> Do you know a property about how to tell if a point lies on the bisector > of an angle, or not? > > If you know this theorem, apply it to two of the bisectors, in particular > their intersection, and then argue that it is on the other bisector. And > since an angle can only have one bisector, then the three meet at a point > (the formal word for this phenomenon is "concurrent" and this point of > concurrency is called the "incenter". It is one of over 100 centers of a > triangle.
This incenter has this name because it is the center of the circle inscribed in the triangle, i.e. touching the sides. There are three very related points in a triangle, which can be found by taking alternative bisectors for two of the angles. These alternative bisectors can be found by lengthening the sides of the triangle, resulting in a 'cross'. In this cross you see that there are in fact two angles that can be bisected. The two alternative bisectors are perpendicular.
In this way you get three _excenters_, centers of three excircles. You should try to find out what that means. If you want to know more about the numerous triangle centers check out the www-page of Clark Kimberling: http://www.evansville.edu/~ck6/tcenters/index.html