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'. |