Geometry Forum Project of the 
Month

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COMMENTS: These guys did a great job. They were very methodical and considered many possibilities. They also managed to define the problem well enough to write a computer program to find the answers. Their explanation goes through all the aspects of the problem they considered and what constraints there were on each option. Realizing that each triangle could just as well include the origin certainly helped narrow things down. In evaluating the solutions, I also found it very useful to have the points included, instead of just the lengths of the sides. Overall, excellent job, and well-written explanation.
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MY METHOD:  I decided to look at the points A, B, and C where one of the
following is true:<P>

        AB=BC and the slope of CA isn't 0, undefined, 1, or -1<br>
        BC=CA and the slope of AB isn't 0, undefined, 1, or -1<br>
     or CA=AB and the slope of BC isn't 0, undefined, 1, or -1<P>

I started by letting A=(0,0), B=(x1,y1), and C=(x2, y2) and then defining
the distances between points as the lengths of the segments and the
slopes, etc.  I figure that with the problem this well defined, you can
run something through the computer that checks all possible combinations
of (x1,y1) and (x2,y2).  But I didn't get that far, so maybe one of you
wants to do it?
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3 July 1995