Re: Internal Bisectors of a Triangle


Maury Barbato wrote:
I wrote:

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

I found the wonderful page
http://www.cut-the-knot.org/triangle/index.shtml
where many constructions are listed. I found here
a simple proof of the impossibility of the construction
of a triangle given the lengths of its bisectors.
However, I didn't yet understand what are the answer to
my questions.

(I) Given three arbitrary lenghts, is there a triangle
whose angle bisectors have the given lengths?

(II) Is it unique?

My Best Regsrds,
Maury

Just to clarify, by the "length" of an angle bisector do you mean the
distance from the vertex to the intersection of the bisector with the
opposite side of the triangle? (Rather than, for example, the distance
from the vertex to the point of intersection of the bisectors.)

.

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