We start with the given triangle. 

The steps 16 establish the incenter and are identical to those in
Constructing the Incenter of a 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. Where the two new lines intersect, mark a point as the incenter of the triangle. 

Optional Step 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. 
7. Draw the perpendicular from the incenter to a side of the triangle. Label the point where it meets the side M.
See Constructing a Perpendicular from a Point for this procedure.


8. Place the compasses on the incenter and set the width to point M.
This is the radius of the incircle, sometimes called the inradius of the triangle. 

9. Draw a full circle. 

10. Done. This is the incircle of the triangle 
