Property of Angle Bisectors II
What is this about?
A Mathematical Droodle
Explanation
|Activities|
|Contact|
|Front page|
|Contents|
|Geometry|
|Store|
Copyright © 1996-2012 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.
(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:
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.
|Activities|
|Contact|
|Front page|
|Contents|
|Geometry|
|Store|
Copyright © 1996-2012 Alexander Bogomolny
| 
