>

see

> 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

> the exact rate at which O(n) approaches G. Note also that the sequence

As John pointed out, this part of the argument is erroneous. Sorry for

> O(n), which is on the Euler line, alternates sides around G.

>

this but I wrote in haste after a five hour(1) Departmental meeting.

Michael.- 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

[JHC]

> > 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)

> 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

I'm just looking at a problem of a French examination (Ecole

> isn't analytic.

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 haven't the time to analyze now

Maybe it can help you,

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, 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

> I ask:

>

> 1) is it indeed true that the limit exists?

>

> 2) is it "algebraic" (ie., are its barycentric or normal coordinates

> algebraic functions of a,b,c ?

>

> John Conway

>

>

>

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