- Dear friends,

Given two fixed isogonal points, X and X', and two variable isogonal

points, P and P', what is the locus of P such that X, X', P, P' are

concyclic?

Special cases: X,X' = G,K; O,H; 1st and 2nd Brocard points?

Best regards,

Randy - Dear Randy,

> Given two fixed isogonal points, X and X', and two variable isogonal

I find a bicircular isogonal circum-sextic passing through the in/excenters, X, X', the intersections of (O) and the line XX', their isogonal conjugates at infinity.

> points, P and P', what is the locus of P such that X, X', P, P' are

> concyclic?

>

> Special cases: X,X' = G,K; O,H; 1st and 2nd Brocard points?

A, B, C, X, X' are nodes.

When X lies on (O), the sextic splits into (O), the line at infinity and the pK(X6, X). See

http://bernard.gibert.pagesperso-orange.fr/Tables/table17.html

These sextics seem to be not very prolific in ETC centers, in particular your special cases.

There are 3 bicircular isogonal circum-sextics in CTC but none of them corresponds to this configuration.

Who's going to find a nice one ?

Best regards

Bernard

[Non-text portions of this message have been removed] - Thanks, Bernard and Francisco!

A couple of related problems:

1. What is the locus of the circumcenters?

2. Do ETC centers X(i) and X(j) exist such that X(i), X(j) and their

isogonal conjugates (whether in ETC or not) are concyclic?

Randy

--- In Hyacinthos@yahoogroups.com, Bernard Gibert <bg42@...> wrote:

>

> Dear Randy,

>

> > Given two fixed isogonal points, X and X', and two variable isogonal

> > points, P and P', what is the locus of P such that X, X', P, P' are

> > concyclic?

> >

> > Special cases: X,X' = G,K; O,H; 1st and 2nd Brocard points?

>

> I find a bicircular isogonal circum-sextic passing through the

in/excenters, X, X', the intersections of (O) and the line XX', their

isogonal conjugates at infinity.

> A, B, C, X, X' are nodes.

>

> When X lies on (O), the sextic splits into (O), the line at infinity

and the pK(X6, X). See

>

> http://bernard.gibert.pagesperso-orange.fr/Tables/table17.html

>

> These sextics seem to be not very prolific in ETC centers, in

particular your special cases.

>

> There are 3 bicircular isogonal circum-sextics in CTC but none of them

corresponds to this configuration.

>

> Who's going to find a nice one ?

>

> Best regards

>

> Bernard

>

> [Non-text portions of this message have been removed]

>