External Angle Bisectors
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A Mathematical Droodle

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Explanation

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Copyright © 1996-2018 Alexander Bogomolny

Internal 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

Perhaps curiously, the same is true of the external angle bisectors, i.e., bisectors of the external angles. And the proof is in fact the same.

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Proof

Assume the straight line through C parallel to AD meets AB in E. Then, first of all, ΔAEC is isosceles: AC = AE. This is because

∠AEC = ∠BAD = 90° - ∠BAC/2,

such that

∠ACE = 180° - ∠CAE - ∠AEC = 90° = ∠BAC/2

Therefore, AE = AC, and the required proportion follows from the similarity of triangles BEC and BAD.

This property of angle bisectors is one way to show that one internal and two external 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
  • Property of Angle Bisectors II
  • A Property of Angle Bisectors III
  • 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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    Copyright © 1996-2018 Alexander Bogomolny

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