May 17th, 2012, 04:47 AM  #1 
Member Joined: Dec 2009 Posts: 65 Thanks: 0  Paper problem
Hello! I have an interesting problem, and I don't have any idea how to solve it... It is possible to cut a triangleshaped paper to n (2, 3, 4, 5, ...) isosceles triangleshaped piece of papers? Many thanks, Crouch. 
May 17th, 2012, 07:57 AM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Paper problem
The case n = 2 is not possible in general. Since the original triangle must be cut into two triangles (rather than a triangle and a quadrilateral), you must cut from a point to a side (not a side to a side or a point to a point). Suppose the original triangle has sides of length 10, 10, and 1. Then one of the smaller triangles must have a side of length 10, and hence it must have a second of length 10 to be isosceles (since otherwise it must have two equal sides of length > 5, and you don't have that much 'material'.) But this is not possible without using the whole triangle, leaving nothing for the second triangle.

May 18th, 2012, 02:57 AM  #3 
Member Joined: Dec 2009 Posts: 65 Thanks: 0  Re: Paper problem
Thanks, understood. But can we say something about the very general case? The case of n? I think it's very difficult...

May 19th, 2012, 03:44 PM  #4 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,885 Thanks: 1088 Math Focus: Elementary mathematics and beyond  Re: Paper problem
I don't think any n is possible if the sheet of paper is initially a scalene triangle. If it is initially an isosceles triangle then there any n > 2 isosceles triangles that can be formed, with size as the only limiting factor. 
May 20th, 2012, 02:48 PM  #5 
Senior Member Joined: Feb 2012 Posts: 144 Thanks: 16  Re: Paper problem
a scalene triangle can be cut into 4 isoceles triangles: first cut it into two rightangled triangles.

May 22nd, 2012, 08:23 PM  #6  
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,885 Thanks: 1088 Math Focus: Elementary mathematics and beyond  Re: Paper problem Quote:
Can you provide another counterexample?  
May 23rd, 2012, 02:29 AM  #7 
Senior Member Joined: Feb 2012 Posts: 144 Thanks: 16  Re: Paper problem
mmmh, I may not understand correctly the problem... to me an isosceles triangle is a triangle with to equal sides. And to divide a triangle in two triangles is to draw a line from one vertex to the opposite edge. If I am right, then join the vertex where the right angle to the middle point of the opposite edge. This middle point is equidistant from the three vertices, hence the two triangles are isosceles.

May 23rd, 2012, 02:32 AM  #8  
Senior Member Joined: Feb 2012 Posts: 144 Thanks: 16  Re: Paper problem Quote:
 
May 23rd, 2012, 08:37 AM  #9  
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,885 Thanks: 1088 Math Focus: Elementary mathematics and beyond  Re: Paper problem Quote:
The construction you give above must be a 306090 triangle, no?  
May 23rd, 2012, 10:42 AM  #10 
Member Joined: Dec 2009 Posts: 65 Thanks: 0  Re: Paper problem
Mehoul, you are not right. In general, you can't cut a triangle to two isosceles triangles. In the second comment is proved, that we can't cut a triangle to two isosceles triangles. Any idea for the general case? 

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