RE: [EMHL] The orthic limit.
> 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 sequenceAs 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.
- Dear John
> > If A[n],B[n],C[n] is the n-th orthic triangle (with A=A), wecan
> > 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 thedefinition
> could be extended in some way to circumvent it. I admit thatthat's
> rather unlikely, but would like to see a proof that the orthic limitI'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); B=(1, cot t); C=(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
- Dear friends
Owing to google, i just see this reference with the keywords
orthic limit pedal triangle
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
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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