Dao Thanh Oai Vietnamese describes two circles passing through

the triangle centers:

X(13), X(16), X(110), X(399), X(1338), X(2381)

X(14), X(15), X(110), X(399), X(1337), X(2380)See:

https://www.facebook.com/photo.php?fbid=1440405696182504I observe that:

The centers X(13), X(16), X(1388) are isogonal conjugates

of X(15), X(14), X(1337), resp. lying on the Neuberg cubic and the

common point X(399) lies on the same cubic as well.

Do we have more points on the Neuberg cubic with the same

property?

Or, do we have analogous circles related to other isogonal pivotal (or not?)

cubics? ie

Let X,Y,Z be three, non-isogonal conjugate in pairs, points

and X*,Y*,Z* their isog. conjugates resp.

Let U,V be the intersection points of the two circles XYZ, X*Y*Z* (real or not).

Is there an isogonal pivotal cubic passing through

X,Y,Z (and X*,Y*,Z*) and through one of the points U or V ?

Or the isogonal pivotal cubic (if exists) passing through X,Y,Z, X*,Y*,Z*

passes necessarily through one of the intersection points of the

circles XYZ and X*Y*Z* ??

APH