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X(99), X(512)

A', B', C' reflections of A, B, C in X(99)

A", B", C" their isogonal conjugates

U, V, W feet of the trilinear polar of the root R, the reflections of A", B", C" in X(99)

All isogonal central nK are circular and have their center N on the circumcircle. They form the class CL001 of cubics. The real asymptote is the perpendicular at N to the Simson line of N. As such, they are van Rees focal cubics. See Special isocubics §3.4.2 and Central cubics in the glossary. See also Z+(O) = CL025 and K087.

The figure above shows K084, the central isogonal focal nK centered at the Steiner X(99) with its real asymptote perpendicular at the Steiner point to the Brocard axis.

The root of K084 is X(11052), the intersection of the lines X(2)X(39) and X(620)X(690).

The inflexional tangent at X(99) passes through X(2142), the cevian quotient of X(512) and X(99).

The polar conic of X(512) is the rectangular hyperbola centered at X(99) passing through X(39) and whose asymptotes are the lines X(99)X(511) and X(99)X(512). It contains the midpoints of AA", BB", CC" and the two centers of anallagmaty E1, E2. These two points also lie on one of the bisectors of the asymptote and inflexional tangent.

K084 is the locus of foci of inconics centered on the line X(99)-X(512) i.e. on the asymptote.

It is also the locus of point M such that the line passing through the isogonal conjugate M* of M and the reflection M' of M in X(99) is perpendicular to the Brocard axis.