Angle Bisectors & the Incenter
Another special triangle segment is an angle bisector. It is a segment that cuts an angle of the triangle into two equal parts; it has one endpoint at a vertex and the other on the opposite side. How many angle bisectors can there be in a triangle? What kind of triangle would you have if one of the angle bisectors and one of the altitudes are the same line?
The point of concurrency of the three angle bisectors is called the incenter of a triangle. Each point on an angle bisector is the same distance from the sides of an angle. Since the incenter is on all three angle bisectors, it is equally distanct from all three sides. Remember that the center of a circle is the same distance from all points on the circle, so a circle can be drawn inside the triangle just touching the three sides. It is called an inscribed circle. Using Geometer's Sketchpad can you create the inscribed circle?