Angle Bisector
An angle bisector is a ray that divides an angle into two congruent angles of equal measure.
Formula
\text{angle}_1 = \text{angle}_2 = (\text{original angle}) / 2
Definition
An angle bisector is a ray from the vertex of an angle that cuts it exactly in half, dividing it into two congruent angles ("bisect" means to cut into two equal parts). The Angle Bisector Theorem states that in a triangle, the bisector of an angle divides the opposite side in the ratio of the adjacent sides, and more generally, the bisector of angle $AOB$ is the locus of points equidistant from rays $OA$ and $OB$.
Example
If you have a $60^\circ$ angle and draw its bisector, each half is $30^\circ$; folding one side of an angle onto the other on paper creates a crease that is the bisector. In triangle $ABC$, the bisector from $A$ meets $BC$ at $D$, giving $BD/DC = AB/AC$: if $AB=6$ and $AC=4$, then $BD/DC=3/2$, so with $BC=10$, $BD=6$ and $DC=4$. For a triangle with $a=5$, $b=7$, $c=8$, $A=60^\circ$, the bisector length from $A$ is $t_a = (2\cdot7\cdot8/(7+8))\cos(30^\circ) = (112/15)(\sqrt{3}/2) = 56\sqrt{3}/15$.
Key Insight
The three angle bisectors of any triangle always meet at a single point, the incenter, the center of the circle that fits inside the triangle; the incenter is equidistant from all three sides, and Area $= r \times s$ (where $s$ is the semi-perimeter) connects it to the triangle's measurements. Because the bisector is a locus of points equidistant from two lines, it pairs naturally with the perpendicular bisector (locus equidistant from two points); together these two loci generate the classical construction toolkit and the four classic triangle centers: incenter, circumcenter, centroid, and orthocenter.