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K465

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X(3), X(4), X(5), X(1154), X(1157), X(1263), X(14072), X(14979), X(14980)

Qa, Qb, Qc : reflections of A, B, C in X(5) i.e. centers of the Johnson circles

P = X(14979) = X(3)X(1291) /\ X(5)X(476) /\ X(30)X(930), etc, on the circumcircle

Q = X(14980) = reflection of P in X(5), on the circle QaQbQc.

The cubics K026 and K044 contain the centers of the three Johnson circles. See a definition here.

They generate a pencil of central cubics with center X(5), the nine point center. This pencil contains one and only one circular cubic which is K465. Thus, K465 is a central focal cubic with center and singular focus X(5).

The isogonal transform of K465 is K466.

Locus properties

• locus of M such that the nine-points circles of MAQa, MBQb, MCQc are coaxial, together with the line at infinity (see Hyacinthos #27292).

• locus of pivots of circular pKs whose orthic line passes through X(195).