RE: [EMHL] The orthic limit.

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• ... see ... As John pointed out, this part of the argument is erroneous. Sorry for this but I wrote in haste after a five hour(1) Departmental meeting.
Message 1 of 9 , Jun 5, 2002
>
> One last point: If G is the (common) centre of gravity of the nth
> circle and O(n) its circumcirle then, as GO(n+1)= (1/2)* GO(n), we
see
> the exact rate at which O(n) approaches G. Note also that the sequence
> O(n), which is on the Euler line, alternates sides around G.
>

As John pointed out, this part of the argument is erroneous. Sorry for
this but I wrote in haste after a five hour(1) Departmental meeting.

Michael.
• Dear John [JPE] ... can ... [JHC] ... definition ... that s ... I m just looking at a problem of a French examination (Ecole Centrale : May 2002). The problem
Message 2 of 9 , Jun 5, 2002
Dear John

[JPE]
> > If A[n],B[n],C[n] is the n-th orthic triangle (with A[0]=A), we
can
> > see, by induction that
> > B[n]C[n] = R|sin(2^n A)|/2^(n-1)
> > and A[n]A[n+1] = R|sin(2^n B)sin(2^n C)|/2^(n-1)

[JHC]

> I was aware of this behavior, but thought that maybe the
definition
> could be extended in some way to circumvent it. I admit that
that's
> rather unlikely, but would like to see a proof that the orthic limit
> isn't analytic.

I'm just looking at a problem of a French examination (Ecole
Centrale : May 2002).
The problem studies the particular case of an isosceles triangle
A[0]=(0,0); B[0]=(1, cot t); C[0]=(1, -cot t) (in rectangular
cartesian coordinates) where 0<t<Pi.
The sequence converges to(x(t),0) where
x(t) = 1 + sum((-1/2)^n (sin(2^n t)/sin(t))^2, n = 1..infinity)
or (using Fourier expansion)
x(t) sin(t)^2 = 1/3 + sum((-1/2)^n cos(2^n t), n=1..infi)
Then, it is proved that x is continuous on (0,Pi) and admits a
derivative at no point.
The whole problem is available at
http://csmp.ecp.fr/CentraleSupelec/2002/PC/sujets/math1.pdf
Friendly. Jean-Pierre
• Dear friends Owing to google, i just see this reference with the keywords orthic limit pedal triangle www.mathstat.usouthal.edu/~hitt/research/markov.pdf I
Message 3 of 9 , Jun 5, 2002
Dear friends

Owing to google, i just see this reference with the keywords
orthic limit pedal triangle
www.mathstat.usouthal.edu/~hitt/research/markov.pdf
I haven't the time to analyze now

regards from the rainy south east of the France

Arnaud PASCAL, teacher in highschool

[Non-text portions of this message have been removed]
• I think that this problem is solved in The Sequence of Pedal Triangles , by J.G. Kingston and J.L. Synge, American Mathematical Monthly volume 95 number 7,
Message 4 of 9 , Jun 5, 2002
I think that this problem is solved in "The Sequence of Pedal Triangles", by
J.G. Kingston and J.L. Synge, American Mathematical Monthly volume 95 number
7, 1988.

Quim Castellsaguer
Barcelona (Spain)

En/Na John Conway ha escrit:

> Let A1 B1 C1 be the orthic triangle of A B C, A2 B2 C2 the
> orthic triangle of A1 B1 C1, and so on. Then I think it's pretty
> obvious that the triangles An Bn Cn tend to a point, which I'll
> call the orthic limit point of A B C.
>
> Of course this is only one of a number of such limits, but it's
> about the simplest whose location isn't immediately obvious. May