Angle Bisector
GeometryAn 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 that cuts an angle exactly in half. The two smaller angles it creates are equal. "Bisect" means to cut into two equal parts.
Example
If you have a $60^\circ$ angle and draw an angle bisector, each half is $30^\circ$. On a piece of paper, you can fold one side of an angle onto the other - the crease is the angle bisector.
Key Insight
Bisectors appear in triangles in an important way: the three angle bisectors of any triangle always meet at a single point called the incenter - the center of the circle that fits inside the triangle.
Definition
An angle bisector is a ray from the vertex of an angle that divides the angle into two congruent (equal-measure) angles. 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.
Example
In triangle $ABC$, the bisector from $A$ to side $BC$ hits $BC$ at point $D$. Then $BD/DC = AB/AC$. If $AB = 6$ and $AC = 4$, then $BD/DC = 6/4 = 3/2$. If $BC = 10$, then $BD = 6$ and $DC = 4$.
Key Insight
The incenter of a triangle (where the three angle bisectors meet) is equidistant from all three sides. This distance is the inradius $r$, and the area formula Area $= r \times s$ (where $s$ is the semi-perimeter) connects the incenter to the triangle's measurements.
Definition
The angle bisector of angle $AOB$ is the locus of points equidistant from rays $OA$ and $OB$. In a triangle with sides $a, b, c$, the length of the bisector from vertex $A$ to side $BC$ is: $t_a = (2bc/(b+c))\cos(A/2)$. The incenter $I$ has position vector $(aA + bB + cC)/(a+b+c)$ in barycentric coordinates.
Example
For triangle with sides $a=5$, $b=7$, $c=8$ and $A = 60^\circ$: bisector length $t_a = (2 \cdot 7 \cdot 8/(7+8))\cos(30^\circ) = (112/15)(\sqrt{3}/2) = 56\sqrt{3}/15$.
Key Insight
The angle bisector as a locus of equidistant points connects it to the perpendicular bisector (locus equidistant from two points). Together, these two fundamental loci (equidistant from two lines vs. equidistant from two points) generate the classical construction toolkit and the four triangle centers: incenter, circumcenter, centroid, orthocenter.