Angle bisector of a triangle
by Bernat Requena Serra · Published 6 November, 2020 · Updated 14 January, 2021
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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 (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 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 (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.
Lines Associated with a Triangle
Altitude of a Triangle
Median of a Triangle
Perpendicular Bisector of a Triangle
Tags: triangle
AUTHOR: Bernat Requena Serra
MATHEMATICS – GEOMETRY – TRIANGLE
Area of a Triangle
Area of a Triangle Using the Base and Height
Perimeter of a Triangle
Basic Elements of a Triangle
Heron's Formula
Types of Triangles
Equilateral Triangle
Area of an Equilateral Triangle
Perimeter of an Equilateral Triangle
Isosceles Triangle
Area of an Isosceles Triangle
Perimeter of an Isosceles Triangle
Scalene Triangle
Area of a Scalene Triangle
Perimeter of a Scalene Triangle
Right Triangle
Area of a Right Triangle
Perimeter of a Right Triangle
Basic Elements of a Right Triangle
Hypotenuse
The Geometric Mean Theorem
The Leg Rule
Triangle Center
Centroid of a Triangle
Orthocenter of the Triangle
Circumcenter of a Triangle
Incenter of a Triangle
Points, Lines, and Circles Associated with a Triangle
Euler Line
Similarity of Triangles
Thales's Theorems
Circle, Semicircle and Annulus
Solid Figures
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