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∑ a^2 x (c^4y^2 + b^4z^2) + 2 (∑a^4 SA) x y z = 0 

X(523), X(1576) feet of the Lemoine axis 

For any point W different of K, there is one and only one nK0 with pole W which is a circular cubic. For example, with W = G, we find the cubic K197. This circular cubic is a focal cubic if and only if W lies on K214. K214 is a nK with pole X(32), root X(6) and it passes through X(523), X(1576). These two points are poles of decomposed cubics. In fact, a circular nK0 with pole on the line at infinity (resp. on the circumconic (C) with perspector X32) must split into the line at infinity and a circumconic passing through X2 (resp. the circumcircle and a line passing through X6). One asymptote of K214 is always real and is perpendicular to the Euler line. The two others are parallel to those of the circumconic (C). They are real if and only if ABC is obtusangle as in the figure below. The "last" common point of K214 and (C) is X(1576), the X(32)isoconjugate of X(523). The tangents to K214 at A, B, C are the sidelines of the tangential triangle i.e. K214 is tritangent to the circumcircle.


The osculating circles at A, B, C have the same radius namely R / 8 cosA cosB cosC. The triangle formed by their centers is homothetic at X(3) to ABC. This property is actually true for any nK(X32, X6, ?) but with a different radius. See K229 for another example.
The isogonal transform of K214 is the Tucker cubic nK(X2, X2, X110), the locus of point X such that the cevian triangles of X and X(110) have the same (algebraic) area.
K214 is a member of CL064. See also CL028. K214 is analogous to K721. 
