Angle Bisector Theorem

Date: 10/11/2002 at 02:45:00
From: Jeffrey S. Pankewich
Subject: Euclidian Geometry theorem

Dr. Math,

I am studying to be a middle school math teacher, and I am taking a 
Foundations of Geometry course. We have come to a theorem that our 
group just can't figure out. The theorem (IX. 16) states, "The 
bisector of an interior angle of a triangle divides the opposite side 
internally into two segments which are proportional to the adjacent 
sides."  If you could provide any help, we would greatly appreciate 
it.

Thank you,
Jeffrey S. Pankewich

Date: 10/11/2002 at 13:02:08
From: Doctor Peterson
Subject: Re: Euclidian Geometry theorem

Hi, Jeffrey.

As often happens, this becomes easy if you draw some extra lines, and 
get a few little insights. Rather than show you, let me ask a couple 
questions so you can enjoy the "ahah!" moment yourself.

Here's a picture illustrating the theorem:
  

What's interesting about the angle bisector? If you recall that it is 
used in finding the incenter of the triangle, or have studied loci, 
you will know that every point on the angle bisector is equidistant 
from the two legs of the angle. There's only one point in our picture 
on this bisector, other than vertex C itself. Draw some line segments 
from that point, related to the fact I just mentioned, and mark what 
you know about them.

Now look at the triangles ACD and BCD. What do you know about their 
AREAS?

If you need more help, please write back and show me how far you got.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/