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Property of Angle Bisectors II: What is this about?
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Angle bisectors divide the opposite side in the ratio of the adjacent sides. More accurately, if, in 
ABC, 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.
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Then triangles ACD and ABE are similar. In particular,
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AEB = ADC.
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From where (and if necessary passing to the supplementary angles)
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BED = BDE.
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Which implies that DBE 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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