Re: Triangles Inscribed in a Circle
- From: Thomas Mautsch <mautsch@xxxxxxx>
- Date: 5 Sep 2006 00:56:50 +0100
Maury Barbato wrote:
consider an isosceles triangle ABC, with AC=BC, inscribedequal) has a greater area and and a longer perimeter than
in a circle C. Then move the vertex A (or B) on the
circle to obtain A'C=A'B (respectively B'A= B'C). The new
triangle A'BC (or AB'C, but these two triangles are
ABC. Now you can repeat the procedure, moving B or C
(respectively A or C), to obtain a new isosceles
triangle, and so on. The sequence of inscribed triangles
has increasing area and perimeter, and I think it
converges to an equilateral triangle inscribed in the
circle. What do you think?
This result would allow to give a pure geometric proof
of the well-known fact that the equilateral triangle
is the triangle of maximum area and perimeter among all
the triangle inscribed in a given circle.
In news:<1156857484.474839.6880@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>
schrieb "The Qurqirish Dragon" <qurqirishd@xxxxxxx>:
My previous post, which asserted the conclusion actually
proved that the equilateral triangle has the maximum area.
No. You only showed
that to every inscribed triangle that is not equilateral,
there exists an inscribed triangle with larger area.
But you did not mention that one also has to assert
that there *is* an inscribed triangle
for which the area takes on a maximum value.
Otherwise, you could apply your style of "proof"
to show that 0 is the maximum number in the half-open interval [0,1):
For every value x in (0,1), the number x*(2-x) also lies in (0,1),
and is larger than x itself - so it can't be the maximal value;
on the other hand for the value x = 0, x*(2-x) stays at 0.
Would you conclude that 0 will be the maximum in the interval [0,1)?
[ ... ]
There were two questions to be answered:
1) what triangle has the maximum area
(and perimeter, which as you noted I did not show)?
That's another misunderstanding: I wanted to point out to *Maury*
that finding the inscribed triangle of maximal perimeter
is slightly more complicated than finding
the inscribed triangle of maximal area. - The reason being,
that for the second problem
you want to find the parallel line
farthest away from a given chord of the given circle
which still intersects the circle,
while for the first problem
you want to find the largest ellipse
amongst those with focal points in the endpoints of a given chord
which still intersects the given circle...
2) Is there a procedure based on the OP idea of "moving a point along
the circumfrence" that will generate this triangle?
Once you have answered 1, you can use that result to answer 2- which is
what I did. If you think that my answer/proof for part 1 is incorrect,
then that is a seperate matter- obviously a flawed statement cannot be
used in the proof of another (unless you are doing a proof by
contradiction, where the flaw is the manner of proof).
Maybe I missed out one of your messages,
but Maury asked for a proof of 2), and wanted to deduce 1) from 2);
and in the messages I have seen,
you also seemed to argue along these lines.
So, what was your proof of 1)?
Sorry for the late answer.
Thomas
.
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