Property of Angle Bisectors II
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A Mathematical Droodle


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Explanation

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Copyright © 1996-2018 Alexander Bogomolny

Angle bisectors divide the opposite side in the ratio of the adjacent sides. More accurately,

If, in ΔABC, AD is an angle bisector of angle A, then

AB/AC = DB/DC.


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(The proof below was suggested by Prof. W. McWorter.)

Proof

Let E on AD (or its extension) be such that

∠ACB = ∠ABE.

Then triangles ACD and ABE are similar. In particular,

∠AEB = ∠ADC.

From where (and if necessary passing to the supplementary angles)

∠BED = ∠BDE.

Which implies that ΔDBE is isosceles:

(*) BD = BE.

On the other hand, similarity of triangles ACD and ABE implies

AB/AC = BE/DC,

which combined with (*) gives the desired proportion. (The proof is different from a more standard one.)

This property of angle bisectors is one way to show that the three angle bisectors in a triangle meet in a point. The result is an immediate consequence of Ceva's theorem.


Related material
Read more...

Angle Bisector

  • Angle Bisector
  • Angle Bisector Theorem
  • All about angle bisectors
  • Angle Bisectors in Ellipse
  • Angle Bisectors in Ellipse II
  • Angle Bisector in Equilateral Trapezoid
  • Angle Bisector in Rectangle
  • Property of Angle Bisectors
  • A Property of Angle Bisectors III
  • External Angle Bisectors
  • Projections on Internal and External Angle Bisectors
  • Angle Bisectors On Circumcircle
  • Angle Bisectors in a Quadrilateral - Cyclic and Otherwise
  • Problem: Angle Bisectors in a Quadrilateral
  • Triangle From Angle Bisectors
  • Property of Internal Angle Bisector - Hubert Shutrick's PWW
  • Angle Bisectors Cross Circumcircle
  • For Equality Choose Angle Bisector
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