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

>


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