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X(1), X(99), X(512), X(2142), X(2143), X(5539), X(7315) excenters Brocard points Ω1 and Ω2 details and other points below 

K021 is the isogonal circular pK with pivot X(512) and singular focus X(98). Locus properties :




K021 contains the intersections of the sidelines of the antimedial triangle GaGbGc and the tangential triangle KaKbKc. These points are labelled Ab, Ac, Ba, Bc, Ca, Cb and lie on the cevian lines of the two Brocard points. 

X(5539) is the third point of the line X(1)X(99) on K021. It is the cevian quotient of the pivot X(512) and X(1). X(7315) is the isogonal conjugate of X(5539). These two points are weak points hence K021 contains their six extraversions. The figure shows the extraversions A1, B1, C1 of X(5539). Note that ABC and A1B1C1 are perspective at X(7315). 



The locus of of point M such that the cevian triangles of M and its isogonal conjugate M* have opposite (algebraic) areas is the nonpivotal isogonal cubic with root X(39) that passes through the six already mentioned points Ab, Ac, Ba, Bc, Ca, Cb. This cubic and K021 generate a pencil that contains the two decomposed cubics which are the union of the sidelines of the anticevian triangles of G and K. These two cubics are mentioned in a paper by C. E. Noble namely : An Anallagmatic Cubic, AMM vol. 55, Jan. 1948, pp. 714. The trilinear equations given there are (by+cz)(cz+ax)(ax+by) ± (bx+ay)(cx+az)(cy+bz) = 0, and the cubics are called "minussign cubic" for K021 and "plussign cubic" for the other. 
