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Triangle Proof: Parallel Sides and Proportionality

Date: 07/27/2004 at 09:24:25
From: Sue
Subject: Ratio

How do I prove that a line which cuts two sides of a triangle
proportionately is parallel to the third side?

I can prove that a straight line drawn parallel to one side of a
triangle will divide the other two sides proportionately, but I really
have no idea how to prove the converse.



Date: 07/28/2004 at 09:15:09
From: Doctor Willae
Subject: Re: Ratio

Sue,

Thanks for writing to Dr. Math.  When dealing with geometric proofs
like this, I like to draw things out.  And, when I do, I like to make
it as simple as possible.  So, for example, when I draw a triangle for
your problem, I'm going to put one vertex at the origin and another on
the x-axis.  Let's take a look (and please forgive the bad ASCII art):

                 (B, C)
                    *
                   * *
                  *   *
                 *     *
                *       *
               *         *
              *           *
             *             *
            *               * 
           *                 *
          *                   *     
         ***********************
      (0, 0)                  (A, 0)

OK, here's what's going on.  As I said, I've placed one vertex at the 
origin (0, 0).  I put the second vertex along the x-axis at some point 
(A, 0).  I can't make any assumptions about the third vertex without 
losing generality, so it's just at some point, (B, C).

Now let's add in the cutting line.  Let P be the fraction of the side
that's cut, as measured from point (B,C). (This implies that P must 
be in the range [0, 1].)

                      (B, C)
                         *
                        * *
                       *   *
                      *     *
                     *       *
                    *         *
                   *           *
                  *             *
                 *               * 
 (P * B, P * C) +++++++++++++++++++  (A + P * (B - A), P * C)
               *                   *     
              ***********************
           (0, 0)                 (A, 0)

All that remains to show is that the slope of the cutting line matches
the slope of the third side.  They're both zero so you're good to go.

Let me know if you have any more questions about this.

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



Date: 07/30/2004 at 22:51:03
From: Sue
Subject: Thank you (Ratio)

Doctor Willae - 

Thank you very much for your help.  I really appreciate it.
Associated Topics:
High School Triangles and Other Polygons

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