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May 23rd, 2012, 11:57 AM   #11
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Re: Paper problem

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Originally Posted by Crouch
In the second comment is proved, that we can't cut a triangle to two isosceles triangles.
Or rather, that there are some triangles that can't be cut into two isosceles triangles. I don't know that it's impossible in general.

Edit: In fact it is possible in some cases, depending on how you interpret "isosceles". Consider a right triangle with sides of length 2, and 1, and bisect the side of length 2 through the opposite vertex. The resulting triangles have sides with lengths 1, and 1; 1, 1, and 1.
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May 23rd, 2012, 01:32 PM   #12
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Re: Paper problem

ok, take any right-angled triangle. Complete it to make a rectangle. One of the sides of the original triangle is a diagonal of that rectangle. Now draw the other diagonal. Do you agree that the two diagonals cut the rectangle in 4 isosceles triangles? and that two of these isosceles triangles belong to the original right-angled triangle?

is that a number theoretic question by the way?
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May 23rd, 2012, 02:19 PM   #13
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Re: Paper problem

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Originally Posted by mehoul
ok, take any right-angled triangle. Complete it to make a rectangle. One of the sides of the original triangle is a diagonal of that rectangle. Now draw the other diagonal. Do you agree that the two diagonals cut the rectangle in 4 isosceles triangles? and that two of these isosceles triangles belong to the original right-angled triangle?
Yes, I agree.
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May 23rd, 2012, 03:04 PM   #14
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Re: Paper problem

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is that a number theoretic question by the way?
Probably not, unless you're restricting at least some of the sides to be rational or integral.
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May 24th, 2012, 12:05 AM   #15
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Re: Paper problem

there is also a dissection of some scalene triangles into 3 isosceles triangles. But it doesn't work for any triangle. Take a triangle ABC and consider the center D of the circle containing its three vertices. If this point lies inside the triangle, then draw a line from D to each of A,B,C. The three triangles ABD, BCD and CAD are isosceles. In the case of a right angle triange D lies on one side and there are only two triangle (plus a flat triangle).
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