We start with the given triangle. 

1. Place the compasses' point on any of the triangle's vertices.
Adjust the compasses to a medium width setting. The exact width is not important. 

2. Without changing the compasses' width, strike an arc across each adjacent side. 

3. Change the compasses' width if desired, then from the point where each arc crosses the side,
draw two arcs inside the triangle so that they cross each other, using the same compasses' width for each. 

4. Using the straightedge, draw a line from the vertex of the triangle to where the last two arcs cross. 

5. Repeat all of the above at any other vertex of the triangle. You will now have two new lines drawn. 

6. Done. Mark a point where the two new lines intersect. This is the incenter of the triangle. 

7. (Optional) Repeat steps 14 for the third vertex. This will convince you that the three angle bisectors
do, in fact, always intersect at a single point. But two are enough to find that point. 
