Angle Bisector TheoremDate: 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/ |
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