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Subject: "Build ABC from OHI"     Previous Topic | Next Topic
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Conferences The CTK Exchange College math Topic #696
Reading Topic #696
jack202
Member since Sep-30-08
Sep-30-08, 12:52 PM (EST)
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"Build ABC from OHI"
 
   Looking at
https://www.cut-the-knot.org/triangle/index.shtml

I was asking myself how to find the vertices of ABC, with
straightedge and compass, knowing the positions of O,H,I.

(Incidentally, I underline that (A,a,R) does not fix
a triangle, since the sine law holds.)

There are some remarkable facts.
By calling N the midpoint of OH we have that

1) IN = R/2 - r due to Feuerbach's theorem
2) OI^2 = 2R(R/2-r) due to Euler's theorem

so we know (R,r) and can easily draw the circumcircle, the incircle,
the 9-points circle, the polar circle, the ortic axis and so on.

Moreover, due to Poncelet's lemma, if we take a point U on
the circumcircle, draw the tangents to the incircle and intersect
them with the circumcircle, we determine a chord VW that is
also tangent to the incircle.

(Incidentally, I've noted that one of the proof of the Butterfly
theorem on this site proves Poncelet's lemma too.)

For all this UVW triangles, I,O,R and r are fixed, so, for example,
when P travels on the circumcircle, the nine-point-center N(UVW)
travels on a circumference with center I and radius (R/2-r).

However, the map that brings U into N(UVW) is not a "simple"
map between circumferences: it is 3-to-1, and its inversion
is a cubic problem, generally not solvable with straightedge
and compass.

Otherwise, we can build G (centroid), Na (Nagel point)
and F (tangency point of the incircle and the 9-point-circle)
and try to intersect the Feuerbach hyperbola (we know
its center F and three points on it: I,H,Na) with the circumcircle,
but this is a fourth (or third?!) degree problem.

This considerations raise the question of the possibility,
or, more probably, the impossibility of the construction.

Using trilinears, one may say that it is hard to determine
unsymmetric things like the coordinates of the vertices A<1,0,0>
B<0,1,0> C<0,0,1> dealing with proper (symmetric) centers of the
triangle (there are also a good number of reformulations in terms
of trigonometric functions of the angles in A,B,C, or symmetric
functions of the lengths of the sides).

So things become even more hard, and I ask you:

1) Does the (I,O,H) problem admit a simpler reverse construction
(lower degree) than mine?

2) WHEN, given three proper centers of ABC, is it possible
to find (A,B,C) with straightedge and ruler?



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alexbadmin
Charter Member
2285 posts
Oct-01-08, 08:13 AM (EST)
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1. "RE: Build ABC from OHI"
In response to message #0
 
   A sorry "I do not know" is my answer to both your questions.

The answer to the second one may well be known to the experts. I would chack with the Hycinthos mailing list at

https://tech.groups.yahoo.com/group/hyacinthos/

They will know if anybody at all does.

I would very much appreciate if, after learning more about you question, you continued your thread here.

(I am going to add an entry at that table of triangle constructions accompanied by your message. Does your name has an apostrophe after D?)


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jack202
Member since Sep-30-08
Oct-01-08, 09:08 AM (EST)
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2. "RE: Build ABC from OHI"
In response to message #1
 
   Thank you very much, I've a pending subscription to Hyacinthos
at the moment, and I'm willing to continue the discussion.

There's a simple algebraic reformulation of the problem in terms
of symmetric functions of the ex-radii (I'll post it later).
Does this imply that we are dealing with an "intrinsic-3rd-degree"
problem? Does it'suffice to state the impossibility of the euclidean
reverse construction?

I'd like to discuss another question, too, different (cyclic)
but close to the first one (symmetric).

Knowing the lenghts of the bisectors, is it possible to rebuild ABC?
(With straighedge and ruler, I mean.)

And: Yes, my surname has an apostrophe after the "D". ;-)


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alexbadmin
Charter Member
2285 posts
Oct-01-08, 09:10 AM (EST)
Click to EMail alexb Click to send private message to alexb Click to view user profileClick to add this user to your buddy list  
3. "RE: Build ABC from OHI"
In response to message #2
 
   >Knowing the lenghts of the bisectors, is it possible to
>rebuild ABC?

See

https://www.cut-the-knot.org/triangle/TriangleFromBisectors.shtml


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