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K732

too complicated to be written here. Click on the link to download a text file.

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.