Angle Bisector of a Triangle

Learn what an angle bisector in a triangle is, how it is constructed, its properties, and the incentre explained using simple diagrams and student-friendly notes.

1. What is an Angle Bisector?

An angle bisector of a triangle is a line segment that divides an angle into two equal angles. Every angle of a triangle has its own bisector.

      A
     / \
    /   \
   B-----C
     \|
      D  (AD bisects angle A)

If \(AD\) is the angle bisector of \(\angle A\), then:

\( \angle BAD = \angle DAC \)

2. Definition of Angle Bisector

Definition: In \( \triangle ABC \), the segment that divides angle \(A\) into two equal angles is called the angle bisector of angle A. Similarly, we can draw bisectors from angles \(B\) and \(C\).

3. Properties of Angle Bisectors

  • Every triangle has three angle bisectors.
  • Each angle bisector divides its opposite side in a special ratio.
  • The three angle bisectors always meet at one point.
  • This point is the same distance from all three sides of the triangle.

3.1. Angle Bisector Theorem

The Angle Bisector Theorem states that the bisector of an angle in a triangle divides the opposite side in the ratio of the adjacent sides.

In \(\triangle ABC\), if \(AD\) is the bisector of \(\angle A\), then:

\( \dfrac{BD}{DC} = \dfrac{AB}{AC} \)

This theorem is widely used for solving problems involving unknown side lengths.

4. Point of Concurrency: Incentre

The point where the three angle bisectors meet is called the incentre of the triangle.

  • The incentre is always located inside the triangle.
  • It is the centre of the triangle’s incircle.
  • The incircle is the largest circle that fits exactly inside the triangle and touches all three sides.

4.1. Incentre Properties

The incentre has two special properties:

  • It is equidistant from all three sides of the triangle.
  • A perpendicular dropped from the incentre to any side will be a radius of the incircle.

5. Constructing an Angle Bisector

To construct the angle bisector of \( \angle A \):

  1. Put the compass at \(A\) and draw an arc cutting both arms of the angle.
  2. From those two intersection points, draw arcs that intersect each other.
  3. Draw a line from \(A\) to the intersection of the arcs—this is the bisector.

6. Why Angle Bisectors Are Useful

Angle bisectors are important because they help in:

  • Constructing the incircle of a triangle.
  • Solving triangle geometry problems using ratios.
  • Proving relationships among sides and angles.
  • Dividing a triangle proportionally for design and analysis.

They are powerful tools in both elementary and advanced geometry.