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Topic: Dissecting a heronian triangle (Read 1973 times) |
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JocK
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Dissecting a heronian triangle
« on: Aug 18th, 2005, 12:20pm » |
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Can you dissect a Heronian triangle into three Heronian triangles?
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solving abstract problems is like sex: it may occasionally have some practical use, but that is not why we do it.
xy - y = x5 - y4 - y3 = 20; x>0, y>0.
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Barukh
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Yes.
Moreover, I think Heronian triangle can be dissected into any number of Heronian triangles.
JocK, do you want to strenghthen the requirements?
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Grimbal
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Re: Dissecting a heronian triangle
« Reply #2 on: Aug 19th, 2005, 1:41am » |
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If you know a way to dissect a heronian triangle into 2 heronian triangles, it is trivial to dissect one into n heronian triangles.
Whatever a heronian triangle is... 
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Grimbal
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Re: Dissecting a heronian triangle
« Reply #3 on: Aug 19th, 2005, 4:12am » |
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Funny. I reread the question and I see it asks to cut the triangle in 3. I would have sworn it said to cut it in 2, which explains my remark.
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Grimbal
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Re: Dissecting a heronian triangle
« Reply #4 on: Aug 19th, 2005, 4:20am » |
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This should work:
If you have a triange ABC, where AB<AC, construct a similar A'B'C', mirror reversed and scaled down, such that
A' = A, C' = B, B' is on AC.
This cuts the triangle in 2. Repeat to have 3 pieces.
All sides should be rational, as are the surfaces.
It doesn't work for the equilateral triangle, but it isn't heronian anyway.
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JocK
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Re: Dissecting a heronian triangle
« Reply #5 on: Aug 19th, 2005, 1:24pm » |
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on Aug 18th, 2005, 11:12pm, Barukh wrote:
JocK, do you want to strenghthen the requirements? |
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Sure...  ... the follow-up question is:
Can you dissect a Heronian triangle in three Heronian triangles such that the resulting triangles have equal area?
(The much harder follow-up question I leave for later...  )
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| « Last Edit: Aug 19th, 2005, 1:28pm by JocK » |
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solving abstract problems is like sex: it may occasionally have some practical use, but that is not why we do it.
xy - y = x5 - y4 - y3 = 20; x>0, y>0.
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Barukh
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I don't know what is the much harder follow-up, but this equal-area variation is hard enough (at least for me).
The more I look at this problem, the more I think that the answer is "no", but cannot prove it. What I succeeded to show is that certain dissections (e.g. depicted below) are impossible.
Any comments?
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| « Last Edit: Aug 30th, 2005, 3:55am by Barukh » |
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