the two sides; conversely, every interior point equidistant from the sides lies on the bisector (Angle bisector test).
• The three internal bisectors of a triangle are concurrent. They meet at the incenter, which is equidistant from the three sides and is the center of the inscribed circle (Three angle bisectors meet (incenter)).
• Internal and external bisectors. An angle and its adjacent supplementary angle each have a bisector, and those two bisectors are perpendicular.