I fooled around with the concept of an angle bisector, and I (thought I) found out (and some websites confirmed this, but now I'm in doubt) that the angle bisector of a vertex is the collection of points equidistant from the 2 sides of the vertex it's bisecting. However, how is this possible. Wouldn't this mean that the bisector would divide the opposite side in 2 equally long segments? But this would create a median, and obviously a median and an angle bisector are different things. Can anyone help?
Best thing would be to draw a picture. However let me clarify: If you consider a triangle (I guess you considered one), then the angle bisector is indeed the line equidistant from the two sides whose intersection is the vertex. However, unless the sides are equal, then it will not bisect the opposite side. To get what I mean, draw a scalene triangle, bisect an angle, measure and convince yourself.