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Cutting a Triangle into Two Congruent Triangles


Date: 10/06/98 at 14:59:33
From: David Auerbach
Subject: Cutting a triangle into two congruent triangles

I need to know how to cut a triangle into two congruent equilateral 
triangles with the minimum number of cuts. We have tried trial and 
error and we got nothing.

I would also like to know how to cut a square into two congruent 
squares with the minimum cuts, because, even though we solved it, 
our math teacher says there might be a better way.

Thanks for the help,
-David


Date: 10/06/98 at 16:33:17
From: Doctor Rob
Subject: Re: Cutting a triangle into two congruent triangles

For the squares, two diagonal cuts should suffice, from corner to 
opposite corner.

The triangle is a nice problem. Start with the triangle like this:

          
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Cut 1:  Vertical, through C, intersecting AB at D:

        
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Flip the left triangle over, and move it so that AC coincides with CB.
That makes a rectangle:

            
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Now cut with a line through D making a 45-degree angle with BD, and 
meeting BP at E:

             
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Now move triangle BDE so that BD coincides with PC, to form a 
parallelogram:

             
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Now cut with a line through E making the 120-degree angle DEG, and
meeting CF at point G.

             
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Translate triangle EFG so that EF coincides with CD:

           
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Finally, cut along line EH.  Then you have two equilateral triangles 
DEH and GEH, each half the size of the original.

           
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This took four cuts.  I have no proof that this is minimum, but I am 
fully convinced that it is.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Geometry
High School Triangles and Other Polygons

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