Angle Bisector Theorem

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An important consequence of the BPT is the following theorem.

Angle Bisector Theorem: In a triangle, the angle bisector of any angle will divide the opposite side in the ratio of the sides containing the angle.

Consider the figure below, in which AD is the bisector of \(\angle A\). Our theorem tells us that BD:DC = AB:AC.

Angle Bisector of a triangle

Proof: Draw a ray CX parallel to AD, and extend BA to intersect this ray in E:

Line parallel to angle bisector of a triangle

By the BPT, BD:DC = BA:AE. Now, we will prove that AE = EC. For that, consider the figure above once again, with the angles marked, as shown below:

corresponding and alternate interior angles

We have:

  • \(\angle1\) = \(\angle 2\) (corresponding angles)

  • \(\angle 3\) = \(\angle 4\) (alternate interior angles)

But AD is the bisector of \(\angle BAC\), which means that \(\angle 1\) = \(\angle 3\). Thus, \(\angle 2\) = \(\angle 4\), which further implies that AC = AE. Finally, this means that:

\[\frac{{BD}}{{DC}} = \frac{{BA}}{{AE}} = \frac{{BA}}{{AC}}\]

The angle bisector AD thus divides the opposite side BC in the ratio of the sides (AB and AC) containing that angle.

 

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