Online Course Discussion Forum

How to prove the converse of the Angle Bisector Theorem

 
 
Picture of Dan Wei
How to prove the converse of the Angle Bisector Theorem
by Dan Wei - Saturday, 24 July 2021, 2:13 PM
 

This question appeared on my homework and I have no idea how to start. Help would be appreciated. Thanks!

 
Picture of Dr. Kevin Wang
Re: How to prove the converse of the Angle Bisector Theorem
by Dr. Kevin Wang - Monday, 26 July 2021, 12:35 AM
 

Let's phrase the converse of the Angle Bisector Theorem as follows:

In $\triangle ABC$, let $D$ be a point on side $\overline{BC}$ such that $\dfrac{BD}{DC}=\dfrac{AB}{AC}$.  Prove that $\overline{AD}$ bisects $\angle BAC$.

To prove this result, you may consider the "phantom point" method.  Construct the angle bisector $\overline{AD'}$ of $\angle BAC$, where $D'$ is on $\overline{BC}$.  Then try to prove that $D'$ and $D$ are in fact the same point.