Exterior Angle Bisector Theorem

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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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