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X(3), X(4), X(5), X(54)

S reflection of X(54) in X(5)

infinite points of pK(X6, X54)

common points of pK(X6, S) and the circumcircle

foci of the MacBeath conic (O, H and two other imaginary)

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

A variable line L passing through X54 is reflected about X5 in L' (passing through X382). The isogonal transform of L meets L' at two points on K526.

K526 is a central cubic with center X5, the nine point center.

K526 is spK(X54, X5) in CL055.