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22016TWO CIRCLES - CUBICS

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  • Antreas Hatzipolakis
    Nov 26, 2013
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    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=1440405696182504

    I 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