Interior Angle Bisector Theorem
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Interior Angle Bisector Theorem : The angle bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle.Given : A ΔABC in which AD is the internal bisector of ∠A and meets BC in D.
Prove that : BD / DC = AB / AC
Construction : Draw CE || DA to meet BA produced in E.

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1) CE || DA | 1) By construction |
2) ∠2 = ∠3 | 2) Alternate interior angles |
3) ∠1 = ∠4 | 3) Corresponding angles |
4) AD is the bisector | 4) Given |
5) ∠1 =∠2 | 5) Definition of angle bisector |
6) ∠3= ∠4 | 6) From (2) and (3) |
7) AE = AC | 7) In ΔACE, side opposite to equal angles are equal |
8) BD / DC = BA / AE | 8) In ΔBCE DA || CE and by BPT theorem |
9) BD / DC = AB / AC | 9) From (7) |
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Examples
1) In the given figure, AD is the bisector of ∠A. If BD = 4 cm,
DC = 3 cm and AB = 6 cm, find AC.

Solution :
In Δ ABC, AD is the bisector of ∠A.
∴ BD / DC = AB / AC ( Angle bisector theorem)
⇒ 4 / 3 = 6 / AC
⇒ 4 AC = 18
⇒ AC = 18 / 4 = 4.5 cm.
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2) AD is a median of ΔABC. The bisector of ∠ADB and ∠ADC meet AB and AC in E and F respectively.
Given : In ΔABC, AD is the median and DE and DF are angle bisectors of ∠ADB and ∠ADC respectively.
Prove that : EF || BC

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1) DE is the angle bisector of ∠ADB | 1) Given |
2) ∴ AD / DB = AE / EB | 2) By interior angle bisector theorem |
3) DF is the angle bisector of ∠ADC | 3) Given |
4) AD / DC = AF / FC | 4) By angle bisector theorem |
5) AD is the median | 5) Given |
6) BD = DC | 6) By definition of median |
7) AD / DB = AF / FC | 7) From (6) |
8) AE / EB = AF / FC | 8) From (2) and (7) |
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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