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:

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, B_b=1.47 cm and B_c=3.32 cm.

Lines Associated with a Triangle

• Altitude of a Triangle
• Median of a Triangle
• Perpendicular Bisector of a Triangle