Exterior Angle Bisector Theorem

Exterior angle bisector theorem : The external bisector of an angle of a triangle divides the opposite side externally in the ratio of the sides containing the angle.

Given : A ΔABC, in which AD is the bisector of the exterior ∠A and intersects BC produced in D.

Prove that : BD / CD = AB / AC

Construction : Draw CE || DA meeting AB in E.


Statements
Reasons
1) CE || DA 1) By construction
2) ∠1 = ∠3 2) Alternate interior angle
3) ∠2 = ∠4 3) Corresponding angle (CE ||DA and BK is a transversal
4) AD is a bisector of ∠A 4) Given
5) ∠1 = ∠2 5) Definition of angle bisector
6) ∠3 = ∠4 6) Transitivity (from 2 and 4)
7) AE = AC 7) If angles are equal then side opposite to them are also equal
8) BD / CD = BA/EA 8) By Basic proportionality theorem(EC ||AD)
9) BD /CD = AB/AE 9) BA = AB and EA = AE
10) BD /CD = AB /AC 10) AE = EC and from(7)

Examples

1) In the given figure, AE is the bisector of the exterior ∠CAD meeting BC produced in E. If AB = 10 cm, AC = 6 cm and BC = 12 cm, find CE.


Given : AB = 10 cm, AC = 6 cm and BC = 12 cm

By exterior angle bisector theorem

BE / CE = AB / AC

(12 + x) / x = 10 / 6

6( 12 + x ) = 10 x [ by cross multiplication]

72 + 6x = 10x
<
72 = 10x – 6x

72 = 4x

x = 72/4

x = 18

CE = 18 cm

2) The bisector of interior ∠A of ΔABC meets BC in D, and the bisector of exterior ∠A meets BC produced in E. Prove that BD / BE = CD / CE.

Given : In ΔABC, AD and AE are respectively the bisectors of the interior and exterior angles at A.

Prove that : BD/BE = CD/CE


Statements
Reasons
1) AD is the internal bisector of ∠A 1) Given
2) AB / AC = BD / DC 2) By Internal angle bisector theorem
3) AE is an external bisector of ∠A 3) Given
4) AB/AC = BE / CE 4) By External angle bisector theorem
5) BD / DC = BE / CE 5) From (2) and (4)
6) BD/BE = CD/CE 6) By alternendo
( a/b=c/d ⇒ a/c = b/d)

Similarity in Triangles

• Similarity in Geometry
• Properties of similar triangles
• Basic Proportionality Theorem(Thales theorem)
• Converse of Basic Proportionality Theorem
• Interior Angle Bisector Theorem
• Exterior Angle Bisector Theorem
• Proofs on Basic Proportionality
• Criteria of Similarity of Triangles
• Geometric Mean of Similar Triangles
• Areas of Two Similar Triangles

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