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points Ua, Ub, Uc mentioned in the Neuberg cubic page. See also table 16 and table 18.

their isogonal conjugates

The points Ua, Ub, Uc and their isogonal conjugates Ua*, Ub*, Uc* are described in the page K001.

They also lie on another isogonal cubic which is K732, the non-pivotal nK with root R732, the intersection of the lines X(2)X(6) and X(5)X(49), now X(14389) in ETC (2017-09-09).

Hence K001 and K732 are two isogonal cubics meeting at nine known points namely A, B, C, Ua, Ub, Uc, Ua*, Ub*, Uc*. They generate a pencil of cubics stable under isogonal conjugation : the isogonal transform of one member of the pencil is another member of the pencil. K001 and K732 are the "fixed" members.

K732 meets the line at infinity (L∞) and the circumcircle (O) at the same points as nK0(X6, X5306). The points on (L∞) are those of the sidelines of UaUbUc and the points on (O) are Za = UbUc* /\ Ub*Uc, Zb and Zc likewise. These points are the images of Ua, Ub, Uc under the homothety h(X376, -1/2) and also the images of Ua*, Ub*, Uc* under the homothety h(O, -1/2).

K732cer153 K732cer155

K732 is the locus of M such that M and its isogonal conjugate M* are conjugated in the fixed circle C1 with center X(550) and squared radius 4R^2 - ON^2 where N is X(5).

The polar line L(M) of M in C1 passes through M*.

K732 is the locus of M such that the pedal circle C(M) of M and M* is orthogonal to the fixed circle C2 with center X(140) and squared radius 7(R^2 - ON^2)/4.