Sum

Show that the angle bisectors of a triangle are concurrent

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#### 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.

Concept: Concurrency Theorems

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