Assignment # 4: Part 15

Angle Bisectors in a Triangle


  Our objective is o prove the concurrency of angle bisectors in a triangle. We will do that by constructing two of them and looking at the point of intersection.

Now, we want to drop some perpendiculars (in red) from D to sides AB, AC, and CB. This will create the right angles that we need to make congruent triangles.

 

Let's prove some congruence. We will use the equals sign (=) not literally, but as a means of congruence ...

  • Angle GBD = Angle IBD
  • Angle BGD = Angle BID
  • BD = BD
  • So Triangle BGD = Triangle BID by AAS

The same method can used to show that Triangle CID = Triangle CHD.

  • GD = DI
  • DI = DH
  • So GD = DH

We are ready to go ahead.

 

Now, we make a segment AD.

  • AD = AD
  • Angle DGA = Angle DHA = Right Angle
  • GD = FD

So by Hypotanuse-Leg Theorem, Triangle GDA = Triangle HDA.

From that, we know that Angle GAD = Angle FAD, so the angle bisector does pass through point 'D'.

Therefore, angle bisectors are concurrent.

Return to Pawel's 6680 Class Page