## Σάββατο, 14 Σεπτεμβρίου 2013

### PRIZE (Re: ORTHOCENTER - REFLECTIONS - CONCURRENT CIRCLES)

[APH]: In fact we can take any point P (instead of H) and any points O1,O2,O3 on the circumcircles of PBC,PCA,PAB, resp.

Then the circumcircles of the triangles

AO2O3, BO3O1, CO1O2

are concurrent.

Anopolis #850

For a proof I offer the book:

R. G. SANGER: SYNTHETIC PROJECTIVE GEOMETRY (1939)

APH

## Πέμπτη, 12 Σεπτεμβρίου 2013

### CONCYCLIC

RE: [EGML] CONIC - LOCUS

Fri Sep 6, 2013 10:28 pm

Dear Antreas:

They are concyclic for all points P.

The center of their circle is the complement of the isotomic of P.

Regards

De: Anopolis@yahoogroups.com [mailto:Anopolis@yahoogroups.com] En nombre de Antreas Hatzipolakis

Enviado el: Jueves, 05 de Septiembre de 2013 06:28 p.m.

Para: anopolis@yahoogroups.com

Asunto: [EGML] CONIC - LOCUS

Let ABC be a triangle and A'B'C' the cevian triangle of P.

Denote:

Ab, Ac = the intersections of the circles with diameters BC, AA'

(near to B,C, resp.)

Bc, Ba = the intersections of the circles with diameters CA, BB'

Ca, Cb = the intersections of the circles with diameters AB, CC'

For P = H, the six points are concyclic.

For which P's are the six points lying on a conic?

APH

Anopolis #956

## Πέμπτη, 5 Σεπτεμβρίου 2013

### INTEGER SEQUENCES

In the book:

Marc Barbut, Bernard Monjardet: Ordre et Classification: Algebre et Combinatoire. Tome II. Hachette Universite, Paris, 1970.

are listed some integer sequences.

In pp. 44-46 the following:

1,3,19,219,4231,130023,6129859,.... It is A001035 in OEIS (The On-Line Encyclopedia of Integer Sequences)

1,4,29,355,6942,209527,9535241..... It is A000798 in OEIS

1,3,9,33,139,.... It is A001930 in OEIS 1,3,13,75,541,4683,47293,... It is A000670 in OEIS

In p. 101 the following:

1,2,5,,15,52,203,876,... It is A056273 in OEIS

0,1,3,7,15,31,63,.... It is A000225 in OEIS

0,0,1,6,25,90,301,... It is A000392 in OEIS

In pp. 165-166 the following:

1,4,18,166,7579,7828352,2414682040996,... It is A007153 in OEIS

1,2,4,12,81,... It is A001206 in OEIS

BIBLIOGRAPHY:

For pages 1-78:

For pages 83-166:

## Τετάρτη, 4 Σεπτεμβρίου 2013

### PRIZE FOR CONCURRENT CIRCLES CONJECTURE

This "theorem" (conjecture) is still unproved (quoetd below).

Seiichi Kirikami has computed the coordinates for (P, P*) = (G, K)

Available here: Hyacinthos #21992

I offer the following books for proofs:

1. For an analytic proof by computing the homogeneous coordinates of the concurrence points:

RICHARD HEGER: ELEMENTE DER ANALYTISCHEN GEOMETRIE IN HOMOGENEN COORDINATEN (1872)

2. For a synthetic proof:

FRANK MORLEY & F. V. MORLEY: INVERSIVE GEOMETRY

3. For any other proof (by complex numbers, etc):

C. ZWIKKER: THE ADVANCED GEOMETRY OF PLANE CURVES AND THEIR APPLICATIONS.

Antreas

--- In Anopolis@yahoogroups.com, "Antreas" wrote:

>

> Let ABC be a triangle and P,P* two isogonal conjugate points.

> Denote: H1,H2,H3 = the orthocenters of PBC, PCA, PAB, resp. and

> Ha,Hb,Hc = the orthocenters of P*BC, P*CA, P*AB, resp.

>

> The circumcircles of:

> (1) H1HbHc, H2HcHa, H3HaHb

> (2) HaH2H3, HbH3H1, HcH1H2

>

> are concurrent.

>

> Figure: Here

>

> If P = (x:y:z), which are the points of concurrences?

>

> APH

>