|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 ...
The same method can used to show that Triangle CID = Triangle CHD.
We are ready to go ahead.
Now, we make a segment AD.
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'.