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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.

- 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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