Terminology: angle bisector, circle.
Required: Some proficiency with the Geometer's Sketchpad Application.
1. Select the segment tool and draw a triangle.
2. Go to the selection tool and select an angle. Go to construct and construct the angle bisector. Construct all three angle bisectors this way.
3. Select two of the angle bisectors, go to construct and find the intersection of the two lines. This gives us the intersection of all three angle bisectors. Grab one of the vertices and drag it around the screen to convince yourself that the three angle bisectors always meet in a single point.
4. Use the point tool to draw a point somewhere in the plane. (It doesn't matter where, but do not use a point that already exists as part of your figure.) Select first the intersection point of the three angle bisectors, then (while holding down the shift key) select the point you just drew. Go to construct and draw a circle given the center and a point. You can now expand the circle until it is tangent to the triangle. We have now created the inscribed circle for a triangle.
5. If you deform the triangle by taking one of the vertices and draging it along, you will have to adjust the inscribed circle. But you will see that the intersection of the three angle bisectors is always the center of the inscribed circle.