22016TWO CIRCLES - CUBICS
- Nov 26, 2013cubics? ieOr, do we have analogous circles related to other isogonal pivotal (or not?)property?Do we have more points on the Neuberg cubic with the sameof X(15), X(14), X(1337), resp. lying on the Neuberg cubic and theI observe that:Dao Thanh Oai Vietnamese describes two circles passing throughthe 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)
The centers X(13), X(16), X(1388) are isogonal conjugates
common point X(399) lies on the same cubic as well.
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 throughX,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 thecircles XYZ and X*Y*Z* ??APH