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∑ (a^4 - b^2c^2)(c^2 y + b^2 z) y z = 0

X(2), X(99), X(385), X(804)

vertices of the first anti-Brocard triangle T1

A', B', C' : vertices of the third Brocard triangle of T1

centers Ωa, Ωb, Ωc of the Apollonius circles

infinite points of the Steiner ellipse

The first anti-Brocard triangle T1= a1b1c1 is defined here.

There is one and only one isogonal nK0 with respect to T1 which is a circum-cubic of ABC. It is K740 which is also a nK0 with respect to ABC, namely nK0(X385, X6). K740 is the G-Hirst inverse of K017 = nK0(X6, X385).

K740 is actually the first Brocard cubic K017 for T1. The root with respect to T1 is X(14931) = X(2)X(99) /\ X(6)X(1916) /\ X(98)X(3098), etc.

K740 is also a nK with respect to A'B'C' and then its root is X(8289). The lines BC, B'C', b1c1 concur at Ωa, the center of the A-Apollonius circle and likewise for the other analogous lines.

See the related K699 and also CL022.

Note that the three triangles ABC, T1, A'B'C' are two by two triply perspective with each time two perspectors which are the brocardians X(1916)' and X(1916)" of X(1916), these points on the circumcircle.

The three remaining perspectors are the collinear points X(1916), X(4027), X(5989). See figure.

Locus properties :

• K740 is the locus of point P such that P and its X(385)-isoconjugate P* are conjugated with regard to the circumcircle.

• K740 is the locus of point P such that the pedal triangles of P and P* are parallelogic.