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