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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 circum-conic (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 circum-conic (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.