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Topic: The original geometry problem
Replies: 5   Last Post: Dec 7, 2010 11:34 PM

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Philippe 92

Posts: 44
Registered: 5/10/06
Re: The original geometry problem
Posted: Dec 7, 2010 10:43 AM
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Hi Mary,

1) No, the 18-gon is not _compass_and_straightedge_ constructible
from scratch. However it is constructible with other tools.
I won't extend this topic any further, construct the 18-gon is
equivallent to angle trisection.
A n-gon is constructible iff n = 2^k product of distinct Fermat
primes. (that is 3, 5, 17, 257, 65537, and no others are known)

2) IMHO, an angle is "given" iff it is so mentionned in the problem
assignement ;-)
All others are deduced, either obviously like when two angles
of a triangle are given, then the third is deduced. Or most
often after a more or less complicated proof, which is precisely
the aim of the problem !
Also note that giving an angle extends compass and straight
edge possibilities. The given angle may be not constructible !
The construction is now no more "from scratch" but "from a
given figure". From a _given_ 40 deg angle in the asignement,
we can easily construct the 18-gon.

About angles which are "guessed", this corresponds to what you wrote:
"Assume that the unknown angle measures xx degrees. Calculate..."
Guessing the result often gives a hint to solving.
However just "calculate... " doesn't generally suffice to solve
Sujeet's puzzles, as several values are consistent with simple
relations. We must use the "geometric HaHa trick", or deeper
calculations, to get a contradiction or a direct proof.

Best Regards.

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