
Re: to prove: rays bisecting 3 angles of a triangle meet at a single point
Posted:
Dec 28, 1997 5:34 AM


Michael Keyton wrote:
> 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.
new bisector   \  / \  / \  / \  / \/ X old bisector /\ /  \ /  \ /  \
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 wwwpage of Clark Kimberling: http://www.evansville.edu/~ck6/tcenters/index.html
Regards, Floor van Lamoen

