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

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