# Sequences related to curves (was: Forum Geometricorum)

Antreas P. Hatzipolakis xpolakis at otenet.gr
Mon Apr 23 22:29:15 CEST 2001

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On Mon, 09 Apr 2001, Paul Yiu wrote:

>The following paper has been published in Forum Geometricorum. It can be
>viewed at
>
>http://forumgeom.fau.edu/FG2001volume1/FG200109index.html
>
>The Editors
>Forum Geometricorum
>
>---------------------
>Antreas P.Hatzipolakis, Floor van Lamoen, Barry Wolk, and Paul Yiu,
>Concurrency of four Euler lines,
>Forum Geometricorum, 1 (2001) 59 -- 68.
>
>Abstract:   Using tripolar coordinates, we prove that if P is a point in
>the plane of triangle ABC such that the Euler lines of triangles PBC, APC
>and ABP are concurrent, then their intersection lies on the Euler line of
>triangle ABC. The same is true for the Brocard axes and the lines joining
>the circumcenters to the respective incenters. We also prove that the locus
>of P for which the four Euler lines concur is the same as that for which
>the four Brocard axes concur. These results are extended to a family L_n of
>lines through the circumcenter. The locus of P for which the four L_n lines
>of ABC, PBC, APC and ABP concur is always a curve through 15 finite real
>points, which we identify.

The reader of this paper may wonder about the sequence Sn of the orders of
the curves Cn (which are related to the lines Ln).

Jean-Pierre Ehrmann and Barry Wolk found that Sn = 2n - 1 for n even,
S1 = 14, S3 = 46 and conjectured that in general Sn = 16n - 2 for n odd.

So, the sequence Sn is: 14, 3, 46, 7, 78, .....

Speaking about sequences related to curves:

In the the monumental work:

H. Brocard et T. Lemoyne: Courbes geometriques remarquables
(courbes speciales) Planes & Gauches. Tome I.
Paris: Albert Blanchard, 1967 [First publ. in 1919]

(the only volume, out of three, that I have)

one can find some remarkable sequences.

Here is one (page 411):

0, 4, 36, 132, 340, .....

Formula:  (m-1)m(m^2 + 3m - 6)/2

This gives "le nombre des coniques qui passent par trois points et
sont bitangentes a une courbe generale d'ordre m"

The authors refer to Steiner, Cr[elle's J.], t. 49, p. 273.

More sequences:

In page 135 we find the following ones (related to "Cayleyenne" curve):

3(m-2)(5m -11)

3(m-1)(m-2)

9(m-2)(5m-13)(5m^2 - 19m + 16)/2

18(m-2)(2m-5)

9(m-2)^2 * (m^2 - 2m - 1)

In page 375:

"le nombre des tangentes doubles d'ordre m est en general"
m(m-2)(m-3)(m+3)/2

Antreas

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