Re: Internal Bisectors of a Triangle

matt271829-news@xxxxxxxxxxx wrote:

Maury Barbato wrote:
Hello,
the Italian Mathematician O. Chisini proved in his
work "Sulla costruzione di un triangolo date le
tre
bisettrici, Periodico di Mat., (4), 1, 1921, 43-51
and
108-121" that the construction of a triangle, given
its three internal bisectors, can't be made only with
ruler
and compass.

By "internal bisectors" do you mean the perpendicular
bisectors of the
triangle's sides? If so, I'm a little surprised that
you can't
construct a triangle having given bisectors using
ruler and compass. It
looks to me like you can do the calculations with no
more than
addition, subtraction, multiplication and division of
lengths, which
should mean you can do it with ruler and compass, no?


Surely this contruction should be possible. But I don't
know your reasons: what calculations do you refer?

I don't think you can mean the angle bisectors
either, as a
ruler-and-compass construction is possible in that
case (by
coincidence, see
http://groups.google.com/group/alt.math.undergrad/brow
se_frm/thread/9634317e4e965288).


Maybe I have made a mistake, or maybe you mean
something else.


Sorry, I have made a howler. In the problem studied by
Chisini is totally different: to construct a triangle
give the lengths of its three angle bisectors. I think
the existence is not ensured, but maybe the uniqueness
is. What do you think about?

I have now two more elementary questions:
(I) given three distinct straight lines concurrent
in a
point I, is there a triangle having these lines as
internal bisectors?
(II) if such a triangle exists, is it essentially
unique
(that is every other triangle which solves the
problem
can be obtained by the first by a homothety with
center
O)?
Thank you very much for your help.
Maury


Many other problems like that studied by Chisini can be
put. E.g., one can assign the lenghts of the three
perpendicular bisectors (the segmnents which join the
circumcenter to the midpoints of each side). Do you know
a bibliography about them?

Thank you very much for your help.
Maury
.

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