Answer 1

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.