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X(3), X(15), X(16), X(23), X(111), X(187), X(5000), X(5001), X(6104), X(6105)

inversive images in the circumcircle of X(13), X(14), X(368), foci of the Steiner in-ellipse, vertices of the fourth Brocard triangle

centers of Apollonian circles (inversive images in the circumcircle of the vertices of the second Brocard triangle)

Let P be a point and Pa, Pb, Pc its inversive images in the Apollonian circles C_a, C_b, C_c respectively. ABC and PaPbPc are perspective if and only if P lies on the circumcircle or on the Apollonius quartic Q049. The inversive image of Q049 in the circumcircle is K018, the second Brocard cubic.

Q049 is a bicircular quartic.