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Question
Show that the angle bisectors of a triangle are concurrent
Solution
Given: ABC is a triangle. AD, BE and CF are the angle bisector of ∠A, ∠B, and ∠C.
To Prove: Bisector AD, BE and CF intersect
Proof: The angle bisectors AD and BE meet at O.
Assume CF does not pass through O. By angle bisector theorem.
AD is the angle bisector of ∠A
`"BD"/"DC" = "AB"/"AC"` ...(1)
BE is the angle bisector of ∠B
`"CE"/"EA" = "BC"/"AB"` ...(2)
CF is the angle bisector ∠C
`"AF"/"FB" = "AC"/"BC"` ...(3)
Multiply (1) (2) and (3)
`"BD"/"DC" xx "CE"/"EA" xx "AF"/"FB" = "AB"/"AC" xx "BC"/"AB" xx "AC"/"BC"`
So by Ceva’s theorem.
The bisector AD, BE and CF are concurrent.
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