Property of Angle Bisectors II
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A Mathematical Droodle
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Copyright © 1996-2018 Alexander BogomolnyAngle 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.
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