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X(6), X(5467), X(9178)

centers of the 3 Apollonian circles (inflexion points)

K229 = nK(X32, X6, X6) = cK(#X6, X6) is a circum-conico-pivotal cubic (See Special Isocubics ยง8.3). Compare K229 with K228 isogonal circum-conico-pivotal cubic and K015 isotomic circum-conico-pivotal cubic.

It is an unicursal cubic with singularity at K (Lemoine point) and pole X(32). For any point M on K229, the line through M and its isoconjugate M* envelopes the circumcircle. K229 is tritangent at A, B, C to this circle.


Locus property :

Let M be a variable point on the circumcircle. The trilinear polar of M and the tangent at M to the circumcircle meet at M'. The locus of M' is K229.

The osculating circles at A, B, C have the same radius namely R/8.

The triangle formed by their centers is the image of ABC under the homothety h(X3, 9/8).

This property is actually true for any nK(X32, X6, ?) although with a different radius. See K214 for another example.


The isogonal transform of K229 is the Tucker cubic K015. K015 and K229 are two members of CL064.