Angle bisector of a triangle
There are three angle bisectors (Ba, Bb and Bc), depending on the angle at which it starts. We can find the length of the angle bisector by using this formula:
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:
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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 (Bb) 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.
Let ABC de a triangle in which a=3 cm, b=4 cm and c=2 cm. Find the angle bisectors Ba, Bb y Bc.
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, Ba=2.45 cm, Bb=1.47 cm and Bc=3.32 cm.
AUTHOR: Bernat Requena Serra