Angle bisector of a triangle

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Drawing of the three angle bisectors of a triangle and the incenter

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:

Formula of the three angle bisectors of the triangle

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

Drawing of the three angle bisectors of a triangle, the incenter and the inscribed circle

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:

Formula for the inradius inscribed in the triangle with center at the incenter

Drawing of circumscribed and inscribed circles for the relationship between their radii

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:

Formula of the relationship between the radii of the circumscribed and inscribed circle in a triangle.

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

Drawing the 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:

Triangle Bisector Theorem Formula

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

Calculations in the Triangle Bisector Theorem

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:
    Equations of two lines in the Triangle Bisector Theorem
  • Then, the angle bisector which lies on the side of the origin (where the two lines cross), can be defined by the equation:
    Equation which lies on the side of the origin in the Triangle Bisector Theorem

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

Example 1 of a triangle, its three bisectors and the incenter

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.

SOLUTION:

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

Formula of the three angle bisectors of the triangle

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

Calculation of the semiperimeter in example 1

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

Solution in example 1

Therefore, Ba=2.45 cm, Bb=1.47 cm and Bc=3.32 cm.

Lines Associated with a Triangle


AUTHOR: Bernat Requena Serra

YEAR: 2020


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