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.
On 27 Dec 1997, Eileen Stevenson wrote:
> I have a geometry puzzle that I cannot seem to prove. > I believe that if you bisect each of the 3 angles of a triangle with 3 > rays, those rays will meet at a single point. > But I can't prove it. Can anyone help? > > ( related: there are 2 other interesting points determined by a triangle: > 1. The center: If you bisect a side of a triangle & draw a line > passing through the bisecting point and the oposite corner of the triangle, > then do the same with the other 2 sides, all three will intersect at a > single point, the center of the triangle. This is generally NOT the same > point as I am asking about above. > > 2. The center of the circle defined by the triangle. 3 points > determine a circle. If you bisect the 3 sides with perpendicular lines, > they will meet at a point that determines the center of the unique circle > specified by the triangle. > > I can prove these 2, & will if anyone cares, but not my first queston. > Any help is appriciated. > tia. >