Re: Internal Bisectors of a Triangle

matt271829-news@xxxxxxxxxxx wrote:


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.)


Surely the length of an angle bisector is the distance
from the vertex to the intersection of the bisector with
the opposite side of the triangle.
I have studied Chisini's work. I'm quite sure now that
the answer to (II) is no. Maybe, (I) has a positive
answer, but I'm not sure. The equation studied by Chisini
is quite complicated: it is an equation (in two variables) of degree 10!!! He studies it geometrically,
using the means of algebraic geometry, whose I have no
idea. So ...

My Best Regards,
Maury
x,y
.

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