Tamil Nadu Board of Secondary EducationSSLC (English Medium) Class 10th

Show that the angle bisectors of a triangle are concurrent - Mathematics

Sum

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.

Is there an error in this question or solution?

APPEARS IN

Samacheer Kalvi Mathematics Class 10 SSLC Tamil Nadu State Board
Chapter 4 Geometry
Exercise 4.4 | Q 9 | Page 198