# Angle bisector of a triangle

The **angle bisector** of a triangle is a line segment that bisects one of the vertex angles of a triangle, and ends up on the corresponding opposite side.

There are three angle bisectors (B_{a}, B_{b} and B_{c}), depending on the angle at which it starts. We can find the length of the angle bisector by using this formula:

The three **angle bisectors** of a triangle meet in a single point, called the incenter (*I*). This point is always inside the triangle.

The incenter (*I*) of a triangle is the center of its inscribed circle (also called, **incircle**).

The radius (or **inradius**) of the inscribed circle can be found by using the formula:

The relationship between the radius *R* of the circumcenter *O* (point where the perpendicular bisectors of the triangle‘s sides converge) and the radius *r* of the incenter *I* (point of concurrency of the angle bisectors) is:

Download this **calculator** to get the results of the formulas on this page. Choose the initial data and enter it in the upper left box. For results, press ENTER.

Triangle-total.rar or Triangle-total.exe

Note. Courtesy of the author: **José María Pareja Marcano**. Chemist. Seville, Spain.

## Angle Bisector Theorem

The **angle bisector theorem** states than in a triangle Δ *ABC* the ratio between the length of two sides adjacent to the vertex (side *AB* and side *BC*) relative to one of its bisectors (*B _{b}*) is equal to the ratio between the corresponding segments where the angle bisector divides the opposite side (segment

*AP*and segment

*PC*).

In other words, an angle bisector of a triangle divides the opposite side in two segments that are proportional to the other two sides of the triangle.

So, by the **angle bisector theorem**:

In addition, but not included in this theorem, it’s also true that:

A procedure for finding the **equation** of the **angle bisector** is based on the following:

- If we have the
**equations of two lines**that cross**at one point**: - Then, the
**angle bisector**which lies on the side of the origin (where the two lines cross), can be defined by the equation:

**Note**: in a triangle these two lines would pass through the two sides which form the angle that is divided by the bisector angle.

## Exercise

Let *ABC* de a triangle in which *a*=3 cm, *b*=4 cm and *c*=2 cm. Find the angle bisectors B_{a}, B_{b} y B_{c}.

**SOLUTION:**

We’re going to solve this exercise using the angle bisector’s formula:

Since we already know what all the sides of the triangle measure, we only have to find the semiperimeter (*s*):

Then, we can substitute the values in the angle bisector’s formula:

Therefore, ** B_{a}=2.45 cm**,

**and**

*B*=1.47 cm_{b}**.**

*B*=3.32 cm_{c}