In this article, we’ll learn how to determine the equation of the bisector of the angle between two lines whose equations are given. For example, suppose that the equations of lines g and h are and , respectively. As Figure 1 shows, they are intersected lines.

**Figure 1**

In the figure, the green and black lines represent the lines g and h, respectively.

An * angle bisector* is the line or line segment that divides the angle into two equal parts. In Figure 1, the red line is an angle bisector. It divides the angle formed by the green and black lines. What is the equation of the angle bisector? This article addresses the question.

Let’s first derive the formula …

Given:

- line g with equation A
_{1}x + B_{1}y + C_{1}= 0, - line h with equation A
_{2}x + B_{2}y + C_{2}= 0, - line k, which is a bisector of the angle between g and h

To be sought: the equation of k

**Answer**

Let P_{0}(x_{0},y_{0}) be any point that lies on k. Let A be on line g such that and B on line h such that . Consequently, = distance from P_{0} to line g and = distance from P_{0} to line h. (See Figure 2.)

**Figure 2**

In Figure 2, O is the point of intersection of the lines g and h. Note that in ΔOAP_{0} is of an equal length as in ΔOBP_{0} and ∠AOP_{0} = ∠BOP_{0}. As a consequence, ΔOAP_{0} ΔOBP_{0}. Therefore, . By applying the formula for the distance between a point and a line, we have the following.

Since P_{0}(x_{0},y_{0}) is any point in k, the last equation above holds for all points on the line k. As a result, the equation of k can be expressed as:

……………………………………………………………………………………….. (*)

Note that from (*) it seems that there are actually two possible bisectors.

**For every pair of intersected lines there are always two angle bisectors.**

The other angle bisector is represented by the yellow line in Figure 3.

**Figure 3**

It was said earlier that there were two angle bisectors for each pair of intersected lines. Furthermore, it can be proved that the angle bisectors are perpendicular to each other.

Now we are ready to solve the problem at the beginning of this article.

Given:

- line g with equation ,
- line h with equation ,
- line k, which is a bisector of the angle between g and h

To be sought: the equation of k

**Answer**

Note that the lines g and h can be expressed in the form of Ax + By + C = 0 as follows.

g ≡ 7x – 2y = 0

h ≡ x – 2y = 0

Substituting A_{1} = 7, B_{1} = -2, C_{1} = 0, A_{2} = 1, B_{2} = -2, and C_{2} = 0 into (*), we get:

There are two angle bisectors. Their equations are determined as follows.

**First angle bisector**: (See Figure 4)

**Figure 4**

**Second angle bisector**: (See Figure 5)

**Figure 5**