An angle

bisector is a

line or a

ray that divides an

angle into two equal angles.
The method for constructing an angle bisector with an unmarked ruler and a compass is detailed in

proposition 9 of Euclid's Elements: Book I.

For any

triangle, the

angle bisectors of all three angles coincide at the

incenter (see

incenter for proof).

Given any triangle ABC, and an angle bisector PB of angle ABC where P lies on AC, the following about the ratio of lengths is true.

|A-P| |P-C|
------- = -------
|A-B| |B-C|
A P C
*--------------*-------------*
| / /
| / /
| / /
| / /
| / /
| / /
| / /
| / /
| / /
| / /
| / /
| / /
| / /
|//
*
B