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X(3), X(4), X(1316), X(2698), X(2782), X(9513), X(14251), X(14382), X(39641), X(39642)

Brocard points Ω1, Ω2

E = X(14251) on the line Ω1, Ω2

X(39641), X(39642) are the imaginary foci of the Brocard ellipse

four foci of the MacBeath inconic : X(3), X(4) and two imaginary

This isogonal focal is the locus of foci of inscribed conics centered on the line X(5)X(39). The singular focus F is X(2698). See also K164.

K166 = nK(X6, X3569, X3). Also, K166 is spK(X2782, X5) as in CL055 and SpK(X3, L) as in CL056 where L is the Brocard axis.

Hence, if (L1), (L2) are two lines secant at O and symmetric about the Brocard axis, and if (C1), (C2) are the rectangular circum-hyperbolas which are their isogonal transforms, then {M1, N1} = (L1) ∩ (C2) and {M2, N2} = (L2) ∩ (C1) are two pairs of points on K166 such that {M1, M2} and {N1, N2} are two pairs of isogonal conjugate points.

K166 is a spK with orthic line passing through X(5), X(39) and X(2782) at infinity, hence the lines (L1), (L2) above can be replaced by two parallels passing through X(2782), symmetric about the orthic line, and the same properties apply.

K166 is obviously the locus of foci of inscribed conics centered on this orthic line.

K166 is an isoptic cubic, locus of M such that the oriented segments [O, Ω1] and [Ω2, H] are seen from M under the same angle.