**Angle Bisector Theorem :**

The internal (external) bisector of an angle of a triangle divides the opposite side internally (externally) in the ratio of the corresponding sides containing the angle.

To know more about proof, please visit the page "Angle bisector theorem proof".

**Example 1 :**

In a triangle MNO, MP is the external bisector of
angle M meeting NO produced at P. IF MN = 10 cm, MO = 6 cm, NO - 12 cm,
then find OP.

**Solution :**

MP is the external bisector of angle M by using angle bisector theorem in the triangle MNO we get,

(NP/OP) = (MN/MO)

NP = NO + OP

= 12 + OP

(12 + OP)/OP = 10/6

6 (12 + OP) = 10 OP

72 + 6 OP = 10 OP

72 = 10 OP - 6 OP

4 OP = 72

OP = 72/4

= 18 cm

**Example 2 :**

In a quadrilateral ABCD, the bisectors of ∠B and ∠D intersect on AC at E. Prove that (AB/BC) = (AD/DC).

**Solution :**

Here DE is the internal angle bisector of angle D.

by using internal bisector theorem, we get

(AE/EC) = (AD/DC) ----- (1)

Here BE is the internal angle bisector of angle B.

(AE/EC) = (AB/BC) ----- (2)

from (1) and (2) we get,

(AB/BC) = (AD/DC)

Hence proved.

**Example 3 :**

The internal bisector of ∠A of triangle ABC meets BC at D and the external bisector of ∠A meets BC produced at E. Prove that (BD/BE) = (CD/CE).

**Solution :**

In triangle ABC, AD is the internal bisector of angle A.

by using angular bisector theorem in triangle ABC

(BD/DC) = (AB/AC) ----- (1)

In triangle ABC, AE is the internal bisector of angle A.

(BE/CE) = (AB/AC) ----- (2)

from (1) and (2) we get,

(BD/DC) = (BE/CE)

(BD/BE) = (DC/CE)

Hence proved.

**Example 4 :**

ABCD is a quadrilateral with AB = AD. If AE and AF are internal bisectors of ∠BAC and ∠DAC respectively,then prove that the sides EF and BD are parallel.

**Solution:**

In triangle ABC, AE is the internal bisector of ∠BAC

by using bisector theorem, we get

(AB/AC) = (BE/EC) ----- (1)

In triangle ADC, AF is the internal bisector of ∠DAC

by using bisector theorem, we get

(AD/AC) = (DF/FC)

Since lengths of AD and AB are equal, we may replace AB instead of AD.

(AB/AC) = (DF/FC) ----- (2)

from (1) and (2) we get

(DF/FC) = (BE/EC)

So, EF and BD are parallel by using converse of "Thales theorem".

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