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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). See Table 76.

The isogonal transform of K465 is K466, its inverse in the circumcircle is K1351, two other focal cubics. See Table 73 for properties and other related cubics.

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).

• locus of isopivots of circular pKs passing through X(5). The locus of pivots is K1350. See K050, K060, K067.