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X(6), X(15), X(16), X(23), X(111), X(2854)

vertices of the 4th anti-Brocard triangle

K640 is the locus of M such that the Euler line of the pedal triangle of M is parallel to the line GK.

K640 is a focal cubic with focus X(23), the inverse of G in the circumcircle. The polar conic of X(23) is the circle passing through X(6), X(23), X(111), X(381), X(671), X(2080).

The isogonal transform of K640 is the orthopivotal cubic K065. See Orthopivotal Cubics.

K640 belongs to the pencils of cubics generated by

• K018 and the union of the circumcircle and the Brocard axis which also contains K730. All these cubics have their singular focus on the circle passing through X(3), X(23), X(99), X(111), X(2079), X(2930).

• K148 and K292 which also contains K641.

K640 is connected with K018 in another way since it is actually K018 for the 4th anti-Brocard triangle T4 = a4b4c4.

It follows that K640 is the locus of :

• contacts of tangents drawn from X(23) to the circles passing through X(6) and X(111),

• foci of conics inscribed in T4 with center on the line passing through X(6) and X(111).

Recall that the circumcircle of T4 is the circle passing through X(23) with center the reflection O' of O in X(111).

Note that O' lies on the (green) polar conic of X(23) as above.