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X(1), X(3), X(4), X(952), X(953), X(3109)

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

This isogonal strophoid is the locus of foci of inscribed conics centered on the line X(1)X(5). It is a member of the class CL003 of cubics. See also K040, K086 and K164. For other circum-strophoids passing through H (or any given point), see Q038. See also Table 69.

It is also the locus of point P such that a bisector of (OP,HP) passes through I. Hence X(953) is the only point (apart A, B, C) on the circumcircle verifying this property. X(953) is in fact the singular focus of the strophoid.

K165 is nK(X6, X10015, X1) or cK(#X1, X10015) where X10015 is the intersection of the line X(1)X(676) with the trilinear polar of X(264), the isotomic conjugate of O.

K165 is spK(X952, X1) in CL055.

The inverse (in the circumcircle) of K165 is K274. See Table 73 for properties and other related cubics.